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1102.3922

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in this paper , we have generated a collection of evolutionary models for brown dwarfs and very - low - mass stars for different atmospheric metallicities , with and without clouds . these models employ realistic atmosphere boundary conditions that allow us to consistently predict , given a detailed opacity model , the evolution of the object s radius . we have sought to demonstrate that with transit or eclipse radius measurements one is testing a multi - parameter theory , and not a universal radius@xmath4mass relation . the spread in radius at a given mass can be as large as @xmath00.1 to @xmath00.25 ( or @xmath010% to @xmath025% ) , with higher - metallicity , higher - cloud - thickness atmospheres resulting quite naturally in larger radii , all else being equal . for each 0.1 dex increase in atmospheric [ fe / h ] , the radius is expected to increase by @xmath01% to @xmath02.5% , depending upon age and mass . therefore , in order to constrain the hydrogen - helium equation of state one must control for the metallicity and cloud model . if the goal is to test structural theory and the viability of a suite of theoretical models , absent measurements of , for example , the metallicity , and a good constraint on the age , any radius measurement is of correspondingly limited utility . conversely , one should expect a range of radii for the natural range of metallicities and possible cloud properties expected for substellar objects and vlm stars in the solar neighborhood . in addition , we have calculated the effect of helium fraction on brown dwarf and vlm radii and find that , while for smaller masses and older ages radius decreases with increasing helium fraction ( as expected ) , for more massive brown dwarfs and a wide range of ages it increases with helium fraction . this runs counter to common lore , which expects that higher mean molecular weights universally result in smaller radii . we find that the increase in radius in going from @xmath2 to @xmath3 can be as large as @xmath00.025 ( @xmath02.5% ) . furthermore , we suggest that properly including the trace of heavy elements in the core eos should further augment this effect . we do not suggest that the cloud model we constructed for this investigation is definitive , nor uniquely applicable . rather , we engineered a cloud model , and its corresponding opacities and optical depths , to be representative of the generic effect of the clouds we know reside in brown dwarf atmospheres . we have sought merely to demonstrate that the presence or absence of clouds has an important effect on the radius of a brown dwarf . we note that many other cloud models can be constructed which may prove in the long run to be more viable . nevertheless , the qualitative effect of clouds that we have highlighted is robust . similarly , and more straightforwardly , atmospheric metallicity has a direct and clear effect on brown dwarf radii that needs to be accounted for in any interpretation of radius measurements . ten to twenty - five percent variations in radius exceed any reasonable error stemming from uncertainities in the equation of state alone and serve to emphasize that measurements of brown dwarf radii constrain a collection of effects , importantly including the atmosphere and condensate cloud models . without an independent measure of the atmospheric metallicity , constraints on the surface clouds , and a good age estimate , a measurement of a brown dwarf radius may be more difficult to interpret than previously thought . increasing the atmospheric metallicity of the vlm stars we studied in this paper from 0.0 to 0.5 increases their radii by @xmath04% . if we increase their atmospheric metallicitiy from -0.5 to 0.5 , their radii increase by @xmath010% . the latter percentage is slightly above what is often quoted as the discrepancy in radius between measurement and theory in the vlm regime ( morales et al . 2009 ; carter et al . 2011 ) . though we do not in this paper discuss models in the @xmath00.2 to 0.25 band , the clear implication of the systematic behavior we have derived in the stellar realm is that opacity due to higher metallicity might naturally account for the apparent radius anomalies in some eclipsing vlm systems . be that as it may , the effect of metallicity and clouds on the radii of brown dwarfs and vlms is straightforward and natural . therefore , we suggest it is necessary to incorporate these extra degrees of freedom into any interpretation of brown dwarf and vlm radius measurements and into any attempts usefully to constrain their equation of state . we acknowledge useful conversations with dave spiegel and nikku madhudsudhan , and ivan hubeny for his general support of the cooltlusty code . we also acknowledge support in part under nasa atp grant nnx07ag80 g , hst grants hst - go-12181.04-a and hst - go-12314.03-a , and jpl / spitzer agreements 1417122 , 1348668 , 1371432 , and 1377197 . baglin , a. , auvergne , m. , boisnard , l. , lam - trong , t. , barge , p. , catala , c. , deleuil , m. , michel , e. , & weiss , w. 2006 , 36th cospar scientific assembly , 36 , 3749 . bakos , g.a . , lzr , j. , papp , i. , sri , p. , & green , e.m . 2002 , , 114 , 974 johnson , j.a . , apps , k. , gazak , j.z . , crepp , j. , crossfield , i.j . , howard , a.w . , marcy , g.w . , morton , t.d . , chubak , c. , and isaacson , h. 2010 , submitted to ( arxiv:1008.4141 ) koch , d.g . 2010 , , 713 , l79 konacki , m. , torres , g. , jha , s. , & sasselov , d. 2003 , nature , 421 , 507 morales , j.c . 2009 , , 691 , 1400 pollacco , d. et al . 2004 , , 385 , 1576 pont , f. , melo , c.h.f . , bouchy , f. , udry , s. , queloz , d. , mayor , m. , & santos , n.c . 2005 , , 433 , l21 pont , f. , moutou , c. , bouchy , f. , behrend , r. , mayor , m. , udry , s. , queloz , d. , santos , n. , & melo , c. 2006 , , 447 , 1035 saumon , d. , chabrier , g. , & van horn , h. 1995 , , 99 , 713 = 0.0 [ dashed ] and 0.5 [ solid ] , shown in red ) and 2 ) clear atmospheres ( at [ fe / h ] = 0.0 [ dashed ] , shown in black ) . ( see section [ method ] for a discussion of the cloud prescription . ) the four masses shown are @xmath74 , @xmath75 , @xmath76 , and @xmath77 solar mass ( or approximately @xmath90 , @xmath91 , @xmath92 , and @xmath93 jupiter masses ) . the helium mass fraction ( @xmath1 ) for each model is here set to @xmath94 , i.e. , roughly the solar value . this figure shows that the radius of a brown dwarf is an increasing function of metallicity and is larger for models with clouds , all else being equal . three of the brown dwarfs which we highlight in this paper ( lhs 6343c , wasp-30b , and corot-15b ) have masses within this range and are included ( in blue ) on the figure , with putative observational error bars . the formal errors in the age are the most uncertain . as this figure indicates , before @xmath01 gyr , the radius evolves quickly , but afterwards significantly decelerates its shrinkage . ] , @xmath95 , @xmath96 , and @xmath97 gyr . notice the slight peak near @xmath04 m@xmath81 , the decrease in radius with increasing mass in the brown dwarf regime " at greater masses , and the monotonic radius shrinkage with increasing age . the curves start to rise near the main - sequence edge ( near @xmath075 m@xmath81 ) and ogle-122b and ogle-123b are clearly stars . six objects ( four brown dwarfs , two vlms ) are shown in blue ( with putative mass and age error bars ) : lhs 6343c ( johnson et al . 2010 ) , wasp-30b ( anderson et al . 2011 ) , corot-3b ( deleuil et al . 2008 ) , corot-15b ( bouchy et al . 2011 ) , ogle - tr-122b ( pont et al . 2005 ) , and ogle - tr-123b ( pont et al . 2006 ) . the quoted age of lhs 6343 is between one and five billion years . wasp-30b is suspected to be younger ( see fig.[fig : evolution ] ) . pont et al ( 2005,2006 ) suggest that the ogle objects might be younger than 0.5 gyrs , and , hence , their measured radii might be consistent with these models . the age of corot-3 is not well constrained . therefore , with the suggested age range of @[email protected] gyr for the star corot-15 ( bouchy et al . 2011 ) , corot-15b is the only object in this set for which a solution using these heritage models may be problematic . see the text for a discussion . ] . the heritage models from burrows et al . ( 1997 ) ( in green ) are included for comparison . cloudy - atmosphere models are shown in red and clear - atmosphere models are shown in black . the clear - atmosphere models have metallicities ( [ fe / h ] ) of -0.5 , 0.0 , and + 0.5 , while the cloudy - atmosphere models have metallicities of 0.0 and + 0.5 . a general trend is that the presence of clouds retards shrinkage , as does higher metallicity . the model with [ fe / h ] = 0.5 and a cloudy atmosphere generally has the largest radii and the variation in radii among the models shown can be [email protected] r@xmath81 at a given mass and age . ] , but highlighting the differences between models with two different helium fractions , @xmath2 ( blue / aqua ) and @xmath3 ( magenta ) . the gas - phase metallicity is solar for all models shown . as a reference , the model from burrows et al . ( 1997 ) is also plotted ( in green ) . the solid lines represent models with clouds , while the dashed lines are those with clear atmospheres . the solid ( cloudy ) models are generally larger than the corresponding dashed ( clear ) models . interestingly , the magenta ( @xmath3 ) models are larger than the blue / aqua ( @xmath2 ) models in the mass range above @xmath055 - 60 at later ages ( 3.0 and 5.0 gyr ) and above @xmath035 - 40 at earlier ages ( 0.5 and 1.0 gyr ) . this is counter to common lore , which suggests that planets with a higher molecular weight and lower electron fraction should be smaller . this is true only for cold planets . note that the burrows et al . ( 1997 ) models were calculated for @xmath2 ( saumon & marley 2008 ) . see text for a discussion . ] = 0.5 ( top two panels ) and [ fe.h ] = 0.0 ( bottom two panels ) . all the models shown are for a helium fraction , @xmath1 , of 0.28 . on each panel are shown isochrones at 0.5 , 1.0 , 3.0 , and 5.0 gyrs . superposed on all panels is the data point for lhs 6343c at @xmath89 and @xmath98 ( johnson et al . 2010 ) . the metallicity of lhs 6343c s primary , lhs 6343a , is suggested by anderson et al . ( 2011 ) to be [ fe / h ] = @xmath12 , i.e. super - solar , but johnson et al . ( 2010 ) quote a value of @xmath13 . clear models for [ fe / h ] = 0.0 ( solar ) and [ fe / h ] = 0.5 metallicities fit for ages greater than @xmath02 gyrs , but the cloudy model with @xmath83 and [ fe / h ] = 0.0 also fits for late ages . models with @xmath82 fit slightly better , but given the systematic observational uncertainties , nothing substantive can be said about @xmath1 for lhs 6343c . the new model with clouds and super - solar metallicity does not fit for ages less than @xmath07 gyr . as fig.[burrows97 ] demonstrates , the burrows et al . ( 1997 ) models fit well for the suggested age of @xmath02 gyrs . see the text for a discussion of the implications of all these model comparisons . ] , but for wasp-30b and with the [ fe / h ] = 0.5 models at the bottom and the [ fe / h ] = 0.0 models at the top . ( note that a slightly different range of radii on the ordinate is used . ) as in fig . [ lhs6343 ] , all the models shown are for a helium fraction of 0.28 and isochrones at 0.5 , 1.0 , 3.0 , and 5.0 gyrs are plotted . superposed on all panels is the data point for wasp-30b at @xmath20 and @xmath21 ( anderson et al . 2011 ) . the metallicity of wasp-30 is quoted to be [ fe / h ] = @xmath22 , basically solar . a variety of models and age@xmath4metallicity combinations fit the wasp-30b data . clear models with [ fe / h ] = 0.0 fit well for @xmath01@xmath42 gyrs . clear models with [ fe / h ] = 0.5 fit well for ages from @xmath02 to @xmath03 gyrs . cloudy models with [ fe / h ] = 0.0 fit well for ages of @xmath88 gyrs and our cloudy model with [ fe / h ] = 0.5 still fits near ages of @xmath05 gyrs . cloudy models with @xmath82 fit at slightly younger ages . as fig . [ burrows97 ] suggests , the heritage models from burrows et al . ( 1997 ) fit wasp-30b for an age near @xmath01 gyr . see the text for a discussion of these fits and conclusions concerning wasp-30b . ] , but for corot-15b and with slightly different ranges for the radius and mass axes . the mass of corot-15b is measured to be @xmath89 and its radius is measured to be @xmath15 ( bouchy et al . 2011 ) . corot-15 has an estimated metallicity of @xmath16 , again basically solar . bouchy et al . ( 2011 ) suggest that its parent star has an age in the range @[email protected] gyr . our solar - metallicity ( [ fe / h ] = 0.0 ) models can fit the lower age range to @xmath01-@xmath86 to @xmath01.5-@xmath86 , with the best fit for the [ fe / h ] = 0.0 cloudy model . however , our [ fe / h ] = 0.5 models fit the suggested age range better , with the clear [ fe / h ] = 0.5 models fitting an age of @xmath01 gyr within @xmath01-@xmath86 and the cloudy [ fe / h ] = 0.5 models fitting anywhere in the suggested age range . cloudy models with [ fe / h ] = 0.0 and clear models with [ fe / h ] = 0.5 fit corot-15b almost equally well . figure [ burrows97 ] suggests that the burrows et al . ( 1997 ) solar - metallicity models would fit only for very young ages less than @xmath00.5 gyrs . see the text for details . ] , [ wasp30 ] , and [ corot15 ] , but for corot-3b and for a lower brown - dwarf mass range between 5 and 40 . the clear models are on the left panels and the cloudy models are on the right panels . the top models are for [ fe / h ] = 0.0 and the bottom models are for [ fe / h ] = 0.5 . the measured mass of corot-3b is @xmath17 and its measured radius is @xmath18 ( deleuil et al . see text for a discussion . ] = 0.0 ( top panels ) and 0.5 ( bottom panels ) and for clear ( left panels ) and cloudy ( right panels ) models . the data for ogle - tr-122b and ogle - tr-123b are taken from pont et al . ( 2005 ) and pont et al . ( 2006 ) , respectively . since these objects have measured masses of @xmath24 ( ogle - tr-122b ) and @xmath25 ( ogle - tr-123b ) , the plots are for a mass range from 60 to 120 . the measured radii are @xmath26 and @xmath27 for ogle - tr-122b and ogle - tr-123b , respectively , and the plotted radius range is 0.8 to 1.6 . see text for a discussion of the issues involved . ]

employing realistic and consistent atmosphere boundary conditions , we have generated evolutionary models for brown dwarfs and very - low - mass stars ( vlms ) for different atmospheric metallicities ( [ fe / h ] ) , with and without clouds . we find that the spread in radius at a given mass and age can be as large as% to% , with higher - metallicity , higher - cloud - thickness atmospheres resulting quite naturally in larger radii . for each 0.1 dex increase in [ fe / h ] , radii increase by% to.5% , depending upon age and mass . we also find that , while for smaller masses and older ages brown dwarf radii decrease with increasing helium fraction ( ) ( as expected ) , for more massive brown dwarfs and a wide range of ages they increase with helium fraction . the increase in radius in going from to can be as large as.025 (.5% ) . therefore , we suggest that opacity due to higher metallicity might naturally account for the apparent radius anomalies in some eclipsing vlm systems . ten to twenty - five percent variations in radius exceed errors stemming from uncertainities in the equation of state alone .

employing realistic and consistent atmosphere boundary conditions , we have generated evolutionary models for brown dwarfs and very - low - mass stars ( vlms ) for different atmospheric metallicities ( [ fe / h ] ) , with and without clouds . we find that the spread in radius at a given mass and age can be as large as% to% , with higher - metallicity , higher - cloud - thickness atmospheres resulting quite naturally in larger radii . for each 0.1 dex increase in [ fe / h ] , radii increase by% to.5% , depending upon age and mass . we also find that , while for smaller masses and older ages brown dwarf radii decrease with increasing helium fraction ( ) ( as expected ) , for more massive brown dwarfs and a wide range of ages they increase with helium fraction . the increase in radius in going from to can be as large as.025 (.5% ) . furthermore , we find that for vlms an increase in atmospheric metallicity from 0.0 to 0.5 dex , increases radii by% , and from -0.5 to 0.5 dex by% . therefore , we suggest that opacity due to higher metallicity might naturally account for the apparent radius anomalies in some eclipsing vlm systems . ten to twenty - five percent variations in radius exceed errors stemming from uncertainities in the equation of state alone . this serves to emphasize that transit and eclipse measurements of brown dwarf radii constrain numerous effects collectively , importantly including the atmosphere and condensate cloud models , and not just the equation of state . at all times , one is testing a multi - parameter theory , and not a universal radiusmass relation .

1105.6223

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stability of two - dimensional ( 2d ) plasma crystals is a fundamental problem of complex ( dusty ) plasmas . such crystals which are monolayers of hexagonally ordered monodisperse microparticles can be ( routinely ) created in a rf plasma @xcite : particles get negatively charged in a plasma and therefore the electric force exerted on them in the sheath above a horizontal electrode can compensate for gravity , thus providing a stable levitation . there are several mechanisms operating in complex plasmas that can result in the melting of 2d crystals . these mechanisms can generally be divided into two categories _ generic _ and _ plasma - specific_. generic mechanisms of melting are those operating in any ( classical ) system with a given pair interaction between particles , provided the interaction can be described by a hamiltonian . the melting in 2d systems can either be a two - step process ( which involves consecutive unbinding of weakly interacting dislocation and disclination pairs , respectively , with the intermediate hexatic phase ) @xcite or a one - step process ( where the hexatic phase is preempted by the formation of dislocation chains ) @xcite . these generic melting mechanisms can operate in very different 2d systems including complex plasmas @xcite . plasma - specific melting mechanisms can only operate in complex plasmas . such mechanisms are associated with the energy exchange between microparticles and ambient plasma and can be considered as a result of the system openness . for instance , 2d plasma crystals can be strongly perturbed by single particles moving above or below the monolayer @xcite , or they can melt due to fluctuations of particle charges @xcite . the most universal among the plasma - specific mechanisms is that associated with the _ wake - mediated _ interaction between microparticles : in the presence of strong plasma flow the screening cloud around each charged particle becomes highly asymmetric @xcite . these clouds are usually referred to as `` plasma wakes '' @xcite and play the role of a `` third body '' in the interparticle interaction , making it nonreciprocal @xcite . under certain conditions , this makes the system non - hamiltonian and provides effective conversion of the energy of flowing ions into the kinetic energy of microparticles @xcite . the wake - induced mechanism of melting of crystalline monolayers was discovered theoretically a decade ago by ivlev and morfill @xcite . based on a simple model of a particle chain , it was shown that the longitudinal in - plane and transverse out - of - plane dust - lattice ( dl ) wave modes are no longer independent they are coupled due to the wake - mediated interactions . when the modes intersect they become modified by the coupling and form a _ hybrid _ mode in a narrow vicinity of the crossing . this can trigger the _ mode - coupling instability _ which causes the melting . this melting mechanism had received strong confirmation later on , when the instability threshold predicted by the theory was compared @xcite with the experimental observations by u. konopka and numerical simulations by g. joyce . however , the direct comparison of theory and experiment became possible only recently , after an experimental method of measuring the out - of - plane mode was developed @xcite and therefore the essential fingerprint of the mode - coupling instability the hybrid mode became observable . in recent experiment by coudel _ _ @xcite an implementation of this method unambiguously demonstrated that the melting indeed occurs due to the resonance coupling between the longitudinal in - plane and transverse out - of - plane modes . the variation of the wave modes with the experimental conditions , including the emergence of the hybrid branch , revealed exceptionally good agreement with the theory of mode - coupling instability generalized for 2d case by zhdanov _ _ @xcite . the theory of mode - coupling instability @xcite predicts a number of distinct fingerprints to be observed upon the instability onset : along with the emergence of hybrid mode mentioned above , these are a critical angular dependence the hybrid mode first appears only for wave vectors oriented along one of the principal lattice axes , a mixed polarization the two dl modes that form the hybrid mode are no longer purely longitudinal and transverse close to the merging point , and distinct thresholds the instability sets in only when ( i ) the particle number density in the monolayer is high enough or / and vertical confinement eigenfrequency is low enough ( so that the two dl modes can cross and form the hybrid mode ) and ( ii ) the gas pressure is low enough ( so that the growth rate of the hybrid mode exceeds the damping rate ) . all these fingerprints have been mentioned in our previous theoretical publications @xcite , some of them were also illustrated in the followup experimental paper @xcite . however , their discussion was apparently too concise to address all these important properties in necessary detail . the need for in - depth discussion of the wake - mediated mode coupling became evident now , when new publications have appeared where some essential properties of the instability were misinterpreted ( see , e.g. , recent experimental paper @xcite and the subsequent erratum @xcite by liu _ et al . _ ) . therefore , in this paper we summarize the key features of the mode coupling and provide a detailed discussion , analyze the critical dependence on experimental parameters , and highlight the outstanding issues .

experiments with two - dimensional ( 2d ) plasma crystals are usually carried out in rf plasma sheaths , where the interparticle interactions are modified due to the presence of plasma wakes . the theory predicts a number of distinct fingerprints to be observed upon the instability onset , such as the emergence of a new hybrid mode , a critical angular dependence , a mixed polarization , and distinct thresholds . in this paper we summarize these key features and provide their detailed discussion , analyze the critical dependence on experimental parameters , and highlight the outstanding issues .

experiments with two - dimensional ( 2d ) plasma crystals are usually carried out in rf plasma sheaths , where the interparticle interactions are modified due to the presence of plasma wakes . the wake - mediated interactions result in the coupling between wave modes in 2d crystals , which can trigger the mode - coupling instability and cause melting . the theory predicts a number of distinct fingerprints to be observed upon the instability onset , such as the emergence of a new hybrid mode , a critical angular dependence , a mixed polarization , and distinct thresholds . in this paper we summarize these key features and provide their detailed discussion , analyze the critical dependence on experimental parameters , and highlight the outstanding issues .

1008.0905

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in this paper , we study schrdinger eigenvalue problems with real and complex polynomial potentials in the complex plane under various decaying boundary conditions . we provide explicit asymptotic formulas relating the index @xmath11 to a series of fractional powers of the eigenvalue @xmath12 ( see theorem [ main_thm1 ] ) . also , we recover the polynomial potentials from asymptotic formula of the eigenvalues ( see theorem [ thm_112 ] and corollary [ cor6 ] ) as well as applications to the so - called @xmath9-symmetric hamiltonians ( see theorems [ main_thm2 ] and [ thm_111 ] ) . for integers @xmath0 and @xmath13 , we consider the schrdinger eigenvalue problem @xmath14u(z)=\lambda u(z),\quad\text{for some $ \lambda\in{\mathbb c}$},\ ] ] with the boundary condition that @xmath15 where @xmath6 is a polynomial of degree at most @xmath7 of the form @xmath16 if a nonconstant function @xmath17 satisfies with some @xmath18 and the boundary condition , then we call @xmath19 an _ eigenvalue _ of @xmath20 and @xmath17 an _ eigenfunction of @xmath20 associated with the eigenvalue @xmath19_. sibuya @xcite showed that the eigenvalues of @xmath20 are the zeros of an entire function of order @xmath21 and hence , by the hadamard factorization theorem ( see , e.g. , @xcite ) , there are infinitely many eigenvalues . we call the entire function the stokes multiplier ( or the spectral determinant ) , and the algebraic multiplicity of an eigenvalue @xmath19 is the order of the zero @xmath19 of the stokes multiplier . also , the geometric multiplicity of an eigenvalue @xmath19 is the number of linearly independent eigenfunctions associated with the eigenvalue @xmath19 , that is @xmath22 for every eigenvalue @xmath19 @xcite . we number the eigenvalues @xmath23 in the order of nondecreasing magnitudes , counting their algebraic multiplicities . we will show that the magnitude of large eigenvalues is strictly increasing ( see , lemma [ monoton ] ) and hence , there is a unique way of ordering large eigenvalues , but this is not guaranteed for small eigenvalues . however , how we order these small eigenvalues will not affect results in this paper . throughout this paper , we will use @xmath12 to denote the eigenvalues of @xmath20 without explicitly indicating their dependence on the potential and the boundary condition . also , we let @xmath24 be the coefficient vector of @xmath25 . the anharmonic oscillators @xmath20 with the various boundary conditions are considered in @xcite . when @xmath26 is even and @xmath27 , @xmath28 is a schrdinger operator in @xmath29 ( see , e.g. , @xcite ) . this is self - adjoint if the potential @xmath30 is real on the real line , and non - self - adjoint if the potential is non - real . some particular classes of @xmath31 have been studied extensively in recent years in the context of theory of @xmath9-symmetry @xcite . the @xmath20 is @xmath9-symmetric if the potential @xmath32 satisfies @xmath33 , @xmath34 , that is equivalent to @xmath35 . in this paper , we will generalize results in @xcite ( where @xmath36 is studied ) to @xmath13 and introduce some new results . these results are consequences of the following asymptotic expansion of the eigenvalues . [ main_thm1 ] for each integer @xmath0 and @xmath13 , there exists an integer @xmath37 such that the eigenvalues @xmath38 of @xmath20 satisfy @xmath39 where @xmath40 and @xmath41 are defined in and , respectively . also , we obtain the partial reality of the eigenvalues for @xmath9-symmetric @xmath20 . [ main_thm2 ] if @xmath20 is @xmath9-symmetric , then eigenvalues are all real with at most finitely many exceptions . if @xmath3 is an eigenfunction associated with the eigenvalue @xmath19 of a @xmath9-symmetric @xmath20 , then @xmath42 is also an eigenfunction associated with @xmath43 . in corollary [ monoton ] , we will show that @xmath44 for all large @xmath11 and @xmath45 . thus , @xmath12 are real and positive for all large @xmath11 since @xmath46 . many @xmath9-symmetric operators have real eigenvalues only @xcite . however , there are some @xmath9-symmetric @xmath20 that produce a finite number of non - real eigenvalues @xcite . when @xmath20 is self - adjoint , the spectrum is real . conversely , when the spectrum is real , what can we conclude about @xmath20 ? the next theorem provides a necessary and sufficient condition for @xmath20 to have infinitely many real eigenvalues . [ thm_111 ] suppose that @xmath10 . then @xmath20 with the potential @xmath47 has infinitely many real eigenvalues if and only if @xmath20 with the potential @xmath48 for some @xmath49 is @xmath9-symmetric . the next theorem reveals an interesting feature of the eigenvalues as a sequence : [ thm_112 ] suppose that @xmath10 . let @xmath38 and @xmath50 be the eigenvalues of @xmath20 with the potentials @xmath32 and @xmath51 , respectively . suppose that @xmath52 as @xmath53 . then @xmath54 for some @xmath49 and @xmath55 for all @xmath56 after , if needed , small eigenvalues are reordered . the asymptotic expansions of the eigenvalues of @xmath20 with @xmath57 have been studied in , for example , @xcite . maslov @xcite computed the first three terms of asymptotic expansions of @xmath58 , where @xmath12 are the eigenvalues of @xmath59 helffer and robert @xcite considered @xmath60 where @xmath61 are positive integers and where @xmath62 is a _ real _ polynomial of degree at most @xmath63 . they obtained the existence of asymptotic expansions of the eigenvalues to all orders , and suggested an explicit way of computing the coefficients of the asymptotic expansion . in particular , for the case when the potential is @xmath64 , @xmath65 , helffer and robert @xcite computed the first nine terms of the asymptotic expansion of @xmath58 . also , fedoryuk @xcite considered with complex polynomial potentials and with for @xmath57 and computed the first term in the asymptotic expansion . also , sibuya @xcite computed the first term in the asymptotic expansion for @xmath66 . this paper is organized as follows . in section [ sec_cor ] , we define @xmath40 and some other notations . also , we invert , expressing @xmath12 as a series of fractional powers of the index @xmath11 and prove theorems [ thm_111 ] and [ thm_112 ] , and other interesting direct and inverse spectral results . in section [ prop_sect ] , we introduce some properties of the solutions of the differential equation in , due to hille @xcite and sibuya @xcite . we study the asymptotic of the stokes multiplier associated with @xmath36 in section [ sec_4 ] and treat the general case @xmath20 in section [ sec_5 ] . in section [ asymp_eigen ] , we relate the eigenvalues of @xmath20 with the zeros of the stokes multiplier . we prove theorem [ main_thm1 ] for @xmath67 in section [ sec_7 ] and for @xmath68 in section [ sec_8 ] .

for integers and , we study the eigenvalue problemsu(z)=\lambda u(z)$ ] with the boundary conditions that decays to zero as tends to infinity along the rays in the complex plane , where is a polynomial of degree at most . then we show that if the eigenvalue problem is-symmetric , then the eigenvalues are all real and positive with at most finitely many exceptions . moreover , we show that when , the eigenvalue problem has infinitely many real eigenvalues if and only if its translation or itself is-symmetric . also , we will prove some other interesting direct and inverse spectral results . _ 2010 _ mathematics subject classification _ : 34l40 , 34l20 , 81q12 _ + _ key words : anharmonic oscillators , asymptotics of the eigenvalues ,-symmetry _ = 18pt

for integers and , we study the eigenvalue problemsu(z)=\lambda u(z)$ ] with the boundary conditions that decays to zero as tends to infinity along the rays in the complex plane , where is a polynomial of degree at most . we provide asymptotic expansions of the eigenvalues . then we show that if the eigenvalue problem is-symmetric , then the eigenvalues are all real and positive with at most finitely many exceptions . moreover , we show that when , the eigenvalue problem has infinitely many real eigenvalues if and only if its translation or itself is-symmetric . also , we will prove some other interesting direct and inverse spectral results . _ 2010 _ mathematics subject classification _ : 34l40 , 34l20 , 81q12 _ + _ key words : anharmonic oscillators , asymptotics of the eigenvalues ,-symmetry _ = 18pt

math0411143

i

in this paper , we study schrdinger operators with any polynomial potential of degree @xmath0 with complex coefficients , under decaying boundary conditions along two rays to infinity in the complex plane , and provide asymptotic expansions of the eigenvalue counting functions and the eigenvalues . then we will use these to reconstruct some coefficients of polynomial potentials from asymptotic expansions of the eigenvalues , and to show that all but finitely many eigenvalues of every @xmath8-symmetric oscillator with a polynomial potential are real and positive . for integers @xmath0 fixed and @xmath9 , we are considering the non - standard eigenvalue problems @xmath10u(z,\lambda)=\lambda u(z,\lambda),\quad\text{for some $ \lambda\in{\mathbb c}$},\ ] ] with the boundary condition that @xmath11 where @xmath12 is a polynomial of degree at most @xmath13 of the form @xmath14 the anharmonic oscillators @xmath15 with the various boundary conditions are considered in @xcite . the most studied case is when @xmath16 is even and @xmath17 , for example , see @xcite . in this case , @xmath15 is an schrdinger operator in @xmath18 . this is self - adjoint if all the coefficients of the polynomial @xmath19 in @xmath4 are real , and non - self - adjoint if a coefficient of @xmath19 is non - real . the case when @xmath20 has been studied extensively in recent years in the context of @xmath8-symmetric theory @xcite . throughout this paper , we use the integer @xmath0 for the degree of the polynomial potential and integer @xmath21 with @xmath1 . we will use @xmath22 , depending on the potential and the boundary condition , to denote the eigenvalues of @xmath15 , without explicitly indicating their dependence on the potential and the boundary condition . also , we let @xmath23 be the coefficient vector of @xmath24 . if a nonconstant function @xmath25 satisfies with some @xmath26 and the boundary condition , then we call @xmath27 an _ eigenvalue _ of @xmath15 and @xmath25 an _ eigenfunction of @xmath15 associated with the eigenvalue @xmath27_. also , the _ geometric multiplicity of an eigenvalue @xmath27 _ is the number of linearly independent eigenfunctions associated with the eigenvalue @xmath27 . for each @xmath21 fixed , we number the eigenvalues @xmath28 in the order of nondecreasing magnitudes , counting their `` algebraic multiplicities '' , where the integer @xmath29 , depending on the potential and the boundary condition , is due to our method of proof of theorem [ main_thm1 ] . in theorem [ main_thm1 ] we show that for every large @xmath30 , there exists @xmath22 satisfying below . however , we do not know the number of eigenvalues `` near '' zero , and this is why we need to have @xmath29 in numbering the eigenvalues . before we state our main theorems , we first introduce some known facts by sibuya @xcite about the eigenvalues @xmath22 of @xmath15 . [ main2 ] the eigenvalues @xmath7 of @xmath15 have the following properties . 1 . the set of all eigenvalues is a discrete set in @xmath31 . the geometric multiplicity of every eigenvalue is one . infinitely many eigenvalues , accumulating at infinity , exist . when @xmath20 the eigenvalues have the following asymptotic expansion @xmath32\quad\text{as $ n$ tends to infinity},\quad n \in { \mathbb n}.\ ] ] this paper contains results on direct and inverse spectral probelms , and their applications to @xmath8-symmetric potential problems . theorem [ main_thm1 ] below is the main result , regarding asymptotic expansions of `` eigenvalue counting functions '' . the other results below in the introduction are deduced from theorem [ main_thm1 ] . here , we first introduce the following theorem , regarding asymptotic expansions of a kind of eigenvalue counting functions , where we use multi - index notations with @xmath33 also , we use @xmath34 , @xmath35 and @xmath36 . [ main_thm1 ] for @xmath37 , the eigenvalues @xmath22 of @xmath15 satisfy @xmath38 where @xmath39 is the largest integer that is less than or equal to @xmath40 , and where the error @xmath41 term is uniform on any compact set of @xmath37 , and @xmath42 where @xmath43 @xmath44 and @xmath45 where @xmath46 is the beta function . we obtain by investigating the asymptotic expansions of an entire function whose zeros are the eigenvalues . sibuya @xcite got by using the first order asymptotic expansion of the entire function . next , we let @xmath47 , @xmath48 , be the eigenvalue counting function , that is , @xmath47 is the number of eigenvalues @xmath27 of @xmath15 such that @xmath49 . then the following theorem on an asymptotic expansion of the eigenalue counting function is a consequence of theorem [ main_thm1 ] . [ main_thm2 ] let @xmath37 be fixed . suppose that @xmath50 for @xmath51 . then @xmath47 has the asymptotic expansion @xmath52 where the error @xmath53 is uniform for any compact set of @xmath37 . in theorem [ ineq_eq ] below , we show that @xmath54 for all large @xmath30 . suppose that @xmath55 . then since @xmath56 we see from theorem [ eigen_asy ] below that @xmath57 . thus , @xmath58 hence , replacing @xmath59 in by @xmath60 , and solving the resulting equation for @xmath61 complete the proof . next , we improve the asymptotic expansion of the eigenvalues @xmath7 of @xmath62 . in particular , we will prove the following , which essentially invert to get @xmath22 in terms of @xmath61 . [ eigen_asy ] for each @xmath37 , there exist some constants @xmath63 , @xmath64 , such that @xmath65 where the error term is uniform for any compact set of @xmath37 , and where @xmath66 and @xmath67 , @xmath68 , are defined recurrently by @xmath69 and @xmath70 where @xmath71 . we note for the first summation in the definition of @xmath72 that @xmath73 implies @xmath74 whenever @xmath75 . also , for the second summation , we point out that @xmath76 implies @xmath74 whenever @xmath77 . the asymptotic expansions of the eigenvalues of @xmath15 with @xmath78 and @xmath20 have been studied by a number of people . for example , maslov @xcite computed the first three terms of asymptotic expansions of @xmath79 , where @xmath22 are the eigenvalues of @xmath80 helffer and robert @xcite considered @xmath81 where @xmath82 are positive integers and where @xmath83 is a _ real _ polynomial of degree at most @xmath84 . they obtained existence of asymptotic expansions of eigenvalues to all orders , and suggested an explicit way of computing the coefficients of the asymptotic expansion . in particular , for the case when the potential is @xmath85 , @xmath86 , helffer and robert @xcite computed the first nine terms of the asymptotic expansion of @xmath79 . fedoryuk @xcite considered with complex polynomial potentials and with for @xmath78 , and showed the existence of asymptotic expansions of the eigenvalues to all orders . note that there appear to be typographical errors in @xcite . for example , when @xmath16 is even ( and @xmath17 ) the leading coefficient of the asymptotic expansion of @xmath22 in @xcite is @xmath87 , which is different from @xmath88 found in @xcite and again in theorem [ eigen_asy ] above . next , we point out some differences between work of fedoryuk @xcite and the present work . fedoryuk @xcite showed the existence of asymptotic expansion of the eigenvalues to all orders while we do not . on the other hand , we treat all decaying boundary conditions with @xmath1 , while fedoryuk @xcite studied the case @xmath78 only . moreover , we computed more coefficients @xmath67 explicitly , and our methods are different from fedoryuk s . here , we introduce results on inverse spectral problems , but first the following corollary is an easy consequence of theorems [ main_thm1 ] and [ eigen_asy ] , regarding how the coefficients of the asymptotic expansions depend on @xmath37 . [ corollary_1 ] let @xmath68 be a fixed integer . then we have the following . 1 . @xmath89 and @xmath72 are polynomials in @xmath90 . 2 . @xmath89 and @xmath72 do not depend on @xmath91 . 3 . if @xmath92 is a multiple of @xmath16 , then @xmath93 , and @xmath72 does not depend on @xmath94 . 4 . if @xmath92 is not a multiple of @xmath16 , then @xmath89 and @xmath72 depend linearly on @xmath94 . statements on @xmath89 are direct consequences of the definition of @xmath89 in theorem [ main_thm1 ] . one can use statements on @xmath89 and induction on @xmath95 to prove statements on @xmath72 . next , one can reconstruct some coefficients of the polynomial potential from the asymptotic expansion of the eigenvalues . let @xmath96 be a fixed integer . suppose that @xmath97 is known whenever @xmath98 and @xmath99 is a multiple of @xmath16 . if @xmath92 is a multiple of @xmath16 , then the asymptotic expansions of the eigenvalues @xmath7 of @xmath15 of type with an error term @xmath100 uniquely and explicitly determine @xmath94 . from the asymptotic expansion of the eigenvalues , one gets @xmath101 that are explicit polynomials in @xmath102 . then since we know @xmath97 if @xmath99 is a multiple of @xmath16 , corollary [ corollary_1 ] says that we can find all @xmath102 . one says that @xmath15 is @xmath8-symmetric if the potential @xmath103 satisfies @xmath104 , @xmath105 , that is equivalent to @xmath106 . here , we prove the partial reality of the eigenvalues @xmath22 of @xmath8-symmetric @xmath15 . but first , we show the following theorem , regarding monotonicity of modulus of @xmath22 for all large @xmath30 . [ ineq_eq ] for each @xmath37 there exists @xmath107 such that @xmath108 if @xmath109 . see theorem 3 in @xcite for the proof of the case when @xmath20 . one can see that proof of theorem 3 in @xcite can be easily adapted for the cases when @xmath110 . now we are ready to prove the following theorem on the partial reality of the eigenvalues @xmath22 of @xmath15 . [ main_theorem ] suppose that @xmath106 . then all but finitely many eigenvalues of @xmath15 are real and positive . hence @xmath50 for all @xmath51 , so that the counting function formula in theorem [ main_thm2 ] is valid . when @xmath15 is @xmath8-symmetric ( i.e. , @xmath106 ) , @xmath111 is an eigenfunction associated with an eigenvalue @xmath27 if and only if @xmath112 is an eigenfunction associated with the eigenvalue @xmath113 . thus , the eigenvalues either appear in complex conjugate pairs , or else are real . so theorem [ ineq_eq ] implies theorem [ main_theorem ] . in recent years , these @xmath8-symmetric operators have gathered considerable attention , because ample numerical and asymptotic studies suggest that many of such operators have real eigenvalues only even though they are not self - adjoint . in particular , the differential operators @xmath114 with some polynomial potential @xmath103 and with the boundary condition have been considered in @xcite and references therein . the rigorous proof of reality and positivity of the eigenvalues of @xmath8-symmetric operators with certain classes of polynomial potentials and with the boundary condition for @xmath20 , was given by dorey , dunning and tateo @xcite in 2001 and by the present author @xcite in 2002 . however , there are some @xmath8-symmetric polynomial potentials that produce a finite number of non - real eigenvalues @xcite for some particular classes of polynomial potentials . so without any further restrictions on the real coefficients @xmath97 , theorem [ main_theorem ] is the most general result one can expect about reality of eigenvalues of @xmath8-symmetric operators with polynomial potentials . this paper is organized as follows . in section [ prop_sect ] , we will introduce work of hille @xcite and sibuya @xcite , regarding properties of solutions of . also , we introduce entire functions @xmath115 whose zeros are closely related with the eigenvalues of @xmath15 , due to sibuya @xcite ( c. f. , section [ asymp_eigen ] ) . in section [ sec_4 ] , we then provide asymptotics of the entire function @xmath116 as @xmath117 in the complex plane @xcite , improving the asymptotics of @xmath116 in @xcite . in section [ sec_5 ] , we provide asymptotic expansions of @xmath115 as @xmath117 in @xmath31 . in section [ asymp_eigen ] , we investigate how the zeros of @xmath118 are related with the eigenvalues of @xmath15 . in sections [ sec_7 ] and [ sec_8 ] , we prove theorem [ main_thm1 ] . in section [ sec_9 ] , we prove theorem [ eigen_asy ] . finally , in the appendix we compute @xmath119 in theorem [ main_thm1 ] , that is originally given in terms of certain integrals .

for integers and , we study the eigenvalue problemu(z)=\lambda u(z)$ ] with the boundary conditions that decays to zero as tends to infinity along the rays in the complex plane , where is a polynomial . we provide asymptotic expansions of the eigenvalue counting function and the eigenvalues . then we apply these to the inverse spectral problem , reconstructing some coefficients of polynomial potentials from asymptotic expansions of the eigenvalues . also , we show for arbitrary-symmetric polynomial potentials of degree and all symmetric decaying boundary conditions that the eigenvalues are all real and positive , with only finitely many exceptions . _ preprint . _ = 18pt

for integers and , we study the eigenvalue problemu(z)=\lambda u(z)$ ] with the boundary conditions that decays to zero as tends to infinity along the rays in the complex plane , where is a polynomial . we provide asymptotic expansions of the eigenvalue counting function and the eigenvalues . then we apply these to the inverse spectral problem , reconstructing some coefficients of polynomial potentials from asymptotic expansions of the eigenvalues . also , we show for arbitrary-symmetric polynomial potentials of degree and all symmetric decaying boundary conditions that the eigenvalues are all real and positive , with only finitely many exceptions . _ preprint . _ = 18pt

1504.04789

i

let @xmath22 $ ] denote the set of @xmath3-hlder continuous functions @xmath23\to { \mathbb{r}}$ ] . in 2009 , kahane and katznelson @xcite proved the following result and asked whether it is sharp . for every @xmath0 there exists a function @xmath24 such that if @xmath9 $ ] and @xmath25 is of bounded variation , then the hausdorff dimension satisfies @xmath26 . is the above result the best possible ? we answer this question negatively and determine the optimal bound . let @xmath27}\{\dim_{\mathcal{h}}a : f|_{a } \textrm { is of bounded variation}\},\ ] ] so that the above theorem states @xmath28 , see figure [ f : v ] . [ t : va ] for all @xmath0 we have @xmath29 ; ; ; ; ( 6.7,0 ) ; \to [ 0,1]$ ] with hlder exponent . its graph consists of 6 affine copies of itself.,scaledwidth=90.0% ] kahane and katznelson also asked about dimensions of sets @xmath30 such that the restriction to @xmath30 is hlder continuous . ( see the next section for related results . ) we present two constructions , one deterministic and one stochastic , of functions that are not hlder on any set of high enough dimension . first we consider self - affine functions . these are constructed in definition [ d : sa ] below , see figure [ f:23 ] for illustration . [ t : sa0 ] there is a dense set @xmath31 with the following property . for each @xmath32 there is a self - affine function @xmath33 such that for all @xmath9 $ ] 1 . if @xmath34 is @xmath14-hlder continuous for some @xmath15 , then @xmath35 ; 2 . if @xmath34 is of bounded variation , then @xmath36 . for a stochastically self - affine process , fractional brownian motion ( see definition [ d : fbm ] ) , we prove the following . [ t : fbm0 ] let @xmath0 and let @xmath8\}$ ] be a fractional brownian motion of hurst index @xmath3 . then , almost surely , for all @xmath9 $ ] 1 . if @xmath37 is @xmath14-hlder continuous for some @xmath15 , then @xmath35 ; 2 . if @xmath37 is of bounded variation , then @xmath38 . [ c : fbm ] let @xmath0 and let @xmath8\}$ ] be a fractional brownian motion of hurst index @xmath3 . then @xmath39 let @xmath40 be the zero set of @xmath16 and let @xmath41 : b(t)=\max_{s\in [ 0,t ] } b(s)\}$ ] be the set of record times of @xmath16 . it is classical that , almost surely , @xmath42 , see ( * ? ? ? * chapter 18 ) . for the record , let us state the following , more subtle fact . [ p : r ] almost surely , @xmath43 . we could not find a reference for this in the literature , and include a proof in section [ s : fbm ] . clearly @xmath40 and @xmath44 witness that theorem [ t : fbm0 ] and corollary [ c : fbm ] are best possible . simon @xcite proved that a standard linear brownian motion is not monotone on any set of positive lebesgue measure . theorem [ t : fbm0 ] for @xmath45 with hausdorff dimension in place of upper minkowski dimension is due to balka and peres @xcite . the methods used there do not extend to minkowski dimension or to general exponents @xmath3 . related results in the discrete setting , concerning non - decreasing subsequences of random walks , can be found in @xcite . now we consider higher dimensional brownian motion . let @xmath46 and @xmath23\to { \mathbb{r}}^d$ ] . we say that @xmath47 is _ non - decreasing _ on a set @xmath9 $ ] if all the coordinate functions of @xmath48 are non - decreasing . [ t:2dbm ] let @xmath8\}$ ] be a standard two - dimensional brownian motion . then , almost surely , there exists a compact set @xmath18 $ ] such that @xmath16 is non - decreasing on @xmath49 and @xmath19 . corollary [ c : fbm ] ( or ( * ? ? ? * theorem 1.2 ) ) implies that , almost surely , the @xmath50-dimensional brownian motion @xmath16 can not be non - decreasing on any set of hausdorff dimension larger than @xmath51 . the following problem remains open in all dimensions @xmath46 . let @xmath46 and let @xmath52 be a standard @xmath50-dimensional brownian motion . what is the supremum of the numbers @xmath53 such that , almost surely , @xmath16 is non - decreasing on some set of hausdorff dimension @xmath53 ? finally , we prove restriction theorems for a generic @xmath3-hlder continuous function ( in the sense of baire category ) , see the following section for the details .

for let denote the supremum of the numbers such that every-hlder continuous function is of bounded variation on a set of hausdorff dimension . kahane and katznelson ( 2009 ) proved the estimate and asked whether the upper bound is sharp . the upper bound on is a consequence of the following theorem . let\}$ ] be a fractional brownian motion of hurst index . then , almost surely , there exists no set $ ] such that and is of bounded variation . furthermore , almost surely , there exists no set $ ] such that and is-hlder continuous for some . the zero set and the set of record times of witness that the above theorems give the optimal dimensions . we also prove similar restriction theorems for deterministic self - affine functions and generic-hlder continuous functions . finally , let\}$ ] be a two - dimensional brownian motion . we prove that , almost surely , there is a compact set $ ] such that and is non - decreasing in each coordinate . it remains open whether is best possible .

for let denote the supremum of the numbers such that every-hlder continuous function is of bounded variation on a set of hausdorff dimension . kahane and katznelson ( 2009 ) proved the estimate and asked whether the upper bound is sharp . we show that in fact . let and denote the hausdorff and upper minkowski dimension , respectively . the upper bound on is a consequence of the following theorem . let\}$ ] be a fractional brownian motion of hurst index . then , almost surely , there exists no set $ ] such that and is of bounded variation . furthermore , almost surely , there exists no set $ ] such that and is-hlder continuous for some . the zero set and the set of record times of witness that the above theorems give the optimal dimensions . we also prove similar restriction theorems for deterministic self - affine functions and generic-hlder continuous functions . finally , let\}$ ] be a two - dimensional brownian motion . we prove that , almost surely , there is a compact set $ ] such that and is non - decreasing in each coordinate . it remains open whether is best possible .

hep-th9508172

i

more than twenty years ago , brezin , le guillou and zinn - justin ( bgz ) studied the phase transition of a cubic anisotropic system by means of renormalization group equations @xcite . within a @xmath4-expansion , they found that to lowest nontrivial order in @xmath5 , the only stable fixed point for @xmath6 is the @xmath7-symmetric one , where @xmath8 is the number of field components appearing in the cubic anisotropic model . they interpreted this as an indication that the anisotropy is irrelevant as long as @xmath8 is smaller than four . for @xmath9 , the isotropic fixed point destabilized and the trajectories crossed over to the cubic fixed point . recently , our knowledge of perturbation coefficients of the renormalization group functions of the anisotropic system was extended up to the five - loop level by kleinert and schulte - frohlinde @xcite . since the perturbation expansions are badly divergent , they do not directly yield improved estimates for the crossover value @xmath10 where the isotropic fixed point destabilizes in favor of the cubic one . an estimate using pad approximants @xcite indicates @xmath10 to lie below @xmath11 , thus permitting real crystals to exhibit critical exponents of the cubic universality class . for a simple @xmath12-theory , the pad approximation is known to be inaccurate . in fact , the most accurate renormalization group functions for that theory have been obtained by combining perturbation expansions with large - order estimates and using a resummation procedure based on borel - transformations @xcite@xcite . it is the purpose of this paper to derive the large - order behavior of the renormalization group functions for the anisotropic @xmath13-theory . in a forthcoming paper we will combine these results with the five - loop perturbation expansion of kleinert and schulte - frohlinde to derive the precise value for the crossover value @xmath10 . for the simple @xmath13-theory , the large - order behavior of perturbation coefficients has been derived by lipatov @xcite , bgz @xcite and others @xcite@xcite in a number of papers . the generalization to the @xmath7-symmetric case was given in @xcite . an equivalent method for calculating the large - order behavior is based on the observation that for a negative coupling constant green functions possess an exponentially small imaginary part due to the fact that the ground state is unstable @xcite . the imaginary part is associated with the tunneling decay rate of the ground state . it determines directly the large - order behavior of the perturbation coefficients via a dispersion relation in the complex coupling constant plane . in the semiclassical limit , the imaginary part of all green functions can be calculated with the help of classical solutions called _ instantons_. for a massless @xmath13-theory in @xmath14 space dimensions , these instantons can be found analytically . the imaginary part is a consequence of a negative frequency mode in the spectrum of the fluctuation operator , whose determinant enters the one - loop correction to the instanton contribution . mckane , wallace and de alcantara bonfim @xcite found a way to continue the results of the @xmath13-theory in @xmath14 to a field theory in @xmath15 dimensions . they proposed an extended dimensional regularization scheme for nonperturbative renormalizing the imaginary parts of vertex functions . in the present work we have to extend this scheme to the case of a @xmath16-theory with cubic anisotropy , where the energy functional has the following form : @xmath17 \,\ , .\ ] ] for @xmath18 , the corresponding model in quantum mechanics was first studied by banks , bender and wu @xcite who used multidimensional wkb - techniques to derive the large - order behavior of the perturbation expansion for the ground state energy . in @xmath19 janke @xcite presented a more efficient calculation using a path integral approach . in the present work , this approach will be generalized to quantum field theory and extended by a careful discussion of the region near the isotropic limit @xmath20 . this is important , since the infrared - stable cubic fixed point is expected to appear very close to the @xmath7-symmetric one . in fact , it will be sufficient to give the quantum - field theoretical generalization of @xcite in terms of an expansion about the isotropic case in powers of @xmath1 . the paper is organized as follows . the method is developed by treating first the case @xmath21 . in we derive the feynman rules for the power series expansion of all green functions around the isotropic limit . in we calculate the small - oscillation determinants for the transversal and longitudinal fluctuations . in we use the extended renormalization scheme of @xcite to find the full ( real and imaginary ) vertex functions , and derive the renormalization constants to one loop . in we calculate the imaginary parts of the renormalization - group functions and thus the large - order behavior of the perturbation coefficients . in , finally , we extend the results to the physically relevant case @xmath22 .

for the anisotropic$]-theory with we calculate the imaginary parts of the renormalization - group functions in the form of a series expansion in , around the isotropic case . the vertex functions in the presence of instantons are renormalized with the help of a nonperturbative procedure introduced for the simple-theory by mckane et al .

for the anisotropic$]-theory with we calculate the imaginary parts of the renormalization - group functions in the form of a series expansion in , around the isotropic case . dimensional regularization is used to evaluate the fluctuation determinants for the isotropic instanton near the space dimension . the vertex functions in the presence of instantons are renormalized with the help of a nonperturbative procedure introduced for the simple-theory by mckane et al .

astro-ph0110540

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the recent discovery of extrasolar planets on orbits very close to their parent stars ( mayor & queloz 1995 ; marcy et al . 2000 ; vogt et al . 2000 ; butler 2001 ) has raised a number of questions about the formation mechanisms of such systems . these close - in planetary companions are all presumed to be gas giants , typically with masses of the order of jupiter mass @xmath5 g. it is very unlikely that such planets were formed at their present locations ( boss 1995 ) : current theories ( mizuno 1980 ; bodenheimer & pollack 1986 ) predict that giant planets were formed by gas accretion onto massive ( @xmath6 ) rocky core which themselves are the result of accumulation of a large number of icy planetesimals . the most favorable conditions for this process exist beyond the so - called `` snow line '' ( hayashi 1981 ; sasselov & lecar 2000 ) which is estimated to lie several au from the star , far larger than the actual orbital radii of the known extrasolar planets . the most popular theory explaining this paradox presumes that giant planets were indeed formed far outside their present locations but then migrated inwards as a result of tidal interaction with a gaseous nebula ( ward 1997a ) . if a planet is fully immersed in a gas disk then the migration process works as follows : planetary gravitational torques produce density waves carrying angular momentum in the surrounding gas disk . for the planet to migrate inwards ( outwards ) two conditions must be fulfilled : ( 1 ) interaction with the outer ( inner ) part of the disk should be stronger than with the inner ( outer ) one , and ( 2 ) , the waves must not return to the planet . in a finite disk the only way to fulfill the second condition is for the waves to dissipate so that their angular momentum gets transferred to the disk flow . it was demonstrated by ward ( 1986 ) that in keplerian disks conditions are usually such that the planet tends to migrate inwards . the typical timescale of this process , @xmath7 yr for a jupiter - mass planet , is very short compared with the lifetime of the nebula itself ( @xmath8 yr ) . this short timescale presents a significant problem for the migration scenario because the migrating planet is likely to drift right into its parent star in very short time . various mechanisms to _ stop _ migration have been proposed : formation of a cavity in the inner disk by magnetospheric activity of the central star ( shu et al . 1994 ) , resonant interaction with another planet in the system ( masset & snellgrove 2001 ) , spin - orbital interaction with the central star ( trilling et al . 1998 ) , etc . on the other hand , there is a process almost inevitable in the course of giant planet formation gap formation in the disk near the planet which can effectively _ slow down _ planet migration . when the planet is not very massive its gravity can not strongly affect the disk surface density distribution in its vicinity and strong interaction with the gas disk through lindblad resonances ( goldreich & tremaine 1980 , hereafter gt80 ) leads to the rapid migration described above ( so called `` type i migration '' , ward 1997a ) . as the planetary mass grows the drift speed increases but at some point the strength of the torque exerted on the surrounding gas becomes so large that planet simply repels the gas and a gap forms . this strongly diminishes the tidal interaction because it is usually dominated by high - order lindblad resonances which now lie inside the gap . thus the orbital evolution of the planet becomes tied to the evolution of the disk and it migrates on the viscous timescale of the disk which could be much longer ( @xmath9 yr or even longer , depending on the viscosity in the disk ) . this stage is called `` type ii migration '' and its existence might help to explain the survival of planetary systems in the course of their orbital evolution or at least significantly alleviate this problem ( ward 1997 ) . gap formation also provides a reason for the existence of a maximum mass of extrasolar giant planets ( nelson et al . 2000 ) . in view of all this a very important question arises : how massive should a planet be in order to open a gap ? the answer depends not only on the conditions in the nebula ( surface density , temperature , viscosity ) but also on the dissipation mechanism of planet - induced density waves and the mobility of the planet . if the viscosity in the disk is absent and the planet is not drifting there is no mechanism which could oppose tidal torques and a gap is opened by an arbitrarily small perturber . this conclusion changes if planetary migration is taken into account self - consistently because a low - mass planet is able to drift through the gap before gap fully forms ( ward & hourigan 1989 , hereafter wh89 ) . the planet must slow down its migration somehow for the gap to be opened . in their study wh89 demonstrated that the disk surface density is enhanced in front of the drifting planet and is reduced behind it . this effect _ diminishes _ the difference between the torques produced by the planet in the disk outside and inside its orbit and acts to _ slow down _ the migration . for some mass of the planet its steady drift becomes impossible , the planet halts , and gap formation ensues . thus , in this picture there is a minimum mass for gap opening even in an inviscid disk . assuming that wave dissipation is a very rapid process and that angular momentum is immediately transferred to the disk wh89 estimated this critical mass to be @xmath10 here @xmath11 is a surface density of the disk , @xmath12 is its geometric thickness ( here @xmath13 is the sound speed and @xmath14 is the angular orbital frequency ) , and @xmath15 is the distance from the central star . typical protosolar nebular parameters were summarized by hayashi ( 1980 ) in the form of the minimum mass solar nebula ( mmsn ) model : @xmath16 using ( [ hayashi ] ) we obtain that @xmath17 and @xmath18 g at @xmath19 au which is about the mass of mercury . of course , if the disk possesses nonzero viscosity this mass would increase because viscous diffusion opposes gap formation . lin & papaloizou ( 1993 , hereafter lp93 ) present a different point of view on the planetary mass required to open a gap . they also assume damping of the density waves to be local but require mass of the planet to be high enough for the waves to be strongly nonlinear from the beginning and shock immediately transferring angular momentum to the fluid . they showed that this minimum mass corresponds to the case when the hill s radius of the planet @xmath20 ( @xmath21 is the mass of the planet , @xmath22 is the mass of the central star ) is comparable to the vertical disk scaleheight yielding @xmath23 as the relevant mass for opening a gap [ the numerical estimate in equation ( [ m1 ] ) is made for mmsn parameters given in equation ( [ hayashi ] ) ] . another way to look at this restriction is to notice that at @xmath24 the pressure gradient in the disk in the vicinity of the planet becomes so high that epicyclic frequency becomes imaginary and rayleigh s instability develops . thus , this restriction could be considered as an upper limit on the mass of the planet that does not open a gap . values of the gap opening mass estimated by these two approaches differ by a huge factor . indeed : @xmath25 where @xmath2 is the toomre stability parameter ( binney & tremaine 1987 ) and @xmath26 is the epicyclic frequency ( @xmath27 for keplerian rotation law ) ; using mmsn parameters given by equation ( [ hayashi ] ) we obtain that @xmath28 and @xmath29 at @xmath19 au meaning that @xmath30 . ward & hourigan ( wh89 ) assume that the damping mechanism is independent of the planetary mass and essentially linear , such as viscous dissipation ( takeuchi et al . 1996 ) or radiative damping ( cassen & woolum 1996 ) . however these mechanisms are probably ineffective in cold , weakly ionized and optically thick systems such as protoplanetary disks ( hawley & stone 1998 ; goodman & rafikov 2001 , hereafter gr01 ) ; in this case tides raised by low - mass planets could propagate much further than just a fraction of disk scale length and gap opening requires significantly more massive perturbers ( ward 1997a ) . requiring waves to be strongly nonlinear and damp immediately as a necessary condition for gap formation is also probably too radical . indeed , density waves produced by the planet can still evolve due to weak nonlinearity and are able to form a weak shock even if the planet is less massive than @xmath0 ( gr01 ) . this mechanism can lead to a gap formation by lower mass planets than lp93 assumed . clearly to obtain a reliable estimate of the critical planetary mass one must use a realistic wave damping prescription . goodman & rafikov ( gr01 ) have considered nonlinear evolution of the density waves produced by low - mass planets in two - dimensional disks using the shearing sheet approximation and assuming the background surface density and sound speed to be constant . they have found that a shock is formed quite rapidly ( depending on @xmath31 , typically after travelling several disk scaleheights from the planet but still not immediately ) because the radial wavelength of the perturbation constantly decreases while its amplitude increases ( as a consequence of angular momentum flux conservation ) thus facilitating wavefront breaking . after wave shocks its angular momentum is gradually transferred to the disk fluid leading to surface density evolution ( thus violating the constant surface density assumption ) . rafikov ( 2001 ) generalized this analysis by taking into consideration the effects of spatial variations of the surface density and sound speed in the disk as well as the cylindrical geometry of the problem . realistic prescription of the global angular momentum dissipation was provided under the condition that the surface density and sound speed vary on scales larger than the wavelength of the perturbation . in this paper we study the criterion for gap opening by planetary tides by determining under which conditions a steady - state solution for the disk surface density perturbation is no longer possible ( method used in wh89 ) . planet migration is taken into account self - consistently . tidal perturbations are supposed to be damped nonlocally by weak nonlinearity and their angular momentum is assumed to be transferred to the disk , as described quantitatively in rafikov ( 2001 ) . for the gap opening the assumption of varying surface density in rafikov ( 2001 ) is the most important one because it allows us to solve the problem self - consistently ( angular momentum transfer depends on how the surface density is distributed radially ) . after developing the general analytical apparatus in [ geneq ] we first explore the case of an inviscid disk in [ visc - less ] to highlight the most important physical mechanisms relevant for gap formation . then we generalize consideration to include the disk viscosity in [ viscous ] . finally , in [ disc ] we discuss our results .

density waves launched by the planet are assumed to be damped as a result of their nonlinear evolution leading to shock formation and its subsequent dissipation . as a consequence wave angular momentum is transferred to the disk , leading to evolution of its surface density . planetary migration is an important ingredient of the theory ; effects of the planet - induced surface density perturbations on the migration speed are considered . an analytical limit on the planetary mass necessary to open a gap in an inviscid disk is derived .

gap formation in a gas disk triggered by disk - planet tidal interaction is considered . density waves launched by the planet are assumed to be damped as a result of their nonlinear evolution leading to shock formation and its subsequent dissipation . as a consequence wave angular momentum is transferred to the disk , leading to evolution of its surface density . planetary migration is an important ingredient of the theory ; effects of the planet - induced surface density perturbations on the migration speed are considered . a gap is assumed to form when a stationary solution for the surface density profile is no longer possible in the frame of reference migrating with the planet . an analytical limit on the planetary mass necessary to open a gap in an inviscid disk is derived . the critical mass turns out to be smaller than mass for which planetary hill s radius equals disk scaleheight by a factor of at least ( is the toomre stability parameter ) depending on the strength of the migration feedback . in viscous disks the critical planetary mass could vary from to , depending on the disk viscosity . this implies that a gap could be formed by a planet with mass depending on the disk aspect ratio , viscosity , and planet s location in the nebula .

1612.00209

i

a self - similar set @xmath0 is defined as the limit set of an iterated function system ( ifs ) of contractive similitudes , and the iteration is addressed by a symbolic space of finite words which forms a tree . the topological boundary of the tree is a cantor - type set . however the tree does not capture all the geometric and analytic properties of @xmath0 . by incorporating more information of @xmath0 onto the tree , kaimanovich @xcite first explored the concept of `` augmented tree '' on the sierpinski gasket such that it is a hyperbolic graph in the sense of gromov ( @xcite@xcite ) and its hyperbolic boundary is homeomorphic to the gasket . lau and wang ( @xcite@xcite ) extended this idea on more general self - similar sets satisfying the open set condition ( osc ) or the weak separation condition . recently they @xcite completed the previous studies by removing some superfluous conditions , and obtained that for any ifs , the augmented tree is always a hyperbolic graph . moreover , the hyperbolic boundary is hlder equivalent to the @xmath0 . the setup of augmented trees connects fractal geometry , graph theory and markov chains , it has been frequently used to study the random walks on self - similar sets and the induced dirichlet forms ( @xcite@xcite@xcite@xcite@xcite ) . on the other hand , the author and his coworkers made a first attempt to apply augmented trees to the study on lipschitz equivalence of self - similar sets . in a series of papers ( @xcite@xcite@xcite ) , we investigated in detail the lipschitz equivalence of totally disconnected self - similar sets with equal contraction ratio and their hyperbolic boundaries . recall that two metric spaces @xmath1 and @xmath2 are said to be _ lipschitz equivalent _ , write @xmath3 , if there exists a bi - lipschitz map @xmath4 , i.e. , @xmath5 is a bijection and there exists a constant @xmath6 such that @xmath7 the lipschitz equivalence of cantor sets was first considered in @xcite and @xcite . for its extension on self - similar sets , it has been undergoing rapid development recently ( @xcite@xcite@xcite@xcite@xcite@xcite@xcite@xcite@xcite@xcite@xcite@xcite ) . however , most of the studies are based on the nice geometric structure of self - similar sets such as cantor sets or totally disconnected self - similar sets with osc . there are few results on the non - totally disconnected self - similar sets ( @xcite@xcite ) or self - similar sets without osc . in this paper , we unify our previous approaches on augmented trees and consider the lipschitz equivalence of more general self - similar sets which allow non - equal contraction ratios and substantial overlaps . let @xmath8 be an ifs on @xmath9 where @xmath10 has a contraction ratio @xmath11 , let @xmath0 be the self - similar set of the ifs satisfying @xmath12 . let @xmath13 and @xmath14 be the symbolic space ( by convention , @xmath15 ) . for @xmath16 , we denote by @xmath17 , and @xmath18 . let @xmath19 . we also define a new symbolic space @xmath20 in the special case that all @xmath21 are equal , @xmath22 . in general , @xmath23 is a proper subset of @xmath24 . if @xmath25 , we denote the length by @xmath26 , and say that @xmath27 lies in level @xmath28 . the @xmath23 has a natural tree structure according to the standard concatenation of finite words , we denote the edge set by @xmath29 ( @xmath30 for vertical ) . we also define a horizontal edge for a pair @xmath31 in @xmath32 if @xmath33 and @xmath34 where @xmath35 is a fixed constant , and denote this set of edges by @xmath36 ( @xmath37 for horizontal ) . let @xmath38 , then the graph @xmath39 is an augmented tree in the sense of kaimanovich ( @xcite@xcite ) . lau and wang @xcite already showed that the augmented tree @xmath39 is a gromov hyperbolic graph and its hyperbolic boundary @xmath40 under a visual metric is hlder equivalent to the @xmath0 . based on this , our approach to the lipschitz equivalence of self - similar sets is to lift the consideration to the augmented tree @xmath41 . we define a _ horizontal component _ of @xmath23 to be the maximal connected horizontal subgraph @xmath42 in some level with respect to @xmath36 . let @xmath43 be the set of all horizontal components of @xmath23 . for @xmath44 , we use @xmath45 to denote the union of @xmath42 and its all descendants , with the subgraph structure inherited from @xmath39 . we say that @xmath46 are equivalent if @xmath45 and @xmath47 are graph isomorphic . we call @xmath39 _ simple _ if there are only finitely many equivalence classes in @xmath43 . similarly , we can define a simple tree for @xmath48 if there are only finitely many equivalence classes of vertices . obviously , if @xmath39 is simple then @xmath49 is simple . we use @xmath50 ( resp . @xmath51 ) to denote the incidence matrix of @xmath39 ( resp . @xmath49 ) , which encodes graph relation of the equivalence classes . let a vector @xmath52 represent the finite classes in @xmath53 by means of the classes of vertices . ( see definition [ def of simple tree ] and remark [ b - u - remark ] . ) the following is our first main result [ th - main ] suppose an augmented tree @xmath54 is simple , and suppose the incidence matrix @xmath50 is @xmath55-rearrangeable . then there exists a near - isometry between the augmented tree @xmath54 and the tree @xmath56 , so that @xmath57 . the concept of rearrangeable matrix was initiated by the author ( @xcite@xcite@xcite ) in studying the lipschitz equivalence . a lot of nonnegative integer matrices including primitive matrices are rearrangeable . it is a combinatorial device to arrange the vertices and edges of the augmented tree properly to construct the near - isometry , which is stronger than the rough isometry in literature . in the paper , we extend the original definition of rearrangeable matrix so as to solve more general situations ( please see definition [ def rearrangeable ] ) . for an ifs with substantial overlaps ( i.e. , without the osc ) , it may happen that @xmath58 for @xmath59 . in this case , the augmented tree @xmath39 may be no longer simple . but we can modify the augmented tree by identifying @xmath60 for @xmath61 and @xmath58 , and let @xmath62 denote the quotient space with the induced graph . moreover , we can further reduce the graph @xmath63 into a tree @xmath64 ( see section 2 ) . then following the same proof of theorem [ th - main ] , we have [ th - quotient ] suppose the quotient space @xmath65 of an augmented tree is simple , and suppose the incidence matrix @xmath50 is @xmath55-rearrangeable . then there exists a near - isometry between the quotient space @xmath65 and the reduced tree @xmath66 , so that @xmath67 . an ifs is called _ strongly separated _ @xcite if @xmath68 for @xmath69 , in this case the @xmath0 is called _ dust - like _ ( @xcite@xcite ) . by reducing the lipschitz equivalence on the augmented trees in theorems [ th - main ] and [ th - quotient ] to the self - similar sets , we have our second main result [ th - main2 ] under the assumption of theorem [ th - main ] , the associated self - similar set @xmath0 is lipschitz equivalent to a dusk - like self - similar set with the same contraction ratios as for @xmath0 . under the assumption of theorem [ th - quotient ] , the associated self - similar set @xmath0 is lipschitz equivalent to a cantor - type set ( may be not a dust - like self - similar set ) . the first part of theorem [ th - main2 ] is applied for the self - similar sets satisfying the osc ; while the second part is applied for the self - similar sets satisfying the weak separation condition ( wsc ) @xcite . the wsc contains many important overlapping cases , it has been studied extensively in connection with the multifractal structure of self - similar measures ( see @xcite@xcite@xcite and the references therein ) . finally , we provide a criterion for the hyperbolic boundary ( or the associated self - similar set ) to be totally disconnected ( i.e. , a cantor - type set ) from the perspective of augmented trees which also answers an open question raised in @xcite . [ th - main3 ] let @xmath54 be an augmented tree of bounded degree . then the hyperbolic boundary @xmath40 ( or the self - similar set @xmath0 ) is totally disconnected if and only if the sizes of horizontal components are uniformly bounded . the theorem is still valid if we replace @xmath23 by its quotient space @xmath70 . the paper is organized as follows : in section 2 , we recall basic results on hyperbolic graphs and introduce the notion of simple augmented tree ; moreover , we state theorem [ th - main ] and prove theorem [ th - quotient ] there . in section 3 , we generalize the concept of rearrangeable matrix and prove theorem [ th - main ] . in section 4 , we prove theorem [ th - main2 ] and give two examples to illustrate our results . theorem [ th - main3 ] will be showed in section 5 . we include some remarks and open questions in section 6 .

given an iterated function system ( ifs ) of contractive similitudes , the theory of gromov hyperbolic graph on the ifs has been established recently . in the paper , we introduce a notion of simple augmented tree which is a gromov hyperbolic graph . by generalizing a combinatorial device of rearrangeable matrix , we show that there exists a near - isometry between the simple augmented tree and the symbolic space of the ifs , so that their hyperbolic boundaries are lipschitz equivalent . we then apply this to consider the lipschitz equivalence of self - similar sets with or without assuming the open set condition . moreover , we also provide a criterion for a self - similar set to be a cantor - type set which completely answers an open question raised in .

given an iterated function system ( ifs ) of contractive similitudes , the theory of gromov hyperbolic graph on the ifs has been established recently . in the paper , we introduce a notion of simple augmented tree which is a gromov hyperbolic graph . by generalizing a combinatorial device of rearrangeable matrix , we show that there exists a near - isometry between the simple augmented tree and the symbolic space of the ifs , so that their hyperbolic boundaries are lipschitz equivalent . we then apply this to consider the lipschitz equivalence of self - similar sets with or without assuming the open set condition . moreover , we also provide a criterion for a self - similar set to be a cantor - type set which completely answers an open question raised in . our study extends the previous works .

astro-ph9610096

i

each specific model for the development of cosmogonic structure ( e.g. the hot dark matter [ hdm ] or cold dark matter [ cdm ] scenarios ) has one free parameter , the amplitude of the density power spectrum . in the light of the cobe observations ( smoot et al . 1992 ; gorski et al . 1994 ) , it occurs , for the first time , that this parameter is fixed ( @xmath0 ) by the determination on the @xmath1 scale in the linear regime . with its amplitude fixed , a secure determination of the potential fluctuation on any other scale provides a test for a particular cosmological model . any single conflict between the theory and reality can falsify the former . the most leverage is obtained for tests which are made on angular scales as far away as possible from the cobe measurements . but they should not be so small as to be greatly influenced by the difficulty inherent in modeling the physics of the gaseous , baryonic components ( @xmath2 ) . thus , critical tests are best made on scales @xmath3 . gravitational lensing measures _ directly _ the fluctuations in the gravitational potential along random lines of sight to distant objects . in contrast , the conventional tools for comparing cosmogonic theories with observations rely on either galaxy density or galaxy velocity information , both of which unavoidably suffer from the uncertain situations with regard to density or velocity bias of galaxies over the underlying mass distribution , hampering our attempts to understand the more fundamental " question concerning the mass evolution and distribution . hence gravitational lensing provides a powerful , independent test of cosmogonic models , because of its unique ability to directly measure the gravitational potential fluctuations . the sampled properties of cosmic structure on certain scales are also more fair " than other conventional measures owing to the fact that lines of sight to distant objects are _ random _ with regard to the foreground matter distribution . we present here a method to determine gravitational lensing effects for realistic mass distributions from the weak to the strong lensing regime . first , we fill the universe densely " with adjacent mass cubes , taken from cosmological simulations , along the line of sight , where inside each cube the matter is projected onto the middle plane . then , we study the lensing properties of the intervening matter by shooting " light rays through the lens ( mass ) planes representing matter distributions along redshifts ( cf . figure 1 for a schematic version of our actual scheme [ which uses of order 100 lens planes ] ) . the gravitational lens effects of many lens planes have been studied by several authors in the past ( e.g. , kochanek & apostolakis 1988 ; schneider & weiss 1988a , b ; jaroszyski 1989 , 1991 ; jaroszyski et al . 1990 ; lee & paczyski 1990 ; babul & lee 1991 ; bartelmann & schneider 1991 ) . these authors have typically used point lenses or isothermal spheres which were randomly distributed on individual lens planes . the number of lens planes used ranges from 2 to 32 . most of these studies involved the regime of weak lensing , i.e. slight effects on the magnification of sources , but not multiple imaging . rauch ( 1991 ) studied the microlensing effect of three - dimensionally distributed point lenses on high redshift supernovae . a complementary approach to the one described here , a semi - analytical method using the press - schechter formalism was first used by narayan and white ( 1987 ) , and more recently applied by kochanek ( 1995 ) to various cosmological models . a very preliminary attack on the problem of studying the lensing properties of different cosmological models has been presented by cen et al . ( 1994 , hereafter cgot ) . in that work no ray tracing was done . rather it was simply checked whether or not mass accumulations were greater than the critical level ( cf . turner , ostriker & gott 1984 , henceforth tog ) at which multiple imaging will occur . the mathematical and physical treatment here is far superior to that adopted in cgot ; the essential conclusions for the standard cdm scenario , however , are unchanged . some of these conclusions have already been presented in wambsganss et al . ( 1995 ) . here we provide the mathematical basis of that method as well as some tests of the method . we use cosmological simulations for the standard cobe - normalized @xmath4 , cdm universe as an example to illustrate the method . a first report on the results has been published elsewhere ( wambsganss et al . 1995 ) . in principle , mass distributions from any cosmological model can be treated with this method . the ultimate goal of this project is to provide a quantitative comparison of the lensing properties of different cosmological models ( e.g. , cdm with @xmath4 ; low density ( open ) cdm models ; mixed dark matter [ mdm ] models ; flat models with a cosmological constant ; isocurvature models etc ) with observations . for example , we anticipate that different cosmological models , which are typically tuned to give approximately the right properties when being compared with our local universe at redshift @xmath5 , will show different lensing properties because they reach the final state ( @xmath5 ) from quite different paths . thus , the typical lens will appear at different redshifts in different models . for example , in mdm models , where structure forms late , the lenses will have small redshift , but in isocurvature models , where the first structures form shortly after recombination , the average lens redshift will be much larger . for many questions ( e.g. the fraction of multiply imaged quasars as a function of angular separation , magnification ratio , redshift ) one needs a large number of simulations with different realizations of the matter distribution , in order to give quantitative answers on a good statistical basis . for other applications , e.g. detailed studies of lensing by clusters of galaxies and the effects of chance alignments with foreground or background matter , a specific realization must be studied in detail . while it is in principle possible to simulate , numerically in one simulation , the entire path from observer to source ( very large simulations with elongated geometry have been done ; e.g. , park & gott 1991 ) , technical limitations preclude this at the present time . for now we perform many independent simulations for different volumes along the line of sight ( since repeated structures are obviously to be avoided ) . we have developed an approximate technique for this which we term the convolution method " . physically it consists of assuming that separate pieces of an inhomogeneous universe act as if they were part of an homogeneous universe of the same mean density as themselves . details of the method , which is an improved version over that applied in cgot , are presented in the appendix , as well as detailed checks which indicate the accuracy of the approximations involved . although we think this convolution technique describes the properties accurately enough on the scales in which we are interested , we hope in the future to be able to avoid this step and be able to use simulations that have a large enough dynamic range to cover a representative part of the universe and have high enough resolution in order to represent the potential wells of galaxies , groups and clusters appropriately . the analytical techniques presented in this paper for the determination of the lens properties are independent of the details of the method for computing the evolution . we will use the same ray tracing method when later we utilize higher resolution numerical evolution techniques which do not depend on the convolution method . the rest of the paper is organized as follows . the ray - shooting technique is described and discussed in detail in section 2 . as a result of this ray - shooting , we obtain a mapping of the rays from the image plane ( sky plane ) to the source plane . with this mapping it is possible to determine the magnification distribution of background sources , the fraction of the source plane that is multiply imaged , the positions and shapes as well as the topology of the caustics and the corresponding critical lines ( section 3 ) . for a set of given source positions and shapes in the source plane at a given redshift , we determine and analyze the corresponding image configurations as they would appear in the sky , as is shown in sections 4 and 5 . finally in section 6 we give a short summary and describe the planned applications and the types of predictions we will be able to make for the different cosmological models .

gravitational lensing directly measures mass density fluctuations along the lines of sight to very distant objects . no assumptions need to be made concerning bias , the ratio of fluctuations in galaxy density to mass density . , this method is applied here to a standard cold dark matter universe . different cosmological models differ , increasingly with redshift , in their predictions for the mass ( thus gravitational potential ) distributions . our ultimate goal is to apply this method to a number of cosmogonic models and to eliminate some models whose gravitational lensing properties are inconsistent with those observed .

gravitational lensing directly measures mass density fluctuations along the lines of sight to very distant objects . no assumptions need to be made concerning bias , the ratio of fluctuations in galaxy density to mass density . hence , lensing is a very useful tool to study the universe at low to moderate redshifts . we describe in detail a new method to trace light rays from redshift zero through three dimensional mass distribution to high redshift . as an example , this method is applied here to a standard cold dark matter universe . we obtain a variety of results , some of them statistical in nature , others from rather detailed case studies of individual lines of sight " . among the former are the frequency of multiply imaged quasars , the distribution of separation of the multiple quasars , and the redshift distribution of lenses : all that as a function of quasar redshift . we find effects from very weak lensing to highly magnified multiple images of high redshift objects , which , for extended background sources , ( i.e. galaxies ) , range from slight deformations of the shapes through tangentially aligned arclets up to giant luminous arcs . different cosmological models differ , increasingly with redshift , in their predictions for the mass ( thus gravitational potential ) distributions . our ultimate goal is to apply this method to a number of cosmogonic models and to eliminate some models whose gravitational lensing properties are inconsistent with those observed .

1310.2206

i

this paper studies two - channel finite impulse response ( fir ) perfect reconstruction filter banks @xcite . figure [ ds_poly ] depicts the polyphase - with - advance representation of a filter bank @xcite . a lifting factorization @xcite decomposes the polyphase matrices @xmath0 and @xmath1 into upper and lower triangular _ lifting matrices _ , a variant of a well - known result from linear algebra ( e.g. , ( * ? ? ? * proposition vii.2.11 ) ) stating that any matrix over a euclidean domain can be diagonalized by elementary matrices . the construction of such factorizations via the euclidean algorithm was given for general fir perfect reconstruction filter banks in @xcite and was subsequently refined for linear phase filter banks in @xcite . these latter works were motivated by the iso / iec jpeg 2000 standard @xcite , which specifies whole - sample symmetric ( ws , or fir type 1 linear phase ) wavelet filter banks in terms of _ _ half-__sample symmetric ( hs , or fir type 2 ) lifting filters . ( the connection between filter banks and wavelet transforms is well - known and will not be treated here ; see @xcite . ) elementary matrix decompositions are known to be nonunique and it is also known that , in general , lifting factorizations of filter banks are similarly nonunique . nonetheless , this paper introduces a framework for lifting factorizations , called a _ group lifting structure _ , that allows us to prove _ uniqueness _ of lifting steps under carefully stated hypotheses . this is an extension of the group - theoretic approach to lifting introduced in @xcite . a companion paper @xcite applies the theory developed here to the two principal classes of linear phase filter banks , the ws and hs classes shown in figure [ ws - hs ] . it is proven in @xcite that there is _ only one way _ to factor a given ws filter bank into hs lifting filters as specified in @xcite , even though the given filter bank has many other lifting factorizations into different ( e.g. , nonlinear phase ) lifting filters . as shown in @xcite , factoring hs filter banks into linear phase lifting steps is more complicated than in the ws case , but @xcite contains analogous results on the uniqueness of linear phase lifting factorizations for hs filter banks . a follow - on paper currently in preparation studies the group - theoretic structure of lifting factorizations and determines the groups ( up to isomorphism ) arising in unique lifting factorizations . this work is being applied to develop a group - theoretic numerical framework for optimal filter bank design , a goal motivated by the observation that perfect reconstruction filter banks do not form vector spaces . examples of nonunique lifting factorizations are ubiquitous . consider our old friend , the haar filter bank , normalized with a polyphase determinant of 1 and a lowpass dc response of 1 : @xmath2[c]{,c15c , } 1/2 & 1/2 \\ -1 & 1 \end{ieeeeqnarraybox*}\right]\,.\ ] ] the lifting factorization of ( [ haar ] ) in @xcite and ( * ? ? ? * annex h ) is @xmath3[c]{,c15c , } 1 & 1/2 \\ 0 & 1 \end{ieeeeqnarraybox*}\right ] \left [ \begin{ieeeeqnarraybox*}[{\fontsize{11}{11}\selectfont}][c]{,c15c , } 1 & 0 \\ -1 & 1 \end{ieeeeqnarraybox*}\right]\,.\ ] ] the haar matrix has another factorization , however , with different lifting steps and a diagonal gain scaling matrix : @xmath4[c]{,c15c , } 1/2 & 0 \\ 0 & 2 \end{ieeeeqnarraybox*}\right ] \left [ \begin{ieeeeqnarraybox*}[{\fontsize{11}{11}\selectfont}][c]{,c15c , } 1 & 0 \\ -1/2 & 1 \end{ieeeeqnarraybox*}\right ] \left [ \begin{ieeeeqnarraybox*}[{\fontsize{11}{11}\selectfont}][c]{,c15c , } 1 & 1 \\ 0 & 1 \end{ieeeeqnarraybox*}\right]\,.\ ] ] similar observations are made in @xcite regarding nonunique factorizations of the 4-tap/4-tap ( @xmath5 ) orthogonal wavelet filter bank , and that example can be extended to exhibit nonunique lifting factorizations for the entire family of orthogonal daubechies ( @xmath6 ) wavelet filter banks . to appreciate just how fundamentally nonunique lifting factorizations are , equate ( [ haar_lifting_steps ] ) and ( [ alt_haar_lifting_steps ] ) and move the lifting steps from ( [ haar_lifting_steps ] ) over to the right end of ( [ alt_haar_lifting_steps ] ) . use ( * ? ? ? * section 7.3 ) to factor @xmath7 into lifting steps , yielding factorization ( [ identity_lift ] ) . rcl & = & + & & [ identity_lift ] the formulas in ( * ? ? ? * section 7.3 ) can be used to generate infinitely many other lifting factorizations of the identity . in spite of the general nonuniqueness of elementary matrix decompositions , work on the jpeg 2000 standard raised a number of questions regarding uniqueness of _ linear phase _ lifting factorizations . for instance , the lifting factorization of ws filter banks in ( * ? ? ? * theorem 9 ) involves left matrix `` downlifting '' of a ws transfer matrix into hs lifting steps . one is naturally led to ask whether the same sequence of hs lifting filters is obtained for right matrix factorizations or , more generally , for a mix of left and right factorization operations . for example , the software accompanying @xcite uses the euclidean algorithm to compute nine different lifting factorizations of a 7-tap/5-tap cohen - daubechies - feauveau ws wavelet filter bank based on cubic b - splines @xcite . one factorization ( [ cdf7 - 5 ] ) , computed in ( * ? ? ? * section 5.4 ) in synthesis matrix form , uses hs lifting filters exclusively ( the other eight factorizations involve non - symmetric lifting filters ) . l 2 ^ -9/2 + = [ cdf7 - 5 ] + this exact same linear phase lifting is obtained using both left and right matrix downlifting factorization as in @xcite . other questions about unique factorization arise when factoring hs filter banks , which contain an hs lowpass filter and a half - sample _ anti_-symmetric ( ha , or fir type 4 ) highpass filter , as in figure [ ws - hs](b ) . hs filter banks factor using whole - sample antisymmetric ( wa , or fir type 3 ) lifting filters and concentric equal - length hs base filter banks ( * ? ? ? * section vi ) . for instance , a 6-tap/10-tap hs wavelet filter bank , @xmath8 , can be derived from the same halfband spectral factorization as the 9-tap/7-tap cohen - daubechies - feauveau ws wavelet filter bank in @xcite , ( * ? ? ? * figure 8.8 ) . annex h.4.1.2.1 of @xcite has a factorization of @xmath8 in which it is lifted from a concentric 6-tap/6-tap hs base filter bank by a second - order wa lifting step . at the time that @xcite was written , it was unknown whether it was possible to lift @xmath8 from the haar filter bank using wa lifting filters . the same question was raised about a 10-tap/18-tap hs filter bank that is lifted from a 10-tap/10-tap hs base filter bank in ( * ? ? ? * annex h.4.1.2.2 ) by a fourth - order wa lifting filter . it would have simplified the writing of ( * ? ? ? * annex h ) if all hs filter banks could be lifted by wa lifting filters from a single base filter bank such as the haar , but our results will show this to be impossible . our approach , which is algebraic rather than algorithmic , ignores the mechanism used to _ generate _ lifting factorizations ( e.g. , the euclidean algorithm in @xcite or the `` downlifting '' approach in @xcite ) and instead constrains the universe of allowable lifting steps . under suitable hypotheses , we shall prove uniqueness within precisely defined classes of lifting factorizations . for example , using the results of this paper , we prove in @xcite that one always obtains the same hs lifting filters when factoring a ws filter bank into hs lifting steps , regardless of how the factorization is constructed . in @xcite we also prove uniqueness of the wa lifting filters and the equal - length hs base filter bank that arise in the factorization of a suitably normalized hs filter bank . both results follow from theorem [ thm : unique_factorization ] of the present paper , a result that does not depend on linear phase characteristics . as in @xcite , we make extensive use of the group - theoretic structure of lifting factorizations . readers may be surprised by the absence of traditional ring - theoretic methods for establishing unique factorization results over polynomial domains . while ring - theoretic techniques such as grbner bases @xcite have been used for constructing wavelet filter banks , filter banks are elements of _ matrix _ polynomial rings , which are not integral domains . more significantly , perfect reconstruction filter banks ( including lifting matrices ) are _ units _ in the ring of @xmath9 matrices over the laurent polynomials , @xmath10 $ ] . this means that traditional algebraic methods based on ideal theory can tell us nothing about uniqueness of lifting factorizations . of course , it is also surprising that unique factorization results can be proven using group - theoretic techniques since everything divides everything in a group , but this only highlights the importance of carefully formulating the universe of admissible lifting factors for a given class of polyphase matrices . the author considered recasting the problem in terms of rings of _ causal _ matrix polynomials , but doing so would have required reworking much of the basic theory of lifting and would have distracted attention from the results presented here . fortunately , the group - theoretic approach proved adequate for deriving the results of interest , and it probably resulted in a more elementary level of algebraic machinery than would have been used in a ring - theoretic approach . moreover , the technical crux of the group - theoretic approach , which is the proof of an order - increasing property , would probably be required in a ring - theoretic treatment as well . the reader should note that this paper does not require an extensive background in group theory . the basic material group axioms , subgroups , and homomorphisms found in any introductory text on `` abstract algebra '' will suffice ; e.g. , @xcite . daubechies and sweldens @xcite point out that the euclidean algorithm can generate many different lifting factorizations for a given filter bank , and they pose the analysis and exploitation of nonuniqueness as `` an interesting topic for future research . '' nonuniqueness of lifting factorizations was studied in @xcite to find optimal integer - to - integer liftings for a given reversible filter bank . recent research by zhu and wickerhauser @xcite exploits nonuniqueness to construct liftings that use only `` nearest - neighbor '' data ( i.e. , first - order lifting filters of the form @xmath11 or @xmath12 ) in order to minimize memory fetches in hardware implementations . their work shows that some lifting factorizations are much more ill - conditioned than others , which likely affects rate - distortion scalability in source coding applications . as far as the author is aware , however , there are no prior results establishing _ uniqueness _ of lifting steps for _ any _ classes of lifting factorizations . unlike @xcite , which also apply group theory to multirate discrete - time systems , the present paper makes no use of the specialized theory of _ finite _ groups . moreover , our results are not constrained to finite signals of length @xmath13 : the papers @xcite use group - theoretic tools to analyze the multiscale structure of the underlying space of finite - length signals , whereas we employ group - theoretic methods to analyze the algebraic structure of multirate fir filter banks acting on general discrete - time signals with no length constraints . section [ sec : filter ] reviews filter banks and lifting . section [ sec : abelian ] introduces the group - theoretic concepts that provide the framework for our uniqueness results , which are based on the notion of _ group lifting structures_. section [ sec : linearphase ] presents examples of group lifting structures for factoring ws and hs filter banks into linear phase lifting filters , including the elasf class of reversible hs filter banks lifted from the haar filter bank that was defined by m. adams . in section [ sec : uniqueness ] we define the key _ polyphase order - increasing property _ that a group lifting structure must satisfy to ensure unique lifting factorizations , and we prove the main uniqueness theorem . section [ sec : concl ] contains concluding remarks .

examples of group lifting structures are given for linear phase lifting factorizations of the two nontrivial classes of two - channel linear phase fir filter banks , the whole- and half - sample symmetric classes , including both the reversible and irreversible cases . , it is shown that lifting factorizations are highly nonunique . when certain hypotheses developed in the paper are satisfied , however , lifting factorizations generated by a group lifting structure are shown to be unique . a companion paper applies the uniqueness results proven in this paper to the linear phase group lifting structures for whole- and half - sample symmetric filter banks . = 1 * final revision , reformatted for arxiv.org * filter bank , wavelet , unique factorization , polyphase , lifting , linear phase filter , group .

group lifting structures are introduced to provide an algebraic framework for studying lifting factorizations of two - channel perfect reconstruction fir filter banks . the lifting factorizations generated by a group lifting structure are characterized by abelian groups of lower and upper triangular lifting matrices , an abelian group of unimodular gain scaling matrices , and a set of base filter banks . examples of group lifting structures are given for linear phase lifting factorizations of the two nontrivial classes of two - channel linear phase fir filter banks , the whole- and half - sample symmetric classes , including both the reversible and irreversible cases . this covers the lifting specifications for whole - sample symmetric filter banks in parts 1 and 2 of the iso / iec jpeg 2000 still image coding standard . the theory is used to address the uniqueness of lifting factorizations . with no constraints on the lifting process , it is shown that lifting factorizations are highly nonunique . when certain hypotheses developed in the paper are satisfied , however , lifting factorizations generated by a group lifting structure are shown to be unique . a companion paper applies the uniqueness results proven in this paper to the linear phase group lifting structures for whole- and half - sample symmetric filter banks . = 1 * final revision , reformatted for arxiv.org * filter bank , wavelet , unique factorization , polyphase , lifting , linear phase filter , group .

1310.2206

c

a new algebraic framework , the group lifting structure , has been introduced for studying lifting factorizations of fir perfect reconstruction filter banks . the primary focus of the paper is the uniqueness of the matrix factors arising in lifting factorizations . with no constraints on the lifting process , proposition [ prop : nonuniqueness ] shows that any fir perfect reconstruction filter bank can be irreducibly lifted from any other fir perfect reconstruction filter bank in infinitely many different ways . virtually all such factorizations are mathematical pathologies , however , and are of no interest for engineering applications . to impose constraints on the universe of feasible lifting factorizations for a given class of filter banks , we define group lifting structures in terms of abelian groups of lower and upper triangular lifting matrices , an abelian group of gain scaling matrices , and sets ( not necessarily groups ) of base filter banks . if a group lifting structure satisfies a polyphase order - increasing hypothesis and if the groups of lifting matrices are invariant under the action of the gain scaling group , the main result of the paper , theorem [ thm : unique_factorization ] , shows that irreducible group lifting factorizations are unique modulo , at most , rescaling . theorem [ thm : unique_factorization ] is used in @xcite to prove uniqueness results for both reversible and irreversible lifting factorizations of ws and hs filter banks parameterized by the linear phase group lifting structures defined in section [ sec : linearphase ] of the present paper . this scope includes the specification of ws filter banks in jpeg 2000 and m. adams elasf class of reversible hs filter banks . the key to these proofs is establishing the order - increasing property for the associated group lifting structures . i. c. daubechies , _ ten lectures on wavelets _ , ser . cbms - nsf regional conf . series in appl . math . , ( univ . mass.lowell , june 1990).1em plus 0.5em minus 0.4emphiladelphia : soc . indust . , 1992 , no . 61 . c. m. brislawn , `` group lifting structures for multirate filter banks , ii : linear phase filter banks , '' los alamos national lab , tech . laur-09 - 3049 , may 2009 , to appear in _ ieee trans . signal process . _ m. maslen and p. abbott , `` automation of the lifting factorisation of wavelet transforms , '' _ computer physics communications _ , 127 , no . 2 - 3 , pp . 309326 , 2000 . [ online ] . available : http://www.sciencedirect.com / science / article / b6tj5 - 40fgct9-g/2/502cc126% fa01e35997405bd199000fa4[http://www.sciencedirect.com / science / article / b6tj5 - 40fgct9-g/2/502cc126% fa01e35997405bd199000fa4 ] r. foote , g. mirchandani , d. rockmore , d. healy , and t. olson , `` a wreath product group approach to signal and image processing part i : multiresolution analysis , '' _ ieee trans . signal process . _ , vol . 48 , no . 1 , 102132 , jan . 2000 . g. mirchandani , r. foote , d. rockmore , d. healy , and t. olson , `` a wreath product group approach to signal and image processing part ii : convolution , correlation , and applications , '' _ ieee trans . signal process . _ , vol . 48 , no . 3 , pp . 749767 , mar . 2000 . a. zandi , j. d. allen , e. l. schwartz , and m. boliek , `` compression with reversible embedded wavelets , '' in _ proc . data compress . _ 1em plus 0.5em minus 0.4emsnowbird , ut : ieee computer soc . , mar . 1995 , pp . 212221 . d. legall and a. tabatabai , `` subband coding of digital images using symmetric short kernel filters and arithmetic coding techniques , '' in _ intl . conf . speech , signal process.__1em plus 0.5em minus 0.4emnew york city : ieee signal process . soc . , apr . 1988 , pp . 761764 . christopher m. brislawn ( m91sm05 ) received the b.s . degree in 1982 from harvey mudd college , claremont , ca , and the ph.d . degree in 1988 from the university of colorado boulder , both in mathematics . he was a visiting assistant professor at the university of southern california from 1989 to 1990 . he joined los alamos national laboratory ( lanl ) , los alamos , nm , as a postdoc in 1990 and became a permanent staff member in 1993 . currently he is a scientist in group ccs-3 of the computer , computational and statistical sciences division . his research interests include wavelet transforms , digital filter banks , communications coding , and statistical signal processing . from 1991 to 1993 he coauthored the wavelet / scalar quantization specification for compression of digitized fingerprint images with the u.s . federal bureau of investigation . from 1999 to 2003 he served as lanl s principal member in working group l3.2 of the international committee for information technology standards and led a lanl team that participated in writing the iso / iec jpeg 2000 standard ( iso 15444-x ) . his team worked on parts 1 and 2 of the standard , and he coauthored the proposal to create jpeg 2000 part 10 ( extensions for three - dimensional data ) , serving as the first editor of part 10 . in 20072008 he represented lanl on the motion imagery standards board for the national geospatial intelligence agency . in addition to his technical work , dr . brislawn has mentored 15 graduate students and 4 postdocs at lanl and has co - supervised one ph.d . dissertation for the university of texas austin .

group lifting structures are introduced to provide an algebraic framework for studying lifting factorizations of two - channel perfect reconstruction fir filter banks . the lifting factorizations generated by a group lifting structure are characterized by abelian groups of lower and upper triangular lifting matrices , an abelian group of unimodular gain scaling matrices , and a set of base filter banks . this covers the lifting specifications for whole - sample symmetric filter banks in parts 1 and 2 of the iso / iec jpeg 2000 still image coding standard . the theory is used to address the uniqueness of lifting factorizations . with no constraints on the lifting process

group lifting structures are introduced to provide an algebraic framework for studying lifting factorizations of two - channel perfect reconstruction fir filter banks . the lifting factorizations generated by a group lifting structure are characterized by abelian groups of lower and upper triangular lifting matrices , an abelian group of unimodular gain scaling matrices , and a set of base filter banks . examples of group lifting structures are given for linear phase lifting factorizations of the two nontrivial classes of two - channel linear phase fir filter banks , the whole- and half - sample symmetric classes , including both the reversible and irreversible cases . this covers the lifting specifications for whole - sample symmetric filter banks in parts 1 and 2 of the iso / iec jpeg 2000 still image coding standard . the theory is used to address the uniqueness of lifting factorizations . with no constraints on the lifting process , it is shown that lifting factorizations are highly nonunique . when certain hypotheses developed in the paper are satisfied , however , lifting factorizations generated by a group lifting structure are shown to be unique . a companion paper applies the uniqueness results proven in this paper to the linear phase group lifting structures for whole- and half - sample symmetric filter banks . = 1 * final revision , reformatted for arxiv.org * filter bank , wavelet , unique factorization , polyphase , lifting , linear phase filter , group .

0704.2340

i

during the last decade , many radioactive beam facilities have become available in the world . at these facilities , nuclear reactions involving nuclei with large neutron or proton excess can be studied , providing the opportunities to study the properties of nuclear matter under the extreme condition of large isospin asymmetry . this has led to a lot of interests and activities in a new research direction in nuclear physics , namely the isospin physics . the ultimate goal of studying isospin physics is to extract information on the isospin dependence of in - medium nuclear effective interactions as well as the equation of state ( eos ) of isospin asymmetric nuclear matter , particularly its isospin - dependent term , i.e. , the density dependence of the nuclear symmetry energy . there are already extensive reviews on the isospin physics in nuclear physics , and they can be found in , e.g. , refs.@xcite . knowledge about the nuclear symmetry energy extracted from the eos of isospin asymmetric nuclear matter is essential in understanding not only many aspects of nuclear physics , such as heavy - ion collisions induced by radioactive nuclei and the structure of exotic nuclei , but also a number of important issues in astrophysics , such as nucleosynthesis during pre - supernova evolution of massive stars and the cooling of protoneutron stars . although the nuclear symmetry energy at normal nuclear matter density is known to be around @xmath0 mev from the empirical liquid - drop mass formula @xcite , its values at other densities , especially at supra - normal densities , are poorly known @xcite . predictions based on various many - body theories differ widely at both low and high densities @xcite . empirically , the incompressibility of asymmetric nuclear matter is essentially undetermined @xcite , even though the incompressibility of symmetric nuclear matter at its saturation density @xmath1 @xmath2 has been determined to be @xmath3 mev from nuclear giant monopole resonances ( gmr ) @xcite and the eos at densities of @xmath4 has also been constrained by measurements of collective flows in nucleus - nucleus collisions dan02a . theoretical studies of the eos of isospin asymmetric nuclear matter were started by brueckner _ _ @xcite and siemens @xcite in the late 60 s . since then , there have been many studies on this subject based on different many - body theories using various two - body and three - body forces or interaction lagrangians . these many - body theories provide very useful tools for understanding the properties of hot and dense nuclear matter , and they can be roughly classified into three categories : the microscopic many - body approach , the effective - field theory approach , and the phenomenological approach . in the microscopic many - body approach , the nuclear many - body problem is treated microscopically using nucleon - nucleon interactions fitted to high - precision experimental data and is thus free of parameters . the microscopic many - body approach mainly includes the non - relativistic brueckner - hartree - fock ( bhf ) approach @xcite , relativistic dirac - brueckner - hartree - fock ( dbhf ) approach mut87,har87,sum92,hub93,fuc04,ma04,sam05a , self - consistent green s function approach @xcite , and variational many - body approach @xcite . in the effective - field theory approach , an effective interaction is constructed based on the effective - field theory ( eft ) , leading to a systematic expansion of the eos in powers of density ( the fermi momentum ) . the effective - field theory approach can be based on the density functional theory @xcite or on chiral perturbation theory @xcite . since this approach can be linked to low energy qcd and its symmetry breaking , it has the advantage of small number of free parameters and a correspondingly higher predictive power . the phenomenological approach is based on effective density - dependent nuclear forces or effective interaction lagrangians . in these approaches , a number of parameters have to be adjusted to fit the properties of many nuclei . this type of models mainly includes the relativistic mean - field ( rmf ) theory @xcite , relativistic and non - relativistic hartree - fock mil74,bro78,jam81,hor83,bou87,lop88,ber93,wer94,kho96,vau72,bra85,sto07 or thomas - fermi approximations @xcite . these phenomenological approaches allow the most precise description for the properties of finite nuclei . both the phenomenological and eft approaches contain parameters that are fixed by nuclear properties around the saturation density and thus usually give excellent descriptions for the nuclear properties around or below the saturation density . their predictions at the high density region are , however , probably unreliable . in addition , due to different approximations or techniques used in different microscopic many - body approaches , their predictions on the properties of nuclear matter could be very different even for the same bare nucleon - nucleon interaction @xcite . in particular , predictions on the properties of isospin asymmetric nuclear matter , especially the density dependence of the nuclear symmetry energy , are still significantly different for different many - body theory approaches . fortunately , heavy - ion reactions induced by radioactive beams provide a unique opportunity to investigate in terrestrial laboratories the eos of asymmetric nuclear matter , particularly the density dependence of the nuclear symmetry energy . during the past decade , a large amount of theoretical and experimental efforts have been devoted to the study of the properties of isospin asymmetric nuclear matter via heavy - ion reactions @xcite . to extract information about the eos of neutron - rich matter , especially the density dependence of the nuclear symmetry energy , from heavy - ion reactions induced by radioactive beams , one needs reliable theoretical tools . transport models that include explicit isospin - dependent degrees of freedom are especially useful for understanding the role of isospin degree of freedom in the dynamics of central nuclear reactions induced by neutron nuclei at intermediate and high energies and in extracting information about the eos of produced neutron - rich matter . during past two decades , significant progresses have been made in developing semi - classical transport models for nuclear reactions . these semi - classical models mainly include the following two types : the boltzmann - uehling - ulenbeck ( buu ) model @xcite and the quantum molecular dynamical ( qmd ) model @xcite . while it is important to develop practically implementable quantum transport theories , applications of the semi - classical transport models have enabled us to learn a great deal of interesting physics from heavy - ion reactions . in particular , with the development of the radioactive nuclear beam physics , some isospin - dependent transport models @xcite have been successfully developed in recent years to describe the nuclear reactions induced by neutron nuclei at intermediate and high energies . in studying the properties of asymmetric nuclear matter from heavy - ion reactions induced by neutron - rich nuclei , a key task is to identify experimental observables that are sensitive to the density dependence of the nuclear symmetry energy , especially at high densities . because of the fact that the symmetry potentials for neutrons and protons have opposite signs and that they are generally weaker than the nuclear isoscalar potential at same density , most observables proposed so far use differences or ratios of isospin multiplets of baryons , mirror nuclei and mesons , such as the neutron / proton ratio of nucleon emissions @xcite , neutron - proton differential flow @xcite , neutron - proton correlation function @xcite , @xmath5/@xmath6he @xcite , @xmath7 @xcite , @xmath8 @xcite and @xmath9 ratios @xcite , etc .. in addition , in order to reduce the systematical errors , multiple probes taken from several reaction systems using different isotopes of the same element have also been proposed . these multiple probes mainly include double ratio or double differential flow . indeed , recent experimental and theoretical analysis of the isospin diffusion data from heavy - ion reactions has led to significant progress in determining the nuclear symmetry energy at subnormal densities @xcite . based on the same underlying skyrme interactions as the ones constrained by the isospin diffusion data , the neutron - skin thickness in @xmath10pb calculated within the hartree - fock approach is consistent with available experimental data @xcite . this symmetry energy is also consistent with that from a relativistic mean - field model using an accurately calibrated parameter set that reproduces both the giant monopole resonance in @xmath11zr and @xmath10pb , and the isovector giant dipole resonance of @xmath10pb @xcite . it further agrees with the symmetry energy recently obtained from isoscaling analyses of isotope ratios in intermediate - energy heavy ion collisions @xcite . these different studies have provided so far the best phenomenological constraints on the symmetry energy at sub - normal densities . information on the symmetry energy at supra - normal densities , on the other hand , remains inclusive and more efforts are needed to investigate the supra - normal density behavior of the symmetry energy . heavy - ion collisions induced by future high energy radioactive beams to be available at high energy radioactive beam facilities will provide a unique opportunity for determining the symmetry energy at supra - normal densities . in the present paper , we review recent progress on the determination of the nuclear symmetry energy in heavy - ion reactions induced by neutron - rich nuclei . in particular , we review the exciting results on density dependence of the nuclear symmetry energy at subnormal densities determined from recent analysis of the isospin diffusion data in heavy - ion reactions . we also discuss the implications derived from this new information on nuclear effective interactions and the neutron skin thickness of heavy nuclei . in addition , we review theoretical progress in studying the behavior of nuclear symmetry energy at high density from heavy - ion reactions induced by high energy radioactive beams . the paper is organized as follows . in section [ esym ] , we give a brief introduction to the nuclear symmetry energy . we then describe in section ibuu04 the ibuu04 hadron transport model for nuclear reactions induced by radioactive beams at intermediate energies . in section [ diffusion ] , we present the results from the ibuu04 model analysis of the isospin diffusion data in heavy - ion reactions and discuss the stringent constraint they have imposed on the nuclear symmetry energy around the nuclear matter saturation density . based on the constrained symmetry energy from the isospin diffusion data , we discuss in section [ skyrmenskin ] the implications of the isospin diffusion data on nuclear effective interactions and the neutron skin thickness of heavy nuclei . in section [ highdensity ] , we review theoretical progress on studying the behavior of the nuclear symmetry energy at high density in heavy - ion reactions induced by high energy radioactive beams . finally , a summary is given in section [ summary ] .

recent analyses of the isospin diffusion data in heavy - ion reactions have already put a stringent constraint on the nuclear symmetry energy around the nuclear matter saturation density . we review this exciting result and discuss its implications on nuclear effective interactions and the neutron skin thickness of heavy nuclei . in addition , we also review the theoretical progress on probing the high density behaviors of the nuclear symmetry energy in heavy - ion reactions induced by high energy radioactive beams .

heavy - ion reactions induced by neutron - rich nuclei provide a unique means to investigate the equation of state of isospin - asymmetric nuclear matter , especially the density dependence of the nuclear symmetry energy . in particular , recent analyses of the isospin diffusion data in heavy - ion reactions have already put a stringent constraint on the nuclear symmetry energy around the nuclear matter saturation density . we review this exciting result and discuss its implications on nuclear effective interactions and the neutron skin thickness of heavy nuclei . in addition , we also review the theoretical progress on probing the high density behaviors of the nuclear symmetry energy in heavy - ion reactions induced by high energy radioactive beams .

1410.3256

i

superconductivity is a striking physical phenomenon ; it is a manifestation of quantum effects on a macroscopic scale . the most striking features of superconductivity are ( i ) the zero value of the electrical resistivity , ( ii ) the expulsion of the magnetic flux ( meissner effect ) , ( iii ) the interference between macroscopically separated junctions . none of these phenomena can be studied using any magnetic resonance method because the basic properties of superconductivity are the quantum effects on a macroscopic scale , while magnetic resonance is a microscopic tool . therefore , the properties of superconductors from magnetic resonance were not expected to be striking as the above mentioned effects . however , it was expected that they can help to investigate the microscopic aspects of superconductivity . in particular , these are the superconducting coherence phenomena , the superconducting and magnetic correlations . significant role of the nuclear magnetic resonance ( nmr ) studies of superconductors is well known . indeed , one of the first experimental confirmations of the validity of microscopic theory derived by bardeen - cooper - schriffer ( bcs ) @xcite was obtained by nmr . hebel and slichter @xcite have shown experimentally that the nuclear spin relaxation time @xmath1 just below the superconducting transition temperature @xmath0 is shorter than in the normal state . that is the direct evidence of validity of bcs theory . this happens ( see , e.g. , @xcite ) due to the opening up of the superconducting gap when the total number of states is a constant , the density of states of conduction electrons at the fermi level @xmath2 averaged over an energy interval of about @xmath3 ( for @xmath4 ) is larger in the superconducting state than in the normal state . therefore , the shortening of @xmath1 just below @xmath0 is not surprising ; the salient feature of the bcs theory is that for spin independent phenomena such as ultrasonic attenuation , this effect does not occur since the matrix element for the transition becomes smaller and cancels the density of states effect , while for relaxation due to the contact interaction the effect is present . strictly speaking , according to the bcs theory @xmath5 at @xmath6 since density of states @xmath7 is finite , but @xmath8 ^ 2 de$ ] diverges ; since the bcs theory is only an approximation , and the excitation in reality have a finite lifetime , @xmath9 does not become infinite . hebel and slichter @xcite incorporated this lifetime effect by empirical smearing of the density of states function . it is necessary to note that in the first works on nmr in superconductors long nuclear relaxation time allowed to overcome the main difficulty for the observation of the resonance caused by the meissner effect using the method of cycling of dc magnetic field @xcite . formation of nonequilibrium response of the nuclear spin system was performed in the superconducting state and the resonance signal was observed in the normal state . because of a short relaxation time this method can not be used in the case of epr . for a long time the possibility of using epr of conduction electrons for the study of the superconducting state was considered to be impossible since the cooper pairs which are formed by two electrons with opposite spins in the superconducting state have the zero total spin . furthermore , superconductors expel completely the external magnetic field from their inside due to the meissner effect . this makes impossible even the observation of the resonance of a localized moment which can be specially introduced into the superconducting matrices as a special spin probe . however , in type ii superconductors in the mixed state magnetic field penetrates as the abrikosov vortices @xcite . this offered a possibility to use epr for the study of the type ii superconductors . the works concerning the study of superconductors by epr were started more than 40 years ago when its pioneering observation by the group of e. g. kharakhshyan @xcite was done at the kazan physical technical institute of the russian academy of sciences ( in present zavoisky physical technical institute ) . soon after that two groups possessing the most sensitive in the world epr equipment which allow to perform measurements at low and ultralow temperatures , join in the study of epr in superconductors . these are the group of r. orbach in california university and the group of k. baberschke in frei university in berlin . the above groups have published a whole series of papers on the epr study of gd@xmath10 ion in both the normal and superconducting state of different intermetallic compounds @xcite . the detailed analysis of the temperature dependencies of the linewidth and g - value performed in these works allowed to get information concerning the behavior of the electron magnetic susceptibility of a superconductor and the peculiarities of its electronic structure . these studies have been performed using the samples containing a small amount of magnetic impurity in order to exclude the broadening of the resonance line due to the spin - spin interaction . in principle , this information could be obtained using the nmr experiments . as to the conduction electron spin resonance ( cesr ) even in the abrikosov state due to the fast decrease of the elementary excitations above the superconducting gap upon lowering the temperature and large value of the spin - orbit interaction the possibility for observation of cesr remained open for many years . cesr was observed at the later stage by yafet _ they reported the first observation of cesr in both the normal and superconducting state of pure niobium . the resonance displayed a g - value of [email protected] in both states . the linewidth narrows considerably in the superconducting state . the authors showed that this narrowing is a consequence of the coherence effects . the influence of paramagnetic impurities on the properties of superconductors was one of the intensively investigated problems of superconductivity physics in 1970ies . the observation of epr of a localized moment in type ii superconductor added the epr method to the number of physical methods used to study this problem . being a direct detector of the formation of an electronic localized magnetic state , this method yields information on a number of important properties of a superconductor doped with a magnetic impurity . all the possible applications of epr to superconductors have not yet been clarified completely , but the following can already be noted at present . the epr method makes it possible to measure directly the exchange interaction of conduction electrons and the localized states , to obtain detailed information on the spin scattering of the conduction electrons individually for a given kind of impurity , to investigate the collective spin - density oscillations produced at sufficiently high concentrations of the paramagnetic impurity ( the `` electron bottleneck '' effect @xcite ) ( see also appendix ii ) . finally , and apparently most significantly , epr makes it possible to investigate directly the character and the strength of the interactions between the electronic localized states in the superconducting phase and consequently to investigate the problem of the coexistence of magnetic order and superconductivity . this paper presents the results obtained by the group of e. g. kharakhshyan concerning the first observation of epr in type ii superconductor , observation of new type of an exchange interaction between localized moments , magnetic ordering of impurities in the superconducting state and , finally , the phase separation in high-@xmath0 superconductors . all epr measurements were performed using an x - band spectrometer equipped by the home - made helium cryostat ( [email protected] k ) and by the flowing helium gas system ( t=5@xmath12300 k ) . first observation of epr was done using spectrometer re-1306 ( russia ) and all other measurements using epr spectrometer b - er 418@xmath13 ( bruker ) .

historical review on the studies of the electron paramagnetic resonance in superconductors performed in the period from 1970 to 1990 at the kazan physical technical institute of the russian academy of sciences ( group of dr . e. g. kharakhashyan ) in collaboration with kazan state university ( group of prof . b. i. kochelaev ) and with the institute for physical problems of russian academy of sciences ( group of prof . we have observed for first time electron paramagnetic resonance of impurities in a type ii superconductor ; found indication for a long - range exchange interaction between magnetic impurities arising due to the superconducting correlations ; observed the magnetic ordering of impurities in the superconducting state ; and , finally , we found one of the first evidences for heterogeneity of the 1:2:3 high- superconductor which is its natural property

historical review on the studies of the electron paramagnetic resonance in superconductors performed in the period from 1970 to 1990 at the kazan physical technical institute of the russian academy of sciences ( group of dr . e. g. kharakhashyan ) in collaboration with kazan state university ( group of prof . b. i. kochelaev ) and with the institute for physical problems of russian academy of sciences ( group of prof . n. e. alekseevskii ) is presented . we have observed for first time electron paramagnetic resonance of impurities in a type ii superconductor ; found indication for a long - range exchange interaction between magnetic impurities arising due to the superconducting correlations ; observed the magnetic ordering of impurities in the superconducting state ; and , finally , we found one of the first evidences for heterogeneity of the 1:2:3 high- superconductor which is its natural property

0803.2475

i

in this talk i would like to describe some remarkable progress that has been made in the past few years in understanding the structure of gauge boson scattering amplitudes in a particular gauge theory , @xmath0 super - yang - mills theory . while this theory differs in many details from the electroweak and qcd theories whose radiative corrections were the subject of this symposium , there are many common issues , particularly associated with infrared structure . indeed , the understanding of infrared divergences in qcd acquired over the last few decades has proved extremely useful in unraveling some of the structure of @xmath0 super - yang - mills theory . @xmath0 super - yang - mills theory is the most supersymmetric theory possible without gravity . in the free theory , starting from the helicity @xmath2 massless gauge boson ( `` gluon '' ) state , the four supercharges can be used to lower the helicity by @xmath3 units , until the helicity @xmath4 gluon state is reached . if one had more supercharges , one would need spin @xmath5 states , and it is not known how to quantize such theories in a unitary way without including at least spin 2 gravitons . along the way from the helicity @xmath2 to the helicity @xmath4 gluon state , one passes through the 4 massless ( majorana ) spin @xmath6 gluinos , and 6 real ( or 3 complex ) massless spin @xmath7 scalars . in this maximally supersymmetric yang - mills theory ( msym ) , all the massless states are in the adjoint representation of the gauge group , which we will take to be @xmath8 . the interactions are all uniquely specified by the choice of gauge group , and one dimensionless gauge coupling @xmath9 . the theory is an exactly scale - invariant , conformal field theory ; that is , the beta function vanishes identically for all values of the coupling @xcite . here we will consider the t hooft limit of msym , in which the number of colors @xmath10 , with the t hooft parameter @xmath11 held fixed @xcite . in this limit , only planar feynman diagrams contribute . also , the anti - de sitter space / conformal field theory ( ads / cft ) duality @xcite suggests that for @xmath10 the weak - coupling perturbation series in @xmath12 might have some very special properties . the reason is that , according to ads / cft , the strongly - coupled ( large @xmath12 ) limit of the four - dimensional conformal gauge theory has an equivalent description in terms of a weakly - coupled string theory . the intuition is that the perturbative series should know about this simple strong - coupling limit , and organize itself accordingly @xcite . sketches how events such as gluon scattering look in the ads / cft duality @xcite . five - dimensional anti - de sitter space , ads@xmath13 , contains , besides the usual four - dimensional space - time @xmath14 , an additional radial variable @xmath15 , which corresponds to a resolution scale in the four - dimensional theory . large values of @xmath15 correspond to the ultraviolet ( uv ) region ; small values to the infrared ( ir ) . the figure shows a `` big '' glueball state in the ir , and a `` small '' glueball state in the uv . the arrows represent the motion of plane - wave single gluon states in @xmath14 for @xmath16 scattering at @xmath17 . we ll discuss the motion in @xmath15 later . the radius of curvature of ads@xmath13 is proportional to @xmath18 . large @xmath12 means that the space - time is only weakly curved , which makes it much simpler to study the string theory ; higher excitations of the string can usually be neglected . the ads / cft duality is a weak / strong duality . quantities that can be computed at weak coupling in one picture have a strong - coupling description in the other picture . this property makes ads / cft both powerful and difficult to check explicitly although there is certainly convincing evidence in its favor . there are a few quantities that are known ( modulo a few assumptions ) to all orders in @xmath12 ; that is , for which one can interpolate all the way from weak to strong coupling . notable among these is the cusp ( or soft ) anomalous dimension @xmath19 . the qcd version of this quantity crops up a lot in soft - gluon resummation . beisert , eden and staudacher @xcite have given an all - orders proposal for @xmath19 , based on integrability , plus a number of other properties . their proposal is consistent with the first four loops in the weak - coupling expansion @xcite , and also agrees @xcite with the first three terms in the strong - coupling expansion @xcite . in this talk i would like to discuss the evidence for another proposal @xcite , namely that gluon - gluon scattering @xmath16 in msym , for any scattering angle @xmath20 can be fully specified by just three functions of @xmath12 , independent of @xmath20 . one of these three functions is already `` known '' , because it is just @xmath19 . this proposal has received some confirmation at strong coupling , through the work of alday and maldacena @xcite . it was motivated by the structure of ir divergences in gauge theory .

i describe some recent developments in the understanding of gluon scattering amplitudes in super - yang - mills theory in the large- limit . these amplitudes can be computed to high orders in the weak coupling expansion , and also now at strong coupling using the ads / cft correspondence . slac

i describe some recent developments in the understanding of gluon scattering amplitudes in super - yang - mills theory in the large- limit . these amplitudes can be computed to high orders in the weak coupling expansion , and also now at strong coupling using the ads / cft correspondence . they hold the promise of being solvable to all orders in the gauge coupling , with the help of techniques based on integrability . they are intimately related to expectation values for polygonal wilson loops composed of light - like segments . slac pub13167 address = stanford linear accelerator center , stanford university , stanford , ca 94309 , usa

1401.3075

i

on an acyclic multicast network , if there is a linear solution over gf(@xmath0 ) , could we claim that there is a linear solution over every gf(@xmath23 ) with @xmath312 ? it would be tempting to answer it positively because by the result in @xcite , the claim is correct when @xmath0 is no smaller than the number of receivers and moreover , as a consequence of the result in @xcite , the positive answer is affirmed for the special case that the source dimension of the network is equal to 2 . in the present paper , however , we show the negative answer for general cases by constructing several classes of multicast networks with different emphasis . these networks are the first ones discovered in the network coding literature with the property that @xmath14 , the maximum field size for the nonexistence of a linear solution over gf(@xmath14 ) , is larger than @xmath6 , the minimum field size for the existence of a linear solution over gf(@xmath6 ) . the insight of various exemplifying networks established in the present paper is that not only the field size of gf(@xmath0 ) , but also the _ order of the proper multiplicative subgroup _ of gf(@xmath0)@xmath59 affects the networks linear solvability over gf(@xmath0 ) . the results in this paper bring about a new thread on the fundamental study of linear network coding , specific to the case of multicast networks , as discussed below : * besides the field size and multiplicative subgroup orders , it is not clear whether there are some other undiscovered inherent structures in a finite field that affects the linear solvability of a multicast network . it deserves further investigation . * all multicast networks presented in this paper that are linearly solvable over a field gf(@xmath0 ) but not over a larger field gf(@xmath23 ) share a common property that for some values @xmath313 , there is a proper subgroup @xmath127 in gf(@xmath0 ) subject to @xmath251 and @xmath252 , but there does not exist a proper subgroup @xmath314 in gf(@xmath23 ) subject to @xmath315 and @xmath316 . thus , when @xmath317 does not contain any proper subgroup other than @xmath121 , _ i.e. _ , @xmath318 is a prime , we can simply set @xmath319 and @xmath320 , such that for any values @xmath313 subject to @xmath321 and @xmath322 , the conditions @xmath323 and @xmath324 hold as well . this leads us to an ambitious conjecture that if a multicast network is linearly solvable over gf(@xmath0 ) where @xmath325 is a prime , then it is linearly solvable over all finite fields of sizes larger than @xmath0 . * as special cases of the previous conjecture , it is of particular interest to study : i ) whether a multicast network linearly solvable over gf(2 ) ( or over gf(3 ) ) is linearly solvable over all larger finite fields ; ii ) whether a multicast network linearly solvable over gf(2 ) and over gf(3 ) is linearly solvable over all finite fields . these two cases are related to other interesting conjectures in the network coding literature : it was conjectured and partially proved in @xcite that every planar multicast network is linearly solvable over gf(3 ) , and certain special planar multicast networks ( including relay - coface networks and terminal - coface networks ) are always linearly solvable over both gf(2 ) and gf(3 ) . the conjectures presented here will further suggest that these special multicast networks may be linearly solvable over all finite fields . stemming from the above discussions , we end this paper by proposing a number of open problems , all of which , except for the first , we conjecture to have positive answers : * for a multicast network , what is the smallest prime power @xmath0 larger than @xmath14 ( such that the network is linearly solvable over all gf(@xmath23 ) with @xmath312 ) ? * can the gap @xmath326 tend to infinity ? * are there infinitely many prime power pairs @xmath327 with @xmath328 such that each @xmath327 corresponds to @xmath329 of some multicast network ? * if a multicast network is linearly solvable over such a gf(@xmath0 ) that gf(@xmath0)@xmath59 does not contain any proper multiplicative subgroup other than @xmath121 , is it linearly solvable over all larger finite fields than gf(@xmath0 ) ? * if a multicast network is linearly solvable over gf(2 ) ( or over gf(3 ) ) , is it linearly solvable over all larger finite fields ? * if a multicast network is linearly solvable over both gf(2 ) and gf(3 ) , is it linearly solvable over all finite fields ? observe that the network in fig . [ fig : rank_3_networks](a ) can be constructed by algorithm 1 introduced in section [ sec : general_network ] with parameters @xmath192 . then theorem [ thm : general_network_solvability ] implies that the network in fig . [ fig : rank_3_networks](a ) is linearly solvable over gf(@xmath0 ) if and only if there exist @xmath330 , @xmath331 such that @xmath332 under constraint ( [ eqn : gf_7_gf_8_1 ] ) , the cardinality of @xmath333 is no smaller than 3 and no larger than 9 . thus , * when @xmath334 , there does not exist any assignment of @xmath335 from gf(@xmath0)@xmath59 satisfying ( [ eqn : gf_7_gf_8_2 ] ) , and hence the network is not linearly solvable over gf(@xmath0 ) ; * when @xmath336 , there are always ways to assign @xmath335 from gf(@xmath0)@xmath59 subject to ( [ eqn : gf_7_gf_8_1 ] ) and ( [ eqn : gf_7_gf_8_2 ] ) , and hence the network is linearly solvable over gf(@xmath0 ) . it remains to consider the case @xmath0 = 7 , 8 , 9 , or 11 . when @xmath0 = 7 , 9 , or 11 , there is a proper subgroup @xmath127 of order at least 3 in the multiplicative group @xmath317 . we can then assign arbitrary three distinct values in @xmath127 to @xmath337 , @xmath338 , @xmath339 and to @xmath340 , @xmath341 , @xmath342 . in this way , the cardinality of @xmath333 is upper bounded by @xmath343 and thus there are at least three values remained in @xmath344 , which @xmath345 , @xmath346 , @xmath347 can be assigned to . in the last case @xmath0 = 8 , since there is no proper subgroup in gf@xmath348 , the method depicted in the previous paragraph does not work any more and an exhaustive search will verify that there does not exist any assignment of @xmath335 from gf(@xmath0)@xmath59 satisfying ( [ eqn : gf_7_gf_8_1 ] ) and ( [ eqn : gf_7_gf_8_2 ] ) . we can now affirm that the network depicted in fig . [ fig : rank_3_networks](a ) is linearly solvable over every gf(@xmath0 ) with @xmath80 except for @xmath72 . observer that the network in fig . [ fig : rank_3_networks](b ) can be constructed by algorithm 1 introduced in section [ sec : general_network ] with parameters @xmath193 . then theorem [ thm : general_network_solvability ] implies that the network in fig . [ fig : rank_3_networks](b ) is linearly solvable over gf(@xmath0 ) if and only if there is an assignment of @xmath98 , @xmath99 from gf(@xmath0)@xmath59 subject to @xmath349 under condition ( [ eqn : gf_16_gf_17_1 ] ) , the cardinality of @xmath350 is lower bounded by 5 and upper bounded by 25 . thus , * when @xmath351 , there does not exist any assignment of @xmath98 , @xmath99 from gf(@xmath0)@xmath59 satisfying ( [ eqn : gf_16_gf_17_1 ] ) , and hence the network is not linearly solvable over gf(@xmath0 ) ; * when @xmath352 , there are always ways to assign @xmath98 , @xmath99 subject to ( [ eqn : gf_16_gf_17_1 ] ) and ( [ eqn : gf_16_gf_17_2 ] ) , and hence the network is linearly solvable gf(@xmath0 ) . it remains to consider the case that @xmath353 . note that when @xmath0 is not equal to 17 or 32 , there is also a proper subgroup @xmath127 in the multiplicative group gf(@xmath0)@xmath59 such that @xmath127 has order no smaller than 5 and the cardinality of gf(@xmath0)@xmath354 is at least 10 . then , we can respectively assign any 5 distinct elements in @xmath127 to @xmath355 and to @xmath356 , and assign any 10 distinct elements in gf(@xmath0)@xmath357 to @xmath358 . such an assignment obeys conditions ( [ eqn : gf_16_gf_17_1 ] ) and ( [ eqn : gf_16_gf_17_2 ] ) . when @xmath359 , denote by @xmath105 be a primitive element in gf(32)@xmath59 . assign @xmath360 for all @xmath104 , and @xmath361 for all @xmath362 . it is easy to check that such an assignment obeys conditions ( [ eqn : gf_16_gf_17_1 ] ) and ( [ eqn : gf_16_gf_17_2 ] ) . when @xmath363 , in order to make @xmath364 contains at least 10 elements , @xmath355 and @xmath356 should be such assigned that the cardinality of @xmath365 is no larger than 6 . to minimize this cardinality , as many as @xmath366 and @xmath367 should be assigned to a same proper subgroup in gf(17)@xmath59 . an exhaustive search will then verify that it is infeasible to assign @xmath355 and @xmath356 from @xmath368 so that @xmath365 contains no more than 6 elements . we can now assert that the network depicted in fig . [ fig : rank_3_networks](b ) is linearly solvable over every gf(@xmath0 ) with @xmath96 except for @xmath363 . observe that the swirl network can be constructed by algorithm 1 in section [ sec : general_network ] with parameters @xmath194 . then according to theorem [ thm : general_network_solvability ] , the swirl network is linearly solvable over gf(@xmath0 ) if and only if there exist @xmath369 , @xmath370 subject to @xmath371 given arbitrary @xmath369 , @xmath370 , define @xmath372 and @xmath373 , @xmath374 . it is straightforward to check that condition ( [ eqn : swirl_proof_1 ] ) holds if and only if @xmath375 moreover , because @xmath376 condition ( [ eqn : swirl_proof_2 ] ) holds if and only if @xmath377 we can now conclude that the swirl network is linearly solvable over gf(@xmath0 ) if and only if there exist @xmath112 satisfying conditions ( [ eqn : swirl_1 ] ) and ( [ eqn : swirl_2 ] ) .

in an acyclic multicast network , it is well known that a linear network coding solution over gf( ) exists when is sufficiently large . in particular , for each prime power no smaller than the number of receivers , a linear solution over gf( ) can be efficiently constructed . in this work specifically , we prove by construction that : ( i ) for every source dimension no smaller than 3 , there is a multicast network linearly solvable over gf(7 ) but not over gf(8 ) , and another multicast network linearly solvable over gf(16 ) but not over gf(17 ) ; ( ii ) there is a multicast network linearly solvable over gf(5 ) but not over such gf( ) that is a mersenne prime plus 1 , which can be extremely large ; ( iii ) a multicast network linearly solvable over gf( ) and over gf( ) is _ not _ necessarily linearly solvable over gf( ) ; ( iv ) there exists a class of multicast networks with a set of receivers such that the minimum field size for a linear solution over gf( ) is lower bounded by , but not every larger field than gf( ) suffices to yield a linear solution . linear network coding , multicast network , field size , lower bound , mersenne prime .

in an acyclic multicast network , it is well known that a linear network coding solution over gf( ) exists when is sufficiently large . in particular , for each prime power no smaller than the number of receivers , a linear solution over gf( ) can be efficiently constructed . in this work , we reveal that a linear solution over a given finite field does _ not _ necessarily imply the existence of a linear solution over all larger finite fields . specifically , we prove by construction that : ( i ) for every source dimension no smaller than 3 , there is a multicast network linearly solvable over gf(7 ) but not over gf(8 ) , and another multicast network linearly solvable over gf(16 ) but not over gf(17 ) ; ( ii ) there is a multicast network linearly solvable over gf(5 ) but not over such gf( ) that is a mersenne prime plus 1 , which can be extremely large ; ( iii ) a multicast network linearly solvable over gf( ) and over gf( ) is _ not _ necessarily linearly solvable over gf( ) ; ( iv ) there exists a class of multicast networks with a set of receivers such that the minimum field size for a linear solution over gf( ) is lower bounded by , but not every larger field than gf( ) suffices to yield a linear solution . the insight brought from this work is that not only the field size , but also the order of subgroups in the multiplicative group of a finite field affects the linear solvability of a multicast network . linear network coding , multicast network , field size , lower bound , mersenne prime .

1001.1132

i

in this article we summarize the assumptions made and the reasoning that led to the development and construction of our parton bubble model ( pbm)@xcite , which successfully explained the charge - particle - pair correlations in the central ( 0 - 10% centrality ) @xmath11 200 gev au au data@xcite . the pbm is also consistent with the au au central collision hbt results . this is presented and discussed in sec . 4 of ref.@xcite and sec . i and ii of this paper . in sec . iii we discussed extending our model , which was a central collision model , to be able to treat the geomery of bubble production for 0 - 80% collision centralities ( pbme@xcite ) such as measured and analyzed in rhic data@xcite . the bubbles represent a significant substructure of gluonic hot spots formed on the surface of a dense opaque fireball at kinetic freezeout . the origins of these hot spots come from a direct connection between our parton bubble model ( pbm@xcite ) and the glasma flux tube model ( gftm)@xcite ) . in the gftm a flux tube is formed right after the initial collision of the au au system . this flux tube extends over many units of pseudorapidity ( @xmath9 ) . a blast wave gives the tubes near the surface transverse flow in the same way it gave flow to the bubbles in the pbm . this means the transverse momentum ( @xmath2 ) distribution of the flux tube is directly translated to the @xmath2 spectrum . in the pbm this flux tube is approximated by a sum of partons which are distributed over this same large @xmath9 region . initially the transverse space is filled with flux tubes of large longitudinal extent but small transverse size @xmath7@xmath23 . the flux tubes that are near the surface of the fireball get the largest radial flow and are emitted from the surface . as in the parton bubble model these partons shower and the higher @xmath2 particles escape the surface and do not interact . @xmath22 is around 1 gev / c thus the size of the flux tube is about 1/4 fm initially . the flux tubes near the surface are initially at a radius @xmath75 fm . the @xmath10 angle wedge of the flux tube @xmath7 1/20 radians or @xmath7@xmath24 . in sec . v ( also see sec . iv ) we connect the gftm to the pbm : by assuming the bubbles are the final state of a flux tube at kinetic freezeout , and discussing evidence for this connection which is further developed in sec . vi and sec . vii . with the connection of the pbm to the gftm two new predictions become possible for our pbm . the first is related to the fact that the blast wave radial flow given to the flux tube depends on where the tube is initially in the transverse plane of the colliding au au system . the tube gets the same radial boost all along its longitudinal length . this means that there is correlated @xmath2 among the partons of the bubble . in this paper we consider predictions we can make in regard to interesting topics and comparisons with relevant data which exist by utilizing the parton bubble model ( pbm@xcite ) , and its features related to the glasma flux tube model ( gftm)@xcite ) . topic 1 : the ridge is treated in sec . topic 2 : strong cp violation ( chern - simons topological charge ) is treated in sec . we show in sec . vi that if we trigger on particles with 3 to 4 gev / c and correlate this trigger particle with an other charged particle of greater than 1.1 gev / c , the pbm can produce a phenomenon very similar to the ridge@xcite . ( see figs . we then selected charged particles inside the ridge and predicted the correlation that one should observe when compared to the average charged particles of the central au au collisions at @xmath0 = 200 gev@xcite . in sec . vi d ( comparison to data ) : triggered experimental angular correlations showing the ridge were presented at quark matter 2006@xcite . figure 7 shows the experimental @xmath12 vs. @xmath9 ci correlation for 0 - 10% central au au collisions at @xmath0 = 200 ; requiring one trigger particle @xmath2 between 3 to 4 gev / c and an associated particle @xmath2 above 2.0 gev / c . the yield is corrected for the finite @xmath9 pair acceptance . for the pbm generator , we then form a two charged particle correlation between one charged particle with a @xmath2 between 3.0 to 4.0 gev / c and another charged particle whose @xmath2 is greater than 2.0 gev / c . these are the same trigger conditions as in ref.@xcite which is shown in fig . 11 , that shows the corrected pair yield in the central data . 12 shows the correlation function generated by the pbm which does not depend on the number of events analyzed . the two figures were shown to be in reasonable agreement when compared as explained previously in sec . vi d. in fig . 13 we show the ridge signal predicted by the pbm for very similar data but with 0 - 5% centrality . figure 14 shows the extraction of the jet signal . explanations are given in the text . the second prediction is a development of a predictive pionic measure of the strong cp violation . the gftm flux tubes are made up of longitudinal color electric and magnetic fields which generate topological chern - simons charge@xcite through the @xmath3 term that becomes a source of cp violation . the color electric field which points along the flux tube axis causes an up quark to be accelerated in one direction along the beam axis , while the anti - up quark is accelerated in the other direction . so when a pair of quarks and anti - quarks are formed they separate along the beam axis leading to a separated @xmath5 and @xmath6 pair along this axis . the color magnetic field which also points along the flux tube axis ( which is parallel to the beam axis ) causes an up quark to rotate around the flux tube axis in one direction , while the anti - up quark rotates in the other direction . so when a pair of quarks and anti - quarks are formed they will pickup or lose transverse momentum . these changes in @xmath2 will be transmitted to the @xmath5 and @xmath6 pairs . the above pionic measures of strong cp violation are used to form correlation functions based on four particles composed of two pairs which are opposite sign charge - particle - pairs that are paired and binned . these four particle correlations accumulate from bubble to bubble by particles that are pushed or pulled ( by the color electric field ) and rotated ( by the color magnetic field ) in a right or left handed direction . the longitudinal color electric field predicts aligned pairs in a pseudorapidity or @xmath9 measure . the longitudinal color magnetic field predicts anti - aligned pairs in a transverse momentum or @xmath50 measure . the observations of these correlations would be a strong confirmation of this theory . the much larger unlike - sign pairs than like - sign pairs in the pbm and the data ; and the strong dip of the ci correlation at small @xmath9 ( see fig . 4 and sec . vii c for full details ) shows very strong evidence supporting the color electric alignment prediction in the @xmath0 = 200 gev central au au collision data analyses at rhic@xcite . this highly significant dip ( @xmath7@xmath43 ) means that like - sign pairs are removed as one approaches the region @xmath9 = @xmath12 = 0.0 . thus this is very strong evidence for the predicted effect of the color electric field . the color magnetic anti - aligned pairs in the transverse momentum prediction , treated in sec . vii d as of now has not been observed or looked for . however our predicted specific four charged particle correlations can be used to search for experimental evidence for the color magnetic fields . our success in demonstrating strong experimental evidence for the expected color electric field effects from previously published data suggests that the unique detailed correlations we have presented for searching for evidence for the predicted color magnetic field effects should be urgently investigated . if we are lucky and the predicted color magnetic effects can be confirmed experimentally we would have strong evidence for the following : \1 ) cp is violated in the strong interaction in isolated local space time regions where topological charge@xcite is generated . \2 ) the glasma flux tube model ( gftm ) which was evolved from the color glass condensate ( cgc ) would be found to be consistent with a very significant experimental check . \3 ) the parton bubble model event generator ( pbm ) is clearly closely connected to the gftm . the bubble substructure strongly supported by the pbm is likely due to the final state of the flux tube at kinetic freezeout .

these sheets create boost invariant flux tubes of longitudinal color electric and magnetic fields . a blast wave gives the tubes near the surface transverse flow in the same way it gave transverse flow to the bubbles in the pbm . in this paper in the gftm the longitudinal color electric and magnetic fields have a non - zero topological charge density .

in an earlier paper we developed a parton bubble model ( pbm ) for rhic , high - energy heavy - ion collisions . pbm was based on a substructure of a ring of localized bubbles ( gluonic hot spots ) which initially contain 3 - 4 partons composed of almost entirely gluons . the bubble ring was perpendicular to the collider beam direction , centered on the beam , at midrapidity , and located on the expanding fireball surface of au au central collisions ( 0 - 10% ) at=200 gev . the bubbles emitted correlated particles at kinetic freezeout , leading to a lumpy fireball surface . for a selection of charged particles ( 0.8 gev / c 4.0 gev / c ) , the pbm reasonably quantitatively ( within a few percent ) explained high precision rhic experimental correlation analyses in a manner which was consistent with the small observed hbt source size in this transverse momentum range . we demonstrated that surface emission from a distributed set of surface sources ( as in the pbm ) was necessary to obtain this consistency . in this paper we give a review of the above comparison to central au au collisions . the bubble formation can be associated with gluonic objects predicted by a glasma flux tube model ( gftm ) that formed longitudinal flux tubes in the transverse plane of two colliding sheets of color glass condensate ( cgc ) , which pass through one another . these sheets create boost invariant flux tubes of longitudinal color electric and magnetic fields . a blast wave gives the tubes near the surface transverse flow in the same way it gave transverse flow to the bubbles in the pbm . in this paper we also consider the equivalent characteristics of the pbm and gftm and connect the two models . in the gftm the longitudinal color electric and magnetic fields have a non - zero topological charge density . these fields cause a local strong cp violation which effects charged particle production coming from quarks and anti - quarks created in the tube or bubble . # 1#2#3#4#1 * # 2 * , # 3 ( # 4 )

1308.4760

i

the x - ray binary cygnus x-1 was discovered in the early days of x - ray astronomy @xcite , and its compact primary was the first black hole candidate to be established via dynamical observations @xcite . recently , in three sequential papers on cygnus x-1 , we reported accurate values of the source distance @xmath7 @xcite , black hole mass @xmath8 and orbital inclination angle @xmath9 @xcite , and an extreme value for the black hole s spin parameter , @xmath0 ( @xmath10 ; * ? ? ? * ) with @xmath11 , where @xmath12 is the angular momentum of the black hole . ] . we measured the spin of the black hole by fitting the thermal x - ray continuum spectrum of the accretion disk to the thin - disk model of @xcite . the key fit parameter is the radius of the inner edge of the accretion disk , which is equivalent to the radius of the innermost stable circular orbit @xmath13 @xcite . in turn , @xmath14 is directly related to the dimensionless spin parameter @xmath15 @xcite . the continuum - fitting method of measuring spin is simple : it is strictly analogous to measuring the radius of a star whose flux , temperature and distance are known . by this analogy , it is clear that one must have accurate estimates of @xmath7 , @xmath8 and @xmath9 in order to obtain an accurate estimate of @xmath15 by fitting the x - ray spectrum . the robustness of the continuum - fitting method is demonstrated by the dozens or hundreds of independent and consistent measurement of spin that have been obtained for several black holes ( e.g. , * ? ? ? * ) , and through careful consideration of many sources of systematic errors ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) . herein , using the continuum - fitting method @xcite and precisely the same methodologies that are described in gou et al . ( 2011 ; hereafter gou11 ) but using data of much higher quality we confirm our conclusion that cygnus x-1 s black hole is a near - extreme kerr hole , a result that has received support via the independent fe - line method of measuring spin ( see section 7.1 ) . importantly , these new data allow us to obtain a more stringent limit on the spin parameter , namely @xmath2 ( @xmath10 ) . for reliable application of the continuum - fitting method , it is essential that the thermal disk component dominate over the compton power - law component @xcite , which is always present in the spectra of x - ray binaries . it is by this criterion that the present data are of much higher quality than those analyzed in gou11 , as we now explain . the strength of the complicating compton component is parameterized by the scattering fraction @xmath5 , which is the fraction of the thermal seed photons that are scattered into the power - law component @xcite . ideally , @xmath5 is a few percent , while the limit for reliable application of the continuum - fitting method , based on a thorough investigation of two black hole binaries , has been shown to be @xmath16% @xcite . the extreme spin reported in gou11 is based on an analysis of the only three spectra of cygnus x-1 that were then available and suitable for measurement of spin via the continuum - fitting method . one spectrum was marginally within the limit ( @xmath17% ) and the other two were above the limit ( both with @xmath18% ) . herein , we report on spin results for six new spectra , five of which have much more favorable scattering fractions in the range @xmath1910%-19% . each of the six spectra individually confirms the spin limit set by gou11 ( @xmath0 at @xmath1 ) . it is challenging to measure the spin of cygnus x-1 not only because the compton component is always relatively strong for this source ( e.g. , see section 4.3 in * ? * ) , but also for two additional reasons : ( 1 ) it is essential to have spectral data that span a broad energy range , @xmath20 kev , in order to simultaneously constrain the unusually soft thermal component ( @xmath21 kev ) and the compton power - law and reflected components ( see section 2 and figure [ fig : third ] in gou11 ) , and such broadband data are rare ; and ( 2 ) the source dwells in its soft state only a small fraction of the time . in mid-2010 , cygnus x-1 again entered the soft state . seizing this opportunity , we observed the source with _ chandra _ , _ swift _ , _ suzaku _ , and _ rxte _ and obtained the spectra with moderate values of @xmath5 that are mentioned above . the times of these various observations are indicated by arrows in the x - ray light curve shown in figure [ fig : first ] . the paper is organized as follows : in section 2 we describe the observations and data reduction , and in section 3 the data analysis and our spectral model . presented in sections 4 , 5 and 6 respectively are our results , a discussion of their robustness , and a comprehensive analysis of the errors . in section 7 we first discuss spin results obtained using the fe - line method and then compare cygnus x-1 to two other well - studied persistent black hole systems . we offer our conclusions in the final section .

in gou et al . ( 2011 ) , we reported that the black hole primary in the x - ray binary cygnus x-1 is a near - extreme kerr black hole with a spin parameter ( ) . . the earlier work , which was based on an analysis of all three useful spectra that were then available , was possibly biased by the presence in these spectra of a relatively strong compton power - law component : the fraction of the thermal seed photons scattered into the power law was% , while the upper limit for reliable application of the continuum - fitting method is% . five of these spectra are of high quality with in the range 10% to 19% , a regime where the continuum - fitting method has been shown to deliver reliable results . individually , the six spectra give lower limits on the spin parameter that range from to , allowing us to conservatively conclude that the spin of the black hole is ( ) .

in gou et al . ( 2011 ) , we reported that the black hole primary in the x - ray binary cygnus x-1 is a near - extreme kerr black hole with a spin parameter ( ) . we confirm this result while setting a new and more stringent limit : at the ( 99.7% ) level of confidence . the earlier work , which was based on an analysis of all three useful spectra that were then available , was possibly biased by the presence in these spectra of a relatively strong compton power - law component : the fraction of the thermal seed photons scattered into the power law was% , while the upper limit for reliable application of the continuum - fitting method is% . we have subsequently obtained six additional spectra of cygnus x-1 suitable for the measurement of spin . five of these spectra are of high quality with in the range 10% to 19% , a regime where the continuum - fitting method has been shown to deliver reliable results . individually , the six spectra give lower limits on the spin parameter that range from to , allowing us to conservatively conclude that the spin of the black hole is ( ) .

1008.0454

c

as noted above , polarized position angle variations could be caused either by phenomena at the pulsar ( e.g. , precession or magnetospheric adjustments in the emission beam geometry ) ; or by interestellar faraday rotation variability . the observed measurements and upper limits on position angle changes reported in tables [ table : sinefits ] - [ table : outliers ] can be translated ( 1 ) directly into changes or upper limits thereof in projected spin axis or magnetospheric orientation ; or to ( 2 ) changes or upper limits thereof in interstellar magnetic fields via faraday rotation . we now discuss each in turn . much theoretical work suggests that the amplitude of the precession would be smaller than a degree , which would place it below the threshold of measurability for most pulsars in this experiment ( _ cf . _ table [ table : big ] ) , and furthermore that it would quickly damp , at least in the presence of rigid pinning or even the slow creep of superfluid vortices [ e.g. , @xcite ] . nevertheless , there are several observations of periodic changes in pulse shape and/or in arrival times of isolated pulsars in the literature , along with ours , that seem to indicate the presence of precession . in a few of the other cases , the derived amplitude is on the order of one to several degrees and the period is in the few - yr range , which are similar to our results . the best documented example is psr b1828 - 11 , whose arrival times and pulse shape vary periodically and in a correlated fashion , indicating precession with a period on the order of 1 yr and @xmath111 amplitude @xcite , or @xmath112 @xcite or possibly more @xcite . psr b1642 - 03 also exhibits both phenomena @xcite , which have been modeled as stemming from precession with an amplitude of @xmath113 over several yr @xcite.@xmath114 @xcite discovered pulse shape and timing variations in psr b2217 + 47 , which they modeled as induced by precession with a timescale @xmath115 yr . psr b1557 - 50 shows a several - year timing and dispersion measure periodicity ( the latter interpreted as possible frequency - dependent temporal pulse shape variations ) which is modeled as precession with an amplitude of @xmath116 @xcite . @xcite observed quasiperiodic timing and profile variations in psr b0959 - 54 which could be caused by precession with a period @xmath117 2500 d and amplitude @xmath118 . quasiperiodicities on several hundred day timescales in crab nebula pulsar arrival times may also indicate precession @xcite . @xcite observed pulse shape , ppa , and timing variations in the vela pulsar which they suggested could result from free precesssion with a 330-d period ; and @xcite further elaborated the precession model to explain variations in x - ray morphology of the vela pulsar s pulsar wind nebula . @xcite have proposed that the galactic center radio transient gcrt j1745 - 3009 could be a neutron star precessing by @xmath119 , although an experiment like the current one would be insensitive to its 77-min period . finally , the x - ray emitting isolated neutron star rx j0720.4 - 3125 exhibits a wide variety of correlated periodic phenomena indicative of precession on a 7 - 8 yr timescale @xcite . our analysis shows that 19 of our 81 pulsars exhibit ppa variations that are significantly better fit ( at approximately the 3@xmath0 level ) by a sinusoidal ppa function than by a constant ppa . of these , we judge four pulsars to exhibit the most convincing evidence for sinusoidal variations in ppa . ( these are our `` class i '' pulsars : b0523 + 11 , b0611 + 22 , b0656 + 14 , and b2053 + 21 ; see fig . [ fig : classifits ] ) . our results indicate that variations , and even sinusoidal variations , are occasionally present in the pulsar population . as a group , these 19 pulsars display sinusoidal periods mostly grouped around 185 to 450 days , with a peak at 200 days and outliers at approximately 790 , 1050 , and 1250 days . the amplitude of the variation ranges from 1@xmath120 to 8@xmath120 for most of the significant fits , with a peak at 2@xmath120 and an outlier at 12@xmath120 . note that these sinusoidal results do not include the six large ppa excursions , rejected from the fits in table [ table : big ] and detailed in table [ table : outliers ] ; as these large excursions do not carry the signature of precession even if they are real . in [ sec : orientation]b , we briefly discussed rotating magnetospheric currents as a possible source of periodic ppa changes . in the absence of well - developed models , we only note here that such phenomena might provide an alternate explanation for the observed periodic ppa variations . table [ table : outliers ] lists the properties of those few ppa measurements lying far from the mean . as noted above , these could indicate the presence of magnetized variations in the ism , although they might conceivably result instead from an experimental problem . under the first assumption ( magnetized ism variations causing the deviations ) , we calculate and list in table [ table : outliers ] the quantities @xmath121 ( eq . [ eqn : deltafaradayrot ] ) and @xmath122 ( eq . [ eqn : deltarmfrac ] ) . however , the great majority of our pulsars shows no evidence whatsoever of ppa variations above the noise . under the second assumption ( that the few large ppa deviations observed are spurious ) , our measured upper limits on temporal variations of position angle among `` retained sessions '' ( the quantity @xmath57 of table [ table : big ] ) can be interpreted as upper limits on magnetized variations , again via eqs . [ eqn : deltafaradayrot ] and [ eqn : deltarmfrac ] where the quantities @xmath21 and @xmath24 are replaced by their respective statistical analogs , @xmath57 and @xmath123 . table [ table : big ] lists these derived upper limits on @xmath123 and @xmath124 ) . there are occasional reports in the literature of temporal variations in interstellar magnetic fields , some of which may have origins similar to ours . first , there are the cases of the vela and crab nebula pulsars @xcite . the variations of @xmath25 ( and also of @xmath107 ) are so large and frequent that they are almost certainly associated with passage of inhomogeneous nebular snr material across the line of sight . this interpretation is easy to reconcile with these pulsars relative youth and the observed presence of an snr . there are other reports in the literature of @xmath25 variations toward ordinary pulsars on multiyear timescales , sometimes accompanied by @xmath107 variations as well @xcite , which appear to be caused by the passage of a cloud across the line of sight . @xcite measured many pulsar @xmath25s and compared their results with earlier work wherever possible . they noted numerous cases of @xmath25s changing significantly with respect to values measured @xmath125 yr earlier , while the @xmath107 tended to vary far less on similar timescales . yet our sample of pulsars exhibits only rare evidence for @xmath25 variations over the four years of observation , and _ no _ evidence for sustained changes or long - term trends . we can suggest several possible resolutions to the disagreement between the rather common @xmath25 changes in the literature and our finding of essentially steady @xmath25s in our sample : ( 1 ) @xmath25 variations tend to be small on the four - year timescale of our experiment , but grow on decade - long scales . this would have interesting implications for the fluctuation spectrum of interstellar plasma . however , recall that we have some sensitivity to changes on timescales longer than our dataspan , but no evidence was found for variations on scales beyond @xmath126 d. ( 2 ) a subtle nonorthogonal emission mode competition process can lead to spurious apparent @xmath25 variations @xcite , which might grow with time . ( 3 ) one of the previously published multiepoch @xmath25 measurements was incorrect . this latter possibility can not be ruled out , since @xmath25 measurements are difficult , and since the pairs of observations were frequently performed by different groups on different telescopes . further measurements are required in order to determine the correct explanation .

we show that the uncertainties in a single - epoch measurement of position angle is usually dominated by random pulse - to - pulse jitter of the polarized subpulses . even with these uncertainties , we find that the position angle variations in 19 pulsars are significantly better fitted ( at the 3 level ) by a sinusoid than by a constant . such variations could be caused by precession , which would then indicate periods of d and amplitudes of degrees .

in order to study precession and interstellar magnetic field variations , we measured the polarized position angle of 81 pulsars at several - month intervals for four years . we show that the uncertainties in a single - epoch measurement of position angle is usually dominated by random pulse - to - pulse jitter of the polarized subpulses . even with these uncertainties , we find that the position angle variations in 19 pulsars are significantly better fitted ( at the 3 level ) by a sinusoid than by a constant . such variations could be caused by precession , which would then indicate periods of d and amplitudes of degrees . we narrow this collection to four pulsars that show the most convincing evidence of sinusoidal variation in position angle.also , in a handful of pulsars , single discrepant position angle measurements are observed which may result from the line of sight passing across a discrete ionized , magnetized structure . we calculate the standard deviation of position angle measurements from the mean for each pulsar , and relate these to limits on precession and interstellar magnetic field variations .

1103.4237

i

the accretion disk models have been developed much in the past several decades . many disk models have been proposed and some of them are widely adopted in astrophysical studies nowadays , including but not restricted to shakura - sunyaev disk ( ssd ; see shakura & sunyaev 1973 ) , advection dominated accretion flow ( adaf ; see narayan & yi 1994 ; abramowicz et al . 1995 ) , and slim disk ( abramowicz et al . 1988 ) . however , the outflow structure of accretion disks still remains an open problem . the equations that describe the hydrodynamic processes are the navier - stokes equations , which are quite difficult to solve in the case of accretion disks which involve viscosity and radiation . therefore , in most works , some kind of simplification , such as one - zone or polytropic distribution and hydrostatic equilibrium , are usually applied in the vertical direction , and the vertical variation ( @xmath4 dependence in cylindrical coordinates ) of the velocity field is usually neglected . in this way the equations are changed to a set of ordinary differential equations ( odes ) in the radial direction , which can be solved numerically . however , by taking these assumptions , one can not get a clear picture of the vertical structure of accretion flows ; the velocity is always radially inward and no mass will cross the disk surface , displaying no outflow structure . among the exceptions is a work done by narayan & yi in 1995(hereafter ny95 ) , which used self - similar assumptions in the radial direction and solved the structure along the @xmath1 direction in spherical coordinates ( @xmath0 ) . however , in their work they assumed @xmath5 and thus can not get a proper velocity field , and their solutions compose of only pure inflow . they argued that the bernoulli parameter is positive in their solutions so that a bipolar outflow is expected to develop near the vertical axis . blandford & begelman ( 1999 , hereafter bb99 ) relaxed the mass conservation assumption and assumed that the mass inflow rate varies with radius , and obtained solutions with outflow called adiabatic inflow - outflow solutions ( adios ) . their solutions are one - dimensional self - similar solutions that are height - averaged and they also applied the bernoulli parameter to argue the presence of outflow . however , abramowicz et al.(2000 ) pointed out that the positive bernoulli function ( instead of bernoulli parameter , which is defined in convenience of self - similar assumptions ) is not sufficient for outflow ( also see the simulation works done by stone et al . 1999 and yuan & bu 2010 ) . blandford & begelman ( 2004 ) furthered their work and presented some self - similar two - dimensional solutions of radiatively inefficient accretion flows with outflow . they assumed hydrostatic equilibrium in the vertical direction and that convection dominates the heat transport , which may only be applicable in certain cases . xu & chen ( 1997 , hereafter xc97 ) relaxed @xmath5 and obtained two types of solutions with outflow : accretion and ejection solutions . however their solutions require the net accretion rate to be 0 , which is not realistic . xue & wang(2005 , hereafter xw05 ) followed ny95 and solved the disk structure along the @xmath1 direction considering @xmath6 . they arbitrarily set a disk surface , at which @xmath7 , and the sound speed on the surface for their calculation . their solutions display a field of inflow near the equatorial plane with wind blowing out of the upper boundary , however the boundary is set as an input parameter rather than being calculated , and they only investigated several cases of adafs . in 3.2 we will see that according to our calculation , their assumption does nt hold for accretion flows with large @xmath2 value . sadowski et al.(2010 ) abandoned the self - similar assumptions and solved the accretion disk structure in the radial and vertical direction simultaneously . because the navier - stokes equations for accretion disks can not be decoupled intrinsically , they adopted other assumptions , e.g. the disk is not geometrically thick , to derive the equations . as in their work they did nt consider @xmath8 while @xmath9 and @xmath10 were supposed not to vary vertically , they were not able to study outflows . in summary , in the analytical model study , the vertical or @xmath1-direction structure of accretion disks and outflows can still not be dealt with satisfactorily , and many improvements are still needed to be made . on the other hand , observationally there are more and more evidences for outflows in accretion systems , such as sgr a*(marrone et al . 2006 ; xie & yuan 2008 ) , soft x - ray transients(loeb et al . 2001 ) and quasars with blueshifted absorption lines(e.g . , pg1115 + 80 ; chartas et al . 2003 ) . many numerical simulation works have also discovered outflows in their results ( e.g. stone et al . , 1999 ; igumenshchev & abramowicz , 2000 ; okuda et al . 2005 ; ohsuga et al.2005 ; ohsuga & mineshige 2007 ; ohsuga et al . the common existence of outflows in these works inspire us to investigate the vertical structure of accretion flows explicitly , to find solutions which can deal with @xmath6 and positive @xmath9 to get a clear velocity field , and with more reasonable boundary conditions . as a first step , we followed the work done by ny95 and xw05 , using self - similar assumptions in the radial direction and solving the odes along the @xmath1 direction in spherical coordinates ( @xmath0 ) . we used the @xmath2 viscosity prescription and assumed that the @xmath11 component of the viscosity stress tensor is dominant . by neglecting other components of the viscosity stress tensor , the necessary number of boundary conditions is reduced and we will only need the boundary conditions in the equatorial plane , which is obatined by symmetry and is thus quite physical . as we did nt set other arbitrary restrictions to @xmath9 and @xmath6 other than the self - similar assumptions , we can get a velocity field containing positive @xmath9 , to discuss the flow structure with possible outflows . these assumptions are applicable to different kinds of accretion disk models , ranging through ssd , adaf and slim disk , so we also did many calculations with different sets of parameters ( which is very difficult to do in a numerical simulation work as it would be too time - consuming ) , trying to find the flow structure dependence on the parameters , in order to understand the inflow / outflow mechanism more physically . these results can also be helpful to future numerical simulation works when they set the input parameters . it should be noted that there is another branch of researches which investigate accretion flows with an outflow ( usually wind ) plus accretion disk model . in these researches , the configuration of the outflow and accretion disk is usually preset , and the calculations are focused on either the outflow or the accretion disk while the influence of the other part is simplified or parameterized ( e.g. fukue , 1989 ; takahara et al . , 1989 ; kusunose , 1991 ; bb99 ; misra & taam , 2001 ; fukue , 2004 ; xie & yuan , 2008 ) . recently there are also several works done in this way which deal with the outflow and the accretion disk simultaneously ( kawabata & mineshige , 2009 ; dotan & shaviv , 2010 ) . compared with our work , in these studies the accretion disk is usually height - integrated and the configuration of the accretion flow is assumed rather than being calculated , while in our work we solve the full hydrodynamic equations to get the configuration of the accretion flow . our work focuses on studying the general structure of disks , outflows and the physical mechanism behind them , and the results can be complementary to one another . in 2 we present the basic equations and assumptions we used in our calculations . in 3.1 we discuss our numerical methods and present solutions corresponding to typical parameters of ssd , adaf and slim disk . in 3.2 we show the disk structure dependence on different parameters and discuss their physical meanings . 4 we present our summary and discussion .

in order to study the outflows from accretion disks , we solve the set of hydrodynamic equations for accretion disks in the spherical coordinates ( ) to obtain the explicit structure along the direction . using self - similar assumptions in the radial direction , we change the equations to a set of ordinary differential equations ( odes ) about the-coordinate , which are then solved with symmetrical boundary conditions in the equatorial plane , and the velocity field is obtained . the viscosity prescription is applied and an advective factor is used to simplify the energy equation.the results display thinner , quasi - keplerian disks for shakura - sunyaev disks ( ssds ) and thicker , sub - keplerian disks for advection dominated accretion flows ( adafs ) and slim disks , which are consistent with previous popular analytical models . however , we also present the structure dependence on the input parameters and discuss their physical meanings . the caveats of this work and possible improvements in the future are discussed .

in order to study the outflows from accretion disks , we solve the set of hydrodynamic equations for accretion disks in the spherical coordinates ( ) to obtain the explicit structure along the direction . using self - similar assumptions in the radial direction , we change the equations to a set of ordinary differential equations ( odes ) about the-coordinate , which are then solved with symmetrical boundary conditions in the equatorial plane , and the velocity field is obtained . the viscosity prescription is applied and an advective factor is used to simplify the energy equation.the results display thinner , quasi - keplerian disks for shakura - sunyaev disks ( ssds ) and thicker , sub - keplerian disks for advection dominated accretion flows ( adafs ) and slim disks , which are consistent with previous popular analytical models . however , an inflow region and an outflow region always exist , except when the viscosity parameter is too large , which supports the results of some recent numerical simulation works . our results indicate that the outflows should be common in various accretion disks and may be stronger in slim disks , where both advection and radiation pressure are dominant . we also present the structure dependence on the input parameters and discuss their physical meanings . the caveats of this work and possible improvements in the future are discussed .

cond-mat0402574

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our current understanding of critical phenomena results from an intensive interplay between experimental studies of a large variety of physical systems , ground - breaking theoretical developments ( including renormalization group methods , finite - size scaling theory and conformal invariance ) and extensive numerical investigations of model systems . in many cases the systems under investigation are treated as bulk systems , thus neglecting the existence of surfaces which are unavoidable in real physical systems . discarding surfaces in systems with short - range interactions is justifiable when studying bulk critical properties , as the contribution of the surface to extensive quantities is vanishing in the thermodynamic limit . however , a surface breaks the translation symmetry of a system and changes local quantities . thirty years ago , it has been realized that this leads to surface critical behaviour which differs from the bulk critical behaviour . since that time numerous theoretical and experimental studies have been undertaken in order to determine local critical quantities in systems with perfect surfaces . real surfaces , however , are usually not perfectly smooth but display some degree of roughness due to the presence of surface defects as for example steps , islands , or vacancies . impurities are also often encountered at crystalline surfaces and may be viewed as the source of some disorder at the surface . furthermore , experimentalists nowadays create thin films which do not appear in nature , by growing artificial structures on the film surface . all these defects have some impact on magnetic surface quantities . the present work reviews the recent progress achieved in the study of critical phenomena in systems with boundaries . besides semi - infinite systems and films with perfect surfaces , more complex geometries with wedges and corners as well as more realistic surfaces with defects are discussed . the review hereby focuses on the question whether the different types of geometries and/or of surface defects have an impact on the surface critical behaviour . it is therefore complementary to earlier reviews on surface criticality @xcite that exclusively considered flat , perfect surfaces . the thermodynamics of a surface is completely described by the surface free energy per area @xmath0 . singularities occurring in @xmath0 determine the phase diagram of semi - infinite systems . at some phase boundaries of this phase diagram surface and bulk free energies both exhibit singularities , whereas at other boundaries only @xmath0 becomes singular . an example for the former case is the * ordinary transition * where bulk and surface ordering occur at the same temperature , whereas the latter case is encountered at the so - called * surface transition * where the surface layer alone orders , while the bulk remains disordered . this surface transition is encountered at temperatures higher than the bulk critical temperature , the critical fluctuations of the @xmath1-dimensional semi - infinite system then being essentially ( @xmath2)-dimensional , corresponding to a phase transition in @xmath2 dimensions . in section 2 the critical behaviour at perfect surfaces is discussed . at the bulk critical point different surface universality classes are obtained for every bulk universality class . these universality classes are discussed and their differences emphasized . furthermore , recent progress in our understanding of surface critical behaviour in systems with competing interactions is reviewed . surface critical dynamics and the effect of symmetry breaking surface fields are also briefly discussed . section 3 is devoted to more complex geometries with a wedge . wedge - shaped models are very interesting systems where local critical exponents , which change continuously with the wedge opening angle , arise because of the particular geometrical properties of the wedge . however , at a given opening angle , surface critical behaviour at the ordinary transition is still universal and does not depend on microscopic details of the model as for example the lattice type or the strengths of the local interactions . this is completely different at the surface transition , where under certain circumstances non - universal local critical behaviour can be observed . at a fixed opening angle , edge critical exponents then not only depend continuously on the values of the local couplings but also reflect the existence of the disordered bulk . this intriguing behaviour results from the fact that at the surface transition the edge acts like a defect line in a two - dimensional critical system . corner critical behaviour is discussed in this section , too . in the last years a great deal of activity focused on the critical behaviour of wedges in presence of external surface fields . this rapidly developing field is also briefly summarized . section 4 deals with the important issue of thin films and semi - infinite systems with non - perfect surfaces . surfaces are very often naturally rough , due to the growth mechanism or because of erosion effects . adatom islands , vacancy islands or steps are typical defects encountered at real surfaces . on the other hand , specific surface structures , as for example lines of adatoms or regular geometrical patterns , can be created on purpose by using advanced experimental methods . as these surface defects have an impact on local surface quantities , one has to ask the question whether they do change the surface critical behaviour . for semi - infinite systems we must again distinguish between the ordinary transition and the surface transition . at the ordinary transition , common surface defects ( presence of a step , surfaces with uncorrelated roughness , amorphous surface ) are usually irrelevant for the surface critical behaviour , but there are some notable exceptions . at the surface transition , defect structures like steps and additional lines of atoms located at the surface may yield non - universal local critical exponents , as observed in semi - infinite ising models with additional surface structures . interestingly , additional lines located at the surface of thin ising films always lead to a non - universal local critical behaviour . section 5 finally contains concluding remarks .

in the past perfect surfaces have been shown to yield a local critical behaviour that differs from the bulk critical behaviour . on the other hand surface defects , whether they are of natural origin or created artificially , are known to modify local quantities . it is shown furthermore that under certain circumstances non - universal local critical behaviour may be observed at surfaces .

in the past perfect surfaces have been shown to yield a local critical behaviour that differs from the bulk critical behaviour . on the other hand surface defects , whether they are of natural origin or created artificially , are known to modify local quantities . it is therefore important to clarify whether these defects are relevant or irrelevant for the surface critical behaviour . the purpose of this review is two - fold . in the first part we summarise some of the important results on surface criticality at perfect surfaces . special attention is thereby paid to new developments as for example the study of surface critical behaviour in systems with competing interactions or of surface critical dynamics . in the second part the effect of surface defects ( presence of edges , steps , quenched randomness , lines of adatoms , regular geometric patterns ) on local critical behaviour in semi - infinite systems and in thin films is discussed in detail . whereas most of the defects commonly encountered are shown to be irrelevant , some notable exceptions are highlighted . it is shown furthermore that under certain circumstances non - universal local critical behaviour may be observed at surfaces .

cond-mat0402574

c

surface criticality has been the subject of intensive studies in the last thirty years . thereby a large variety of different methods ( analytical , numerical , and experimental ) has been used , yielding a host of interesting results , as reviewed in this work . critical phenomena at perfect surfaces are now in general well understood , at least when dealing with static critical quantities . this is not really the case for dynamic critical behaviour at surfaces for which a coherent picture has not yet emerged . a major problem in this context is the total lack of experimental studies on surface dynamic properties at criticality . the situation is also not very satisfactory from the theoretical point of view , as only selected results on equilibrium and non - equilibrium surface dynamic behaviour at criticality have been published . clearly , this is one of the most important aspects of surface criticality that warrants more attention in the future . wedge - shaped geometries , which can be viewed as generalisations of semi - infinite systems , have also been discussed in detail in this review . it is encouraging that the effect of curved surfaces on the local critical behaviour has been observed in simulations of liquid - vapour transitions near a weakly attractive surface . this may point to possible experimental systems where this kind of problems can be studied . there are indeed a vast number of theoretical predictions for this kind of geometry , and experimental investigations are therefore welcomed . in the last years the focus of research on surface criticality has somehow shifted , as the main emphasis has been on more realistic surfaces . the facts that real surfaces are usually rough , displaying a variety of different surface defects , and that experimental physicists can create artificial structures on top of a surface directly lead to the question whether these quantities have an impact on local critical behaviour . we have presented a comprehensive overview of the field , thereby discussing in detail critical behaviour in semi - infinite systems with surface defects as well as in thin films with additional surface structures . some common surface defects have been shown to be irrelevant for the surface critical behaviour at the ordinary transition . there are however some interesting exceptions , as for example the case of self - affine rough surfaces . the situation is even more complex at the surface transition where in a three - dimensional system the critical fluctuations are of two - dimensional nature . indeed , additional structures as steps or lines of adatoms have been shown in numerical studies to lead to non - universal local critical behaviour where the values of the local critical exponents reflect the strengths of the coupling constants as well as the presence of the disordered bulk . it is worth noting that in thin films this kind of additional surface structures in general leads to non - universal critical behaviour . it is obvious from our overview that a large number of recent studies of the effects of surface defects on the local critical behaviour are either of purely numerical nature or are using rather crude approximations . this is especially the case when dealing with non - perfect surfaces at the surface transition . there is a need for more elaborate analytical approaches , and it is one of the intention of this review to encourage further theoretical ( and also experimental ) investigations of critical phenomena at non - perfect surfaces . it is a pleasure to thank all my collaborators who worked with me on various aspects of surface criticality : f. . bagamry , d. catrein , m .- c . chung , f. igli , m. kaulke , i. peschel , w. selke , and l. turban . i am indebted to w. selke for introducing me to the field of surface critical behaviour and for many years of fruitful collaboration . i also thank h.w . diehl , m. henkel , and a. hller for many inspiring discussions .

the effect of surface defects ( presence of edges , steps , quenched randomness , lines of adatoms , regular geometric patterns ) on local critical behaviour in semi - infinite systems and in thin films is discussed in detail . whereas most of the defects commonly encountered are shown to be irrelevant , some notable exceptions are highlighted .

in the past perfect surfaces have been shown to yield a local critical behaviour that differs from the bulk critical behaviour . on the other hand surface defects , whether they are of natural origin or created artificially , are known to modify local quantities . it is therefore important to clarify whether these defects are relevant or irrelevant for the surface critical behaviour . the purpose of this review is two - fold . in the first part we summarise some of the important results on surface criticality at perfect surfaces . special attention is thereby paid to new developments as for example the study of surface critical behaviour in systems with competing interactions or of surface critical dynamics . in the second part the effect of surface defects ( presence of edges , steps , quenched randomness , lines of adatoms , regular geometric patterns ) on local critical behaviour in semi - infinite systems and in thin films is discussed in detail . whereas most of the defects commonly encountered are shown to be irrelevant , some notable exceptions are highlighted . it is shown furthermore that under certain circumstances non - universal local critical behaviour may be observed at surfaces .

cond-mat0703433

i

since superfluidity was first realized in ultracold trapped two - component atomic fermi gases @xcite , there has been increasing interest in the properties of spin - polarized fermi superfluids @xcite . as in unpolarized fermi gas superfluids , by making use of a feshbach scattering resonance between fermions , the effective scattering length @xmath3 that characterizes low - energy scattering between fermions in different hyperfine states ( i.e. , @xmath4 ) can be continuously tuned from negative to positive values . we use @xmath5 to denote the two species of atomic fermions prepared in different hyperfine states . when @xmath6 , the system is in the bcs region and pairing can occur about the fermi surface between fermions in different hyperfine states . passing through unitarity ( where @xmath7 ) , the scattering length becomes positive and one enters the bec region , characterized by the disappearance of the fermi surface ( the fermion chemical potential becomes negative ) and the appearance of strongly bound molecular pairs . in the spin - polarized fermi superfluid , there is an excess of one of the two - species of fermions : @xmath8 . in the bec region , all of the minority species of fermions are paired up , leaving a gas of the remaining spin @xmath9 fermions . thus , in the bec region , a spin - polarized fermi gas should behave like a bose - fermi mixture of the dimer molecules and unpaired excess fermions . in this paper , we prove that the effective action for a spin - polarized two - component fermi superfluid in the bec region reduces to an effective action for a bose - fermi mixture . the idea that a spin - polarized fermi superfluid reduces to a bose - fermi mixture in the bec region has been explored in refs . we extend those studies by considering quartic pairing fluctuations of the effective bose action for a spin - polarized fermi superfluid . at this order , we show that a spin - polarized fermi superfluid can still be described in terms of an equivalent bose - fermi mixture with a renormalized dimer - fermion interaction that is valid beyond the born approximation value @xmath10 @xcite , where @xmath11 is the reduced mass of the molecular dimer - atomic fermion system . our discussion of the effective action of a spin - polarized fermi gas is based on the functional integration treatment @xcite of the bcs - bec crossover problem developed in refs . @xcite . extending the approach of ref . @xcite , we expand this action in powers of pairing fluctuations about the bcs - type mean - field saddle - point up to quartic order in fluctuations . at the level of quadratic ( gaussian ) fluctuations , we derive the bogoliubov theory of excitations of a dimer molecular condensate in the spin - polarized gas in sec . [ bogspectrum ] . we show how the bogoliubov excitation energy involves a dimer - dimer interaction mediated by the unpaired @xmath9 fermions through a lindhard response function . this result is compared with the bogoliubov excitation spectrum in a bose - fermi mixture in sec . [ gaussianbf ] . in sec . [ quartic ] we discuss the quartic fluctuations of an effective bose action for a bose - fermi mixture as well as for a spin - polarized fermi gas in the bec limit . comparing the two , we prove their equivalence and derive an expression for the momentum- and energy - dependent dimer - fermion interaction in a spin - polarized . in order to make contact with the extensive literature on two - body dimer - fermion scattering in free space @xcite , in sec . [ tbsa ] we show how our theory reproduces the well - known result @xmath1 due to skorniakov and ter - martirosian @xcite for the dimer - fermion scattering length .

in the strong - coupling bec region where a feshbach resonance gives rise to tightly - bound dimer molecules , we show that a spin - polarized fermi superfluid reduces to a simple bose - fermi mixture of bose - condensed dimers and the leftover unpaired fermions . using a many - body functional integral formalism , , we show how the action for a spin - polarized fermi superfluid reduces to one for a bose - fermi mixture .

in the strong - coupling bec region where a feshbach resonance gives rise to tightly - bound dimer molecules , we show that a spin - polarized fermi superfluid reduces to a simple bose - fermi mixture of bose - condensed dimers and the leftover unpaired fermions . using a many - body functional integral formalism , the gaussian fluctuations give rise to an induced dimer - dimer interaction mediated by the unpaired fermions , with the dimer - fermion vertex being given by the ( mean - field ) born approximation . treating the pairing fluctuations to quartic order , we show how the action for a spin - polarized fermi superfluid reduces to one for a bose - fermi mixture . this bose - fermi action includes an expression for the effective dimer - unpaired fermion interaction in a spin - polarized fermi superfluid beyond the born approximation , in the superfluid phase at finite temperatures . in the low - density limit , we show how this dimer - fermion interaction gives the-wave scattering length ( is the-wave fermion scattering length ) , a result first derived by skorniakov and ter - martirosian in 1957 for three interacting fermions . 0 * 0 *

0803.0707

i

let @xmath8=\{1,\ldots , p\}$ ] , and @xmath9 be the set of permutations of @xmath8 $ ] , for @xmath10 . when @xmath10 is even , let @xmath11 be the set of _ pairings _ on @xmath8 $ ] , which are partitions of the set @xmath8 $ ] into disjoint pairs ( subsets of size @xmath2 ) . we refer to the single element of @xmath12 as the _ empty _ pairing . where the context is appropriate , we shall also regard @xmath11 as the conjugacy class of involutions with no fixed points in @xmath9 . in this latter context , each pair becomes a disjoint cycle consisting of that pair of elements . of course , the number of pairings in @xmath11 is @xmath13 , with the empty product convention that @xmath14 . now , for @xmath15 and even , let @xmath16 , in disjoint cycle notation , and let @xmath17 . let @xmath18 be the number of permutations in @xmath19 with @xmath20 cycles in the disjoint cycle representation , for @xmath21 . the generating series for these numbers are given by @xmath22 . harer and zagier [ [ hz ] ] obtained the following result . [ hzthm ] _ ( harer and zagier [ [ hz ] ] ) _ for a positive , even integer @xmath6 , with @xmath23 , @xmath24 other proofs of theorem [ hzthm ] have been given by itzykson and zuber [ [ iz ] ] , jackson [ [ j ] ] , kerov [ [ ke ] ] , kontsevich [ [ k ] ] , lass [ [ l ] ] , penner [ [ p ] ] and zagier [ [ z ] ] ( see also the survey by zvonkin [ [ zv ] ] , section 3.2.7 of lando and zvonkin [ [ lz ] ] and the discussion in section 4 of the paper by haagerup and thorbjornsen [ [ ht ] ] ) . recently , goulden and nica [ [ gn ] ] gave a direct bijective proof of theorem [ hzthm ] . in the present paper , we consider a similar bijective approach to extend this important result of harer and zagier to the case in which the permutation @xmath25 is replaced by a fixed permutation with two cycles in its disjoint cycle representation . some additional notation is required . let @xmath26'=\{1',\ldots , q'\}$ ] , and let @xmath27 be the set of permutations of @xmath8\cup[q]'$ ] , for @xmath28 . let @xmath29 be the set of pairings on @xmath8\cup[q]'$ ] , for @xmath28 , where @xmath30 is even ( we refer to the single element of @xmath31 as the _ empty _ pairing ) . a pair in a pairing is called _ mixed _ if it consists of one element from @xmath8 $ ] and one element from @xmath26'$ ] . where the context is appropriate , we shall also regard @xmath29 as the conjugacy class of involutions with no fixed points in @xmath27 . for @xmath32 , we consider the permutation @xmath33 , and let @xmath34 , and @xmath35 be the number of permutations in @xmath36 with @xmath20 cycles in the disjoint cycle representation , for @xmath37 . consider the generating series @xmath38 the main result of this paper is the following expression for @xmath39 . [ mainthm ] for @xmath40 , with @xmath41 of the same odd - even parity and @xmath42 , we have @xmath43 where @xmath44 note that theorem [ mainthm ] gives a summation of nonnegative terms , since for all choices of summation indices @xmath45 with @xmath46 ( so that @xmath47 is nonzero ) , the difference @xmath48 is nonnegative . the proof of theorem [ mainthm ] is based on a combinatorial model that is developed in section [ sec2 ] . as a consequence , it is sufficient to enumerate a particular graphical object that we call a _ paired array_. we then give two combinatorial reductions , in sections [ sec3 ] and [ sec4 ] , in terms of a simpler class of paired arrays called _ vertical _ paired arrays . these are explicitly enumerated in section [ sec5 ] , which allows us to complete the proof of theorem [ mainthm ] . one of the combinatorial conditions on paired arrays is that two graphs associated with them must be acyclic . because of this , a key component of sections [ sec4 ] and [ sec5 ] is the enumeration of rooted forests which contain a given forest as a subgraph . thus in section [ sec35 ] we give a new bijection for this fundamental combinatorial problem . however , before we turn to our combinatorial model and subsequent reductions , we consider some consequences of theorem [ mainthm ] , and give some comparisons to results in the existing literature . a major reason that harer and zagier s result ( theorem [ hzthm ] ) is important ( as evidenced by so many published proofs ) is that it can be restated as an equivalent geometric problem in terms of maps . map _ is an embedding of a connected graph ( with loops and multiple edges allowed ) in an orientable surface , partitioning the surface into disjoint regions ( called the _ faces _ of the map ) that are homeomorphic to discs ( this is called a two - cell embedding ) . a _ rooted _ map is a map with a distinguished edge and incident vertex ( so , the map is `` rooted '' at that end of the distinguished edge ) . the well - known embedding theorem allows us to consider this as equivalent to a pair of permutations and their product ( see , e.g. , tutte [ [ t ] ] , where the terminology `` rotation system '' is used to describe this triple of permutations ) . from this point of view , the @xmath20th coefficient @xmath18 in the generating series @xmath49 evaluated in theorem [ hzthm ] is equal to the number of rooted maps with @xmath50 vertex , @xmath3 edges and @xmath20 faces ( where @xmath23 , as in theorem [ hzthm ] ) . denoting the genus of the surface in which such a map is embedded by @xmath51 , then the euler - poincar theorem implies that @xmath52 , or that @xmath53 . similarly , theorem [ mainthm ] has a geometric interpretation . let @xmath54 be the conjugacy class of @xmath55 in which there are two disjoint cycles , of lengths @xmath6 and @xmath7 . then the coefficient @xmath35 in the generating series @xmath39 is equal to @xmath56 times the number of rooted maps with @xmath2 vertices ( of degrees @xmath6 and @xmath7 ) , @xmath3 edges ( exactly @xmath57 of which join the two vertices together , plus @xmath58 that are loops at the vertex of degree @xmath6 , plus @xmath59 that are loops at the vertex of degree @xmath7 ) , and @xmath20 faces ( where @xmath42 , as in theorem [ mainthm ] ) . in this case , if we denote the genus of the surface in which such a map is embedded by @xmath51 , then we obtain @xmath60 . of course , since genus is a nonnegative integer , we must have @xmath61 , and indeed the coefficient of @xmath62 in the summation for @xmath39 given in theorem [ mainthm ] is zero , since the summand corresponding to @xmath63 ( which has @xmath64 as a factor ) is itself equal to zero . for the planar case , which corresponds to @xmath65 , the only nonzero summand that contributes to the coefficient of @xmath66 in the summation of theorem [ mainthm ] corresponds to @xmath67 , and this gives immediately that @xmath68 this checks with the straightforward computation that one can make to determine this value by elementary means there are @xmath57 edges between the two vertices ; between the ends of these edges at each vertex is an even number of vertices , joined by loops without crossings ( and there is catalan number of such arrangements for each such even interval ) . this explains the term `` genus '' in the title ; the term `` annular '' is adapted from its usage in mingo and nica [ [ mn ] ] . it refers to an equivalent embedding for a map with two vertices , in an annulus . the ends of the edges incident with one of the vertices ( say the one of degree @xmath6 ) are identified with @xmath6 points arranged around the disc on the exterior of the annulus , and the ends incident with the other vertex are identified with @xmath7 points arranged around the disc on the interior of the annulus . the points corresponding to the two ends of an edge are joined by an arc in the interior of the annulus . we have been able to find one relevant enumerative result ( jackson [ [ j ] ] ) in the literature about such maps , in which the total number of edges is specified , but not the exact number joining the two vertices together . to compare this result to our main result , we must sum over @xmath69 ( since the underlying graph must be connected , then @xmath57 , the number of edges joining the two vertices together , must be positive ) , and thus define @xmath70 then jackson [ [ j ] ] has considered the case @xmath71 , and obtained the following result , restated in terms of our notation ( by applying the proportionality constant @xmath72 ) . [ jint ] _ ( jackson [ [ j ] ] ) _ for @xmath73 , @xmath74 by slightly modifying jackson s [ [ j ] ] integration argument we are able to obtain the following expression for @xmath75 , with arbitrary @xmath4 of the same parity . [ gsint ] for @xmath76 , with @xmath30 even , and @xmath42 , @xmath77 we have checked computationally , with the help of maple , that theorems [ jint ] and [ gsint ] agree with theorem [ mainthm ] , summed over @xmath69 , for a wide range of values of @xmath4 . however , we have been unable to prove this for all @xmath4 , since we have not been able to show that the sum over @xmath69 of the result of theorem [ mainthm ] is equal to the result of theorem [ gsint ] . note that the summation in theorem [ gsint ] can be made symmetrical in @xmath4 ( so the ordering @xmath78 is not required ) by changing the summation variable @xmath20 to @xmath79 . the method employed in jackson [ [ j ] ] for theorem [ jint ] , and in many of the papers listed above that give proofs of theorem [ hzthm ] , is matrix integration . however , we do not see how to adapt the matrix integration methodology to prove our main result , theorem [ mainthm ] , since it does nt seem possible to specify that there are exactly @xmath57 edges joining the two vertices together in the matrix method . the simplicity of our result seems to suggest that an extended theory of matrix integration to allow a specified number of edges between particular vertices might be possible , and worth investigating . the simplicity of the result also suggests that there should be a more direct combinatorial proof than the one presented in this paper .

in the symmetric group on a set of size , let denote the conjugacy class of involutions with no fixed points ( equivalently , we refer to these as `` pairings '' , since each disjoint cycle has length ) . their famous result has been reproved many times , primarily because it can be interpreted as the genus distribution for-cell embeddings in an orientable surface , of a graph with a single vertex attached to loops . in this paper we give a new formula for the cycle distribution when a fixed permutation with two cycles ( say the lengths are , where ) is multiplied by the elements of . it can be interpreted as the genus distribution for-cell embeddings in an orientable surface , of a graph with two vertices , of degrees and . in terms of these graphs , the formula involves a parameter that allows us to specify , separately , the number of edges between the two vertices and the number of loops at each of the vertices .

in the symmetric group on a set of size , let denote the conjugacy class of involutions with no fixed points ( equivalently , we refer to these as `` pairings '' , since each disjoint cycle has length ) . harer and zagier explicitly determined the distribution of the number of disjoint cycles in the product of a fixed cycle of length and the elements of . their famous result has been reproved many times , primarily because it can be interpreted as the genus distribution for-cell embeddings in an orientable surface , of a graph with a single vertex attached to loops . in this paper we give a new formula for the cycle distribution when a fixed permutation with two cycles ( say the lengths are , where ) is multiplied by the elements of . it can be interpreted as the genus distribution for-cell embeddings in an orientable surface , of a graph with two vertices , of degrees and . in terms of these graphs , the formula involves a parameter that allows us to specify , separately , the number of edges between the two vertices and the number of loops at each of the vertices . the proof is combinatorial , and uses a new algorithm that we introduce to create all rooted forests containing a given rooted forest .

1304.5327

i

the optically bright , extended region sh2 - 252 is a part of the gemini ob1 association . this region is mainly composed of two small clusters ngc 2175s and teu 136 and four cregions namely a , b , c and e. in this paper , an extensive survey of the star forming complex sh2 - 252 has been undertaken with an aim to explore its hidden young stellar population , their characteristics , spatial distribution , morphology of the region and finally to understand the star formation scenario of the complex for the first time . _ spitzer_-irac , mips photometry ( 3.6 - 24 @xmath2 m ) are combined with 2mass - nir and optical data sets to identify and classify the ysos by their ir excess emission from their circumstellar material . using the ir c - c criteria , we have identified 577 ysos in the complex , of which , 163 are consistent with class i , 400 are consistent with class ii and 14 are consistent with transition disk ysos , suggesting a moderately rich number of ysos in this region . from the cmd and sed based analyses , majority of the ysos are found to have an age distribution between 0.1 - 5 myr and mass in the range of 0.3 - 3.0 m@xmath83 . spatial distribution of the candidate ysos shows that they are mostly clustered around the sub - regions of the complex such as in a , c , e , f and teu 136 . however , majority of the candidate ysos are distributed in the western part of the complex when compared to the east , suggesting enhanced star formation activity towards its west . using the sed and cmd based age analyses , we derived probable evolutionary status of the sub - regions of sh2 - 252 . our analyses suggest that the region a is the youngest ( @xmath0 0.5 myr ) , the regions b , c and e are of similar evolutionary stage ( @xmath0 1 - 2 myr ) and the small clusters ngc 2175s and teu 136 , located towards the east of sh2 - 252 are slightly evolved ( @xmath0 2 - 3 myr ) . morphology of the region in the _ spitzer _ colour images as well as in the 1.1 mm map shows an almost semi - circular ring - like shape towards the western half of the complex . indeed , we find a molecular shell composed of several clumps distributed around the main ionizing source ( i.e. , hd 42088 ) , suggesting that the expansion of the region is collecting the surrounding material , which gives rise to the semi - circular ring shape . we find several candidate ysos distributed over the semi - circular molecular shell which is an evidence for the star formation activity within the shell . finally , by comparing the age of the ionizing source , fragmentation time of the collected molecular shell and age of the ysos , we suggest collect and collapse scenario as one of the the possible mechanisms responsible for the star formation within the shell . we observed the densest concentration of ysos , ( mostly class i , @xmath0 0.5 myr ) at the western outskirts of the complex , within a molecular clump located between the cregions a and b. the correlation between the molecular clump at this location with large number of class i ysos and the associated water and methanol masers suggest that it is indeed a site of cluster formation in a very early evolutionary stage , sandwiched between the two relatively evolved cregions a and b. we conclude that the region is undergoing a complex star formation activity and there is a strong interplay between the radiation from the expanding region to the surrounding molecular material . the region certainly deserves attention for high resolution molecular line observations to further explore its hidden structure .

in this paper , an extensive survey of the star forming complex sh2 - 252 has been undertaken with an aim to explore its hidden young stellar population as well as to understand the structure and star formation history for the first time . we used 2mass - nir and _ spitzer_-irac , mips photometry to identify and classify the young stellar objects ( ysos ) by their infra - red ( ir ) excess emission . using the ir colour - colour criteria , we identified 577 ysos , of which , 163 are class i , 400 are class ii and 14 are transition disk ysos , suggesting a moderately rich number of ysos in this complex . spatial distribution of the candidate ysos shows that they are mostly clustered around the sub - regions in the western half of the complex , suggesting enhanced star formation activity towards its west . using the spectral energy distribution ( sed ) and optical colour - magnitude diagram ( cmd ) based age analyses , we derived probable evolutionary status of the sub - regions of sh2 - 252 . our analysis shows that the region a is the youngest ( 0.5 myr ) , the regions b , c and e are of similar evolutionary stage ( 1 - 2 myr ) and the clusters ngc 2175s and teu 136 are slightly evolved ( 2 - 3 myr ) . morphology of the region in the 1.1 mm map shows a semi - circular shaped molecular shell composed of several clumps and ysos bordering the western ionization front ( if ) of sh2 - 252 . our analyses suggest that next generation star formation is currently under way along this border and that possibly fragmentation of the matter collected during the expansion of the region as one of the major processes responsible for such stars . we observed the densest concentration of ysos ( mostly class i , 0.5 myr ) at the western outskirts of the complex , within a molecular clump associated with water and methanol masers and we suggest that it is indeed a site of cluster formation at a very early evolutionary stage , sandwiched between the two relatively evolved cregions a and b. [ firstpage ] stars : formation stars : premainsequence infrared : ism regions - ism : individual objects : sh2 - 252

in this paper , an extensive survey of the star forming complex sh2 - 252 has been undertaken with an aim to explore its hidden young stellar population as well as to understand the structure and star formation history for the first time . this complex is composed of five prominent embedded clusters associated with the sub - regions a , c , e , ngc 2175s and teu 136 . we used 2mass - nir and _ spitzer_-irac , mips photometry to identify and classify the young stellar objects ( ysos ) by their infra - red ( ir ) excess emission . using the ir colour - colour criteria , we identified 577 ysos , of which , 163 are class i , 400 are class ii and 14 are transition disk ysos , suggesting a moderately rich number of ysos in this complex . spatial distribution of the candidate ysos shows that they are mostly clustered around the sub - regions in the western half of the complex , suggesting enhanced star formation activity towards its west . using the spectral energy distribution ( sed ) and optical colour - magnitude diagram ( cmd ) based age analyses , we derived probable evolutionary status of the sub - regions of sh2 - 252 . our analysis shows that the region a is the youngest ( 0.5 myr ) , the regions b , c and e are of similar evolutionary stage ( 1 - 2 myr ) and the clusters ngc 2175s and teu 136 are slightly evolved ( 2 - 3 myr ) . morphology of the region in the 1.1 mm map shows a semi - circular shaped molecular shell composed of several clumps and ysos bordering the western ionization front ( if ) of sh2 - 252 . our analyses suggest that next generation star formation is currently under way along this border and that possibly fragmentation of the matter collected during the expansion of the region as one of the major processes responsible for such stars . we observed the densest concentration of ysos ( mostly class i , 0.5 myr ) at the western outskirts of the complex , within a molecular clump associated with water and methanol masers and we suggest that it is indeed a site of cluster formation at a very early evolutionary stage , sandwiched between the two relatively evolved cregions a and b. [ firstpage ] stars : formation stars : premainsequence infrared : ism regions - ism : individual objects : sh2 - 252

1504.05260

i

in the mathematical modelling of epidemic diseases , the fate of the disease can be predicted through the uninfected and infected equilibria and their stability . the basic reproduction number , @xmath1 , represents the average number of new infectives introduced into an otherwise disease - free system by a single infective , and is usually chosen as the bifurcation parameter . if the model involves a forward bifurcation , the uninfected equilibrium is in general globally asymptotically stable @xcite , characterized by @xmath2 , and infection fails to invade in this parameter regime . the threshold @xmath3 defines a bifurcation ( or critical ) point , and when @xmath4 , a stable infected equilibrium emerges . this simple exchange of stability implies that complex dynamics will not typically occur in forward bifurcation . in contrast , backward bifurcation describes a scenario in which a turning point of the infected equilibrium exists in a region where all state variables are positive , and @xmath5 . this induces multiple infected equilibria , disrupting the global stability of the uninfected equilibrium . multiple stable states ( e.g. , bistability ) may likewise appear in @xcite , and yu et al . ( submitted for publication ) . instead of converging globally to the uninfected equilibrium when @xmath2 , the solution may approach an infected equilibrium , depending on initial conditions . in practice , the phenomenon of backward bifurcation gives rise to new challenges in disease control , since reducing @xmath1 such that @xmath2 is not sufficient to eliminate the disease @xcite . instead , @xmath1 needs to be reduced past the critical value given by the turning point @xcite , since the result in yu et al . ( submitted for publication ) shows that the uninfected equilibrium in backward bifurcation is globally stable if @xmath1 is smaller than the turning point . furthermore , an infective outbreak or catastrophe may occur if @xmath1 increases and crosses unity , while the upper branch of the infected equilibrium remains stable @xcite . in addition , oscillation or even recurrent phenomena may occur if uninfected and infected equilibria coexist in a parameter range , and both are unstable @xcite . @xcite predicted oscillations arising from backward bifurcation , and @xcite pointed out that the unstable infected equilibrium `` commonly arises from hopf bifurcation '' , but did not demonstrate oscillations . several mechanisms leading to backward bifurcation have been proposed , such as partially effective vaccination programs @xcite , educational influence on infectives behavior @xcite , the interaction among multi - group models @xcite and multiple stages of infection @xcite . in this study , we will investigate the emergence of backward bifurcation in three simple disease models which have arisen in the study of epidemiology , in - host disease and autoimmunity . in each case , we find that backward bifurcation facilitates the emergence of hopf bifurcation(s ) , and hopf bifurcation in turn underlies a range of complex and clinically relevant dynamical behaviors . a central theme in our investigation is the role of the incidence rate in the epidemiological and in - host disease models . the incidence rate describes the speed at which an infection spreads ; it denotes the rate at which susceptibles become infectives . under the assumptions of mass action , incidence is written as the product of the infection force and the number of susceptibles . for example , if @xmath6 and @xmath7 denote the susceptible and infective population size respectively , a bilinear incidence rate , @xmath8 ( where @xmath9 is a positive constant ) , is linear in each of the state variables : @xmath6 and @xmath7 . the possibility of saturation effects @xcite has motivated the modification of the incidence rate from bilinear to nonlinear . saturation occurs when the number of susceptible contacts per infective drops off as the proportion of infectives increases . a nonlinear incidence rate , therefore , typically increases sublinearly with respect to the growth of the infective population , and may finally reach an upper bound . the development of nonlinear incidence was first investigated in the form @xmath10 , where @xmath9 , @xmath11 , and @xmath12 are positive constants @xcite . other forms of nonlinear incidence have also been analysed , such as @xmath13 @xcite , and @xmath14 @xcite . since the nonlinear incidence functions described above were often developed to incorporate saturation effects , these functions are typically concave at realistic parameter values . @xcite used this feature to derive general results for disease models with concave incidence . they proved that standard epidemiological models with concave incidence functions will have globally asymptotically stable uninfected and infected equilibria for @xmath2 and @xmath4 , respectively . more specifically , denoting the incidence rate function as @xmath15 , where @xmath16 is the population size , the classical sirs model considered in @xcite takes the form @xmath17 where @xmath18 , @xmath19 , and @xmath20 represent the birth / death rate , the recovery rate and the loss of immunity rate , respectively . when @xmath21 , system ( [ paper3_eq1 ] ) becomes an sir model . assuming that the total population size is constant , that is , @xmath22 , the above system can be reduced to a 2-dimensional model : @xmath23 moreover , it is assumed in @xcite that the function @xmath15 , denoting the incidence rate , satisfies the following three conditions : @xmath24 & \frac{\partial f(s,\,i,\,n)}{\partial i}>0 , \quad \frac{\partial f(s,\,i,\,n)}{\partial s}>0 , \quad \forall \ ; s,\,i > 0 \label{paper3_eq3b}\\[0.5ex ] & \frac{\partial^2 f(s,\,i,\,n)}{\partial i^2}\leq 0 , \quad \forall \ ; s,\,i > 0 \label{paper3_eq3c}.\end{aligned}\ ] ] [ paper3_eq3 ] the first two conditions ( [ paper3_eq3a ] ) and ( [ paper3_eq3b ] ) are necessary to ensure that the model is biologically meaningful . the third condition ( [ paper3_eq3c ] ) implies that the incidence rate @xmath15 , is concave with respect to the number of infectives . it is also assumed that @xmath25 evaluated at the uninfected equilibrium is proportional to the basic reproduction number @xmath1 @xcite , and thus should be a positive finite number @xcite . korobeinikov and maini first considered @xmath26 , or @xmath27 , and showed that forward bifurcation occurs in model ( [ paper3_eq2 ] ) with a concave incidence function . they further proved that the uninfected equilibrium @xmath28 and the infected equilibrium @xmath29 are globally asymptotically stable , when @xmath30 and @xmath4 , respectively . in the sections to follow , for an incidence rate function @xmath31 , satisfying ( [ paper3_eq3a ] ) and ( [ paper3_eq3b ] ) , we define @xmath31 as concave , if it satisfies ( [ paper3_eq3c ] ) ; as convex , if @xmath32 , @xmath33 ; and as convex - concave , if there exist @xmath34 , such that @xmath35 , @xmath36 , and @xmath37 , @xmath38 , @xmath39 , for @xmath40 , @xmath41 , @xmath42 . several models closely related to ( [ paper3_eq2 ] ) have been previously studied . for example , by adding a saturating treatment term to model ( [ paper3_eq2 ] ) with a concave incidence rate , @xcite showed that this model may yield backward bifurcation and hopf bifurcation . with an even more sophisticated nonlinear incidence rate function : @xmath13 , where @xmath43 , @xcite proved that a reduced 2-dimensional sirs model could exhibit backward bifurcation , hopf bifurcation , and even bogdanov - takens bifurcation and homoclinic bifurcation . although the choice of @xmath43 was not motivated by a specific physical process , this important result demonstrates that a nonlinear incidence rate can induce backward bifurcation , and further generate complex dynamics in a simple disease model . one of the focal points of our study will be a convex incidence function which arose in a 4-dimensional hiv antioxidant therapy model @xcite . in this model , the infectivity of infected cells was proposed to be an increasing function of the density of reactive oxygen species , which themselves increase as the infection progresses . in @xcite , meaningful parameter values were carefully chosen by data fitting to both experimental and clinical results . in this parameter regime , the model was observed to capture the phenomenon of viral blips , that is , long periods of undetectable viral load punctuated by brief episodes of high viral load . viral blips have been observed clinically in hiv patients under highly active antiretroviral therapy @xcite , and have received much attention in the research literature , both by experimentalists @xcite and mathematicians @xcite . nonetheless , the mechanisms underlying this phenomenon are still not thoroughly understood @xcite . we recently re - examined the model developed in @xcite , with the aim of providing new insight into the mechanism of hiv viral blips @xcite . focusing on the dynamics of the slow manifold of this model , we reduced the dimension of the 4-dimensional model by using quasi - steady state assumptions . after a further generalization and parameter rescaling process , a 2-dimensional in - host hiv model @xcite was obtained , given by @xmath44 where @xmath45 and @xmath46 denote the concentrations of the uninfected and infected cells respectively . the constant influx rate and the death rate of @xmath46 have been scaled to @xmath47 . the death rate of @xmath45 is @xmath48 . the 2-dimensional infection model above ( [ paper3_eq4 ] ) , reduced from the 4-dimensional hiv model @xcite , preserves the viral blips observed in the hiv model . importantly , system ( [ paper3_eq4 ] ) is equivalent to the sir model ( [ paper3_eq2 ] ) , except that the incidence function is convex , as we will show in section [ hopfconvex ] . this equivalence can be demonstrated if we set @xmath49 , @xmath50 , and @xmath51 with @xmath52 and @xmath53 . in this case , system ( [ paper3_eq2 ] ) is rescaled to @xmath54 which takes the same form as system ( [ paper3_eq4 ] ) . therefore , although system ( [ paper3_eq2 ] ) arises in epidemiology and system ( [ paper3_eq4 ] ) was derived as an in - host model , they are mathematically equivalent in this sense . we will refer to both systems ( [ paper3_eq2 ] ) and ( [ paper3_eq4 ] ) as infection models . in previous work @xcite , we analyze the recurrent behavior which emerges in system ( [ paper3_eq4 ] ) in some detail . recurrence is a particular form of oscillatory behavior characterized by long periods of time close to the uninfected equilibrium , punctuated by brief episodes of high infection @xcite . thus hiv viral blips are an example of recurrent behavior , but recurrence is a more general feature of many diseases @xcite . we have demonstrated that the increasing and saturating infectivity function of system ( [ paper3_eq4 ] ) is critical to the emergence of recurrent behaviour . this form of an infectivity function corresponds to a convex incidence rate function in the associated 2-dimensional infection model ( [ paper3_eq4 ] ) , and can likewise induce recurrence in this model . convex incidence has been previously suggested to model ` cooperation effects ' in epidemiology @xcite , or cooperative phenomena in reactions between enzyme and substrate , as proposed by @xcite . the rest of this paper is organized as follows . in section 2 , we study two 2-dimensional infection models , both closely related to system ( [ paper3_eq2 ] ) . we show that system ( [ paper3_eq2 ] ) with either ( a ) a concave incidence rate and saturating treatment term or ( b ) a convex incidence rate as shown in system ( [ paper3_eq4 ] ) , can exhibit backward bifurcation ; we then identify the necessary terms in the system equations which cause this phenomenon . in section 3 , we demonstrate that in both models , backward bifurcation increases the likelihood of a hopf bifurcation on the upper branch of the infected equilibrium . studying system ( [ paper3_eq4 ] ) in greater detail , we illustrate how the location of the hopf bifurcations and their directions ( supercritical or subcritical ) , determine the possible dynamical behaviors , concluding that backward bifurcation facilitates hopf bifurcation(s ) , which then underly the rich behaviours observed in these models . in section 4 , we explore backward bifurcation further , presenting an autoimmune disease model which exhibits negative backward bifurcation , that is , a bifurcation for which the turning point when @xmath5 is located in a region where one or more state variables is negative . although this bifurcation introduces two branches of the infected equilibrium , we demonstrate that , in the biologically feasible area , only forward bifurcation exists in this model . we then present a modification to this autoimmune model , motivated by the recent discovery of a new cell type , which generates a negative backward bifurcation and hopf bifurcation , and allows recurrent behavior to emerge . a conclusion is drawn in section 5 .

in this paper , dynamical systems theory and bifurcation theory are applied to investigate the rich dynamical behaviours observed in three simple disease models . the 2- and 3-dimensional models we investigate have arisen in previous investigations of epidemiology , in - host disease , and autoimmunity . these closely related models display interesting dynamical behaviors including bistability , recurrence , and regular oscillations , each of which has possible clinical or public health implications . in this contribution we demonstrate that backward bifurcation facilitates the appearance of hopf bifurcations , and the varied dynamical behaviors are then determined by the properties of the hopf bifurcation(s ) , including their location and direction . a maple program developed earlier

in this paper , dynamical systems theory and bifurcation theory are applied to investigate the rich dynamical behaviours observed in three simple disease models . the 2- and 3-dimensional models we investigate have arisen in previous investigations of epidemiology , in - host disease , and autoimmunity . these closely related models display interesting dynamical behaviors including bistability , recurrence , and regular oscillations , each of which has possible clinical or public health implications . in this contribution we elucidate the key role of backward bifurcation in the parameter regimes leading to the behaviors of interest . we demonstrate that backward bifurcation facilitates the appearance of hopf bifurcations , and the varied dynamical behaviors are then determined by the properties of the hopf bifurcation(s ) , including their location and direction . a maple program developed earlier is implemented to determine the stability of limit cycles bifurcating from the hopf bifurcation . numerical simulations are presented to illustrate phenomena of interest such as bistability , recurrence and oscillation . we also discuss the physical motivations for the models and the clinical implications of the resulting dynamics .

1002.2125

i

neptune mass extrasolar planets around main sequence stars were first detected five years ago . since then , 27 planets with a projected mass lower than 25 earth masses have been discovered , the lightest one having a projected mass of 1.94 m@xmath2 . among these objects , 17 have a semi major axis smaller than 0.1 astronomical unit ( au ) . they are called hot neptunes or hot super earths . so far , two multiple planet systems with at least two such objects have been observed . one of them is a four planet system around the m dwarf gj 581 ( bonfils et al . 2005 , udry et al . 2007 , mayor et al . the projected masses of the planets are 1.9 , 15.6 , 5.4 and 7.1 m@xmath2 and the periods are 3.15 , 5.37 , 12.93 and 66.8 days , respectively . note that although the eccentricities of the two innermost planets are compatible with zero , those of the next outermost and outermost planets are 0.16 and 0.18 respectively . the second multiple system which we shall focus upon in this paper is that around the k dwarf hd 40307 ( mayor et al . it comprises three planets with projected masses of 4.2 , 6.9 and 9.1 m@xmath2 and periods of 4.31 , 9.62 and 20.46 days , respectively . the eccentricities are all compatible with zero . if we label the planets in a system with successive integers starting from the innermost labelled , 1 , and moving outwards , then , in the system gl 581 , the ratio of the periods of planets 2 and 1 is 2.41 and that of the periods of planets 3 and 2 is 1.70 . we note that these numbers depart from 5/2 and 5/3 only by 4% and 2% , respectively.a in the hd 40307 system , the ratio of the periods of planets 2 and 1 is 2.23 and that of planets 3 and 2 is 2.13 , which depart from 2 by 11.5% and 6.5% , respectively . because departure from exact mean motion resonances in these systems is significant , the importance of resonances for their dynamical evolution has been ruled out ( mayor et al . 2009a , 2009b , barnes et al . we shall address this aspect for the hd 40307 system in this paper . migration due to tidal interaction with the disc is probably the mechanism by which planets end up on short period orbits , as _ in situ _ formation requires very massive discs ( e.g. , raymond et al . planet scatterings may also lead to short period orbits , but only when tidal circularization is efficient . according to kennedy & kenyon ( 2008 ) , scatterings are not a likely way of producing hot super earths , because of the long circularization timescales involved . in a previous paper ( terquem & papaloizou 2007 , see also brunini & cionco 2005 ) , we proposed a scenario for forming hot super earths on near commensurable orbits , in which a population of cores that assembled at some distance from the central star migrated inwards due to tidal interaction with the disc while merging on their way in . we found that evolution of an ensemble of cores in a disc almost always led to a system of a few planets , with masses that depended on the total initial mass of the system , on short period orbits with mean motions that frequently exhibited near commensurabilities and , for long enough tidal circularization times , apsidal lines that were locked together . starting with a population of 10 to 25 planets of 0.1 or 1 m@xmath3 , we ended up with typically between two and five planets with masses of a few tenths of an earth mass or a few earth masses inside the disc inner edge . interaction with the central star led to the initiation of tidal circularization of the orbits which , together with possible close scatterings and final mergers , tended to disrupt mean motion resonances that were established during the migration phase . the system , however , often remained in a configuration in which the orbital periods were close to commensurability . apsidal locking of the orbits , if established during migration , was often maintained through the action of these processes . this scenario has been questioned ( mayor et al . 2009a , 2009b ) because the multiple planet systems of hot super earths detected so far do not exhibit close enough mean motion commensurabilities . in this paper , we argue that _ the system around hd 40307 does actually exhibit the effects of resonances _ through secular effects produced by the action of the resonant angles coupled with the action of tides from the central star , and that such a configuration could be produced if the system formed as described in terquem & papaloizou ( 2007 ) , but with the addition of relatively small perturbations to the orbits arising from close encounters or collisions after the system enters the disc inner cavity . the reason that what are regarded as resonant effects can arise even when departures from strict commensurability are apparently large is that tidal circularization produces small eccentricities which , for first order resonances , can be consistent with resonant angle libration under those conditions ( see murray & dermott 1999 ) . we consider in detail the evolution of this system and other putative similar systems with scaled masses and orbital periods after they form a strict three planet laplace resonance as a result of convergent inward orbital migration . we go on to consider the onset of tidal circularization and how it leads to a separation of the semi - major axes and consequent increasing departure from commensurability . we find that if the strict laplace resonance is broken by a fairly small relative perturbation corresponding to a few parts in a thousand , continuing evolution in a modified form of the three planet resonance could in principle in the case of hd 40307 lead to a system like the observed one . however , this would require masses significantly larger than the minimum estimates and possibly unrealistically efficient tidal dissipation with the tidal dissipation parameter @xmath4 however , weaker tidal effects could be responsible for some of the deviation from strict commensurability , with the rest being produced by a larger perturbation resulting from the processes mentioned above inducing eccentricities of the order @xmath5 note however that tidal evolution could play a more significant role in similar systems with shorter orbital periods . in section [ sec : model ] , we describe the numerical model we use to simulate the evolution of a system of three planets migrating in a disc and evolving under the tidal interaction with the star after they enter the disc inner cavity . in section [ sec : hd40307 ] , we simulate the system around hd 40307 using either the minimum or twice the minimum masses for the planets under varying assumptions about initial eccentricities and the strength of the tidal interaction . we show that three of the four possible resonant angles associated with 2:1 commensurabilities either librate or have long term time averages , indicating evolution of the system towards a resonant state driven by the tidal influence of the star . in a related appendix , we give a semi analytic model for a system in a three planet resonance undergoing circularization with departures from strict mean motion commensurabilities . we note in passing that the discussion may also be applied to a two planet system near a 2:1 resonance . we show that the departure from commensurability increases with time being @xmath6 and derive an expression for the timescale required to attain a given departure that can be used to interpret extrapolate from and generalize the numerical simulations . in section [ sec : simulations ] , we present results of numerical simulations of a three planet system migrating in a disc , establishing a 4:2:1 laplace like resonance , and evolving under tidal effects induced by the star after entering the disc inner cavity . as expected , as the orbits are being tidally circularized , the planets move away from exact commensurabilites , and the period ratios of the outer and inner pairs of planets increases . the system is still resonant in the sense that some of the resonant angles continue to librate . to get a larger period ratio for the inner pair of planets , as in hd 40307 , some disruption of this laplace resonant state is needed . in our simulations , this is achieved by applying an impulse ( that could result from an encounter or collision ) to the system . this leads to departures from exact commensurabilites of the form seen in hd 40307 while still retaining libration of three of the four resonant angles . finally , in section [ sec : discussion ] we discuss and summarize our results .

, we expect that three resonant angles could be involved in this way . however , the relationship between these pairs that occurs in hd 40307 might be produced if the resonance is impulsively modified by an event like a close encounter shortly after the planetary system decouples from the disc . we find this to be in principle possible for a small relative perturbation on the order of a few but then we find that evolution to the present system in a reasonable time is possible only if the masses are significantly larger than the minimum masses and tidal dissipation is very effective . on the other hand we found that a system like hd 40307 with minimum masses and more realistic tidal dissipation could be produced if the eccentricity of the outermost planet was impulsively increased to we remark that the form of resonantly coupled tidal evolution we consider here is quite general and could be of greater significance for systems with inner planets on significantly shorter orbital periods characteristic of for example corot 7 b. planetary systems : formation planetary systems : protoplanetary discs

in this paper , we consider the dynamics of a system of hot super earths or neptunes such as hd 40307 . we show that , as tidal interaction with the central star leads to small eccentricities , the planets in this system could be undergoing resonant coupling even though the period ratios depart significantly from very precise commensurability . in a three planet system , this is indicated by the fact that resonant angles librate or are associated with long term changes to the orbital elements . in hd 40307 , we expect that three resonant angles could be involved in this way . we propose that the planets in this system were in a strict laplace resonance while they migrated through the disc . after entering the disc inner cavity , tidal interaction would cause the period ratios to increase from two but with the inner pair deviating less than the outer pair , counter to what occurs in hd 40307 . however , the relationship between these pairs that occurs in hd 40307 might be produced if the resonance is impulsively modified by an event like a close encounter shortly after the planetary system decouples from the disc . we find this to be in principle possible for a small relative perturbation on the order of a few but then we find that evolution to the present system in a reasonable time is possible only if the masses are significantly larger than the minimum masses and tidal dissipation is very effective . on the other hand we found that a system like hd 40307 with minimum masses and more realistic tidal dissipation could be produced if the eccentricity of the outermost planet was impulsively increased to we remark that the form of resonantly coupled tidal evolution we consider here is quite general and could be of greater significance for systems with inner planets on significantly shorter orbital periods characteristic of for example corot 7 b. planetary systems : formation planetary systems : protoplanetary discs

1002.2125

c

in this paper , we have shown that the planets around the star hd 40307 could be undergoing a resonant interaction , despite departure of the period ratios from very precise commensurability . this is indicated by the fact that three of the four resonant angles librate or are associated with long term changes to the orbital elements . note that such a resonant state can occur without exact commensurability only because the eccentricities are very small , which can be brought about as a result of tidal circularization from a state with initially higher eccentricities ( see section [ initecc ] ) . when this occurs in a system that starts from a near 2:1 commensurability , the difference of the period ratios from two then increases @xmath178 we propose that the planets in this system were in a strict laplace resonance while they migrated through the disc , with all 4 resonant angles librating . exact commensurability was then departed from as indicated above as a result of the tidal interaction with the star , which preserved libration of at least some of the resonant angles , after the planets entered the disc inner cavity . because of the laplace relation , the period ratios evolve in such a way that the ratio for the inner pair of planets is smaller than that of the outer pair of planets . the opposite is observed in hd 40307 . to get the appropriate form for the period ratios , some disruption of the resonance set up in the cavity needs to take place ( this is in contrast to claims by zhou 2009 ) . a close encounter for instance might be a possible cause . an impulsive interaction would be sudden enough that the resonance would not respond adiabatically . if the velocity of the innermost planet is reduced by a factor 0.996 for instance , deviation of the inner period ratio from two gets twice as large as the corresponding deviation for the outer period ratio , while 3 of the 4 resonant angles still librate , similar to the situation in hd 40307 , within the lifetime of the system . however , we found that we could get a period ratio of 2.23 for the inner pair only if the masses are significantly larger than the minimum masses and tidal dissipation is possibly unrealistically effective . from the results obtained in section [ sec : simulations ] , when the laplace resonance is disrupted at an early stage , we estimate that the deviation of the inner period ratio from two is approximately given as a function of time by @xmath179 where @xmath180 @xmath181 and @xmath182 ( @xmath183 and @xmath184 are assumed fixed for the purposes of this discussion ) . accordingly , to attain @xmath185 we would require masses exceeding the minimum estimate by about a factor of two for @xmath186 if the laplace resonance is preserved , from the results obtained in section [ sec : simulations ] , we may also estimate that the deviation of the outer period ratio from two is approximately given as a function of time by @xmath187 . thus it may be possible to get a period ratio of 2.13 for the outer pair of planets with near to minimum masses in a reasonable time if @xmath188 for all the planets , but then the inner period ratio would be only @xmath189 it would thus seem likely that additional stronger dynamical interactions are required to bring the system to its final state . for example we found that if the outer planet was given an eccentricity @xmath190 shortly after decoupling from the disc , period ratios similar to those observed in hd 40307 could be produced on a time scale of a few @xmath191 ( see sections [ initecc ] and [ sec : simulations ] ) . then , the work presented in this paper indicates that tidal circularization would continue through resonant interaction with the period ratios separating after these interactions are over . note that a similar type of scenario may apply to the system of planets around gj 581 , which is currently closer to exact commensurablilty than hd 40307 . indeed , different mean motion resonances can be established while the planets migrate within the disc , depending on the initial relative location of the planets and on the migration timescale . however , as it appears that this system may have been in higher order resonances than hd 40307 , it is possible that these would have been broken completely as a result of circularization . we intend to present a study of gj 581 in a forthcoming paper . in a study of the tidal evolution of the system around hd 40307 , barnes et al . ( 2009 ) have concluded that the planets can not be earth like . according to them , if it were the case , extrapolating back in time the tidal evolution of the eccentricities from current values ( estimated from direct simulations of the observed system ) , the system would have been unstable . the work in this paper does not support such a conclusion . we note that barnes et al . ( 2009 ) neglected the dynamical interactions between the planets , which clearly will have played a role if the current eccentricities are small because resonant interactions coupled with tidal effects have been associated with long term changes to the orbital elements . the results reported in this paper indicate that resonances in multiple low mass planet systems may play more of a role than currently claimed in the literature . significant departure from exact mean motion resonance does not necessarily preclude a resonant state when , for example , tidal effects are important causing eccentricities to be small , as libration of some of the resonant angles in that case may still exist , and be associated with long term evolution of the orbital elements . although the confirmation of resonance effects is problematic for small eccentricities , we expect that further detections and analysis of systems similar to hd 40307 will test the relevance of the scenarios explored in this paper . finally , we comment that , because of the rapid increase of the effectiveness of tidal interactions with decreasing orbital period , the dynamical interactions discussed here would be of much greater importance for systems in which the period of the innermost planet is significantly shorter than in the hd 40307 system . for example , we note that the planet corot7 b , which has a projected mass of 11.12 m@xmath2 , has a period of only 0.85 day ( lger et al . if this short period orbit is a result of migration , it is likely this planet was pushed inwards by a companion which would have been on a commensurable orbit ( terquem & papaloizou 2007 ) . we note in support of this idea that pre - main sequence stars typically have rotation periods of several days ( bouvier et al . 1993 ) , which would imply that the disc inner cavity should extend well beyond the orbit of corot7 b if that were magnetically maintained . in that case , this planet could not have migrated so far inwards without being shepherded by a companion .

in this paper , we consider the dynamics of a system of hot super earths or neptunes such as hd 40307 . we show that , as tidal interaction with the central star leads to small eccentricities , the planets in this system could be undergoing resonant coupling even though the period ratios depart significantly from very precise commensurability . in a three planet system , this is indicated by the fact that resonant angles librate or are associated with long term changes to the orbital elements . in hd 40307 we propose that the planets in this system were in a strict laplace resonance while they migrated through the disc . after entering the disc inner cavity ,

in this paper , we consider the dynamics of a system of hot super earths or neptunes such as hd 40307 . we show that , as tidal interaction with the central star leads to small eccentricities , the planets in this system could be undergoing resonant coupling even though the period ratios depart significantly from very precise commensurability . in a three planet system , this is indicated by the fact that resonant angles librate or are associated with long term changes to the orbital elements . in hd 40307 , we expect that three resonant angles could be involved in this way . we propose that the planets in this system were in a strict laplace resonance while they migrated through the disc . after entering the disc inner cavity , tidal interaction would cause the period ratios to increase from two but with the inner pair deviating less than the outer pair , counter to what occurs in hd 40307 . however , the relationship between these pairs that occurs in hd 40307 might be produced if the resonance is impulsively modified by an event like a close encounter shortly after the planetary system decouples from the disc . we find this to be in principle possible for a small relative perturbation on the order of a few but then we find that evolution to the present system in a reasonable time is possible only if the masses are significantly larger than the minimum masses and tidal dissipation is very effective . on the other hand we found that a system like hd 40307 with minimum masses and more realistic tidal dissipation could be produced if the eccentricity of the outermost planet was impulsively increased to we remark that the form of resonantly coupled tidal evolution we consider here is quite general and could be of greater significance for systems with inner planets on significantly shorter orbital periods characteristic of for example corot 7 b. planetary systems : formation planetary systems : protoplanetary discs

1603.07922

i

if @xmath3 is a finite set of primes , we define the set of _ @xmath0-units _ to be those integers of the shape @xmath4 , with exponents @xmath5 nonnegative integers . the arithmetic of such sets has been frequently studied due to its connections to a wide variety of problems in number theory and arithmetic geometry . in the latter direction , equations of the shape @xmath6 where @xmath7 and @xmath8 are @xmath0-units for certain specific sets @xmath0 , arise naturally when one wishes to make effective a theorem of shafarevich on the finiteness of isomorphism classes of elliptic curves over a number field @xmath9 with good reduction outside a given finite set of primes . by way of a simple example , if we wish to find all elliptic curves @xmath10 with nontrivial rational @xmath11-torsion and good reduction outside @xmath12 , we are led to consider curves @xmath13 of the shape @xmath14 where @xmath15 and @xmath16 are rational integers satisfying @xmath17 for nonnegative integers @xmath5 . writing @xmath18 , we thus seek to solve equation ( [ two ] ) , with @xmath19 and @xmath20 . an algorithm for computing all solutions to equations of the shape ( [ two ] ) , over @xmath21 , can be found in chapter 7 of de weger @xcite , where , for instance , one can find a complete characterization of the solutions to equation ( [ two ] ) in case @xmath22 . this algorithm combines lower bounds for linear forms in complex and @xmath23-adic logarithms with lattice basis reduction for @xmath23-adic lattices . more generally , for a given set of primes @xmath0 , we may consider equations of the shape @xmath24 with @xmath25 an integer , @xmath7 and @xmath8 @xmath0-units , and @xmath26 a nonzero integer . we will call such a quadruple @xmath27 a _ primitive _ solution of ( [ enn ] ) if @xmath28 is @xmath29th - power free . such equations are the main topic of discussion in chapter 9 of shorey and tijdeman @xcite , due to their connections to the problem of characterizing perfect powers in nondegenerate binary recurrence sequences of algebraic numbers . if we write @xmath30 in equation ( [ enn ] ) , with @xmath31 @xmath29th - power free ( whereby there are at most @xmath32 choices for @xmath31 ) , then it follows from the theory of thue - mahler equations that , for a fixed value of @xmath33 , the set of primitive solutions @xmath27 of ( [ enn ] ) is finite ( see ( * ? ? ? * thm . 7.2 ) ) . a stronger statement still is the following ( essentially theorem 9.2 of @xcite ) : [ shti - thm9.2 ] there are only finitely many coprime @xmath0-units @xmath34 and @xmath35 for which there exist integers @xmath36 and @xmath37 such that @xmath38 . the aim of this paper is to illustrate the use of frey - hellegouarch curves in solving equations like ( [ two ] ) and , more generally , ( [ enn ] ) . specifically , we will apply such an approach to provide a new proof of theorem [ shti - thm9.2 ] that _ a priori _ avoids the use of lower bounds for linear forms in logarithms , instead combining frey - hellegouarch curves , modularity and level - lowering , with the aforementioned theorem of shafarevich . in fairness , it must be mentioned , that effective versions of the latter result have typically depended fundamentally on linear forms in logarithms ; for recent papers along these lines , see the work of fuchs , von knel and wstholz @xcite and von knel @xcite . the benefit of our approach is it enables us to , in section [ s : all_solutions ] , explicitly solve equation ( [ enn ] ) for a pair of sets @xmath0 with cardinality @xmath39 . to the best of our knowledge , this is the first time this has been carried out . indeed , it is unclear whether the classical approach to theorem [ shti - thm9.2 ] via only lower bounds for linear forms in logarithms can be made practical with current technology , in any nontrivial situations . the outline of our paper is as follows . section [ sec2 ] introduces our basic notation . in section [ s : finiteness ] , we show how to obtain various finiteness results currently proved with techniques from diophantine approximation , via frey - hellegouarch curves over @xmath40 . philosophically , this bears a strong resemblance to recent work of von knel @xcite and of murty and pasten @xcite . section [ sec4 ] contains explicit details of the connections between frey - hellegouarch curves and modular forms . in sections [ s : n=2 ] and [ s : n=3 ] , we carry out such a `` modular '' approach quite explicitly for exponents @xmath41 and @xmath42 respectively . as an illustration of our methods , we completely solve ( [ enn ] ) for @xmath43 and @xmath44 ( hence recovering de weger s aforementioned result ) , and also for @xmath45 and @xmath44 , for every prime @xmath46 . it should be emphasized that this is not a `` serious '' application of our method , but merely meant as an illustration of a partial converse of the connection between solving equations of the shape ( [ enn ] ) and computing elliptic curves . the reader may wish to omit these sections at first ( and , for that matter , subsequent ) readings . a more interesting result along these lines is due to kim @xcite , where the connection between more general cubic thue - mahler equation and shafarevich s theorem is mapped out . section [ s : all_solutions ] contains , as previously mentioned , the main result of the paper , namely an explicit solution of equation ( [ enn ] ) for the sets @xmath1 and @xmath47 . the techniques we employ to prove these results , besides the aforementioned use of frey - hellegouarch curves and their associated modular forms , are local methods and appeal to computer algebra packages for solving thue and thue - mahler equations ; for the last of these , we rely extensively upon the computational number theory packages magma @xcite , pari @xcite and sage @xcite . we thank rafael von knel and benjamin matschke for pointing out to us a missing solution in a previous version of proposition [ prop : local_obstructions_2 - 3-p - n=2 ] .

our approach is based upon the modularity of galois representations and , for the most part , does not require lower bounds for linear forms in logarithms . its main virtue is that it enables to carry out such a program explicitly , at least for certain small sets of primes ; we do so for and .

in this paper , we develop a new method for finding all perfect powers which can be expressed as the sum of two rational-units , where is a finite set of primes . our approach is based upon the modularity of galois representations and , for the most part , does not require lower bounds for linear forms in logarithms . its main virtue is that it enables to carry out such a program explicitly , at least for certain small sets of primes ; we do so for and .

1102.0738

i

quite a long time ago , wigner @xcite introduced random matrices in the context of nuclear physics . he suggested that the highly - excited energy levels of complex nuclei can locally be well represented by the eigenvalues of a large random matrix . a big nucleus is a rather complex system composed of many strongly interacting quantum particles and it is practically impossible to describe its spectral properties via first principle calculations . the idea of wigner was to model the spectral properties of the complex hamiltonian of such a big nucleus by those of a large random matrix preserving the same symmetry . this was a very successful approach in nuclear physics . since then , the random matrix theory ( rmt ) has gone beyond nuclear physics and has found a wide number of applications in various fields of physics and mathematics including quantum chaos , disordered systems , string theory and even number theory @xcite . a case of special interest is the one of gaussian random matrices ( originally introduced by wigner himself ) where the entries of the matrix are gaussian random variables . depending on the symmetry of the problem , dyson distinguished three classes for the matrix @xmath2 @xcite : @xmath3 the gaussian orthogonal ensemble ( goe ) : @xmath2 is real symmetric . @xmath3 the gaussian unitary ensemble ( gue ) : @xmath2 is complex hermitian . @xmath3 the gaussian symplectic ensemble ( gse ) : @xmath2 is quaternionic hermitian . let us write @xmath4 the adjoint of @xmath2 , i.e. the transpose of @xmath2 for the goe , the complex conjugate transpose for the gue and the quaternionic conjugate transpose for the gse . a gaussian random matrix is a @xmath5 self - adjoint matrix @xmath2 , i.e. @xmath6 distributed according to the law @xmath7 where , for convenience , we have chosen the prefactor @xmath8 of the @xmath9 to be @xmath10 for the goe , @xmath11 for the gue and @xmath12 for the gse . for instance , for the gue we have @xmath11 and @xmath13 as @xmath14 . this means that @xmath2 is a @xmath5 complex hermitian matrix with entries @xmath15 and @xmath16 for @xmath17 that are independent ( real ) random variables distributed according to the same centered gaussian law with variance @xmath18 and the @xmath19 are ( real ) independent gaussian variables with mean @xmath20 and variance @xmath21 . in case of gse , there are @xmath22 eigenvalues , each of them two - fold degenerate and tr in for @xmath12 is defined so that only one of the two fold degenerate eigenvalues in @xmath2 is counted . + self - adjoint matrices can be diagonalized and have real eigenvalues . the joint distribution of eigenvalues of the gaussian ensemble is well known @xcite @xmath23 where @xmath24 is a normalization constant such that @xmath25 ( it depends on @xmath8 ) and the power @xmath8 of the vandermonde term is called the dyson index @xmath26 depending on the ensemble ( resp . goe , gue or gse ) . note that we have chosen the prefactor of @xmath9 term in to be the same as the dyson index @xmath8 just for convenience . this prefactor is not very important as it can be absorbed by rescaling the matrix entries by a constant factor . in contrast , the value of the dyson index @xmath10 , @xmath27 or @xmath28 , characterizing the power of the vandermonde term , plays a crucial role . the normalization constant @xmath24 can be computed using selberg s integral @xcite : @xmath29 $ ] . because of the presence of the vandermonde determinant @xmath30 in eq . , the eigenvalues are strongly correlated random variables , they repel each other . in this paper , our focus is on the statistical properties of the extreme ( maximal ) eigenvalue @xmath31 . had the vandermonde term been not there in the joint distribution , the joint distribution would factorize and the eigenvalues would thus be completely independent random variables , each with a gaussian distribution . for such independent and identically distributed random variables @xmath32 , the extreme value statistics is well understood @xcite and the distribution of the maximum , properly shifted and scaled , belongs to one of the three universality classes gumbel , frechet or weibull ( for large @xmath33 ) depending on the tail of the distribution of individual @xmath34 s . however , in the case of random matrix theory , the eigenvalues @xmath34 s are strongly correlated variables . for strongly correlated random variables there is no general theory for the distribution of the maximum . in case of gaussian random matrices , where the joint distribution is explicitly known , much progress has been made in understanding the distribution of @xmath35 following the seminal work by tracy and widom @xcite . this then provides a very useful solvable model for the extreme value distribution in a strongly correlated system and hence is of special interest . let us first summarize some known properties of the random variable @xmath35 . its average value can be easily obtained from the right edge of the well known wigner semi - circle describing the average density of eigenvalues . for a gaussian random matrix of large size @xmath33 , the average density of eigenvalues ( normalized to unity ) @xmath36 has a semi - circular shape on a finite support @xmath37 $ ] called the wigner semi - circle @xcite : @xmath38 the quantity @xmath39 represents the average fraction of eigenvalues that lie within the small interval @xmath40 $ ] . therefore , eq . means that the eigenvalues of a gaussian random matrix lie on average within the finite interval @xmath37 $ ] . note also that one can rewrite , using the joint distribution in @xmath41 hence the average density of states @xmath42 can also be interpreted as the marginal distribution of one of the eigenvalues ( say the first one ) . thus , the marginal distribution also has the shape of a semi - circle . figure [ fig : denstw ] shows the average density @xmath42 ( @xmath43 here ) . + it then follows that the average value of the maximal eigenvalue @xmath35 is given for large @xmath33 by the upper bound of the density support : @xmath44 however , @xmath35 fluctuates around this average value from one realization to another and has a distribution around its mean value @xmath45 ( see fig . [ fig : denstw ] with @xmath43 ) . what is the full probability distribution of @xmath35 ? from the joint distribution of eigenvalues in eq . ( [ eq : jpdfev ] ) , it is easy to write down formally the cumulative distribution function ( cdf ) of @xmath35 as a multiple integral @xmath46 which can be interpreted as a partition function of a coulomb gas in presence of a hard wall at the location @xmath47 ( see the discussion in section 2 ) . the question is how does @xmath48 behave for large @xmath33 ? it turns out that the fluctuations of @xmath35 around its mean @xmath45 have two scales for large @xmath33 . while typical fluctuations scale as @xmath49 , large fluctuations scale as @xmath50 and their probability distributions are described by different functional forms ( see fig . [ fig : denstw ] with @xmath43 ) . as a function of @xmath51 ( blue dashed line ) . the density has a semi - circular shape ( `` wigner semi - circle '' ) and a finite support @xmath52 $ ] . the maximal eigenvalue has mean value @xmath53 for large @xmath33 and its distribution close to the mean value , over a scale of @xmath54 has the tracy - widom form ( red solid line ) . however , over a scale @xmath55 the distribution has large deviation tails shown by solid green ( left large deviations ) and solid blue ( right large deviations ) lines.,width=453 ] * typical fluctuations : * from an asymptotic analysis of the mutilple integral in eq . ( [ multint ] ) , forrester @xcite , followed tracy and widom @xcite deduced that for large @xmath33 , _ small and typical _ fluctuations of the maximal eigenvalue around its mean value @xmath45 are of order @xmath54 and can be written as @xmath56 where @xmath57 ( for goe and gue ) and @xmath58 ( gse ) and @xmath59 is a random variable characterizing the typical fluctuations . tracy and widom @xcite proved that for large @xmath33 , the distribution of @xmath59 is independent of @xmath33 : @xmath60 . the function @xmath61 depends explicitly on @xmath8 and is called the tracy - widom distribution . for example , for @xmath11 @xcite , @xmath62 \label{f2tw}\ ] ] where @xmath63 satisfies the special case of @xmath64 of the painlev ii equation @xmath65 for @xmath64 , the solution only requires the right tail boundary condition for its unique specification : @xmath66 as @xmath67 , where @xmath68 is the airy function that satisfies the differential equation @xmath69 and vanishes as , @xmath70 as @xmath71 . this solution of the special case @xmath64 of the painlev - ii equation is called the hastings - mcleod solution @xcite . for @xmath11 and @xmath12 , one has @xcite @xmath72^{1/2}\ , \exp\left[\frac{1}{2}\int_x^{\infty } q(z ) dz\right ] \label{f1tw } \\ f_4(x)&= & \left[f_2(x)\right]^{1/2}\ , \cosh\left[\frac{1}{2}\int_x^{\infty } q(z ) dz\right ] . \label{f4tw}\end{aligned}\ ] ] note that @xmath73 is the cumulative probability of the scaled random variable @xmath59 and hence it approaches to @xmath74 as @xmath75 and vanishes to @xmath20 as @xmath76 . the corresponding probability density function ( pdf ) @xmath77 vanishes as @xmath78 in an asymmetric fashion @xmath79 \quad { \rm as}\quad x\to -\infty \label{lefttw } \\ & \sim & \exp\left[- \frac{2\beta}{3}\ , x^{3/2}\right ] \quad { \rm as}\quad x\to \infty \label{righttw}\end{aligned}\ ] ] over the last decade or so , the tracy - widom distribution has appeared in a wide variety of problems ranging from statistical physics and probability theory all the way to growth models and biological sequence matching problems ( for reviews see @xcite ) . these include the longest increasing subsequence or the ulam problem @xcite , a wide variety of ( 1 + 1)-dimensional growth models @xcite , directed polymer in random medium @xcite and the continuum kardar - parisi - zhang equation @xcite , bernoulli matching problem between two random sequences @xcite , nonintersecting brownian motions ( see e.g. @xcite and references therein ) . this distribution has also been measured in a variety of recent experiments , e.g. , in the height distribution of fronts generated in paper burning experiment @xcite , in turbulent liquid crystals @xcite and more recently in coupled fiber laser systems @xcite . * large deviations : * tracy - widom distribution describes the probability of _ typical _ fluctuations of @xmath35 around its mean ( on a scale of @xmath49 ) , but not the _ atypical _ large fluctuations , i.e. , fluctuations of order @xmath80 around the mean value @xmath81 . questions regarding such large / rare fluctuations do arise in various contexts @xcite and have recently been computed @xcite to dominant order for large @xmath33 . as a summary , the probability density of @xmath35 , @xmath82 $ ] , is given for large @xmath33 by : @xmath83 where @xmath61 is the tracy - widom distribution and where @xmath84 and @xmath85 are respectively the left and right large deviation functions describing the tails of the distribution of @xmath35 . the rate function @xmath86 was explicitly computed in @xcite , while @xmath87 was computed in @xcite , both by simple physical methods exploiting the coulomb gas analogy . a more complicated , albeit mathematically rigorous , derivation of @xmath87 in the context of spin glass models can be found in @xcite . these rate functions read @xmath88+\frac{\ln 3}{2 } , \;\ ; \textrm{for $ z<\sqrt{2}$}\nonumber\\ \psi_{+}(z)&=&\frac{z \sqrt{z^2 - 2}}{2 } + \ln\left[\frac{z-\sqrt{z^2 - 2}}{\sqrt{2}}\right],\ ; \;\ ; \;\;\ ; \textrm{for $ z>\sqrt{2}$.}\end{aligned}\ ] ] note that in ref . @xcite , the function @xmath89 was expressed in terms of a complicated hypergeometric function , which however can be reduced to a simple algebraic function as presented above in eq . ( [ eq : psi_-+ ] ) . note also that while @xmath90 depends explicitly on @xmath8 , the rate functions @xmath86 and @xmath89 are independent of @xmath8 . these rate functions only give the dominant order for large @xmath33 in the exponential . in other words , the precise meaning of @xmath91 is that for large @xmath33 : @xmath92 for @xmath93 and @xmath94 for @xmath95 . when @xmath96 approaches @xmath97 ( from below or above ) it is easy to see that the rate functions vanish respectively as @xmath98 note that the physics of the left tail @xcite is very different from the physics of the right tail @xcite . in the former case , the semi - circular charge density of the coulomb gas is _ pushed _ by the hard wall ( @xmath93 ) leading to a reorganization of all the @xmath33 charges that gives rise to an energy difference of @xmath99 @xcite . in contrast , for the right tail @xmath100 , the dominant fluctuations are caused by _ pulling _ a single charge away ( to the right ) from the wigner sea leading to an energy difference of @xmath101 @xcite . the different behaviour of the probability distribution for @xmath93 and @xmath100 leads to a ` phase transition ' strictly in the @xmath102 limit at the critical point @xmath103 in the following sense . indeed , if one scales @xmath51 by @xmath104 and takes the @xmath102 limit , one obtains @xmath105 note that since @xmath106 can be interpreted as a partition function of a coulomb gas ( see eq . ( [ multint ] ) ) , its logarithm has the interpretation of a free energy . since @xmath107 as @xmath108 from below , the @xmath109-rd derivative of the free energy is discontinuous at the critical point @xmath103 . hence , this can be interpreted as a _ third order _ phase transition . however , for finite but large @xmath33 , it follows from that the behavior to the left of @xmath103 smoothly crosses over to the behaviour to the right as one varies @xmath96 through its critical point @xmath103 and the tracy - widom distribution in around the critical point is precisely this crossover function . indeed , if one zooms in close to the mean value @xmath81 by setting @xmath110 ( for @xmath111 ) in the rate functions @xmath112 and @xmath113 in , one expects to recover , by taking large @xmath33 limit , respectively the left and the right tail of the tracy - widom distribution . with this scaling , and using , one finds @xmath114 and thus @xmath115 , which indeed matches the dominant order in the far right tail of the tracy - widom distribution for @xmath111 in . similarly for the left tail ( @xmath116 ) , using , one finds @xmath117 , thus @xmath118 which matches the left tail of the tracy - widom distribution in . more recently , higher order corrections for large @xmath33 have been computed for the left tail of the distribution @xcite using methods developed in the context of matrix models . note that in @xcite a different notation for @xmath8 was used : @xmath119 ( goe ) , @xmath10 ( gue ) and @xmath11 ( gse ) . to avoid confusion , we present below the results in terms of the standard dyson index @xmath120 . @xmath121 where @xmath122 with @xmath86 given in eq . ( dominant order ) . the subleading terms are given by @xcite @xmath123 \label{phi1}\end{aligned}\ ] ] and @xmath124 and ( see eq . ( 4 - 35 ) in ref . @xcite ) @xmath125+\left(\frac{3}{8}-\frac{1}{4 \beta } -\frac{\beta } { 16}\right ) \ln \left [ -2 z+\sqrt{6+z^2 } \right]\nonumber \\ & + & \left(\frac{1}{2 } -\frac{1}{3\beta } -\frac{\beta } { 12}\right ) \ln \left [ z+\sqrt{6+z^2}\right ] \nonumber \\ & + & \left(\frac{-4}{3}+\frac{4}{3 \beta } + \frac{\beta } { 3}\right ) \ln \left[\sqrt{-2 z+\sqrt{6+z^2}}+\sqrt{3}\left(z^2 + 6\right)^{1/4 } \right]\nonumber \\ & + & \frac{5}{3}\left(1-\frac{1}{\beta } -\frac{\beta } { 4}\right)\ln \left[-z+2 \sqrt{6+z^2}+\sqrt{3}\left(z^2 + 6\right)^{1/4 } \sqrt{\sqrt{6+z^2}-2 z } \right]\nonumber \\ & -&\ln \left[\left(-18 + z^2\right)z+\left(6+z^2\right)^{3/2}\right]- \frac{\ln \beta } { 2}-\kappa_{\beta } \label{phi0}\end{aligned}\ ] ] where @xmath126 is a complicated function of @xmath8 . for @xmath127 integer , it reduces to @xcite @xmath128 for instance , for the gue ( @xmath11 ) , we find @xmath129 . for @xmath111 and @xmath28 , the expression in eq . matches the left asymptotics of the tracy - widom distribution , i.e. the asymptotic behaviour of @xmath130 for @xmath131 , see ref . . however , for the right tail of the distribution of @xmath35 , the corrections to dominant order for large @xmath33 are , to our knowledge , not known until now . in fact , one of the results of this paper is to compute these right tail corrections for the gue ( @xmath11 ) . both left and right large deviations are plotted in fig . [ fig : largedev ] for the gue . the left tail is described by @xmath132 in eq . , the right tail is described by our result given in eq . . another result of this paper concerns a simpler and pedestrian derivation of the tracy - widom distribution for the gue case . the original derivation of the tracy - widom law for the distribution of typical fluctuations of @xmath35 @xcite is somewhat complex as it requires a rather sophisticated and nontrivial asymptotic analysis of the fredholm determinant involving airy kernel @xcite . since this distribution appears in so many different contexts , it is quite natural to ask if there is any other simpler ( more elementary ) derivation of the tracy - widom distribution . in this paper , we provide such a derivation for the gue case . our method is based on a suitable modification of a technique of orthogonal polynomials developed by gross and matytsin @xcite in the context of two - dimensional yang - mills theory . in fact , the partition function of the continuum two - dimensional pure yang - mills theory on a sphere ( with gauge group @xmath133 ) can be written ( up to a prefactor ) as a discrete multiple sum over integers @xcite @xmath134 where @xmath135 is the area of the sphere . in the @xmath102 limit , the free energy @xmath136 , as a function of @xmath135 , undergoes a 3rd order phase transition known as the douglas - kazakov transition @xcite at the critical value @xmath137 . for @xmath138 , the system is in the ` strong ' coupling phase while for @xmath139 , it is in the ` weak ' coupling phase . for finite but large @xmath33 , there is a crossover between the two phases as one passes through the vicinity of the critical point . in the so called double scaling limit ( where @xmath140 , @xmath102 but keeping the product @xmath141 fixed ) , the singular part of the free energy satisfies a painlev ii equation @xcite . gross and matytsin ( see also @xcite ) used a method based on orthogonal polynomials to analyse the partition sum in the double scaling limit , as well as in the weak coupling regime ( @xmath139 ) where they were able to compute non - perturbative ( in @xmath1 expansion ) corrections to the free energy . actually , a similar @xmath109-rd order phase transition from a weak to strong coupling phase in the @xmath102 limit was originally noticed in the lattice formulation ( with wilson action ) of the two dimensional @xmath133 gauge theory @xcite and in the vicinity of the transition point the singular part of the free energy was shown to satisfy a painlev ii equation @xcite . note that similar calculations involving the asymptotic analysis of partition functions using orthogonal polynomials were used extensively in the early 90 s to study the double scaling limit of the so called one - matrix model ( for a recent review and developments , see e.g. ref . @xcite ) . in our case , for the distribution of @xmath35 , we need to analyse the asymptotic large @xmath33 behaviour of a multiple _ indefinite _ integral in eq . ( [ multint ] ) , as opposed to the discrete sum in eq . ( [ ympf ] ) . however , we show that one can suitably modify the orthogonal polynomial method of gross and matytsin to analyse the multiple integral in eq . ( [ multint ] ) in the limit of large @xmath33 . in fact , we find a similar third order phase transition ( in the @xmath102 limit ) in the largest eigenvalue distribution @xmath48 as a function of @xmath47 at the critical point @xmath142 . the regime of left large deviation of @xmath48 ( @xmath143 ) is similar to the ` strong ' coupling regime @xmath144 of the yang - mills theory , while the right large deviation tail of @xmath48 ( @xmath145 ) is similar to the ` weak ' coupling regime @xmath146 of the yang - mills theory . for finite but large @xmath33 , the crossover function across the critical point that connects the left and right large deviation tails is precisely the tracy - widom distribution . thus the tracy - widom distribution corresponds precisely to the double scaling limit of the yang - mills theory and one finds the same painlev ii equation . a similar 3rd order phase transition was also found recently in a model of non - intersecting brownian motions by establishing an exact correspondence between the reunion probability in the brownian motion model and the partition function in the @xmath27-d yang - mills theory on a sphere @xcite . the advantage of this orthogonal polynomial method to analyse the maximum eigenvalue distribution is twofold : ( i ) one gets the tracy - widom distribution in a simple elementary way ( basically one carries out a scaling analysis of a pair of nonlinear recursion relations near the critical point and shows that the scaling function satisfies a painlev ii differential equation ) and ( ii ) as an added bonus , we also obtain precise subleading corrections to the leading right large deviation tail @xmath147 . the subleading corrections , in the yang - mills language , correspond to the non - perturbative corrections in the weak coupling regime as derived by gross and matytsin @xcite . more precisely we show that @xmath148 where @xmath87 is given in eq . ( [ eq : psi_-+ ] ) . note that only the leading behaviour @xmath149 $ ] was known before @xcite , but the subleading corrections are , to our knowledge , new results . we also verify that our expression matches the precise right asymptotics of the tracy - widom distribution . figure [ fig : largedev ] shows the distribution of @xmath35 for the gue : close to the mean value it is described by the tracy - widom distribution , whereas the tails are described by the large deviations . the right tail ( right large deviation ) is given by our result in eq . . together with the subleading terms in the left tail in eq . ( [ eq : phigaetan ] ) , our new result in eq . ( [ eq : righttailintro ] ) then provides a rather complete picture of the tail behaviors of the distribution of @xmath35 on both sides of the mean @xmath45 . associated to the distribution @xmath150 of the maximal eigenvalue of a random matrix from the gue for large @xmath33 . close to the mean value @xmath103 , the distribution is a tracy - widom law ( red line ) , it describes the small typical fluctuations around the mean value . atypical large flutuations are described by the large deviations : the left large deviation in green ( @xmath93 ) , the right deviation in blue ( @xmath100).,width=453 ] the rest of the paper is organized as follows . in section [ sec : intro ] , we start with some general notations and scaling remarks for the gue . in section [ sec : orthpol ] , we explain the method of orthogonal polynomials on a semi - infinite interval and derive some basic recursion relations . in section [ sec : largedev ] , we compute the right tail of the distribution of @xmath35 ( dominant order and corrections for the gue ) : it describes atypical large fluctuations of @xmath35 to the right of its mean value . in section [ sec : tw ] , using results of the previous sections and basic scaling remarks , we derive the tracy - widom law ( with @xmath11 for the gue ) that describes small typical fluctuations close to the mean value .

next we focus on the gaussian unitary ensemble ( gue ) and by suitably adapting a method of orthogonal polynomials developed by gross and matytsin in the context of yang - mills theory in two dimensions , we provide a rather simple derivation of the tracy - widom law for gue . our derivation is based on the elementary asymptotic scaling analysis of a pair of coupled nonlinear recursion relations . as an added bonus , this method also allows us to compute the precise subleading terms describing the right large deviation tail of the maximal eigenvalue distribution . in the yang - mills language , these subleading terms correspond to non - perturbative ( in expansion ) corrections to the two - dimensional partition function in the so called ` weak ' coupling regime .

in this paper , we first briefly review some recent results on the distribution of the maximal eigenvalue of a random matrix drawn from gaussian ensembles . next we focus on the gaussian unitary ensemble ( gue ) and by suitably adapting a method of orthogonal polynomials developed by gross and matytsin in the context of yang - mills theory in two dimensions , we provide a rather simple derivation of the tracy - widom law for gue . our derivation is based on the elementary asymptotic scaling analysis of a pair of coupled nonlinear recursion relations . as an added bonus , this method also allows us to compute the precise subleading terms describing the right large deviation tail of the maximal eigenvalue distribution . in the yang - mills language , these subleading terms correspond to non - perturbative ( in expansion ) corrections to the two - dimensional partition function in the so called ` weak ' coupling regime .

math9903089

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in this paper we investigate the large scale geometry of connected nilpotent lie groups equipped with left invariant riemannian metrics by studying their quasi - isometric embeddings into various metric spaces . let @xmath1 be a connected nilpotent lie group with a left invariant riemannian metric and @xmath2 be the induced distance function on @xmath3 . if @xmath4 is a complete metric space , then @xmath5 is an @xmath6-quasi - isometric embedding if , for all @xmath7 , @xmath8 after studying certain invariants of these maps , we prove two main applications : [ nocat0 ] there do not exist quasi - isometric embeddings of a connected nonabelian nilpotent lie group equipped with a left invariant riemannian metric into a @xmath0 metric space . [ nocbb0 ] there do not exist quasi - isometric embeddings of a connected nonabelian nilpotent lie group equipped with a left invariant riemannian metric into a @xmath9 metric space . @xmath10 ( resp . @xmath11 ) metric spaces are spaces of curvature bounded above ( resp . below ) by @xmath12 is the sense of topanogov triangle comparison . thus , a @xmath0 metric space is a generalized space of nonpositive curvature while a @xmath9 metric space is a generalized space of nonnegative curvature . these include simply connected riemannian manifolds of nonpositive and nonnegative curvature respectively . in the literature , @xmath0 spaces are also called hadamard spaces and @xmath11 spaces are called alexandrov spaces ( with curvature bounded below ) . we retain the earlier notion for simplicity and consistency . we will discuss these in detail later in the paper . in @xcite , wolf showed that a connected nonabelian nilpotent lie group equipped with a left invariant riemannian metric must contain 2-planes of both positive and negative curvatures . theorems [ nocat0 ] and [ nocbb0 ] can be viewed as a large scale analogue of this theorem of j. wolf . while the large scale characteristics of such nilpotent lie groups may be interesting in their own right , we shall see that such investigations reduce to problems which are motivated by the now standard arguments used in the proof of mostow s rigidity theorem and its many extensions ( see @xcite , @xcite , @xcite , @xcite , @xcite as well as many others ) . in the proofs of these results , one attempts to show that two candidate spaces ( e.g. two compact constant negative curvature spaces with isomorphic fundamental groups ) are isometric by exhibiting an equivariant quasi - isometry between their universal covers and showing that the existence of such a quasi - isometry must imply the existence of a true equivariant isometry . a common element in many such proofs is to reduce to an examination of the ideal boundaries of the two spaces in question . the quasi - isometries between the two spaces induce quasiconformal maps on the boundaries which then become the focus of the study . in the cases of the papers mentioned above , the tangent cones at points on the ideal boundaries are isometric to connected simply connected graded nilpotent lie group equipped with left invariant carnot - carathodory metrics . thus , a local analysis of the ideal boundaries involves an examination of the geometry of graded nilpotent lie groups with left invariant carnot - carathodory metrics . this provides a natural motivation for considering the geometry of such spaces and , in particular , of studying the quasi - conformal maps between them . it is a natural generalization to instead consider the quasi - conformal embeddings of such spaces . unfortunately , the study of such embeddings seems quite intractable but does provide motivation for various lines of study - for example the study of bilipschitz embeddings of such spaces . as we shall see , the proofs of theorems [ nocat0 ] and [ nocbb0 ] rest on a local analysis of bilipschitz embeddings of graded nilpotent lie groups with left invariant carnot - carathodory metrics into complete metric spaces which are _ locally _ @xmath10 or @xmath11 . for metric spaces which are locally @xmath10 , i.e. for each point there is a closed ball about that point which is itself @xmath10 , we use the notation @xmath13 ( for `` curvature bounded above '' ) . note that , for a lower curvature bound , these distinctions are not necessary . in particular , we prove an intermediate theorem from which theorems [ nocat0 ] and [ nocbb0 ] follow : [ inter ] let @xmath14 be a connected simply connected graded nilpotent lie group equipped with a left invariant carnot - carathodory metric and @xmath15 be an open set . then @xmath16 does not admit a bilipschitz embedding into any @xmath13 metric space or into any @xmath11 metric space . to prove the main theorems from this intermediate theorem , we study some bilipschitz embedded invariants of graded nilpotent lie groups with left invariant carnot - carathodory metrics and use them to build local obstructions to quasi - isometric embeddings of nonabelian nilpotent lie groups with left invariant riemannian metrics into various metric spaces . next we give a brief outline of the argument and the paper and show how theorems [ nocat0 ] and [ nocbb0 ] follow from theorem [ inter ] . let @xmath3 be a connected simply connected nilpotent lie group with a left invariant riemannian metric @xmath17 and let @xmath2 be the induced distance function on @xmath3 . if @xmath5 is some @xmath6-quasi - isometric embedding of @xmath3 into a complete metric space @xmath18 , we first consider asymptotic cones of @xmath3 and @xmath18 and a map between them , denoted @xmath19 , derived from @xmath20 and the coning procedure . the asymptotic cone of a metric space @xmath4 is a limit metric space of the pointed dilated spaces @xmath21 for some sequence @xmath22 . since such spaces do not necessarily converge in the gromov - hausdorff topology , we use gromov s ultrafilter construction , choosing a nonprincipal ultrafilter @xmath23 ( see section [ tangcones ] for a definition ) to form the asymptotic cone @xmath24 . for precise details of the construction , we refer the reader to either @xcite or @xcite . for our purposes , there are a few keys pieces of information . first , if @xmath1 is a nilpotent lie group with a left invariant riemannian metric , then pansu ( @xcite ) proved that the asymptotic cone is unique and isometric to @xmath25 , a graded nilpotent lie group equipped with a left invariant carnot - carathodory metric ( see section [ cc ] for definitions ) . second , an asymptotic cone of a @xmath0 space is also @xmath0 and an asymptotic cone to a @xmath9 space is also @xmath9 . these facts ( and references to their proofs ) are reviewed in section [ curv ] . third , the quasi - isometric embedding @xmath20 gives rise to an l - bilipschitz map @xmath19 between the cones @xmath25 and @xmath26 . the construction of this map depends on the choice of ultrafilter . thus , to prove theorems a and b , we must prove instead theorem [ inter ] . the paper is devoted to proving this intermediate theorem . the main technical result in the paper is that , in an appropriate sense , such an l - bilipschitz map is differentiable in certain directions almost everywhere . this is a generalization of a classical theorem of rademacher . our proof follows the same lines as a covering argument used in kleiner s proof of the differentiability of lipschitz maps from @xmath27 into metric spaces ( in @xcite ) , which is a special case of differentiability theory of section 1.9 in @xcite . in @xcite , kirchheim shows the metric differentiability of lipschitz maps @xmath28 where the targets are complete metric spaces . our differentiability result is an extension of kirchheim s work . the interested reader should also consult @xcite and @xcite for results concerning the differentiability of quasiconformal maps between carnot - carathodory spaces . to prove theorem [ inter ] , we construct a tangent cone of @xmath19 at a point of differentiability . using this tangent cone , we can compare the local geometry of the two asymptotic cones . the local geometry of connected graded nilpotent lie groups with left invariant carnot - carathodory metrics is well understood and the differentiability allows us to `` push forward '' this structure to a tangent cone to the asymptotic cone of @xmath18 . when this asymptotic cone has additional structure , this allows us to measure the compatibility of these two objects . in the case of theorem [ inter ] , this structure provides estimates on the rate of growth of the spread between two tangent cone geodesics which show that they spread apart sublinearly but do not remain a bounded distance from one another . comparing this to the spread of geodesics in the tangent cone to either a @xmath0 or @xmath9 space , we derive a contradiction which proves theorem [ inter ] . sections [ nilrev ] and [ cc ] are reviews of the constructions mentioned above and discuss , respectively , nilpotent lie groups and carnot - carathodory metrics . section [ secdiff ] contains the proof of the limited differentiability of bilipschitz embeddings of connected graded nilpotent lie groups with left invariant carnot - carathodory metrics into complete metrics spaces . the main goal of this section is to prove theorem [ kscc ] . section [ tangcones ] reviews the tangent cone construction and interprets theorem [ kscc ] as a statement about maps between tangent cones . section [ curv ] reviews the definition and some properties of @xmath0 and @xmath9 spaces . in section [ end ] we prove theorem c , that there do not exist bilipschitz embeddings of a nonabelian connected graded nilpotent lie group equipped with a carnot - carathodory metric into either a @xmath13 or @xmath11 space . the author wishes to thank bruce kleiner for suggesting this line of work as well as for many helpful discussions . thanks are also due to chris croke for helpful discussions of this work and for many hours of help with the preparation of this document .

in this paper , we prove results concerning the large scale geometry of connected , simply connected nilpotent lie groups equipped with left invariant riemannian metrics . precisely , we prove that there do not exist quasi - isometric embeddings of such a nilpotent lie group into either a metric space or an alexandrov metric space with curvature bounded below . the main technical aspect of this work is the proof of a limited metric differentiability of lipschitz maps between connected graded nilpotent lie groups equipped with left invariant carnot - carathodory metrics and complete metric spaces . [ section ] [ pro]lemma [ pro]sublemma [ pro]theorem [ pro]definition [ pro]corollary

in this paper , we prove results concerning the large scale geometry of connected , simply connected nilpotent lie groups equipped with left invariant riemannian metrics . precisely , we prove that there do not exist quasi - isometric embeddings of such a nilpotent lie group into either a metric space or an alexandrov metric space with curvature bounded below . the main technical aspect of this work is the proof of a limited metric differentiability of lipschitz maps between connected graded nilpotent lie groups equipped with left invariant carnot - carathodory metrics and complete metric spaces . [ section ] [ pro]lemma [ pro]sublemma [ pro]theorem [ pro]definition [ pro]corollary

1307.8325

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we address the issue of front propagation in a reaction - transport equation of kinetic type , @xmath1 here , the density @xmath2 describes a population of individuals in a continuum setting , and @xmath3 is the macroscopic density . the subset @xmath4 is the set of all possible velocities . individuals move following a velocity - jump process : they run with speed @xmath5 , and change velocity at rate 1 . they instantaneously choose a new velocity following the probability distribution @xmath6 . unless otherwise stated , we assume in this paper that @xmath7 is symmetric and @xmath8 satisfies the following properties : @xmath9 , and @xmath10 in addition , individuals are able to reproduce , with rate @xmath11 . new individuals start with a random velocity chosen with the same probability distribution @xmath6 . we could have chosen a different distribution without changing the main results , but we do not for the sake of clarity of the presentation . finally , we include a quadratic saturation term , which accounts for local competition between individuals , regardless of their speed . the main motivation for this work comes from the study of pulse waves in bacterial colonies of _ escherichia coli _ @xcite . kinetic models have been proposed to describe the run - and - tumble motion of individual bacteria at the mesoscopic scale @xcite . several works have been dedicated to derive macroscopic equations from those kinetic models in the diffusion limit @xcite . recently it has been shown that for some set of experiments , the diffusion approximation is not valid , so one has to stick to the kinetic equation at the mesoscopic scale to carefully compare with data @xcite . there is one major difference between this motivation and model . pulse waves in bacterial colonies of _ e. coli _ are mainly driven by chemotaxis which create macroscopic fluxes . growth of the population can be merely ignored in such models . in model however , growth and dispersion are the main reasons for front propagation , and there is no macroscopic flux due to the velocity - jump process since the distribution @xmath8 satisfies @xmath12 . for the sake of applications , we also refer to the growth and branching of the plant pathogen _ phytophthora _ by mean of a reaction - transport equation similar to @xcite . there is a strong link between and the classical fisher - kpp equation @xcite . in case of a suitable balance between scattering and growth ( more scattering than growth ) , we can perform the parabolic rescaling @xmath13 in , @xmath14 the diffusion limit yields @xmath15 , where @xmath16 is solution to the fisher - kpp equation ( see @xcite for example ) , @xmath17 we recall that for nonincreasing initial data decaying sufficiently fast at @xmath18 , the solution of ( [ kpp ] ) behaves asymptotically as a travelling front moving at the minimal speed @xmath19 @xcite . in addition , this front is stable in some weighted @xmath0 space @xcite . therefore it is natural to address the same questions for . we give below the definition of a travelling wave for equation . [ def : deftw ] we say that a function @xmath2 is a smooth travelling front solution of speed @xmath20 of equation if it can be written @xmath21 , where the profile @xmath22 satisfies @xmath23 in fact the profile @xmath24 is a solution to the stationary equation in a moving frame , @xmath25 the existence of travelling waves in reaction - transport equations has been adressed by schwetlick @xcite for a similar class of equations . first , the set @xmath7 is bounded and @xmath8 is the uniform distribution over @xmath7 . second , the nonlinearity can be chosen more generally ( either monostable as here , or bistable ) , but it depends only on the macroscopic density @xmath26 ( * ? ? ? ( 4 ) ) . for the monostable case , using a quite general method he was able to prove existence of travelling waves of speed @xmath27 for any @xmath28 , a result very similar to the fisher - kpp equation . we emphasize , that although the equations differ between schwetlick s work and ours , they coincide as far as the linearization in the regime of low density @xmath29 is concerned . on the contrary to schwetlick , we do not consider a general nonlinearity and we restrict to the logistic case , but we consider general velocity kernels @xmath30 . more recently , the rescaled equation ( [ eq : kinkpp2 ] ) has been investigated by cuesta , hittmeir and schmeiser @xcite in the parabolic regime @xmath31 . using a micro - macro decomposition , they construct possibly oscillatory travelling waves of speed @xmath32 for @xmath33 small enough ( depending on @xmath34 ) . in addition , when the set of admissible speeds @xmath7 is bounded , and @xmath35 they prove that the travelling wave constructed in this way is indeed nonnegative . lastly , when @xmath8 is the measure @xmath36 for some @xmath37 , equation ( [ eq : kinkpp ] ) is analogous to the reaction - telegraph equation for the macroscopic density @xmath26 ( up to a slight change in the nonlinearity however ) . this equation has been the subject of a large number of studies in the applied mathematics community @xcite . recently , the authors prove the existence of a minimal speed @xmath38 such that travelling waves exist for all speed @xmath39 @xcite . moreover these waves are stable in some @xmath0 weighted space , with a weight which differs from the classical exponential weight arising for the fisher - kpp equation . as the reaction - telegraph equation involves both parabolic and hyperbolic contributions , the smoothness of the wave depends on the balance between these contributions . in fact there is a transition between a parabolic ( smooth waves ) and a hyperbolic regime ( discontinuous waves ) , see remark [ rem : two velocities ] below . the authors also prove the existence of supersonic waves , having speed @xmath40 ( see remark [ rem : supersonic ] ) . the aim of the present paper is to investigate the existence and stability of travelling waves for equation ( [ eq : kinkpp ] ) for arbitrary kernels @xmath8 satisfying ( [ eq : hypm ] ) . for the existence part , we shall use the method of sub- and supersolutions , which do not rely on a perturbation argument . the stability part relies on the derivation of a suitable weight from which we can build a lyapunov functional for the linearized version of . the crucial assumption for the existence of travelling waves is the boundedness of @xmath7 . we prove in fact that in the case @xmath41 there exists no ( positive ) travelling wave . we then investigate the spreading rate for some particular choices of @xmath8 ( gaussian distribution , cauchy s distribution ) . unfortunately we are only able to give partial answer to this last question . in the last stage of writting of this paper , we realized that this issue was already addressed by mndez , campos and gmez - portillo for a slightly different equation admitting the same linearization near the front edge @xcite . our results are in agreement with their predictions . [ existence - tw ] assume that the set @xmath7 is compact , and that @xmath42 satisfies . let @xmath43 . there exists a speed @xmath44 such that there exists a travelling wave @xmath24 solution of of speed @xmath27 for all @xmath45 . the travelling wave is nonincreasing with respect to the space variable : @xmath46 . moreover , if @xmath47 then there exists no travelling wave of speed @xmath48 . the minimal speed @xmath38 is given through the following implicit dispersion relation : for each @xmath49 there is a unique @xmath50 such that @xmath51 then we have the formula @xmath52 [ rem : two velocities ] in the special case of two possible velocities only @xcite , corresponding to @xmath53 , two regimes have to be distinguished , namely @xmath54 and @xmath55 . in the case @xmath56 the travelling wave with minimal speed vanishes on a half - line . there , the speed of the wave is not characterized by the linearized problem for @xmath57 . note that this case is not contained in the statement of theorem [ existence - tw ] since it is assumed that @xmath42 . this makes a clear difference between the case of integrable @xmath8 and the case of a measure with atoms . [ rem : supersonic ] we expect that travelling waves exist for any @xmath39 , although this seems to contradict the finite speed of propagation when @xmath58 . in fact _ supersonic _ waves corresponding to @xmath58 should be driven by growth mainly , as it is the case in a simplified model with only two speeds @xcite . a simple argument to support the existence of such waves consists in eliminating the transport part , and seeking waves driven by growth only , @xmath59 . integrating with respect to @xmath60 yields a logistic equation for @xmath61 , @xmath62 , which as a solution connecting @xmath63 and @xmath64 for any positive @xmath27 . however these waves are purely artificial and we do not address this issue further . we now define @xmath65 and investigate the dependence of the minimal speed with respect to the velocity kernel @xmath66 . in the following proposition , we give some general bounds on the minimal speed . [ prop : bound on c * ] assume that @xmath7 is symmetric and that @xmath67 for all @xmath5 . the minimal speed satisfies the following properties , 1 . _ [ scaling ] _ for @xmath68 , define @xmath69 , then @xmath70 2 . _ [ rearrangement ] _ denote @xmath71 the schwarz decreasing rearrangement of the function @xmath8 ( see @xcite for a definition of this notion ) and @xmath72 the schwarz increasing rearrangement of the density distribution @xmath8 , then @xmath73 3 . _ [ comparison ] _ if @xmath54 then @xmath74 on the other hand , if @xmath55 then @xmath75 4 . _ [ diffusion limit ] _ in the diffusion limit @xmath13 we recover the kpp speed of the wave , @xmath76 in the case of a bounded set of velocities , we prove that for suitable initial data @xmath77 , the front spreads asymptotically with speed @xmath38 , in a weak sense . [ prop : spreadingbounded ] assume that @xmath7 is bounded and that @xmath78 . let @xmath79 such that @xmath80 for all @xmath81 . let @xmath82 be the solution of the cauchy problem . then 1 . if there exists @xmath83 such that @xmath84 for all @xmath85 and @xmath5 , then for all @xmath86 , @xmath87 2 . if there exists @xmath88 and @xmath89 such that @xmath90 for all @xmath91 and @xmath5 , then for all @xmath92 , @xmath93 where @xmath38 is the minimal speed of existence of travelling waves given by theorem [ existence - tw ] we also establish linear and nonlinear stability in suitable weighted @xmath0 spaces . the keypoint is to derive a correct weight which enables to build a lyapunov functional for the linear problem . we construct a semi - explicit weight @xmath94 , but we believe it is not the optimal one in some sense ( see remark [ rk : phi not optimal ] ) . let @xmath24 be a travelling wave of speed @xmath27 , and let @xmath95 the perturbation of @xmath24 in the moving frame . neglecting the nonlinear contributions , we are led to investigate the linear equation @xmath96 [ linstability ] there exists a weight @xmath97 such that the travelling front of speed @xmath98 is linearly stable in the weighted space @xmath99 . more precisely , the following lyapunov identity holds true for any solution @xmath100 of the linear equation , @xmath101 the weight @xmath97 is explicitly given in definition [ eq : def weight phi ] . using a comparison argument , in the spirit of @xcite , together with the explicit formula of the dissipation for the linearized system , we prove a nonlinear stability result . [ nonlinstability ] let @xmath102 $ ] . let @xmath103 be a travelling wave with speed @xmath98 . let @xmath82 be a solution of . suppose that the initial data satisfies @xmath104 then the following lyapunov identity holds true for the perturbation @xmath105 , @xmath106 \left\vert u \right\vert^2 e^{- 2 \phi ( z , v ) } d z dv \leq 0\,,\ ] ] where @xmath97 denotes the same weight as in proposition [ linstability ] . we expect that nonlinear stability holds true for any @xmath107 $ ] . however this would require to redefine the weight @xmath97 , since we believe it is not the optimal one . boundedness of @xmath7 is a crucial hypothesis in order to build the travelling waves . we believe that it is a necessary and sufficient condition . we make a first step to support this conjecture by investigating the case @xmath41 . we first prove infinite speed of spreading of the front under the natural assumption @xmath108 . as a corollary there can not exist travelling wave in the sense of definition [ def : deftw ] . note that there exist travelling waves with less restrictive conditions than definition [ def : deftw ] , at least in the diffusive regime @xcite . these fronts are expected to oscillate as @xmath109 . we expect that such oscillating fronts do exist far from the diffusive regime . in the case where @xmath41 and @xmath8 is gaussian , we plotted the dispersion relation in the complex plane @xmath110 , for an arbitrary given @xmath111 . we observed that it selects two complex conjugate roots , supporting the fact that oscillating fronts should exist ( results not shown ) . [ prop : spreadingunbounded ] assume that @xmath112 for all @xmath113 . let @xmath79 such that @xmath80 for all @xmath81 and there exists @xmath88 and @xmath114 such that @xmath90 for all @xmath91 and @xmath5 . let @xmath82 be the solution of the cauchy problem . then for all @xmath111 , @xmath115 we can immediately deduce from this result the non - existence of travelling waves when @xmath116 , by taking such a travelling wave as an initial datum @xmath117 in order to reach a contradiction . assume that @xmath118 for all @xmath113 . then equation does not admit any travelling wave solution . next we investigate specific choices for the distribution @xmath8 , both numerically and theoretically . in the case of a gaussian distribution , we expect a spreading rate following the power law @xmath119 ( see also @xcite ) . to support this guess , we prove in fact that spreading occurs _ at most _ with this rate . for this purpose we build a supersolution which is spreading with this rate . this issue has been addressed by mndez , campos and gmez - portillo @xcite in a physical paper for a slightly different equation , where the nonlinearity lies in the diffusion kernel instead of the growth rate . they conjectured that , as for the kpp equation , the front speed is determined through the linearization of the equation near the unstable steady state @xmath64 . we believe that the linearization should give the power law of the propagation , but it is not clear to us that it will give the exact location of the transition . however , it turns out that the linearized equations in @xcite and in the present paper are the same . then , performing some fourier - laplace transform of the solution , mendez et al derived heuristically the power law of the propagation . in particular , for a gaussian kernel , they found out that the spreading rate is @xmath119 . [ thm : unbounded ] let @xmath120 for all @xmath113 . let @xmath79 such that @xmath80 for all @xmath81 . assume that there exists @xmath121 such that @xmath122 let @xmath82 be the solution of the cauchy problem . then for all @xmath123 , one has @xmath124 in the case of a cauchy distribution @xmath125 , we obtain faster spreading rate under similar assumptions ( see proposition [ supersolunbounded - cauchy ] ) , namely for all @xmath123 , one has @xmath126 the phenomenon of accelerating fronts have raised a lot of attention in the literature of reaction - diffusion equations . this phenomenon occurs for the fisher - kpp equation when the initial datum decays more slowly than any exponential @xcite ; for a variant of the fisher - kpp equation where the diffusion operator is replaced by a nonlocal dispersal operator with fat tails @xcite , or by a nonlocal fractional diffusion operator @xcite . recently , accelerating fronts have been conjectured to occur in a reaction - diffusion - mutation model which extends the fisher - kpp equation to a population with heterogeneous diffusion coefficient @xcite . there is some subtlety hidden behind this phenomenon of infinite speed of spreading . in fact the diffusion limit of the scattering equation ( namely @xmath127 ) towards the heat equation makes no difference between bounded or unbounded velocity sets ( see @xcite and the references therein ) . however very low densities behave quite differently , which can be measured in the setting of large deviations or wkb limit . this can be observed even in the case of a bounded velocity set . in @xcite the large deviation ( wkb ) limit of the scattering equation is performed . it differs largely from the classical eikonal equation obtained from the heat equation . the case of unbounded velocities is even more complicated @xcite . to conclude , let us emphasize that low densities are the one that drive the front here ( pulled front ) . so the diffusion limit is irrelevant in the case of unbounded velocities , since very low density of particles having very large speed makes a big difference .

the model describes particles moving according to a velocity - jump process , and proliferating thanks to a reaction term of monostable type . the minimal speed of propagation of waves is obtained from an explicit dispersion relation . we construct the waves using a technique of sub- and super- solutions and prove their stability in a weighted space . in case of an unbounded velocity set , we prove a superlinear spreading and give partial results concerning the rate of spreading associated to particular initial data .

in this paper , we study the existence and stability of travelling wave solutions of a kinetic reaction - transport equation . the model describes particles moving according to a velocity - jump process , and proliferating thanks to a reaction term of monostable type . the boundedness of the velocity set appears to be a necessary and sufficient condition for the existence of positive travelling waves . the minimal speed of propagation of waves is obtained from an explicit dispersion relation . we construct the waves using a technique of sub- and super- solutions and prove their stability in a weighted space . in case of an unbounded velocity set , we prove a superlinear spreading and give partial results concerning the rate of spreading associated to particular initial data . it appears that the rate of spreading depends strongly on the decay at infinity of the stationary maxwellian . * key - words : * kinetic equations , travelling waves , dispersion relation , superlinear spreading .

1308.1259

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performance of low - density parity - check ( ldpc ) codes under iterative decoding algorithms in the error floor region is closely related to the problematic structures of the code s tanner graph @xcite , @xcite , @xcite , @xcite , @xcite , @xcite , @xcite , @xcite , @xcite , @xcite , @xcite , @xcite , @xcite . following the nomenclature of @xcite , here , we collectively refer to such structures as _ trapping sets_. the most common approach for classifying the trapping sets is by a pair @xmath2 , where @xmath3 is the size of the trapping set and @xmath4 is the number of odd - degree ( unsatisfied ) check nodes in the subgraph induced by the set in the tanner graph of the code . among the trapping sets , the so - called _ elementary trapping sets ( ets ) _ are known to be the main culprits @xcite , @xcite , @xcite , @xcite , @xcite , @xcite . these are trapping sets whose induced subgraph only contains check nodes of degree one or two . for a given ldpc code , the knowledge of dominant trapping sets , i.e. , those that are most harmful , is important . such knowledge can be used to estimate the error floor @xcite , to modify the decoder to lower the error floor @xcite , @xcite , @xcite , or to design codes with low error floor @xcite , @xcite . while the knowledge of dominant trapping sets is most helpful in the design and analysis of ldpc codes , attaining such knowledge is generally a hard problem @xcite . much research has been devoted to devising efficient search algorithms for finding small ( dominant ) trapping sets , see , e.g. , @xcite , @xcite , @xcite , @xcite , @xcite , @xcite , @xcite , @xcite , and to the ( partial ) characterization of such sets @xcite , @xcite @xcite , @xcite , @xcite , @xcite . asymptotic analysis of trapping sets has also been carried out in @xcite , @xcite , @xcite , @xcite , @xcite , @xcite . @xcite studied the characterization of small @xmath2 trapping sets of size up to 8 ( @xmath9 ) and @xmath10 in ldpc codes from steiner triple systems ( sts ) . sts ldpc codes are a special category of regular ldpc codes with variable node degree 3 . @xcite showed that for a regular ldpc code with variable node degree @xmath11 and check node degree @xmath12 , and girth @xmath13 , no @xmath14 trapping set with @xmath15 and @xmath16 can exist . they also studied the trapping sets of euclidean geometry ( eg ) ldpc codes and provided some bounds on the size and the number of unsatisfied check nodes of the trapping sets of such codes @xcite . an eg - ldpc code with parameter @xmath17 is a regular ldpc code of length @xmath18 , with variable node degree @xmath19 and check node degree @xmath17 . a consequence of the bounds derived in @xcite is that for the case where @xmath20 , there is no trapping set of size smaller than the minimum distance of the code , i.e. , @xmath21 , with less than @xmath22 unsatisfied check nodes . a subset of trapping sets , called _ absorbing sets _ , for array - based ldpc codes with variable node degrees 2 , 3 and 4 , were studied in @xcite , @xcite . absorbing sets are trapping sets in which each variable node is connected to more satisfied than unsatisfied check nodes in the induced subgraph of the set @xcite , @xcite . array - based ldpc codes are a subclass of ( regular ) protograph ldpc codes which are constructed by lifting a fully - connected base graph using cyclic permutations . the analysis in @xcite and @xcite was focused on minimal absorbing sets , i.e. , the ones with the smallest size and with the smallest number of unsatisfied check nodes for a given size . @xcite studied the topological structure of trapping sets of size up to @xmath23 in regular ldpc codes with variable node degree 3 , and proposed a hierarchical search method to find them . the study of the graphical structure of trapping sets so far has been mainly limited to structured codes , codes with certain variable node degrees , and to relatively small trapping sets . in this work , for the category of variable - regular ldpc codes with a certain variable node degree and a given girth , we study the topological structure of @xmath2 etss for given values of @xmath3 and @xmath4 , and find all the non - isomorphic structures of such etss . a careful examination of these structures , which are independent of the check node degree distribution of the code , reveals that for relatively small values of @xmath3 and @xmath4 , the structures are all layered supersets ( lss ) of small cycles , i.e. , they can be characterized by a nested sequence of etss which starts from a short cycle and grows to the ets one node at a time . the lss property lends itself to a simple search algorithm that starts from short cycles of the code s tanner graph as input and can find all the etss with lss property in a guaranteed fashion . although the general approach discussed here can be applied to any category of variable - regular ldpc codes with arbitrary variable node degree @xmath0 and girth @xmath1 and to any class of etss with arbitrary values of @xmath3 and @xmath4 , the results presented here are for @xmath6 , @xmath7 , and @xmath8 . one of the main advantages of the results presented here is that they are applicable to specific codes , rather than just to an ensemble or a category of codes . in particular , the search algorithm based on lss property can be used to efficiently find the dominant etss of a code in a guaranteed fashion . this , for example , would imply having a faster and more accurate estimation of error floor for the code under consideration using techniques such as importance sampling . moreover , the results presented here can be used in the design of ldpc codes with low error floor . this can be achieved by avoiding certain dominant etss in the tanner graph of the code . in such a context , this work can help in identifying the dominant etss . it has been known that dominant trapping sets of ldpc codes have a close relationship with short cycles in the code s tanner graph @xcite , @xcite , @xcite . this work takes a rigorous step in establishing such a relationship . in general , in comparison with existing results on characterization of trapping sets such as @xcite , @xcite , @xcite , the results presented here are more general in terms of being applicable to both structured and random codes , and to cover a wider range of variable node degrees and trapping set classes . the remainder of this paper is organized as follows . basic definitions and notations are provided in section ii . in section iii , we present and discuss the lss property . in section iv , we develop the algorithm which guarantees to find all the etss with the lss property starting from the short cycles of the tanner graph . sections v , vi , vii , and viii present the results for variable - regular ldpc codes with variable node degrees 3 , 4 , 5 and 6 , respectively . as part of the material presented in these sections , we provide the lengths of short cycles that are required for the proposed algorithm to find all the @xmath2 etss with lss property in a guaranteed fashion , for different values of @xmath3 and @xmath4 . section ix is devoted to discussions and conclusions .

for the set of ldpc codes with a given variable node degree and girth , we identify all the non - isomorphic structures of an arbitrary class of etss , where is the number of variable nodes and is the number of odd - degree check nodes in the induced subgraph of the ets . our study leads to a simple characterization of dominant classes of etss ( those with relatively small values of and ) based on short cycles in the tanner graph of the code . for such classes of etss , we prove that any set in the class is a layered superset ( lss ) of a short cycle , where the term `` layered '' is used to indicate that there is a nested sequence of etss that starts from the cycle and grows , one variable node at a time , to generate . this characterization corresponds to a simple search algorithm that starts from the short cycles of the graph and finds all the etss with lss property in a guaranteed fashion .

in this paper , we study the graphical structure of elementary trapping sets ( ets ) of variable - regular low - density parity - check ( ldpc ) codes . etss are known to be the main cause of error floor in ldpc coding schemes . for the set of ldpc codes with a given variable node degree and girth , we identify all the non - isomorphic structures of an arbitrary class of etss , where is the number of variable nodes and is the number of odd - degree check nodes in the induced subgraph of the ets . our study leads to a simple characterization of dominant classes of etss ( those with relatively small values of and ) based on short cycles in the tanner graph of the code . for such classes of etss , we prove that any set in the class is a layered superset ( lss ) of a short cycle , where the term `` layered '' is used to indicate that there is a nested sequence of etss that starts from the cycle and grows , one variable node at a time , to generate . this characterization corresponds to a simple search algorithm that starts from the short cycles of the graph and finds all the etss with lss property in a guaranteed fashion . specific results on the structure of etss are presented for , and in this paper . the results of this paper can be used for the error floor analysis and for the design of ldpc codes with low error floors .

1308.1259

c

in this paper , we investigated the structure of elementary trapping sets ( ets ) of left - regular ldpc codes . we developed an approach to find _ all _ the non - isomorphic structures of a given @xmath2 class of etss , where @xmath3 is the size and @xmath4 is the number of unsatisfied check nodes of the ets . for left - regular ldpc codes with left degrees @xmath6 , and girths @xmath7 , we studied such structures and demonstrated that an overwhelming majority of them are layered supersets ( lss ) of short cycles in the tanner graph of the code . in particular , we proved that for any category of left - regular ldpc codes with given @xmath0 and @xmath1 , there exist integers @xmath125 and @xmath191 such that all the classes of @xmath2 etss with @xmath192 and @xmath193 , are lsss of short cycles . this implies that for any category of left - regular ldpc codes , the dominant etss are all lsss of short cycles . the lss characterization of dominant etss is particularly important as it corresponds to a simple algorithm that can find _ all _ such etss in a _ guaranteed _ fashion starting from the short cycles of the graph . for any class of @xmath2 etss , the lengths of the required short cycles were provided in this paper . one important contribution of this paper is the approach developed to exhaustively find all the non - isomorphic structures of a given class of @xmath2 etss for arbitrary values of @xmath3 and @xmath4 and for left - regular ldpc codes of arbitrary left degree and girth . in a more general context , the database of such structures can be very helpful in the analysis and the design of ldpc codes with low error floor . in particular , one can use this information to find all the etss of a certain class in a guaranteed fashion regardless of whether those etss satisfy the lss property or not . to the best of our knowledge , the results presented in tables [ d3g6 ] - [ d6g6 ] are the most comprehensive results available so far on the structure of etss of regular ldpc codes . s. abu - surra , d. declercq , d. divsalar , and w. ryan , trapping set enumerators for specific ldpc codes , " proc . theory and applications workshop , _ san diego , ca , jan . 31- feb . 5 , 2010 , pp . 1 - 5 . s. abu - surra , w. e. ryan , d. divsalar , and w. ryan , ensemble trapping set enumerators for protograph - based generalized ldpc codes , " proc . theory and applications workshop , _ san diego , ca , jan . 27feb . 1 2008 , pp . 6365 . e. cavus and b. daneshrad , a performance improvement and error floor avoidance technique for belief propagation decoding of ldpc codes , " proc . _ 16th ieee international symposium pers . indoor mobile radio communications , _ los angeles , ca , sept . 2005 , vol . 2386 - 2390 . qiuju diao , ying yu tai , shu lin , and khaled a. s. abdel - ghaffar , trapping set structure of finite geometry ldpc codes , " proc . _ ieee international symposium on information theory ( isit12 ) _ , 2012 , pp . 3088 - 3092 . l. dolecek , z. zhang , v. anantharam , m.wainwright , and b. nikolic , analysis of absorbing sets for array - based ldpc codes , " proc . _ interntional conf . communications , _ glasgow , scotland , jun . 24 - 28 , 2007 , pp . 6261 - 6268 . l. dolecek , z. zhang , m. wainwright , v. anantharam , and b. nikolic , evaluation of the low frame error rate performance of ldpc codes using importance sampling , " proc . _ ieee inform . theory workshop _ , lake tahoe , ca , sept . 2 - 6 , 2007 , pp . 202 - 207 . l. dolecek , p. lee , z. zhang , v. anantharam , b. nikolic , and m. wainwright , predicting error floors of ldpc codes : deterministic bounds and estimates , " _ ieee journal on selected areas in communications _ , vol . 27 , no . 6 , pp . 908 - 917 , aug . 2009 . l. dolecek , z. zhang , v. anantharam , m. j. wainwright , and b. nikolic , analysis of absorbing sets and fully absorbing sets of array - based ldpc codes , " _ ieee trans . inform . theory _ 181 - 201 , jan . 2010 . c. di , d. proietti , i. e. telatar , t. j. richardson , and r. l. urbanke , finite - length analysis of low - density parity - check codes on the binary erasure channel , " _ ieee trans . inform . theory _ 48 , no . 6 , pp . 1570 - 1579 , june 2002 . m. ivkovic , s. k. chilappagari , and b. vasic , eliminating trapping sets in low - density parity - check codes by using tanner graph covers , " _ ieee trans . inform . theory _ 3763 - 3768 , aug . 2008 . s. laendner and o. milenkovic , algorithmic and combinatorial analysis of trapping sets in structured ldpc codes , " proc . _ ieee international conference on wireless networks , communications and mobile computing _ , hawaii , usa , june 13 - 16 , 2005 , pp . 630 - 635 . s. laendner , t. hehn , o. milenkovic and j.b . huber , characterization of small trapping sets in ldpc codes from steiner triple systems , " proc . _ 6th international symposium on turbo codes & iterative information processing , _ brest , france , sept . 6 - 10 , 2010 , pp . 93 - 97 . c. koller , a. graell i amat , j. kliewer , and d. j. costello , on trapping sets for repeat accumulate accumulate codes , " proc . _ theory and applications workshop _ , san diego , la jolla , ca , feb . 8 - 13 , 2009 . c. koller , a. graell i amat , j. kliewer , and d. j. costello , trapping set enumerators for repeat multiple accumulate code ensembles , " proc . _ ieee international symposium on information theory ( isit09 ) _ , seoul , korea , jun . - jul . 2009 . g. b. kyung and c .- c . wang , finding the exhaustive list of small fully absorbing sets and designing the corresponding low error - floor decoder , " _ ieee trans . communications _ , vol . 60 , pp . 1487 - 1498 , june . 2012 . y. mao and a.h . banihashemi , a heuristic search for good low - density parity - check codes at short block lengths , " proc . _ ieee international conference on communications _ , helsinki , finland , june 2001 , pp . 41 - 44 . e. pusane , d. j. costello , and d. g.m . mitchell , trapping set analysis of protograph - based ldpc convolutional codes , " proc . _ ieee international symposium on information theory ( isit09 ) _ , seoul , korea , jun . - jul . 2009 , pp . 561565 . c .- c . wang , on the exhaustion and elimination of trapping sets : algorithms & the suppressing effect , " proc . _ ieee international symposium on inform . nice , france , june 24 - 29 , 2007 , pp . 2271 - 2275 . h. xiao and a. h. banihashemi , estimation of bit and frame error rates of low - density parity - check codes on binary symmetric channels , " _ ieee trans . 2234 - 2239 , dec . 2007 . h. xiao and a. h. banihashemi , error rate estimation of low - density parity - check codes on binary symmetric channels using cycle enumeration , " _ ieee trans . communications _ , vol . 1550 - 1555 , june 2009 . h. xiao , a. h. banihashemi and m. karimi , error rate estimation of low - density parity - check codes decoded by quantized soft - decision iterative algorithms , " _ ieee trans . communications _ , vol . 474 - 483 , feb . 2013

in this paper , we study the graphical structure of elementary trapping sets ( ets ) of variable - regular low - density parity - check ( ldpc ) codes . the results of this paper can be used for the error floor analysis and for the design of ldpc codes with low error floors .

in this paper , we study the graphical structure of elementary trapping sets ( ets ) of variable - regular low - density parity - check ( ldpc ) codes . etss are known to be the main cause of error floor in ldpc coding schemes . for the set of ldpc codes with a given variable node degree and girth , we identify all the non - isomorphic structures of an arbitrary class of etss , where is the number of variable nodes and is the number of odd - degree check nodes in the induced subgraph of the ets . our study leads to a simple characterization of dominant classes of etss ( those with relatively small values of and ) based on short cycles in the tanner graph of the code . for such classes of etss , we prove that any set in the class is a layered superset ( lss ) of a short cycle , where the term `` layered '' is used to indicate that there is a nested sequence of etss that starts from the cycle and grows , one variable node at a time , to generate . this characterization corresponds to a simple search algorithm that starts from the short cycles of the graph and finds all the etss with lss property in a guaranteed fashion . specific results on the structure of etss are presented for , and in this paper . the results of this paper can be used for the error floor analysis and for the design of ldpc codes with low error floors .

astro-ph9910032

i

hierarchical clustering scenarios predict that early - type galaxies we see today in large virialized structures and in low density environments have formed at different epochs ( ellis 1998 and references therein ) . cluster objects formed through major merging events at red - shift @xmath3 , whereas early - type galaxies , today at the border of large structures , have formed significantly later , at @xmath4 . the morphology of early - type galaxies shows many relationships of different kind with the surrounding environment . the population of the early - type galaxies in clusters appears to be homogeneous contrary - wise to the large heterogeneity shown by objects inhabiting the low density media . although the bulk of galaxies visible today in low density environments are already in place at @xmath5 0.5 ( griffith at al . 1994 ) , i.e nearly half the hubble time , a large number of them show peculiarities such as signatures of interaction , fine structures etc . ( see schweizer 1992 ; reduzzi et al . 1996 and references therein ) which may hint at more recent activity . in the above framework , early - type galaxies are currently interpreted as the final product of major / minor merging events ( schweizer 1992 ; barnes 1996 ) . nevertheless , both from observational and theoretical points of view , evidence has grown over the years that different mechanisms have also played a significant role in shaping their final structure . in clusters , galaxy encounters are fast enough to make mergers less probable than _ harassments _ ( moore et al . 1996 ) . in low density environments encounters are most likely to result in a merger ( barnes & hernquist 1992 ) but dissipative mechanisms ( bender 1997 ) and _ weak - interaction _ events ( thomson 1991 ) could also contribute to the structural evolution of a galaxy . this paper is the fourth of a series dedicated to the study of typical galaxies in low density environments , i.e. galaxies showing signatures of present / past interactions . the sample is composed of 21 shell - galaxies and 30 members of interacting pairs most of which show fine structures . among them shell - galaxies represent a class of objects exhibiting signatures of past interactions , i.e. minor / major mergers ( schweizer 1992 ) or weak - interactions ( thomson & wright 1990 ) and thomson ( 1991 ) . pair - members are instead objects with ongoing interaction . longhetti et al . ( 1998a ) measured 19 line - strength indices in the nuclear regions of the galaxies in our sample . sixteen indices belong to the group defined by worthey ( 1992 ) and g93 and include h@xmath0 , mg2 and some fe features . all indices were transformed into the lick ids system . furthermore , three indices , particularly sensitive to recent star formation ( rose 1984 , 1985 ; leonardi & rose 1996 ) , i. e. @xmath64000 , h@xmath7/fei and caii(h+k ) , were added to the list . longhetti et al . ( 1998b ) derived the inner kinematics of the sample and corrected the line strength indices for central velocity dispersion ( longhetti et al . 1998a ) . the comparison of these galaxies with those in virgo and fornax ( rampazzo et al . 1999a ) singled out a group of pair - galaxies in our sample with peculiar behavior in the ( log r@xmath8 , @xmath9 ) plane and the @xmath10 space . these galaxies , apparently missing among cluster early - type objects , are tidally stretched and most likely in early stages of interaction . this finding makes evident the large scatter of line strength indices as compared to galaxy scale properties . as an example , the correlation between galaxy shapes ( as measured by the a4/a parameter ) and ages ( as deduced from h@xmath0 ) that was advanced by de jong & davies ( 1997 ) is not confirmed by the rampazzo et al . ( 1999a , b ) study . the purpose of this paper is to cast light on the past star formation ( sf ) history of shell- and pair - galaxies in lde as traced by their line strength indices , with particular attention to the role played by dynamical interactions in triggering star formation . the comparison with the results obtained for galaxies in denser environments ( burstein et al . 1984 ; pickles 1985 ; rose 1995 ; bower et al . 1990 ; g93 ) will help us to understand the effects of the environment on the formation / evolution of early - type galaxies . in addition to this , we address the problem of the origin of shell structures by analyzing the evolutionary history of the stellar populations hosted by the nucleus of the interacting galaxy . numerical simulations of dynamical interactions among galaxies yield still contrasting explanations for the occurrence of shells ( weil & hernquist 1993 ; thomson 1991 ) . in this study , we seek to assess the duration of the shell phenomenon by means of the age of the last episode of star formation as inferred from the nuclear indices . the paper is organized as follows . in sect . 2 we compare indices for single stellar populations ( ssps ) obtained using _ fitting functions _ by different authors ( buzzoni et al . 1992 , 1994 ; worthey 1992 ; idiart et al . 1995 ) and a unique source of isochrones ( bertelli et al . 1994 ) . by doing so , we are able to quantify the uncertainties affecting line strength indices calculations . based on this preliminary comparison , we adopt the fitting functions of worthey ( 1992 ) . in sect . 3 we introduce the sample of galaxies of g93 adopted as the reference template . in sect . 4 , firstly we compare the observational line strength indices for the nuclear region of galaxies in our sample with the theoretical predictions , and secondly we compare our sample of shell- and pair - galaxies with that of _ normal _ elliptical galaxies by g93 . notes on individual galaxies in the h@xmath0 vs. [ mgfe ] diagram are reported in sect . a tentative explanation of the distribution of galaxies in the h@xmath0 vs. [ mgfe ] diagram is presented in sect . 6 both for the present sample and the template . the interpretation is based on statistical simulations of the observations . finally , sect . 7 summarizes the results . rrr rrr rrr rr & @xmath6% & @xmath11% & @xmath6% & @xmath11% & @xmath6% & @xmath11% & @xmath6% & @xmath11% & @xmath6% & @xmath11% + & & & t@xmath123gyr & t@xmath123gyr & t@xmath133gyr & t@xmath133gyr & t@xmath121gyr & t@xmath121gyr & t@xmath131gyr & t@xmath131gyr + & & & & & & & & & + h@xmath0 & & & -34.72 & 40.36 & & & & & & + mg2 & 12.72 & 12.94 & 9.45 & 9.45 & 22.97 & 23.53 & 10.31 & 10.32 & 27.28 & 6.92 + mgb & 15.50 & 15.76 & 11.19 & 11.23 & 29.17 & 29.87 & 12.39 & 12.44 & 34.65 & 8.76 + & & & & & & & & & + h@xmath0 & -6.34 & 11.19 & -4.29 & 4.75 & -6.99 & 10.31 & -2.51 & 3.85 & -8.40 & 2.64 + mg2 & 96.88 & 168.29 & -1.82 & 3.20 & -837.16 & -1006.74 & 2.75 & 11.27 & -386.13 & -107.19 + fe5270 & 36.49 & 76.69 & -7.89 & 8.21 & 284.63 & 354.45 & -5.11 & 9.09 & 1837.04 & 520.67 + fe5335 & 6.52 & 10.54 & -0.80 & 2.65 & 24.30 & 24.74 & 0.39 & 5.79 & 28.63 & 29.23 + [ t1 ] 70mml rrr rrr + age & h@xmath0 & mg2 & mgb & fe52 & fe53 & mgfe + 7.70 & 5.10 & -0.57 & 0.44 & 0.64 & 0.44 & 0.49 + 7.85 & 5.51 & -0.58 & 0.45 & 0.52 & 0.33 & 0.44 + 8.00 & 5.72 & -0.56 & 0.49 & 0.48 & 0.28 & 0.43 + 8.95 & 5.09 & 0.06 & 0.97 & 0.83 & 0.61 & 0.84 + 9.00 & 4.82 & 0.06 & 1.05 & 0.93 & 0.71 & 0.93 + 9.18 & 4.01 & 0.08 & 1.27 & 1.27 & 0.96 & 1.19 + 9.70 & 2.41 & 0.12 & 1.94 & 1.77 & 1.47 & 1.77 + 9.78 & 2.27 & 0.13 & 2.03 & 1.83 & 1.53 & 1.85 + 9.85 & 2.14 & 0.13 & 2.12 & 1.89 & 1.59 & 1.92 + 9.90 & 2.02 & 0.14 & 2.19 & 1.95 & 1.64 & 1.98 + 9.95 & 1.98 & 0.14 & 2.24 & 1.95 & 1.65 & 2.01 + 10.00 & 1.91 & 0.14 & 2.28 & 1.99 & 1.69 & 2.05 + 10.04 & 1.84 & 0.15 & 2.33 & 2.02 & 1.74 & 2.09 + 10.08 & 1.78 & 0.15 & 2.37 & 2.06 & 1.78 & 2.14 + 10.18 & 1.71 & 0.15 & 2.42 & 2.06 & 1.81 & 2.16 + + 7.70 & 5.25 & 0.06 & 0.61 & 0.88 & 0.91 & 0.74 + 7.85 & 5.65 & 0.05 & 0.65 & 0.84 & 0.87 & 0.75 + 8.00 & 6.00 & 0.05 & 0.73 & 0.81 & 0.86 & 0.78 + 8.95 & 4.44 & 0.10 & 1.65 & 1.78 & 1.54 & 1.65 + 9.00 & 4.06 & 0.12 & 1.82 & 1.94 & 1.69 & 1.82 + 9.18 & 3.31 & 0.15 & 2.25 & 2.24 & 1.99 & 2.18 + 9.70 & 1.94 & 0.22 & 3.37 & 2.84 & 2.54 & 3.01 + 9.78 & 1.83 & 0.23 & 3.49 & 2.90 & 2.60 & 3.10 + 9.85 & 1.76 & 0.23 & 3.56 & 2.92 & 2.61 & 3.14 + 9.90 & 1.68 & 0.24 & 3.66 & 2.97 & 2.66 & 3.21 + 9.95 & 1.61 & 0.25 & 3.74 & 3.02 & 2.71 & 3.27 + 10.00 & 1.54 & 0.25 & 3.83 & 3.07 & 2.75 & 3.34 + 10.04 & 1.49 & 0.26 & 3.91 & 3.12 & 2.80 & 3.40 + 10.08 & 1.45 & 0.26 & 3.96 & 3.15 & 2.83 & 3.44 + 10.18 & 1.37 & 0.27 & 4.08 & 3.21 & 2.89 & 3.53 + + 7.70 & 6.21 & 0.06 & 0.75 & 0.85 & 1.14 & 0.86 + 7.85 & 6.46 & 0.06 & 0.75 & 0.92 & 1.21 & 0.90 + 8.00 & 6.74 & 0.06 & 0.75 & 0.96 & 1.25 & 0.91 + 8.95 & 3.81 & 0.16 & 2.30 & 2.57 & 2.36 & 2.38 + 9.00 & 3.50 & 0.17 & 2.50 & 2.71 & 2.48 & 2.55 + 9.18 & 2.79 & 0.21 & 3.02 & 3.03 & 2.78 & 2.96 + 9.70 & 1.70 & 0.30 & 4.34 & 3.60 & 3.31 & 3.87 + 9.78 & 1.60 & 0.31 & 4.51 & 3.67 & 3.38 & 3.99 + 9.85 & 1.50 & 0.33 & 4.67 & 3.75 & 3.47 & 4.11 + 9.90 & 1.47 & 0.33 & 4.72 & 3.76 & 3.47 & 4.13 + 9.95 & 1.41 & 0.34 & 4.84 & 3.81 & 3.52 & 4.21 + 10.00 & 1.36 & 0.34 & 4.94 & 3.86 & 3.57 & 4.28 + 10.04 & 1.34 & 0.35 & 4.95 & 3.87 & 3.58 & 4.29 + 10.08 & 1.32 & 0.35 & 4.99 & 3.88 & 3.58 & 4.32 + 10.18 & 1.24 & 0.36 & 5.09 & 3.96 & 3.66 & 4.41 + [ t2 ]

( 1998a , b ) for a sample of 51 early - type galaxies located in low density environments ( lde ) and showing signatures of fine structures and/or interactions . the sample contains 21 shell - galaxies and 30 members of interacting pairs . looking at the three indices in common , i.e. mg2 , fe5270 , and h , the calibrations by buzzoni et al . ( 1995 ) fitting functions result to be systematically lower than those obtained from using worthey ( 1992 ) calibrations . secondly , we discuss the properties of the galaxies in our sample by comparing them both with theoretical single stellar populations ( ssps ) and the _ normal _ galaxies of the gonzlez ( 1993 : g93 ) sample .

in this paper we analyze the line - strength indices in the lick - system measured by longhetti et al . ( 1998a , b ) for a sample of 51 early - type galaxies located in low density environments ( lde ) and showing signatures of fine structures and/or interactions . the sample contains 21 shell - galaxies and 30 members of interacting pairs . firstly we perform a preliminary comparison between three different sources of calibrations of the line strength indices , namely buzzoni et al . ( 1992 , 1994 ) , worthey ( 1992 ) , worthey et al . ( 1994 ) and idiart et al . ( 1995 ) , derived from stars with different effective temperature , gravity , and metallicity . looking at the three indices in common , i.e. mg2 , fe5270 , and h , the calibrations by buzzoni et al . ( 1992 , 1994 ) , worthey ( 1992 ) and worthey et al . ( 1994 ) lead to mutually consistent results . the calibration of h by idiart et al . ( 1995 ) can be compared with the previous ones only for a limited range of ages , in which good agreement is found . mg2 and mgb indices predicted by the idiart s et al . ( 1995 ) fitting functions result to be systematically lower than those obtained from using worthey ( 1992 ) calibrations . secondly , we discuss the properties of the galaxies in our sample by comparing them both with theoretical single stellar populations ( ssps ) and the _ normal _ galaxies of the gonzlez ( 1993 : g93 ) sample . the analysis is performed by means of several diagnostic planes . in the , mg2 , fe5270 and fe5335 space , _ normal _ , shell- and pair - galaxies have a different behavior . first of all , normal and pair - galaxies follow the universal vs. mg2 relation , whereas shell - galaxies lie above it ; secondly the fe versus mg2 relation of normal , shell- and pair - galaxies is flatter than the theoretical expectation . this fact hints for enhancement of-elements with respect to solar partition in galaxies with strong fe indices and/or high velocity dispersion , mass and luminosity in turn . in the vs. h plane _ normal _ galaxies seem to follow a nice relation suggesting that objects with shallow gravitational potential have strong h values ( youth signature ? ) , whereas shell- and pair - galaxies scatter all over the plane . a group of galaxies with deep gravitational potential and strong h is found . is this a signature of recent star formation ? in the h vs. [ mgfe ] plane,[multiblock footnote omitted ] which is perhaps best suited to infer the age of the stellar populations , the peculiar galaxies in our sample show nearly the same distribution of the _ normal _ galaxies in the g93 sample . there is however a number of peculiar galaxies with much stronger h . does this mean that the scatter in the h vs. [ mgfe ] plane , of normal , shell- and pair - galaxies has a common origin , perhaps a secondary episode of star formation ? we suggest that , owing to their apparent _ youth _ , shell- and pair - galaxies should have experienced at least one interaction event after their formation . the explanation comes natural for shell- and pair - galaxies where the signatures of interactions are evident . it is more intrigued in _ normal _ galaxies ( perhaps other causes may concur ) . noteworthy , the distribution in the h vs. [ mgfe ] plane of _ normal _ , shell- and pair - galaxies is confined within a narrow strip that runs significantly steeper than the path followed by aging ssps . this feature is explained as due to metal enrichment always accompanying star formation . shell - galaxies encompass the whole range of ages inferred from the h vs. [ mgfe ] plane , indicating that among them recent and old interaction / acquisition events are equally probable . if shells are formed at the same time at which the rejuvenating event took place , shells ought to be long lasting phenomena .

astro-ph9910032

i

in this paper firstly we have compared the line strength indices for ssps that one would obtain using different calibrations in literature , namely worthey ( 1992 ) , worthey et al . ( 1994 ) , buzzoni et al . ( 1992 , 1994 ) and idiart et al . ( 1995 ) . secondly , with the aid of the worthey ( 1992 ) and worthey et al . ( 1994 ) calibrations , the sample of shell- and pair - galaxies by longhetti et al . ( 1998a , b ) , and the sample of g93 for _ normal _ elliptical galaxies have been systematically analyzed , looking at the position of all these galaxies in various diagnostic planes . the aim is to cast light on the star formation history that took place in these systems with particular attention to those ( shell- and pair - objects ) for which the occurrence of dynamical interaction is evident . finally , from comparing normal to dynamically interacting galaxies we attempt to understand the reasons for their similarities and differences . \(1 ) the various calibrations for the line strength indices as a function of basic stellar parameters ( effective temperature , gravity and metallicity ) lead to quite different results . specifically , the buzzoni et al . ( 1992 , 1994 ) calibrations for mg2 and fe5270 agree with those by worthey ( 1992 ) and worthey et al . ( 1994 ) only for ssps older than 3 gyr . for h@xmath0 the agreement is also good if one excludes all ages younger than about 1 gyr . idiart s et al . ( 1995 ) calibrations can be compared with the others only for a limited range of ssp ages . for the mgb and mg2 indices we find a roughly constant offset , that could be attributed to different properties ( g.e . , metallicity , gravity ) of the calibrating sample of stars . for the purposes of this study we have adopted worthey ( 1992 ) and worthey et al . ( 1994 ) as the reference calibrations . \(2 ) the comparison of the mg and fe indices ( specifically mg2 , fe5270 and fe5335 ) and the velocity dispersion @xmath1 of normal , shell- and pair - galaxies suggests firstly a different behavior of shell - galaxies with respect to pair - objects and secondly that strong - lined galaxies are likely to have super - solar [ mg / fe ] abundance ratios . various kinds of star formation histories leading to super - solar [ mg / fe ] are examined at the light of current understanding of the mechanisms of galaxy formation . none of these is however able to give a self - consistent explanation of the [ @xmath2]-enhancement problem . \(3 ) the same galaxies are analyzed in the h@xmath0 vs. [ mgfe ] plane and compared to the g93 set of data . the shell- and pair - objects have the same distribution in this diagnostic plane as the _ normal _ galaxies even if they show a more pronounced tail toward high h@xmath0 values . \(5 ) as shell- and pair - galaxies share the same region of the h@xmath0 vs. [ mgfe ] plane occupied by _ normal _ galaxies , we suggest that a common physical cause is at the origin of their distribution . the star formation history in these objects is investigated with the aid of very simple galaxy models . we find that the tail at high h@xmath0 values can be ascribed to secondary bursts of star formation which , in the case of shell- and pair - galaxies , can be easily attributed to interaction / acquisition events whose signatures are well evident in their morphology . \(6 ) a typical model where the burst is superimposed to an otherwise old and coeval population is however not able to reproduce the smooth distribution of galaxies in the h@xmath0 vs. [ mgfe ] plane . this kind of model would predict an outstanding clump at low h@xmath0 values , contrary to what is observed . models in which the bulk of the star formation happened over a significant fraction of the hubble time ( 4 gyr @xmath36 t@xmath39 @xmath36 16 gyr ) better match the observed diagram . \(7 ) in this context , the peculiar , smooth and almost vertical distribution of galaxies ( normal , shell- pair - objects ) in the h@xmath0 vs. [ mgfe ] plane is interpreted as the trace of the increase of the _ average _ metallicity accompanying all star forming events . this could be the signature of a metal enrichment happening on a cosmic scale . \(8 ) although deciphering the position of galaxies in the h@xmath0 vs [ mgfe ] plane to infer the age of the constituent stellar populations is a difficult task due to possible blurring caused by the secondary stellar activity , still we may draw some conclusions for the duration of the shell phenomenon . specifically , since shell - galaxies can be found in the same region of old normal galaxies , we may say that shells are a morphological feature able to persist for long periods of time , much longer than the star forming activity that likely accompanied their formation . among current dynamical models in which shell - structures can be formed , the _ weak interaction _ mechanism of thomson & wright ( 1990 ) and thomson ( 1991 ) naturally predicts long - lived shells without particular hypotheses on the type of encounters . ml acknowledges the kind hospitality of the astronomical observatory of brera ( milan ) and padua during her phd thesis and the support by the european community under tmr grant erbfmbi - ct97 - 2804 . cc wishes to acknowledge the friendly hospitality and stimulating environment provide by mpa in garching where this paper has been completed during leave of absence from the astronomy department of the padua university . this study has been financially supported by the european community under tmr grant erbfmrx - ct96 - 0086 .

in this paper we analyze the line - strength indices in the lick - system measured by longhetti et al . firstly we perform a preliminary comparison between three different sources of calibrations of the line strength indices , namely buzzoni et al . ( 1992 , 1994 ) , worthey ( 1992 ) , worthey et al . ( 1992 , 1994 ) , worthey ( 1992 ) and worthey et al . ( 1995 ) can be compared with the previous ones only for a limited range of ages , in which good agreement is found . normal _ , shell- and pair - galaxies have a different behavior . normal _ galaxies seem to follow a nice relation suggesting that objects with shallow gravitational potential have strong h values ( youth signature ? ) , whereas shell- and pair - galaxies scatter all over the plane . a group of galaxies with deep gravitational potential and strong h is found . which is perhaps best suited to infer the age of the stellar populations , the peculiar galaxies in our sample show nearly the same distribution of the _ normal _ galaxies in the g93 sample . does this mean that the scatter in the h vs. [ mgfe ] plane , of normal , shell- and pair - galaxies has a common origin , perhaps a secondary episode of star formation ? we suggest that , owing to their apparent _ youth _ , shell- and pair - galaxies should have experienced at least one interaction event after their formation . the explanation comes natural for shell- and pair - galaxies where the signatures of interactions are evident . it is more intrigued in _ normal _ galaxies ( perhaps other causes may concur ) . noteworthy , the distribution in the h vs. [ mgfe ] plane of _ normal _ , shell- and pair - galaxies is confined within a narrow strip that runs significantly steeper than the path followed by aging ssps . this feature is explained as due to metal enrichment always accompanying star formation . shell - galaxies encompass the whole range of ages inferred from the h vs. [ mgfe ] plane , indicating that among them recent and old interaction / acquisition events are equally probable . if shells are formed at the same time at which the rejuvenating event took place , shells ought to be long lasting phenomena .

in this paper we analyze the line - strength indices in the lick - system measured by longhetti et al . ( 1998a , b ) for a sample of 51 early - type galaxies located in low density environments ( lde ) and showing signatures of fine structures and/or interactions . the sample contains 21 shell - galaxies and 30 members of interacting pairs . firstly we perform a preliminary comparison between three different sources of calibrations of the line strength indices , namely buzzoni et al . ( 1992 , 1994 ) , worthey ( 1992 ) , worthey et al . ( 1994 ) and idiart et al . ( 1995 ) , derived from stars with different effective temperature , gravity , and metallicity . looking at the three indices in common , i.e. mg2 , fe5270 , and h , the calibrations by buzzoni et al . ( 1992 , 1994 ) , worthey ( 1992 ) and worthey et al . ( 1994 ) lead to mutually consistent results . the calibration of h by idiart et al . ( 1995 ) can be compared with the previous ones only for a limited range of ages , in which good agreement is found . mg2 and mgb indices predicted by the idiart s et al . ( 1995 ) fitting functions result to be systematically lower than those obtained from using worthey ( 1992 ) calibrations . secondly , we discuss the properties of the galaxies in our sample by comparing them both with theoretical single stellar populations ( ssps ) and the _ normal _ galaxies of the gonzlez ( 1993 : g93 ) sample . the analysis is performed by means of several diagnostic planes . in the , mg2 , fe5270 and fe5335 space , _ normal _ , shell- and pair - galaxies have a different behavior . first of all , normal and pair - galaxies follow the universal vs. mg2 relation , whereas shell - galaxies lie above it ; secondly the fe versus mg2 relation of normal , shell- and pair - galaxies is flatter than the theoretical expectation . this fact hints for enhancement of-elements with respect to solar partition in galaxies with strong fe indices and/or high velocity dispersion , mass and luminosity in turn . in the vs. h plane _ normal _ galaxies seem to follow a nice relation suggesting that objects with shallow gravitational potential have strong h values ( youth signature ? ) , whereas shell- and pair - galaxies scatter all over the plane . a group of galaxies with deep gravitational potential and strong h is found . is this a signature of recent star formation ? in the h vs. [ mgfe ] plane,[multiblock footnote omitted ] which is perhaps best suited to infer the age of the stellar populations , the peculiar galaxies in our sample show nearly the same distribution of the _ normal _ galaxies in the g93 sample . there is however a number of peculiar galaxies with much stronger h . does this mean that the scatter in the h vs. [ mgfe ] plane , of normal , shell- and pair - galaxies has a common origin , perhaps a secondary episode of star formation ? we suggest that , owing to their apparent _ youth _ , shell- and pair - galaxies should have experienced at least one interaction event after their formation . the explanation comes natural for shell- and pair - galaxies where the signatures of interactions are evident . it is more intrigued in _ normal _ galaxies ( perhaps other causes may concur ) . noteworthy , the distribution in the h vs. [ mgfe ] plane of _ normal _ , shell- and pair - galaxies is confined within a narrow strip that runs significantly steeper than the path followed by aging ssps . this feature is explained as due to metal enrichment always accompanying star formation . shell - galaxies encompass the whole range of ages inferred from the h vs. [ mgfe ] plane , indicating that among them recent and old interaction / acquisition events are equally probable . if shells are formed at the same time at which the rejuvenating event took place , shells ought to be long lasting phenomena .

1701.06237

i

let @xmath0 be a given space - time domain in @xmath1 , denoted by @xmath2 in this domain we consider a continuity equation of the form : @xmath3 in the space of probability measures , with the constraint that the support of @xmath4 lies in the closure of @xmath0 . when @xmath5 , this constraint yields the co - normal boundary data on the lateral boundary of @xmath0 . the first - order system , @xmath6 , will be formulated using a projection operator : we will show that this system can be obtained as the vanishing viscosity limit as @xmath7 . the above system describes the density of moving particles which are confined to some region and flow with a velocity field @xmath8 inside of the domain . one part of the velocity field is generated from interactions between different particles represented by the interaction potential @xmath9 , given by @xmath10 this type of problem arises in many applications with various interaction kernel @xmath9 , such as in swarming models with @xmath11 and in models of chemotaxis with @xmath12 , see @xcite for more references . at the same time , the particles are subject to an external potential @xmath13 . both @xmath14 are assumed to be smooth and @xmath15-convex . more assumptions will be presented in section [ sub2.1 ] and [ sub3.1 ] . for the diffusion term , the model takes into account random movements of the particles . in the first part of this paper , we consider @xmath6 . let @xmath16 be the speed of the boundary ( with positive sign if the boundary is expanding ) and @xmath17 be the unit space outer normal for @xmath18 . we set @xmath19 if @xmath20 are not on the boundary . for simplicity sometimes we only write @xmath21 . then we define a projection operator @xmath22 as follows ( see figure 1 ) : @xmath23 note at @xmath24 in the interior , @xmath22 is an identity map on @xmath25 . we refer readers to @xcite where they defined a similar projection operator on stationary domains . operator , scaledwidth=40.0% ] set @xmath26 . we formulate the equation as : @xmath27 here @xmath28 is a probability measure on @xmath29 with finite second moment . first we assume it is compactly supported , and later we will also consider probability measures with exponential decay properties . our results of part one are mainly motivated by the previous work by carrillo , slepcev and wu @xcite , where they show the well - posedness of equation in stationary , non - convex domains with compactly supported initial data . our main contributions are two - fold . first we generalize the well - posedness result to general space - time domains and allow non - compactly supported initial data . second , we show that can be obtained as the limit as @xmath30 of the diffusion equation given with the co - normal boundary data , imposing the additional condition that the domain is bounded and spatially convex . this result is significant since it provides a natural justification for the first - order system . let us outline the proof for the well - posedness of with compactly supported initial data and illustrate the difficulties involving time - dependent domains . as in @xcite , we use particle approximations . take a sequence @xmath31 which converges to @xmath28 in 2-wasserstein metric ( see preliminaries [ 1.1 ] or readers can refer to @xcite ) where @xmath32 is a sum of dirac masses . the corresponding solutions @xmath33 to equation can be found by solving an ode system . it is not hard to show that @xmath34 converges to some @xmath35 by stability properties of solutions to . but due to the discontinuity of operator @xmath36 with respect to @xmath37 on the boundary , it does not immediately follow that @xmath38 is a solution . to resolve this difficulty , we use the gradient flow structure of the problem to show that the limit @xmath38 is a weak solution to . to be more precise , we take the limit of the curve of maximal slope " ( see inequality ) . the novelty in this paper is that , for space - time domains an extra term ( the last term in ) appears in the curve of maximal slope " : @xmath39 where @xmath40 . intuitively @xmath41 comes from the moving boundary constraints and the possible situation that particles attempt to move out of the domain with potential velocity @xmath42 but end up moving with the boundary with velocity @xmath43 . according to the definition of @xmath22 , it is not hard to see that @xmath41 is only nontrivial if @xmath4 is singular with mass concentrating on the boundary . alternatively if the domain is stationary , this term vanishes , since if the boundary speed @xmath44 , @xmath45 is to the normal direction while @xmath46 is tangential . this additional term requires careful analysis when dealing with the limit as @xmath47 : see the proof of theorem [ thmu ] , where we study the integral on the boundary of positive speed and negative speed separately . the associate energy to will be used which is given as the sum of potential and interaction energy : @xmath48 note later we will only do the integration within some @xmath49 which is the support of @xmath38 . for non - compactly supported initial data with exponential decay property ( see condition ( r ) in section [ sub2.4 ] ) , existence of solutions can be done similarly by the approximation method . uniqueness of solutions satisfying exponential decay is guaranteed by proving a stability estimate . the requirement below that @xmath50 , as well as the exponential decay condition , are essential in the proof of theorem [ converge ] , as shown by the examples in theorem [ example ] . now we summarize the main theorem in part one . we use @xmath51 to denote the 2-wasserstein distance between probability measures . the notion of weak solutions will be defined in section [ 1.1 ] . [ a ] assume conditions ( o1)(c1)-(c3 ) hold ( see section [ sub2.1 ] for details ) . let @xmath28 be a probability measure supported in @xmath29 with finite second moment and fix any @xmath52 . \(a ) ( theorem [ thmu ] ) suppose @xmath28 is compactly supported . then there is a weak solution @xmath35 to equation and @xmath38 is compactly supported for @xmath53 . if @xmath54 are two solutions with compact support in @xmath55 $ ] , then there exists @xmath56 such that @xmath57 \(b ) ( theorem [ converge ] ) suppose @xmath28 satisfies the exponentially decay property in condition ( r ) . then there exists a weak solution @xmath35 of equation and @xmath38 satisfies ( r ) for @xmath53 . if @xmath54 are two solutions with initial data @xmath58 and @xmath54 satisfy ( r ) in @xmath55 $ ] , then for any @xmath59 , if @xmath60 and @xmath61 are small enough , we have @xmath62 \(c ) ( theorem [ example ] ) for non - convex unbounded domains , examples can be found that the @xmath63 above can not be improved to @xmath64 . furthermore for @xmath28 with less decay , even the above stability property does not hold . in the second part , we consider the case @xmath5 : @xmath65 here @xmath66 is the lateral boundary of @xmath0 . the co - normal boundary condition above guarantees the mass preservation . in this case the associated energy @xmath67 is given by @xmath68 @xmath69 in the first term , @xmath70 is the probability density function of @xmath4 if @xmath4 is absolutely continuous with respect to euclidean measure . we set @xmath71 if @xmath4 is not absolutely continuous . with the convergence of @xmath7 in mind , we show the existence of solutions by discrete - time gradient - flow ( jko ) solutions ( see @xcite ) . for this purpose , technically we further require @xmath72 to be @xmath73 and bounded below . in time - dependent domain , the scheme is slightly different from the standard version that we minimize each movement among probability measures with support contained in @xmath0 ( see ) . to obtain the continuum time limit of the discrete - time solutions , we show the uniform boundedness of the second moment and the boundedness of @xmath74 along solutions from the discrete scheme . this is one part that the analysis for problems on stationary domains that can not be directly carried over for the time - dependent domains . the problem is solved in proposition [ prop 2 ] , the proof of which is inspired by the work of di marino , maury and santambrogio @xcite who encountered the same problem . also let us mention that solutions obtained in this way inherit the gradient flow structure which will be important later . after establishing the well - posedness of weak solutions , we send @xmath30 . we show that if the domain is bounded and spatially convex , equations are indeed the vanishing viscosity approximation of the first order equation in the first part . this convergence justifies the formulation of equation , in addition to the derivation via particle system . in @xcite , the vanishing viscosity limit problem in the whole domain was studied in the case when @xmath75 and @xmath76 is the newtonian potential . their proof heavily relies on the specific choice of kernel @xmath9 , and also the fact that the domain is @xmath77 which eliminates the task of determining the limiting boundary condition . now let us give the main theorem of the second part of this paper . [ mainthmb ] assume conditions ( c1)-(c4)(o1)(o2 ) hold ( see sections [ sub2.1 ] , [ sub3.1 ] for details ) , @xmath28 is an absolutely continuous ( with respect to lebesgue measure ) probability measure supported in @xmath78 with finite second moment , and @xmath79 is any fixed positive number . then \(a ) ( theorem [ existence ] ) there exists a weak solution @xmath35 to equation . + ( b ) ( theorem [ uniqueness ] ) suppose @xmath80 is bounded for each @xmath81 and the density of @xmath28 is in @xmath82 . then there exists a unique weak solution @xmath35 to equation with density @xmath83 a.e . if @xmath85 are two solutions with initial data @xmath86 , then there exits @xmath56 such that their densities satisfy @xmath87.\ ] ] \(c ) ( theorem [ convergence ] ) suppose @xmath80 is bounded and convex for each @xmath81 . let @xmath88 be the weak solutions to equations and @xmath35 be the weak solution to equation . then for all @xmath60 @xmath89 we conclude by outlining the strategy for proving the convergence as @xmath7 . write @xmath90 . since the domain is convex , it can be shown that @xmath91 where @xmath92 is an optimal transport map from @xmath93 to @xmath94 defined in section [ 1.1 ] below , @xmath95 is the identity map on @xmath96 , @xmath97 and @xmath98 belong to the frchet subdifferential of @xmath99 at @xmath100 respectively . next from the convexity conditions on @xmath72 , we can show that the functionals @xmath101 are uniformly @xmath102-convex for some @xmath102 . by the variational inequalities in gradient flow theory ( section 10.1.1 @xcite ) , we have the following inequalities at each time @xmath60 : @xmath103 where @xmath104 can be any probability measures supported in @xmath49 . formally we take @xmath105 and add up the above two inequalities which then , combining with equality , leads to @xmath106 thus it remains to show that @xmath107 and @xmath108 converge to @xmath109 . however we point out that it is possible @xmath110 for some @xmath84 , since in general even with smooth initial data , @xmath4 can concentrate mass in finite time as discussed in @xcite . to overcome this problem we look for a @xmath111 to replace @xmath4 which is close to @xmath4 in some sense , and @xmath112 is bounded point - wisely by @xmath113 for some @xmath114 . if such @xmath115 exists , using that the domain is bounded , we obtain @xmath116 . then we encounter another problem when we take @xmath117 in the variational inequality . a key step is to show that @xmath118 are close not only in 2-wasserstein metric but also in pseudo - wasserstein distance with base @xmath119 ( the definition will be given in [ 1.1 ] ) . to do this we will use properties of generalized geodesics in probability measure space with pseudo - wasserstein metric as well as properties of optimal transport plan . we write @xmath17(or simply @xmath120 ) as the spatial normal vector and @xmath16(or @xmath121 ) as the speed of the boundary at @xmath18 . throughout the paper , we fix a @xmath52 which is assumed to be large . we say a constant is universal if it only depends on @xmath79 and constants in conditions ( o1)(o2)(c1)-(c4 ) ( @xmath122 and bounds about @xmath123 ) . we denote by @xmath56 a constant which may depend on universal constants and @xmath28 , possibly changing from one estimate to another . a spatial ball in @xmath96 centered at @xmath124 with radius @xmath125 is denoted by @xmath126 , and we may simply write @xmath127 if @xmath124 is the origin . given a probability measure @xmath4 , we write @xmath128 as the second moment of @xmath4 . the set of all probability measures on @xmath129 with finite second moment is denoted by @xmath130 . the set of absolutely continuous ( with respect to lebesgue measure ) probability measures with finite second moment is written as @xmath131 . for @xmath132 , we usually write @xmath133 where @xmath70 is its density . for probability measures supported in @xmath129 , we will think of them as measures in @xmath134 , extended by @xmath109 outside @xmath129 . now we give the definition of weak ( measure ) solutions to equations and . a locally absolutely continuous ( in wasserstein metric ) curve @xmath135 is a weak solution to with initial value @xmath136 if @xmath137 and for all @xmath138 : @xmath139 and for each @xmath81 , @xmath140 a locally absolutely continuous curve @xmath141 is a weak solution to with initial value @xmath142 if for all @xmath138 : @xmath143 and for each @xmath81 , @xmath144 now we discuss wasserstein metric , and we refer readers to @xcite for details . suppose @xmath145 are measurable subsets of @xmath134 and @xmath146 . a plan between @xmath147 is any borel measure @xmath148 on @xmath149 which has @xmath94 as its first marginal and @xmath93 as its second marginal . we write @xmath150 . from optimal transport theory , there exists an optimal transport plan @xmath151 such that @xmath152 the above quantity is defined to be the _ 2-wasserstein distance _ between @xmath147 ( the kantorovich s formulation ) . throughout this paper we use this distance for probability measures with notation @xmath51 unless otherwise stated . and later by wasserstein distance ( metric ) we always mean 2-wasserstein distance ( metric ) . we denote the set of optimal transport plans between @xmath94 , @xmath93 by @xmath153 . let @xmath154 , a measurable function @xmath155 transports @xmath93 onto @xmath156 if @xmath157 for all measurable @xmath158 , and we write @xmath159 . if @xmath160 , then for any @xmath161 there is an optimal transport map @xmath162 such that @xmath163 ( with reference to @xcite ) . and we have , in monge s formulation , @xmath164 given @xmath165 . let @xmath166 be an optimal transport maps from @xmath4 to @xmath94 and @xmath93 respectively . then the _ pseudo - wasserstein distance _ with base @xmath4 is defined as @xmath167 by proposition 1.15 @xcite , @xmath168 is a metric on @xmath169 and we always have for any @xmath4 , @xmath170

in this paper we consider a class of continuity equations that are conditioned to stay in general space - time domains , which is formulated as a continuum limit of interacting particle systems . we also consider the vanishing viscosity approximation of the system , given with the co - normal boundary data . if the domain is spatially convex , the limit coincides with the solution of our original system , giving another interpretation to the equation .

in this paper we consider a class of continuity equations that are conditioned to stay in general space - time domains , which is formulated as a continuum limit of interacting particle systems . we study the well - posedness of the solutions and provide examples illustrating that the stability of solutions is strongly related to the decay of initial data at infinity . we also consider the vanishing viscosity approximation of the system , given with the co - normal boundary data . if the domain is spatially convex , the limit coincides with the solution of our original system , giving another interpretation to the equation .

1308.1010

i

there is a vast literature on @xmath2-body vortex problems , much of which is reference in the book by newton @xcite . the geostrophic vortex models are described in many geophysics texts including @xcite . there has been some work done on two - layer point vortex dynamics on the entire plane . these include the works of young @xcite , and hogg and stommel @xcite@xcite , and which have been primarily numerical investigations . some experimental work has been done by griffiths and hopfinger @xcite . more analytic results can be found in the work of gryanik @xcite and zabusky and mcwilliams @xcite and flierl , polvani and zabusky @xcite . integrable two layer point vortex motion in the plane has been extensively studied by jamaloodeen and newton @xcite@xcite . more recent studies can also be found in the works of koshel et al @xcite@xcite . in these are studied the equilibrium solutions , the vortex collapse problem and the transition to chaotic advection through perturbations of known equilibrium solutions . the two layer vortex problem in domains other than the entire plane has not been as extensively studied as the one layer vortex problem on domains with boundaries . a good exposition of the one layer vortex problem on domains with boundaries can be found in work of flucher and gustafsson@xcite . the work in this study can be understood as applying the techniques for one layer integrable vortex dynamics on domains with boundaries @xcite , and the analytic techniques for integrable two layer vortex dynamics in the unbounded plane @xcite@xcite@xcite@xcite , to integrable two layer vortex dynamics in the upper half plane . this paper is organized as follows . we begin by deriving the equations of motion , by first obtaining the streamfunctions for an ensemble of point vortices . we also obtain the invariants through the use of symmetry and establish the integrability of the two vortex problem in the upper half plane . through analysis of the hamiltonian energy curves we characterize all 2 point vortex motion in the upper half plane for both cases @xmath3 @xmath4 and @xmath3 @xmath5 including conclusions about 2 vortex collapse . we then determine all equilibrium solutions for the 2 point vortex motion , again , for both cases . we proceed by comparing the hamiltonian energy curves for the two layer problem in upper half plane with the one layer problem in the upper half plane . we then present qualitative aspects of streamline topologies for the two layer problem , for both cases @xmath3 @xmath6 and @xmath3 @xmath5 . we conclude with a study of integrable 3 vortex motion by considering symmetrical initial configurations , and seeking conditions to maintain the symmetry of the initial configuration . by enforcing those conditions , which simplify by symmetry of the configuration , we are able to obtain , numerically , rich classes of relative equilibria in both cases @xmath3 @xmath6 and @xmath7 @xmath5 .

in this paper we derive the equations of motion for two - layer point vortex motion on the upper half plane . we study the invariants using symmetry , including the hamiltonian and show that the two vortex problem is integrable . we characterize all two vortex motions for the cases where the vortex strengths are both equal , and when they are opposite . we also make observations concerning the finite - time collapse of two vortices in the half plane . we also study several classes of streamline topologies for two vortices in different layers .

in this paper we derive the equations of motion for two - layer point vortex motion on the upper half plane . we study the invariants using symmetry , including the hamiltonian and show that the two vortex problem is integrable . we characterize all two vortex motions for the cases where the vortex strengths are both equal , and when they are opposite . we also prove that there are no equilibria for the two vortex problem when . we show that there is only one relative equilibrium configuration when and the vortices are in different layers . we also make observations concerning the finite - time collapse of two vortices in the half plane . we then compare the regimes of motion for both cases ( motion on the half plane ) with the case of the two - layer vortex problem on the entire plane . we also study several classes of streamline topologies for two vortices in different layers . we conclude with a hamiltonian study of integrable two - layer 3 vortex motion on the half plane by studying integrable symmetrical configurations and provide a rich class of new relative equilibria .

1107.5180

i

lagrangian transport methods are based on following the individual trajectories obtained by solving the original differential equation ( ode ) for the particle location starting from a set of initial conditions @xmath0 at time @xmath1 : @xmath2 where @xmath3 is the velocity field . the geometrical approach of dynamical systems theory is particularly useful when the flow field has lagrangian coherent structures that separate the flow into distinct regions . then lagrangian lobe dynamics describes the transport process between these regions using stable and unstable manifolds of hyperbolic trajectories as ( moving ) boundaries . the lagrangian methods have been applied successfully to a number of unsteady geophysical flow problems ; for a review , see @xcite . if the flow is steady , i.e. , @xmath4 , the invariant manifolds are stationary and no transport occurs between the regions . on the contrary , eulerian - based methods are mainly concerned with the amount of transport across stationary ( eulerian ) boundaries without computing individual trajectories . an advantage of eulerian methods is that they tend to be much less elaborate than lagrangian methods in terms of computational implementation . the choice for the eulerian boundaries is generally flexible , unlike the lagrangian methods . from the dynamical systems point of view , a parallel development of a method that computes transport across the eulerian boundary has yet to take place . in this paper we begin the development of such a method . the method makes use of the interaction between the reference ( mean ) state and the unsteady variability ( eddy ) as the fundamental mechanism of transport . hence we refer to it as the * t*ransport * i*nduced by the * m*ean-*e*ddy interaction ( time ) . using a streamline of the reference state as the boundary across which we consider transport , time can be thought as a hybrid of lagrangian and eulerian methods . like the eulerian method , the boundary is stationary . like the lagrangian method , the boundary is kinematically defined and there is no time in the steady flow without the unsteady eddy component in the velocity . in certain situations we are able to describe the geometrical relationship of time along the eulerian boundaries with lagrangian lobe dynamics . we require no assumption of incompressibility in our theoretical framework . therefore the ideas and techniques of the time method can be applied to two - dimensional compressible flow or three - dimensional volume - preserving flow which can be represented as special classes of two - dimensional flows , such as the shallow - water model . remarks concerning incompressibility are provided throughout the paper as special cases . extensions to three - dimensional flow are possible @xcite , but there is more complexity in the geometry of the transport , and this is will be the topic of a future publication . the outline of this paper is as follows . in section [ sec : bg ] , we provide a brief mathematical background and introduce the notion of a kinematically - defined eulerian boundary ; readers who are familiar with elementary dynamical systems theory may omit this section without significant loss of continuity by referring back to the notation and definitions as necessary . a brief glossary is also provided in table [ tbl : glssry ] . the time method is defined in section [ sec : fnc ] , along with the two types of time functions . these functions , along with the notion of pseudo - lobes , are further explored in section [ sec : chr ] . an application of the time method is carried out in section [ sec : application ] for the inter - gyre transport in the double - gyre ocean circulation model and a comparison with lagrangian transport theory is presented . appendix [ sec : p ] provides details of perturbation theory , and appendix [ sec : l ] compares the time method with the lagrangian transport methods . while in this paper we focus on introducing and developing the two time functions that estimate the amount and the geometry of transport , in the companion paper @xcite , we expand the time method further as a diagnostic tool for transport processes by analyzing in detail the influence of the mean - eddy interaction .

the ( eulerian ) boundaries across which we consider ( lagrangian ) transport are kinematically defined by appropriately chosen streamlines of the mean flow . by evaluating the impact of the mean - eddy interaction on transport , the time method can be used as a diagnostic tool for transport processes that occur during a specified time interval along a specified boundary segment . we introduce two types of time functions : one that quantifies the accumulation of flow properties and another that measures the displacement of the transport geometry . an application of the time method is carried out for inter - gyre transport in the wind - driven oceanic circulation model and a comparison with the lagrangian transport theory is made . and eulerian transport , lagrangian transport , mean - eddy interaction , dynamical systems approach , wind - driven ocean circulation 47.10.fg , 47.11.st , 47.27.ed , 47.51.+a , 92.05.-x , 92.10.a- , 92.10.ab , 92.10.ah 92.10.ak , 92.10.lq , 92.10.ty , 92.60.bh

in this paper we develop a method for the estimation of * t*ransport * i*nduced by the * m*ean-*e*ddy interaction ( time ) in two - dimensional unsteady flows . the method is built on the dynamical systems approach and can be viewed as a hybrid combination of lagrangian and eulerian methods . the ( eulerian ) boundaries across which we consider ( lagrangian ) transport are kinematically defined by appropriately chosen streamlines of the mean flow . by evaluating the impact of the mean - eddy interaction on transport , the time method can be used as a diagnostic tool for transport processes that occur during a specified time interval along a specified boundary segment . we introduce two types of time functions : one that quantifies the accumulation of flow properties and another that measures the displacement of the transport geometry . the spatial geometry of transport is described by the so - called pseudo - lobes , and temporal evolution of transport by their dynamics . in the case where the time functions are evaluated along a separatrix , the pseudo - lobes have a relationship to the lobes of lagrangian transport theory . in fact , one of the time functions is identical to the melnikov function that is used to measure the distance , at leading order in a small parameter , between the two invariant manifolds that define the lagrangian lobes . we contrast the similarities and differences between the time and lagrangian lobe dynamics in detail . an application of the time method is carried out for inter - gyre transport in the wind - driven oceanic circulation model and a comparison with the lagrangian transport theory is made . and eulerian transport , lagrangian transport , mean - eddy interaction , dynamical systems approach , wind - driven ocean circulation 47.10.fg , 47.11.st , 47.27.ed , 47.51.+a , 92.05.-x , 92.10.a- , 92.10.ab , 92.10.ah 92.10.ak , 92.10.lq , 92.10.ty , 92.60.bh

1107.5180

i

we have developed a mathematical framework for the estimation of * t*ransport * i*nduced by the * m*ean-*e*ddy interaction ( time ) for flow properties and fluid particles with emphasis on two - dimensional unsteady geophysical flows , without the assumption of incompressibility . the time method estimates the amount of lagrangian transport across the kinematically - defined eulerian boundary by the appropriately chosen streamline of the reference flow ( section [ sec : bg ] ) . the time method is a hybrid combination of lagrangian and eulerian methods and is based on the dynamical systems approach . it enables on to analyze unique features of transport that neither the lagrangian nor eulerian methods can provide . by considering two different aspects of transport , we obtain the accumulation function for flow properties , as well as the displacement distance and area functions for fluid particles ( section [ sec : fnc ] ) . the dynamical systems approach leads to the useful characteristics such as invariance , independence and coherency of the geometry ( section [ sec : chr ] ) . in the companion paper @xcite , we develop a framework for the analysis of the transport process in which these characteristics play a key role . the notion of pseudo - lobes is developed to describe the geometry associated with time . when a heteroclinic is used as the eulerian boundary and the time interval for transport to take place is set over a bi - infinite time interval , the pseudo - lobes are geometrically closely related to the lagrangian lobes of the associated invariant manifolds . the novel turnstile mechanisms for lagrangian transport can be carried over in the time method by taking the mirror image of the pseudo - lobes in the upstream region of the heteroclinic connection ( appendix [ sec : l ] ) . an application to an oceanic problem and a comparison with the lagrangian lobe dynamics studied by @xcite ( section [ sec : application ] ) . the time method is designed to augment and supplement lagrangian and eulerian transport methods by providing the unique capability to analyze the underlying transport processes , as it will be shown in @xcite . the method can be applied to more genercal cases than the application presented in this paper ; the method itself does not require the time periodicity of the flow filed or a heteroclinic connection in the reference state . various applications and extensions of the eulerian transport theory , including three - dimensionality @xcite , brings a new point of view and direction to transport studies in geophysical flows .

in this paper we develop a method for the estimation of * t*ransport * i*nduced by the * m*ean-*e*ddy interaction ( time ) in two - dimensional unsteady flows . the method is built on the dynamical systems approach and can be viewed as a hybrid combination of lagrangian and eulerian methods . where the time functions are evaluated along a separatrix , the pseudo - lobes have a relationship to the lobes of lagrangian transport theory .

in this paper we develop a method for the estimation of * t*ransport * i*nduced by the * m*ean-*e*ddy interaction ( time ) in two - dimensional unsteady flows . the method is built on the dynamical systems approach and can be viewed as a hybrid combination of lagrangian and eulerian methods . the ( eulerian ) boundaries across which we consider ( lagrangian ) transport are kinematically defined by appropriately chosen streamlines of the mean flow . by evaluating the impact of the mean - eddy interaction on transport , the time method can be used as a diagnostic tool for transport processes that occur during a specified time interval along a specified boundary segment . we introduce two types of time functions : one that quantifies the accumulation of flow properties and another that measures the displacement of the transport geometry . the spatial geometry of transport is described by the so - called pseudo - lobes , and temporal evolution of transport by their dynamics . in the case where the time functions are evaluated along a separatrix , the pseudo - lobes have a relationship to the lobes of lagrangian transport theory . in fact , one of the time functions is identical to the melnikov function that is used to measure the distance , at leading order in a small parameter , between the two invariant manifolds that define the lagrangian lobes . we contrast the similarities and differences between the time and lagrangian lobe dynamics in detail . an application of the time method is carried out for inter - gyre transport in the wind - driven oceanic circulation model and a comparison with the lagrangian transport theory is made . and eulerian transport , lagrangian transport , mean - eddy interaction , dynamical systems approach , wind - driven ocean circulation 47.10.fg , 47.11.st , 47.27.ed , 47.51.+a , 92.05.-x , 92.10.a- , 92.10.ab , 92.10.ah 92.10.ak , 92.10.lq , 92.10.ty , 92.60.bh

1004.1617

i

the rotational spectra of many commonly observed molecules such as hcn , hnc , nh@xmath0 , n@xmath1h@xmath2 , and c@xmath3o exhibit hyperfine structure from the splitting of the rotational energy levels by electric quadrupole and magnetic dipole interactions induced by the nuclear moments of atoms such as n or @xmath3o with non - zero spin . hyperfine lines reduce the effective optical depth of the rotational transition by spreading the emission out over a wider bandwidth . because estimates of the density , temperature , and molecular abundance depend on the optical depth , the hyperfine structure should be taken into account in analyzing spectral line observations properly treated , the hyperfine structure is quite useful . the observed relative intensities of pairs of hyperfine lines constrain the optical depth independently of the molecular abundance and independently of the spatial coupling of the telescope beam with the cloud structure ( beam filling factor ) . in contrast , optical depth determination from the brightness ratios of spectral lines of isotopologues such as @xmath5co and @xmath6co requires knowledge of the isotopic abundance ratios , and furthermore the lines may be at sufficiently different frequencies that the observing beam may be differently coupled to the structure of the cloud . numerical techniques such as @xmath4iteration that alternately solve the equations of statistical equilibrium to determine the level populations and the equation of radiative transfer to determine the mean radiation field are able to predict line intensities over a broad range of conditions including varying temperature and density and non - lte excitation . however , these calculations require knowledge of both the radiative and collisional rates for all transitions . this presents a problem in the case of the hyperfine lines . for most molecules , the radiative rates , einstein @xmath7 , are known for all the transitions including hyperfine transitions , but the collisional rates are known only as the net rates between rotational levels . these net rates represent the weighted sum of the rates of all the individual hyperfine transitions between the rotational levels . collisional rates between the individual hyperfine levels themselves have been calculated for only three molecules : hcn ( monteiro & stutzki 1986 ) , nh@xmath0 ( chen , zhang & zhou 1998 ) , and n@xmath1h@xmath2 ( daniel et al 2005 ) , and even then for only a limited number of hyperfine levels . two approximations have been suggested for modeling the emission from molecules with unknown hyperfine collisional rates . the first approximation , `` hyperfine statistical equilibrium '' ( _ hse _ ) , assumes that the the hyperfine levels within each rotational level are populated in proportion to their statistical weights @xcite . the second approximation , the _ proportional _ approximation , assumes that the collisional rates between the individual hyperfine levels are proportional to the total rate between their rotational levels and the statistical degeneracy of the final hyperfine level of the transition @xcite . in this paper , we discuss and evaluate these two approximations , and compute sample n@xmath1h@xmath2 spectra from each method . because the collisional rates for the hyperfine transitions of n@xmath1h@xmath2are known @xcite we can compare n@xmath1h@xmath2 spectra produced using the approximate collisional rates of the _ proportional _ method against spectra produced using the exact " rates determined from the numerical quantum mechanical calculations . we also show how the collisional rates for the elastic ( @xmath8 ) rotational transitions may be determined by extrapolation from the inelastic rates . these elastic rates are required in the _ proportional _ approximation in order to determine the collisional rates for hyperfine transitions within the same rotational state . however , the elastic rates are generally not included in compilations of calculated rate coefficients .

in this paper we discuss two approximate methods previously suggested for modeling hyperfine spectral line emission for molecules whose collisional transitions rates between hyperfine levels are unknown . hyperfine structure is seen in the rotational spectra of many commonly observed molecules such as hcn , hnc , nh , nh , and co . the intensities of these spectral lines can be modeled by numerical techniques such asiteration that alternately solve the equations of statistical equilibrium and the equation of radiative transfer . however , these calculations require knowledge of both the radiative and collisional rates for all transitions . for most commonly observed radio frequency spectral lines , only the net collisional rates between rotational levels are known . for such cases , two approximate methods have been suggested . the first method , hyperfine statistical equilibrium ( _ hse _ ) , distributes the hyperfine level populations according to their statistical weight , but allows the population of the rotational states to depart from local thermodynamic equilibrium ( lte ) . the second method , the _ proportional _ method approximates the collision rates between the hyperfine levels as fractions of the net rotational rate apportioned according to the statistical degeneracy of the final hyperfine levels . we compare simulations of nh hyperfine lines made with approximate and more exact rates and find that satisfactory results are obtained .

in this paper we discuss two approximate methods previously suggested for modeling hyperfine spectral line emission for molecules whose collisional transitions rates between hyperfine levels are unknown . hyperfine structure is seen in the rotational spectra of many commonly observed molecules such as hcn , hnc , nh , nh , and co . the intensities of these spectral lines can be modeled by numerical techniques such asiteration that alternately solve the equations of statistical equilibrium and the equation of radiative transfer . however , these calculations require knowledge of both the radiative and collisional rates for all transitions . for most commonly observed radio frequency spectral lines , only the net collisional rates between rotational levels are known . for such cases , two approximate methods have been suggested . the first method , hyperfine statistical equilibrium ( _ hse _ ) , distributes the hyperfine level populations according to their statistical weight , but allows the population of the rotational states to depart from local thermodynamic equilibrium ( lte ) . the second method , the _ proportional _ method approximates the collision rates between the hyperfine levels as fractions of the net rotational rate apportioned according to the statistical degeneracy of the final hyperfine levels . the second method is able to model non - lte hyperfine emission . we compare simulations of nh hyperfine lines made with approximate and more exact rates and find that satisfactory results are obtained .

1008.1984

i

the _ very _ center of our galaxy houses the variable source named sgr a * , first discovered in the radio as a compact source @xcite . the fact that this source is motionless ( to better than 1 km / s ) at the dynamical center of the galaxy @xcite , and its coincidence with the common focus of elliptical orbits of stars tracked over the last decade and a half , clearly associates it with a supermassive black hole of @xmath4 @xcite . sgr a * has been detected across the electromagnetic spectrum , at radio , submm , nir and x - ray wavelengths . at nir and x - ray wavelengths @xcite the emission is highly variable ( factors up to @xmath2160 and 27 in the x - ray and nir respectively ; @xcite , this work ) compared to the comparatively steady emission at longer wavelengths . nir peaks are detected more often than in the x - ray ( peaks occur @xmath21 and 4 times a day for x - ray and nir variable emission , respectively ; @xcite ) . some nir flares have been detected without any accompanying x - ray flare @xcite . however , when both nir and x - ray exhibit increases in emission , the peaks in emission occur simultaneously ( e.g. , * ? ? ? * ; * ? ? ? the near - infrared lightcurves from sgr a * exhibit @xmath51 hour long increases in emission that are often called ` flares ' in the literature . a number of these have exhibited very suggestive substructural features with timescales of @xmath6 minutes @xcite , possibly quasi - periodic oscillations ( qpos ) . however the existence of qpos and even the use of the term flare to describe the nir variability of sgr a * has been questioned by @xcite ( see also @xcite and @xcite ) who argue that there is no true quasi - periodicity , just a variability process with a featureless red noise power spectrum ( e.g. a power spectrum @xmath7 where @xmath8 is frequency ) . a stochastic source with a red noise power spectrum has higher variability at longer timescales and could potentially be responsible for the structures on longer timescales seen in the real lightcurves . the authors suggest that apparent flare peaks may simply be the highest observed flux excursions in such a purely stochastic source and are not isolated events . a main reason for the two rather contrasting interpretations of the variable emission from sgr a * has been that the nature of the faint emission from sgr a * and its relationship to the high flux emission is uncertain . the nir emission from the galactic center is dominated by the central cluster of bright stars , and adaptive optics at 8-meter class telescopes is required in order to separate the faint source sgr a * from the closest s - stars ( even at this resolution sgr a * is still on occasion confused with a relatively bright star ) . additional , faint stars may be present very close to sgr a * which have not yet been tracked and identified as stars from astrometric monitoring programs ( e.g. * ? ? ? * ) . while the dramatic high flux variability can be unambiguously attributed to the black hole , when a faint source is detected at the position of sgr a * , it is not necessarily clear that the source is sgr a * , faint stars , or a combination of both . accordingly , it is not clear whether sgr a * continues to emit at all at low fluxes , whether it exhibits a ` quiescent state ' ( a non- or weakly active low state ) , or whether the low flux emission continues to vary constantly with the same statistical properties as the high flux emission . cccccc & & & & & + _ brightest states _ & & _ peak fluxes _ & & & + & & & & & + gen+03 & 15 jun 2003 & 13.2 ( 10.5 + 2.7 ) & 1.10 & 0.76 & 11.0 + & & & & & + gen+03 & 16 jun 2003 & 10 ( 7.3 + 2.7 ) & 1.10 & 0.76 & 8.4 + & & & & & + mey+07 & 6 oct 2003 & 22@xmath9 & 1.2 & 0.76 & 20.1 + & & & & & + tri+07/mey+06 & 31 may 2006 & 16/23 ( + s17 ) & - & - & ( 16.7 ) 13.5@xmath10 + & & & & & + hor+07 & 31 jul 2005 & 11.6 & @xmath51.06@xmath11 & 0.52 & 6.4 + & 2 may 2006 & 26.8 & & & 14.8 + & 17 jul 2006 & 6.8 & & & 3.7 + & & & & & + do+08 & 3 may 2006 & 0.8 & @xmath51.06@xmath11 & - & 8.5 + & 20 jun 2006 & 0.65 & & & 6.9 + & 21 jun 2006 & 0.4 & & & 4.2 + & 17 jul 2006 & 0.3 & & & 3.2 + & 18 may 2007 & 0.6 & & & 6.4 + & 12 aug 2007 & 0.2 & & & 2.1 + & & & & & + eck+08 & 15 may 2007 & 24 ( + s17 ) & 0.76 & 0.76 & ( 13.9 ) 10.7 + & & & & & + this work & 5 aug 2008 & 30.7 ( + s17 ) & - & - & ( 30.7 ) 27.5 + & & & & & + & & & & & + _ faintest states _ & & & & & + & & & & & + hor+02 & 9 may 2001 & @xmath12 mjy & @xmath51.06@xmath11 & - & @xmath13 + & & ( upper limit , not dered ) & & & + & & & & & + sch+02 & 5 jun 2006 & @xmath14 mjy & ? & 0.76 & @xmath15 + & & & & & + do+08 & 2006 : may 3 , & @xmath16 mjy & + & jun 20 , 21 , jul 17 & & ( median flux , not dered ) & 1.06@xmath17 & - & 2.0 + & 2007 : may 18 , aug 12 & @xmath18 & 1.06 & - & @xmath19 + & & ( faintest flux , not dered ) & & & + & & & & & + sab+10 & 23 sep 2004 & 2.4 mjy & @xmath20 & 0.76 & 1.4 + & & ( upper limit , & & & + & & no stellar contam . ) & & & + & 23 sep 2004 & 0.9 mjy & & & 0.5 + & & ( upper limit , & & & + & & full stellar contam . ) & & & + in addition to this , an unbiased overview of the properties of the near - infrared emission from sgr a * can be difficult to obtain from the published literature because of publication bias ( bright events have individual interest and are often published alone ) . however , some studies have looked at the statistical properties . @xcite and @xcite presented lightcurves and flux distributions for sgr a * for about @xmath511 hours of 1.6@xmath21 m and @xmath532 hours of 1.7@xmath21 m data observed with nicmos on the hubble space telescope ( hst ) . with the resolution of the hst the close stellar sources are not as well separated from sgr a * as with the vlt or keck telescopes , and the stars s17 and s2 overlap with the sgr a * source in these observations . the flux distributions were fitted with a gaussian at low fluxes , which was attributed to the observational noise on constant sources ( the contribution from s2 , s17 and possible quiescent emission ) and a power - law at high fluxes , which was attributed to transient flares . the best fit models implied that sgr a * was active ( above the noise at low levels ) more than 40% of the time . @xcite presented an analysis of six nights of k-band ( and one l-band ) observations at the keck observatory , using an unbiased set of observations taken between 2005 and 2007 . a source at the position of sgr a * was always detected in this dataset , with an estimated maximum 35% contribution from stellar contamination . these authors reported that the source sgr a * was continuously variable , based on the larger variance of sgr a * compared to stars of similar brightness on five of the six k-band observation nights . this was the data set used to investigate timing properties of sgr a * in which it was claimed the data set was consistent with a featureless red noise power spectrum with no quasi - periodicity . however , with a sum duration for the k - band observations of about 12.1 hours , this data set did not sample well the higher fluxes of sgr a * , i.e. the source was relatively faint compared to publications where variable emission with suggestive quasi - periodic structure have been reported ( * ? ? ? * ; * ? ? ? * ; * ? ? ? * for a comparison of ks - band peak emission from the literature see table [ table2 ] in the appendix ) . although the studies of @xcite , @xcite and @xcite have gone some way towards understanding the statistical properties of sgr a * in the near - infrared , there has not yet been a study on a very large , unbiased dataset of the variability of sgr a * where the rare high fluxes are also well sampled . in this paper , we analyse the ks - band flux distribution of sgr a * for the years 2004 - 2009 from 117 observation nights carried out with the vlt in paranal , chile with the aim of seeking the flux - dependent characteristics of the variability of sgr a * at both high and low fluxes . we do this through investigation of the flux distribution of sgr a*. the dataset of this paper is @xmath512 times larger than the data set of @xcite . in order that our results might be compared with other publications , we give a summary in table [ table2 ] ( in the appendix ) of the brightest and/or notable ks / k-band variable emission reported in the literature , as well as the faintest values or upper limits . all are reported in the literature with different calibrations / extinction values , and we have done our best to determine the corrections to scale them to the calibration and extinction values used in this paper . in section [ section_dataset ] we present our observations and the results of using two different methods of photometry : a six year lightcurve ( aperture photometry ) , and the 2009 data set ( psf photometry ) . in section [ section_results ] we present the flux distribution of sgr a * and our results from various model fits to the flux distribution . in section [ section_discussion ] we discuss our results in the context of two variability states for sgr a * : a low - level lognormally varying _ quiescent state _ and sporadic high flux _ flares_. we summarize in section [ section_conclusions ] .

in this paper we examine properties of the variable source sgr a * in the near - infrared ( nir ) using a very extensive ks - band data set from naco / vlt observations taken 2004 to 2009 . the lognormal distribution has a median flux of.1 mjy , but above 5 mjy the flux distribution is significantly flatter ( high flux events are more common ) than expected for the extrapolation of the lognormal distribution to high fluxes . this flare was a factor 27 increase over the median flux of sgr a * , close to double the brightness of the star s2 , and 40% brighter than the next brightest flare ever observed from sgr a*.

in this paper we examine properties of the variable source sgr a * in the near - infrared ( nir ) using a very extensive ks - band data set from naco / vlt observations taken 2004 to 2009 . we investigate the variability of sgr a * with two different photometric methods and analyze its flux distribution . we find sgr a * is continuously emitting and continuously variable in the near - infrared , with some variability occurring on timescales as long as weeks . the flux distribution can be described by a lognormal distribution at low intrinsic fluxes ( mjy , dereddened with ) . the lognormal distribution has a median flux of.1 mjy , but above 5 mjy the flux distribution is significantly flatter ( high flux events are more common ) than expected for the extrapolation of the lognormal distribution to high fluxes . we make a general identification of the low level emission above 5 mjy as _ flaring _ emission and of the low level emission as the _ quiescent state_. we also report here the brightest ks - band flare ever observed ( from august 5th , 2008 ) which reached an intrinsic ks - band flux of 27.5 mjy ( ) . this flare was a factor 27 increase over the median flux of sgr a * , close to double the brightness of the star s2 , and 40% brighter than the next brightest flare ever observed from sgr a*.

0705.0466

i

the deregulation of energy markets has given rise to various families of contracts . many of them appear as some derivative products whose underlying is some tradable futures ( day - ahead , etc ) on gas or electricity ( see @xcite for an introduction ) . the class of swing options has been paid a special attention in the literature , because it includes many of these derivative products . a common feature to all these options is that they introduce some risk sharing between a producer and a trader , of gas or electricity for example . from a probabilistic viewpoint , they appear as some stochastic control problems modeling multiple optimal stopping problems ( the control variable is the purchased quantity of energy ) ; see @xmath0 @xcite in a continuous time setting . gas storage contracts ( see @xcite , @xcite ) or electricity supply agreements ( see @xcite , @xcite ) are examples of such swing options . indeed , energy supply contracts are one simple and important example of such swing options that will be deeply investigated in this paper ( see below , see also @xcite for an introduction ) . it is worth mentioning that this kind of contracts are slightly different from multiple exercises american options as considered in @xcite for example . in our setting the volumetric constraints play a key role and thus , the flexibility is not restricted to time decisions , but also has to take into account volumes management . designing efficient numerical procedures for the pricing of swing option contracts remains a very challenging question as can be expected from a possibly multi - dimensional stochastic control problem subject to various constraints ( due to the physical properties of the assets like in storage contracts ) . most recent approaches developed in mathematical finance , especially for the pricing of american options , have been adapted and transposed to the swing framework : tree ( or forest " ) algorithms in the pioneering work @xcite , least squares regression mc methods ( see @xcite ) , pde s numerical methods ( finite elements , see @xcite ) . the aim of this paper is to deeply investigate an old question , namely to elucidate the structure of the optimal control in supply contracts ( with firm constraints ) and how it impacts the numerical methods of pricing . we will provide in a quite general ( and abstract ) setting some natural " ( and simple ) conditions involving the local and global purchased volume constraints to ensure the existence of _ bang - bang optimal strategy _ ( such controls usually do not exist ) . it is possible to design _ a priori _ the contract so that their parameters satisfy these conditions . to our knowledge very few theoretical results have been established so far on this problem ( see however @xcite in a markovian framework for contracts with penalized constraints and @xcite , also in a markovian framework ) . this first result of the paper not only enlightens the understanding of the management of a swing contract : it also has some deep repercussions on the numerical methods to price it . as a matter of fact , taking advantage of the existence of a bang - bang optimal strategy , we propose and analyze in details ( when the underlying asset has a markovian dynamics ) a quantized dynamic programming procedure to price any swing options whose volume constraints satisfy the bang - bang " assumption . furthermore some _ a priori _ error bounds are established . this procedure turns out to be dramatically efficient , as emphasized in the companion paper @xcite where the method is extensively tested with assets having multi - factor gaussian underlying dynamics and compared to the least squares regression method . [ [ the - abstract - swing - contract - with - firm - constraints ] ] the abstract swing contract with firm constraints + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + the holder of a supply contract has the right to purchase periodically ( daily , monthly , etc ) an amount of energy at a given unitary price . this amount of energy is subject to some lower and upper local " constraints . the total amount of energy purchased at the end of the contract is also subject to a global " constraint . given dynamics on the energy price process , the problem is to evaluate the price of such a contract , at time @xmath1 when it is emitted and during its whole life up to its maturity . to be precise , the owner of the contract is allowed to purchase at times @xmath2 , @xmath3 a quantity @xmath4 of energy at a unitary _ strike price _ @xmath5 . at every date @xmath2 , the purchased quantity @xmath4 is subject to the firm _ local " constraint _ , @xmath6 whereas the global purchased quantity @xmath7 is subject to the ( firm ) global constraint @xmath8\qquad ( 0<q_{\min } \le q_{\max}<+\infty).\ ] ] the strike price process @xmath9 can be either deterministic ( even constant ) or stochastic , @xmath0 indexed on past values of other commodities ( oil , etc ) . usually , on energy markets the price is known through future contracts @xmath10 where @xmath11 denotes the price at time @xmath12 of the forward contract delivered at maturity @xmath13 . the available data at time @xmath14 are @xmath15 ( in real markets this is of course not a continuum ) . the underlying asset price process , temporarily denoted @xmath16 , is often the so - called day - ahead " contract @xmath17 which is a tradable instrument or the spot price @xmath18 which is not . all the decisions about the contract need to be adapted to the filtration of @xmath19 @xmath20 @xmath21 , @xmath3 ( with @xmath22 ) . this means that the price of such a contract is given at any time @xmath23 , by @xmath24,j = k,\ldots , n-1 , \sum_{j = k}^{n-1}q_j\!\in [ q^k_{\min } , q^k_{\max}]\right\}\end{aligned}\ ] ] where @xmath25 , @xmath26 denote the _ residual global constraints _ and @xmath27 denotes the ( deterministic ) interest rate . this pricing problem clearly appears as a stochastic control problem . in the pioneering work by @xcite , this type of contract was computed by using some forests of ( multinomial ) trees . a natural variant , at least for numerical purpose , is to consider a penalized version of this stochastic control . thus , in @xcite , a penalization @xmath28 with @xmath29 is added ( @xmath30 is negative outside @xmath31 $ ] and zero inside ) . as concerns more sophisticated contracts ( like storages ) , the holder of the contract receive a quantity @xmath32 when deciding @xmath33 . when dealing with gas this is due to the storing constraints since injecting or withdrawing gas from its storing units induce fixed costs ( and physical constraints ( pressure , etc ) ) . as concerns the underlying asset dynamics , it is commonly shared in finance to assume that the traded asset has a markovian dynamics ( or is a component of a markov process like with stochastic volatility models ) . the dynamics of physical assets for many reasons ( some of them simply coming from history ) are often modeled using some more deeply non - markovian models like long memory processes , etc . all these specific features of energy derivatives suggest to tackle the above pricing problem in a rather general framework , trying to avoid as long as possible to call upon markov properties . this is what we do in the first part of the paper where the general setting of a swing option defined by an abstract sequence of @xmath34-adapted payoffs is deeply investigated as a function of its global constraints @xmath35 ( when the local constraints are normalized @xmath20 @xmath33 is @xmath36$]-valued for every @xmath37 ) . we show that this premium is a concave , piecewise affine , function of the global constraints , affine on triangles of the @xmath38 , @xmath39 , @xmath40 and @xmath41 , @xmath42 , @xmath43 . we also show that for integral valued global constraints , the optimal controls are always bang - bang @xmath20 the _ a priori _ @xmath36$]-valued optimal purchased quantities @xmath44 are in fact always equal to @xmath14 or @xmath45 . such a result can be extended in some way to any couple of global constraints when all the payoffs are nonnegative . then , when there is an underlying markov structure process " , we propose an optimal quantization based on numerical approach to price efficiently swing options . this markov structure process " can be the underlying traded asset itself or a higher dimensional hidden markov process : such a framework comes out in case of multi - factor processes having some long - memory properties . optimal quantization was first introduced as a numerical method to solve nonlinear problem arising in mathematical finance in a series of papers @xcite devoted to the pricing and hedging of american style multi - asset options . it has also been applied to stochastic control problem , namely portfolio optimization in @xcite . the purely numerical aspects induced by optimal quantization , with a special emphasis on the gaussian distribution , have been investigated in @xcite . see @xcite for a survey on numerical application of optimal quantization to finance . for other applications ( to automatic classification , clustering , etc ) , see also @xcite . in this paper , we propose a quantized backward dynamic programming to approximate the premium of a swing contract . we analyze the rate of convergence of this algorithm and provide some _ a priori _ error bounds in terms of quantization errors . we illustrate the method by computing the whole graph of the premium viewed as a function of the global constraints , combining the affine property of the premium and the quantized algorithm in toy model " : the future prices of gas are modeled by a two factor gaussian model . an extensive study of the pricing method by optimal quantization is carried out from both a theoretical and numerical point of view in @xcite . the paper is organized as follows . in the section below we detail the decomposition of swing options into a swap contract and a normalized swing option . in section [ abstractswing ] , we precisely describe our abstract setting for normalized swing options with firm constraints and the variable of interest ( global constraints , local constraints , etc ) . in section [ progdynabst ] , we establish the dynamic programming formula satisfied in full generality by the premium as a function of the global constraints ( this unifies the similar results obtained in markov settings , see @xcite , @xcite , etc ) and we show this is a concave function with respect to the global constraints . then , in section [ piecaffine ] , we prove in our abstract framework that the premium function is piecewise affine and that the optimal purchased quantities satisfy a @xmath14-@xmath45 " or bang - bang principle ( theorem [ main ] ) . a special attention is paid to the @xmath46-period model which provides an intuitive interpretation of the results . in section [ swingquant ] , after some short background on quantization and its optimization , we propose a quantization based backward dynamic programming formula as a numerical method to solve the swing pricing problem . then we provide some error bounds for the procedure depending on the quantization error induced by the quantization of the markov structure process . @xmath47 the lipschitz coefficient of a function @xmath48 is defined by @xmath49_{\rm lip}:=\sup_{x\neq y}\frac{|f(x)-f(y)|}{|x - y|}\le + \infty$ ] . the coefficient @xmath50_{\rm lip}$ ] is finite if and only if @xmath51 lipschitz continuous . @xmath47 the canonical euclidean norm on @xmath52 will be denoted @xmath53 .

we show , for a fully general payoff process , that the premium , solution to a stochastic control problem , is concave and piecewise affine as a function of the global constraints of the contract . when the payoff process is driven by an underlying markov process , we propose a quantization based recursive backward procedure to price these contracts . a priori error bounds are established , uniformly with respect to the global constraints . _

in this paper we investigate a class of swing options with firm constraints in view of the modeling of supply agreements . we show , for a fully general payoff process , that the premium , solution to a stochastic control problem , is concave and piecewise affine as a function of the global constraints of the contract . the existence of bang - bang optimal controls is established for a set of constraints which generates by affinity the whole premium function . when the payoff process is driven by an underlying markov process , we propose a quantization based recursive backward procedure to price these contracts . a priori error bounds are established , uniformly with respect to the global constraints . _ key words : swing option , stochastic control , optimal quantization , energy . _

math0509603

i

in this paper we give a multifractal analysis for stern brocot intervals , continued fractions and certain diophantine growth rates . we apply and extend the multifractal formalism for average growth rates of @xcite to obtain a complete multifractal description of two dynamical systems originating from the set of real numbers . recall that the process of writing an element @xmath0 of the unit interval in its regular continued fraction expansion @xmath1=\cfrac{1}{a_{1}(x)+\cfrac{1}{a_{2}(x)+\cfrac{1}{a_{3}(x)+\cdots}}}\ ] ] can be represented either by a uniformly hyperbolic dynamical system which is based on an infinite alphabet and hence has infinite topological entropy , or by a non - uniformly hyperbolic dynamical system based on a finite alphabet and having finite topological entropy . obviously , for these two systems the standard theory of multifractals ( see e.g. @xcite ) does not apply , and therefore it is an interesting task to give a multifractal analysis for these two number theoretical dynamical systems . there is a well known result which gives some information in the generic situation , that is for a set of full @xmath2-dimensional lebesgue measure @xmath3 . namely with @xmath4 $ ] referring to the @xmath5-th approximant of @xmath0 , we have for @xmath3-almost every @xmath6 , @xmath7 note that by employing the analogy between regular continued fraction expansions of real numbers and geodesics on the modular surface , the number @xmath8 can be interpreted as the hyperbolic length associated with the approximant @xmath9 . also , the parameter @xmath5 represents the word length associated with @xmath9 with respect to the dynamical system on the infinite alphabet , whereas @xmath10 can be interpreted as the word length associated with @xmath9 with respect to the dynamical system on the finite alphabet . there are two classical results by khintchin and lvy @xcite , @xcite , @xcite , @xcite which allow a closer inspection of the limit @xmath11 . that is , for @xmath3-almost every @xmath6 we have , with @xmath12 , @xmath13 clearly , dividing the sequence in @xmath14 by the sequence in @xmath15 leads to the sequence in @xmath11 . therefore , if we define the level sets @xmath16 then these classical results by lvy and khintchin imply for the hausdorff dimensions ( @xmath17 ) of these level sets @xmath18 a natural question to ask is what happens to this relation between these hausdorff dimensions for prescribed non - generic limit behavior . our first main results in this paper will give an answer to this question . namely , with @xmath19 referring to the golden mean , we show that for each @xmath20 $ ] there exists a number @xmath21 such that , with the convention @xmath22 and @xmath23 , @xmath24 furthermore , for the dimension function @xmath25 given by @xmath26 we show that @xmath25 can be expressed explicitly in terms of the legendre transform @xmath27 of a certain pressure function @xmath28 , referred to as the stern brocot pressure . for the function @xmath28 we obtain the result that it is real - analytic on the interval @xmath29 and vanishes on the complement of this interval . we then show that the dimension function @xmath25 is continuous and strictly decreasing on @xmath30 $ ] , that it vanishes outside the interval @xmath31 , and that for @xmath20 $ ] we have @xmath32 before we state the main theorems , let us recall the following classical construction of stern brocot intervals ( cf . @xcite , @xcite ) . for each @xmath33 , the elements of the @xmath5-th member of the stern brocot sequence @xmath34 are defined recursively as follows . * @xmath35 and @xmath36 ; * @xmath37 for @xmath38 ; * @xmath39 , for @xmath40 . with this ordering of the rationals in @xmath41 $ ] we define the set @xmath42 of stern brocot intervals of order @xmath5 by @xmath43 clearly , for each @xmath33 we have that @xmath42 represents a partition of the interval @xmath44 . the first members in this sequence of sets are the following , and it should be clear how to continue this list using the well known method of mediants . @xmath45 as already mentioned above , our multifractal analysis will make use of the stern brocot pressure function @xmath28 . this function is defined for @xmath46 by @xmath47 in here , @xmath48 refers to the euclidean length of the interval @xmath49 . we will see that @xmath28 is a well defined convex function ( cf . proposition [ pro : analyticpropertiesp ] ) . also , note that we immediately have that@xmath50 the following theorem gives the first main results of this paper . in here , @xmath27 refers to the legendre transform of @xmath28 , given for @xmath51 by @xmath52 . [ thm : main ] _ ( see fig . [ cap : the - stern brocot - pressure ] ) _ 1 . the stern brocot pressure @xmath28 is convex , non - increasing and differentiable throughout @xmath53 . furthermore , @xmath28 is real analytic on the interval @xmath29 and is equal to @xmath54 on @xmath55 . 2 . for every @xmath20 $ ] , there exist @xmath56 and @xmath21 related by @xmath57 such that , with the conventions @xmath58 and @xmath22 , @xmath59 furthermore , the dimension function @xmath25 is continuous and strictly decreasing on @xmath30 $ ] , it vanishes outside the interval @xmath31 , and for @xmath20 $ ] we have@xmath60 where @xmath61 . also , for the left derivative of @xmath25 at @xmath62 we have @xmath63 . theorem [ thm : main ] has some interesting implications for other canonical level sets . in order to state these , recall that the elements of @xmath42 cover the interval @xmath44 without overlap . therefore , for each @xmath6 and @xmath64 there exists a unique stern brocot interval @xmath65 containing @xmath0 . the interval @xmath66 is covered by two neighbouring intervals from @xmath67 , a left and a right subinterval . if @xmath68 is the left of these then we encode this event by the letter @xmath69 , otherwise we encode it by the letter @xmath70 . in this way every @xmath6 can be described by a unique sequence of nested stern brocot intervals of any order that contain @xmath0 , and therefore by a unique infinite word in the alphabet @xmath71 . it is well known that this type of coding is canonically associated with the continued fraction expansion of @xmath0 ( see section 2 for the details ) . in particular , this allows to relate the level sets @xmath72 and @xmath73 to level sets given by means of the stern brocot growth rate @xmath74 of the nested sequences @xmath75 , and to level sets of certain diophantine growth rates @xmath76 and @xmath77 . these growth rates are given by ( assuming that these limits exist ) @xmath78 @xmath79 [ thm:2 ] let @xmath80 be given . if one of the limits in @xmath81 exists then also the other two do exist , and @xmath82 furthermore , @xmath83 exists if and only if @xmath84 exists , and if one of these exists then @xmath85 by theorem [ thm : main ] , it therefore follows that for each @xmath20 $ ] , @xmath86 note that the level sets @xmath87 have already been under consideration in @xcite . there they were introduced in terms of homological growth rates of hyperbolic geodesics ( see remark [ rem : vergleich ] ( 2 ) ) . clearly , theorem [ thm : main ] and proposition [ thm:2 ] consider the dynamical system associated with the finite alphabet , a system which is closely related to the farey map . now , our second main result gives a multifractal analysis for the system based on the infinite alphabet , and this system is closely related to the gauss map . in here , the relevant pressure function is the _ diophantine pressure _ @xmath88 , which is given by @xmath89 } q_{k}\left(\left[a_{1},\ldots , a_{k}\right]\right)^{-2\theta } , \textrm { for } \theta>\frac{1}{2}.\ ] ] we remark that a very detailed analysis of the function @xmath88 can be found in @xcite . our second main result is the following . [ thm : main3 ] _ ( see fig . [ cap : diaophantine - pressure ] ) _ the function @xmath88 has a singularity at @xmath90 , and @xmath88 is decreasing , convex and real - analytic on @xmath91 . furthermore , for @xmath92 we have@xmath93 also , the dimension function @xmath94 is real - analytic on @xmath95 , it is increasing on @xmath96 $ ] and decreasing on @xmath97 . in particular , @xmath94 has a point of inflexion at some point greater than @xmath98 and a unique maximum equal to @xmath2 at @xmath98 . additionally , @xmath99 , @xmath100 and @xmath101 . the paper is organized as follows . in section 2 we first recall two ways of coding elements of the unit interval . one is based on a finite alphabet and the other on an infinite alphabet , and both are defined in terms of the modular group . these codings are canonically related to regular continued fraction expansions , and we end the section by commenting on a 1 - 1 correspondence between stern brocot sequences and finite continued fraction expansions . in section 3 we introduce certain cocycles which are relevant in our multifractal analysis . in particular , we give various estimates relating these cocycles with the geometry of the modular codings and with the sizes of the stern brocot intervals . this will then enable us to prove the first part of proposition [ thm:2 ] . section 4 is devoted to the discussion of several aspects of the stern brocot pressure and its legendre transform . in section 5 we give the proof of theorem [ thm : main ] , which we have split into the parts _ the lower bound _ , _ the upper bound _ , and _ discussion of boundary points of the spectrum_. finally , in section 6 we give the proof of theorem [ thm : main3 ] by showing how to adapt the multifractal formalism developed in section 4 and 5 to the situation here . throughout , we shall use the notation @xmath102 to denote that for two non - negative functions @xmath103 and @xmath104 we have that @xmath105 is uniformly bounded away from infinity . if @xmath102 and @xmath106 , then we write @xmath107 . one immediately verifies that the results of theorem [ thm : main ] and proposition [ thm:2 ] can be expressed in terms of the farey map @xmath108 acting on @xmath41 $ ] , and then @xmath25 represents the multifractal spectrum of the measure of maximal entropy ( see e.g. @xcite ) . likewise , the results of theorem [ thm : main3 ] can be written in terms of the gauss map @xmath109 , and then in this terminology @xmath110 describes the lyapunov spectrum of @xmath109 . for the definitions of @xmath108 and @xmath109 and for a discussion of their relationship we refer to remark [ rem : fg ] . since the theory of multifractals started through essays of mandelbrot @xcite @xcite , frisch and parisi @xcite , and halsey et al . @xcite , there has been a steady increase of the literature on multifractals and calculations of specific multifractal spectra . for a comprehensive account on the mathematical work we refer to @xcite and @xcite . essays which are closely related to the work on multifractal number theory in this paper are for instance @xcite , @xcite , @xcite , @xcite , @xcite , @xcite and @xcite . we remark that brief sketches of some parts of theorem [ thm : main3 ] have already been given in @xcite . the results there do for instance not cover the boundary points of the spectra . furthermore , note that for the @xmath77spectrum partial results have been established in @xcite ( corollary 2 ) .

in this paper we obtain multifractal generalizations of classical results by lvy and khintchin in metrical diophantine approximations and measure theory of continued fractions . we give a complete multifractal analysis for stern brocot intervals , for continued fractions and for certain diophantine growth rates . in particular , we give detailed discussions of two multifractal spectra closely related to the farey map and to the gauss map .

in this paper we obtain multifractal generalizations of classical results by lvy and khintchin in metrical diophantine approximations and measure theory of continued fractions . we give a complete multifractal analysis for stern brocot intervals , for continued fractions and for certain diophantine growth rates . in particular , we give detailed discussions of two multifractal spectra closely related to the farey map and to the gauss map .

1501.07896

c

while the word chaos is widely used in science and mathematics , there are a variety of ways of defining it . thus , for this 25th anniversary issue of the journal chaos , we are motivated to review issues that arise when attempting to formulate a generally applicable definition of chaos , and to advocate a particular entropy - based definition that seems to us to be especially apt . we also relate our proposed definition to previous definitions . intuitively , perhaps the two most prominent ( not necessarily independent ) attributes of what scientists commonly think of as chaos are the presence of complex orbit structure and extreme sensitivity of orbits to small perturbations . indeed , in the paper by li and yorke@xcite where the term chaos was introduced in its now widely accepted nonlinear dynamics context , the term was motivated by the simultaneous presence of unstable periodic orbits of all periods , as well as an uncountable infinity of non - periodic orbits . thus , li and yorke s introduction of this terminology was motivated by the chaos attribute of complex orbit structure . on the other hand , lorenz@xcite was concerned with weather forecasting and accordingly focused on the chaos attribute of temporally exponential increase of the sensitivity of orbit locations to small initial perturbations . as we will discuss , these two attributes can be viewed as `` two sides of the same coin '' . we think of a definition of chaos as being `` good '' if it conforms to common intuitive notions of chaos ( such as complex orbit structure and orbit sensitivity ) and , at the same time , has the following three desirable features : * _ generality _ : the definition should work for almost all the examples that typical readers of this journal are likely to judge as chaotic . * _ simplicity _ : the definition should be fairly concise and not too technical . * _ computability _ : the definition should allow a practical , straightforward computational implementation for discerning the existence of chaos in a model . considering the issue of generality , one would like a definition of chaos to be applicable not only to attractors , but also to non - attracting sets , often called repellers . with respect to chaotic repellers@xcite , we note that they are central to the physically relevant topics of fractal basin boundaries@xcite , chaotic transients , and chaotic scattering@xcite , occurring , for example in fluid dynamics@xcite , celestial mechanics@xcite , chemistry@xcite , and atomic physics@xcite . furthermore , again considering the issue of generality , due to their common occurrence in applications , we desire that our definition of chaos be applicable to non - autonomous dynamical systems ( i.e. , systems that are externally forced by time dependent inputs ) , including external inputs that are temporally quasi - periodic@xcite , stochastic@xcite , or are themselves chaotic . here physical examples include quasi - periodic forcing of atmospheric jets@xcite , quasi - periodic forcing of stellar luminosity variation by two superposed stellar modal oscillations@xcite , advective transport in fluids with temporally and spatially irregular flow fields@xcite , and phase synchronism of chaos by noisy or chaotic drives@xcite . we emphasize that , when considering externally forced systems , we are interested in identifying chaos in the response of the system to a particular realization of the forcing , not in characterizing whether the forcing is chaotic . an important point for consideration of non - periodically forced chaotic systems is that the notion of a compact invariant set , which is typically used in definitions of chaos for autonomous systems ( including poincar maps of periodically forced systems ) , may not be appropriate or convenient for situations with non - periodic forcing . furthermore , in practice , it may be difficult to locate or detect an invariant set that is not an attractor . thus , rather than defining chaos for an invariant set , we will instead consider a notion of chaos for the dynamics within any given bounded positive - volume subset @xmath0 of the state space . we call such a set @xmath0 a _ restraining region_. for autonomous systems , chaos for an invariant set can be detected by taking @xmath0 to be a neighborhood of the desired invariant set . in our opinion , the currently most satisfactory way of defining chaos for autonomous systems is by the existence of positive topological entropy or metric entropy . we note , however , that the standard definitions of these entropies are quite difficult to straightforwardly implement in a numerical procedure . in addition , while generalizations to the original definitions of topological and metric entropy have been proposed , we view it as desirable to have a relatively simple definition that is applicable very broadly . however , we do not address here the question of identifying chaos in experimental data , which presents additional challenges , especially in the cases of non - attracting sets and externally forced systems . motivated by the considerations above , in sec . [ sec2 ] we introduce and discuss the definition of an alternate entropy quantity that we call `` expansion entropy '' . the expansion entropy of an @xmath1-dimensional dynamical system on a restraining region @xmath0 is the difference between two asymptotic exponential rates : first , the maximum over @xmath2 of the rate at which the system expands @xmath3-dimensional volume within @xmath0 ; and second , the rate at which @xmath1-dimensional volume leaves @xmath0 ( this rate is @xmath4 for an invariant set ) . _ we define chaos as the existence of positive expansion entropy on a given restraining region . _ expansion entropy generalizes ( to nonautonomous systems and noninvariant restraining regions ) a quantity that was formulated by sacksteder and shub@xcite in the case of an autonomous system on a compact manifold . in this restricted case , by the results of kozlovski@xcite , expansion entropy is equal to topological entropy for infinitely differentiable maps . in sec . [ sec2.6 ] we present examples of the application of our definition of expansion entropy to various systems , and also provide illustrative numerical evaluations of expansion entropy for some of these examples . section [ sec3 ] discusses topological entropy and previous work on computation of this quantity . section [ sec4 ] discusses issues that arise in previous non - entropy - based definitions of chaos .

in this paper we propose , discuss and illustrate a computationally feasible definition of chaos which can be applied very generally to situations that are commonly encountered , including attractors , repellers and non - periodically forced systems . this definition is based on an entropy - like quantity , which we call `` expansion entropy '' , and we define chaos as occurring when this quantity is positive . we also present example illustrations , discuss computational implementations , and point out issues arising from attempts at giving definitions of chaos that are not entropy - based . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ toward the end of the 19th century , poincar demonstrated the occurrence of extremely complicated orbits in the newtonian dynamics of three gravitationally attracting bodies . this complexity is now called chaos and has received a vast amount of attention since poincar s early discovery . in spite of this abundant past and current work , there is still no broadly applicable , convenient , generally accepted definition of the term chaos . in this paper , we advocate a particular entropy - based definition that appears to be very simple , while , at the same time , is readily accessible to numerical computation , and can be very generally applied to a variety of often - encountered situations , including attractors , repellers , and non - periodically forced systems . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

in this paper we propose , discuss and illustrate a computationally feasible definition of chaos which can be applied very generally to situations that are commonly encountered , including attractors , repellers and non - periodically forced systems . this definition is based on an entropy - like quantity , which we call `` expansion entropy '' , and we define chaos as occurring when this quantity is positive . we relate and compare expansion entropy to the well - known concept of topological entropy , to which it is equivalent under appropriate conditions . we also present example illustrations , discuss computational implementations , and point out issues arising from attempts at giving definitions of chaos that are not entropy - based . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ toward the end of the 19th century , poincar demonstrated the occurrence of extremely complicated orbits in the newtonian dynamics of three gravitationally attracting bodies . this complexity is now called chaos and has received a vast amount of attention since poincar s early discovery . in spite of this abundant past and current work , there is still no broadly applicable , convenient , generally accepted definition of the term chaos . in this paper , we advocate a particular entropy - based definition that appears to be very simple , while , at the same time , is readily accessible to numerical computation , and can be very generally applied to a variety of often - encountered situations , including attractors , repellers , and non - periodically forced systems . we also review and compare various previous definitions of chaos . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

hep-ph0201075

i

nowadays the standard model ( sm ) of elementary particle physics is well established . some parts of it ( e.g. the properties of the @xmath8 boson ) have been tested to an accuracy far below the percent level mostly at the cern large - electron - positron collider ( lep , geneva ) , at the slac linear collider ( slc , stanford ) and at the fermilab tevatron ( chicago ) . up to now no significant deviation between theory and experiment has been found . for other parts of the sm , related to cp violation and quark mixing , the @xmath9 factories like babar at slac , belle at kek ( tsukuba ) or hera - b at desy ( hamburg ) will provide deeper insight , and significant improvements in the determination of the corresponding parameters will be obtained . currently mainly the scalar sector of the sm eludes from direct experimental observation . this affects both the generation of the particle masses and the existence of the higgs boson itself . probably run ii of the tevatron and certainly the large hadron collider ( lhc ) at cern will provide more sureness not only in connection to the higgs sector but also to possible extensions of the sm . once the higgs boson would be discovered it immediately would become subject to precision measurements . in particular at a future @xmath2 linear collider such as desy tesla , a precise study of its properties would be possible . in the recent years there has been an enormous development in the evaluation of radiative corrections . it is fair to say that the major part of it was initiated by the fundamental works of t hooft and veltman in 1972 @xcite where dimensional regularization ( see also @xcite ) was established as a powerful tool in the evaluation of multi - loop diagrams . since that time a whole industry has been formed to develop techniques for the computation of complicated feynman integrals . at one - loop order the procedure of the computation has been systematically studied quite some time ago @xcite . nevertheless also nowadays it is not completely straightforward to evaluate an arbitrary one - loop diagram in particular if many legs and complicated momentum configurations are involved . one can easily imagine that at two and more loops one arrives quite soon at the limit where the occuring mathematical expressions can not be solved . at two - loop order certain classes of diagrams can still be treated by either using a combination of analytical simplifications and fast numerical routines , like in the case of two - point function with several non - zero masses @xcite , or applying purely analytical methods , like in the case of massless digrams with four external legs @xcite . however , at three - loop order it is essentially only possible to solve one - scale integrals . a systematic study at four or more loops is still missing . qcd , the field theoretical realization of the strong interaction , constitutes an important part of the sm and also of most of its extensions . at low energies the coupling constant of qcd , @xmath10 , is large and perturbative calculations are not possible . however , due to the phenomenon of asymptotic freedom the value of @xmath10 gets smaller with raising energy and perturbation theory is an appropriate tool to evaluate radiative corrections . up to now the vast majority of the multi - loop calculations have been performed in the framework of qed and qcd . one reason is certainly that the calculations are simpler as compared to the full sm since there are less parameters . furthermore , there is a strong hierarchy both in the quark and lepton masses which also simplifies the calculations . on the other hand , the higher - order corrections are indeed necessary . in qed there exist precise experiments which require high theoretical precision and although the coupling constant is quite small sometimes high loop orders are necessary . for example , in the case of the anomalous magnetic moment of the electron , four - loop corrections are needed to match the experimental precision . in qcd the coupling is roughly a factor of ten bigger . nevertheless it is often still small enough to perform a perturbative expansion . however , the higher order terms are significant and can not be neglected in the cases where high precision is required . in this work some recent developments in the calculation of multi - loop diagrams are reviewed . thereby we will mention the most important methods which have been used in the computation of higher order quantum corrections and explain a few selected ones in greater detail . at the same time , we discuss the present theoretical status of important physical quantities . in particular , the renormalization group functions in the modified minimal subtraction scheme @xcite ( @xmath7 ) are provided up to the four - loop order . as is well - known , four loop running must be accompanied by three - loop matching at quark thresholds . the corresponding decoupling relations are presented in section [ sec : dec ] . the hadronic higgs decay is closely connected to the decoupling relations as we will show is section [ sec : dim4 ] . parts of the quantum corrections can be computed in the framework of an effective lagrangian where the coefficients can be determined from the decoupling relations . the origin of this miraculous connection lies in the use of the dimension - four operators , which constitute an important ingredient of the effective lagrangian . another application of the dimension - four operators are the quartic mass corrections to the cross section @xmath11 , which is also discussed in section [ sec : dim4 ] . the last issue is again picked up in section [ sec : pade ] , where also qcd corrections to the production of hadrons in @xmath2 annihilation are computed . putting together all terms one arrives at a complete picture up to the quartic mass corrections of order @xmath1 . the main purposes of section [ sec : pade ] are practical applications of asymptotic expansions for details . ] . besides the diagonal correlators also the non - diagonal ones are considered . the fermi constant , @xmath6 , the mass of the @xmath8 boson , @xmath12 , and the electromagnetic coupling constant , @xmath13 , are the best known parameters of the sm . @xmath12 has been measured at lep with an accuracy of a few per mille to be @xmath14 gev . in this review we want to discuss quantum corrections to the other two parameters . an essential ingredient to the running of the electromagnetic coupling from @xmath15 to @xmath16 is provided by the cross section @xmath11 . the correction terms discussed earlier have been used to obtain so - called theory - driven results . the different approaches are discussed . as further applications we present the status of the qed corrections to the muon decay and the relation between the @xmath7 and the on - shell quark mass . let us in the following discuss the individual issues in more detail . as far as radiative corrections are concerned , a crucial role is played by the renormalization group functions . in particular the functions @xmath17 and @xmath18 governing the running of the coupling and the quark masses comprise a significant part of the higher quantum corrections . they are in particular very important to re - sum large logarithms to all orders in perturbation theory . only a few years ago the four - loop terms of order @xmath0 have been evaluated for @xmath17 @xcite and @xmath18 @xcite . the latter has been computed by two groups using completely independent methods . both methods are based on the fact that the pole part of a logarithmically divergent diagram is independent of the masses or momenta . in @xcite this is exploited together with the technique of infra - red re - arrangement ( irr ) in order to obtain a factorization of the four - loop integrals into massless three - loop and massive one - loop ones . in refs . @xcite all lines were assigned to the same mass , and all external momenta were set to zero , which leads to a special class of bubble diagrams . from them only the pole parts have to be computed . in section [ sec : rge ] we want to review both methods and explicitly demonstrate the way they work . in this review special emphasis is put on the construction of effective theories in the framework of qcd . in section [ sec : dec ] an effective qcd lagrangian is constructed for the case where one of the quarks is much heavier than the others . the construction is made explicit by specifying the relations between the parameters in the full and effective theories . these relations provide at the same time the well - known decoupling constants which have to be applied in qcd every time a particle threshold is crossed . the most prominent example for their necessity is probably the computation of @xmath19 from @xmath20 or vice versa . in the latter case five quarks are active whereas in the former one only three quarks are present in the effective qcd lagrangian . in section [ sec : dim4 ] a slightly different point of view is adopted . here the scalar operators of dimension four are considered in qcd . in a first step they are used to construct an effective lagrangian describing the coupling of an intermediate - mass higgs boson to quarks and gluons . the top quark is considered as heavy and manifests itself in the coefficient functions of the effective lagrangian . once the latter has been found , the imaginary part of the higgs boson correlator in the effective theory leads to the total decay rate . as a central result of section [ sec : dim4 ] we derive a low - energy theorem which considerably simplifies the computation of the coefficient functions as they are related to the decoupling constants of qcd evaluated in section [ sec : dec ] . in section [ sub : hggbfm ] the background field method is introduced as a convenient tool for the computation of higher - order corrections . as an example , the coefficient functions describing the decay of the higgs boson into gluons is also computed in this framework . in the second part of section [ sec : dim4 ] another important application of the scalar dimension four operators is discussed , namely the quartic mass corrections to the cross section @xmath11 . mass corrections of order @xmath21 and @xmath22 are obtained relatively easy as in qcd there are no non - trivial operators of dimension less than four . however , the quartic corrections require the inclusion of the dimension - four operators with all their renormalization and mixing properties . we will explain the techniques and present results obtained recently at order @xmath1 . the last part of this review , section [ sec : pade ] , is devoted to the discussion of some results obtained with the help of asymptotic expansion accompanied with conformal mapping and pad approximation . this method has been developed in the recent years and has been applied successfully to a number of important processes . the underlying idea is the following : only in rare cases it is possible to compute three - loop diagrams involving more than one scale . however , if a certain hierarchy exists between the scales it is promising to apply an asymptotic expansion . this effectively reduces the number of scales present in the integrals which are subsequently significantly simplified . in particular we will discuss the corrections of order @xmath3 to the photon polarization function . its imaginary part is directly connected to the physical quantity @xmath23 . the application of our method leads to the full mass dependence . combining the results with the quartic corrections given in section [ sub : as3m4 ] , one obtains a prediction for @xmath24 up to and including @xmath25 . only recently also the non - diagonal current correlator formed by a massive and a massless quark has been computed . in this application special emphasis lies on the extraction of information about the threshold behaviour which has some relevance in the framework of heavy - quark effective qcd . as an application of the knowledge of @xmath24 to high perturbative order , we discuss the evaluation of @xmath26 . the electromagnetic coupling is defined at vanishing momentum transfer . however , its evolution to high energies constitutes the dominant part of the radiative corrections to electroweak observables . the accurate determination of @xmath26 is thus essential for any precise test of the theory . at the same time the indirect determination of the masses of heavy , hitherto unobserved particles , e.g. the higgs boson or supersymmertic particles , depends critically on this parameter . of particular importance in this context is the hadronic vacuum polarization . it is nearly as large as the leptonic contribution , but can not yet be computed perturbatively . however , it may be related through dispersion relations to the cross section for hadron production in electron - positron annihilation . the integrand can thus be obtained from data , phenomenological models and/or perturbative qcd , whenever applicable . in section [ subsub : delal ] we will discuss the developments in the evaluation of @xmath26 which took place in the recent two to three years due to the knowledge of the complete mass dependence of @xmath24 at order @xmath3 ( cf . section [ subsub : r ] ) . @xmath6 is defined through the muon lifetime , and the decay of the muon , as a purely leptonic process , is rather clean both experimentally and theoretically . the one - loop corrections of order @xmath13 were computed more than 40 years ago @xcite , whereas only recently the two - loop corrections of order @xmath4 have been evaluated @xcite . the large gap in time shows that this calculation is highly non - trivial . the inclusion of the two - loop terms greatly reduced the relative theoretical error of @xmath27 which was an estimate of the size of the missing corrections . the remaining error on @xmath6 now reads @xmath28 and is of pure experimental nature . upcoming experiments will further improve the accuracy of the muon lifetime measurement and therefore the @xmath29 corrections to the muon decay are very important and constitute a crucial ingredient from the theoretical side . in section [ sub : mudec ] we discuss the results obtained with the help of asymptotic expansion . in the sm the quark masses have still relatively big uncertainties . this is mainly due to the confinement property of qcd which prevents the production of free quarks . it is also important to have a convenient definition of the quark mass in order to perform a comparison between theory and experiment . recently there has been quite some activity connected to the precise determination of the bottom- and top - quark masses . the bottom - quark mass is determined with the help of qcd sum rules where a proper mass definition helps to reduce the error . in the case of the top quark , studies have been performed for an @xmath2 collider with a center - of - mass energy in the threshold region of top - quark - pair production . an energy scan which provides the measurement of the total production cross section would provide an error of about 100 mev in the top - quark mass . also here a special mass definition has to be employed . in both cases the three - loop on - shell@xmath7 conversion formula is needed in order to obtain the corresponding @xmath7 quark mass . the latter is important for processes not connected to the threshold . in section [ sub : msos ] these issues are discussed in detail .

furthermore the connection to observables involving a scalar higgs boson is worked out in detail . an all - order low energy theorem is derived which establishes a relation between the coefficient functions in the hadronic higgs decay and the decoupling constants . we review the radiative corrections of a higgs boson into gluons and quarks and present explicit results up to order and , respectively . in this review special emphasis is put on the applications of asymptotic expansions . a method is described which combines expansion terms of different kinematical regions with the help of conformal mapping and pad approximation . this method allows us to proceed beyond the present scope of exact multi - loop calculations . as far as physical processes are concerned , we review the computation of three - loop current correlators in qcd taking into account the full mass - dependence . in particular , we concentrate on the evaluation of the total cross section for the production of hadrons in annihilation . the knowledge of the complete mass dependence at order has triggered a bunch of theory - driven analyses of the hadronic contribution to the electromagnetic coupling evaluated at high energy scales . are considered which contribute to the order qed corrections to the decay . its relevance for the determination of the fermi constant is discussed . finally the calculation of the three - loop relation between the and on - shell quark mass definitions is presented and physical applications are given . to complete the presentation , some technical details are presented in the appendix , where also explicit analytical results are listed .

in this review some recent multi - loop results obtained in the framework of perturbative quantum chromodynamics ( qcd ) and quantum electrodynamics ( qed ) are discussed . after reviewing the most advanced techniques used for the computation of renormalization group functions , we consider the decoupling of heavy quarks . in particular , an effective method for the evaluation of the decoupling constants is presented and explicit results are given . furthermore the connection to observables involving a scalar higgs boson is worked out in detail . an all - order low energy theorem is derived which establishes a relation between the coefficient functions in the hadronic higgs decay and the decoupling constants . we review the radiative corrections of a higgs boson into gluons and quarks and present explicit results up to order and , respectively . in this review special emphasis is put on the applications of asymptotic expansions . a method is described which combines expansion terms of different kinematical regions with the help of conformal mapping and pad approximation . this method allows us to proceed beyond the present scope of exact multi - loop calculations . as far as physical processes are concerned , we review the computation of three - loop current correlators in qcd taking into account the full mass - dependence . in particular , we concentrate on the evaluation of the total cross section for the production of hadrons in annihilation . the knowledge of the complete mass dependence at order has triggered a bunch of theory - driven analyses of the hadronic contribution to the electromagnetic coupling evaluated at high energy scales . the status is summarized in this review . in a further application four - loop diagrams are considered which contribute to the order qed corrections to the decay . its relevance for the determination of the fermi constant is discussed . finally the calculation of the three - loop relation between the and on - shell quark mass definitions is presented and physical applications are given . to complete the presentation , some technical details are presented in the appendix , where also explicit analytical results are listed . ( to appear in physics reports )

1304.1719

i

in this work we have studied @xmath0 decay and scattering processes mediated by the higgs with quarks in the initial and final states using the formalism of non - equilibrium quantum field theory . starting from the kadanoff - baym equations for the lepton propagator we have derived the corresponding quantum - corrected boltzmann and rate equations for the total lepton asymmetry . as compared to the canonical ones the latter are free of the notorious double - counting problem and ensure that the asymmetry automatically vanishes in thermal equilibrium . to compute the collision term we have taken into account one- and two - loop contributions to the lepton self - energy and used the extended quasiparticle approximation for the higgs propagator . the impact of the sm gauge interactions on the collision term has been approximately taken into account in the form of effective thermal masses of the higgs , leptons and quarks . we find that the inclusion of the effective masses and quantum - statistical terms suppresses the washout reaction densities of the decay and scattering processes with respect to the conventional ones , where these effects are neglected , in the whole relevant range of temperatures . for the @xmath210 process the ratio of the improved and conventional washout reaction densities slowly approaches a constant value close to unity at low temperatures . interestingly enough , for the @xmath224 processes this ratio decreases even at low temperatures . finally for @xmath225 process the ratio slowly approaches unity at low temperatures . as far as the _ cp_-violating reaction densities are concerned , we find that for the scattering processes the ratio of the improved and the conventional ones is greater than unity at high temperatures but is smaller than unity at intermediate and low temperatures because of the thermal masses and quantum - statistical effects . for the three - body decay this ratio is smaller than unity in the whole relevant range of temperatures . we expect that the effects studied here can induce a @xmath226 correction to the total generated asymmetry . for a detailed phenomenological analysis it is necessary to include further phenomena such as flavour effects and process with gauge bosons in the initial and final states . the work of a.k . has been supported by the german science foundation ( dfg ) under grant ka-3274/1 - 1 `` systematic analysis of baryogenesis in non - equilibrium quantum field theory '' . t.f . acknowledges support by the imprs - ptfs . we thank a. hohenegger for useful discussions .

in this work we study the contribution to leptogenesis from decay and scattering processes mediated by the higgs with quarks in the initial and final states using the formalism of non - equilibrium quantum field theory . starting from fundamental equations for correlators of the quantum fields we derive quantum - corrected boltzmann and rate equations for the total lepton asymmetry improved in that they include quantum - statistical effects and medium corrections to the quasiparticle properties . to compute the collision term we take into account one- and two - loop contributions to the lepton self - energy and use the extended quasiparticle approximation for the higgs two - point function .

in this work we study the contribution to leptogenesis from decay and scattering processes mediated by the higgs with quarks in the initial and final states using the formalism of non - equilibrium quantum field theory . starting from fundamental equations for correlators of the quantum fields we derive quantum - corrected boltzmann and rate equations for the total lepton asymmetry improved in that they include quantum - statistical effects and medium corrections to the quasiparticle properties . to compute the collision term we take into account one- and two - loop contributions to the lepton self - energy and use the extended quasiparticle approximation for the higgs two - point function . the resulting _ cp_-violating and washout reaction densities are numerically compared to the conventional ones .

math-ph0505010

c

in this work we have studied the scalar ( gauss ) curvature of weinhold metric for a thermodynamical systems with two thermodynamical degrees of freedom . we get criteria for the positivity , nullity and negativity of scalar curvature in terms of _ hessian surface _ of the thermodynamical potential , found scalar curvature for a general thermodynamical systems with two thermodynamical degrees of freedom . we have studied relation of the signature change of weinhold metric and the scalar curvature to the curves of phase transition of these systems . as examples we have considered the systems with the heat capacity @xmath1 constant , in particular the ideal and van der waals gases , and the berthelot gas . results obtained here suggest a kind of duality relation between the constitutive surface of a 2d thermodynamical system in the gibbs space ( space with coordinates @xmath599 in the case of internal energy ) and its hessian surface . relations between the convexity properties of both surfaces , curvature and signature of thermodynamical metric , extremal properties of corresponding thermodynamical potential and the phase transitions in the thermodynamical system present interesting and , in our opinion , highly promising direction of the future work . 99 m. akvis , v. goldberg , _ differential geometry of varieties with degenerate gauss maps _ , n.y . , berlin , springer verlag , 2004 . v. arnold , _ mathematical methods of classical mechanics _ , pringer verlag . r.s.berry , p.salamon , e.ihrig , _ a group of coordinate transformations which preserve the metric of weinhold _ , j.math.phys . 24(10 ) , 1983,2515 - 2520 . einstein manifolds _ , springer verlag , berlin , 1987 . h.b.callen , _ thermodynamics _ , whiley , 1985 . c. caratheodory,_unterschungen uber die grundlagen der thermodynamyk _ , gesammelte mathematische werke , b.2 . , munchen , 1955 , s. 131 - 177 . j. gibbs , _ the collected works _ , v.1 , thermodynamics , yale univ . press , 1948 . h. heemeyer , _ pseudoriemannsche faserbundel und ihre anwendung in der allegemeinrealitivistischen nichtgleichgewichtsthermodynamik _ , diss . berlin , wissenschaft und technik verlag , berlin , 1995 . r. hermann , _ geometry , physics and systems _ , dekker , n.y . , 1973 . h. janyszek , _ on the riemannian metric structure in the classical statisticsl equilibrium thermodynamics _ , reports of mathematical physics , v.24 , no.1 , 1986 , pp . 1 - 10 . h. janyszek , _ riemannian geometry and stability of thermodynamical equilibrium systems _ , journal of physics , a : math . v.23 , 1991 , pp . 477 - 490 . h. janyszek , r. mrugala _ geometrical structure of the state space in classical statisticsl and phenomenological thermodynamics _ , reports of mathematical physics , v.27 , no.2 , 1989 , pp . 145 - 159 . h. janyszek , r. mrugala _ riemannian geometry and the thermodynamics of model magnetic systems _ , physical review , a , v.39 , no.12 , 1989 , pp . 6515 - 6523 . a. knapp , _ lie groups beyond the introduction _ , birkhauser , boston , 1996 . k. kaviani , a.d.rezaie , _ riemannian geometry of the pauli paramagnetic gas _ , preprint , arxiv : cond - mat/9812054v1 , 3dec . d. kondepudi , i. prigogine,_modern thermodynamics _ , wiley and sons , 2001 . a. mlodziewskii , _ geometrical thermodynamics _ , mgu , moscow , 1956 ( in russian ) . r. mrugala , _ geometrical formulation of equilibrium phenomenological thermodynamics _ , reports of mathematical physics , v.14,no.3 , 1978 , pp.419 - 427 . r. mrugala , _ on equivalence of two metrics in classical thermodynamics _ , physica a , v.125,1984,pp.631 - 639 . r. mrugala , j.d . nulton , j.c . schoen , p. salamon , _ contact structure in thermodynamic theory _ , reports of mathematical physics , v.29 , no.1 , 1991 , pp.109 - 121 . r. mrugala , _ on a riemannian metric on contact thermodynamic spaces _ , reports of mathematical physics , v.38 , no.3 , 1996 , pp.339 - 348 . f.j.millero , a.lo surdo , c.shin , _ the apparent molal volumes and adiabatic compressibilities of aqueous amino acids at 25 c. _ , j. phys . chem , vol . 82 , no . 7 , 1978 , 784 - 792 . j. nulton , p.salamon,_geometry of ideal gas _ , physical review a , v.31,n.4,pp . 2520 - 2524 , 1985 . e. paci , m. marchi , _ instrinsic compressibility and volume compression in solvated proteins by molecular dynamics simulation at high pressure _ , proc . natl.acad.sci . m. postnikov , _ geometry vi . riemannian geometry _ , encyclopedia of mathematical sciences , v.91 , springer , 2001 . a.pogorelov , _ differential geometry _ , groningen , p.noordhoff , 1962 . s. preston , _ notes on the geometrical structures of homogeneous thermodynamics _ , manuscript , 2004 . s. preston , j.vargo , _ geometrical properties of mrugala metric _ , in preparation . g.ruppeiner , _ thermodynamics : a riemannian geometric model _ , phys . review a. 20(4 ) , 1979 , 1608 - 1613 . g. ruppeiner,_riemannian geometry in thermodynamic fluctuation theory _ , reviews of modern physics , v.67,n.3,1995 , pp.605 - 659 . g. ruppeiner,_riemannian geometry approach to critical points : general theory _ review e , v.57,n.5 , 1998 , pp.5135 - 5145 . p.salamon , j.nulton , e. ihrig,_on the relation between entropy and energy versions of thermodynamic length _ phys . , 80 , 436 ( 1984 ) . f. weinhold , _ metric geometry of equilibrium thermodynamics_,p . i - v , journal of chemical physics , v.63 , n.6,2479 - 2483 , 2484 - 2487 , 2488 - 2495,2496 - 2501,1976 , v.65,n.2,pp.559 - 564,1976 .

in this work[multiblock footnote omitted ] the curvature of weinhold ( thermodynamical ) metric is studied in the case of systems with two thermodynamical degrees of freedom . cases of systems with the constant , including ideal and van der waals gases , and that of berthelot gas are discussed in detail .

in this work[multiblock footnote omitted ] the curvature of weinhold ( thermodynamical ) metric is studied in the case of systems with two thermodynamical degrees of freedom . conditions for the gauss curvature to be zero , positive or negative are worked out . signature change of the weinhold metric and the corresponding singular behavior of the curvature at the phase boundaries are studied . cases of systems with the constant , including ideal and van der waals gases , and that of berthelot gas are discussed in detail .

1310.3840

i

in the last two decades , ultracold quantum gases have been the subject of many theoretical and experimental investigations @xcite . among the many systems that have been studied , ultracold fermi gases have received wide attention @xcite . due to the experimental controllability achieved with ultracold gases , quantum many - body phenomena such as fermionic superfluidity can be studied in great detail in these systems . by controlling and tuning the interaction strength between fermions in different states using feshbach resonances @xcite , it has become possible to study the crossover from a bardeen - cooper - schrieffer ( bcs ) superfluid state of weakly interacting cooper pairs , to a bose - einstein condensate ( bec ) of strongly coupled molecules @xcite . aside from the interaction strength , another important parameter that can be tuned is the population imbalance between fermions in different states . this parameter is of importance because spin - imbalance frustrates the bcs superfluid pairing mechanism . in the bcs state , pairing between fermions occurs at the fermi surface . however , a population imbalance will create a gap between the fermi surfaces of the two spin states , making the bcs state energetically less favorable . theoretically it was predicted that at a certain critical spin - imbalance , known as the clogston - chandrasekhar limit @xcite , a first order phase transition from the bcs state to the normal state would occur . by preparing a fermi gas in one hyperfine state , and using a radio - frequency sweep to create a mixture of two hyperfine states ( labeled spin - up and spin - down ) , the transition from a superfluid to a normal gas , induced by spin - imbalance , was demonstrated experimentally @xcite . at this point , the question remains whether a non - uniform superfluid can exist in a spin - imbalanced 3d fermi gas . the most prominent example of non - uniform superfluidity is the fulde - ferrell - larkin - ovchinnikov ( fflo ) state , which was proposed independently by fulde and ferrell ( ff ) @xcite and by larkin and ovchinnikov ( lo ) @xcite in 1964 . the fflo state differs from the bcs state in that it has cooper pairs with non - zero momentum , which in position space results in an oscillating superfluid order parameter . it was suggested that this exotic superfluid state could exist at non - zero polarization . in part of the literature , a further distinction is made between the ff state and the lo state : the former has one momentum component , whereas the latter is the superposition of two momentum components of equal magnitude but opposite sign . in this paper , we will focus on the ff state but we will henceforth call this the fflo state , bearing in mind that we mean the superfluid state with one momentum component . following the success of creating a spin - imbalanced fermi gas , the theoretical investigation of the fflo superfluid state was intensified . the first studies focused on the three - dimensional ( 3d ) fermi gas , at the saddle - point ( mean - field ) level @xcite , and found that the fflo state is only present in a very small sliver of the ground - state phase diagram @xcite . the 1d case has also received wide theoretical attention , and has proven to be a promising setup for detecting the fflo state . in 1d , the presence of the fflo state in the ground - state phase diagram is much larger compared to the 3d case @xcite . following these theoretical predictions , the first indirect experimental evidence for the fflo state was found in a 1d fermi gas by the hulet group at rice university @xcite . inspired by this success , several new experimental detection techniques have been proposed @xcite , both for the 1d and for the 3d case . however , in the latter case , the fflo state still eludes experimental observation . to acquire a better understanding of the fflo state in a 3d fermi gas , it is necessary to go beyond the mean - field level , which , while resulting in quantitatively correct results in the limit of weak interaction ( bcs limit ) at temperature zero , offers a qualitative description at best for temperatures above zero or for stronger interactions . up till now , little attention has been devoted to this subject and to the effect of fluctuations on the fflo state in general . one important exception is the work by radzihovsky , @xcite in which a low - energy model for the fulde - ferrell state and for the larkin - ovchinnikov state is developed , with an in - depth focus on the nature of the emerging goldstone modes for the latter state . in this paper , we contribute to this subject by explicitly studying the effect of phase fluctuations on the presence of the fflo state in the phase diagram of a 3d fermi gas . our main motivation is the following : the fflo state is characterized by a momentum component @xmath0 , which means that the rotational symmetry of the system is spontaneously broken by this state . the momentum component @xmath0 results in an oscillating phase of the order parameter in position space . because of this , fluctuations of the phase of the order parameter are equivalent to fluctuations in the direction of the momentum @xmath0 . since a 3d fermi gas exhibits spherical symmetry , these fluctuations cost zero energy . hence , one would expect these fluctuations to proliferate and destabilize the fflo state . our main point of interest is to see whether the region of fflo in the phase diagram of a 3d fermi gas vanishes due to phase fluctuations , which would help to understand why this state has not been observed experimentally in 3d . to the best of our knowledge , this specific problem has not yet been studied in literature . the rest of this paper is organized as follows . in section [ hydrodynamic effective action ] we derive a hydrodynamic effective action , starting from the partition function of a 3d fermi gas with spin - imbalance , within the path - integral adiabatic approximation . in section [ fluctuation action ] we perform an expansion of the action up to second order in the fluctuation field , which leads to the fluctuation part of the action . from the fluctuation action , the fluctuation free energy is readily derived . subsequently , in section [ phase diagram ] , using this free energy , we calculate the phase diagram of the system and determine whether corrections emerge by taking into account phase fluctuations . finally in section [ conclusions ] we draw conclusions .

in ultracold fermi gases , the effect of spin - imbalance on superfluidity has been the subject of intense study . one of the reasons for this is that spin - imbalance frustrates the bardeen - cooper - schrieffer ( bcs ) superfluid pairing mechanism , in which fermions in different spin states combine into cooper pairs with zero momentum . in 1964 , it was proposed that an exotic superfluid state called the fulde - ferrell - larkin - ovchinnikov ( fflo ) state , in which the cooper pairs have nonzero momentum , could exist in a spin - imbalanced fermi gas . at the saddle - point ( mean field ) level , it has been shown that the fflo state only occupies a very small sliver in the ground state phase diagram of a 3d fermi gas . however , a question that remains to be investigated is : what is the influence of phase fluctuations around the saddle point on the fflo state ? in this work we show that phase fluctuations only lead to relatively small quantitative corrections to the presence of the fflo state in the saddle - point phase diagram of a 3d spin - imbalanced fermi gas . starting from the partition function of the system , we calculate the effective action within the path - integral adiabatic approximation . the action is then expanded up to second order in the fluctuation field around the saddle point , leading to the fluctuation free energy . using this free energy

in ultracold fermi gases , the effect of spin - imbalance on superfluidity has been the subject of intense study . one of the reasons for this is that spin - imbalance frustrates the bardeen - cooper - schrieffer ( bcs ) superfluid pairing mechanism , in which fermions in different spin states combine into cooper pairs with zero momentum . in 1964 , it was proposed that an exotic superfluid state called the fulde - ferrell - larkin - ovchinnikov ( fflo ) state , in which the cooper pairs have nonzero momentum , could exist in a spin - imbalanced fermi gas . at the saddle - point ( mean field ) level , it has been shown that the fflo state only occupies a very small sliver in the ground state phase diagram of a 3d fermi gas . however , a question that remains to be investigated is : what is the influence of phase fluctuations around the saddle point on the fflo state ? in this work we show that phase fluctuations only lead to relatively small quantitative corrections to the presence of the fflo state in the saddle - point phase diagram of a 3d spin - imbalanced fermi gas . starting from the partition function of the system , we calculate the effective action within the path - integral adiabatic approximation . the action is then expanded up to second order in the fluctuation field around the saddle point , leading to the fluctuation free energy . using this free energy , we calculate corrections due to phase fluctuations to the bcs - fflo transition in the saddle - point phase diagram . at temperatures at which the fflo state exists , we find only small corrections to the size of the fflo area . our results suggest that fluctuations of the phase of the fflo order parameter , which can be interpreted as an oscillation of its momentum vector , do not cause an instability of the fflo state with respect to the bcs state .

1210.4833

i

the main goal of this paper is to study the representation theory of infinitesimal cherednik algebras @xmath2 , a deformation of the representation theory of @xmath1 with infinitely many deformation parameters @xmath4 . namely , @xmath1 can be represented as @xmath5 , where @xmath6 are the natural representations of @xmath0 on vectors and covectors . in this representation of @xmath1 , the elements of @xmath7 commute with each other , as do the elements of @xmath8 . the commutation relations of @xmath0 with @xmath6 are given by the usual action of matrices on vectors and covectors , while commutators of @xmath7 with @xmath8 produce elements of @xmath0 . to pass to the deformation @xmath2 , one needs to change only the last relation : commutators of @xmath7 and @xmath8 will now be not just elements of @xmath0 but rather some polynomial @xmath9 of them , where @xmath10 are the deformation parameters mentioned above and @xmath11 are basis polynomials introduced in @xcite . this deformation turns out to be very interesting , since it unifies the representation theory of @xmath1 with that of degenerate affine hecke algebras ( @xcite,@xcite ) and of symplectic reflection algebras ( @xcite ) . the main results of this paper are the following . in section [ section : shapform ] , we generalize a classical result from the representation theory of kac - moody algebras by computing the determinant of the contravariant ( or shapovalov ) form , thus determining when the verma module over @xmath2 is irreducible . this proof requires knowledge of the quadratic central element and its action on the verma module . in section [ section : gln ] , we find the quadratic central element of @xmath2 ; this extends the work of tikaradze @xcite , who proved using methods of homological algebra that the center of @xmath2 is a polynomial algebra in @xmath12 generators , but did not get any explicit formulas for these generators . in section [ section : finite ] , we provide a complete classification and character formulas for finite dimensional representations of @xmath2 , generalizing chmutova s unpublished work . in sections [ section : poisson ] to [ section : sp2n ] , we introduce poisson analogues of the infinitesimal cherednik algebras , compute their poisson center , and use them to give a second proof of the formula for the quadratic central element of @xmath2 . we also provide an analogous formula for the center of the poisson analogue of @xmath3 . finally , in section [ section : kostant ] , we investigate an analogue of kostant s theorem for @xmath3 . @xmath13

infinitesimal cherednik algebras are continuous analogues of rational cherednik algebras , and in the case of , are deformations of universal enveloping algebras of the lie algebras . in the first half of this paper , we compute the determinant of the shapovalov form , enabling us to classify all irreducible finite dimensional representations of . in the second half , we investigate poisson - analogues of the infinitesimal cherednik algebras and generalize various results to , including kostant s theorem .

infinitesimal cherednik algebras are continuous analogues of rational cherednik algebras , and in the case of , are deformations of universal enveloping algebras of the lie algebras . in the first half of this paper , we compute the determinant of the shapovalov form , enabling us to classify all irreducible finite dimensional representations of . in the second half , we investigate poisson - analogues of the infinitesimal cherednik algebras and generalize various results to , including kostant s theorem .

1504.04048

i

thermoelectric effects in nanoscale structures have been frequently studied with the aim to improve the efficiency of devices . @xcite in addition , it has been recently demonstrated that thermopower measurements can serve as an interesting tool to characterize complex scenarios due to coherences and interactions in nanoscale systems . @xcite here the intrinsic advantage of thermoelectric measurements is that they probe asymmetries around the fermi level and therefore can easily provide information about excited states . @xcite close to a degeneracy of energy levels in a quantum dot , interference and correlation effects can play an important role for transport and lead to pronounced quantum mechanical phenomena . for a spin - degenerate quantum dot ( qd ) the conductance experiences an enhancement for low temperatures due to the kondo effect . @xcite the thermopower of the single and multiple quantum dots in the presence of the kondo effect was examined both theoretically @xcite and experimentally . @xcite on the other hand , in the case of two degenerate levels with equal spin , at degeneracy , conductance can be suppressed due to electron correlation and interference effects . @xcite such a spin - polarized two - level model was widely used to interpret the phenomena of phase lapses of transmission in aharonov - bohm interferometer containing a qd . @xcite additionally , due to their small size the quantum dots also can exhibit coulomb blockade effect . @xcite the thermopower of the usual coulomb blockade sequential tunneling peaks and the cotunneling signal were addressed in refs . . in ref . a system of two spin - polarized degenerate levels was realized in an insb nanowire qd , where different @xmath0-factors allow to control level crossings for the same spin by a magnetic field . the experiment showed , in good agreement with supporting calculations , that the suppression cuts as a canyon through the standard conductance plot for different parities of the level couplings . a more detailed analysis of the conductance spectrum can be found in ref . . a related two - level system was optimized for achieving high thermoelectric performance in ref . . in this work we further elaborate on thermoelectric properties and focus on the fingerprint of conductance suppression in the thermopower signal . ( color online ) schematic of the system : a quantum dot with single particle levels @xmath1 and @xmath2 coupled to the leads via tunneling barriers . the leads have a temperature difference @xmath3 which can give rise to a current flowing through the dot . ] the paper is organized as follows . in section [ sec : model ] the model for the spin - polarized two - level quantum dot is introduced . results for conductance and thermopower are presented in section [ sec : results ] . here we focus on the zeros of the thermopower , which are relatively easy to extract experimentally . we start in section [ subsec : uzero ] with a discussion of the non - interacting case , where results are obtained using transmission formalism . here we show that up to five zeros in the thermopower can be found if the gate voltage is varied . furthermore , we establish the role of temperature and level broadening for the existence of multiple zeros . the impact of finite qd charging energy is addressed in section [ subsec : ufinite ] , where calculations are performed by the second order von neumann ( 2vn ) approach . this reveals the full scenario for the canyon of conductance suppression . concluding remarks are given in section [ sec : concl ] .

interference effects in quantum dots between different transport channels can lead to a strong suppression of conductance , which cuts like a canyon through the common conductance plot [ phys . rev . lett . * 104 * , 186804 ( 2010 ) ] . in the present work , we consider the thermoelectric transport properties of the canyon of conductance suppression using the second - order von neumann approach . this demonstrates that thermoelectric measurements are an interesting complimentary tool to study complex phenomena for transport through confined systems .

interference effects in quantum dots between different transport channels can lead to a strong suppression of conductance , which cuts like a canyon through the common conductance plot [ phys . rev . lett . * 104 * , 186804 ( 2010 ) ] . in the present work , we consider the thermoelectric transport properties of the canyon of conductance suppression using the second - order von neumann approach . we observe a characteristic signal for the zeros of the thermopower . this demonstrates that thermoelectric measurements are an interesting complimentary tool to study complex phenomena for transport through confined systems .

1511.03695

i

gamma - ray burst ( grb ) pulse characteristics remain poorly understood even though pulses are the most common structures in grb light curves : there is little agreement concerning the physical mechanisms responsible for producing them . pulse characteristics are difficult to extract due to a mixture of strong spectral - temporal evolution , instrumental and sampling biases , confusion caused by overlapping pulses , and low signal - to - noise measurements . understanding pulse behaviors is important , as it is difficult to adequately model grb progenitors and outflow without first understanding the mechanism by which pulses liberate energy , and it is difficult to understand pulse mechanisms without first identifying pulse behaviors . data driven analysis approaches are providing new insights into grb pulse characteristics . these approaches have both advantages and disadvantages over standard theoretical modeling techniques : they can identify behaviors ( typically clustering and/or correlations ) that theoretical models are not trying to explain and thus may not recognize , but by asking questions about pulse behaviors they do not necessarily couch these questions in the form of specific theoretical models . data driven approaches have allowed the identification of important pulse characteristics even as theoretical models have been unable to explain these behaviors . as an example , consider empirical light curve models , which can be used to provide an accurate representation of grb pulse behaviors in the absence of identified pulse physics . several empirical models are able to adequately explain the general shapes of grb pulse light curves ( e.g. see the introduction in @xcite ) . among these , the simple model of @xcite is useful because it has only four free parameters which allow reasonable fits over the wide range of observed pulse characteristics . empirical approaches demonstrate that grb pulse light curves can be generally represented by a _ hard - to - soft _ spectral evolution . pulses are generally hardest at the first instant they appear , and they continue to soften during the pulse rise , past the pulse peak , and through the pulse decay @xcite . although hard - to - soft evolution is the most common pulse evolution ( e.g. @xcite ) , _ intensity tracking _ pulse evolution has been identified in a smaller number of pulses ( e.g. @xcite ) . it is difficult to estimate what fraction of pulses exhibit intensity tracking behaviors , since there are many cases where overlapping hard - to - soft pulses appear to have been misclassified as intensity tracking @xcite . hard - to - soft spectral evolution is so central to grb pulse structure that it can be used as a defining characteristic in rigorous statistical modeling techniques @xcite . the harder a pulse is , the more pronounced the evolution typically is . a small percentage of pulses appear to harden during the pulse rise , but these _ intensity tracking _ pulses are typically among the very softest , suggesting that instrumental biases might be in part responsible for the initial soft emission . several of the well - known correlated spectral - temporal grb pulse properties are simply a result of this hard - to - soft spectral decay : rapid hardness evolution produces short duration pulses with short spectral lags while slow hardness evolution produces long duration ones with long lags . the relationship between pulse asymmetry and hard - to - soft evolution is not as clear as the one between duration , lag , and evolution . asymmetry is anti - correlated with hardness and correlated with duration across a large batse grb pulse sample such that asymmetric pulses are generally longer and softer than symmetric pulses @xcite . this implies that asymmetric pulses in general do not undergo a significant hardness evolution , since the softening is mild and occurs over a long time interval . however , some of the faintest , softest , longest grb pulses are symmetric and undergo very little hard - to - soft evolution , while some asymmetric pulses are quite hard . to further confuse matters , the bright , short pulses in classical short grbs are also symmetric yet very hard ; undergoing a rapid hard - to - soft evolution . without an obvious connection between hard - to - soft evolution and pulse asymmetry , the possibility exists that asymmetry and hard - to - soft evolution are independent or semi - independent pulse characteristics . grb pulses ( at least those belonging to the long and intermediate grb classes ) exhibit additional intensity fluctuations that overlay the hard - to - soft monotonic pulse component @xcite ; these fluctuations can be seen in the residuals obtained by subtracting the @xcite modeled monotonic pulse shape from the light curve . these fluctuations , which have the appearance of three peaks separated by two valleys , are more easily observed in high signal - to - noise pulses ; this makes bright pulses look like they have variable , multi - peaked light curves , in contrast to the smooth monotonic light curves of fainter pulses . when combined with spectral softening , bright pulses appear to be composed of at least three peaks , with the first one ( during or immediately preceding the pulse rise ) generally being the hardest , the second ( and brightest ) one being of intermediate hardness , and the last one ( during the long decay ) being soft . hard - to - soft evolution and phased non - monotonic intensity fluctuations can be used to help determine if a grb emission episode is composed of a single pulse or of several merged pulses . a single pulse , at least in the energy ranges of batse and fermi , has pulse residual parameters that correlate with the pulse asymmetry . the residual light curve waves can be fitted by a simple four - parameter empirical model @xcite this empirical model has several interesting features : 1 ) the basis of the model is a truncated bessel function of the first kind , @xmath0 that begins at the true pulse peak , 2 ) the wave can be modeled by a mirror image of itself in time , with the time - reflected part occurring during the pulse rise and the time - forward part occurring during the pulse decay , 3 ) the pulse decay portion is a time - stretched version of the mirrored part on the pulse rise , and 4 ) most if not all of the pulse residual fit parameters correlate with the pulse asymmetry . at the present time , no physical mechanism is known that can provide these observed model characteristics . it is the reproducibility of the light curve residual fluctuations , combined with the overall hard - to - soft spectral evolution , which demonstrates that the fluctuations all combine to form a single evolving pulse with a non - monotonic shape . all pulse light curves seem to undergo multiple intensity increases and decreases . however , the definition of a pulse being composed of several monotonically rising and falling regions may be a semantic one : a single pulse might indeed be composed of three separate , linked , smaller pulses . if we do choose to believe that each fluctuation identifies a separate pulse , then the coupling between these fluctuations and the overall hard - to - soft evolution indicates that multiple pulses always combine to form similar , temporally - linked , complex shapes that need to be described by a single smooth hardness evolution . the grbs used to measure pulse residual fluctuations have thus far all been those observed by batse and fermi s gbm experiment . since these instruments have spectral responses favoring detection of higher energy photons ( 25 kev to 2 mev ) over lower ones , it is reasonable to wonder whether these residual features are present at lower energies observed by swift , or if they are primarily a high energy phenomenon . to this end , we examine a sample of single episodic bursts observed by swift .

isolated swift gamma - ray burst ( grb ) pulses , like their higher - energy batse counterparts , emit the bulk of their pulsed emission as a hard - to - soft component that can be fitted by the empirical pulse model . this signal is overlaid by a fainter , three - peaked signal that can be modeled by an empirical wave - like function : the two fits combine to reproduce grb pulses with distinctive three - peaked shapes . isolated grb pulses are dominated by hard - to - soft evolution ; this is more pronounced for asymmetric pulses than for symmetric ones .

isolated swift gamma - ray burst ( grb ) pulses , like their higher - energy batse counterparts , emit the bulk of their pulsed emission as a hard - to - soft component that can be fitted by the empirical pulse model . this signal is overlaid by a fainter , three - peaked signal that can be modeled by an empirical wave - like function : the two fits combine to reproduce grb pulses with distinctive three - peaked shapes . the precursor peak appears on or before the pulse rise and is often the hardest component , the central peak is the brightest , and the decay peak converts exponentially decaying emission into a long , soft , power - law tail . accounting for systematic instrumental differences , the general characteristics of the fitted pulses are remarkably similar . isolated grb pulses are dominated by hard - to - soft evolution ; this is more pronounced for asymmetric pulses than for symmetric ones . isolated grb pulses can also exhibit intensity tracking behaviors that , when observed , are tied to the timing of the three peaks : pulses with the largest maximum hardnesses are hardest during the precursor , those with smaller maximum hardnesses are hardest during the central peak , and all pulses can re - harden during the central peak and/or during the decay peak . since these behaviors are essentially seen in all isolated pulses , the distinction between `` hard - to - soft '' and `` intensity - tracking '' pulses really no longer applies . additionally , the triple - peaked nature of isolated grb pulses seems to indicate that energy is injected on three separate occasions during the pulse duration : theoretical pulse models need to account for this .

astro-ph0606277

i

@xcite first pointed out that star formation in giant molecular clouds ( gmcs ) happens surprisingly slowly . comparing the mass of gmcs in the galaxy with the total galactic star formation rate implies that no more than @xmath0 of the gas can form stars for each cloud free - fall time . this result is sufficiently surprising that numerous theories have been proposed to explain it , ranging from the idea that strong magnetic fields ( e.g. * ? ? ? * ) or turbulence ( e.g. * ? ? ? * ) within clouds inhibit star formation to the idea that galactic - scale gravitational instability regulates star formation ( e.g. * ? ? ? * ) to the idea that gmcs are , contrary to most observational estimates to date @xcite , actually gravitationally unbound ( e.g. * ? ? ? following @xcite , we define the dimensionless star formation rate per free - fall time @xmath1 as the fraction of an object s mass that it converts into stars per free - fall time at the mean density of that object . the @xcite argument shows that @xmath2 for gmcs . this provides a powerful constraint on models of gmcs . for example , it rules out the early gmc model of @xcite , a simplified version of which is that gmcs are spheres of gas of density @xmath3 that are in free - fall collapse . the clouds reach a singularity in a time @xmath4^{1/2}$ ] , at which point their mass is converted into stars . for a population of such clouds @xmath5 , which is inconsistent with the @xcite result . an important observational question , and a crucial test for theories of how star formation is regulated , is to what densities and length scales @xmath1 remains much smaller than unity . in other words , is there a density at which something like the @xcite free - fall collapse model becomes reasonable ? as an example , consider observing a star - forming region using a molecular tracer sensitive to gas at densities of @xmath6 @xmath7 , where @xmath8 is the number density of hydrogen nuclei , to estimate the total mass of gas at such high densities . this is larger than the mean density of `` cores '' seen both observationally ( e.g. * ? ? ? * ) and in simulations of star formation regulated by turbulence ( e.g. * ? ? ? * ) , and is roughly the density at which models of magnetically - regulated star formation predict that gas will completely decouple from the magnetic field and enter free - fall collapse ( e.g. * ? ? ? . at such high densities protostellar outflows can probably stop at most half the gas from reaching a star @xcite , and the thermal pressure in gas at @xmath9 @xmath7 is considerably higher than the typical ram pressure of the turbulence in gmcs , so gas at such high densities is largely impervious to external perturbations . thus , regardless of the model of star formation one adopts , one would expect that almost all of the gas at densities @xmath10 @xmath7 is part of gravitationally - bound , collapsing objects that have largely decoupled from the background turbulent flow . in the absence of effective internal support or external disturbance , order unity of the gas at such high densities is likely to be incorporated into a star within one free - fall time . for this reason , essentially all models of star formation predict that the total mass of gas at densities @xmath10 @xmath7 , divided by the free - fall time of this gas , should yield a value comparable to the total star formation rate in the region over which the mass is measured . instead of @xmath11 as for gmcs , i.e. slow star formation , one would obtain @xmath12 , i.e. rapid star formation . however , different models make different predictions about the shape of the curve of @xmath1 versus density in between @xmath0 at the characteristic density of gmcs , @xmath13 @xmath7 , and @xmath14 at a density @xmath10 @xmath7 . these different predictions correspond to different models of the physical scale at which gas both decouples from the background flow and ceases to be supported by internal feedback mechanisms , and thus transitions from slow to rapid star formation . at one extreme , magnetic regulation models such as those of @xcite , neglecting for the moment turbulent enhancement of the ambipolar diffusion rate ( e.g. * ? ? ? * ) , predict that star formation only becomes rapid once gas decouples from the magnetic field , a process that does not even begin until densities of @xmath15 @xmath7 . at the other extreme , @xcite argue that star clusters form from gas clumps at densities of @xmath16 @xmath7 that undergo a free - fall collapse in which at least @xmath17 of their mass is converted into stars @xcite , so star formation should be rapid at this density or higher . in this model , the decoupling scale corresponds to the transition from globally unbound structures ( gmcs ) to globally bound structures ( protoclusters ) . thus , extending the @xcite calculation of star formation rate divided by free - fall time to higher densities , in hopes of identifying a scale at which there is a transition from slow to rapid star formation , provides a means of distinguishing between models of how star formation is regulated . it is important at this point to differentiate the concepts of the _ rate _ and _ efficiency _ of star formation , and the _ lifetime _ of star - forming clouds . unfortunately these terms are often confused in the literature , and there are no standard definitions , so we describe here the definitions we use in this paper . the star formation rate @xmath18 is the easiest to define , since it is simply the instantaneous conversion rate of gas into stars within some volume . if we pick a density threshold @xmath3 , we can then define the dimensionless star formation rate per free - fall time for the gas above that density threshold , @xmath19 $ ] , where @xmath20 is the mass of material inside the volume of density @xmath3 or greater . in contrast , the lifetime @xmath21 of star - forming cloud is somewhat more ambiguous . we take it to mean the total duration during which a cloud is visible in a tracer sensitive to densities of @xmath3 or more . note that our definition neglects the complication that something that starts as a single cloud may in the course of its evolution break up into multiple pieces , so the visible lifetime and the dynamical lifetime may be different . finally , by the efficiency @xmath22 we mean the fraction of gas mass @xmath20 that is converted into stars by a cloud over its lifetime @xmath23 . again , we neglect the complication that clouds are not closed boxes , so @xmath20 is likely to be time - dependent due to continuing accretion or mass loss . ideally @xmath22 should be computed using a lagrangian definition of the cloud mass , i.e. all the mass that reached density @xmath3 or more at some point . however , this is generally not a direct observable . roughly speaking , the rate , efficiency , and lifetime are related by @xmath24 . consequently , one can define a rough time - averaged value for @xmath1 in a cloud @xmath25 $ ] , i.e. the mass fraction converted into stars divided by the cloud lifetime in free - fall times . one could also describe @xmath26 as being the efficiency per free - fall time , since it measures a fraction of mass converted into stars . however , we refer to it as a rate because it is measured in amount per unit time , whereas efficiency in the literature most commonly refers to the total fraction of mass converted into stars , not the amount per unit time . moreover , since the time - averaged definition is ambiguous anyway due to uncertainties in exactly what is meant by the efficiency and the lifetime , we will generally use the instantaneous value of @xmath1 , which is well defined and , as we show , directly observable . the reason for making all these definitions explicit it to point out that observational constraints on @xmath1 by themselves do not directly constraint @xmath27 or @xmath28 , and vice versa . numerous authors have used various observational techniques to estimate @xmath27 in clusters and gmcs ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? however , with the exception of [ clustertime ] , we will not discuss the question of cloud lifetime at all in this paper . when we refer to rapid versus slow star formation , what we mean is @xmath12 or @xmath29 , not is @xmath30 or @xmath31 . these questions are conceptually distinct . as an example , note that in the @xcite picture of gmcs as collapsing spheres , the evolution time scale is always the cloud free - fall time , @xmath32 . however , the model still gives @xmath5 , and we would therefore describe it as a rapid star formation . in contrast , @xcite argue that star formation lasts only one crossing time of a molecular cloud , so again @xmath33 , but that during this time only @xmath0 of the mass turns into stars . we would describe this as slow star formation , since @xmath34 , even though the cloud evolution time is similar to that in the @xcite model . the @xcite argument rules out the @xcite free - fall collapse model for gmcs , but is consistent with the @xcite model . thus , we emphasize that _ in this paper we remain completely agnostic on the question of molecular cloud lifetime_. in this paper we consider star formation in several classes of object . infrared dark clouds ( irdcs ) are regions of high extinction seen in absorption against the galactic infrared background @xcite . irdcs are clearly associated with star formation , and in at least some cases irdcs have massive protostars embedded within them @xcite . several authors have suggested ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) that irdcs are the progenitors of star clusters . within irdcs , at still higher densities , are dense molecular clumps . these objects may be observed in a variety of molecular transitions with high critical densities , and we consider two here : hcn(1 - 0 ) @xcite and cs(5 - 4 ) @xcite . molecular clumps seen in these two transitions are often associated with water masers and other signs of massive , clustered star formation . our goal is to determine @xmath1 for each of these increasingly dense gas tracers . we also determine this quantity for the orion nebula cluster using a completely different method , which provides an independent check on our estimates . the remainder of this paper proceeds as follows : in [ sfrest ] we use a variety of observations to derive @xmath1 for our objects and construct a plot of @xmath1 versus characteristic density to search for signs of a transition from slow to fast star formation . in [ implications ] , we compare our results to theoretical models for the regulation of star formation , and also point out some implications for the process of star cluster formation . finally , in [ conclusions ] we summarize our conclusions , and suggest directions for future work .

it has been known for more than 30 years that star formation in giant molecular clouds ( gmcs ) is slow , in the sense that only of the gas forms stars every free - fall time . this result is entirely independent of any particular model of molecular cloud lifetime or evolution . here we find no evidence for a transition from slow to rapid star formation in structures covering three orders of magnitude in density . this has important implications for models of star formation , since competing models make differing predictions for the characteristic density at which star formation should transition from slow to rapid . the data are inconsistent with models that predict that star clusters form rapidly and in free - fall collapse .

it has been known for more than 30 years that star formation in giant molecular clouds ( gmcs ) is slow , in the sense that only of the gas forms stars every free - fall time . this result is entirely independent of any particular model of molecular cloud lifetime or evolution . here we survey observational data on higher density objects in the interstellar medium , including infrared dark clouds and dense molecular clumps , to determine if these objects form stars slowly like gmcs , or rapidly , converting a significant fraction of their mass into stars in one free - fall time . we find no evidence for a transition from slow to rapid star formation in structures covering three orders of magnitude in density . this has important implications for models of star formation , since competing models make differing predictions for the characteristic density at which star formation should transition from slow to rapid . the data are inconsistent with models that predict that star clusters form rapidly and in free - fall collapse . magnetic- and turbulence - regulated star formation models can reproduce the observations qualitatively , and the turbulence - regulated star formation model of krumholz & mckee quantitatively reproduces the infrared - hcn luminosity correlation recently reported by gao & solomon . slow star formation also implies that the process of star cluster formation can not be one of global collapse , but must instead proceed over many free - fall times . this suggests that turbulence in star - forming clumps must be driven , and that the competitive accretion mechanism does not operate in typical cluster - forming molecular clumps .

hep-ph9701393

i

the violation of baryon number ( b ) in the hot , symmetric phase of electroweak theory plays a crucial role in scenarios for electroweak baryogenesis . the rate of b violation is tied , through the electroweak anomaly , to the the rate of topological transitions of the electroweak gauge fields . for the symmetric phase , this rate is not calculable by any perturbative method . in this paper , i address whether the topological transition rate can , in principle , be extracted from lattice simulations . the discussion will reveal some surprising features of real - time thermal field theory on the lattice . it has long been appreciated that , at finite temperature , topological transitions in real time are not directly related to topological transitions in euclidean time @xcite . as a result , there is no apparent way to measure the real - time thermal transition rate in a standard , euclidean - time , lattice simulation of quantum field theory . several years ago , ambjrn and krasnitz @xcite cleverly implemented the observation @xcite that a full simulation of _ quantum _ field theory is not actually required . topological transitions occur in the symmetric phase through large configurations which are essentially classical . indeed , _ all _ long - distance bosonic physics in a hot , ultrarelativistic plasma is effectively classical because of bose enhancement . the number of quanta per mode in low - energy modes is given by the bose distribution @xmath2 and , by the correspondence principle , this is the classical regime . unlike quantum field theory , real - time simulations of classical field theory are tractable : just evolve the classical equations of motion . classical thermal field theory has an ultraviolet catastrophe , historically famous in the context of black - body radiation . because every mode has energy @xmath3 by the classical equipartition theorem , and because there are an infinite number of modes per unit volume in continuum field theory , the energy density is infinite . in quantum field theory , in contrast , the ultraviolet contribution is cut off at momenta of order @xmath4 . ambjrn and krasnitz reasoned that the details of short - distance physics should not affect the long - distance physics of topological transitions . in their simulations , they put their classical system on a spatial lattice and then evolved it in continuous real time . the ultraviolet catastrophe was cut off by the lattice spacing , which they progressively made smaller and smaller . son , yaffe , and i @xcite have recently pointed out that short - distance effects do not decouple as cleanly as ambjrn and krasnitz hoped . in particular , the short - distance modes cause damping of the long - distance dynamics , and this damping affects the transition rate . we showed that damping reduces the transition rate by a factor of @xmath5 , where @xmath6 is the weak fine structure constant . because the damping is caused by short - distance physics , it is not universal , and so a comparison of theories with different short - distance physics becomes non - trivial . in the classical lattice theory used by ambjrn and krasnitz , for example , damping reduces the rate instead by a factor of @xmath7 where @xmath8 is the lattice spacing . as i shall discuss , there is an even more serious problem : because damping is dominated by short - distance physics , it knows about the anisotropies of the lattice . as we shall see explicitly , and as was first noted by bodeker _ et al . _ @xcite , the effective long - distance physics of the classical lattice theory is not even rotationally invariant . this is in striking contrast to the familiar situation of euclidean - time simulations , where rotational invariance is always recovered in the continuum limit . i shall assume that our analysis of damping in ref . @xcite is correct and will eventually be borne out by numerical simulations at sufficiently weak coupling . in this paper , i focus on the natural follow - up question : given that the real quantum theory and the classical lattice theory have different long - distance physics , is there any way to measure the real topological transition rate ? the answer is yes in principle but an exact calculation requires a careful choice of lattice action and the numerical extraction of a somewhat awkward limit . but i shall also propose a rough numerical measure of the suitability of generic lattice actions and argue that even results from simple , canonical actions should be in the right ballpark if properly interpreted . i shall not address at all the complicated problem of how one measures topological transitions on the lattice in the first place , which has a long , difficult history for euclidean - time quantum simulations and a shorter but still confusing one for real - time classical simulations . i simply focus on dynamics and assume the measurement problem will eventually be solved , using cooling or some other technique . section 2 of this paper outlines my method for taking a measurement of the topological transition rate in a theory with one ultraviolet cut - off ( _ e.g. _ a classical spatial lattice theory ) and using it to predict the rate in the same theory with a different ultraviolet cut - off ( _ e.g. _ the real , continuum , quantum field theory ) . the procedure will require , however , that the long - distance dynamics be rotationally invariant in both cases , and this poses a problem for the lattice that will eventually be dealt with . section 2 rests on a very rough and schematic discussion of the effective long - distance dynamics , and i return in section 3 to do a better job of reviewing the correct long - distance theory . i review the derivation of `` hard thermal loop '' effective theory but add a small twist . the usual discussions in the literature are based on the assumption that hard ( _ i.e. _ high - momentum ) excitations in the plasma move at the speed of light . this is not true for lattice theories , and i show how the usual results easily generalize . in section 4 , as a warm - up example of a rotational - invariant classical theory , i consider continuum classical thermal field theory with the ultraviolet regulated by higher - derivative interactions . in section 5 , i turn to the canonical definition of the classical theory on a simple cubic spatial lattice . by explicit computation , the damping at long distances is shown to be anisotropic and to have an interesting structure of cusp and logarithmic singularities in its angular distribution . i discuss the origin of these singularities . then i argue that measurements of the topological transition rate in simple lattice theories can still ( if properly interpreted ) be used to estimate the real transition rate , and i propose a rough measure of the systematic error arising from the anisotropy of the lattice . in section 6 , i propose that the real transition rate could in principle be measured arbitrarily well from lattice simulations by implementing a lattice version of the higher - derivative continuum theory discussed in section 4 . in section 7 , i briefly discuss some possibilities of alternative lattice theories that may be more rotational - invariant than the canonical one , yet easier to implement than my proposal of section 6 . section 8 explores two quantities important to thermal field theory but slightly tangential to the main thrust of this paper : the plasma frequency and the debye mass . i compute both for classical , thermal gauge theory on the lattice . the results provide one test of whether lattice simulations of a given size are indeed in the small coupling limit , and i comment on the application of this test to the simulations of ref . section 9 offers my conclusions and a summary of what remains to be done .

it has recently been argued that the rate per unit volume of baryon number violation ( topological transitions ) in the hot , symmetric phase of electroweak theory is of the form in the weak - coupling limit , where is a non - perturbative numerical coefficient . over the past several years , there have been attempts to extract the rate of baryon number violation from real - time simulations of classical thermal field theory on a spatial lattice . unfortunately , the coefficient will not be the same for classical lattice theories and the real quantum theory . ( this is unlike euclidean - time measurements , where rotational invariance is always recovered in the continuum limit . ) i then propose how this violation of rotational invariance can be eliminated and the real b violation rate measured by choosing an appropriate lattice hamiltonian . i also propose a rough measure of the systematic error to be expected from using simpler , unimproved hamiltonians . as a byproduct of my investigation , the plasma frequency and debye mass are computed for classical thermal field theory on the lattice .

it has recently been argued that the rate per unit volume of baryon number violation ( topological transitions ) in the hot , symmetric phase of electroweak theory is of the form in the weak - coupling limit , where is a non - perturbative numerical coefficient . over the past several years , there have been attempts to extract the rate of baryon number violation from real - time simulations of classical thermal field theory on a spatial lattice . unfortunately , the coefficient will not be the same for classical lattice theories and the real quantum theory . however , by analyzing the appropriate effective theory on the lattice using the method of hard thermal loops , i show that the _ only _ obstruction to precisely relating the rates in the real and lattice theories is the fact that the long - distance physics on the lattice is not rotationally invariant . ( this is unlike euclidean - time measurements , where rotational invariance is always recovered in the continuum limit . ) i then propose how this violation of rotational invariance can be eliminated and the real b violation rate measured by choosing an appropriate lattice hamiltonian . i also propose a rough measure of the systematic error to be expected from using simpler , unimproved hamiltonians . as a byproduct of my investigation , the plasma frequency and debye mass are computed for classical thermal field theory on the lattice . * v * -20pt

1009.2483

i

[ [ section ] ] consider a family @xmath8 over the open disk , satisfying a suitable condition of local triviality over @xmath9 . in @xcite , verdier defines a ` specialization morphism ' for constructible functions , producing a function @xmath10 on the central fiber @xmath3 of the family for every constructible function @xmath11 on @xmath5 . the key property of this specialization morphism is that it commutes with the construction of chern classes of constructible functions in the sense of macpherson ( @xcite ) ; cf . theorem 5.1 in verdier s note . the specialization morphism for constructible functions is induced from a morphism at the level of constructible sheaves @xmath12 , by taking alternating sums of ranks for the corresponding complex of nearby cycles @xmath13 . the main purpose of this note is to give a more direct description of the specialization morphism ( in the algebraic category , over algebraically closed fields of characteristic @xmath14 ) , purely in terms of constructible functions and of resolution of singularities , including an elementary proof of the basic compatibility relation with chern classes . we will assume that @xmath5 is nonsingular away from @xmath3 , and focus on the case of the specialization of constant functions ; by linearity and functoriality properties , this suffices in order to determine @xmath15 in the situation considered by verdier . on the other hand , the situation we consider is more general than the specialization template recalled above : we define a constructible function @xmath2 for _ every _ proper closed subscheme @xmath3 of a variety @xmath5 ( such that @xmath1 is nonsingular ) , which agrees with verdier s specialization of the constant function @xmath4 when @xmath3 is the fiber of a morphism from @xmath5 to a nonsingular curve . the definition of @xmath2 ( definition [ maindef ] ) is straightforward , and can be summarized as follows . let @xmath16 be a proper birational morphism such that @xmath17 is nonsingular , and @xmath18 is a divisor with normal crossings and nonsingular components , and for which @xmath19 is an isomorphism . then define @xmath20 to be @xmath21 if @xmath22 is on a single component of @xmath23 of multiplicity @xmath21 , and @xmath14 otherwise ; and let @xmath2 be the push - forward of @xmath24 to @xmath3 . readers who are familiar with verdier s paper @xcite should recognize that this construction is implicit in 5 of that paper , if @xmath3 is the zero - locus of a function on @xmath5 . our contribution is limited to the realization that the weak factorization theorem of @xcite may be used to adopt this prescription as a _ definition , _ that the properties of this function follow directly from the standard apparatus of intersection theory , and that this approach extends the theory beyond the specialization situation considered by verdier ( at least in the algebraic case ) . denoting by @xmath25 the chern - schwartz - macpherson class of a constructible function , we prove the following : * theorem i. * let @xmath26 be an effective cartier divisor . then @xmath27 an expression for @xmath28 in terms of the basic ingredients needed to define @xmath2 as above may be given as soon as @xmath29 is a regular embedding ( cf . remark [ regemb ] ) . in fact , with suitable positions , theorem i holds for arbitrary closed embeddings @xmath0 ( theorem [ spethm ] ) . theorem i reproduces verdier s result when @xmath3 is a fiber of a morphism from @xmath5 to a nonsingular curve ; in that case ( but not in general ) @xmath30 may be replaced with @xmath4 , as in verdier s note . the definition of @xmath2 is clearly compatible with smooth maps , and in particular the value of @xmath2 at a point @xmath22 may be computed after restricting to an open neighborhood of @xmath22 . thus , verdier s formula for the specialization function in terms of the euler characteristic of the intersection of a nearby fiber with a ball ( 4 in @xcite ) may be used to compute @xmath2 if @xmath3 is a divisor in @xmath5 , over @xmath31 . from the definition it is clear that the function @xmath2 is birationally invariant in the following weak sense : * theorem ii . * let @xmath32 be a proper birational morphism ; let @xmath33 , and assume that @xmath34 restricts to an isomorphism @xmath35 . then @xmath36 ( here , @xmath37 is the push - forward of constructible functions . ) in fact , the whole specialization _ morphism _ commutes with arbitrary proper maps , at least in verdier s specialization situation ( @xcite , corollary 3.6 ) . it would be desirable to establish this fact for the morphism induced by @xmath2 for arbitrary @xmath3 , by the methods used in this paper . the definition summarized above yields a natural decomposition of the constructible function @xmath38 ( and hence of its chern class @xmath28 ) according to the multiplicities of some of the exceptional divisors , see remarks [ distdecfun ] and [ distdeccla ] . in the specialization situation , this decomposition matches the one induced on the milnor fiber by monodromy , as follows from the description of the latter in @xcite . as schrmann pointed out to me , an analogous description in the more general case considered here may be found in @xcite , theorem 3.2 . [ [ section-1 ] ] in the basic specialization situation , in which @xmath3 is the zero - scheme of a function @xmath39 on @xmath5 and @xmath5 is nonsingular , let @xmath40 be the _ singularity subscheme _ of @xmath3 ( i.e. , the ` critical scheme ' of @xmath39 ) . one can define a constructible function @xmath41 on @xmath3 by @xmath42 in this case ( and over @xmath31 ) , the function @xmath2 agrees with verdier s specialization function @xmath43 ( here we use notation as in @xcite , cf . especially proposition 5.1 ) . the function @xmath41 is @xmath14 outside of @xmath40 , so may be viewed as a constructible function on @xmath40 . in fact , it has been observed ( cf . e.g. , @xcite ) that the function @xmath41 is a specific linear combination of local euler obstructions , and in particular it is determined by the scheme @xmath40 and can be generalized to arbitrary schemes . kai behrend denotes this generalization @xmath44 in @xcite . the definition of @xmath2 given in this paper yields an alternative computation of @xmath41 when @xmath40 is the critical scheme of a function , and theorem ii describes the behavior through modifications along @xmath3 of this function : if @xmath32 is as in theorem ii , then @xmath45 provided that @xmath5 and @xmath46 are nonsingular , @xmath3 and hence @xmath33 are hypersurfaces , and @xmath40 , @xmath47 are their singularity subschemes . as @xmath2 is defined for arbitrary @xmath0 , there may be a generalization of ( * ) linking behrend s function and @xmath48 when @xmath3 is not necessarily a hypersurface ; it would be interesting to have statements analogous to ( * * ) , holding for more general @xmath3 . in [ wmather ] we comment on the relation between @xmath7 and the ` weighted chern - mather class ' of the singularity subscheme @xmath40 of a hypersurface @xmath3 ; the degree of this class is a donaldson - thomas type invariant ( @xcite , 4.3 ) . we also provide an explicit formula for the function @xmath41 in terms of a resolution of the hypersurface @xmath3 . it would be interesting to extend these results to the non - hypersurface case . in a different vein , j. schrmann has considered the iteration of the specialization operator over a set of generators for a complete intersection @xmath0 ( @xcite , definition 3.6 ) . it would be a natural project to compare schrmann s definition ( which depends on the order of the generators ) with our definition of @xmath2 ( which is independent of the order , and may be extended to arbitrary @xmath0 ) . schrmann also points out that the _ deformation to the normal cone _ may be used to reduce an arbitrary @xmath0 to a specialization situation ; this strategy was introduced in @xcite , and is explained in detail in @xcite , 1 . again , it would be interesting to establish the precise relation between the resulting specialization morphism and the function @xmath2 studied here . [ [ section-2 ] ] we include in [ mot ] a brief discussion of a ` motivic ' invariant @xmath6 , also defined for any closed embedding @xmath0 into a variety , still assumed for simplicity to be nonsingular outside of @xmath3 . this invariant can be defined in the quotient of the grothendieck ring of varieties by the ideal generated by the class of a torus @xmath49 , or in a more refined relative ring over @xmath3 . the definition is again extremely simple , when given in terms of a resolution in which the inverse image of @xmath3 is a divisor with normal crossings ; the proof that the invariant is well - defined also follows from the weak factorization theorem . as its constructible function counterpart , the class @xmath6 admits a natural ` monodromy ' decomposition ( although a milnor fiber is not defined in general in the situation we consider ) , see remark [ distdecmot ] . when @xmath5 is nonsingular and @xmath3 is the zero - locus of a function on @xmath5 , @xmath6 is a poor man s version of the denef - loeser motivic milnor / nearby fiber ( @xcite , 3.5 ) ; it is defined in a much coarser ring , but it carries information concerning the topological euler characteristic and some other hodge - type data . we note that @xmath6 is _ not _ the image of the limit of the _ naive _ motivic zeta function of denef - loeser , since it does carry multiplicity information , while @xmath50 discards it ( see for example @xcite , corollary 3.3 ) . the limit of the ( non - naive ) denef - loeser motivic zeta function @xmath51 encodes the multiplicity and much more as actual monodromy information , and in this sense it lifts the information carried by our @xmath6 . it would be interesting to define and study an analogous lift for more general closed embeddings @xmath0 , and possibly allowing @xmath5 to be singular along @xmath3 . the approach of @xcite could be used to unify the constructions of @xmath2 and @xmath6 given in this paper , and likely extend them to other environments , but we will not pursue such generalizations here since our aim is to keep the discussion at the simplest possible level . likewise , ` celestial ' incarnations of the milnor fiber ( in the spirit of @xcite , @xcite ) will be discussed elsewhere . [ [ section-3 ] ] in our view , the main advantages of the approach taken in this paper are the simplicity afforded by the use of the weak factorization theorem and the fact that the results have a straightforward interpretation for any closed embedding @xmath0 , whether arising from a specialization situation or not . these results hold with identical proofs over any algebraically closed field of characteristic zero . we note that the paper @xcite of van proeyen and veys also deals with arbitrary closed embeddings with nonsingular complements , as in this note . a treatment of verdier specialization over arbitrary algebraically closed fields of characteristic zero , also using only the standard apparatus of intersection theory , was given by kennedy in @xcite by relying on the lagrangian viewpoint introduced by c. sabbah @xcite . fu ( @xcite ) gives a description of verdier s specialization in terms of normal currents . we were motivated to take a new look at verdier s specialization because of applications to string - theoretic identities ( cf . @xcite , 4 ) . also , verdier specialization offers an alternative approach to the main result of @xcite . the main reason to allow @xmath5 to have singularities along @xmath3 is that this typically is the case for specializations arising from pencils of hypersurfaces in a linear system , as in these applications . see [ pencils ] for a few simpler examples illustrating this point . [ [ section-4 ] ] i am indebted to m. marcolli for helpful conversations , and i thank j. schrmann and w. veys for comments on a previous version of this paper .

let be a closed embedding , with nonsingular . we define a constructible function on , agreeing with verdier s specialization of the constant function when is the zero - locus of a function on . our definition is given in terms of an embedded resolution of ; the independence on the choice of resolution is obtained as a consequence of the weak factorization theorem of . the main property of is a compatibility with the specialization of the chern class of the complement . with the definition adopted here , this is an easy consequence of standard intersection theory . it recovers verdier s result when is the zero - locus of a function on .

let be a closed embedding , with nonsingular . we define a constructible function on , agreeing with verdier s specialization of the constant function when is the zero - locus of a function on . our definition is given in terms of an embedded resolution of ; the independence on the choice of resolution is obtained as a consequence of the weak factorization theorem of . the main property of is a compatibility with the specialization of the chern class of the complement . with the definition adopted here , this is an easy consequence of standard intersection theory . it recovers verdier s result when is the zero - locus of a function on . our definition has a straightforward counterpart in a motivic group . the function and the corresponding chern class and motivic aspect all have natural ` monodromy ' decompositions , for for any as above . the definition also yields an expression for kai behrend s constructible function when applied to ( the singularity subscheme of ) the zero - locus of a function on .

1306.3350

i

let @xmath0 be a closed orientable surface of genus @xmath1 equipped with the area form , and let @xmath2 be the identity component of the group of area - preserving diffeomorphisms of @xmath0 . denote by @xmath7 and @xmath8 the unit @xmath9-disc in a plane and the standard @xmath9-sphere respectively . in @xcite gambaudo and ghys gave a construction of quasi - morphisms on @xmath10 and on @xmath11 from quasi - morphisms on the artin pure braid group @xmath12 and on the pure sphere braid group @xmath13 respectively . in this paper we are going to extend their construction to the case of @xmath2 , when @xmath14 , and to the case of @xmath4 , when @xmath15 . here @xmath4 is the group of hamiltonian diffeomorphisms of @xmath0 . more precisely , we are going to show that every non - trivial homogeneous quasi - morphism on the full surface braid group @xmath16 defines a non - trivial homogeneous quasi - morphism on the group @xmath2 . as a consequence of this construction , we construct two injective homomorphisms @xmath5 , which are bi - lipschitz with respect to the word metric on @xmath6 and the autonomous and fragmentation metrics on @xmath4 respectively . in particular , we generalize the results of the author and kedra proved for the case of @xmath10 , see @xcite . a quasi - morphism on @xmath4 is called calabi if it equals to the calabi homomorphism on every diffeomorphism supported in a displaceable disc . they have various applications in dynamics and symplectic geometry , see e.g. @xcite . calabi quasi - morphisms on @xmath4 were constructed by entov - polterovich @xcite ( in case when @xmath17 ) and by py @xcite ( in case when @xmath18 ) . in this paper we give a new topological construction of infinitely many linearly independent calabi quasi - morphisms on @xmath4 for every @xmath14 . let us start with a definition . a function @xmath19 from a group @xmath20 to the reals is called a _ quasi - morphism _ if there exists a real number @xmath21 such that @xmath22 for all @xmath23 . the infimum of such @xmath24 s is called the _ defect _ of @xmath25 and is denoted by @xmath26 . if @xmath27 for all @xmath28 and all @xmath29 then @xmath25 is called _ homogeneous_. any quasi - morphism @xmath25 can be homogenized by setting @xmath30 the vector space of homogeneous quasi - morphisms on @xmath20 is denoted by @xmath31 . the space of homogeneous quasi - morphisms on @xmath20 modulo the space of homomorphisms on @xmath20 is denoted by @xmath32 . for more information about quasi - morphisms and their connections to different brunches of mathematics , see @xcite . for every pair of points @xmath33 let us choose a geodesic path @xmath34\to { { \mathbf \sigma}}_g$ ] from @xmath35 to @xmath36 . let @xmath37 be an isotopy from the identity to @xmath38 and let @xmath39 be a basepoint . for @xmath40 we define a loop @xmath41\to{{\mathbf \sigma}}_g$ ] by @xmath42\\ f_{3t-1}(y ) & \text { for } t\in \left [ \frac13,\frac23\right ] \\ s_{f(y)z}(3t-2 ) & \text { for } t\in \left [ \frac23,1\right ] . \end{cases}\ ] ] let @xmath43 be the configuration space of all ordered @xmath3-tuples of pairwise distinct points in the surface @xmath0 . it s fundamental group @xmath44 is identified with the pure surface braid group @xmath45 . let point @xmath46 in @xmath43 be a base point . for almost each point @xmath47 the @xmath3-tuple of loops @xmath48 is a based loop in the configuration space @xmath43 . let @xmath14 . then the group @xmath2 is simply - connected ( see e.g. ( * ? ? ? * section 7.2 ) ) . hence the based homotopy class of this loop does not depend on the choice of the isotopy @xmath49 . let @xmath50 be an element represented by this loop . let @xmath15 . in this case the group @xmath51 is not simply - connected . however , the group @xmath52 is simply - connected ( see e.g. ( * ? ? ? * section 7.2 ) ) . hence , for every hamiltonian isotopy @xmath49 between @xmath53 and @xmath54 , the based homotopy class of the loop @xmath48 does not depend on the choice of the isotopy @xmath49 . let @xmath55 be a homogeneous quasi - morphism , where @xmath16 is a full braid group on @xmath3 strings on a surface @xmath0 . for @xmath14 we define the map @xmath56 by @xmath57 for @xmath15 the map @xmath58 is defined as in . [ t : gen - gg ] let @xmath14 . then the function @xmath59 is a well defined homogeneous quasi - morphism . let @xmath15 . then the function @xmath60 is a well defined homogeneous quasi - morphism . note that when @xmath61 the group @xmath62 . in this case theorem [ t : gen - gg ] was proved by l. polterovich in @xcite and by gambaudo - ghys in @xcite . we must add that when @xmath15 and @xmath61 the polterovich quasi - morphism @xmath63 becomes trivial on @xmath52 . by theorem [ t : gen - gg ] , in case when @xmath14 , the above construction defines linear map @xmath64 and in case when @xmath15 it defines linear map @xmath65 [ t : prop - gg - map ] let @xmath66 . the map @xmath67 is injective for each @xmath68 . in case when @xmath15 , one can similarly show that the map @xmath69 is injective . in case when @xmath17 , theorem [ t : prop - gg - map ] was proved by ishida @xcite . recall that @xmath0 is a compact orientable surface of genus @xmath1 . let @xmath70 be a smooth function with zero mean . there exists a unique vector field @xmath71 such that @xmath72 it is easy to see that @xmath71 is tangent to the level sets of @xmath73 . let @xmath74 be the time - one map of the flow @xmath75 generated by @xmath71 . the diffeomorphism @xmath74 is area - preserving and every diffeomorphism arising in this way is called _ autonomous_. such a diffeomorphism is relatively easy to understand in terms of its generating function . let @xmath4 be the group of hamiltonian diffeomorphisms of @xmath0 . since it is a simple group @xcite , every hamiltonian diffeomorphism of @xmath0 is a composition of finitely many autonomous diffeomorphisms . the set of such diffeomorphisms is invariant under conjugation . hence we define the conjugation - invariant _ autonomous norm _ on the group @xmath4 by @xmath76 where each @xmath77 is autonomous . the associated metric is defined by @xmath78 since the autonomous norm is conjugation - invariant , the autonomous metric is bi - invariant . [ t : main ] let @xmath14 . then for every natural number @xmath79 there exists an injective homomorphism @xmath80 which is bi - lipschitz with respect to the word metric on @xmath6 and the autonomous metric on @xmath4 . * remarks . * 1 . the study of bi - invariant metrics on groups of geometric origin was initiated by burago , ivanov and polterovich @xcite . the autonomous metric is a particular case of such metrics . moreover , it is the most natural word metric on the group of hamiltonian diffeomorphisms of a symplectic manifold . it is particulary interesting in the two - dimensional case due to the following observation . + let @xmath81 be a morse function and let @xmath74 be a hamiltonian diffeomorphism generated by @xmath73 . after cutting the surface @xmath0 along critical level sets we are left with finite number of regions , so that each one of them is diffeomorphic to annulus . by arnold - liouville theorem @xcite , there exist angle - action symplectic coordinates on each one of these annuli , so that @xmath74 rotates each point on a regular level curve with the same speed , i.e. the speed depends only on the level curve . it follows that a generic hamiltonian diffeomorphism of @xmath0 may be written as a finite composition of autonomous diffeomorphisms , such that each one of these diffeomorphisms is `` almost everywhere rotation '' in right coordinates , and hence relatively simple . of course this decomposition is nor unique , neither canonical . however , it is plausible that it might be useful in dynamical systems . 2 . let @xmath82 be the group of smooth compactly supported area - preserving diffeomorphisms of the open unit disc @xmath83 in the euclidean plane . in @xcite together with kedra we showed that for every positive number @xmath84 there exists an injective homomorphism @xmath85 which is bi - lipschitz with respect to the word metric on @xmath86 and the autonomous metric on @xmath82 . we should mention that gambaudo and ghys showed that the diameter of the group of area - preserving diffeomorphisms of the 2-sphere equipped with the autonomous metric is infinite as well , see ( * ? ? ? * section 6.3 ) . 3 . let @xmath87 be a closed symplectic manifold . it is interesting to know whether the autonomous metric is unbounded on the group of hamiltonian diffeomorphisms @xmath88 , and if yes then can one prove theorem [ t : main ] in the case when the dimension of @xmath89 is greater than two ? let @xmath90 be a closed symplectic manifold , and denote by @xmath88 the group of hamiltonian diffeomorphisms of @xmath89 . let @xmath91 be a collection of regions such that * for each @xmath92 the region @xmath93 is diffeomorphic to a ball of the same dimension as @xmath89 . * all elements in @xmath91 have the same volume @xmath94 . every region of volume @xmath94 in @xmath89 , which is diffeomorphic to a @xmath95-dimensional ball , lies in @xmath91 . since @xmath88 is a simple group @xcite , every diffeomorphism in @xmath88 is a composition of finitely many diffeomorphisms , such that each such diffeomorphism is supported in some ball @xmath96 . the set of such diffeomorphisms is invariant under conjugation . we define the conjugation - invariant _ fragmentation norm _ on the group @xmath88 with respect to the cover @xmath91 by @xmath97 where the support of @xmath77 lies in some @xmath98 . the associated metric is defined by @xmath99 . since the fragmentation norm is conjugation - invariant , the fragmentation metric is bi - invariant . recently , fragmentation metric received a considerable attention , since , in particular , it is related to the question about the simplicity of the group of compactly supported area - preserving homeomorphisms of an open 2-disc , see e.g. @xcite . [ t : frag - metric ] * 1 . * let @xmath0 be any closed surface of genus @xmath14 . then there exists an injective homomorphism @xmath100 which is bi - lipschitz with respect to the word metric on @xmath101 and the fragmentation metric on @xmath4 . let @xmath90 be a closed symplectic manifold of dimension greater then two . suppose that there exists an embedding of the free group on two generators @xmath102 such that the image of the induced map @xmath103 is infinite dimensional . then for every natural number @xmath79 there exists an injective homomorphism @xmath104 which is bi - lipschitz with respect to the word metric on @xmath6 and the fragmentation metric on @xmath88 . it is known that every non - elementary word - hyperbolic group @xmath20 contains @xmath105 such that the induced map @xmath106 is infinite dimensional , see @xcite . let @xmath90 be a closed negatively curved symplectic manifold of dimension greater then two . then @xmath107 contains bi - lipschitz embedded finitely generated free abelian group of an arbitrary rank . let @xmath90 be a symplectic manifold . recall that a homogeneous quasi - morphism @xmath108 on @xmath88 is called calabi if its restriction to a subgroup @xmath109 , where @xmath110 is a displaceable ball in @xmath89 , coincides with calabi homomorphism on @xmath109 , see e.g. @xcite . here we focus on the case of surfaces , but the works quoted above deal also with groups of hamiltonian diffeomorphisms of higher dimensional symplectic manifolds . in @xcite entov and polterovich constructed a calabi quasi - morphism on a group of hamiltonian diffeomorphisms of the two - sphere . their construction uses quantum homology . they asked whether one can construct a calabi quasi - morphism on @xmath4 , where @xmath111 . around 2006 pierre py gave a positive answer to this question , see @xcite . let @xmath14 . in this paper we give a different construction of an infinite family of calabi quasi - morphisms on @xmath4 . let @xmath112 be an open disc in the euclidean plane and denote by @xmath113 the group of compactly supported hamiltonian diffeomorphisms of @xmath112 . let @xmath114 be a homomorphism , which takes value @xmath115 on the generator of the braid group @xmath116 on two generators , which is cyclic . let @xmath117 be the induced , by gambaudo - ghys construction , homomorphism from @xmath113 to the reals . the calabi homomorphism @xmath118 may be defined as follows : @xmath119 for the proof of this fact see e.g. @xcite . it turns out that the generalized gambaudo - ghys construction allows us to construct an infinite family of linearly independent calabi quasi - morphisms , i.e. we prove the following [ t : calabi - qm ] let @xmath14 . the space of calabi quasi - morphisms on @xmath4 is infinite dimensional . we should note that the above theorem is not entirely new , because py construction @xcite implies a proof of this theorem as well . nevertheless , we think that this theorem is interesting , since our construction is entirely different from the one given by pierre py .

let be a closed orientable surface of genus and let be the identity component of the group of area - preserving diffeomorphisms of . in this work we present the extension of gambaudo - ghys construction to the case of a closed hyperbolic surface , i.e. we show that every non - trivial homogeneous quasi - morphism on the braid group on strings of defines a non - trivial homogeneous quasi - morphism on the group . as a consequence let be the group of hamiltonian diffeomorphisms of . as an application of the above construction we construct two injective homomorphisms , which are bi - lipschitz with respect to the word metric on and the autonomous and fragmentation metrics on . in addition , we construct a new infinite family of calabi quasi - morphisms on .

let be a closed orientable surface of genus and let be the identity component of the group of area - preserving diffeomorphisms of . in this work we present the extension of gambaudo - ghys construction to the case of a closed hyperbolic surface , i.e. we show that every non - trivial homogeneous quasi - morphism on the braid group on strings of defines a non - trivial homogeneous quasi - morphism on the group . as a consequence we give another proof of the fact that the space of homogeneous quasi - morphisms on is infinite dimensional . let be the group of hamiltonian diffeomorphisms of . as an application of the above construction we construct two injective homomorphisms , which are bi - lipschitz with respect to the word metric on and the autonomous and fragmentation metrics on . in addition , we construct a new infinite family of calabi quasi - morphisms on .

math0209111

i

let @xmath0 be a compact symplectic manifold , and suppose an @xmath1-dimensional torus @xmath2 acts on @xmath0 in a hamiltonian fashion . let @xmath7 be a moment map . the components of the moment map are ( equivariant ) morse - bott functions on @xmath0 , and hence may be used to determine the ( equivariant ) topology of @xmath0 . now suppose that @xmath3 is an anti - symplectic involution on @xmath0 that anti - commutes with the @xmath2 action : @xmath8 for all @xmath9 and @xmath10 . we assume that @xmath11 is nonempty . we call the fixed point set @xmath12 of @xmath3 the _ real locus _ of @xmath0 . the real locus is a lagrangian submanifold of @xmath0 . the motivating example of a real locus is a complex variety @xmath0 under complex conjugation @xmath3 . in this case , the real locus is the set of real points on the variety . by ( [ eq : anticommute ] ) the subgroup of @xmath2 of elements of order 2 acts on @xmath13 . we call this subgroup @xmath14 , the _ real torus _ in @xmath2 . it is immediate that @xmath15 . duistermaat @xcite proved that the real locus has full moment image , that is , that @xmath16 . moreover , he showed that the components of the moment map are morse - bott functions for the real locus , when using @xmath17 coefficients , and so we can understand the topology of @xmath13 via the moment map . biss , guillemin and the second author @xcite proved that these moment map components can also be used to understand the equivariant topology of the real locus with respect to the restricted @xmath14 action . in addition , sjamaar and oshea have generalized the results of duistermaat for hamiltonian actions of nonabelian lie groups @xcite . the principle behind these results is that real loci should behave in a fashion similar to the symplectic manifolds of which they are submanifolds . we show that this philosophy applies in the context of symplectic reductions . given a compact hamiltonian @xmath2-space @xmath0 , kirwan @xcite proved that when @xmath2 acts freely on @xmath18 , the inclusion map @xmath19 induces a surjection in equivariant cohomology with rational coefficients : @xmath20 the map @xmath21 is called the kirwan map . in @xcite the authors note that under reasonable assumptions about the torsion of the fixed point sets and the group action , this map is surjective over the integers as well . for @xmath22 one can state the result as follows . for any prime @xmath23 , assume that _ either _ there is no @xmath23-torsion in the @xmath24-cohomology of the fixed point set @xmath25 , _ or _ every point not in @xmath25 has a free @xmath26 action . then the map ( [ eq : surjection ] ) is surjective with integer coefficients . for higher dimensional @xmath2 , the authors assume the action is quasi - free to prove kirwan s surjectivity over @xmath24 , although this is stronger than necessary . one natural question is , what is the kernel of @xmath21 ? this question was answered in @xcite and refined by the first author in the case of rational coefficients @xcite . suppose that @xmath0 has the additional structure of an anti - symplectic involution , compatible with torus action as specified above . suppose also that @xmath27 is nonempty . in this article we prove a surjectivity result for real loci analogous to kirwan s result ( theorem 1 ) . we also use tolman and weitsman s equivariant morse theoretic methods to compute the kernel of this real version of the kirwan map ( theorem 2 ) . thus the main contribution of this article is to complete the program begun by duistermaat in showing that the real locus of a hamiltonian @xmath2-space has a @xmath28-action which behaves as if it were itself a hamiltonian @xmath2-space . not only is the equivariant cohomology ring of @xmath13 described by restrictions to fixed points , but there is a well - defined notion of real reduction " and the major theorems about the cohomology rings of reduced spaces go through in the real case . the proofs we present are straightforward extensions of the work of atiyah - bott , kirwan , and tolman - weitsman to the real case . we then explore the ramifications of real reduction and these theorems in a series of examples . the key to kirwan s proof of surjectivity is the analysis of the function @xmath29 in the case of @xmath30 , this is not a morse function , but it is morse - bott except at @xmath31 , the function s minimum value . for @xmath32 , there are a finite number of critical values of @xmath33 where @xmath34 is not morse - bott , in addition to the minimum . moreover , @xmath35 kirwan uses the critical sets of @xmath36 to argue inductively ( on the critical sets ) that the equivariant cohomology of @xmath0 surjects onto the ordinary cohomology of @xmath37 . it is clear that surjectivity for general values @xmath38 of @xmath33 follows by shifting the moment map by an appropriate constant . these arguments easily go through when the function is restricted to the real locus , as we will see . a fundamental step in the proof of surjectivity and the computation of the kernel is a lemma due to atiyah and bott @xcite . this lemma describes the local topology , in and around a fixed point component of @xmath25 . under the hypothesis of no 2-torsion ( definition [ def:2torsion ] ) we show that this lemma still holds for @xmath28-cohomology with @xmath17 coefficients . we then apply the lemma and an inductive argument to both the question of surjectivity and to the computation of the kernel for real loci . for the rest of this article , let @xmath0 be a compact , connected symplectic manifold , endowed with a hamiltonian action of a compact torus @xmath2 and moment map @xmath33 . let @xmath3 be an anti - symplectic involution , anti - commuting with the torus action , and let @xmath39 be the set of order 2 elements , plus the identity . recall that @xmath14 acts on @xmath13 . we begin with a definition . [ def:2torsion ] let @xmath40 be a critical point of @xmath33 . let @xmath41 be the maximal connected subtorus of @xmath2 fixing @xmath23 , @xmath42 the weight lattice of @xmath2 , and @xmath43 the weight lattice of @xmath44 . let @xmath45 be the connected component of @xmath46 containing @xmath23 . we denote by @xmath47 the weights of the @xmath44 action on @xmath48 , the fiber over @xmath23 of the normal bundle to @xmath45 in @xmath0 , where @xmath49 . we say that @xmath23 is a _ 2-torsion point _ if @xmath50 for some @xmath51 . suppose @xmath0 has a symplectic involution @xmath3 with real locus @xmath13 . suppose @xmath23 is critical for @xmath33 , fixed by @xmath44 , and @xmath45 is the connected component of @xmath46 containing @xmath23 . if @xmath23 lies in @xmath13 , the @xmath44 action on the normal bundle to @xmath45 in @xmath0 restricts to an @xmath52 action on the normal bundle @xmath53 to @xmath54 in @xmath13 . let @xmath55 be the weights of the irreducible representations obtained by this action on the fiber over @xmath23 . then @xmath23 is a 2-torsion point if and only if @xmath56 is trivial for some @xmath57 . the functions @xmath58 and a family of perturbations ( see section [ se : kernel ] ) have critical sets fixed by various subtori of @xmath2 . tolman and weitsman prove @xmath59 is a surjection provided that @xmath2 acts quasi - freely . as they note , one needs to ensure that the negative normal bundles to all critical sets of these functions have nontrivial equivariant euler classes . for the real locus case , no 2-torsion points " plays the same role as quasi - free " does in the symplectic case . at the critical sets of this family of functions , the negative normal bundles have top stiefel - whitney classes which are nontrivial in the @xmath60-component of @xmath61 for each connected component @xmath45 of the critical set . this amounts to the requirement that , over any point in @xmath45 , the negative normal bundle splits into one - dimensional _ nontrivial _ representations of @xmath14 . the condition that @xmath13 contain no @xmath62-torsion points ensures this for the family of perturbations of @xmath58 . the first main theorem is the surjectivity analogue of for real loci . [ th : surjectivity ] suppose @xmath0 is a compact symplectic manifold with a hamiltonian @xmath63-action , and that @xmath13 is the real locus of @xmath0 . let @xmath33 be a moment map on @xmath0 and @xmath38 a regular value of @xmath33 . suppose further that @xmath63 acts freely on @xmath18 and that @xmath13 contains no @xmath62-torsion points . then the _ real kirwan map _ in @xmath14-equivariant cohomology with @xmath17 coefficients @xmath64 induced by inclusion , is a surjection . we omit the superscript @xmath1 in @xmath63 when the dimension is clear . the hypothesis that the real locus have no @xmath62-torsion points is reasonably strong . real loci of toric varieties and coadjoint orbits in type @xmath65 satisfy this hypothesis , for example , but the real loci of maximal coadjoint orbits in type @xmath66 do not . the next task we complete is to specify the kernel of @xmath67 . the kernel is precisely the real analog of the kernel found in @xcite . for every @xmath68 , let @xmath69 let @xmath70 denote the fixed point set , and let @xmath71 finally , define the ideal @xmath72 [ th : kernel ] suppose @xmath0 is a compact symplectic manifold with a hamiltonian @xmath2-action , an anti - symplectic involution @xmath3 anti - commuting with @xmath2 , and real locus @xmath13 . let @xmath33 be a moment map on @xmath0 and @xmath38 a regular value of @xmath33 such that @xmath2 acts freely on @xmath18 and that @xmath13 contains no @xmath62-torsion points . then the kernel of the real kirwan map is the ideal @xmath73 , so there is a short exact sequence @xmath74 where the cohomology is taken with @xmath17 coefficients . the remainder of this paper is organized as follows . in section [ se : top ] , we review @xmath75-equivariant cohomology with specific attention to the case of @xmath17 coefficients and @xmath76 . here we make explicit the thom isomorphism theorem and the @xmath17 version of the atiyah - bott lemma . in section [ se : morsekirwan ] , we study morse - kirwan theory on real loci . in section [ se : surjectivity ] , we discuss reduction , an induced anti - symplectic involution on the symplectic reduction , and we prove the surjectivity theorem ( theorem [ th : surjectivity ] ) . in section [ se : kernel ] , we prove theorem [ th : kernel ] , the description of the kernel of the real kirwan map . finally , in section [ se : egs ] , we work out several pertinent examples . in particular , we present an example of a symplectic reduction which has the same cohomology as its real locus , with degrees divided in half . as @xmath77 does not in general have a torus action , this example generalizes duistermaat s original work on real loci , in which he uses the @xmath2 action to make an analogous statement for @xmath0 and its real locus . the authors would like to thank victor guillemin , robert kleinberg , dan dugger and reyer sjamaar for useful comments during the preparation of this paper .

let be a compact , connected symplectic manifold with a hamiltonian action of a compact-dimensional torus . suppose that is equipped with an anti - symplectic involution compatible with the-action . duistermaat introduced real loci , and extended several theorems of symplectic geometry to real loci . in this paper , we extend another classical result of symplectic geometry to real loci : the kirwan surjectivity theorem . in addition , we compute the kernel of the real kirwan map . these results allow us to show that a symplectic reduction has the same ordinary cohomology as its real locus , with degrees halved .

let be a compact , connected symplectic manifold with a hamiltonian action of a compact-dimensional torus . suppose that is equipped with an anti - symplectic involution compatible with the-action . the real locus of is the fixed point set of . duistermaat introduced real loci , and extended several theorems of symplectic geometry to real loci . in this paper , we extend another classical result of symplectic geometry to real loci : the kirwan surjectivity theorem . in addition , we compute the kernel of the real kirwan map . these results are direct consequences of techniques introduced by tolman and weitsman . in some examples , these results allow us to show that a symplectic reduction has the same ordinary cohomology as its real locus , with degrees halved . this extends duistermaat s original result on real loci to a case in which there is not a natural hamiltonian torus action .

0804.1887

i

local regularity and multifractal analysis have become unavoidable issues in the past years . indeed , physical phenomena exhibiting wild local regularity properties have been discovered in many contexts ( turbulence flows , intensity of seismic waves , traffic analysis , .. ) . from a mathematical viewpoint , the multifractal approach is also a fruitful source of interesting problems . consequently , there is a strong need for a better theoretical understanding of the so - called multifractal behaviors . in this article , we investigate the relations between multifractal properties and time subordination for continuous functions . the most common functions or processes used to model irregular phenomena are monofractal , in the sense that they exhibit the same local regularity at each point . let us recall how the local regularity of a function is measured . let @xmath5 . for @xmath6 and @xmath7 , @xmath1 is said to belong to @xmath8 if there are a polynomial @xmath9 of degree less than @xmath10 $ ] and a constant @xmath11 such that , locally around @xmath12 , latexmath:[\[\label{defpoint } exponent of @xmath1 at @xmath12 is @xmath14 the singularity spectrum of @xmath1 is then defined by @xmath15 ( @xmath16 stands for the hausdorff dimension , and @xmath17 by convention ) . hence , a function @xmath0 is said to be monofractal with exponent @xmath18 when @xmath19 for every @xmath20 . for monofractal functions @xmath1 , @xmath21 , while @xmath22 for @xmath23 . sample paths of brownian motions or fractional brownian motions are known to be almost surely monofractal with exponents less than 1 . for reasons that appear below , * we focus on monofractal functions associated with an exponent @xmath24 $ ] . * more complex models had to be used and/or developed , for at least three reasons : the occurrence of intermittence phenomena ( mainly in fluid mechanics ) , the presence of oscillating patterns ( for instance in image processing ) , or the presence of discontinuities ( in finance or telecommunications ) . such models may have multifractal properties , in the sense that the support of their singularity spectrum is not reduced to a single point . among these processes , whose local regularity varies badly from one point to another , let us mention mandelbrot multiplicative cascades and their extensions @xcite , ( generalized ) multifractional brownian motions @xcite and lvy processes @xcite ( for discontinuous phenomena ) . starting from a monofractal process as above in dimension 1 , a simple and efficient way to get a more elaborate process is to compose it with a time subordinator , i.e. an increasing function or process . mandelbrot , for instance , showed the pertinency of time subordination in the study of financial data @xcite . from a theoretical viewpoint , it is also challenging to understand how the multifractal properties of a function are modified after a time change @xcite . a natural question is to understand the differences between the multifractal processes above and compositions of monofractal functions with multifractal subordinators . a function @xmath0 is said to be the composition of a monofractal function with a time subordinator ( cmt ) when @xmath1 can be written as @xmath25 where @xmath26 is monofractal with exponent @xmath27 and @xmath28 is an increasing homeomorphism of @xmath4 . in this article , we prove that if a continuous function @xmath0 has a `` homogeneous multifractal '' behavior ( in a sense we define just below ) , then @xmath1 is cmt . hence , @xmath1 is the composition of a monofractal function with a time subordinator , and shall simply be viewed as a complication of a monofractal model . this yields a deeper insight into the understanding of multifractal behaviors of continuous functions , and gives a more important role to the multifractal analysis of positive borel measures ( which are derivatives of time subordinators ) . we explain in section [ self ] and [ multi ] how this decomposition can be used to compute the singularity spectrum of the function @xmath1 . let us begin with two cases where a function @xmath1 is obviously cmt : \1 . if @xmath1 is the integral of any positive borel measure @xmath29 , then @xmath30 , where the identity @xmath31 is monofractal and @xmath1 is increasing . remark that in this case , @xmath1 may even have exponents greater than 1 . any monofractal function @xmath32 can be written @xmath33 , where @xmath34 is monofractal and @xmath35 is undoubtably an homeomorphism of @xmath4 . these two simple cases will be met again below . to bring general answers to our problem and thus to exhibit another class of cmt functions , we develop an approach based on the oscillations of a function @xmath36 . for every subinterval @xmath37 , consider the oscillations of order 1 of @xmath1 on @xmath38 defined by @xmath39 * in the sequel , we assume that @xmath1 is continuous and for every non - trivial subinterval @xmath38 of @xmath4 , @xmath40 . * this entails that @xmath1 is nowhere locally constant , which is a natural assumption for the results we are looking for . it is very classical that the oscillations of order 1 characterize precisely the pointwise hlder exponents strictly less than 1 ( see section [ prel ] ) . let us introduce the quantity that will be the basis of our construction . for every @xmath41 , @xmath42 , we consider the dyadic intervals @xmath43 , so that @xmath44 , the union being disjoint . for every @xmath41 and @xmath42 , for simplicity we set @xmath45(@xmath46 since @xmath1 is @xmath47 ) . for every @xmath41 , let @xmath48 be the unique real number such that @xmath49 we then define the intrinsic monofractal exponent of @xmath1 @xmath50 as @xmath51 this quantity @xmath50 characterizes the asymptotic maximal values of the oscillations of @xmath1 on the whole interval @xmath4 . this exponent is the core of our theorem , because it gives an upper limit to the maximal time distortions we are allowed to apply . it is satisfactory that @xmath50 has a functional interpretation . indeed , if @xmath1 can be decomposed as ( [ decomp ] ) , then the exponent of the monofractal function @xmath2 shall not depend on the oscillation approach nor on the dyadic basis . in section [ secgen ] we explain that @xmath52 where @xmath53 and @xmath54 are respectively the besov space and _ oscillation space _ on the open interval @xmath55 ( see jaffard in @xcite for instance ) . for multifractal functions @xmath1 satisfying some multifractal formalism , the exponent @xmath50 can also be read on the singularity spectrum of @xmath1 . indeed ( see section [ secgen ] ) , @xmath50 corresponds to the inverse of the largest possible slope of a straight line going through 0 and tangent to the singularity spectrum @xmath56 of @xmath1 . these remarks are important to have an idea _ a priori _ of the monofractal exponent of @xmath2 in the decomposition @xmath57 . they also give an intrinsic formula for @xmath50 . let us come back to the two simple examples above : \1 . for the integral @xmath1 of any positive measure @xmath29 , @xmath58 , hence @xmath59 , which corresponds to the monofractal exponent of the identity @xmath60 from the oscillations viewpoint . the first difficulties arise for the monofractal functions @xmath32 . when @xmath32 is monofractal of exponent @xmath61 , then we do nt have necessarily @xmath62 . we always have @xmath63 ( see lemma [ lem0 ] in section [ prel ] ) , but it is always possible to construct wild counter - examples . nevertheless , we treat in details the examples of the weierstrass functions and the sample paths of ( fractional ) brownian motions in section [ mono ] , for which the exponent @xmath64 meets our requirements . unfortunately , the knowledge of @xmath50 is not sufficient to get relevant results . for instance , consider a function @xmath1 that has two different monofractal behaviors on @xmath65 and @xmath66 $ ] . such an @xmath1 can be obtained as the continuous juxtaposition of two weierstrass function with distinct exponents @xmath67 : we have @xmath68 , and @xmath1 can not be written as the composition of a monofractal function with a time subordinator . this is a consequence of lemma [ lemmonof ] , which asserts that two monofractal functions @xmath69 and @xmath70 of disctinct exponents @xmath71 and @xmath72 never verify @xmath73 for any continuous increasing function @xmath28 ( indeed , such an @xmath3 would `` dilate '' time everywhere , which is impossible ) . we need to introduce a homogeneity condition * c1 * to get rid of these annoying and artificial cases . this condition heuristically imposes that the oscillations of any restriction of @xmath1 to a subinterval of @xmath4 have the same asymptotic properties as the oscillations of @xmath1 on @xmath4 . * condition * c1 * : * + let @xmath74 , and @xmath75 . let @xmath76 be the function @xmath77 where @xmath78 is the canonical affine contraction which maps @xmath4 to @xmath79 . condition * c1 * is satisfied for @xmath1 when there is a real number @xmath18 such that for every @xmath74 and @xmath75 , @xmath80 . hence @xmath76 is a renormalized version of the restriction of @xmath1 to the interval @xmath79 . remark that @xmath81 does not depend on the normalization factor @xmath82 . although self - similar functions are good candidates to satisfy * c1 * , a function @xmath1 fulfilling this condition does not need at all to possess such a property . in order to guarantee that @xmath1 is cmt , we strengthen the convergence toward @xmath81 . * condition * c2 * : * + assume that condition * c1 * is fulfilled . there are two positive sequences @xmath83 and @xmath84 and two real numbers @xmath85 with the following property : 1 . @xmath83 and @xmath84 are positive non - increasing sequences that converge to zero , and @xmath86 for some @xmath87 . 2 . for every @xmath74 and @xmath88 , the sequence @xmath89 converges to @xmath90 ( it is not only a liminf , it is a limit ) with the following convergence rate : for every @xmath91 $ ] , @xmath92 assuming that @xmath93 is a limit is of course a constraint , but not limiting in practice , since this condition holds for most of the interesting functions or ( almost surely ) for most of the sample paths of processes . similarly , the decreasing behavior ( [ eq1 ] ) is not very restrictive : such a behavior is somehow expected for a @xmath94 function . the convergence speed ( [ eq1 ] ) is a more important constraint , but the convergence rate we impose on @xmath83 toward 0 is extremely slow , and is realized in the most common cases , as shown below . [ maintheo ] let @xmath95 be a continuous function . assume that @xmath1 satisfies * c1 * and * c2*. then @xmath1 is cmt and the function @xmath2 in ( [ decomp ] ) is monofractal of exponent @xmath50 . such a decomposition is of course not unique : if @xmath1 is cmt and @xmath96 is @xmath97 and strictly increasing , then @xmath98 , where @xmath99 is still a monofractal function of exponent @xmath100 and @xmath101 is an increasing function . nevertheless , if two decompositions ( [ decomp ] ) exist respectively with functions @xmath69 , @xmath70 , @xmath102 and @xmath103 , then @xmath69 and @xmath70 are necessarily monofractal with the same exponent @xmath50 . this is again a consequence of lemma [ lemmonof ] . an important consequence of theorem [ maintheo ] is that the ( possibly ) multifractal behavior of @xmath1 is contained in the multifractal behavior of @xmath3 . more precisely , since @xmath3 is an increasing continuous function from @xmath4 to @xmath4 , @xmath3 is the integral of a positive measure , say @xmath29 , on @xmath4 . the local regularity of @xmath29 is classically quantified through a local dimension exponent defined for every @xmath104 by @xmath105 where @xmath106 stands for the ball ( here an interval ) with center @xmath107 and radius @xmath108 , and @xmath109 is the diameter of the set @xmath110 ( @xmath111 ) . the singularity spectrum of @xmath29 is then @xmath112 it is very easy to see that if @xmath113 , then @xmath114 . hence for every @xmath115 , @xmath116 , i.e. there is a direct relationship between the singularity spectrum of @xmath1 and the one of @xmath29 . as a conclusion , theorem [ maintheo ] increases the role of the multifractal analysis of measures , since for the functions satisfying * c1 * and * c2 * , their multifractal behavior is ruled exclusively by the behavior of @xmath29 . as an application of theorem [ maintheo ] , we will prove the following theorem [ thself ] , which relates the so - called self - similar functions @xmath1 introduced in @xcite with the self - similar measures naturally associated with the similitudes defining @xmath1 . let us recall the definition of self - similar functions . let @xmath117 be a lipschitz function on @xmath118 $ ] ( we suppose that the lipschitz constant @xmath119 equals 1 , without loss of generality ) , and let @xmath120 be @xmath121 contractive similitudes satisfying : 1 . for every @xmath122 , @xmath123 ( open set condition ) , 2 . @xmath124 ( the intervals @xmath125 form a covering of @xmath4 ) . we denote by @xmath126 the ratios of the non trivial similitudes @xmath127 . by construction @xmath128 . let @xmath129 be @xmath121 non - zero real numbers , which satisfy @xmath130 [ defiself ] a function @xmath131 is called self - similar when @xmath1 satisfies the following functional equation @xmath132 relation ( [ cond1 ] ) ensures that @xmath1 exists and is unique @xcite . let us consider the unique exponent @xmath133 such that @xmath134 this @xmath135 is indeed greater than 1 , since @xmath136 and @xmath137 for all @xmath138 by ( [ cond1 ] ) . with the probability vector @xmath139 and the similitudes @xmath140 can be associated the unique self - similar probability measure @xmath29 satisfying @xmath141 [ thself ] let @xmath1 be defined by ( [ defself ] ) . then , either @xmath1 is a @xmath142-lipschitz function for some constant @xmath87 ( expliciteley found in section [ self ] ) , or @xmath1 is cmt and there is a monofractal function @xmath2 of exponent @xmath143 such that @xmath144),\ ] ] where @xmath29 is the self - similar measure ( [ defmu ] ) naturally associated with the parameters used to define @xmath1 . the multifractal analysis of @xmath1 follows from the multifractal analysis of @xmath29 , which is a very classical problem ( see @xcite ) . the paper is organized as follows . in section [ proof ] , theorem [ maintheo ] is proved , by explicitly constructing the monofractal function @xmath2 and the time subordinator @xmath3 . section [ secgen ] contains the possible extensions of theorem [ maintheo ] , the explanation of the heuristics ( [ form ] ) , and the discussion for exponents greater than 1 . in section [ mono ] , [ self ] and [ multi ] , we detail several classes of examples to which theorem [ maintheo ] applies . first we prove that the usual monofractal functions @xmath1 with exponents @xmath61 verify * c1 * and * c2*. we prove theorem [ thself ] in section [ self ] . finally we explicitly compute and plot the time subordinator and the monofractal function for a classical family of multifractal functions @xmath145 which include bourbaki s and perkin s functions . let us finish by the direct by - products and the possible extensions of this work : the reader can check that the proof below can be adapted to more general contexts : * the dyadic basis can be replaced by any @xmath146-adic basis . * if @xmath147 converges to zero ( without any given convergence rate ) , then ( under slight modifications of @xmath148 ) the same result holds true . we focused on a simpler case , but in practice , a convergence rate @xmath86 shall always be always obtained . * the fact the the quantities @xmath149 are limits is only used at the beginning of the proof . in fact , only the existence of the scale @xmath150 $ ] such that ( [ eq1 ] ) and ( [ eq1 ] ) hold true at scale @xmath150 $ ] is determinant . in particular , the conditions may be relaxed : we could treat the case where the @xmath149 are only liminf ( and not limits ) . again , in practice they are often limits , this is why we adopted this viewpoint .

let be a continuous function . we show that if is `` homogeneously multifractal '' ( in a sense we precisely define ) , then is the composition of a monofractal function with a time subordinator ( i.e. is the integral of a positive borel measure supported by ) . when the initial function is given , the monofractality exponent of the associated function is uniquely determined . we study in details a classical example of multifractal functions , for which we exhibit the associated functions and . this provides new insights into the understanding of multifractal behaviors of functions . [ 1994/12/01 ]

let be a continuous function . we show that if is `` homogeneously multifractal '' ( in a sense we precisely define ) , then is the composition of a monofractal function with a time subordinator ( i.e. is the integral of a positive borel measure supported by ) . when the initial function is given , the monofractality exponent of the associated function is uniquely determined . we study in details a classical example of multifractal functions , for which we exhibit the associated functions and . this provides new insights into the understanding of multifractal behaviors of functions . [ 1994/12/01 ]

0907.1177

i

let @xmath0 be a simply connected semisimple algebraic group over an algebraically closed field @xmath1 of characteristic 0 ; all @xmath0-modules considered in the following will be supposed to be rational . an algebraic @xmath0-variety is said to be _ spherical _ if it is normal and if it contains an open @xmath10-orbit , where @xmath11 is a borel subgroup ; a subgroup @xmath12 is said to be _ spherical _ if the homogeneous space @xmath7 is so : any spherical variety can thus be regarded as an open embedding of a spherical homogeneous space , namely its open @xmath0-orbit . important classes of spherical varieties are that of toric varieties and that of symmetric varieties : toric varieties are those spherical varieties whose open orbit is an algebraic torus ; symmetric varieties are those spherical varieties whose generic stabilizer @xmath9 is such that @xmath13 , where @xmath14 is an algebraic involution and where @xmath15 is the set of its fixed points . other important classes of spherical varieties are that of flag varieties and the more general one of wonderful varieties : a _ wonderful variety _ ( of _ rank _ @xmath16 ) is a smooth projective @xmath0-variety having an open @xmath0-orbit which satisfies following properties : * the complement of the open @xmath0-orbit is the union of @xmath16 smooth prime divisors having a non - empty transversal intersection ; * any orbit closure equals the intersection of the prime divisors containing it . a spherical subgroup @xmath9 is said to be _ wonderful _ if @xmath7 possesses a wonderful completion ( which is unique , if it exists ) . by @xcite every self - normalizing symmetric subgroup is wonderful ; more generally every self - normalizing spherical subgroup is wonderful by @xcite . however many natural examples of embeddings of a spherical homogeneous space do not need to be normal . for instance , consider a simple @xmath0-module @xmath2 ( in which case we will call @xmath17 a _ simple projective space _ ) possessing a line @xmath18 $ ] fixed by a spherical subgroup . then consider the orbit @xmath19 \subset { \mathbf p}(v)$ ] , which is spherical , and take its closure @xmath20 } \subset { \mathbf p}(v)$ ] , which generally is not normal ; denote @xmath21 its normalization . the aim of this work is the study of the orbits of compactifications which arise in such a way , and as well the study of the orbits of their normalizations . in @xcite it has been proved that any spherical subgroup which occurs as the stabilizer of a point in a simple projective space is wonderful . if @xmath22 is the wonderful completion of @xmath19 $ ] , then the morphism @xmath19 \to x$ ] extends to @xmath22 and thus we get a morphism @xmath23 : examining such morphism we get a description of the set of orbits of @xmath5 and of @xmath21 . moreover this leads to a combinatorial criterion to establish whether or not two orbits in @xmath22 map onto the same orbit in @xmath5 , which in particular implies that different orbits in @xmath5 are never @xmath0-equivariantly isomorphic . our main theorem ( theorem [ teorema caso stretto ] ) is a combinatorial criterion for @xmath24 to be bijective ; this is done under the assumption that @xmath22 is _ strict _ , i.e. that all isotropy groups of @xmath22 are self - normalizing : strict wonderful varieties , introduced in @xcite , are those wondeful varieties which can be embedded in a simple projective space ; they form an important class of wonderful varieties which generalize the symmetric ones of @xcite . the condition of bijectivity involves the double links of the dynkin diagram of @xmath0 and it is trivially fulfilled whenever @xmath0 is simply laced or @xmath22 is symmetric ; it is easily read off by the _ spherical diagram _ of @xmath22 , which is a useful tool to represent a wonderful variety starting from the dynkin diagram of @xmath0 . main examples of strict wonderful varieties where bijectivity fails arise from the context of _ wonderful model varieties _ introduced in @xcite ; the general strict case is substantially deduced from the model case . a _ model space _ for a connected ( possibly non - simply connected ) semisimple algebraic group @xmath25 is a quasi - affine homogeneous space whose coordinate ring contains every simple @xmath25-module with multiplicity one ; model spaces were classified in @xcite , where it is introduced the _ wonderful model variety _ @xmath26 , whose orbits naturally parametrize up to isomorphism the model spaces for @xmath25 : every orbit of @xmath26 is of the shape @xmath27 , where @xmath28 is a model space , and conversely this correspondence gives a bijection up to isomorphism . in order to illustrate the above mentioned criterion of bijectivity in the case of a wonderful model variety , let s set up some further notation . if @xmath29 is a dominant weight ( w.r.t . a fixed maximal torus @xmath30 and a fixed borel subgroup @xmath31 ) , define the @xmath32 of @xmath29 as the set @xmath33 where @xmath34 is the set of simple roots w.r.t . if @xmath36 is a simple factor of type @xmath37 or @xmath38 , number the associated subset of simple roots @xmath39 starting from the extreme of the dynkin diagram of @xmath40 which contains the double link ; define moreover @xmath41 as the subsets whose element index is respectively even and odd . if they are defined , set @xmath42 @xmath43 or set @xmath44 ( resp . @xmath45 ) otherwise . finally , if @xmath40 is of type @xmath46 , number the simple roots in @xmath47 starting from the extreme of the dynkin diagram which contains a long root . suppose that @xmath19\subset { \mathbf p}(v)$ ] is the open orbit of a wonderful model variety @xmath26 , where @xmath25 is isogenous with @xmath0 ; denote @xmath29 the highest weight of @xmath2 and set @xmath20}$ ] . then the normalization @xmath48 is bijective if and only if the following conditions are fulfilled , for every connected component @xmath49 : * if @xmath50 is of type @xmath37 , then either @xmath51 or @xmath52 ; * if @xmath50 is of type @xmath38 , then @xmath53 ; * if @xmath50 is of type @xmath46 and @xmath54 , then @xmath55 as well . when the generic stabilizer @xmath9 is a self - normalizing symmetric subgroup , compactifications in simple projective spaces were studied in @xcite . under this assumption , an explicit description of the orbits of @xmath5 was given and it was proved that these orbits are equal to those of the normalization of @xmath5 . thus our results generalize those contained in @xcite . in the case of a compactification of the adjoint group @xmath56 ( regarded as a @xmath57-symmetric variety ) obtained as the closure of the orbit of the line generated by the identity in a projective space @xmath58 ( where @xmath2 is a simple @xmath0-module ) , a complete classification of the normality and of the smoothness has been given in @xcite . the paper is organized as follows . in section 1 , we set notations and preliminaries ; in section 2 we give some general results about spherical orbit closures in projective spaces ; in section 3 we recall some results from @xcite about stabilizers of points in simple projective spaces . in section 4 , we describe the orbits of the compactifications @xmath5 and @xmath21 ; in section 5 , we prove the criterion of bijectivity of the normalization map in the strict case ; in section 6 , we briefly consider the non - strict case giving some sufficient conditions of bijectivity and non - bijectivity of the normalization map . + _ aknowledgements . _ i want to thank a. maffei , who proposed me the problem , for all his precious help , and p. bravi for many useful discussions on the subject . as well , i want to thank the referees for their careful reading and useful comments . spherical diagrams have been made with the package _ lunadiagrams _ , made by p. bravi and available at http://www.mat.uniroma1.it/@xmath59bravi/lunadiagrams .

let be a simply connected semisimple algebraic group over an algebraically closed field of characteristic 0 and let be a rational simple-module . if is a spherical orbit and if is its closure , then we describe the orbits of and those of its normalization . if moreover the wonderful completion of is strict , then we give necessary and sufficient combinatorial conditions so that the normalization morphism is a homeomorphism . such conditions are trivially fulfilled if is simply laced or if is a symmetric subgroup .

let be a simply connected semisimple algebraic group over an algebraically closed field of characteristic 0 and let be a rational simple-module . if is a spherical orbit and if is its closure , then we describe the orbits of and those of its normalization . if moreover the wonderful completion of is strict , then we give necessary and sufficient combinatorial conditions so that the normalization morphism is a homeomorphism . such conditions are trivially fulfilled if is simply laced or if is a symmetric subgroup .

1609.08702

i

let @xmath5 be a positive integer . every real number @xmath6 has a base @xmath1 expansion @xmath7 , and this expansion is unique if we adopt the convention that a tail of the coefficients @xmath8 can not be equal to @xmath9 . recall @xmath6 is _ @xmath1-normal _ if for every @xmath10 we have that @xmath11 , where @xmath12 . for a real number @xmath13 , define real functions @xmath14 and @xmath15 by @xmath16 and @xmath17 . we let @xmath0 denote the set of reals @xmath6 which are normal to base @xmath1 . we let @xmath18 let @xmath19 be a function from @xmath20 to @xmath20 . we say that @xmath19 _ preserves @xmath1-normality _ if @xmath21 . we can make a similar definition for preserving normality with respect to the regular continued fraction expansion , @xmath22-expansions , cantor series expansions , the lroth series expansion , etc . several authors have studied @xmath1-normality preserving functions . they naturally arise in h. furstenberg s work on disjointness in ergodic theory@xcite . v. n. agafonov @xcite , t. kamae @xcite , t. kamae and b. weiss @xcite , and w. merkle and j. reimann @xcite studied @xmath1-normality preserving selection rules . the situation for continued fractions is very different . let @xmath23 $ ] be normal with respect to the continued fraction expansion . b. heersink and j. vandehey @xcite recently proved that for any integers @xmath24 , the continued fraction @xmath25 $ ] is never normal with respect to the continued fraction expansion . in 1949 d. d. wall proved in his ph.d . thesis @xcite that for non - zero rational @xmath13 the function @xmath14 is @xmath1-normality preserving for all @xmath1 and that the function @xmath15 is @xmath1-normality preserving for all @xmath1 whenever @xmath13 is rational . these results were also independently proven by k. t. chang in 1976 @xcite . d. d. wall s method relies on the well known characterization that a real number @xmath6 is normal in base @xmath1 if and only if the sequence @xmath26 is uniformly distributed mod @xmath27@xcite . d. doty , j. h. lutz , and s. nandakumar took a substantially different approach from d. d. wall and strengthened his result . they proved in @xcite that for every real number @xmath6 and every non - zero rational number @xmath13 the @xmath1-ary expansions of @xmath28 and @xmath29 all have the same finite - state dimension and the same finite - state strong dimension . it follows that @xmath14 and @xmath15 preserve @xmath1-normality . it should be noted that their proof uses different methods from those used by d. d. wall and is unlikely to be proven using similar machinery . c. aistleitner generalized d. d. wall s result on @xmath15 in @xcite . suppose that @xmath30 is a rational number and that the digits of the @xmath1-ary expansion of @xmath31 are non - zero on a set of indices of density zero . he proved that the function @xmath32 is @xmath1-normality preserving . g. rauzy obtained a complete characterization of @xmath3 in @xcite . m. bernay used this characterization to prove that @xmath33 has zero hausdorff dimension @xcite . one of the main results of this paper , stated in corollary [ maincor ] , is to obtain an exact determination of the descriptive set theoretic complexity of @xmath3 . a significance of this is explained at the end of [ intro : results ] below . m. mends france asked in @xcite if the function @xmath14 preserves simple normality with respect to the regular continued fraction for every non - zero rational @xmath13 . this was recently settled by j. vandehey @xcite who showed that @xmath34 is normal with respect to the continued fraction when @xmath6 is normal with respect to the continued fraction expansion and integers @xmath35and @xmath36 satisfy @xmath37 . work was also done on the normality preserving properties of the functions @xmath14 and @xmath15 for the cantor series expansions by the first and third author in @xcite and additonally with j. vandehey in @xcite . however , these functions are not well understood in this context . in any topological space @xmath38 , the collection of borel sets @xmath39 is the smallest @xmath40-algebra containing the open sets . they are stratified into levels , the borel hierarchy , by defining @xmath41 the open sets , @xmath42 the closed sets , and for @xmath43 we let @xmath44 be the collection of countable unions @xmath45 where each @xmath46 for some @xmath47 . we also let @xmath48 . alternatively , @xmath49 if @xmath50 where @xmath51 where each @xmath47 . we also set @xmath52 , in particular @xmath53 is the collection of clopen sets . for any topological space , @xmath54 . all of the collections @xmath55 , @xmath44 , @xmath56 are pointclasses , that is , they are closed under inverse images of continuous functions . a basic fact ( see @xcite ) is that for any uncountable polish space @xmath38 , there is no collapse in the levels of the borel hierarchy , that is , all the various pointclasses @xmath55 , @xmath44 , @xmath56 , for @xmath57 , are all distinct . thus , these levels of the borel hierarch can be used to calibrate the descriptive complexity of a set . we say a set @xmath58 is @xmath44 ( resp . @xmath56 ) _ hard _ if @xmath59 ( resp . @xmath60 ) . this says @xmath61 is `` no simpler '' than a @xmath44 set . we say @xmath61 is @xmath44-_complete _ if @xmath62 , that is , @xmath63 and @xmath61 is @xmath44 hard . this says @xmath61 is exactly at the complexity level @xmath44 . likewise , @xmath61 is @xmath56-complete if @xmath64 . a fundamental result of suslin ( see @xcite ) says that in any polish space @xmath65 , where @xmath66 , and @xmath67 is the pointclass of continuous images of borel sets . equivalently , @xmath68 iff @xmath61 can be written as @xmath69 where @xmath70 is borel ( for some polish space @xmath71 ) . similarly , @xmath72 iff it is of the form @xmath73 for a borel @xmath74 . the @xmath67 sets are also called the _ analytic _ sets , and @xmath75 the _ co - analytic sets_. we also have @xmath76 for any uncountable polish space . h. ki and t. linton @xcite proved that the set @xmath0 is @xmath77-complete . further work was done by v. becher , p. a. heiber , and t. a. slaman @xcite who settled a conjecture of a. s. kechris by showing that the set of absolutely normal numbers is @xmath77-complete . furthermore , v. becher and t. a. slaman @xcite proved that the set of numbers normal in at least one base is @xmath78-complete . k. beros considered sets involving normal numbers in the difference heirarchy in @xcite . let @xmath79 be the set of numbers normal of order @xmath80 in base @xmath1 . he proved that for @xmath5 and @xmath81 , the set @xmath82 is @xmath83-complete ( see @xcite for a definition of the difference hierarchy ) . additionally , the set @xmath84 is shown to be @xmath85-complete . we are interested in determining the exact descriptive set theoretic complexity of @xmath3 and some related sets . the definition of @xmath3 shows that @xmath3 is @xmath75 , since it involves a universal quantification . it is not immediately clear if @xmath3 is a borel set , but this in fact follows from a result of rauzy . specifically , rauzy @xcite characterized @xmath3 in terms of an entropy - like condition called the _ noise_. we recall this condition and associated notation from @xcite . for any positive integer length @xmath86 , let @xmath87 denote the set of functions from @xmath88 to @xmath1 . we call an @xmath89 a _ block function _ of width @xmath86 . as in @xcite we set , for @xmath90 , @xmath91 where @xmath92 is the ( fractional part ) of the base @xmath1 expansion of @xmath6 . we also let for @xmath93 @xmath94 we then define the lower and upper noises @xmath95 , @xmath96 of @xmath6 by : @xmath97 where @xmath98 the upper entropy @xmath96 is defined similarly using @xmath99 where @xmath100 for a fixed @xmath93 we also let @xmath101 and similarly for @xmath102 . rauzy showed that @xmath103 iff it has the maximal possible noise in that @xmath104 . furthermore , @xmath105 iff it has minmal possible noise in that @xmath106 . it is therefore natural to ask for any @xmath107 $ ] , what are the complexities of the lower and upper noise sets associated to @xmath108 . specifically , we introduce the following four sets . let @xmath109 $ ] . let @xmath110 finally , we let @xmath111 thus , @xmath112 , and @xmath113 . the ki - linton result shows that @xmath0 , and thus @xmath114 is @xmath2-complete for any base @xmath1 . recall also the becher - slaman result which shows that the set of reals which are normal to some base @xmath1 forms a @xmath115-complete set . we have the following complexity results . [ mt ] for any @xmath116 , the set @xmath117 is @xmath4-complete and the set @xmath118 is @xmath2-complete . for any @xmath119 $ ] , the set @xmath120 is @xmath2-complete , and the set @xmath121 is @xmath122-complete . for @xmath123 , the set @xmath124 is @xmath4-complete , and the set @xmath125 is @xmath2-complete . as a corollary we obtain the ki - linton result as well as the determination of the exact complexity of @xmath3 . [ maincor ] the sets @xmath0 and @xmath3 are both @xmath2-complete . we have @xmath126 iff @xmath127 , and @xmath105 iff @xmath128 , so the result follows immeduately from theorem [ mt ] . in defining the noise , it is sometimes convenient to use the minor variation @xmath129 that is , the block function predicts the next digit rather than the previous digit . in this case we must start the sum at @xmath130 rather than @xmath27 , but this does not affect any of the limits used in defining @xmath95 or @xmath96 . in proving theorem [ mt ] we will work with @xmath6 in the sequence space @xmath131 where @xmath132 is the set of @xmath133 which do not end in a tail of @xmath9 s . this is a polish space as @xmath134 is a compact polish space ( with the usual product of discrete topologies on @xmath135 ) and @xmath132 is a @xmath136 ( that is , @xmath122 ) subset of @xmath134 . this is permissible as the map @xmath137 given by @xmath138 is continuous . so , for example , given that @xmath139 is @xmath2-complete , where @xmath140 is defined as @xmath118 , except we consider directly @xmath133 , then it follows that @xmath118 is @xmath2-complete . for if @xmath118 were in @xmath141 , then so would be @xmath139 since @xmath142 , that is , @xmath143 . we remark on the significance of complexity classifications such theorem [ mt ] . aside from the intrinsic interest to descriptive set theory , results of this form can be viewed as ruling out the existence of further theorems which would reduce the complexity of the sets . for example , rauzy s theorem reduces the complexity of @xmath3 from @xmath75 to @xmath2 . the fact that @xmath144 is @xmath2-complete tells us that there can not be other theorems which result in a yet simpler characterization of @xmath3 . lastly , we wish to approximate the hausdorff dimension of the sets @xmath145 , and @xmath124 . put @xmath146 . [ hd ] for @xmath147}$ ] we have @xmath148 furthermore @xmath149

let be the set of real numbers which are normal to base . a well - known result of h. ki and t. linton is that is-complete . we show that the set of reals which preserve under addition is also-complete . we use the characteriztion of given by g. rauzy in terms of an entropy - like quantity called the _ noise_. it follows from our results that no further characteriztion theorems could result in a still better bound on the complexity of . we compute the exact descriptive complexity of other naturally occurring sets associated with noise . one of these is complete at the level .

let be the set of real numbers which are normal to base . a well - known result of h. ki and t. linton is that is-complete . we show that the set of reals which preserve under addition is also-complete . we use the characteriztion of given by g. rauzy in terms of an entropy - like quantity called the _ noise_. it follows from our results that no further characteriztion theorems could result in a still better bound on the complexity of . we compute the exact descriptive complexity of other naturally occurring sets associated with noise . one of these is complete at the level . finally , we get upper and lower bounds on the hausdorff dimension of the level sets associated with the noise .

1703.08241

i

numerous applications motivate the study of families of homomorphisms from a discrete group to a lie group . perhaps the most impactful ( and so compelling ) is the relationship in differential geometry to flat bundles and geometric structures ( see ( * ? ? ? * page 163 ) ) . given a manifold @xmath4 with a base point @xmath5 and a lie group @xmath1 , one can construct a pointed principal @xmath1-bundle over @xmath4 as follows . let @xmath6 be the universal cover of @xmath4 and take a homomorphism @xmath7 . the group @xmath8 acts on @xmath9 by deck transformations in the first factor and by @xmath10 in the second . precisely , given @xmath11 and @xmath12 the action is @xmath13 where @xmath14 is the corresponding deck transformation in the deck group @xmath15 . this action is free since it is free on the first factor and so we obtain a principal @xmath1-bundle @xmath16 by considering the quotient by @xmath2 of the projection @xmath17 since @xmath18 . the transition maps are locally constant and hence the bundle is flat . conversely , any flat principal @xmath1-bundle @xmath19 has a holonomy homomorphism @xmath20 , once a base point is stipulated , so that @xmath21 . we see that the collection of flat principal @xmath1-bundles over @xmath22 is in bijective correspondence with the set @xmath23 . changing base points corresponds exactly to conjugating homomorphisms by @xmath1 . hence the conjugation quotient @xmath24 is bijectively equivalent to the collection of flat principal @xmath1-bundles over @xmath4 ( without dependence on the base point ) . using homomorphisms to tie the geometry of a lie group @xmath1 to the topology of a manifold @xmath4 via its fundamental group accounts for many , if not all , of the ways that spaces of homomorphisms appear in geometry , topology , and mathematical physics . examples occur in the study of knot invariants @xcite , hyperbolic geometry @xcite , holomorphic vector bundles @xcite , yang - mills connections @xcite , higgs bundles @xcite , flat principal bundles @xcite , the geometric langlands program @xcite , and supersymmetry @xcite . another important example where spaces of homomorphisms are prominent is the study of locally homogeneous structures on manifolds @xcite . here the problem is , for a given topology on a compact manifold @xmath4 and a model geometry @xmath25 where @xmath1 is a lie group , to classify all the possible ways of introducing the local geometry of @xmath26 onto the topology of @xmath4 . the ehresmann - thurston theorem shows that the moduli space of such marked @xmath27-structures is locally homeomorphic to the space of conjugacy classes of @xmath23 with the quotient topology . here is a simple but illustrative example . if @xmath4 is an annulus then @xmath28 and @xmath29 by the evaluation mapping sending @xmath30 for any @xmath31 . say @xmath32 . then each conjugacy class of a diagonalizable @xmath31 is determined by its characteristic polynomial @xmath33 where @xmath34 . there are only two classes of non - diagonalizable matrices : one with trace @xmath35 and the other with trace @xmath36 . so the space of flat @xmath37-bundles over an annulus , with its closed points parametrized by @xmath38 , is @xmath39 where @xmath40 are double - points " corresponding to the orbits of non - diagonalizable matrices with trace @xmath41 . in particular , the space is not @xmath42 since the closure of @xmath40 is @xmath43 . one way to deal with the topological complications illustrated above is to consider an approximation of the quotient space @xmath44 . concretely , when @xmath2 is finitely presentable ( which means there is a finite rank free group @xmath45 and an epimorphism @xmath46 with finitely generated kernel ) , we can consider : @xmath47 there is a map , depending on @xmath48 , @xmath49 given by @xmath50 . this map , called an _ evaluation map _ , is clearly injective and has as its image the set @xmath51 where @xmath52 is the identity . as @xmath1 is a lie group the image inherits the subspace topology from @xmath53 , and is consequently an analytic subvariety ( algebraic if @xmath1 is algebraic ) . one can show that @xmath54 is an embedding when @xmath55 is given the more intrinsic compact - open topology with respect to the discrete topology on @xmath2 . this observation shows that any two presentations for @xmath2 result in homeomorphic topologies on @xmath55 as analytic varieties . now consider the subspace of _ polystable _ points : @xmath56 , where @xmath57 denotes the conjugation orbit of @xmath10 and @xmath58 is the closure of the orbit in the aforementioned topology on @xmath55 . then the orbit space @xmath59 is @xmath42 , and called the _ moduli space of @xmath1-representations of @xmath2_. in the above example , @xmath60 since @xmath40 correspond to non - closed orbits . here are some facts about @xmath0 . when @xmath1 is real reductive then @xmath0 is hausdorff and when @xmath1 is real algebraic @xmath0 is moreover semi - algebraic ( see the discussion in @xcite ) . in the special case when @xmath1 is compact , then @xmath61 in the special case when @xmath1 is complex affine algebraic , then @xmath0 is homeomorphic to the geometric invariant theory ( git ) quotient @xmath62 with the euclidean topology ( see ( * ? ? ? * proposition 2.3 ) ) and so is naturally identified with the @xmath63-points of an affine algebraic set . moreover , again when @xmath1 is complex affine algebraic @xmath0 is _ homotopic _ to the potentially non - hausdorff quotient @xmath64 ( see ( * ? ? ? * proposition 3.4 ) ) . with these cases in mind , we consider @xmath0 a reasonable approximation to @xmath44 in general . we will focus our attention in this exposition on the case when @xmath32 , so we say a few more words about the case when @xmath1 is complex reductive . in this case , the git quotient is @xmath65^g)$ ] where @xmath66 $ ] is the coordinate ring of the affine algebraic set @xmath67 and @xmath66^g$ ] is the ring of invariant elements in the coordinate ring under the action of conjugation @xmath68 . it is interesting to note @xcite shows that given any affine variety @xmath69 that there is a character variety @xmath0 that is tale locally isomorphic to @xmath69 as schemes . more globally , ( * ? ? ? * theorem 3 ) shows there is a character variety @xmath0 that agrees biregularly with @xmath69 , excepting at most one point in @xmath0 , as varieties . in particular , since points also arise as character varieties , this shows that character varieties generate the grothendieck group of affine varieties defined over @xmath70 . in the case @xmath71 the coordinate ring of @xmath0 is generated by functions of the form @xmath72 where @xmath73 is a word in generic unimodular matrices ( we will say more about this in section [ gen ] ) . this explains why the moduli space of @xmath1-representations is called `` character variety , '' as it is a variety of characters . however , as discussed in @xcite , for some @xmath1 this later fact remains true and for others it does not . on the other hand , the main theorem in @xcite says that a finite quotient of the stable locus of @xmath0 is always in birationally bijective correspondence to a variety of characters , in particular , they have the same grothendieck class . so although the term `` character variety '' is not always completely accurate , it is accurate enough to not abandon . executive summary : the study of these moduli spaces is a mathematical playground where topology , geometry ( algebraic and differential ) , dynamics , ring theory , geometric analysis , and number theory all come together in non - trivial yet tractable ways . the group @xmath2 encodes topology and the lie group @xmath1 encodes geometry making @xmath0 the space of ways in which the geometry of @xmath1 can be imposed on the topology governed by @xmath2 . perhaps an apt slogan for the study of @xmath0 is the `` geometry of geometries . '' in this paper we assume @xmath32 and @xmath2 is any finitely presented group . we provide an explicit algorithm to determine the structure of the variety @xmath0 , and provide a _ mathematica _ notebook that implements our algorithm . the results in this paper are not new . they are an amalgamation of results from vogt @xcite , horowitz @xcite , magnus @xcite , gonzlez - acua and montesinos - amilibia @xcite , brumfiel and hilden @xcite , and drensky @xcite . we put them together to provide a new proof of the structure theorem for these character varieties , and provide an effective algorithm that does not rely on elimination ideal algorithms . the organization of the paper is as follows . in section 2 we describe a generating set for our character varieties , and in section 3 we describe a generating set for the ideal of defining relations in the aforementioned generators . in section 4 we implement our algorithm to compute some concrete examples .

let be the-character variety of where is a rank 1 complex affine algebraic group and is a finitely presentable discrete group . the main results in this paper are not new , although we hope that as a well - referenced exposition with a companion computer program , it will be useful .

let be the-character variety of where is a rank 1 complex affine algebraic group and is a finitely presentable discrete group . we describe an algorithm , which we implement in _ mathematica _ , that takes a finite presentation for and produces a finite presentation of the coordinate ring of . the main results in this paper are not new , although we hope that as a well - referenced exposition with a companion computer program , it will be useful .

0807.0395

i

an immersed loop in the plane might or might not bound an immersed disk , and if it does , the disk it bounds might not be unique . an immersed loop on a surface might not bound an immersed subsurface , but admit a finite cover which does bound an immersed subsurface i.e. it might `` virtually '' bound an immersed surface . most homologically trivial geodesics on hyperbolic surfaces with boundary do not even virtually bound an immersed surface . however , in this paper , we show that _ every _ homologically trivial geodesic in a _ closed _ hyperbolic surface @xmath1 virtually bounds an immersed surface , and every homologically trivial geodesic in a hyperbolic surface @xmath1 with boundary virtually cobounds an immersed surface together with a sufficiently large multiple of @xmath11 . this has implications for the structure of the ( second ) bounded cohomology of free and surface groups , as we explain in what follows . given a group @xmath12 and an element @xmath13 $ ] , the _ commutator length _ of @xmath14 , denoted @xmath15 , is the smallest number of commutators in @xmath12 whose product is @xmath14 , and the _ stable commutator length _ of @xmath14 is the limit @xmath16 . geometrically , if @xmath17 is a space with @xmath18 and @xmath14 is represented by a loop @xmath19 in @xmath17 , the commutator length of @xmath14 is the least genus of a surface mapping to @xmath17 whose boundary maps to @xmath19 . by minimizing number of triangles instead of genus , one can reinterpret @xmath20 as a kind of @xmath21 norm on relative ( @xmath22-dimensional ) homology . technically , if @xmath23 denotes the vector space of real - valued ( group ) @xmath9-boundaries ( i.e. group @xmath9-chains which are boundaries of group @xmath22-chains ; see [ pseudo_norm_subsection ] ) , there is a well - defined @xmath20 pseudo - norm on @xmath23 . the subspace on which @xmath20 vanishes always includes a subspace @xmath24 spanned by chains of the form @xmath25 and @xmath26 , for @xmath27 and @xmath28 , and therefore @xmath20 descends to a pseudo - norm on the quotient @xmath29 , which we abbreviate by @xmath30 or @xmath31 or even @xmath32 in the sequel . in certain special cases ( for example , when @xmath12 is a free group ) , @xmath20 defines an honest norm on @xmath5 , but we will not use this fact in the sequel . more precise definitions and background are given in [ background_section ] . dual ( in a certain sense ) to the space @xmath30 with its @xmath20 pseudo - norm , is the space @xmath33 of _ homogeneous quasimorphisms _ on @xmath12 , i.e. functions @xmath34 for which there is a least real number @xmath35 ( called the _ defect _ ) such that @xmath36 for all @xmath37 and @xmath27 , and @xmath38 for all @xmath28 . the particular form of duality between @xmath20 and @xmath33 is called _ bavard duality _ , which is the equality @xmath39 ( see [ background_section ] for details ) . the defining properties of a homogeneous quasimorphism can be thought of as an infinite family of linear equalities and inequalities depending on elements and pairs of elements in @xmath12 . the @xmath21-@xmath40 duality between @xmath20 and @xmath33 means that computing @xmath20 is tantamount to solving an ( infinite dimensional ) _ linear programming problem _ in group homology ( for an introduction to linear programming , see e.g. @xcite ) . in finite dimensions , @xmath21 and @xmath40 norms are piecewise linear functions , and their unit balls are rational convex polyhedra . broadly speaking , the main discovery of @xcite is that in free groups ( and certain groups derived from free groups in simple ways ) , computing @xmath20 reduces to a _ finite dimensional _ linear programming problem , and therefore the unit ball of the @xmath20 pseudo - norm on @xmath41 is a rational convex polyhedron ; i.e. for every finite dimensional rational vector subspace @xmath42 of @xmath43 , the unit ball of the @xmath20 pseudo - norm restricted to @xmath42 is a finite - sided rational convex polyhedron ( compare with the well - known example of the gromov - thurston norm on @xmath44 of a @xmath45-manifold ; see @xcite ) . in a finite dimensional vector space , a rational convex polyhedron is characterized by its top dimensional faces i.e. those which are codimension one in the ambient space . in an _ infinite _ dimensional vector space , a rational convex polyhedron need not have any faces of finite codimension at all . the codimension of a face of a convex polyhedron in an infinite dimensional vector space is the supremum of the codimensions of its intersections with finite dimensional subspaces . top dimensional faces of the unit ball of the gromov - thurston norm on @xmath44 of a @xmath45-manifold have a great deal of topological significance ( see e.g @xcite , @xcite , @xcite or @xcite for connections with the theories of taut foliations , seiberg - witten equations , quasigeodesic flows , and heegaard floer homology respectively ) . it is therefore a natural question to ask whether the @xmath20 unit polyhedron in @xmath41 has any faces which are codimension one in @xmath41 , and whether some of these faces have any geometric significance . our first two main theorems answer these questions affirmatively . let @xmath6 be a free group , and let @xmath1 be a compact , connected , orientable surface with @xmath2 and @xmath46 . let @xmath47 be the @xmath9-chain represented by the boundary of @xmath1 , thought of as a finite formal sum of conjugacy classes in @xmath6 . then the projective ray in @xmath41 spanned by @xmath11 intersects the unit ball of the @xmath20 norm in the interior of a face of codimension one in @xmath41 . by bavard duality , a face of the @xmath20 norm ball of codimension one is dual to a unique extremal homogeneous quasimorphism , up to elements of @xmath48 ( which vanish identically on @xmath32 ) . it turns out that we can give an explicit description of the extremal quasimorphisms dual to the `` geometric '' faces of the @xmath20 norm ball described in theorem a. if @xmath1 is a compact , connected , orientable surface with @xmath2 , then @xmath1 admits a hyperbolic structure with geodesic boundary . the hyperbolic structure and a choice of orientation determine a discrete , faithful representation @xmath49 , unique up to conjugacy since @xmath8 is free , this representation lifts to @xmath50 where @xmath51 denotes the universal covering group of @xmath52 . there is a unique continuous homogeneous quasimorphism on @xmath51 ( up to scale ) , called the _ rotation quasimorphism _ ( discussed in detail in [ rotation_subsection ] ) . this quasimorphism pulls back by @xmath53 to a homogeneous quasimorphism @xmath54 on @xmath8 , which is well - defined up to elements of @xmath55 . up to scale , this turns out to be the homogeneous quasimorphism dual to the top dimensional face of the @xmath20 norm ball described above : let @xmath6 be a free group , and let @xmath1 be a compact , connected , orientable surface with @xmath2 and @xmath46 . let @xmath4 be the face of the @xmath20 unit norm ball whose interior intersects the projective ray of the class @xmath11 . the face @xmath4 is dual to the extremal homogeneous quasimorphism @xmath54 . these theorems are both proved in 3 . theorem a shows how hyperbolic geometry and surface topology manifest in the abstract ( bounded ) cohomology of a free group . theorem b is a kind of rigidity result , characterizing the rotation quasimorphism associated to a discrete , faithful representation of @xmath8 into @xmath52 amongst all homogeneous quasimorphisms by the property that it is `` extremal '' for @xmath11 . in [ remark_corollary_subsection ] we use these theorems to deduce a short proof of a relative version of rigidity theorems of goldman @xcite and burger - iozzi - wienhard @xcite , that representations of surface groups into certain lie groups of maximal euler class are discrete ( it should be stressed that @xcite contain much more than the narrow result we reprove ) . in light of theorem a and theorem b , it is natural to ask whether the projective class of every element @xmath56 $ ] intersects the interior of a face of the @xmath20 norm ball of finite codimension . in fact , it turns out that this is not the case . we show by an explicit example ( example [ infinite_codimension ] ) that there are many elements @xmath56 $ ] where @xmath6 has rank at least @xmath57 , whose projective classes are contained in faces of the @xmath20 norm ball of infinite codimension . the method of proof is of independent interest . we show that for any homologically trivial geodesic @xmath9-manifold @xmath19 in a hyperbolic surface @xmath1 , there is a surface @xmath58 and an immersion @xmath59 for which @xmath60 $ ] is taken to some multiple of @xmath61 + n[\partial s]$ ] in @xmath62 ; i.e. the @xmath9-cycle @xmath63 `` rationally bounds '' an immersed surface . note that this remains true even if @xmath1 is closed ! explicitly , the statement of the main technical theorem ( proved in [ immersion_theorem_subsection ] ) is as follows : let @xmath1 be a compact , connected orientable surface with @xmath2 , and @xmath64 a finite rational chain in @xmath65 . then for all sufficiently large rational numbers @xmath66 ( depending on @xmath10 ) , the geodesic @xmath9-manifold in @xmath1 corresponding to the chain @xmath67 rationally bounds an immersed surface @xmath68 . the connection with stable commutator length is as follows : from the main theorem of @xcite it follows that in an oriented hyperbolic surface @xmath1 with boundary , a rational @xmath9-chain @xmath10 bounds an immersed surface if and only if @xmath69 , where @xmath54 is as above ( this is proposition [ rot_is_scl ] below ) . hence theorem c implies that every chain @xmath10 in @xmath65 which is projectively close enough to @xmath11 satisfies @xmath69 ; theorems a and b follow . a number of additional corollaries are stated , including a generalization of the main theorem of @xcite . let @xmath12 be a graph of free or ( closed , orientable ) hyperbolic surface groups amalgamated over infinite cyclic subgroups , and let @xmath70 be a nonzero rational class in @xmath71 . let @xmath72,\cdots,[s_m]$ ] be the fundamental classes in @xmath44 of the vertex subgroups which are closed surface groups . then for all sufficiently big integers @xmath73 , some multiple of the class @xmath74 $ ] in @xmath71 is represented by an injective map from a closed hyperbolic surface group to @xmath12 .

the unique homogeneous quasimorphism on dual to ( up to scale and elements of ) is the rotation quasimorphism associated to the action of on the ideal boundary of the hyperbolic plane , coming from a hyperbolic structure on . these facts follow from the fact that every homologically trivial-chain in rationally cobounds an immersed surface with a sufficiently large multiple of . [ section ] [ theorem]lemma [ theorem]proposition [ theorem]corollary [ theorem]conjecture [ theorem]question [ theorem]problem [ theorem]definition [ theorem]construction [ theorem]notation [ theorem]remark [ theorem]example p

let where is a compact , connected , oriented surface with and nonempty boundary . 1 . the projective class of the the chain intersects the interior of a codimension one face of the unit ball in the stable commutator length norm on . 2 . the unique homogeneous quasimorphism on dual to ( up to scale and elements of ) is the rotation quasimorphism associated to the action of on the ideal boundary of the hyperbolic plane , coming from a hyperbolic structure on . these facts follow from the fact that every homologically trivial-chain in rationally cobounds an immersed surface with a sufficiently large multiple of . this is true even if has no boundary . [ section ] [ theorem]lemma [ theorem]proposition [ theorem]corollary [ theorem]conjecture [ theorem]question [ theorem]problem [ theorem]definition [ theorem]construction [ theorem]notation [ theorem]remark [ theorem]example p

astro-ph0407566

r

table [ param.table ] outlines the galactic and satellite parameters varied to produce a model that fits the constraints detailed above . rather than run lengthy n - body simulations to randomly search for a global minimum in this degenerate , multi - dimensional parameter space ( 8 of which are allowed to vary ) , a more efficient , multi - step approach was taken to converge to the best fit to the observational data , relying on physical insight gained both from analytical descriptions of debris dispersal ( tremaine 1993 , johnston 1998 , helmi & white 1999 , johnston , sackett & bullock 2001 ) and from previous modeling of sgr by the authors ( johnston , spergel , & hernquist 1995 , johnston et al . 1999 ) and other groups ( velazquez & white 1995 , ibata et al . 1997 , edelsohn & elmegreen 1997 , g ' omez - flechoso , fux , & martinet 1999 , mart ' inez - delgado et al . 2004 ) . these studies have found that while there is a systematic distance offset for leading / trailing debris inside / outside the orbit of the sgr dwarf ( a reflection of the debris moving to more / less tightly bound orbits see fig . [ xydiag ] ) the line - of - sight velocity remains approximately aligned with that of the satellite s orbit at all orbital phases . hence in 3.1 below we are able to eliminate a wide range of orbits in a variety of galactic potentials through test - particle integrations alone : we use constraint 4 as an upper limit on a possible orbit s apocentric distance and examine how well the line - of - sight velocities along the orbit match the data in constraints 5 and 6 . this technique allows us to find reasonable values for all of the free parameters listed in table [ param.table ] except for sgr s current mass . in 3.2 we describe full - scale simulations of the destruction run for satellites of various masses along the orbits and in the potentials selected in 3.1 . we assume sgr s current angular position , line - of - sight velocity and direction of proper motion to be fixed by constraints 1 , 2 and 3 respectively , and adopt an amplitude for the motion of sgr perpendicular to our line - of - sight ( @xmath49 ) somewhere within @xmath50 times the error bars on the @xcite measurement of @xmath51 km s@xmath22 . the sgr velocity and position relative to the sun are then tranformed to galactocentric coordinates to provide initial conditions for the test particle orbits , assuming some values for the solar distance from the galactic center ( @xmath52 ) and from sgr ( @xmath39 ) . ( note that changing @xmath39 from the assumed value of 24 kpc scales the distances to all of the sgr m giants by the same fractional amount , since these distances are estimated from a color - apparent magnitude relation derived from m giants in sgr s core paper i ) . these orbits are then integrated backwards and forwards in time in the chosen galactic potential ( see 3.1.2 ) and the quality of fit of the orbital path to the m giant position and velocity data quantified ( as described in 3.1.3 ) . we anticipate that sgr s debris will tell us something about the contours of the gravitational potential in the region that its orbit explores ( @xmath44 10 - 50 kpc ) . hence we do not vary all parameters in equations ( 1 ) - ( [ haloeqn ] ) , but instead hold the bulge component fixed and explore the effect of changing the contribution of the disk to the rotation curve through the parameter @xmath53 , as well as the radial length scale , flattening , and overall depth of the halo potential respectively through the parameters @xmath25 , @xmath54 and @xmath24 . as a final check on the generality of our results , we repeat our experiments with the halo component replaced by models of the form proposed by @xcite hereafter referred to as nfw models . in this case , the flattening is introduced in the density @xmath55 , rather than potential contours , and the approximate form of the potential is adopted from @xcite . to explore a similar effective range in @xmath54 and radial gradient as the logarithmic models , @xmath55 and @xmath25 ( the length scale of the nfw potential ) are chosen from a wider range than for their logarithmic counterparts in particular , the range @xmath56 kpc was explored because this encompasses the range of scale lengths ( of order tens of kpc ) found for dark matter halos of similar mass - scale to the milky way in cosmological models of structure formation at the current epoch ( e.g. * ? ? ? the mass scale of the nfw potential is then constrained to match the adopted @xmath24 . a guideline for assessing the goodness of fit of an orbit to the positional data is that the maximum heliocentric distance observed for the leading debris ( @xmath57 , constraint 4 ) must be systematically less than that of the orbit of the sgr core , @xmath58 i.e. @xmath59 . we can also find an upper limit to this ratio since we expect the size of this offset to scale as @xmath60 , where @xmath61 is the mass of the milky way enclosed within the pericenter of the orbit @xcite . for example , if we take this limit as @xmath62 then we might expect to cover all models with @xmath63 i.e. sgr masses up to 10% of the mass of the milky way . since the internal dispersion measured for sgr ( 11 km s@xmath22 , * ? ? ? * ) suggests a mass far less than this we take @xmath64 as a generous range for considering an orbit apogalacticon distance acceptable . orbits with apogalactica outside this range are immediately rejected . we next quantify the fit of orbits that are not already rejected to the trailing and leading velocity data ( constraints 5 and 6 ) through the parameters @xmath65 and @xmath66 : @xmath67_i^2\over{\sigma^2_{\rm a } } } \label{khia}\ ] ] where `` a '' represents the observed data set being considered ( i.e. `` lead '' or `` trail '' ) , @xmath68 is the number of m giants in the data set , @xmath69 is the velocity of an m giant at @xmath70 and @xmath71 is the velocity of the orbit at this @xmath70 . the data compared to in the leading portion of the debris are selected by fitting a 3rd order polynomial to the full data set of velocities as a function of @xmath30 in the range @xmath72 . outliers from the main trend are thrown out using a 2.5-@xmath73 iterative rejection technique until convergence is reached and the weight @xmath74 calculated as the dispersion of the velocities of this final set of @xmath75 stars about the best - fit polynomials . the process is then repeated for stars in the region @xmath76 most sensitive to the trailing debris . the selected stars in both regions are plotted as black squares in figure [ orb.fig ] . clearly , these data sets are not intended to represent a complete sample of sgr stars , but rather as a guide to the general trends of velocities and dispersion in these regions . we also express these quantities as a single parameter to measure the combined goodness - of - fit : @xmath77 note that since test particle orbits only serve as an indication of where the debris should lie , we do not simply search for the parameters corresponding to the minima of these quantities : for example , we do not consider a difference of order @xmath78 ( corresponding to average systematic offsets @xmath79 km s@xmath22 very much less than the dispersion in the data ) between the fit to two different orbits to be very significant . rather , we use more extreme differences to rule out or favor broad regions of parameter space . although the velocity trends in the leading debris ( constraint 6 ) appear to strongly favor galactic models with prolate ( @xmath80 ) halo components @xcite , we have shown in paper iii that the direction of the precession of debris orbits ( as measured by the offset in the poles of best - fit planes to leading and trailing debris constraint 7 ) strongly favors models with oblate halos since prolate models induce precession in the _ opposite _ sense to that observed . because no other adjustment to the potential can change the fundamental sense of precession in prolate _ vs _ oblate potentials , we restrict ourselves to asking whether we can resolve this contradiction between the implications of constraint 6 and constraint 7 by revisiting the fit to the velocity and distance data alone over a much wider range of parameter space than has been considered previously . the aim is to examine whether there are any circumstances in which an orbit in an oblate potential can be found that can fit all the constraints at once . figure [ khi.fig ] plots the minimum values of @xmath65 ( solid lines ) , @xmath66 ( dotted lines ) and @xmath81 ( dashed lines ) obtained as a function of @xmath54 ( left hand panel , logarithmic halo model ) or @xmath82 ( right hand panel , nfw halo model ) when all other parameters are allowed to vary freely within the ranges outlined in table [ param.table ] . the solid lines show that the trailing velocity data have a slight preference for models with oblate halos , although the difference @xmath83 between the minima for models with @xmath84 and @xmath80 is not sufficiently large that we can confidently rule out prolate models with test particle orbits alone , since it corresponds to a velocity offset much less than the dispersion in the data . in contrast , the dotted lines show that leading velocity data strongly prefer prolate halo models , to such an extent that this preference dominates the combined @xmath81 ( dashed lines ) . these results are the same for the logarithmic and nfw models . overall , we conclude that we can not find a single orbit in a static potential model that simultaneously fits the velocity data in the trailing data together with the sense of precession suggested by the offset of the planes of the leading _ vs _ trailing data . the exciting implication of the conclusion of the previous section that no _ single _ orbit and/or potential can fit all the data is that some evolution of sgr s orbit has occurred over the time since debris in the leading portion of the streamer , furthest in @xmath85 from sgr , was released . we discuss some possible culprits for this orbital evolution in 4.3 , but defer a detailed investigation of these effects for future work . for the remainder of this study , we narrow our present analysis to concentrate on the younger portions of the debris , lost within the last 1 - 2 orbits , where ( 1 ) the effect of orbit evolution is negligible , ( 2 ) the modelling can be acheived with the fewest free parameters , and ( 3 ) the interpretation of the data is less ambiguous . the goal is to ask what the younger debris alone can tell us about the galactic potential and sgr s current mass and orbit . these results can subsequently be used as starting points for studies that use the older debris to examine higher order effects such as orbital evolution , evolution of the potential , and/or multi - component models for sgr . we expect debris in the trailing streamer in the range explored by the velocity data to be roughly the same age as that in the early parts of the leading streamer to about the first apocenter ( as demonstrated by * ? ? ? * and see also 3.2 below ) . in these regions , the velocity data can be similarly fit by both oblate and prolate potentials ( as demonstrated by the solid lines in figure [ khi.fig ] ) , and there is no significant offset in the orbital poles between the leading and trailing components . hence , we now drop constraints 6 and 7 on our models since these were derived from regions where orbit evolution could be significant . we continue our discussion of test - particle constraints on the galactic potential and our position relative to the galactic center and sgr using the condition @xmath86 and examining @xmath65 alone . in order to sort through our large parameter space , we first look at parameters that do not appear to be strongly constrained by the data and fix reasonable values for those ( see discussion in a. and b. below ) before going on to look at preferred ranges for the remaining parameters ( in c. ) . in the upper left hand panels of figures [ ktrail_log.fig ] ( for logarithmic halo experiments ) and [ ktrail_nfw.fig ] ( for nfw halo experiments ) we project results in our 7-dimensional parameter space onto the 2-dimensions of @xmath39 and @xmath87 by plotting the minimum value of @xmath65 at each point in this plane when all other parameters are allowed to vary freely . the plots reveals a preference for larger values of the ratio @xmath88 , with the absolute scale ( as set by @xmath39 ) being arbitrary . for consistency with the distance scales adopted earlier in paper i we choose to take @xmath89 kpc ( which also lies within the 2-@xmath73 error bars of the recent measurement by * ? ? ? * ) and set @xmath90 kpc . so long as @xmath91 we expect all subsequent results involving distances ( e.g. scale - length of the halo @xmath25 , or predicted distances to debris ) can be scaled by whatever value @xmath39 is assumed in a given study . with @xmath89 kpc and @xmath90 kpc fixed , the upper right hand panels of figures [ ktrail_log.fig ] and [ ktrail_nfw.fig ] project the remaining five - dimensions of parameter space onto the @xmath24 - @xmath53 plane . for high enough @xmath24 there is no preference for a particular @xmath53 , but models with lower @xmath24 are inconsistent with heavier galactic disks ( i.e. higher @xmath53 ) . figure [ rot.fig ] offers some clue as to why this is the case by plotting rotation curves for only those potentials in which orbits with @xmath92 could be found . these are very flat out to large radii for all models , with circular velocities at 50 kpc in the range 180 - 220 km s@xmath22 ( which corresponds to enclosed masses for the milky way at these radii of @xmath93 in effect , sgr debris velocities are now providing additional evidence for the existence of a dark matter halo to the milky way ) . if enough of the contribution @xmath94 is provided by the disk , then the remaining halo component is simply not massive enough to support such an extended flat rotation curve . ( larger mass halos could be built by allowing @xmath25 an even wider range , but these models would [ i ] have rising rotation curves at the solar circle ; and [ ii ] be inconsistent with scale lengths measured for milky way - sized dark matter halos formed in cosmological models of structure formation see eke , navarro & steinmetz , 2001 ) . since no values of @xmath53 and @xmath24 are at this point clearly preferred , we adopt @xmath95 and @xmath96 km s@xmath22 . the colored lines in the lower panels of figures [ ktrail_log.fig ] and [ ktrail_nfw.fig ] demonstrate that , with @xmath90 kpc , @xmath89 kpc , @xmath97 and @xmath96 km s@xmath22 fixed , particular values for @xmath25 ( which determines the radial gradient of the potential and hence the shape of the rotation curve ) and @xmath49 ( which determines the scale of the orbit within this potential ) are quite strongly preferred , with only a mild dependence on @xmath54 . hence we perform full n - body simulations in potentials with logarithmic halos in which @xmath98 , @xmath99 kpc ( from the minima in the lower left hand panels ) and @xmath49 in the range @xmath100 km s@xmath22 around @xmath101 km s@xmath22 . all three values , @xmath98 , are considered since all represent equally viable fits to the younger debris . clearly , our choices are not unique . the black curves in the lower panels of figures [ ktrail_log.fig ] and [ ktrail_nfw.fig ] outline where the colored lines would fall if all other parameter choices were the same but @xmath102 km s@xmath22 ( dashed lines ) or @xmath103 ( dotted lines ) . in both cases , the scale - length changes significantly in order to maintain the necessary flatness of the rotation curve , and @xmath49 is similarly affected . in addition , our decision to use logarithmic halos rather than nfw halos is arbitrary , since figures [ ktrail_log.fig ] and [ ktrail_nfw.fig ] reveal no preference for either form of the potential , but rather more generally indicate that any model that generates a flat rotation curve out to 50 kpc will suffice . we anticipate that data exploring even larger distances from the galactic center will be able to address whether an nfw ( with a falling rotation curve in this region ) or logarithmic potential is more appropriate . despite these multiple minima in parameter space , we are able to reach some general conclusions at this point : ( i ) sgr debris data prefers models with large values of @xmath88 and flat rotation curves out to 50 kpc , and ( ii ) with all other parameters fixed , sgr orbits in prolate halos will have systematically lower @xmath49 than in spherical or oblate halos . these conclusions offer a tantalizing glimpse of how sgr debris might be used to map out the galactic potential on large scales once parameters such as @xmath52 and @xmath49 are known with more certainty . using the galactic parameters determined in 3.1 above , we now perform fully self - consistent n - body simulations to refine the estimates obtained in 3.1 of sgr s orbital velocity and to determine the mass of the dwarf . these simulations follow the evolution of satellites with a range of initial masses and physical scales ( varied through the parameters @xmath104 and @xmath105 in equation [ [ plummereqn ] ] ) along a small range of plausible orbits within the three models of the galactic potential ( @xmath1 0.9/1.0/1.25 ) discussed in 3.1.5c . in 3.2.1 we find the mass of sgr ( independent of @xmath105 ) that best fits constraints 8 and 9 in each of these three models of the galactic potential , and demonstrate that this best - fit mass is common to all three cases . fixing the satellite mass to this best - fit value , we refine our estimate for sgr s tangential velocity using constraints 4 and 5 in 3.2.2 and summarize the properties of our best - fit models in 3.2.3 . while we do not attempt to model the sgr core in detail , we are nonetheless able to constrain its current total mass under the assumptions that the dwarf is roughly spherical and non - rotating . motivated by previous work ( e.g. , johnston , hernquist , & bolte 1996 , johnston 1998 ) we expect that debris width ( constraint 8) and velocity dispersion ( constraint 9 ) at a given orbital phase primarily reflect the mass within the tidal radius of the satellite on the orbit immediately prior to that debris becoming unbound , and that they do not depend strongly on the internal structure of the satellite ( in our case parameterized by the scale length of the initial plummer model ) . for the same reasons , we do not expect that our results are strongly sensitive to the particle distribution we have adopted . we do expect the internal orbital distribution will independently affect debris morphology , but do not address that issue in this paper . to compare the simulations to the data constraints , we calculate the average radial velocity dispersion @xmath106 and the average dispersion of distances perpendicular to the sgr plane @xmath107 in the trailing tail for m giant data and our numerical simulations . we do not consider leading debris in obtaining our mass estimates since only our prolate halo model successfully matches the _ trend of leading debris , while all three halo models reproduce the trailing debris trend . @xmath106 is calculated in the range @xmath40 - @xmath46 for consistency with the velocity dispersion analysis presented in paper ii , while @xmath107 is calculated in the range @xmath108-@xmath109 since this range of debris longitudes is one for which all sgr stars in the sample , @xmath110 , @xmath111 , @xmath112 , and 13 kpc @xmath113 40 kpc . ] are at a similar distance @xmath114 from the sun ( this minimizes artificial width inflation on the sky due to differential distance errors ) and is also in a region of the galaxy where sample contamination by milky way disk stars is negligible . figure [ mplot ] plots the calculated velocity dispersion ( left - hand panels ) and width ( right - hand panels ) as functions of simulated bound satellite mass for choices of @xmath1 0.9 ( lower panels ) , 1.0 ( middle panels ) , and 1.25 ( upper panels ) . in all panels the m giant dispersion / width is plotted as a solid line with 1-@xmath73 error bars indicated by the hatched regions , while the points in all panels indicate n - body simulation results ( incorporating a 17% artificial distance scatter to simulate the photometric distance errors given in paper i ) for model satellites evolved along the orbits found earlier in 3.1.5c for a variety of choices of initial satellite mass ( @xmath115 @xmath13 - @xmath116 @xmath13 ) and physical scale ( @xmath117 0.2 kpc - 1.5 kpc ) . clearly , similar values of @xmath118 are preferred for models in oblate , spherical , and prolate galactic potentials alike . to quantify more precisely the range of acceptable masses indicated by figure [ mplot ] we fit the data points in each panel with a third - order polynomial with 2.5-@xmath73 rejection criteria iterated to convergence and extrapolate from the resulting power - series coefficients the mass range whose @xmath106 and @xmath107 lie within the 1-@xmath73 uncertainty range around the m - giant measurements . these results , presented in tabular form in table [ mass.table ] , indicate that in all models of the galactic potential considered the present bound mass of the sgr dwarf should not be very different from @xmath119 2 - 5 @xmath120 if the model dwarf is to successfully reproduce the m giant observations . we now fix the initial mass and scale of the model dwarf such that the present - day dwarf has a bound mass in the range found above in 3.2.1 , and endeavor to refine our orbits using the single remaining free parameter @xmath49 . we explore a range of values @xmath100 km s@xmath22 around the values @xmath121 280/270/254 km s@xmath22 chosen from test - particle orbits previously in 3.1.5c . note that it is not possible to fix the final bound mass of the satellite in these simulations , since the change in the orbital path produced by varying @xmath49 will naturally affect the mass - loss history of the model dwarf . however , as demonstrated by figure [ mplot ] ( filled triangles ) these small variations in @xmath49 have only a minor effect on the final mass of the model dwarf . returning to constraint 4 on the average apogalacticon distance of leading debris , the average distance of observed leading sgr debris ( @xmath37 ) is calculated from the 2mass database by averaging over the distances of all stars in the range @xmath122 - @xmath123 with heliocentric distances @xmath124 kpc @xmath113 60 kpc and subject to the restrictions @xmath125 , @xmath110 , @xmath111 kpc , @xmath126 kpc ( this combination of restrictions was chosen to separate sgr leading arm stars most clearly from the underlying disk population ) . figure [ davgplot ] plots the average apogalacticon distance of the m giants as a solid line with 1-@xmath73 error bars indicated by the hatched region , along with the values calculated from the simulated data ( again incorporating a 17% distance uncertainty ) for the simulations with fixed initial mass and physical scale but varying @xmath49 ( filled triangles ) . simulations with a range of initial masses and physical scales whose present bound mass falls within the acceptable range found in the previous section are also plotted ( filled squares and crosses ) : these points are difficult to distinguish since @xmath118 and @xmath28 are not the primary factors governing the behavior of @xmath37 , demonstrating the minor variation in @xmath37 permitted by the remaining uncertainty in satellite mass . while figure [ davgplot ] shows a strong correlation between leading debris distance and orbital velocity however , the relatively large uncertainty in the m giant debris distance allows us only to place constraints on the dwarf velocity to within about @xmath100 km s@xmath22 . a more compelling velocity constraint can be obtained by again using constraint 5 , that the trailing arm velocities match those observed for m giants . we calculate the average offset of the centroid of simulated trailing debris velocities - @xmath46 . ] from the m giant centroid and plot these offsets as a function of the tangential velocity of the dwarf in figure [ voffsetplot ] . well defined minima corresponding to the best fits to the velocity data are obtained for specific velocities in each choice of the galactic potential , and are fairly insensitive to the remaining uncertainties in satellite mass ( filled squares and crosses ) . we therefore conclude that the best choices of tangential velocity for the model dwarf are @xmath121 275 - 280/265 - 270/250 - 260 km s@xmath22 ( note that , in this case , the best - fit test particle orbits obtained in 3.1.5c actually picked out the best orbits for the n - body simulations ) . although each of these estimates are reasonably consistent with the observed value @xmath127 280 @xmath128 20 km s@xmath22 measured by ibata et al . ( 2001 ) , it is interesting to note that the ibata et al . ( 2001 ) measurement appears to slightly favor oblate models of the galactic halo over prolate models at the 1-@xmath73 level for our current choice of @xmath129 220 km s@xmath22 . note , however , that a higher value of @xmath24 will systematically shift these estimates of @xmath49 to higher velocities ( see figs . 4 & 5 , dashed line in lower right - hand panels ) , resulting in better agreement of estimates of @xmath49 in prolate halos with the ibata et al . ( 2001 ) measurement . based upon figures [ mplot ] , [ davgplot ] , and [ voffsetplot ] , simulations with @xmath119 2.6 - 5.0/2.5 - 5.3/2.5 - 5.5 @xmath120 and @xmath121 275 - 280/265 - 270/255 - 260 km s@xmath22 best fit our constraints , and these models are hereafter referred to as our `` best - fit models''srm4n / sgr/ to aid future comparisons of these models with new observations and new disruption models . ] . although the uncertainty in the galactic potential gives rise to uncertainties in @xmath49 considerably greater than the ranges given here , within a given potential @xmath49 can be constrained to within about @xmath128 5 km s@xmath22 . our best - fit models have a maximum extent of bound material @xmath130 along the semi - major axis , within which we calculate a luminosity for sgr of @xmath131 using data presented in paper i. the mass - to - light ratio of sgr in these models should therefore be @xmath132 19 - 36/18 - 38/18 - 39 @xmath133 . while the 500 maximum extent for bound material is somewhat dependent on the adopted internal structure of the satellite , it is on the order of the true tidal radius previously pointed out ( 4.3.3 of paper i ) as required to avoid sgr having a quite extraordinary ( and unlikely ) bound mass , and is also of order the observed _ minor _ axis dimension ( i.e. , 0.35 times that of the 1801 major axis radius ) of the limiting radius of the fitted king profile to the central satellite . these orbits have periods of 0.85/0.88/0.87 gyr with perigalactica and apogalactica of 10 - 16/14/14 - 19 kpc and 56 - 58/59/56 - 59 kpc respectively , and a present space velocity @xmath134 @xmath135 ( 238 , -42 , 222)/(235 , -40 , 213)/(231 , -37 , 198 ) km s@xmath22 , corresponding to @xmath136(230 , 75 , 222)/(227 , 73 , 213)/(224 , 69 , 197 ) km s@xmath22 and @xmath137 ( 171 , 272 , -65)/(171 , 263 , -63)/(171 , 247 , -59 ) km s@xmath22 with respect to the galactic standard of rest . these velocities will scale roughly with the assumed value of @xmath24 , although will also depend systematically upon @xmath54 , @xmath25 , and @xmath53 . figure [ simplot1 ] plots simulated sgr debris for our best - fit models along with the m giant distance and velocity data from papers i , ii and v , and demonstrates visually that our models generally fit the m giant observations well . the m giant data is clearly traced by debris released during the last two pericentric passages of the model dwarf ( yellow and magenta points ) and possibly by debris released three pericentric passage ago ( cyan points ) , although there appear to be far fewer m giants corresponding to cyan points than magenta or yellow . this corresponds to m giants becoming unbound from the sgr dwarf over the last 1.5 - 2.5 gyr consistent with constraint 10 , that the debris age be younger than the typical age of an m giant star . note , however , that as predicted by the orbits in 3.1 models in oblate and spherical halo potentials fail to fit the leading velocity trend ( particularly for cyan points ) , while the model orbiting in a prolate potential both reproduces this velocity trend and provides a more convincing fit to the apparent trend of m giant distances at @xmath138 - @xmath139 . note also the presence of cyan and green debris within a few kpc of the sun over a wide range of @xmath30 for oblate and spherical halo models - this is a consequence of the leading streamer diving almost directly through the solar neighborhood in these two models . conclusive proof of the presence or absence of sgr debris around the sun would provide a significant additional constraint on the models . the density of stars in the trailing stream for the best - fit models is plotted as a function of @xmath30 in figure [ denplot ] , and is similar in structure to the density of the m giant stream ( constraint 11 , plotted in fig . 13 of paper i ) , with a break in the slope of the density profile around @xmath140 degrees and a relatively constant density thereafter ( we only consider this first break in the observed density profile since we expect this to depend primarily upon satellite mass ) . the details of the run of density along the trailing streamer will depend on the internal light distribution of the satellite . however , since we consider only single - component models in this paper , we omit further consideration of the density profile and internal structure of the dwarf at this time .

we compare this data set to both test particle orbits and n - body simulations of satellite destruction run within a variety of rigid milky way potentials and find that the mass of the milky way within 50 kpc of its center should be in order for any sgr orbit to simultaneously fit the velocity gradient in the sgr trailing debris and the apocenter of the sgr leading debris . in light of this discrepancy , we consider constraints from the younger portions of the debris alone within three models of the flattening of the galactic potential ( 0.90/1.0/1.25 , i.e. oblate / spherical / prolate ) in our further n - body simulations . based upon the velocity dispersion and width along the trailing tidal stream we estimate the current bound mass of sgr to be independant of the form of the galactic potential ; this corresponds to a range of mass to light ratios - 36 for the sgr core . models with masses in this range best fit the apocenter of leading sgr tidal debris when they orbit with a radial period of roughly 0.85 gyr and have perigalactica and apogalactica of about 15 kpc and 60 kpc respectively . these distances will scale with the assumed distance to the sgr dwarf and the assumed depth of the galactic potential .

m giants recovered from the two micron all - sky survey ( 2mass ) have recently been used to map the position and velocity distributions of tidal debris from the sagittarius ( sgr ) dwarf spheroidal galaxy entirely around the galaxy . we compare this data set to both test particle orbits and n - body simulations of satellite destruction run within a variety of rigid milky way potentials and find that the mass of the milky way within 50 kpc of its center should be in order for any sgr orbit to simultaneously fit the velocity gradient in the sgr trailing debris and the apocenter of the sgr leading debris . orbital pole precession of young debris and leading debris velocities in regions corresponding to older debris provide contradictory evidence in favor of oblate / prolate galactic halo potentials respectively , leading us to conclude that the orbit of sgr has evolved over the past few gyr . in light of this discrepancy , we consider constraints from the younger portions of the debris alone within three models of the flattening of the galactic potential ( 0.90/1.0/1.25 , i.e. oblate / spherical / prolate ) in our further n - body simulations . based upon the velocity dispersion and width along the trailing tidal stream we estimate the current bound mass of sgr to be independant of the form of the galactic potential ; this corresponds to a range of mass to light ratios - 36 for the sgr core . models with masses in this range best fit the apocenter of leading sgr tidal debris when they orbit with a radial period of roughly 0.85 gyr and have perigalactica and apogalactica of about 15 kpc and 60 kpc respectively . these distances will scale with the assumed distance to the sgr dwarf and the assumed depth of the galactic potential . the density distribution of debris along the orbit in these models is consistent with the m giant observations , and debris at all orbital phases where m giants are obviously present is younger ( i.e. was lost more recently from the satellite ) than the typical age of a sgr m giant star .

astro-ph0407566

c

as noted in 2.2 , most other sgr detections around the sky fall within the m giant - traced tails ( see fig . 17 of paper i ) , so that our best - fit models also provide a good match to these other data . in this section , we compare our predictions for older sgr debris ( green points ) not traced by the m giants with observations of older tracers . in figure [ simplot2 ] , carbon star data and @xmath141 . ] ( open boxes ) are plotted for comparison with our best - fit sgr models ( colored points ) . while some of the carbon stars appear consistent with both m giant and simulated debris , many others have distances and velocities that differ substantially from the m giant and model distributions , and attempts to fit simulation models to these carbon stars will likely produce results that differ noticeably from our own best - fit models and the m giant data . although some of this discrepancy could be due to the uncertain distance scale for the carbon stars ( see 8.3 of paper i ) , it is also possible that these stars could trace debris older than the @xmath44 2.5 gyr old m giant stream , since carbon stars can have larger ages ( 5 - 6 gyr ) than m giants . the open triangles near @xmath142 in figure [ simplot2 ] represent data for a set of metal - poor , k - giant stars first pointed out by @xcite . using semi - analytical modeling , @xcite proposed that these stars represent debris stripped from sgr three pericentric passages ago ( corresponding to cyan - colored points in our model ) . indeed , our model suggests that these points may plausibly be fit by cyan or green debris ( i.e. debris from 3 - 4 pericentric passages ago ) in the @xmath1 0.90 leading streamer that is currently raining down from the north galactic pole onto the solar neighborhood , although the interpretation of these data is uncertain in models where @xmath1 1.0 or 1.25 . we also note an interesting comparison with possible sgr red clump stars detected in a pencil - beam survey by majewski et al . ( 1999 ) at @xmath143 , and for which the radial velocity data are plotted in figure [ simplot2 ] ( top panel , solid triangles ) . these stars at @xmath144 exhibit a range of line - of - sight velocities from 0 to 150 km s@xmath22 , which closely matches the predicted range of velocities of simulated leading tidal debris wrapped almost @xmath5 in orbital longitude from the sgr dwarf ( cyan and green points ) for simulations where @xmath1 1.0 or 1.25 . the degree of this agreement is highly sensitive to the mass of the model satellite : simulations with present mass @xmath145 predict a larger dispersion in velocities than observed by majewski et al . , while simulations with mass @xmath146 predict a smaller dispersion than observed . it is tempting therefore to point to these data as further evidence in favor of the satellite mass estimates determined earlier in 3.2.1 however , the distance to these stars is measured to be roughly 20 kpc ( majewski et al . 1999 ) about half that of the cyan - green leading debris whose velocities they reproduce so well and therefore , while they are interesting to compare to model data , their true origin and interpretation remains unclear . recently , the discovery of an overdensity of a - colored stars in the sloan digital sky survey with apparent magnitude @xmath147 at @xmath148 degrees and within 15 kpc of sgr s nominal orbital plane was announced ( newberg et al . these authors estimate an average heliocentric distance of 83 kpc to these stars , but note that other detections in directions which overlap the m giant stream suggest that their adopted distance scale is 12.5% larger than that used to calibrate the m giants in paper i. the open circle in figure [ simplot2 ] ( left - hand panels ) plot the average of their data , with the distance rescaled to 73 kpc so that the m giant and sdss distance scales match . figure [ simplot2 ] suggests that it is plausible to identify the sdss detection with debris of age @xmath44 1.5 - 2.5 gyr ( i.e. cyan - colored points ) in the trailing sgr stream , although future radial velocity measurements could help determine whether this identification is correct or if the newberg et al . feature is instead a part of some older , more distant section of the stream or even halo substructure unrelated to sgr . newberg et al . ( 2003 ) also note a hint of precession in the sgr stream by comparing their detections of leading and trailing debris closer to sgr s core , in agreement with our own results presented in paper iii . unfortunately , the angular extent of the 83 kpc debris has not yet been mapped accurately enough to pinpoint the angular position of the centroid of the debris ; such a measurement could in the future provide a strong constraint on the flattening of the galactic potential . previous attempts to model the orbit and disruption history of the sgr dwarf ( e.g. velazquez & white 1995 , johnston , hernquist & bolte 1996 , ibata et al . 1997 , edelsohn & elmegreen 1997 , ibata & lewis 1998 , g ' omez - flechoso , fux & martinet 1999 , johnston et al . 1999 , helmi & white 2001 , ibata et al . 2001 , mart ' inez - delgado et al . 2004 ) have made considerable progress in constraining models of the dwarf using only the previously available pencil - beam detections of satellite debris . in this paper we have presented the first model based upon a complete all - sky view of the satellite s tidal streams , and in this section we review and compare some of the predictions of these earlier models to those of our own best - fit models . we first consider those results for which the majority of simulations by different groups have generally converged . almost all simulations agree that the radial period of the sgr dwarf should be about 3/4 gyr : in this work we find a period for our best - fit models of 0.85/0.88/0.87 gyr , in reasonable agreement with previous estimates of 0.76 gyr ( velazquez & white 1995 , ibata et al . 1997 ) , 0.7 gyr ( ibata & lewis 1998 ) , 0.55 - 0.75 gyr ( johnston et al . 1999 ) , 0.85 gyr ( helmi & white 2001 ) , and 0.74 gyr ( mart ' inez - delgado et al . there is a little more spread in the estimates proposed by different groups for the perigalacticon and apogalacticon distances of the dwarf s orbit : previous estimates include ( respectively ) 10 kpc and 52 kpc ( velazquez & white 1995 ) , 15 kpc and 60 kpc ( ibata & lewis 1998 ) , 15 kpc and 70 kpc ( g ' omez - flechoso , fux & martinet 1999 ) , 13 kpc and 41 kpc ( johnston et al . 1999 ) , 16 kpc and 60 kpc ( ibata et al . 2001 ) , and 12 kpc and 60 kpc ( mart ' inez - delgado et al . 2004 ) . with the 2mass database it is possible to measure the apogalacticon of leading tidal debris directly , and we match this constraint best by using models for sgr that have orbits with perigalacticon and apogalacticon distances of 10 - 16/14/14 - 19 kpc and 56 - 58/59/56 - 59 kpc respectively . we note , however , that the distance scale assumed for the m giants in paper i is not yet secure , and that the estimated size of sgr s orbit may scale accordingly . among those areas in which common values among the disruption models presented by various groups have not yet been found , perhaps foremost is the @xmath149 component of the galactic @xmath134 velocity of the sgr dwarf . some simulations ( e.g. , ibata et al . 1997 ) have simply set @xmath150 km s@xmath22 ( thereby assuming a polar orbit ) since this component was so poorly known . now that we have an accurate measurement of sgr s orbital pole ( paper i ) , we are able to predict the direction of its motion more precisely . based on our best - fit model , we predict that the proper motion of the sgr dwarf should be @xmath151 mas yr@xmath22 and @xmath152 mas yr@xmath22 in the solar rest frame km s@xmath22 relative to the lsr , for which we adopt a rotation velocity of 220 km s@xmath22 ( 3.1.5b ) . ] . the direction of this proper motion prediction ( @xmath153 ) is expected to be fairly robust within potentials with each choice of @xmath54 . however , the amplitude of the proper motion will depend on the exact form of the galactic potential , and hence should be revised once other fundamental galactic parameters such as @xmath52 and @xmath24 are known more precisely . conversely , as more accurate measurements of sgr s proper motion become available it will be possible to refine constraints on the galactic rotation curve . a second area of debate concerns the present bound mass of the sgr dwarf , for which estimates range from @xmath154 ( mart ' inez - delgado et al . 2004 ) to @xmath155 ( ibata et al . helmi & white ( 2001 ) find an intermediate value for a purely stellar satellite model with initial mass @xmath156 . as demonstrated in 3.2.1 , we find that a range of final masses @xmath157 - @xmath158 yield tidal tails whose thickness and velocity dispersion are consistent with m giant measurements in oblate , spherical , and prolate models of the galactic potential . using figure [ mplot ] we conclusively rule out models with a mass far outside this range ( such as that of mart ' inez - delgado et al . 2004 ) , since models with very high or low masses will not be able to produce tidal tails with the observed thickness and dispersion . visual inspection of the figures in mart ' inez - delgado et al . ( 2004 ) appears to contradict this statement . however , these authors simulation embeds the model satellite in a 40,000 particle live halo , which is probably responsible for the width of the debris stream : earlier work ( johnston , spergel & haydn 2002 ) has found that significant heating of a sgr - like debris stream can occur in a simulation using a live halo , even in a halo model realized with @xmath159 particles . as another consequence of the smaller satellite mass used in their model , mart ' inez - delgado et al . ( 2004 ) predict leading debris at @xmath160 ( corresponding to @xmath161 ) to be composed of stars which have been unbound from the satellite for 5 gyr or more ( since debris from lower - mass satellites takes longer to spread along the orbit ) , in contrast to the roughly 2 gyr found by our own analysis . as demonstrated in figure [ simplot1 ] , the sgr m giants which have an estimated age of 2 - 3 gyr are visible to at least this point in the leading tidal stream . as mart ' inez - delgado et al . ( 2004 ) point out ( and we discuss in paper i ) , stellar populations formed in the densest central regions of the satellite should not be immediately reflected in the tidal streams , and it will take some time for these stars to be present in any quantity in the outer regions of the satellite . hence , it is unlikely that the m giant population became mixed into the outer regions of sgr within a small fraction of a gyr , and we consider the mean age estimate of 5 gyr for this section of the tidal stream to be too high . in paper iii we showed that only galactic potentials with oblate halos could reproduce the precession of the orbital plane apparent in the leading _ vs _ trailing data sets . in contrast , @xcite demonstrated that only galactic potentials with prolate halos could reproduce the velocity trends in the leading debris . in this paper ( 3.1 ) we explore a much wider variety of galactic potentials than has been considered previously but fail to find a single orbit that can fit both the velocity trends and sense of precession . our conclusion is that the assumption of non - evolution of the orbit over the time - period that the debris explores is incorrect . since simulated debris in the region with troublesome velocities is cyan and green ( lost 2 and 3 orbits ago respectively ) , we estimate the timescale over which the evolution has taken place to be @xmath162 gyrs . we can get some idea of the physical scale of the evolution necessary by looking at the difference between the orbits in prolate , spherical and oblate potentials that is responsible for the difference in the velocity trend . figure [ orbxz.fig ] plots the orbits shown in figure [ orb.fig ] in galactic coordinates with the region corresponding to the leading debris velocity data shown as bold along each curve . as the potential moves from prolate to oblate , the orbit passes progressively nearer the sun and line - of - sight velocities more closely reflect the full motion along the orbit . this explains why the simulated line of sight velocities in this region become more extreme with the oblateness of the potential . figure [ orbxz.fig ] also suggests that observed debris velocities in the leading region might be accounted for even in an oblate or spherical potential if the pericenter of sgr s orbit has decreased by a factor of order unity within the last 2 - 3 gyrs ( from visual inspection of the figure ) since such a decrease in pericenter of the sgr _ dwarf _ over time could shift older sgr _ debris _ out to greater distances from the sun corresponding to the greater pericenter of the dwarf on the passage on which the debris became unbound . three factors could contribute to this evolution : an encounter with a large lump in the milky way potential , either dark or luminous ( e.g. such as the large magellanic cloud , see * ? ? ? * for a full description of this idea ) : we consider this unlikley since we would expect the signature of such an event to be a sudden change in sgr s orbit , and a corresponding sudden change in the velocities along its debris , rather than the smooth trends seen . global evolution of the galactic potential : we also consider this unlikely since : ( i ) the evolution would have to be very large in order to bring the pericenter inwards by a factor of two in such a short amount of time ; and ( ii ) any global evolution would affect both sgr s and the debris orbits similarly . dynamical friction : if we re - arrange equation [ 7 - 27 ] from @xcite we can find the mass necessary @xmath163 for a circular orbit at @xmath164 kpc ( i.e. to represent an orbit with of order unity larger pericenter than sgr today ) to decay to the center of the galaxy over a time period @xmath165 gyrs in a galaxy with a flat rotation curve and @xmath166 km s@xmath22 : @xmath167 @xcite estimate @xmath168 for the combined large and small magellanic clouds . since we expect @xmath169 , and know the current mass of sgr to be 2 - 5 @xmath120 , we expect @xmath170 to be the relevant range for our own estimate and hence @xmath171 . moreover , we consider this only a lower limit on the necessary mass since sgr s orbit is not circular . ( see * ? ? ? * ; * ? ? ? * for a general discussion of dynamical friction acting on sgr over a hubble time . ) although dynamical friction seems like the most favourable explanation for the orbit evolution it does require sgr to be an order of magnitude more massive just 2 gyrs ago and debris lost at that time in our mass - follows - light models would have a correspondingly larger dispersion in velocity ( by a factor of order @xmath172 ) and distances . since the observed velocity dispersion in the debris in the discrepant , leading portion of the stream is actually quite similar to that seen in our simulations ( @xmath173 km s@xmath22 , see fig . [ simplot1 ] ) this suggests that , in order to fit the data , in addition to dropping our assumption of a single orbit for sgr , we will also have to move beyond modelling sgr as a _ single component _ system . hence , while the mean trend in the leading streamer will tell us how much total mass needs to have been lost from sgr , the low dispersion offers the additional opportunity of constraining how much more tightly bound the light matter is compared to the dark matter . a study of these combined effects is in progress .

m giants recovered from the two micron all - sky survey ( 2mass ) have recently been used to map the position and velocity distributions of tidal debris from the sagittarius ( sgr ) dwarf spheroidal galaxy entirely around the galaxy . the density distribution of debris along the orbit in these models is consistent with the m giant observations , and debris at all orbital phases where m giants are obviously present is younger ( i.e. was lost more recently from the satellite ) than the typical age of a sgr m giant star .

m giants recovered from the two micron all - sky survey ( 2mass ) have recently been used to map the position and velocity distributions of tidal debris from the sagittarius ( sgr ) dwarf spheroidal galaxy entirely around the galaxy . we compare this data set to both test particle orbits and n - body simulations of satellite destruction run within a variety of rigid milky way potentials and find that the mass of the milky way within 50 kpc of its center should be in order for any sgr orbit to simultaneously fit the velocity gradient in the sgr trailing debris and the apocenter of the sgr leading debris . orbital pole precession of young debris and leading debris velocities in regions corresponding to older debris provide contradictory evidence in favor of oblate / prolate galactic halo potentials respectively , leading us to conclude that the orbit of sgr has evolved over the past few gyr . in light of this discrepancy , we consider constraints from the younger portions of the debris alone within three models of the flattening of the galactic potential ( 0.90/1.0/1.25 , i.e. oblate / spherical / prolate ) in our further n - body simulations . based upon the velocity dispersion and width along the trailing tidal stream we estimate the current bound mass of sgr to be independant of the form of the galactic potential ; this corresponds to a range of mass to light ratios - 36 for the sgr core . models with masses in this range best fit the apocenter of leading sgr tidal debris when they orbit with a radial period of roughly 0.85 gyr and have perigalactica and apogalactica of about 15 kpc and 60 kpc respectively . these distances will scale with the assumed distance to the sgr dwarf and the assumed depth of the galactic potential . the density distribution of debris along the orbit in these models is consistent with the m giant observations , and debris at all orbital phases where m giants are obviously present is younger ( i.e. was lost more recently from the satellite ) than the typical age of a sgr m giant star .

1006.0946

i

quantum chromodynamics ( qcd ) , as the gauge field theory of the strong interaction , reliably predicts scattering cross sections involving short distance partonic interactions . however , the vast majority of hadron - hadron scatterings take place through long - distance strong interactions , where no hard scales are present and perturbative qcd calculations are not possible . prominent among these soft interactions are diffractive processes , in which the interacting hadrons remain intact or dissociate into low mass hadronic systems via an exchange which has vacuum quantum numbers , often referred to as a ` pomeron ' @xcite . following the observation of diffractive @xmath63 collisions in which a hard scale is provided by high transverse momentum jets @xcite , it has become possible to describe some classes of diffractive processes in terms of partonic interactions @xcite . more recently , diffractive deep - inelastic scattering ( ddis ) processes at hera @xcite , of the type @xmath64 , have been studied in detail and have led to a new level of understanding of the properties and structure of the diffractive exchange . developing this microscopic description of diffraction in terms of qcd and parton dynamics is a step towards a more complete understanding of the strong interaction . in the framework of a collinear factorisation theorem @xcite for hard scattering in semi - inclusive processes such as ddis , diffractive parton distribution functions ( dpdfs ) may be defined . the dpdfs have similar properties to the standard parton distribution functions ( pdfs ) of the proton , but with the constraint that there be a leading proton present in the final state . this condition may be satisfied equivalently by the experimental signatures of either a leading proton @xcite or the presence of a large gap in the rapidity distribution of final state hadrons , separating an unobserved outgoing proton from the remainder of the hadronic final state @xcite . in various extractions using recent ddis data @xcite , the dpdfs have been found to be dominated by gluons . to good approximation they exhibit a ` proton vertex factorisation ' property , whereby they vary only in normalisation with the four - momentum of the final state proton , the normalisation being well modelled using regge phenomenology @xcite . given a knowledge of the dpdfs , perturbative qcd calculations are applicable to other ddis observables . such calculations have been successful in the prediction of jet @xcite and heavy quark @xcite production in ddis at hera . in both cases , next - to - leading order ( nlo ) qcd predictions using the dpdfs from @xcite describe the measured cross sections well . however , as has long been anticipated @xcite , dpdf - based predictions for hard diffractive processes such as dijet production in @xmath63 scattering fail by around an order of magnitude to describe the data @xcite . this factorisation breaking is generally attributed to absorptive corrections , corresponding to the destruction of the rapidity gap due to multiple interactions within a single event . such effects are possible in @xmath63 scattering , where a beam remnant is present , in contrast to the electron scattering case in ddis at hera . a diversity of models of the absorptive corrections has developed @xcite , several of which reproduce the approximate @xmath65 ` rapidity gap survival probability ' observed in single diffractive @xmath63 scattering . the issues of dpdf applicability and rapidity gap survival can be studied in @xmath6 collisions at hera in hard diffractive ` photoproduction ' , where the virtuality @xmath66 of the exchange photon is close to zero . under these circumstances , the photon can develop an effective partonic structure via @xmath67 fluctuations and further subsequent splittings @xcite . in a leading order picture , there are thus two classes of hard photoproduction : ` direct ' interactions , where the photon enters the hard scatter as a structureless object and ` resolved ' interactions , where the photon interacts via its partonic structure and only a fraction @xmath68 of its four - momentum participates in the hard subprocess . resolved photoproduction interactions can be further divided into a ` hadron - like ' contribution and an ` anomalous ' or ` point - like ' contribution , the latter arising from the inhomogeneous term in the dglap equation for the photon @xcite . interactions involving the hadron - like component resemble hadron - hadron scattering to a large extent and are therefore widely expected to exhibit gap destruction effects . the rapidity gap survival probability for these hadron - like processes has been estimated in a phenomenological model to be @xmath69 @xcite . the point - like contribution to photon structure is expected to be subject to smaller absorptive corrections than the hadron - like part @xcite . in a recent model @xcite a survival probability of around @xmath70 was obtained for diffractive dijet photoproduction , depending slightly on the jet transverse energies ( @xmath71 ) . previous h1 measurements of diffractive dijet photoproduction @xcite have found cross sections to be smaller than nlo theoretical predictions , suggesting rapidity gap survival probabilities of around @xmath72 with little dependence on @xmath68 . a recent zeus measurement at somewhat larger @xmath71 @xcite yielded a larger survival probability , compatible with unity . it has been proposed that this apparent discrepancy may be resolved if the rapidity gap survival probability depends on the scale of the hard interaction , an idea which is supported to some extent by data @xcite . neither h1 nor zeus data provide any evidence for the expected @xmath68 dependence of the rapidity gap survival probability . a measurement of the ratio of the diffractive to the inclusive dijet photoproduction cross sections was proposed in @xcite as a means of evaluating the gap survival probability . this ratio is expected to be relatively insensitive to the model of the photon parton densities and also offers cancellations of experimental systematics and higher order qcd corrections . a similar ratio was measured by the cdf collaboration @xcite as a means of extracting effective @xmath73 dpdfs for comparison with hera predictions and assessment of gap survival probabilities . this paper reports diffractive dijet photoproduction cross section measurements based on a positron - proton scattering data sample with luminosity about a factor three larger than that previously published by h1 @xcite . the larger sample makes double - differential measurements possible , giving greater detail on the dynamics of gap survival and allowing studies of the correlations between the kinematic variables . the hypothesis of an @xmath71 dependent rapidity gap survival probability is tested . the ratio of the diffractive to the inclusive dijet photoproduction cross sections is also extracted for the first time .

measurements are presented of single and double - differential dijet cross sections in diffractive photoproduction based on a data sample with an integrated luminosity of . ratios of the diffractive to the inclusive dijet cross sections are measured for the first time and are compared with monte carlo models .

measurements are presented of single and double - differential dijet cross sections in diffractive photoproduction based on a data sample with an integrated luminosity of . the events are of the type , where the hadronic system contains at least two jets and is separated by a large rapidity gap from the system , which consists of a leading proton or low - mass proton excitation . the dijet cross sections are compared with qcd calculations at next - to - leading order and with a monte carlo model based on leading order matrix elements with parton showers . the measured cross sections are smaller than those obtained from the next - to - leading order calculations by a factor of about . this suppression factor has no significant dependence on the fraction of the photon four - momentum entering the hard subprocess . ratios of the diffractive to the inclusive dijet cross sections are measured for the first time and are compared with monte carlo models . # 1#2#3#4#1 * # 2 * ( # 3 ) # 4 desy 10 - 043issn 0418 - 9833 + march 2010 * diffractive dijet photoproduction + in collisions at hera * h1 collaboration submitted to f.d . aaron , c. alexa , v. andreev , s. backovic , a. baghdasaryan , e. barrelet , w. bartel , k. begzsuren , a. belousov , j.c . bizot , v. boudry , i. bozovic - jelisavcic , j. bracinik , g. brandt , m. brinkmann , v. brisson , d. bruncko , a. bunyatyan , g. buschhorn , l. bystritskaya , a.j . campbell , k.b . cantun avila , k. cerny , v. cerny , v. chekelian , a. cholewa , j.g . contreras , j.a . coughlan , j. cvach , j.b . dainton , k. daum , m. dek , b. delcourt , j. delvax , e.a . de wolf , c. diaconu , m. dobre , v. dodonov , a. dossanov , a. dubak , g. eckerlin , v. efremenko , s. egli , a. eliseev , e. elsen , a. falkiewicz , l. favart , a. fedotov , r. felst , j. feltesse , j. ferencei , d .- j . fischer , m. fleischer , a. fomenko , e. gabathuler , j. gayler , s. ghazaryan , a. glazov , l. goerlich , n. gogitidze , m. gouzevitch , c. grab , a. grebenyuk , t. greenshaw , b.r . grell , g. grindhammer , s. habib , d. haidt , c. helebrant , r.c.w . henderson , e. hennekemper , h. henschel , m. herbst , g. herrera , m. hildebrandt , k.h . hiller , d. hoffmann , r. horisberger , t. hreus , f. huber , m. jacquet , x. janssen , l. jnsson , a.w . jung , h. jung , m. kapichine , j. katzy , i.r . kenyon , c. kiesling , m. klein , c. kleinwort , t. kluge , a. knutsson , r. kogler , p. kostka , m. kraemer , j. kretzschmar , a. kropivnitskaya , k. krger , k. kutak , m.p.j . landon , w. lange , g. latovika - medin , p. laycock , a. lebedev , v. lendermann , s. levonian , k. lipka , b. list , j. list , n. loktionova , r. lopez - fernandez , v. lubimov , a. makankine , e. malinovski , p. marage , ll . marti , h .- u . martyn , s.j . maxfield , a. mehta , a.b . meyer , h. meyer , j. meyer , s. mikocki , i. milcewicz - mika , f. moreau , a. morozov , j.v . morris , m.u . mozer , m. mudrinic , k. mller , th . naumann , p.r . newman , c. niebuhr , a. nikiforov , d. nikitin , g. nowak , k. nowak , j.e . olsson , s. osman , d. ozerov , p. pahl , v. palichik , i. panagoulias , m. pandurovic , th . papadopoulou , c. pascaud , g.d . patel , e. perez , a. petrukhin , i. picuric , s. piec , h. pirumov , d. pitzl , r. plaakyt , b. pokorny , r. polifka , b. povh , v. radescu , n. raicevic , a. raspiareza , t. ravdandorj , p. reimer , e. rizvi , p. robmann , r. roosen , a. rostovtsev , m. rotaru , j.e . ruiz tabasco , s. rusakov , d. lek , d.p.c . sankey , m. sauter , e. sauvan , s. schmitt , l. schoeffel , a. schning , h .- c . schultz - coulon , f. sefkow , r.n . shaw - west , l.n . shtarkov , s. shushkevich , t. sloan , i. smiljanic , y. soloviev , p. sopicki , d. south , v. spaskov , a. specka , z. staykova , m. steder , b. stella , g. stoicea , u. straumann , d. sunar , t. sykora , g. thompson , p.d . thompson , t. toll , t.h . tran , d. traynor , p. trul , i. tsakov , b. tseepeldorj , j. turnau , k. urban , a. valkrov , c. valle , p. van mechelen , a. vargas trevino , y. vazdik , m. von den driesch , d. wegener , e. wnsch , j. ek , j. zlek , z. zhang , a. zhokin , h. zohrabyan , and f. zomer

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a detailed description of the h1 detector can be found elsewhere @xcite . here , a brief account is given of the detector components most relevant to the present analysis . the h1 coordinate system is defined such that the origin is at the nominal @xmath6 interaction point and the polar angle @xmath133 and the positive @xmath134 axis correspond to the direction of the outgoing proton beam . the region @xmath135 , which has positive pseudorapidity @xmath136 , is referred to as the ` forward ' hemisphere . the _ ep _ interaction point in h1 is surrounded by a central tracking region , which includes silicon strip detectors as well as two large concentric drift chambers . these chambers cover a pseudorapidity region of @xmath137 and have a transverse momentum resolution of @xmath138 . they also provide triggering information . the central tracking detectors are surrounded by a finely segmented liquid argon ( lar ) sampling calorimeter covering @xmath139 . its resolution is @xmath140 for electrons and photons and @xmath141 for hadrons , as measured in test beams @xcite . the central tracker and lar calorimeter are placed inside a large superconducting solenoid , which produces a uniform magnetic field of @xmath142 t. the backward region @xmath143 is covered by a lead - scintillating fibre calorimeter ( spacal ) with electromagnetic and hadronic sections . information from the central tracker and the lar and spacal calorimeters is combined using an energy flow algorithm to obtain the hadronic final state ( hfs ) @xcite . the hadronic energy scale is known to @xmath144 for this analysis @xcite . photoproduction events are selected by tagging positrons scattered through very small angles , corresponding to quasi - real photon emission , using a crystal erenkov calorimeter at @xmath145 ( electron tagger ) . the luminosity is measured via the bethe - heitler bremsstrahlung process @xmath146 , the final state photon being detected in another crystal calorimeter at @xmath147 . a set of drift chambers around @xmath148 comprise the forward muon detector ( fmd ) . the proton remnant tagger ( prt ) is a set of scintillators surrounding the beam pipe at @xmath149 . these detectors , used together with the most forward part of the lar , are efficient in the identification of very forward energy flow and are used to select events with large rapidity gaps near to the outgoing proton direction . the analysis is based on a sample of integrated luminosity @xmath0 , collected by h1 in 1999 and 2000 with proton and positron beam energies of @xmath150 and @xmath151 , respectively . the events are triggered on the basis of a scattered positron signal in the electron tagger and at least three high transverse momentum tracks in the drift chambers of the central tracker . the event inelasticity @xmath80 and hence the invariant mass @xmath83 of the photon - proton system are reconstructed using the scattered positron energy @xmath152 measured in the electron tagger according to @xmath153 where @xmath154 is the positron beam energy . the geometric acceptance of the electron tagger limits the measurement to @xmath155 and intermediate values of @xmath80 . the reconstructed hadronic final state objects ( section [ sec : h1 detector ] ) are subjected to the @xmath156 longitudinally invariant jet algorithm @xcite , applied in the laboratory frame with parameters @xmath157 and @xmath158 . to facilitate comparisons with the nlo calculations , different cuts are placed on the transverse energies @xmath121 and @xmath159 of the leading and next - to - leading jets , respectively . as well as these variables , the jet properties are studied in terms of the variables @xmath160 obtained from the laboratory frame pseudorapidities of the jet axes . with @xmath161 and @xmath162 denoting the four - momenta of the two jets , hadron level estimators of the dijet invariant mass and of @xmath68 are obtained from @xmath163 where the sums labelled ` _ hfs _ ' and ` _ jets _ ' run over all hadronic final state objects and those included in the jets , respectively . the diffractive event selection is based on the presence of a large forward rapidity gap . the pseudorapidity of the most forward cluster in the lar calorimeter with energy above @xmath164 mev is required to satisfy @xmath165 . the activity in the prt and the fmd is required not to exceed that typical of noise levels as obtained from randomly triggered events . these requirements ensure that the analysed sample is dominated by elastically scattered protons at small @xmath166 , with a small admixture of events with leading neutrons and low @xmath89 baryon excitations , collectively referred to here as ` proton dissociation ' contributions . the diffractive kinematics are reconstructed using @xmath167 where @xmath168 is the proton beam energy . a cut on @xmath91 is applied to ensure good containment of the system @xmath2 and to suppress sub - leading exchange contributions . a hadron level estimator for the momentum fraction @xmath96 is obtained using @xmath169 the kinematic range in which the diffractive dijet measurement is performed is specified in table [ tab : kinrange ] . the inclusive measurement phase space is defined by these conditions , with the requirements relaxed on the diffractive variables @xmath91 , @xmath170 , @xmath89 and @xmath90 . except where measurements are made explicitly as a function of @xmath171 , the @xmath172 region is excluded from the diffractive analysis . this improves the reliability of the comparison between data and theoretical predictions , since the dpdf sets used are not valid at the largest @xmath96 values . after applying all selection criteria , about @xmath173 out of roughly @xmath174 inclusive dijet photoproduction events are used in the diffractive analysis . a more detailed description of the analysis can be found in @xcite . @xmath175 & & @xmath176 + @xmath177 & & @xmath178 + @xmath179 & & @xmath180 + + @xmath181 & & @xmath182 + @xmath183 & & @xmath184 + the diffractive differential cross section is measured in each bin @xmath101 of a variable @xmath81 using the formula @xmath185 here , @xmath186 is the raw number of reconstructed events passing the selection criteria listed in section [ sec : selection ] and @xmath187 is the trigger efficiency , obtained by reference to an independently triggered sample and parameterised as a function of the multiplicity of charged particle tracks . the trigger efficiency averaged over the full measurement range is @xmath188 . the non - diffractive background contribution obtained from the pythia mc simulation is denoted @xmath189 and does not extend beyond the few percent level for any of the measured data points . the factor @xmath190 corrects the measurement for detector effects , including migrations between bins , to the level of stable hadrons . it is calculated from the rapgap mc and has an average value of @xmath191 for the diffractive analysis , most of the losses being due to the limited electron tagger acceptance of @xmath192 integrated over the measured @xmath80 range . the bin width is denoted @xmath193 , @xmath194 is the luminosity of the data sample and @xmath195 , evaluated using the diffvm mc , corrects the measurement to the chosen range of @xmath123 and @xmath90 ( table [ tab : kinrange ] ) . in the inclusive analysis the cross section is obtained analogously to equation [ xsecformula ] except for the @xmath196 and @xmath197 terms , which are not relevant . uncertainties are evaluated for all significant sources of possible systematic bias . these sources are summarised for the diffractive analysis below , together with their corresponding influences on the total diffractive cross section . [ [ energy - scale ] ] energy scale : + + + + + + + + + + + + + the energy scale of the hfs measurement is tested using the momentum balance constraint between the precisely reconstructed positron and the hfs in neutral current dis events . dedicated data and mc samples are analysed and found to agree to better than @xmath144 . the effect of a relative @xmath144 change in the energy of the hfs between the data and the mc is a @xmath198 shift to the total diffractive dijet cross section . this arises mainly from changes in the migration corrections across the minimum @xmath71 values and the maximum @xmath91 value of the measurement . the energy scale uncertainties are thus highly correlated between the bins of the differential cross section measurements . [ [ large - rapidity - gap - selection ] ] large rapidity gap selection : + + + + + + + + + + + + + + + + + + + + + + + + + + + + + a fraction of the events in the kinematic range of the analysis ( table [ tab : kinrange ] ) give rise to hadronic activity in the forward detectors or at pseudorapidities beyond those allowed by the @xmath199 cut in the lar calorimeter . corrections for this inefficiency of the large rapidity gap selection are made using the rapgap mc simulation . the uncertainties in the correction factors are assessed through a study of forward energy flow in a sample of dijet photoproduction events with leading protons tagged in the h1 forward proton spectrometer @xcite . rapgap is found to describe these migrations to within 10% @xcite , which translates into a @xmath200 uncertainty on the measured total cross section and uncertainties which are correlated between bins of the differential distributions . [ [ proton - dissociation ] ] proton dissociation : + + + + + + + + + + + + + + + + + + + + the model dependence uncertainty on the proton dissociation correction factor ( @xmath197 in equation [ xsecformula ] ) is obtained by varying the elastic and proton dissociation cross sections and the proton dissociation @xmath89 and @xmath90 dependences in the diffvm mc samples , following @xcite . the largest effect arises from varying the ratio of the proton - elastic to the proton - dissociative cross sections between @xmath72 and @xmath118 . the resulting uncertainty on the measured cross section is @xmath201 . [ [ model - dependence ] ] model dependence : + + + + + + + + + + + + + + + + + the influence of the model assumptions on the acceptance and bin migration corrections ( @xmath202 in equation [ xsecformula ] ) , is determined in the diffractive analysis by varying the kinematic distributions in the rapgap simulation within the limits allowed by maintaining an acceptable description of the uncorrected data . the following variations are implemented by reweighting each mc event according to the value of generator level kinematic variables , leading to the quoted systematic uncertainties on the total cross section . * the @xmath91 distribution is reweighted by @xmath203 , leading to a @xmath204 uncertainty . * the @xmath171 distribution is reweighted by @xmath205 , leading to a @xmath206 uncertainty . * the @xmath207 distribution is reweighted by @xmath208 , leading to a @xmath209 uncertainty . * the @xmath121 distribution is reweighted by @xmath210 , leading to a @xmath211 uncertainty . * the @xmath90 distribution is reweighted by @xmath212 , leading to a @xmath213 uncertainty . * the @xmath80 distribution is reweighted by @xmath214 , leading to a @xmath215 uncertainty . [ [ electron - tagger - acceptance ] ] electron tagger acceptance : + + + + + + + + + + + + + + + + + + + + + + + + + + + a dedicated procedure external to this analysis is used to obtain the electron tagger acceptance @xcite . the integrated acceptance over the full @xmath80 range is known to @xmath216 , which affects the cross section normalisation . [ [ trigger - efficiency ] ] trigger efficiency : + + + + + + + + + + + + + + + + + + + the procedure for parameterising the trigger efficiency ( @xmath187 in equation [ xsecformula ] ) leads to a @xmath122 uncertainty . this covers the observed deviations of the parameterisation from the measured efficiencies as a function of all variables relevant to the analysis . this uncertainty is treated as being uncorrelated between data points . [ [ luminosity ] ] luminosity : + + + + + + + + + + + the measurement of the integrated luminosity has an uncertainty of @xmath217 . this translates directly into a @xmath217 normalisation uncertainty on the measured cross sections . [ [ non - diffractive - background ] ] non - diffractive background : + + + + + + + + + + + + + + + + + + + + + + + + + + + a @xmath218 normalisation variation is applied to the non - diffractive background contribution given by the pythia mc model ( @xmath196 in equation [ xsecformula ] ) . the effect of this change is correlated between the data points and leads to a @xmath219 uncertainty on the total cross section . [ [ forward - detector - noise ] ] forward detector noise : + + + + + + + + + + + + + + + + + + + + + + + fluctuations in the fmd noise , leading to losses in the large rapidity gap event selection , are evaluated for each run using randomly triggered events . the standard deviation in the run - by - run distribution of the correction factors is used to derive a @xmath220 normalisation uncertainty on the measured cross sections . noise in the prt detector is negligible . a similar procedure is followed to evaluate the systematic uncertainties in the inclusive dijet analysis . the uncertainties associated with the large rapidity gap selection and the model dependence are no longer relevant . instead , comparisons between the pythia and herwig mcs are used determine a @xmath221 model dependence uncertainty on the acceptance correction when integrated over the full phase space studied . the inclusive cross section systematics are dominated by a contribution at the @xmath222 level from the hfs energy scale uncertainty . however , when forming the ratio of diffractive to inclusive cross sections , this error source cancels to good approximation , the residual uncertainty being less than @xmath223 . the largest remaining contribution to the systematic uncertainty on the cross section ratio arises from the model dependence . the total systematic uncertainty on each data point is formed by adding the individual contributions in quadrature . in the figures and tables that follow , the systematic uncertainties are separated into two categories : those which are uncorrelated between data points ( the model dependence and trigger efficiency ) and those which lead to correlations between data points ( all other sources ) .

de wolf , c. diaconu , m. dobre , v. dodonov , a. dossanov , a. dubak , g. eckerlin , v. efremenko , s. egli , a. eliseev , e. elsen , a. falkiewicz , l. favart , a. fedotov , r. felst , j. feltesse , j. ferencei , d .- j . thompson , t. toll , t.h .

measurements are presented of single and double - differential dijet cross sections in diffractive photoproduction based on a data sample with an integrated luminosity of . the events are of the type , where the hadronic system contains at least two jets and is separated by a large rapidity gap from the system , which consists of a leading proton or low - mass proton excitation . the dijet cross sections are compared with qcd calculations at next - to - leading order and with a monte carlo model based on leading order matrix elements with parton showers . the measured cross sections are smaller than those obtained from the next - to - leading order calculations by a factor of about . this suppression factor has no significant dependence on the fraction of the photon four - momentum entering the hard subprocess . ratios of the diffractive to the inclusive dijet cross sections are measured for the first time and are compared with monte carlo models . # 1#2#3#4#1 * # 2 * ( # 3 ) # 4 desy 10 - 043issn 0418 - 9833 + march 2010 * diffractive dijet photoproduction + in collisions at hera * h1 collaboration submitted to f.d . aaron , c. alexa , v. andreev , s. backovic , a. baghdasaryan , e. barrelet , w. bartel , k. begzsuren , a. belousov , j.c . bizot , v. boudry , i. bozovic - jelisavcic , j. bracinik , g. brandt , m. brinkmann , v. brisson , d. bruncko , a. bunyatyan , g. buschhorn , l. bystritskaya , a.j . campbell , k.b . cantun avila , k. cerny , v. cerny , v. chekelian , a. cholewa , j.g . contreras , j.a . coughlan , j. cvach , j.b . dainton , k. daum , m. dek , b. delcourt , j. delvax , e.a . de wolf , c. diaconu , m. dobre , v. dodonov , a. dossanov , a. dubak , g. eckerlin , v. efremenko , s. egli , a. eliseev , e. elsen , a. falkiewicz , l. favart , a. fedotov , r. felst , j. feltesse , j. ferencei , d .- j . fischer , m. fleischer , a. fomenko , e. gabathuler , j. gayler , s. ghazaryan , a. glazov , l. goerlich , n. gogitidze , m. gouzevitch , c. grab , a. grebenyuk , t. greenshaw , b.r . grell , g. grindhammer , s. habib , d. haidt , c. helebrant , r.c.w . henderson , e. hennekemper , h. henschel , m. herbst , g. herrera , m. hildebrandt , k.h . hiller , d. hoffmann , r. horisberger , t. hreus , f. huber , m. jacquet , x. janssen , l. jnsson , a.w . jung , h. jung , m. kapichine , j. katzy , i.r . kenyon , c. kiesling , m. klein , c. kleinwort , t. kluge , a. knutsson , r. kogler , p. kostka , m. kraemer , j. kretzschmar , a. kropivnitskaya , k. krger , k. kutak , m.p.j . landon , w. lange , g. latovika - medin , p. laycock , a. lebedev , v. lendermann , s. levonian , k. lipka , b. list , j. list , n. loktionova , r. lopez - fernandez , v. lubimov , a. makankine , e. malinovski , p. marage , ll . marti , h .- u . martyn , s.j . maxfield , a. mehta , a.b . meyer , h. meyer , j. meyer , s. mikocki , i. milcewicz - mika , f. moreau , a. morozov , j.v . morris , m.u . mozer , m. mudrinic , k. mller , th . naumann , p.r . newman , c. niebuhr , a. nikiforov , d. nikitin , g. nowak , k. nowak , j.e . olsson , s. osman , d. ozerov , p. pahl , v. palichik , i. panagoulias , m. pandurovic , th . papadopoulou , c. pascaud , g.d . patel , e. perez , a. petrukhin , i. picuric , s. piec , h. pirumov , d. pitzl , r. plaakyt , b. pokorny , r. polifka , b. povh , v. radescu , n. raicevic , a. raspiareza , t. ravdandorj , p. reimer , e. rizvi , p. robmann , r. roosen , a. rostovtsev , m. rotaru , j.e . ruiz tabasco , s. rusakov , d. lek , d.p.c . sankey , m. sauter , e. sauvan , s. schmitt , l. schoeffel , a. schning , h .- c . schultz - coulon , f. sefkow , r.n . shaw - west , l.n . shtarkov , s. shushkevich , t. sloan , i. smiljanic , y. soloviev , p. sopicki , d. south , v. spaskov , a. specka , z. staykova , m. steder , b. stella , g. stoicea , u. straumann , d. sunar , t. sykora , g. thompson , p.d . thompson , t. toll , t.h . tran , d. traynor , p. trul , i. tsakov , b. tseepeldorj , j. turnau , k. urban , a. valkrov , c. valle , p. van mechelen , a. vargas trevino , y. vazdik , m. von den driesch , d. wegener , e. wnsch , j. ek , j. zlek , z. zhang , a. zhokin , h. zohrabyan , and f. zomer

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r

cross sections are measured integrated over the full kinematic range specified in table [ tab : kinrange ] and also single- and double - differentially as a function of a variety of variables which are sensitive to the overall event structure , the hard subprocess and the presence of remnants of the virtual photon and the diffractive exchange . the measured differential cross sections , which correspond to averages over the specified measurement intervals , are given numerically in tables [ table1d ] and [ table2d ] , where the experimental uncertainties and hadronisation corrections applied to the nlo calculations are also listed . tables [ table1dratios ] and [ table2dratios ] contain the ratios of the measurements to the nlo calculations , obtained using the fr framework ( section [ nlo ] ) and the h1 2006 fit b dpdfs ( referred to in the following as fr fit b ) . the total diffractive dijet positron - proton cross section integrated over the full measured kinematic range is @xmath224 the ratio of this result to the corresponding fr fit b nlo prediction is @xmath225 where the statistical and systematic uncertainties originate from the measurement . the scale uncertainty corresponds to the effect of simultaneously varying the renormalisation and factorisation scales from their central values , @xmath114 , by a factor of two in either direction . this large ( @xmath226 ) scale uncertainty arises due to the relatively low @xmath121 range of this analysis , and is the limiting factor in the comparison between data and theory . the dpdf uncertainty is obtained using the method of @xcite , by propagating the eigenvector decomposition of the fit uncertainties . if the h1 2006 fit b dpdfs are replaced by the h1 2007 fit jets dpdfs , the result is @xmath227 , which is inside the quoted dpdf uncertainty . using zeus dpdf sj , a compatible result of @xmath228 is obtained . adding all uncertainties in quadrature , the ratio result in equation [ sigmatot : ratio ] implies at the @xmath229 level that the nlo qcd calculation , neglecting any gap destruction effects , yields a larger diffractive dijet photoproduction cross section than that measured . it confirms the result of a previous h1 analysis in a very similar kinematic range @xcite and is broadly as expected from theoretical calculations of rapidity gap survival probabilities @xcite . figure [ fig : sigma single ] shows the diffractive dijet cross section measured single - differentially in @xmath207 , @xmath121 , @xmath230 , @xmath171 , @xmath231 , @xmath232 , @xmath83 , @xmath95 and @xmath233 , in the phase space defined in table [ tab : kinrange ] . in figure [ fig : sigma single](d ) , the @xmath234 requirement is relaxed and the cross section measured for the region @xmath172 is shown without theoretical comparisons , since the dpdfs are not defined ( see section [ sec : selection ] ) . to allow a more detailed shape comparison between the data and the predictions , ratios of the measured differential cross sections to the fr fit b calculations are plotted in figure [ fig : sigma single ratio ] . these ratios may be taken as measurements of the dependence of the rapidity gap survival probability on the kinematic variables . figures [ fig : sigma single ] and [ fig : sigma single ratio ] show that the suppression by around a factor of @xmath4 of the data with respect to the fr fit b nlo calculations has at most a weak dependence on the kinematic variables . notably , within the uncertainties there is no dependence on @xmath207 ( figure [ fig : sigma single ratio](a ) ) , in contrast to theoretical predictions for the rapidity gap survival probability @xcite . the largest dependence of the central values of the measured ratios on any of the variables appears in the cross section differential in @xmath121 ( figure [ fig : sigma single ratio](b ) ) . although not well established by the current data , this dependence is compatible with previous data @xcite . the @xmath121 dependence is investigated further in section [ double : diff ] . the measured cross sections in figure [ fig : sigma single ] are also compared with a prediction obtained using the rapgap mc generator ( section [ sec : mc ] ) , which does not contain any model of rapidity gap destruction . the shapes of the measured cross sections are well described and the normalisation is only slightly lower than that of the data . however the scale uncertainty in this model is rather large and the same model undershoots diffractive dijet measurements in dis @xcite , where factorisation is expected to hold . in @xcite , the zeus collaboration presented an analysis of diffractive dijet photoproduction data with @xmath235 , which is most readily compared with the second and third @xmath121 intervals in figures [ fig : sigma single](b ) and [ fig : sigma single ratio](b ) . however , even for @xmath235 , a direct comparison between h1 and zeus data is not possible , since the zeus analysis covers a wider @xmath80 range and cuts on the second jet at an @xmath159 value of @xmath236 , larger than the value used here . an indirect comparison can be made on the basis of ratios of the data to nlo theoretical calculations using the h1 fit b dpdfs . zeus obtains a result of around @xmath237 for this ratio , which is compatible with the result for @xmath235 obtained here , within the large combined uncertainties . range are well within the experimental uncertainties alone . ] as discussed in detail in @xcite , the error bands on the dpdfs extracted from inclusive diffraction alone do not include uncertainties due to parton parameterisation choices and thus do not reflect the full uncertainties , particularly in the large @xmath171 region . to give a complementary indication of the possible range of variation , comparisons between the ratios obtained with the h1 2006 fit b dpdfs , the h1 2007 fit jets dpdfs and zeus dpdf sj fit are shown for a subset of variables ( @xmath207 , @xmath121 and @xmath171 ) in figure [ fig : ratiodpdfs ] . the zeus dpdfs lead to ratios which are uniformly @xmath238 larger than those obtained with h1 2006 fit b , with no strong dependence on any of the kinematic variables . the deviation of the h1 2007 fit jets result from the h1 2006 fit b result extends beyond the dpdf error band for @xmath239 , which is correlated with a somewhat stronger dependence of the ratio of data to theory on @xmath121 and a slightly different shape at low @xmath207 . according to the rapgap model , approximately half of the cross section in the kinematic range studied arises from each of the direct and resolved photon - induced contributions . the decomposition of photoproduction processes into direct and resolved interactions is not uniquely defined beyond lo . when modelling rapidity gap survival probabilities in the following , the resolved photon contribution is defined to correspond exactly to that which is calculated using the photon structure function . following the calculation using an absorptive model of a gap survival probability of @xmath69 for the hadron - like component of resolved photoproduction @xcite , previous h1 data @xcite were compared in @xcite with predictions in which the full resolved photon contribution was suppressed by this factor , the direct photon contribution being left unsuppressed . in a later analysis @xcite , this procedure was extended to nlo . the conclusions of these previous studies are confirmed in figures [ fig : sigma single res supp ] and [ fig : sigma single ratio res supp ] through a similar comparison of the current data with nlo calculations in which the resolved photon contribution is globally suppressed by a factor of 0.34 . the overall normalisation of this calculation is in good agreement with the data . however , the shapes of some of the differential distributions are not well reproduced . in particular , there is a variation by more than a factor of two in the ratio of data to theory as a function of @xmath207 ( figure [ fig : sigma single ratio res supp]a ) . the distinction between point - like and hadron - like resolved photon interactions recently developed in @xcite leads to a significantly weaker predicted suppression in the kinematic range of the current analysis . the data are compared with this refined ` kkmr ' model under the approximation of completely neglecting hadron - like resolved photon contributions , which , according to the authors , become dominant only for @xmath240 @xcite , beyond the range of the current analysis . the rapidity gap survival probabilities obtained in @xcite for point - like photon interactions using the grv ho photon pdfs are applied to all resolved photon interactions . interactions involving quarks and gluons from the photon are thus suppressed by factors of @xmath241 and @xmath242 , respectively . the quark - initiated contribution is dominant throughout the measured range , such that the rapidity gap survival probability in the model is approximately @xmath243 for resolved photon interactions and @xmath244 for direct photon interactions . figure [ fig : sigma single res supp ] shows the comparison between the measured single differential cross sections and the nlo qcd predictions , with the resolved photon contribution scaled according to the kkmr model . the corresponding ratios of data to theoretical predictions are shown in figure [ fig : sigma single ratio res supp ] . the overall normalisation of the kkmr - based calculation is larger than that of the data , but is compatible within the large uncertainties . many of the distributions studied are well described in shape ( @xmath121 , @xmath171 , @xmath232 , @xmath83 and @xmath95 ) . the data thus agree with the prediction @xcite that the @xmath121 dependence of the data / theory ratio flattens if the resolved photon contribution alone is suppressed . however , there remains a variation in the ratio of data to the kkmr model with @xmath207 and to a lesser extent with @xmath231 , @xmath91 and @xmath233 . a comparison of figures [ fig : sigma single ratio ] and [ fig : sigma single ratio res supp ] shows that the shapes of the differential cross sections are generally better described with a global suppression factor than with a survival probability applied to resolved photon interactions only . to study further the dynamics of rapidity gap suppression and their dependence on the nature of the photon interaction , cross sections are measured double differentially in two regions of @xmath207 , which are enriched with either resolved ( @xmath245 ) or direct ( @xmath246 ) photon processes . using the rapgap mc model with the grv - g lo photon pdfs , the @xmath245 region is estimated to contain @xmath247 resolved photon interactions integrated over the measurement region ( table [ tab : kinrange ] ) , with a @xmath248 direct photon contribution for @xmath246 . in figures [ fig : sigma et xg 1](a)-(c ) , measurements are presented of the double - differential dijet cross section @xmath249 for three @xmath121 ranges in the resolved and direct photon - enriched @xmath207 intervals . the data are compared with the fr fit b calculations and with the rapgap mc predictions . due to kinematic constraints , the resolved - enriched cross section at low @xmath207 falls most rapidly as @xmath121 increases . there is a suggestion that this @xmath121 dependence in the resolved - enriched region is stronger for the nlo qcd theory than for the data . in the direct - enriched high @xmath207 region , the cross section falls more slowly with @xmath121 and the dependence in the data is similar to that predicted by the nlo calculation . these features are illustrated further in figures [ fig : sigma et xg 1 ] ( d)-(e ) , where the ratios of the data to the nlo theory from figures [ fig : sigma et xg 1 ] ( a)-(c ) are presented as a function of @xmath121 in the resolved and direct photon - enriched @xmath207 regions , respectively . the significance of the @xmath121 dependence in the resolved - enriched region ( figure [ fig : sigma et xg 1 ] ( d ) ) is evaluated through a @xmath250 test . all uncertainties are taken into account in this procedure , though the main contribution comes from the statistical and uncorrelated systematic uncertainties on the data , the remaining uncertainties changing only the normalisation of the ratio to first approximation . a test of the hypothesis that there is no @xmath121 dependence yields a @xmath250 value of 1.36 , with two degrees of freedom , corresponding to an @xmath121 variation at the 73% confidence level . the suppression of the data relative to the nlo prediction in the direct - enriched large @xmath207 region is , within errors , independent of @xmath121 ( figure [ fig : sigma et xg 1 ] ( e ) ) . figures [ fig : sigma et xg 1 ] ( d)-(e ) thus indicate that any @xmath121 dependence of the data - to - theory ratio in figure [ fig : sigma single ratio](b ) is driven primarily by resolved photon interactions . an @xmath121 dependence of the gap survival probability is predicted in the kkmr model , due to variations in the size of the @xmath251 dipole produced by the point - like photon splitting , and hence in the absorptive correction . however the predicted effect is small ( @xmath252 as @xmath121 changes from @xmath253 to @xmath254 ) . figures [ fig : sigma single ratio](d)-(e ) also indicate that when @xmath121 becomes large , the suppression in the direct region may be stronger than that in the resolved region , which is not expected in any model . the large uncertainties permit statistical fluctuations in the data or small inadequacies in the theory as possible explanations . in figure [ fig : sigma zp xg 1 ] , the cross section is shown double differentially in @xmath171 and @xmath207 . the measured cross section is compared with the nlo theory as a function of @xmath171 in two bins of @xmath207 in figures [ fig : sigma zp xg 1 ] ( a)-(b ) and the ratios of data to theoretical predictions are shown in figures [ fig : sigma zp xg 1 ] ( c)-(d ) . the nlo calculations describe the measured shapes rather well , with no evidence for any variation of the suppression factor between any of the measurement ranges . the gap survival probability in a region where there are small or no remnants of either the photon or the diffractive exchange ( highest @xmath171 bin in figure [ fig : sigma zp xg 1]d ) is thus similar to that where both remnants are significant ( lowest @xmath171 bin in figure [ fig : sigma zp xg 1]c ) . this remains the case when the h1 2007 fit jets dpdfs are used in place of h1 2006 fit b. in both figures [ fig : sigma et xg 1 ] and [ fig : sigma zp xg 1 ] , the rapgap mc prediction gives a satisfactory description of the shapes of the double differential cross sections , the normalisation being slightly lower than that of the data . measurements of ratios of diffractive to inclusive dijet photoproduction cross sections have been proposed @xcite as a further test of gap survival issues . their potential advantages over straight - forward diffractive measurements lie in the partial cancellations of some experimental systematics and of theoretical uncertainties due to the photon structure and factorisation and renormalisation scale choices . the sensitivity to absorptive effects of diffractive - to - inclusive ratios is thus potentially superior to that of pure diffractive cross sections . for the ratio extraction presented here , inclusive dijet cross sections are measured using data collected in the same period as the diffractive sample . the experimental method and systematic error treatment for the inclusive case is described in section [ sec : procedure ] . it is identical to the diffractive measurement method , with the exception of the large rapidity gap requirements . at the relatively low transverse energies studied in the present analysis , underlying event effects have a large influence on jet cross sections in inclusive photoproduction @xcite . here , the pythia and herwig mc models are used to correct the inclusive data for detector effects , with mi included as described in section [ sec : mchad ] . the two models agree rather well on the corrections to be applied to the data . the average of the results with the two models is therefore used to calculate the corrections and the uncertainty . the latter is taken from the difference between the results with the two models and is relatively small ( @xmath221 when integrated over the full measured range ) . the ratios of diffractive to inclusive single - differential dijet cross sections are given numerically in table [ tabledifftoincl ] and are shown in figure [ fig : r ] as a function of @xmath207 , @xmath121 , @xmath231 , @xmath232 , @xmath95 and @xmath83 . due to the partial or complete cancellations of some error sources when forming the ratio , the correlated uncertainties are reduced compared with those for the diffractive distributions . since they give adequate descriptions of the diffractive and inclusive data , respectively , the rapgap and pythia mc models are used to assess the relative sensitivity of the diffractive - to - inclusive ratio to the gap survival and mi effects . with no mi effects included in the pythia model , the description of the inclusive data is poor and the ratio of rapgap to pythia exceeds the data by a factor of around 1.5 . as expected , this factor becomes smaller as @xmath121 increases . however , the shape of the prediction also differs from that of the ratio data for most of the other variables studied , in particular @xmath207 . the inclusion of the pythia mi model changes the predicted inclusive cross sections , and hence the ratios , substantially . the ratio of rapgap to pythia then gives an improved description of the shapes of the distributions . the mi effects alter the predicted ratio by a factor of 0.5 at low @xmath207 , where the resolved photon remnant is most important . as expected , there is little effect in the direct photon - dominated large @xmath207 region . the normalisation of the ratio of the models when mi are included is smaller than that of the data . this partially reflects the rapgap description of the diffractive data ( figure [ fig : sigma single ] ) and is partially due to an overshoot in the pythia description of the inclusive data . the fractional reduction in the predicted inclusive cross section when mi are introduced in the pythia model is comparable to the magnitude of the gap survival suppression factor in the diffractive data ( section [ sec : difdata ] ) . the uncertainties in modelling the mi are large and difficult to quantify . the precision with which gap survival issues can be unfolded from mi complications in the ratio of diffractive to inclusive data is correspondingly poor . therefore no strong conclusions can be drawn with our current understanding of mi , despite the relatively good precision of the data .

aaron , c. alexa , v. andreev , s. backovic , a. baghdasaryan , e. barrelet , w. bartel , k. begzsuren , a. belousov , j.c . fischer , m. fleischer , a. fomenko , e. gabathuler , j. gayler , s. ghazaryan , a. glazov , l. goerlich , n. gogitidze , m. gouzevitch , c. grab , a. grebenyuk , t. greenshaw , b.r . grell , g. grindhammer , s. habib , d. haidt , c. helebrant , r.c.w . maxfield , a. mehta , a.b . naumann , p.r . newman , c. niebuhr , a. nikiforov , d. nikitin , g. nowak , k. nowak , j.e . papadopoulou , c. pascaud , g.d . ruiz tabasco , s. rusakov , d. lek , d.p.c . sankey , m. sauter , e. sauvan , s. schmitt , l. schoeffel , a. schning , h .- c . schultz - coulon , f. sefkow , r.n . shtarkov , s. shushkevich , t. sloan , i. smiljanic , y. soloviev , p. sopicki , d. south , v. spaskov , a. specka , z. staykova , m. steder , b. stella , g. stoicea , u. straumann , d. sunar , t. sykora , g. thompson , p.d . tran , d. traynor , p. trul , i. tsakov , b. tseepeldorj , j. turnau , k. urban , a. valkrov , c. valle , p. van mechelen , a. vargas trevino , y. vazdik , m. von den driesch , d. wegener , e. wnsch , j. ek , j. zlek , z. zhang , a. zhokin , h. zohrabyan , and f. zomer

measurements are presented of single and double - differential dijet cross sections in diffractive photoproduction based on a data sample with an integrated luminosity of . the events are of the type , where the hadronic system contains at least two jets and is separated by a large rapidity gap from the system , which consists of a leading proton or low - mass proton excitation . the dijet cross sections are compared with qcd calculations at next - to - leading order and with a monte carlo model based on leading order matrix elements with parton showers . the measured cross sections are smaller than those obtained from the next - to - leading order calculations by a factor of about . this suppression factor has no significant dependence on the fraction of the photon four - momentum entering the hard subprocess . ratios of the diffractive to the inclusive dijet cross sections are measured for the first time and are compared with monte carlo models . # 1#2#3#4#1 * # 2 * ( # 3 ) # 4 desy 10 - 043issn 0418 - 9833 + march 2010 * diffractive dijet photoproduction + in collisions at hera * h1 collaboration submitted to f.d . aaron , c. alexa , v. andreev , s. backovic , a. baghdasaryan , e. barrelet , w. bartel , k. begzsuren , a. belousov , j.c . bizot , v. boudry , i. bozovic - jelisavcic , j. bracinik , g. brandt , m. brinkmann , v. brisson , d. bruncko , a. bunyatyan , g. buschhorn , l. bystritskaya , a.j . campbell , k.b . cantun avila , k. cerny , v. cerny , v. chekelian , a. cholewa , j.g . contreras , j.a . coughlan , j. cvach , j.b . dainton , k. daum , m. dek , b. delcourt , j. delvax , e.a . de wolf , c. diaconu , m. dobre , v. dodonov , a. dossanov , a. dubak , g. eckerlin , v. efremenko , s. egli , a. eliseev , e. elsen , a. falkiewicz , l. favart , a. fedotov , r. felst , j. feltesse , j. ferencei , d .- j . fischer , m. fleischer , a. fomenko , e. gabathuler , j. gayler , s. ghazaryan , a. glazov , l. goerlich , n. gogitidze , m. gouzevitch , c. grab , a. grebenyuk , t. greenshaw , b.r . grell , g. grindhammer , s. habib , d. haidt , c. helebrant , r.c.w . henderson , e. hennekemper , h. henschel , m. herbst , g. herrera , m. hildebrandt , k.h . hiller , d. hoffmann , r. horisberger , t. hreus , f. huber , m. jacquet , x. janssen , l. jnsson , a.w . jung , h. jung , m. kapichine , j. katzy , i.r . kenyon , c. kiesling , m. klein , c. kleinwort , t. kluge , a. knutsson , r. kogler , p. kostka , m. kraemer , j. kretzschmar , a. kropivnitskaya , k. krger , k. kutak , m.p.j . landon , w. lange , g. latovika - medin , p. laycock , a. lebedev , v. lendermann , s. levonian , k. lipka , b. list , j. list , n. loktionova , r. lopez - fernandez , v. lubimov , a. makankine , e. malinovski , p. marage , ll . marti , h .- u . martyn , s.j . maxfield , a. mehta , a.b . meyer , h. meyer , j. meyer , s. mikocki , i. milcewicz - mika , f. moreau , a. morozov , j.v . morris , m.u . mozer , m. mudrinic , k. mller , th . naumann , p.r . newman , c. niebuhr , a. nikiforov , d. nikitin , g. nowak , k. nowak , j.e . olsson , s. osman , d. ozerov , p. pahl , v. palichik , i. panagoulias , m. pandurovic , th . papadopoulou , c. pascaud , g.d . patel , e. perez , a. petrukhin , i. picuric , s. piec , h. pirumov , d. pitzl , r. plaakyt , b. pokorny , r. polifka , b. povh , v. radescu , n. raicevic , a. raspiareza , t. ravdandorj , p. reimer , e. rizvi , p. robmann , r. roosen , a. rostovtsev , m. rotaru , j.e . ruiz tabasco , s. rusakov , d. lek , d.p.c . sankey , m. sauter , e. sauvan , s. schmitt , l. schoeffel , a. schning , h .- c . schultz - coulon , f. sefkow , r.n . shaw - west , l.n . shtarkov , s. shushkevich , t. sloan , i. smiljanic , y. soloviev , p. sopicki , d. south , v. spaskov , a. specka , z. staykova , m. steder , b. stella , g. stoicea , u. straumann , d. sunar , t. sykora , g. thompson , p.d . thompson , t. toll , t.h . tran , d. traynor , p. trul , i. tsakov , b. tseepeldorj , j. turnau , k. urban , a. valkrov , c. valle , p. van mechelen , a. vargas trevino , y. vazdik , m. von den driesch , d. wegener , e. wnsch , j. ek , j. zlek , z. zhang , a. zhokin , h. zohrabyan , and f. zomer

1205.6323

i

gravitational microlensing is an important method to detect extrasolar planets @xcite . the method is sensitive to planets not easily accessible to other methods , in particular cool and small planets at or beyond the snow line @xcite , and free - floating planets @xcite . the snow line represents the location in the protoplanetary disk beyond which ices can exist @xcite and thus the surface density of solids is highest @xcite . according to the core accretion theory of planet formation @xcite , the snow line plays a crucial role because giant planets are thought to form in the region immediately beyond the snow line . therefore , microlensing planets can provide important constraints on planet formation theories , in particular by measuring the mass function beyond the snow line @xcite . a major component of current planetary microlensing experiments is being carried out in survey and follow - up mode , where survey experiments are conducted in order to maximize the event rate by monitoring a large area of the sky one or several times per night , while follow - up experiments are focused on events alerted by survey observations to densely cover planet - induced perturbations . in this mode , high - magnification events are important targets for follow - up observations . this is because the source trajectories of these events always pass close to the central perturbation region and thus the sensitivity to planets is extremely high @xcite . in addition , the time of the perturbation can be predicted in advance so that intensive follow - up observation can be prepared . this leads to an observational strategy of monitoring high - magnification events as intensively as possible , regardless of whether or not they show evidence of planets . as a result , the strategy allows one to construct an unbiased sample to derive the frequency of planets beyond snow line @xcite . for the alternative low - magnification channel of detection , see for instance @xcite . in this paper , we report the discovery of a giant planet detected from the analysis of the light curve of a high - magnification microlensing event moa 2010-blg-477 . due to the high magnification of the event , the perturbation was very densely covered , enabling us to place constraints on the physical parameters from the higher - order effects in the lensing light curve induced by finite source effects as well as the orbital motion of both the lens and the earth . we provide the most probable physical parameters of the planetary system , corresponding to a jupiter - mass planet orbiting a k dwarf at about 2 au , the system lying at about 2 kpc from earth .

microlensing detections of cool planets are important for the construction of an unbiased sample to estimate the frequency of planets beyond the snow line , which is where giant planets are thought to form according to the core accretion theory of planet formation . in this paper , we report the discovery of a giant planet detected from the analysis of the light curve of a high - magnification microlensing event moa 2010-blg-477 .

microlensing detections of cool planets are important for the construction of an unbiased sample to estimate the frequency of planets beyond the snow line , which is where giant planets are thought to form according to the core accretion theory of planet formation . in this paper , we report the discovery of a giant planet detected from the analysis of the light curve of a high - magnification microlensing event moa 2010-blg-477 . the measured planet - star mass ratio is and the projected separation is in units of the einstein radius . the angular einstein radius is unusually large mas . combining this measurement with constraints on the `` microlens parallax '' and the lens flux , we can only limit the host mass to the range . in this particular case , the strong degeneracy between microlensing parallax and planet orbital motion prevents us from measuring more accurate host and planet masses . however , we find that adding bayesian priors from two effects ( galactic model and keplerian orbit ) each independently favors the upper end of this mass range , yielding star and planet masses of and at a distance of kpc , and with a semi - major axis of au . finally , we show that the lens mass can be determined from future high - resolution near - ir adaptive optics observations independently from two effects , photometric and astrometric .

1205.6323

r

in order to select the best among the three competing models ( standard , parallax only , orbital motion and parallax ) , a simple comparison of @xmath61 values is not enough , because more refined models use more parameters . taking into account these additional parameters by normalizing the @xmath61 estimate by the number of degree of freedom is not the proper way to select the best model . a vast literature exists about model selection , and an application of different criteria to astrophysics is described in @xcite . a simple way to take care of the larger number of parameters is to use the akaike information criterion ( aic ) @xcite , which introduces a penalty to the @xmath61 by adding twice the number of additional parameters . different criteria , such as the bayesian information criterion ( bic ) @xcite or the deviation information criterion , introduced by @xcite , can also be used . formulas are given below , where @xmath169 is the number of additional parameters , @xmath170 the number of data points , and @xmath171 the @xmath61 of the average parameter set @xmath172 . @xmath173 as there are 7 parameters in the standard model , 9 in the parallax only model and 11 in the orbital motion and parallax model , we see that our observed difference in @xmath61 of 8.6 in the parallax only model is only marginally significant according to the aic ( the expected difference is 4 ) , while the observed difference of 54.4 ( for the @xmath174 ) or 50.7 ( for the @xmath137 ) is clearly an improvement of the orbital motion and parallax model over the standard one ( the expected difference is 8) . similar results are obtained using dic , while the difference between models returned by bic is less significant . however , it is important to note that @xmath169 strictly corresponds to the number of additional parameters only in the case of a linear regression problem . here , we clearly have non - linear fits , so we should compute an `` effective '' number of parameters , which is difficult to estimate . the above conclusion should therefore not be taken as a quantitative one . a confirmation of the detection of second - order effects comes from the fact that in the orbital motion and parallax model , the degeneracy between @xmath137 and @xmath174 models is clearly broken . it is instructive to plot both second - order effects vs each other , separately for the @xmath137 and @xmath174 solutions . this is done in figure [ fig : secorder ] , where the orbital motion @xmath175 is plotted vs. the parallax effect @xmath38 . if we remember that the ratio of the projected kinetic to potential energy must be smaller than 1 to get bound orbits , and that this ratio is proportional to @xmath176 , where the proportionality constant depends on @xmath38 as given by equation [ eqn : gamma0 ] , it is easy to interpret these diagrams . when @xmath38 increases , the proportionality factor decreases , so that if @xmath175 remains small enough , the bound orbit condition is respected . in the @xmath174 diagram ( left side ) , small values of @xmath38 ( say below 0.17 ) have a proportionality factor larger than 1 . as @xmath175 is nearly 1 for these solutions , they are ruled out , and the light curve confirms it , as few small @xmath61 solutions ( black and red points ) lie there . larger @xmath38 are also excluded , as they correspond to large @xmath175 values , although the light curve would favor such solutions . the only surviving region in this diagram is around @xmath177 , corresponding to lens masses of half a solar mass , where the proportionality factor is about 0.4 and @xmath175 slightly exceeds 1 . in the @xmath137 diagram ( right side ) , although this is slightly disfavored by the light curve ( @xmath178 for the best chain without the circular orbit constraint ) , there is a region where @xmath175 is about constant at 0.7 for @xmath38 varying from 0 to 0.4 . the bound solutions correspond to the larger values of the @xmath38 domain ( smaller proportionality factor ) , and they therefore agree with the range found in the @xmath174 diagram . we therefore conclude that both solutions agree , and give bound orbits when the lens mass is about half solar , corresponding to a lens distance of about 1.6 kpc . if we now move to the post - bayesian analysis , we see that this solution favored by the light curve has some tension with the galactic model constraint , because nearby lenses are rarer than more distant ones . but if we move to more distant lenses , we get many chains with unbound orbits or high eccentricities . by the way , the @xmath174 solution , which has a lower @xmath61 than the alternate @xmath137 solution , is also the one where more chains correspond to unbound orbits . there is therefore a tension between galactic and keplerian priors , and the issue will only be solved photometrically , by measuring the light coming from the source and the lens . this will be the subject of a forthcoming article about this event . we conclude by giving the 1-d distributions of lens mass , lens distance , and planet orbit semi - major axis . the mass function for the lenses involved in these plots include main - sequence stars , brown dwarfs , but also white dwarfs , neutron stars and black holes , which may have large masses without violating the lens flux limit constraint . for the ms and bd stars , we adopt the following slopes of the present - day mass function @xmath179 : @xmath180 between 0.03 and 0.7 @xmath28 , @xmath181 between 0.7 and 1.0 @xmath28 , and @xmath182 above . for the remnants , we adopt gaussian distributions , whose mean value , standard deviation , and fraction of total mass with respect to ms and bd stars below @xmath183 are given in table [ tab : massdist ] . lllr wd & 0.6 & 0.07 & 22/69 + ns & 1.35 & 0.04 & 6/69 + bh & 5.0 & 1.0 & 3/69 for details about the choice of these numbers , please refer to @xcite . in each diagram ( see figure [ fig:1dplots1 ] for mass and flux , and figure [ fig:1dplots2 ] for distance and semi - major axis ) , the black curves show the full mass function , while the red curves show the mass function truncated at @xmath184 . for ms stars , this limit is imposed by the lens flux constraint , and will be refined once we obtain the adaptive optics photometry of the individual stars in the field . wd at this mass are extremely rare ; jovian planets around pulsars ( ns ) have not been found , despite very extensive searches ; and super - jupiter planets orbiting bh are a priori unlikely . let us first consider the lens mass distribution : the no - flux - limit ( black ) curve shows a huge spike at expected ns position and a smaller bump corresponding to bh . note that for these bumps , the @xmath137 solution dominates , despite its @xmath61 handicap . this is because the galactic model very strongly favors distant lenses , primarily because of the volume factor , and this overwhelms the modest preference of the light curve for nearby lenses . because @xmath23 is roughly fixed , these distant lenses are massive . this preference is much stronger in the @xmath137 solution , which can be seen in its rapid rise beginning at @xmath185 . note that the wd peak ( at @xmath186 ) is clearly visible , especially in the @xmath174 solution . the lens distance distribution basically looks at this same situation from the standpoint of distance . the new notable feature is that both ms and ns peaks are in the disk , while the bh bump is in the bulge . and the semi - major axis distribution peaks at about 2 - 3 au . > from these diagrams , we can estimate a most probable value of lens mass , distance and semi - major axis , and an asymmetric standard deviation read at 50% of the distribution corresponding to the red bold curves . we get a star and planet mass of @xmath4 and @xmath5 , respectively , at a distance of @xmath6 kpc , and with a semi - major axis of @xmath7 au . as a final note , it could be said that more complex models are worth exploring : the geometry of the caustic crossing , where the source passes close to the three - cusps tail of the caustic , is extremely sensitive to a third body ( second planet or binary companion to the lens star ) . a similar geometry where two planets were detected is described in @xcite . these models could be investigated in a forthcoming paper , once we get the lens flux measurement from adaptive optics . we acknowledge the following sources of support : creative research initiative program ( 2009 - 0081561 ) of national research foundation of korea ( ch ) ; grants jsps20340052 and jsps22403003 for moa ; czech science foundation grant gacr p209/10/1318 ; the french polar institute ( ipev ) and the italian antarctic programme ( pnra ) for the logistics and data transmission at concordia . ogle project has received funding from the european research council under the european community s seventh framework programme ( fp7/2007 - 2013 ) / erc grant agreement no . part of the computer work was performed using hpc resources from calmip ( grant 2011-p1131 ) . e. bachelet gratefully acknowledges the chungbuk national university for a one - month stay where most of this work was prepared . gaudi and a. gould acknowledge support from nsf ast-1103471 . gaudi , a. gould , and r.w . pogge acknowledge support from nasa grant nng04gl51 g . work by j.c . yee is supported by the national science foundation graduate research fellowship under grant no . 2009068160 . t.c . hinse acknowledges support from the krcf young scientist research fellowship program in south korea . the planet collaboration acknowledges the financial support of anr holmes and pnps grants . astep was financed through the help of anr , ipev , cnrs , observatoire de la cte dazur .

however , we find that adding bayesian priors from two effects ( galactic model and keplerian orbit ) each independently favors the upper end of this mass range , yielding star and planet masses of and at a distance of kpc , and with a semi - major axis of au .

microlensing detections of cool planets are important for the construction of an unbiased sample to estimate the frequency of planets beyond the snow line , which is where giant planets are thought to form according to the core accretion theory of planet formation . in this paper , we report the discovery of a giant planet detected from the analysis of the light curve of a high - magnification microlensing event moa 2010-blg-477 . the measured planet - star mass ratio is and the projected separation is in units of the einstein radius . the angular einstein radius is unusually large mas . combining this measurement with constraints on the `` microlens parallax '' and the lens flux , we can only limit the host mass to the range . in this particular case , the strong degeneracy between microlensing parallax and planet orbital motion prevents us from measuring more accurate host and planet masses . however , we find that adding bayesian priors from two effects ( galactic model and keplerian orbit ) each independently favors the upper end of this mass range , yielding star and planet masses of and at a distance of kpc , and with a semi - major axis of au . finally , we show that the lens mass can be determined from future high - resolution near - ir adaptive optics observations independently from two effects , photometric and astrometric .

astro-ph0405329

i

the formation of stars and planetary systems is one of the fundamental problems in astrophysics . much of the work over the past decades has examined the formation of low mass stellar systems because these objects sometimes form in isolation and are therefore easier to study individually . one of the earliest stages of stellar birth that has been the focus of numerous investigations is the creation of a centrally concentrated molecular core from a portion of a low - density parent giant molecular cloud ( gmc ) . theoretical models account for the condensation as occurring possibly via the slow diffusion of magnetic flux occurring over long timescales ( @xmath4 myr ) ( mouschovias 1999 ; lizano & shu 1989 ) or the dissipation of turbulence on shorter timescales ( stone , ostriker , & gammie 1998 ; mac low et al . 1998 ; myers & lazarian 1998 ; nakano 1998 ) . observations of isolated pre - stellar molecular cores , such as l1544 in the taurus molecular cloud , have provided fertile ground for comparison to these theories ( caselli et al 2002 ; ciolek & basu 2000 ; williams et al 1999 ; tafalla et al 1998 ) . however , it is now recognized that most stars form in groups from small aggregates to large clusters and it is not clear that all theories developed for isolated star formation are easily applicable to the larger scales and simultaneity required for star cluster formation ( ballesteros - paredes , hartmann , & vzquez - semadeni 1999 ) . expanding to larger scales opens the question as to whether the formation of stars , both isolated and clustered , might perhaps be intimately related to the formation of the gmc itself . in a previous paper ( hartmann , ballesteros - paredes , & bergin 2001 @xmath5 hbb01 ) we pointed out that the great majority of molecular cloud complexes in the solar neighborhood appear to be forming young stars , and that the ages of the stellar populations in these clouds are typically @xmath6 2 myr ; stellar associations of ages @xmath7 10 myr are devoid of molecular gas . the calculations indicating that mhd turbulence damps rapidly ( stone et al . 1998 , mac low et al . 1998 , padoan & nordlund 1999 ) also favor the suggestion of short cloud lifetimes , since there is no need for a continuous regeneration of mhd turbulence ; additional support is found through arguments related to to cloud crossing times ( elmegreen 2000 ; hbb01 ) . these results place significant empirical constraints on the mechanism(s ) of nearby molecular cloud formation ( see * ? ? ? * ; * ? ? ? * ; * ? ? ? * for a review ) . for example , it is difficult to reconcile for the transient nature of local clouds with models in which complexes are built up by the coalescence of smaller molecular clouds , or theories in which molecular gas is mostly moved around from one place to another ( see elmegreen 1993 and references therein ) , because these processes are likely to take much more than @xmath8 myr to occur . moreover , short cloud lifetimes also place severe constraints on the processes leading toward fragmentation of the gmc and the condensation towards star formation . in hbb01 we suggested that chemical transformations of local gas are essential to understanding the observational constraints . we suggested that most clouds are formed by large scale flows in the diffuse atomic medium , and that they appear as molecular clouds only when the column density becomes high enough to shield the molecular gas from the dissociating effects of the interstellar radiation field ( isrf ) . we further noted that self - gravity is likely to become important for column densities comparable to that needed for shielding ( also franco & cox 1986 ) , which would explain the rapid onset of star formation after molecular cloud formation . finally , we suggested that dispersal of star - forming gas is accompanied by a reduction in shielding , so that the gas may revert to an atomic state some time before it is completely physically removed from the neighborhood . these suggested chemical transformations lessen , but do not necessarily eliminate , the difficulty of making clouds `` fast enough '' . the formation of h@xmath0 from atomic gas , which generally must precede the formation of co , is not instantaneous . in addition , while h@xmath0 formation places important constraints on the problem , it must also be examined in the context of a model that incorporates the effects of h@xmath0 ( and co ) self - shielding from the ultraviolet ( uv ) radiation field with extinction by dust grains , each with its own associated timescale . following the molecular evolution is critical to an understanding of cloud formation because of observational bias ; molecular clouds are essentially defined not by h@xmath0 emission but through co emission . in addition , possible atomic precursors are difficult to identify against the galactic h i background ( ballesteros - paredes et al . 1999 ) . in this paper we explore the formation of molecular gas in plane - parallel shocks , starting with atomic ( neutral ) ( warm ) gas . our model incorporates all the relevant heating and cooling mechanisms appropriate for the interstellar medium ( ism ) , including chemical processes relevant to the transformation of atomic gas to molecular form . koyama & inutsuka ( 2000 ) considered a similar problem , and did include the important effects of shielding from uv radiation , but only examined maximum column densities of standard h i clouds ( @xmath9 ) . thus , they only achieve molecular hydrogen fractions of a few percent , insufficient to follow cloud formation . we explore a range of shock velocities comparable to the flows expected in the diffuse atomic medium ( ballesteros - paredes , hartmann , & vzquez - semadeni 1999 , and references therein ) ; the parameter ranges are also appropriate for models in which gmc formation is induced by galactic spiral density waves or energetic supernova . moreover , by following the primary cooling lines in the atomic shock and the post - shock evolution we can predict fluxes for key transitions of c ii , c i , o i , co and other species spanning a range of initial ram pressures . this should aid in the search for the progenitors of molecular clouds . finally as a by - product of this work we provide an analytic solution for the formation of molecules behind a shock and discuss the relation between gas temperature and extinction , both of which may be of some use to mhd modeling . in 2 we describe the model with results and a parameter study provided in 3 . in 4 we outline the observational possibilities for the detection of forming molecular clouds via both emission and absorption lines . section 5 summarizes the implications of these results for star formation in the local neighborhood and in 6 we present our conclusions .

motivated by our previous paper , in which we argued for the formation of molecular clouds from large - scale flows in the diffuse galactic interstellar medium , we examine the formation of molecular gas behind shocks in atomic gas using a one - dimensional chemical / dynamical model . in our analysis however , our predictions suggest that the detection of the pre - co stages will be challenging . finally , we provide an analytic solution for time - dependent formation which may be of use in numerical hydrodynamic calculations .

motivated by our previous paper , in which we argued for the formation of molecular clouds from large - scale flows in the diffuse galactic interstellar medium , we examine the formation of molecular gas behind shocks in atomic gas using a one - dimensional chemical / dynamical model . in our analysis we place particular emphasis on constraints placed on the dynamical evolution by the chemistry . the most important result of this study is to stress the importance of shielding the molecular gas from the destructive effects of uv radiation . for shock ram pressures comparable to or exceeding typical local interstellar medium pressures , self - shielding controls the formation time of molecular hydrogen but co formation requires shielding of the interstellar radiation field by dust grains . we find that for typical parameters the molecular hydrogen fractional abundance can become significant well before co forms . the timescale for ( co ) molecular cloud formation is not set by the h formation rate on grains , but rather by the timescale for accumulating a sufficient column density or extinction , . the local ratio of atomic to molecular gas ( 4:1 ) , coupled with short estimates for the lifetimes of molecular clouds ( 3 - 5 myr ) , suggests that the timescales for accumulating molecular clouds from atomic material typically must be no longer than about 12 - 20 myr . based on the shielding requirement , this implies that the typical product of pre - shock density and velocity must be . in turn , depending upon the shock velocity , this implies shock ram pressures which are a few times the typical estimated local turbulent gas pressure , and comparable to the total pressures ( gas plus magnetic plus cosmic rays ) . coupled with the rapid formation of co once shielding is sufficient , flow - driven formation of molecular clouds in the local interstellar medium can occur sufficiently rapidly to account for observations . we also provide detailed predictions of atomic and molecular emission and absorption that track the formation of a molecular cloud from a purely atomic medium , with a view toward helping to verify cloud formation by shock waves . however , our predictions suggest that the detection of the pre - co stages will be challenging . finally , we provide an analytic solution for time - dependent formation which may be of use in numerical hydrodynamic calculations .

astro-ph0405329

r

our standard model assumes an initial shock velocity of @xmath44 km s@xmath40 impacting gas with a temperature of @xmath45 k and a density of @xmath46 ( @xmath47 28000 @xmath14 k ) . this shock speed is consistent with that induced by either spiral density waves @xcite or turbulence driven collisions between flows ( ballesteros - paredes et al 1999 ) . a supernova will likely induce a series of shocks with a range of ram pressures covered by this standard solution and the additional runs that explore parameter space . in all calculations presented below the _ atomic _ model is used to examine the gas evolution while the temperature exceeds 1000 k. when gas temperature in the post - shock evolution decays to below this value the _ molecular _ model is used to examine the subsequent evolution . figure [ basic](a - f ) presents the physical and chemical evolution of the post - shock gas . here panels ( a - c ) show the major heating and cooling terms along with the primary physical parameters ( density , temperature , optical extinction ) as a function of time . panels ( e - f ) show the chemical abundances ( relative to total h ) of important molecular and atomic species as a function of time . in this figure t = 0 represents the onset of the shock . after the shock the gas temperature is quite high ( @xmath48 k ) with the primary cooling is via [ c ii ] , [ o i ] , [ si ii ] , and [ fe ii ] line emission ( hydrogen line emission is also an important coolant in the initial stages ) . at early times ( @xmath49 yr ) the heating processes which dominate the cold neutral medium ( cnm ) , photoelectric and x - ray heating , are unimportant . at t = 10@xmath50 yrs the temperature drops sharply at the cooling timescale set by the overall cooling rate . for this case ( @xmath51 and an initial density of 1 @xmath14 ) the cooling timescale is @xmath52 yr ( spitzer 1978 ) . when the gas cools the density rises due to pressure equilibrium . at this point the solution reaches a stable high density atomic phase where photoelectric heating is balanced by [ c ii ] cooling . this stable `` plateau '' is essentially the wolfire et al . ( 1995 ) cnm solution and is roughly at the temperature of the cnm as measured by @xcite . the plateau exists for @xmath53 yr until the onset of h@xmath0 formation which is followed by the formation of co. at this point co begins to dominate the cooling reducing the temperature to @xmath4 k. photoelectric heating dominates throughout most of the post shock evolution until a@xmath54 mag , whereupon cosmic - ray heating is more important . prior to the formation of co there is a period where the abundance of neutral carbon rises and it contributes to the cooling , but it is not the dominant carbon reservoir . the timescale of this process is controlled by the formation time of h@xmath0 on grains and the slow buildup of shielding due to grains and h@xmath0 molecules downstream in the post - shock gas . at the density of the stable plateau ( n @xmath55 @xmath14 ; t = 23 k ) the h@xmath0 formation time is @xmath56 yr . the onset of molecular formation occurs at much later times and therefore _ the evolution is controlled by the shielding of uv radiation . _ in our standard case the shielding of h@xmath0 is dominated by self - shielding with only small contributions from dust absorption . for co the shielding is dominated through the uv photon absorption by dust grains . in the following we will examine different initial ram pressures through changes in the shock speed given the same initial density and temperature as the standard model . we also examine a few cases with a constant shock velocity but varying initial densities , to verify the similarity of results for similar ram pressures for the molecular gas . in figure [ phys_ev ] we present the results from models with @xmath57 ( @xmath45 k ; @xmath46 ) , which have ram pressures ranging from 1.4 36 @xmath58 @xmath14 k. the h@xmath0 and co abundance as a function of time and visual extinction are given in figure [ abun_ev ] . rather than show the detailed evolution ( as in figure [ basic ] ) we present the temporal evolution of salient parameters : density , temperature , and extinction . we note that the cooling time to t @xmath59 k is extremely short due to collisional excitation of h i. thus the much higher immediate post - shock temperatures in the faster shock models do not appear in figure [ phys_ev ] . in figure [ phys_ev ] the cooling timescale to reach the stable temperature plateau decreases with increasing ram pressure . similarly the timescales of molecular formation decrease ( figure [ abun_ev ] ) . both effects are due to the density of the solution plateau increasing with ram pressure . for h@xmath0 , in each case , except for the 10 @xmath60 shock , self - shielding dominates over dust absorption . the reverse is true for co , dust absorption dominates over self - shielding , except for a 50 @xmath60 shock , where self - shielding becomes more important at a@xmath61 mag . in figures [ phys_ev_ic ] and [ abun_ev_ic ] we examine models with the same shock speed ( @xmath62 ) but different initial densities . we also show the @xmath63 , @xmath46 case as it has similar ram pressure as the @xmath62 , @xmath64 solution . here we see similar effects as illustrated by models with increasing shock velocity , all timescales ( cooling , molecular formation ) shorten with increasing density . models with similar ram pressure ( compare p / k = 32000 @xmath14 k with p / k = 29000 @xmath14 k ) have slightly different evolution in the physical and chemical properties seen in fig . 4 and 5 . indeed the model with the slightly smaller ram pressure has faster physical and chemical evolution . in this model the temperature decay occurs more quickly because of the higher initial density , which increases the cooling through collisionally excited lines . subsequently , h@xmath0 forms earlier , and the higher @xmath65 allows for faster co formation . overall the differences between these two cases are not large and are magnified by the log scale . however , this points out that the effects of mass conservation and pressure conservation are not equivalent . solutions with higher densities will have faster dynamical evolution through increased cooling and quicker _ molecular _ evolution due to the decrease in the time to reach full shielding .

we place particular emphasis on constraints placed on the dynamical evolution by the chemistry . the most important result of this study is to stress the importance of shielding the molecular gas from the destructive effects of uv radiation . for shock ram pressures comparable to or exceeding typical local interstellar medium pressures , self - shielding controls the formation time of molecular hydrogen but co formation requires shielding of the interstellar radiation field by dust grains .

motivated by our previous paper , in which we argued for the formation of molecular clouds from large - scale flows in the diffuse galactic interstellar medium , we examine the formation of molecular gas behind shocks in atomic gas using a one - dimensional chemical / dynamical model . in our analysis we place particular emphasis on constraints placed on the dynamical evolution by the chemistry . the most important result of this study is to stress the importance of shielding the molecular gas from the destructive effects of uv radiation . for shock ram pressures comparable to or exceeding typical local interstellar medium pressures , self - shielding controls the formation time of molecular hydrogen but co formation requires shielding of the interstellar radiation field by dust grains . we find that for typical parameters the molecular hydrogen fractional abundance can become significant well before co forms . the timescale for ( co ) molecular cloud formation is not set by the h formation rate on grains , but rather by the timescale for accumulating a sufficient column density or extinction , . the local ratio of atomic to molecular gas ( 4:1 ) , coupled with short estimates for the lifetimes of molecular clouds ( 3 - 5 myr ) , suggests that the timescales for accumulating molecular clouds from atomic material typically must be no longer than about 12 - 20 myr . based on the shielding requirement , this implies that the typical product of pre - shock density and velocity must be . in turn , depending upon the shock velocity , this implies shock ram pressures which are a few times the typical estimated local turbulent gas pressure , and comparable to the total pressures ( gas plus magnetic plus cosmic rays ) . coupled with the rapid formation of co once shielding is sufficient , flow - driven formation of molecular clouds in the local interstellar medium can occur sufficiently rapidly to account for observations . we also provide detailed predictions of atomic and molecular emission and absorption that track the formation of a molecular cloud from a purely atomic medium , with a view toward helping to verify cloud formation by shock waves . however , our predictions suggest that the detection of the pre - co stages will be challenging . finally , we provide an analytic solution for time - dependent formation which may be of use in numerical hydrodynamic calculations .

astro-ph0405329

c

our model calculations raise an obvious question : when should we consider that a `` molecular cloud '' has been formed ? as shown in figures 1 and 3 , @xmath3 can form at considerably earlier times and lower column densities than co , especially for low ram pressures / shock velocities . thus low - pressure clouds could become molecular , in terms of the dominant constituent , before they become co clouds . however , @xmath3 is difficult to detect , and is not usually a criterion for defining a molecular cloud . we consider the problem of @xmath3 detection in the next subsection ; here we concentrate on the detection of co. it is useful to distinguish between what we call the `` accumulation '' timescale , the length of time it takes to accumulate @xmath86 from the atomic gas and the cloud becomes `` detectable '' in co , from other timescales , for instance the evolution of the co abundance . for purposes of discussion we have made the somewhat arbitrary decision that the cloud appears when co j=10 emission can be detected at a level of 1 k km / s which defines the accumulation timescale . in figure [ co_flux ] we provide the intensity of the j=10 transition of co as a function of time with the dashed line denoting an integrated intensity of 1 k km / s . based on this definition , the cloud accumulation timescale ranges from @xmath87 yr for a 50 @xmath60 shock to @xmath88 yr for a 10 @xmath60 shock ( both with an initial density of 1 ) . as noted before , we find that the column density or extinction is the most important parameter in determining the formation of co. based on the right - hand panel of figure [ co_flux ] , the above definition is roughly equivalent to molecular cloud `` formation '' at @xmath86 mag for ram pressures between @xmath89 @xmath14 k. threshold will increase by an amount of @xmath90 . here we list the uv enhancement factor g@xmath91 as a function of time to denote the fact that effects of radiation will diminish as the shock front moves away from the star . ] the above threshold of @xmath86 mag is based on a 1-dimensional calculation and it is useful to discuss how this might change in a more realistic 3-dimensional geometry . chemically the transition from cii / ci / co will be seen at a level where the co photodissociation becomes ineffective . in 1-d this threshold is found at 0.7 mag with little contribution from self - shielding . in a three - dimensional calculation for a clumpy cloud , there will be two , competing , effects : first , uv radiation will propagate more freely into the cloud ; and second , clumping may increase the local column densities and thus promote the transition to molecular gas . our 1-d requirement of @xmath86 would thus translate into an average over solid angle of the extinction of the diffuse ultraviolet radiation field . the time to reach this threshold is no longer a simple calculation of the @xmath92 product , though we expect that the transition to molecular gas will occur roughly when the average density / column density reaches values comparable to those in our calculations . we also note that in addition to clumping , overall contraction of the cloud in the direction perpendicular to the shock front will promote rapid molecular gas formation by increasing the density and the shielding . recent numerical simulations of finite self - gravitating sheets ( burkert , a. , & hartmann , l. 2004 , in preparation ) suggest that global gravitational collapse of flattened clouds is likely and rapid . in hbb01 we showed that the stellar population ages of nearby molecular clouds imply cloud lifetimes of no more than about 3 - 5 myr . generally , slightly older associations are not immediately next to current sites of star formation , indicating that molecular material is not simply being pushed around but that there is some cycling between the atomic and molecular states . the ratio of atomic to molecular gas in the solar neighborhood is estimated to be @xmath93 @xcite . this implies that the timescale for turning atomic gas into molecular material , on average , is about @xmath94 myr . ( if only a fraction of the atomic gas cycles through molecular stages , the timescale for the conversion of atomic to molecular gas in specified regions must be shorter . ) in our terminology the cloud accumulation timescale must therefore be @xmath95 20 myr ; the cloud lifetime would then be defined by the time from when a cloud is detectable in co emission until the gas is dissipated by star formation . thus , the one - dimensional constraint on the accumulation timescale timescale @xmath96 implied by the ratio of atomic to molecular gas , coupled with the requirement that @xmath86 for co formation , constrains the possible average pre - shock parameters , @xmath97 can this constraint reasonably be satisfied ? the one - dimensional rms turbulent velocity of cold h i in the local interstellar medium is @xmath98 ( boulares & cox 1990 ) ; thus it seems reasonable to take a typical shock velocity @xmath99 . accumulation timescales of 10 - 20 myr then require typical pre - shock densities @xmath100 . this is a few times larger than the average density of the ism , but it would not be particularly surprising if molecular clouds were preferentially formed from initially slightly higher densities . moreover , it has been increasingly accepted that the density fluctuations in the ism ( clouds ) are produced primarily by compressions due to a supersonic turbulent velocity field ( von weiszacker 1951 ; sasao 1973 ; elmegreen 1993 ; padoan 1995 ; ballesteros - paredes , vzquez - semadeni & scalo 1999 ) . in such an environment , the production of the density fluctuations are likely the result of a succession of compression events , so that denser structures are formed by compressions within previously compressed , larger ones , rather than a single , very strong one ( vzquez - semadeni 1994 ) . this scenario naturally explains the density probability density function observed in numerical simulations of isothermal and polytropic flows @xcite , and here it naturally provides the necessary conditions ( 24 @xmath14 ) for the pre - shock gas in our calculations . as already pointed out , ram pressure is the most important parameter for co formation ; the ram pressures for the above parameters are a few times the average turbulent gas pressure ( gas plus magnetic plus cosmic rays ) in the ism ( boulares & cox 1990 ) ; again , this does not seem to be an unreasonable constraint for making the highest density regions in the solar neighborhood . once the shielding column density is achieved , co formation is rapid . additional increases in density due to subsidiary shocks and/or gravitational contraction will yield to even more rapid co formation . at a column density corresponding to @xmath101 , comparable to the shielding length estimated here , the characteristic growth time for gravitational contraction in ( subsonic ) gas at @xmath102 k is of the order of 1 myr . thus , once molecular gas is formed , star formation can ensue rapidly , at least in regions where supersonic turbulence has been dissipated ( hbb01 ) . we conclude that our results are consistent with the picture presented in hbb01 of rapid formation of molecular clouds from atomic material , as long as the starting densities are typically a few times the average interstellar density . here we consider whether the presence of some molecular hydrogen in the pre - shock gas would shorten either the accumulation timescale or the timescale for co abundances to rise . indeed , because h@xmath0 does not emit for typical conditions in the atomic cnm it might be possible to hide a significant molecular component . to examine this question we have examined solutions with a substantial h@xmath0 fraction in the pre - shock gas . this is only performed using the _ molecular _ model because the _ atomic _ model does not include chemical processes linked to h@xmath0 . one limitation is that the cooling via h@xmath0 emission and shock dissociation of h@xmath0 are not included . however , the addition of h@xmath0 cooling will only shorten the cooling time to reach the stable plateau solution . as we will show below , the chemical evolution ( i.e. co formation ) is dominated by the shielding timescale which depends primarily on the initial shock parameters . in figure 10 we present the h@xmath0 and co abundances as a function of time and extinction in our standard model with an initial h@xmath0 fraction of 0.0 , 0.125 , and 0.25 . the sharp rise of the h@xmath0 abundance at t = 10@xmath50 yr is due to the non - inclusion of h@xmath0 in the _ atomic _ model . at this time the abundance of co shows a sharp spike which is due to rapid co formation from the pre - existing h@xmath0 . however , co molecules are quickly dissociated , due to the lack of uv shielding . what is striking in these plots is that even an a priori presence of h@xmath0 molecules a necessary requirement for co formation has little effect on the co chemical evolution . this effect is discussed in 3 , because the co formation requires dust shielding the evolution can not proceed until sufficient dust column exists . this result is robust provided the pre - shock gas is exposed to the isrf with a@xmath103 mag . in figure 11 we present solutions with higher initial extinction and an initial h@xmath0 fraction of @xmath104 . the left - hand panels show the evolution of the co concentration , while the right - hand panels present the co j=10 emission , which we have used to define the cloud accumulation timescale ( @xmath105 = time where emission reaches a level of 1 k @xmath60 ) . with this definition even pre - shock gas with a@xmath106 mag will only accumulate on timescales a factor of two shorter than the model starting with unshielded gas . moreover the differences in the emitted j=10 intensity between these solutions are negligible unless @xmath107 mag . in sum , the presence of h@xmath0 in the pre - shock gas would not hasten the timescales for cloud formation , unless the gas is already significantly shielded from uv radiation and co is already effectively in existence . let us now place these results in an observational context . allen and co - workers have argued for a pervasive molecular component on the basic of the relative placement of dust lanes , radio continuum , and h i emission in external spiral galaxies @xcite . in the standard picture of cloud formation h i emission would appear in front of the spiral shock wave as traced by the radio continuum emission . instead these authors found that the h i emission is observed downstream of the shock , which is interpreted as the result of photodissociation of h@xmath0 by young stars . this led @xcite to theorize that there may be a significant reservoir of molecular gas in the low density cnm which would allow for fast cloud and star formation . they argue that this inter - arm gas could be hidden by having the temperature colder than 10 k , producing only weak co emission . however , these results must be placed in the context of studies of h@xmath0 in absorption and co in emission in our own galaxy . the most extensive initial study of h@xmath0 in absorption was performed by the copernicus satellite and @xcite found that the average h@xmath0 fraction within 500 pc of the sun is @xmath108 . more recent fuse observations have demonstrated that h@xmath0 absorption is pervasive , even in sight lines well out of the galactic plane towards background extragalactic sources @xcite . in addition , an analysis of x - ray absorption spectra along similar lines of sight by @xcite finds that the required x - ray absorption column exceeds the h i column estimated by 21 cm emission . this excess is difficult to account for via other atomic components , but could be due to molecular gas . the uv and x - ray observations point to the potential ubiquitous presence of h@xmath0 in the galaxy with fractions of h@xmath0 towards these high galactic latitude lines of sight ranging from @xmath610% @xcite to @xmath109% @xcite of the h@xmath0 fraction estimated in the local solar neighborhood . while there is important evidence for the presence of some h@xmath0 in the low density cnm , the evidence for co is less substantial . for instance , the surface density of molecular gas estimated by co emission towards high galactic latitudes is only 1% of the value derived in the local neighborhood . this is well below that seen for the h@xmath0 fraction and hints that co may not be fully tracing h@xmath0 in the low density ism . this is not terribly surprising , because the column density needed for equivalent co self - shielding is three orders of magnitude larger than that needed for h@xmath0 formation ( lee et al . 1996 ; draine & bertoldi 1996 ) . given the possibility of a significant present of h@xmath0 in the ism one might question whether significant reservoir of h@xmath0 molecular clouds exist that are `` inert '' in the sense that they will never create co. there are two possibilities that can be examined in this regard . first the @xcite results suggest an abundance of weak shocks are active in the galactic ism . these shocks will evolve on slow times for the creation of both h@xmath0 and co. thus in the @xmath110 myr required to make h@xmath0 there is an additional @xmath110 myr before co creation ( see fig . for such a weak shock there will be a significant amount of time spent in the molecular but pre - co phase . the overall lifetime of such systems may be limited by the large distance that would be transversed . for example , a @xmath6 5 km s@xmath40 shock will travel a distance @xmath111500 pc and therefore has an increased likelihood of passing by a massive star that will certainly destroy the co and perhaps h@xmath0 . an alternate picture would emerge if the shock evolution were limited in some fashion ( perhaps by a strong tangential b - field ) . this will halt the density evolution but column will continue to build up behind the shock . thus , the combination of ensuing extinction evolution , h@xmath0 molecules , and cosmic ray ionization eventually leads to co formation . however , if the density evolution were stunted at a values below @xmath112 then the h@xmath0 phase would last longer in a similar fashion as the weak shock case . however , the importance of such inert clouds might be limited by passing hot stellar wind or supernova bubbles which could disrupt or evaporate them . in sum we can not discount the possibility that there might be a hidden reservior of h@xmath0 molecular gas in the ism . in this picture with ubiquitous h@xmath0 , and little co , cloud formation timescales will depend primarily on the timescale to accumulate a sufficient shielding column density , not the h@xmath0 formation time . hence the solution is similar to that found in the traditional case where cloud formation is modeled from the atomic - molecular hydrogen transition . complex mhd numerical simulations of structure formation in the ism , such as those performed by @xcite , typically neglect the chemical considerations when computing the evolution . this is easily understood given the extreme computational complexities . in our simulations we have included the chemistry but with a simpler dynamical prescription . in this process we have identified two areas where this work can have useful impact on numerical simulations of cloud formation and evolution . our results point to the key importance of the shielding of uv photons in the overall physical evolution . in figure 12 we show the temperature evolution as a function of the mass surface density ( visual extinction ) . the initial spikes below 1 g cm@xmath28 are the shock and initial cool down . the following evolution shows the gradual temperature cool down as the shielding column increases . for all shocks the gas temperature , at a given surface density , is nearly identical , to within a factor of @xmath113 , and declines in a linear fashion in all cases except for the strongest shock . this difference is due to the lack of atomic carbon cooling which typically contributes to the cooling at a@xmath114 1 ( in the fast shock quick co formation via self - shielding reduces the influence of c i ) . the similarities of these solutions and the linear decay suggests that tracking the shielding in each cube of a mhd simulation can be used to provide a simple estimate of the gas temperature and hence pressure . another potential aid to mhd simulations is provided in the appendix where we show that there exists an analytical solution for the time - dependence of h@xmath0 formation ( co formation can be treated via equilibrium calculations ) . this analytical solution requires only knowledge of the local gas density and extinction . under those conditions computationally intensive chemical calculations need not be performed , while still keeping the capability to create a realistic simulation that includes the chemical formation of the two most important molecular species : h@xmath0 and co. if this can adapted into mhd models then it provides a method to use simulations to predict maps of molecular emission , which can be readily compared to observations .

we find that for typical parameters the molecular hydrogen fractional abundance can become significant well before co forms . the timescale for ( co ) molecular cloud formation is not set by the h formation rate on grains , but rather by the timescale for accumulating a sufficient column density or extinction , . the local ratio of atomic to molecular gas ( 4:1 ) , coupled with short estimates for the lifetimes of molecular clouds ( 3 - 5 myr ) , suggests that the timescales for accumulating molecular clouds from atomic material typically must be no longer than about 12 - 20 myr . based on the shielding requirement , this implies that the typical product of pre - shock density and velocity must be . in turn , depending upon the shock velocity , this implies shock ram pressures which are a few times the typical estimated local turbulent gas pressure , and comparable to the total pressures ( gas plus magnetic plus cosmic rays ) . coupled with the rapid formation of co once shielding is sufficient , flow - driven formation of molecular clouds in the local interstellar medium can occur sufficiently rapidly to account for observations .

motivated by our previous paper , in which we argued for the formation of molecular clouds from large - scale flows in the diffuse galactic interstellar medium , we examine the formation of molecular gas behind shocks in atomic gas using a one - dimensional chemical / dynamical model . in our analysis we place particular emphasis on constraints placed on the dynamical evolution by the chemistry . the most important result of this study is to stress the importance of shielding the molecular gas from the destructive effects of uv radiation . for shock ram pressures comparable to or exceeding typical local interstellar medium pressures , self - shielding controls the formation time of molecular hydrogen but co formation requires shielding of the interstellar radiation field by dust grains . we find that for typical parameters the molecular hydrogen fractional abundance can become significant well before co forms . the timescale for ( co ) molecular cloud formation is not set by the h formation rate on grains , but rather by the timescale for accumulating a sufficient column density or extinction , . the local ratio of atomic to molecular gas ( 4:1 ) , coupled with short estimates for the lifetimes of molecular clouds ( 3 - 5 myr ) , suggests that the timescales for accumulating molecular clouds from atomic material typically must be no longer than about 12 - 20 myr . based on the shielding requirement , this implies that the typical product of pre - shock density and velocity must be . in turn , depending upon the shock velocity , this implies shock ram pressures which are a few times the typical estimated local turbulent gas pressure , and comparable to the total pressures ( gas plus magnetic plus cosmic rays ) . coupled with the rapid formation of co once shielding is sufficient , flow - driven formation of molecular clouds in the local interstellar medium can occur sufficiently rapidly to account for observations . we also provide detailed predictions of atomic and molecular emission and absorption that track the formation of a molecular cloud from a purely atomic medium , with a view toward helping to verify cloud formation by shock waves . however , our predictions suggest that the detection of the pre - co stages will be challenging . finally , we provide an analytic solution for time - dependent formation which may be of use in numerical hydrodynamic calculations .

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molecular motors are protein molecules that drive a wide range of intra - cellular activities including transport of molecular cargo @xcite . there are many similarities between collective molecular motor transport and vehicular traffic @xcite . in recent years non - equilibrium statistical mechanics has found unusual application in research on traffic flow of various different types of objects starting from objects as small as molecular motors to macroscopic objects like vehicles @xcite . analytical as well as numerical techniques of the statistical physics are being used to understand rich variety of physical phenomena exhibited by traffic systems . some of these phenomena , observed under different circumstances , include phase transitions , criticality and self - organized criticality , metastability and hysteresis , phase - segregation , etc . a common modeling strategy is to represent the motile objects ( e.g. , a vehicle or a molecular motor ) by a self - propelled particle , ignoring its structural details , and then treating the traffic as a system of interacting particles driven far from equilibrium . these models belong to a class of non - equilibrium systems called _ driven - diffusive lattice gases _ @xcite . in most of these traffic models the dynamics of the particles is formulated using the language of _ cellular automata _ ( ca ) @xcite . to our knowledge , the first model for molecular motor traffic was formulated in 1968 in the context of collective movement of ribosomes on messenger rna track @xcite . in recent years several groups have independently developed a class of minimal generic models for traffic of molecular motors which move on tracks that are filamentary proteins . all these models are essentially extensions of the totally asymmetric simple exclusion process ( tasep ) @xcite which is one of the simplest models of driven diffusive lattice gas systems . in these models @xcite the molecular motors are represented by particles whereas the sites for the binding of the motors with the tracks are represented by a one - dimensional discrete lattice . just as in tasep , the motors are allowed to hop forward , with probability @xmath0 , provided the site in front is empty . however , unlike tasep , the particles can also get `` attached '' to an empty lattice site , with probability @xmath1 , and `` detached '' from an occupied site , with probability @xmath2 from any site except the end points . parmeggiani et al . @xcite demonstrated a novel phase where low and high density regimes , separated from each other by domain walls , coexist . they interpreted this spatial organization as traffic jam of molecular motors . none of the models of molecular motor traffic mentioned above distinguish between kinesins and dyneins which form the two superfamilies of motor proteins that move on the same type of tracks , namely , microtubules . on the other hand , detailed experiments over the last two years have established that , in contrast to kinesins , dyneins can take steps of four different sizes depending on the opposing force or hindrance . one of the aims of this paper is to introduce a minimal model that distinguishes between these two features of kinesin and dynein motors . in this paper we begin by investigating the aggressive driving model ( adm ) , a stochastic ca model for traffic flow that is closely related to the nagel - schreckenberg ( nasch ) model @xcite . one of the reasons for studying this model is that it allows natural extensions so as to capture the essential features of dynein motor traffic including the unique features of dynein stepping ( which we shall explain in section [ dyneinexperiment ] ) . besides , the adm model is an interesting model of vehicular traffic in its own right and is also related to the fukui - ishibashi ( fi ) model @xcite . however , in contrast to the fi model , it still shows spontaneous jam formation . we investigate the properties of the adm by approximate analytical calculations as well as by computer simulations . then , we use an extended version , which we refer to as the dynein traffic model ( dtm ) , for a quantitative desciption of intra - cellular traffic of dynein motors . the paper is organized as follows . in the next section we describe the adm and the method of simulation . in section [ sec3 ] we investigate the properties of the adm with periodic boundary conditions and we describe the analytical theories for calculating its flow properties . we present a comparison of the adm with nasch model at the end of section [ sec3 ] . in section [ sec4 ] we investigate the density profiles and phase diagram of the adm with open boundary conditions . in section [ sec5 ] we describe the experimentally observed hindrance - dependence of the step sizes of dynein motors and introduce the dynein traffic model ( dtm ) . we present the results for the dtm with periodic boundary conditions in section [ sec6 ] and those under open boundary conditions in section [ sec7 ] . finally we summarize the main results and the conclusions in section [ sec8 ] .

we begin by investigating the properties of the aggressive driving model ( adm ) , a simple cellular automata - based model of vehicular traffic , a unique feature of which is that it allows a natural extension to capture the essential features of dynein motor traffic . we first calculate several collective properties of the adm , under both periodic and open boundary conditions , analytically using two different mean - field approaches as well as by carrying out computer simulations . al . lett . * 90 * , 086601 ( 2003 ) ) which can be regarded as a minimal model for traffic of a closely related family of motor proteins called kinesin .

motivated by recent experimental results for the step sizes of dynein motor proteins , we develope a cellular automata model for intra - cellular traffic of dynein motors incorporating special features of the hindrance - dependent step size of the individual motors . we begin by investigating the properties of the aggressive driving model ( adm ) , a simple cellular automata - based model of vehicular traffic , a unique feature of which is that it allows a natural extension to capture the essential features of dynein motor traffic . we first calculate several collective properties of the adm , under both periodic and open boundary conditions , analytically using two different mean - field approaches as well as by carrying out computer simulations . then we extend the adm by incorporating the possibilities of attachment and detachment of motors on the track which is a common feature of a large class of motor proteins that are collectively referred to as cytoskeletal motors . the interplay of the boundary and bulk dynamics of attachment and detachment of the motors to the track gives rise a phase where high and low density phases separated by a stable domain wall coexist . we also compare and contrast our results with the model of parmeggiani et . al . ( phys . rev . lett . * 90 * , 086601 ( 2003 ) ) which can be regarded as a minimal model for traffic of a closely related family of motor proteins called kinesin . finally , we compare the transportation efficiencies of dynein and kinesin motors over a range of values of the model parameters .

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in this paper we have first investigated the properties of the aggressive driving model ( adm ) which is a simple cellular automata model for vehicular traffic . one of the motivations for considering this model is that the rule for aggressive driving can be naturally extended to capture the special features of step sizes of dynein motors and , therefore , the adm is ideally suited for extending so as to study intra - cellular molecular motor traffic by dynein motors . the adm shows different behavior for @xmath56 and @xmath28 . for @xmath56 the model is identical to the nasch model with @xmath56 which has perfect particle - hole symmetry . this symmetry is broken for @xmath28 . the fundamental diagram of this model in the special limit @xmath73 has a form which is quite different from that of the nasch model in the limit @xmath73 . we have also shown few distance headway and time - headway distributions . we have calculated the fundamental diagram using two different mean - field approaches , namely , site - oriented mean - field approach ( somf ) and car - oriented mean - field approach ( comf ) . a simple somf theory shows a poor agreement with the simulation data . however , an improved mean field theory , namely comf , shows good agreement with the numerical data obtained from computer simulations . we compare our adm with the nagel - schreckenberg model which captures essential features of normal driving . we have also investigated the density profiles and phase diagrams of this model replacing the periodic boundary conditions by open boundary conditions . the density profile of this model with open boundary conditions shows periodic structures in the free - flowing regime whose period of oscillation depends on the maximum attainable velocity @xmath24 . we have extended the adm to develope a dynein traffic model ( dtm ) which is a model of intra - cellular molecular motor traffic from cell periphery towards the nucleus of the cell . we have investigated the properties of this model with periodic and open boundary conditions . under open boundary conditions , dtm shows an unusual feature where low and high density phases separated by a static domain wall coexist over a range of parameter values which can be interpreted as a traffic jam of molecular motors . this is in sharp contrast to the phase diagram of the adm which does not exhibit such coexistence of congested and free - flowing regions . the occurrence of the phase is , thus intimately related to the competition between the hopping and the kinetics of attachment / detachment of the motors on the track . finally , we have compared the efficiencies of dynein and kinesin motors for different values of parameters . for very large values of the parameters @xmath218 and @xmath201 system is found in the high density phase and , in that case , one observes practically no difference between the efficiencies of kinesin and dynein motors . to our knoweledge , our dtm is the first model of traffic - like collective transport of _ dynein _ motors on filamentary microtubule tracks . a model that incorporates both the species of dyneins and kinesin motors , which move in opposite directions along the same track , may provide deep insight into experimentally observed bidirectional traffic @xcite . * acknowledgements * : this work has been supported ( through dc ) , in part , by the council of scientific and industrial research ( csir ) of the government of india . dc also thanks max - planck institute for physics of complex systems , dresden , and prof . frank jlicher for hospitality during a short visit when a part of this manuscript was prepared .

motivated by recent experimental results for the step sizes of dynein motor proteins , we develope a cellular automata model for intra - cellular traffic of dynein motors incorporating special features of the hindrance - dependent step size of the individual motors . finally , we compare the transportation efficiencies of dynein and kinesin motors over a range of values of the model parameters .

motivated by recent experimental results for the step sizes of dynein motor proteins , we develope a cellular automata model for intra - cellular traffic of dynein motors incorporating special features of the hindrance - dependent step size of the individual motors . we begin by investigating the properties of the aggressive driving model ( adm ) , a simple cellular automata - based model of vehicular traffic , a unique feature of which is that it allows a natural extension to capture the essential features of dynein motor traffic . we first calculate several collective properties of the adm , under both periodic and open boundary conditions , analytically using two different mean - field approaches as well as by carrying out computer simulations . then we extend the adm by incorporating the possibilities of attachment and detachment of motors on the track which is a common feature of a large class of motor proteins that are collectively referred to as cytoskeletal motors . the interplay of the boundary and bulk dynamics of attachment and detachment of the motors to the track gives rise a phase where high and low density phases separated by a stable domain wall coexist . we also compare and contrast our results with the model of parmeggiani et . al . ( phys . rev . lett . * 90 * , 086601 ( 2003 ) ) which can be regarded as a minimal model for traffic of a closely related family of motor proteins called kinesin . finally , we compare the transportation efficiencies of dynein and kinesin motors over a range of values of the model parameters .

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the optical and uv emission of radio - quiet active galactic nuclei ( agns ) is dominated by a quasi - thermal component , the big blue bump " ( shields 1978 , malkan & sargent 1982 , malkan 1983 ) . the big blue bump is usually interpreted as thermal emission from a geometrically thin , optically thick , accretion disk around a massive black hole ( malkan 1983 , sun & malkan 1989 , laor & netzer 1989 : ln89 ) . this model makes two key predictions . first , the large jump in the absorptive opacity at the lyman edge combined with the vertical temperature gradients in the disk insure that some feature will be present at the lyman edge ( kolykhalov & sunyaev 1984 , ln89 ) . for the same reason , some have predicted that the polarization of the disk should decrease at frequencies above the lyman edge ( laor , netzer & piran 1990 ; but see blaes & agol 1996 for another view ) . second , variations in different portions of the optical / ultraviolet band can be associated with fluctuations in the conditions at particular radii . if fluctuations move radially through the disk , there should be time - delays between variations seen at different wavelengths which correspond to signal travel times within the disk . unfortunately , while this picture is very attractive , it suffers from severe problems , both theoretical and observational . from the theoretical point of view , it is generally expected that the viscous stress is proportional to the pressure ( the @xmath9-model " introduced by shakura & sunyaev 1973 : ss73 ) , including the radiation pressure when the disk is optically thick . unfortunately , if this is true , when the accretion rate is more than a small fraction of the eddington rate , the disk is thermally unstable throughout its inner region ( shakura & sunyaev 1976 ) . for this reason , the most detailed disk spectral calculations made to date ( ln89 ) supposed that the stress was instead proportional to the geometric mean of the gas and radiation pressures . in addition , neither of the two key predictions just described is confirmed by observations . partial lyman edges have been detected in very few agn ( antonucci , kinney & ford 1989 ; koratkar , kinney & bohlin 1992 ) and the polarization of two quasars with partial absorption edges _ rises _ across the lyman edge ( koratkar , 1995 ) . intensive monitoring of the type 1 seyfert galaxy ngc 5548 revealed that variations from the optical to the far - uv are tightly correlated . the signal travel speed through a conventional disk that would be required to coordinate these variations was found to be at least @xmath10 , far higher than any of the expected signal speeds ( krolik 1991 ) . moreover , variations in the soft x - ray flux have been found to be almost as tightly correlated with ultraviolet fluctuations ( clavel , 1992 ) . these two findings suggest that the big blue bump may actually be due to reprocessed x - ray emission ( rokaki , collin - souffrin , & magnan 1993 ) . other evidence also points to the importance of x - ray reprocessing in the formation of agn continua . in many type 1 seyfert galaxies , there is a spectral feature in the vicinity of 10 30 kev which is readily interpreted as compton reflection " of x - rays from a cool , optically thick surface occupying roughly half of the solid angle around the x - ray source ( pounds 1990 ) . it would be very natural to interpret this reflector as an accretion disk ( lightman & white 1988 ) ( but a number of other interpretations are also possible : nandra & george 1994 ; krolik , madau , & @xmath11 1994 ; ghisellini , haardt & matt 1994 ) . further support for the thought that the disk is subjected to intense x - ray irradiation comes from the discovery that the fe k@xmath9 profiles in many agn are extremely broad ( as summarized by nandra 1997 ) . recently popular models for the production of the x - rays are also consistent with this picture in which a large fraction of the emitted x - rays strike a nearby cool surface , possibly the accretion disk ( haardt & maraschi 1993 : hm93 ; haardt , maraschi & ghisellini 1994 ; pietrini & krolik 1995 ; stern 1995 ) . these models suppose that a large part of the disk s dissipation takes place in a small amount of mass at or above the disk s surface . the very large heating rate per unit mass in an optically thin gas creates a very hot corona which cools by inverse compton scattering . roughly half the x - ray flux generated strikes the disk , and is reprocessed into uv photons which then return to the corona as seeds for inverse compton scattering . fits to the broad - band x - ray spectra of agn are at least qualitatively consistent with this sort of thermal comptonization model ( e.g. zdziarski 1995 ) . in its simplest form , the corona model predicts that the observed uv flux should be roughly equal to the observed intrinsic x - ray flux ( i.e. the x - ray flux after subtracting any components due to reprocessing ) . whether this prediction is consistent with the observations is uncertain . the uv / x - ray flux ratio is conventionally parameterized by @xmath12 , the spectral index of a power - law interpolated between 2500 and 2 kev , as measured in the agn rest - frame . depending on the sample definition , the observed distribution of @xmath12 varies somewhat , but typically it ranges from @xmath13 to @xmath14 , with a tendency for lower luminosity agn to have smaller @xmath12 ( stocke et al . 1991 ; wilkes et al . 1994 ; green et al . 1995 ) . the flux at at 2500 is @xmath15 times the flux at 2 kev so that , if we compare at these two frequencies alone , we would conclude that the uv luminosity exceeds the x - ray luminosity in most agn . however , to find the ratio of the total uv and x - ray fluxes , substantial and very uncertain bolometric corrections must be applied to both . we can not observe low - redshift agn in the rest - frame euv , but composite high - redshift quasar spectra ( e.g. zheng et al . 1996 ) suggest that the spectrum rolls over sharply at wavelengths shorter than @xmath16 . if this shape is general , the uv bolometric correction ( relative to @xmath17 at 2500 ) might be as small as @xmath18 , but there is likely to be a considerable dispersion from case to case . the x - ray bolometric correction is similarly uncertain . those low luminosity agn for which osse has been able to obtain spectra typically show power - laws with indices @xmath19 across the hard x - ray range , supplemented by a substantial bump at a few tens of kev which is generally believed to be due to reprocessing ( zdziarski et al . if that power - law extends from 0.5 to 200 kev , the bolometric correction relative to @xmath17 at 2 kev is 6.5 . taking these bolometric corrections at face value , we would conclude that the uv and x - ray fluxes in low luminosity agn ( where the evidence for the corona model is best ) are indeed close to being equal , while the uv flux is typically rather greater than the intrinsic x - ray flux in higher luminosity agn . however , it is clear from the character of this estimate that it is very weakly based . the situation is made still cloudier by the possibility that relativistic effects may direct the majority of the emitted x - rays toward the disk , reducing the observed ratio between x - rays and uv ( martocchia and matt 1996 ) . it is our goal to begin the process of testing these coronal models by comparing the ultraviolet spectrum they predict with observations . in this paper we develop the techniques necessary to compute the predicted uv spectra , and apply them to the simplest version of the model : that in which _ all _ the dissipation takes place in a slab - like corona resting directly on top of the accretion disk . in future work we will examine other variations , such as models in which the dissipation is shared between the disk proper and the corona , and models with more complicated coronal geometry . an immediate consequence of this picture in which most of the dissipation takes place in a corona , and not inside the disk proper , is that radiation pressure support in the bulk of the disk is far weaker than in a conventional disk . the radiation force is simply the product of the radiation flux and the opacity ; where there is no outgoing flux ( because there is no internal heat generation ) , there can be no radiation force . as a result , such disks collapse to a state of much greater density ( and also column density ) than would be predicted by conventional models ( svensson & zdziarski 1995 : sz95 ) . they are then supported primarily by gas pressure , and ( in the @xmath9-model ) are thermally stable . it is also an immediate corollary that these disks are very nearly isothermal zero outward flux in an optically thick environment implies constant temperature . some analogous work has been done in the past . a number of groups ( e.g. ross & fabian 1993 ; @xmath20 1994 ) have computed the spectra of x - ray irradiated disks , but all have assumed the disks are radiation pressure supported . because disks in which most dissipation is in a corona are much denser than conventional radiation pressure - supported disks , the character of the uv spectrum they produce is quite different from what has been predicted by assuming a conventional disk structure . others have tried to predict the uv spectra radiated by dense lumps heated by x - rays ( guilbert & rees 1988 , ferland & rees 1988 ) . in [ sec : isothermal disk ] we solve the structure equations for x - ray irradiated disks . the solution of the radiative transfer and vertical structure equations in the x - ray heated skin is described in [ sec : xray skin ] and our results are summarized in [ sec : results ] the gravitational stability of the cold disk is discussed in [ sec : gravitational instability ] we conclude in [ sec : conclusions ]

this x - ray heated skin has two layers : a radiation pressure supported region in which the uv flux is created , and , immediately above this layer , a warmer zone , optically thin to uv radiation , formed where the x - ray ionization parameter is large . in the lower layer this finding raises questions for many otherwise plausible models in which x - ray irradiation plays a major role . rev . rev . , phys . mod . soc . soc . phys . rev . astr . rad . sp . sci . sci . rev . sp .

motivated by recent work indicating that the uv continuum in agn may be produced by reradiation of energy absorbed from x - rays irradiating an accretion disk , we present a calculation of the vertical structures and ultraviolet spectra of x - ray irradiated accretion disks around massive non - rotating black holes . after finding the radial dependence of vertically - integrated quantities for these disks , we solve the equations of hydrostatic equilibrium , energy balance , and frequency - dependent radiation transfer as functions of altitude . to solve the last set of equations , we use a variable eddington factor method . we include electron scattering , free - free , and hi , hei , and heii bound - free opacities and the corresponding continuum cooling processes . the incident x - ray flux heats a thin layer of material 3 - 4 scale heights above the midplane of the disk . this x - ray heated skin has two layers : a radiation pressure supported region in which the uv flux is created , and , immediately above this layer , a warmer zone , optically thin to uv radiation , formed where the x - ray ionization parameter is large . in the lower layer the gas pressure is nearly independent of altitude but the temperature increases upward . the fraction of the incident hard x - ray flux which emerges in the uv falls with increasing ( the accretion rate in eddington units ) . at frequencies below the lyman edge the slope of the continuum ( ) varies from -1.6 to 0.8 as increases from 0.001 to 1 . here is the mass of the central black hole in units of . in all cases examined ( and ) , the lyman edge appears in emission . the amplitude of the lyman edge feature increases with but is relatively independent of . the amplitude of the lyman edge emission feature increases with disk inclination . compton scattering in disk coronae can smooth the lyman edge feature only if , where is the thomson depth of the coronae . while the overall spectral shape predicted by x - ray irradiation may be compatible with observations , the lyman edge emission feature it predicts is not . this finding raises questions for many otherwise plausible models in which x - ray irradiation plays a major role . # 1#2#3#4#4 19#3 phys . rev . d , # 1 , # 2 # 1#2#3#4#4 19#3 phys . lett . , # 1 , # 2 # 1#2#3#4#4 19#3 phys . rev . lett . , # 1 , # 2 # 1#2#3#4#4 19#3 phys . rev . , # 1 , # 2 # 1#2#3#4#4 19#3 phys . rep . , # 1 , # 2 # 1#2#3#4#4 19#3 phys . fluids , # 1 , # 2 # 1#2#3#4#4 19#3 proc . phys . soc . , # 1 , # 2 # 1#2#3#4#4 19#3 nucl . phys . , # 1 , # 2 # 1#2#3#4#4 19#3 mod . phys . lett . , # 1 , # 2 # 1#2#3#4#4 19#3 ap . j. , # 1 , # 2 # 1#2#3#4#4 19#3 astr . j. , # 1 , # 2 # 1#2#3#4#4 19#3 acta astr . , # 1 , # 2 # 1#2#3#4#4 19#3 rev . mod . phys . , # 1 , # 2 # 1#2#3#4#4 19#3 nuovo cimento c , # 1 , # 2 # 1#2#3#4#4 19#3 sov . phys . jetp , # 1 , # 2 # 1#2#3#4#4 19#3 sov . ast . aj , # 1 , # 2 # 1#2#3#4#4 19#3 pub . ast . soc . japan , # 1 , # 2 # 1#2#3#4#4 19#3 pub . ast . soc . pacific , # 1 , # 2 # 1#2#3#4#4 19#3 ann . phys . ( ny ) , # 1 , # 2 # 1#2#3#4#4 19#3 yad . fiz . , # 1 , # 2 # 1#2#3#4#4 19#3 sov . j. nucl . phys . , # 1 , # 2 # 1#2#3#4#4 19#3 ast . ap . , # 1 , # 2 # 1#2#3#4#4 19#3 ann . rev . astr . ap . , # 1 , # 2 # 1#2#3#4#4 19#3 m.n.r.a.s . , # 1 , # 2 # 1#2#3#4#4 19#3 j. de physique , # 1,#2 # 1#2#3#4#4 19#3 j. quant . spec . rad . transfer , # 1,#2 # 1#2#3#4#4 19#3 j.e.t.p . lett . , # 1,#2 # 1#2#3#4#4 19#3 ap . j. ( letters ) . , # 1,#2 # 1#2#3#4#4 19#3 ap . j. ( supp . ) . , # 1,#2 # 1#2#3#4#4 19#3 ap . lett . , # 1,#2 # 1#2#3#4#4 19#3 ap . sp . sci . , # 1,#2 # 1#2#3#4#4 19#3 nature , # 1,#2 # 1#2#3#4#4 19#3 sp . sci . rev . , # 1,#2 # 1#2#3#4#4 19#3 adv . sp . res . , # 1,#2

astro-ph9609094

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the inner edge of the x - ray heated skin is 3 - 4 scale heights from the midplane of the isothermal disk . the thickness of the skin increases slightly with radius but remains @xmath118 of the height of the inner edge ( fig . [ fig : structure]a ) . the thin disk approximation is valid at all radii and the geometry of the disk can be determined from the isothermal disk solution ( [ sec : isothermal disk ] ) . the density profile in the isothermal disk is gaussian so the gas pressure at the inner edge of the skin is much smaller ( typically by @xmath119 four orders of magnitude ) than the central disk pressure . this pressure is still much larger than the pressure in a fully radiation pressure - supported disk , however . throughout most of the skin , the radiation force of the uv photons is great enough to balance gravity and the force of the incident x - ray flux . the gas pressure is therefore nearly constant for @xmath120 ( see fig . [ fig : structure]b ) . both the x - ray and uv fluxes drop to zero at @xmath121 and a gas pressure gradient forms to support the disk against gravity . the gas pressure profile joins smoothly to the isothermal disk solution at @xmath107 . the radiative cooling rate is equal to the x - ray heating rate when the gas is in radiative equilibrium . therefore , the gas temperature must be greater than the uv radiation temperature because the cooling rate is @xmath122 , where @xmath123 and @xmath124 is the frequency integrated mean intensity of the uv radiation field . the actual gas temperature profile is determined by three competing effects . first , the mean intensity , and @xmath125 , decrease from the bottom of the skin to the surface because of the net outward flux . second , the x - ray heating rate increases from zero at the bottom of the skin to its maximum value at the top . therefore , @xmath126 at @xmath107 and the difference between the two temperatures increases steadily until @xmath127 at the upper edge of the skin . third , the gas temperature gradient changes when an opacity jump , such as the lyman edge , becomes transparent ( mihalas 1978 , p. 206 ) . the gradient can even change sign in some cases , but not in all cases ( see fig . [ fig : structure ] ) . the relative importance of these three effects depends upon the local gas pressure and temperature . we find that @xmath38 generally increases from the midplane outward but the increase is not always monotonic . an example of a typical temperature profile is presented in fig . [ fig : structure]c . the gas is optically thick at all frequencies when @xmath121 and the temperature gradient @xmath128 the gradient is slightly greater than 0 ( @xmath61 increases inward ) at @xmath129 , and slightly less than 0 ( @xmath61 increases outward ) for @xmath130 . the x - ray heating rate increases rapidly at @xmath131 and @xmath38 rises to maintain thermal equilibrium . in this example , the photosphere for frequencies below the lyman edge is at @xmath132 , so the temperature gradient changes sign at this point . the sign flips back at the photosphere for frequencies just above the lyman edge . the gradient remains positive at smaller @xmath96 and the thermal runaway starts at @xmath133 . the upper boundary of the uv reprocessing region is set by the thermal runaway . the temperature runs away , causing the code to fail to converge , when the x - ray ionization parameter @xmath134 where @xmath135 is the x - ray flux at @xmath96 . when @xmath136 exceeds this critical value , bremsstrahlung and recombination can not cool the gas , and its temperature rises toward the compton temperature ( krolik , mckee & tarter 1981 ; voit & shull 1988 ; ferland & rees 1988 ) . the exact mass column of the thermal runaway ( @xmath108 ) depends upon @xmath0 , @xmath3 and @xmath137 , but in general @xmath138 . @xmath108 is never smaller than @xmath139 gm @xmath51 because the skin is optically thin to x - rays above that point and the gas pressure does nt change much over such a small column density . on the other hand , @xmath108 is always less than about one compton depth because the x - rays are exponentially attenuated beyond that point . typically @xmath108 rises sharply at larger radii . a good example of this effect is seen in the model @xmath140 and @xmath141 ( see fig . [ fig : rsig mdot ] ) . the reason for this sharp rise is that heii recombination provides much of the cooling at columns @xmath142 at smaller radii , but the lower temperatures at larger radii mean that little heiii is formed there . within the framework of the model it is possible to estimate @xmath143 , the fraction of the incident x - ray flux which is reradiated in the uv this quantity is determined by how much x - ray flux is either absorbed or reflected by the material in the thermal runaway layer at column densities smaller than @xmath108 . we estimate @xmath143 by the somewhat crude prescription of assuming that the opacity of the warm layer is unchanged by the thermal runaway . a more careful treatment would allow for the reduction in photoelectric opacity due to the higher ionization states found there . nonetheless , we believe our global estimate of @xmath143 is not too bad because the shielding effect is only of major importance when @xmath108 is at least @xmath144 gm @xmath51 . when @xmath108 is that large , it hardly matters whether the disk proper is shadowed by photoelectric opacity or compton scattering . the general trend of @xmath143 is to decrease with increasing @xmath0 because @xmath108 increases . the gas pressure in the skin is @xmath145 on the other hand , @xmath146 at fixed @xmath137 . consequently , @xmath147 at the upper boundary of the skin . thus , as @xmath0 increases , the x - ray flux is able to trigger a thermal runaway at larger column densities and over a broader range of radii . when @xmath148 the x - ray flux is never large enough to cause a runaway ( see fig . [ fig : rsig mdot ] ) . in table [ table : uv fraction ] we list @xmath149 where @xmath150 is the x - ray flux at a mass column @xmath96 in the ring at radius @xmath137 . we find that @xmath143 is 1.0 at @xmath151 and decreases to 0.62 at @xmath141 . the column density of the runaway is insensitive to @xmath3 and @xmath143 varies only slightly when @xmath3 changes by three orders of magnitude . we caution that these numerical values also depend on the specific x - ray spectral shape we have chosen . the material above @xmath108 might influence the emergent spectrum by radiating soft x - rays and by comptonizing the uv flux leaving the disk atmosphere . we can estimate the gas temperature there as @xmath152 ( begelman , mckee & shields 1988 ) , where @xmath153 are the mean intensities of the uv and incident x - rays , respectively , and @xmath154 are the corresponding compton temperatures . although we discount the effect of hard x - rays for heating the disk , they are effective for compton heating , at least up to @xmath155 kev , beyond which the klein - nishina reduction in the scattering rate curtails their contribution . if the spectrum is @xmath156 over many orders of magnitude in photon energy , @xmath157 . as we have already estimated , @xmath158 . the uv and x - ray intensities are comparable in the skin , so the additional cooling provided by scattering of the uv photons will reduce the gas temperature to @xmath159 , roughly independent of radius out to the point where the heating is insufficient to maintain a corona at high temperature . the emission measure ( @xmath160 ) of the soft x - ray corona is dominated by the regions at larger radius where @xmath161 gm @xmath51 . we estimate @xmath162 where @xmath163 is the thickness of the corona . we expect @xmath164 and so the associated bremsstrahlung luminosity ( @xmath165 erg s@xmath166 ) is much smaller than the total disk luminosity unless @xmath167 . similarly , the importance of comptonization to the spectrum depends on the compton @xmath168-parameter , @xmath169 , where @xmath170 is the thomson depth of the layer . because @xmath170 is at most @xmath144 and we expect @xmath171 , this , too , is unlikely to alter the emergent spectrum . we elaborate on the effects of comptonization on the uv spectrum in [ subsubsec : comptonization ] we calculated integrated spectra for face - on disks over a range of @xmath0 and @xmath3 chosen to facilitate comparisons with previous calculations of disk spectra , namely luminosities ranging from 0.003 to 0.3 of the eddington luminosity and central masses of @xmath172 to @xmath173 . in each case , we divided the disk into annuli spaced at ( roughly ) equal intervals in @xmath174 and the local structure and spectrum were calculated . the radii ranged from the inner radius @xmath175 to some outer radius at which the gas temperature was so low that the contribution from the outermost radius to the flux near the lyman edge became negligible . this occurs at @xmath176 where @xmath177 ( eq . [ eq : teff ] ) . finally , the local spectra are added using the method described by ln89 , which accounts for limb - darkening , doppler boosting , and relativistic aberration of the emergent flux . we neglect the general relativistic bending of light rays in the gravitational field of the central black hole ( cunningham 1975 ) . a variety of face - on spectra are plotted in ( figs . [ fig : mass spectra ] , [ fig : mdot spectra ] ) . note that we define @xmath178 as @xmath179 the luminosity per solid angle at an angle @xmath180 from the disk axis . in the simplest treatment of predicted accretion disk spectra , the dynamics are approximated as newtonian , and the spectra of individual rings are assumed to be blackbodies . these approximations lead to a temperature profile @xmath181 and an integrated spectrum whose shape is roughly @xmath182 , where @xmath183 is the temperature of the innermost ring . it is clear from these figures that such an approximation is not a very good description of our predicted spectra . at low frequencies , there is considerable spectral curvature , and strong lyman edge features substantially modify the shape of the exponential cut - off at high frequencies . three different effects alter the spectral shape at frequencies below the lyman edge . first , relativistic effects flatten the temperature profile in the inner rings ( novikov & thorne 1973 ) . this softens the spectrum relative to the newtonian approximation . second , the presence of a large emission feature at the lyman edge implies that a smaller fraction of the reprocessed flux emerges at frequencies below the lyman edge . this depresses the general level of the near - uv continuum . third , at frequencies below the lyman edge , the mass column of the uv photosphere increases with increasing frequency . the flux at lower frequencies then comes from hotter gas because the gas temperature typically increases as @xmath96 decreases . this softens the continuum spectrum relative to the black body approximation . the low - frequency spectrum becomes harder with increasing hard x - ray flux , @xmath2 . the disk temperature is roughly proportional to the effective temperature of the x - ray flux , so the peak of the disk spectrum shifts from @xmath184 hz in the coldest disks to @xmath185 hz in the hottest . to quantify this effect , we define @xmath186 at @xmath187 . the value of this quantity for each model we computed is displayed in table [ table : uv fraction ] . the smallest @xmath188 we found was -1.6 ( @xmath189 ) ; the largest was 0.8 ( @xmath190 ) . as the previous paragraphs explain , some , but not all , these effects are due to the external irradiation . in particular , the temperature gradient effect acts the opposite way when the dissipation is inside the disk rather than outside . an improved calculation of the spectrum of a conventional disk with internal dissipation methods yields a predicted @xmath191 ( sincell & krolik 1996 ) . we found a strong lyman edge emission emission feature over the entire parameter range we studied ( see figs . [ fig : localspec2 ] , [ fig : mass spectra ] and [ fig : mdot spectra ] ) . this , too , is due to the temperature increasing upward ( [ fig : structure]d , [ fig : structure]c ) . the larger opacity above the edge than below causes the lyman continuum photosphere to lie higher in the atmosphere than the photosphere at frequencies just below the edge . the higher temperature at the lyman continuum photosphere naturally leads to a greater emergent flux , i.e. an emission feature ( fig . [ fig : localspec2 ] ) . additional emission features appear at the hei and heii edges in some of the integrated spectra . the smearing of the edges is caused by doppler boosting of radiation emitted at small @xmath137 . the amplitude of the lyman edge feature increases with the central mass ( fig . [ fig : mass spectra ] ) because @xmath192 ( eq . [ eq : skin gas pressure ] ) . low pressure gas cools less efficiently because both the bremsstrahlung and photoionization opacities are proportional to @xmath193 . inefficient cooling leads to larger temperature inversions in the skin . the ratio @xmath194 remains roughly constant because the ratio of the opacities is independent of @xmath193 . therefore , @xmath195 increases with @xmath3 and the amplitude of the lyman edge feature increases . the amplitude of the edge feature is nearly independent of @xmath0 ( fig . [ fig : mdot spectra ] ) . we incorporated an iterative calculation of the variable eddington factors ( auer & mihalas 1970 ) into the @xmath196 , @xmath140 model . the solution for the eddington factors ( @xmath67 ) and the mean intensity is equivalent to the exact solution for the angular distribution of the spectral intensity . the angular distribution of the emergent flux can be determined by a formal solution of the transfer equation with the known source function . the surface eddington factors ( eq . [ eq : surface eddington factor ] ) can not be computed iteratively because we do not have a solution at @xmath109 . the accuracy of the analytic approximation for the eddington factors ( eq . [ eq : analytic fedd ] ) was tested by comparison with the exact solution . the radiation field is isotropic when @xmath197 and @xmath198 , in good agreement with the analytic approximation . however , when @xmath199 we find that @xmath200 . the approximate @xmath67 assumes that the radiation is collimated perpendicular to the accretion disk so that @xmath201 . fortunately , the errors introduced by our assumed @xmath67 are small . the approximate solution underestimates the true gas temperature by @xmath202 at @xmath203 and overestimates the temperature by @xmath118 at larger @xmath96 . the gas pressure is underestimated by @xmath118 at all @xmath96 . the differences in the uv spectrum depend on frequency and never exceed @xmath204 . the approximate solution overestimates the mean intensity below the lyman edge and above the heii edge and underestimates @xmath60 between these frequencies . as a consequence , the amplitude of the lyman edge feature is larger in the exact solution . the observed flux from an accretion disk around a massive black hole depends upon the disk inclination angle . first , as discussed in the previous paragraph , the emergent flux has an intrinsic angular distribution in the local rest frame ( see below ) . second , there are a number of relativistic effects acting upon photons emitted from material orbiting near the black hole . relativistic orbital motion boosts their frequencies and collimates their directions , while the strong gravitational field bends their trajectories and imposes a gravitational redshift on their frequencies as viewed at large distance ( cunningham 1975 ) . the boosting and beaming due to orbital motion tend to be stronger than the gravitational effects , so the net result is to strengthen the radiation emitted close to the equatorial plane from the innermost rings of the disk ( cunningham 1975 , laor , netzer & piran 1990 ) . these effects are stronger in rotating black holes than in non - rotating black holes because the disk extends closer to the event horizon in the former case ( laor , 1990 ) . for this reason , the impact of general relativistic effects is small in our ( non - rotating black hole ) models . we calculated the integrated spectrum of the @xmath205 , @xmath140 disk for three disk inclination angles . for each inclination , the disk is divided into several annuli ( [ subsubsec : face on disk ] ) and each annulus subdivided into several sectors . the radiative transfer equation is solved for the spectral intensity at the appropriate rest - frame inclination angle using the exact source function and the results are summed over all annuli and sectors . as before , doppler boosting , relativistic aberration and the gravitational redshift are included , but we neglect the general relativistic bending of the light rays by the central black hole . at low frequencies , the emergent intensity is limb - darkened ( fig . [ fig : limb disk ] ) . due to the low opacity below the lyman edge , the photosphere for these frequencies tends to lie relatively deep inside the atmosphere , where the temperature gradient is small or positive ( i.e. temperature decreases upward ) . consequently , the angular dependence is determined largely by the usual limb - darkening influence of scattering . however , at frequencies well above the lyman edge , the intensity is more nearly isotropic . several effects combine to explain this change . because of the greater opacity , the photosphere in this range lies higher in the atmosphere , where the temperature tends to increase outward . therefore , photons traveling well away from the axis can only depart from relatively high altitude , where the temperature is comparatively high . in addition , because these frequencies are radiated almost exclusively by the innermost rings of the disk , relativistic boosting and beaming are especially important . the decreasing degree of limb - darkening with increasing frequency causes the feature at the lyman edge to strengthen with disk inclination . we expect the relative limb - brightening of the ionizing continuum to increase with central mass and , if @xmath206 is approximately constant , disk luminosity . this is because temperature gradients increase with @xmath3 ( [ subsubsec : face on disk ] ) . doppler broadening of the lyman edge increases with disk inclination . the typical line of sight to an edge - on disk is nearly parallel to the disk rotation velocity so the doppler boost is maximized . the lyman edge is formed at @xmath207 , where @xmath208 , so the edge is spread over a range @xmath209 ( fig . [ fig : limb disk ] ) . the peak of the emergent flux shifts upward by the same amount . an edge - on disk will not appear to have an emission feature at the lyman edge , but the continuum slope becomes harder going from frequencies below the lyman edge to higher frequencies . this result is consistent with the conclusions of laor ( 1992 ) . the reprocessed uv radiation is inverse compton scattered by the relativistic electrons in the coronae . we calculate the comptonized uv spectrum for scattering by a relativistic maxwell - boltzmann distribution of electrons @xmath210 where @xmath211 , the coronal temperature is @xmath212 and @xmath213 is the second order modified bessel function . this distribution is normalized so that @xmath214 and we assume a thomson depth , @xmath8 , for the corona . we further assume that each photon has a probability @xmath215 of being scattered in the corona and that each photon is scattered at most one time . the spectrum of the scattered radiation is calculated using the formulae for isotropic electron and photon distributions ( rybicki & lightman 1979 ) . the small corrections due to anisotropy of the incident uv radiation field were neglected ( haardt 1993 ) . higher order scatterings determine the shape of the high energy spectrum ( pozdnyakov , sobol & sunyaev 1976 ) so we plot the results over a limited range of frequencies near the lyman edge ( figs . [ fig : compton t ] and [ fig : compton tau ] ) . the spectrum of the comptonized uv flux is determined primarily by the thomson depth of the corona , @xmath8 . inverse compton scattering in an optically thin corona has a negligible effect on the spectrum of the uv continuum because few photons are scattered ( fig . [ fig : compton t ] ) . comptonization in an optically thick corona ( @xmath7 ) smooths the spectrum at all frequencies ( fig . [ fig : compton tau ] ) , preferentially scattering photons from low to high energy . as a consequence , the spectrum of the non - ionizing flux becomes steeper and the ionizing continuum hardens to an approximate power - law . because the width of the lyman edge is dominated by relativistic boosting , inverse compton scattering in an optically thick corona may reduce its amplitude but does not significantly broaden it ( fig . [ fig : compton tau ] ) . the amplitude of the emission feature reaches a minimum when @xmath216 . the actual value of @xmath217 depends upon the parameters of the corona and the input uv spectrum but the range of values is easily understood . each input photon has a probability @xmath218 of being scattered in the corona and the scattered photons are redistributed over a frequency range @xmath219 . thus , very little redistribution occurs when @xmath220 and the scattered photons emerge at much higher energies when @xmath221 . the total optical depth of the corona is approximately the sum of the contributions from the soft x - ray corona ( discussed in [ subsec : vertical structure ] ) and the hard x - ray corona , where the gravitational potential energy is released . in [ subsec : vertical structure ] we demonstrated that a thermal runaway occurs for @xmath222 for @xmath223 and @xmath224 at larger radii . this range of column densities corresponds to @xmath225 . the lyman edge is formed at @xmath226 , where @xmath227 . the optical depth ( @xmath228 ) and geometry of the hard x - ray corona are controversial , but may be estimated through the use of thermal comptonization models . hm93 find @xmath229 assuming a uniform slab ; zdziarski ( 1995 ) found @xmath230 for the same geometry ; pietrini & krolik ( 1995 ) confirmed the zdziarski ( 1995 ) model , but also suggested that either clumping of the corona or a physical separation between it and the reprocessing surface would permit a match to the observed x - ray slope with larger @xmath228 . stern , ( 1995 ) prefer a model in which the corona is clumped , but close to the accretion disk , and has @xmath231 . if the corona is highly clumped , it can have little effect on the bulk of the uv photons . these modeling efforts indicate that a smooth corona must be rather optically thin , so it seems unlikely that the mean total compton depth overlying the rings producing the lyman edge can be any greater than a few tenths in most cases .

the incident x - ray flux heats a thin layer of material 3 - 4 scale heights above the midplane of the disk . the fraction of the incident hard x - ray flux which emerges in the uv falls with increasing ( the accretion rate in eddington units ) . at frequencies below the lyman edge

motivated by recent work indicating that the uv continuum in agn may be produced by reradiation of energy absorbed from x - rays irradiating an accretion disk , we present a calculation of the vertical structures and ultraviolet spectra of x - ray irradiated accretion disks around massive non - rotating black holes . after finding the radial dependence of vertically - integrated quantities for these disks , we solve the equations of hydrostatic equilibrium , energy balance , and frequency - dependent radiation transfer as functions of altitude . to solve the last set of equations , we use a variable eddington factor method . we include electron scattering , free - free , and hi , hei , and heii bound - free opacities and the corresponding continuum cooling processes . the incident x - ray flux heats a thin layer of material 3 - 4 scale heights above the midplane of the disk . this x - ray heated skin has two layers : a radiation pressure supported region in which the uv flux is created , and , immediately above this layer , a warmer zone , optically thin to uv radiation , formed where the x - ray ionization parameter is large . in the lower layer the gas pressure is nearly independent of altitude but the temperature increases upward . the fraction of the incident hard x - ray flux which emerges in the uv falls with increasing ( the accretion rate in eddington units ) . at frequencies below the lyman edge the slope of the continuum ( ) varies from -1.6 to 0.8 as increases from 0.001 to 1 . here is the mass of the central black hole in units of . in all cases examined ( and ) , the lyman edge appears in emission . the amplitude of the lyman edge feature increases with but is relatively independent of . the amplitude of the lyman edge emission feature increases with disk inclination . compton scattering in disk coronae can smooth the lyman edge feature only if , where is the thomson depth of the coronae . while the overall spectral shape predicted by x - ray irradiation may be compatible with observations , the lyman edge emission feature it predicts is not . this finding raises questions for many otherwise plausible models in which x - ray irradiation plays a major role . # 1#2#3#4#4 19#3 phys . rev . d , # 1 , # 2 # 1#2#3#4#4 19#3 phys . lett . , # 1 , # 2 # 1#2#3#4#4 19#3 phys . rev . lett . , # 1 , # 2 # 1#2#3#4#4 19#3 phys . rev . , # 1 , # 2 # 1#2#3#4#4 19#3 phys . rep . , # 1 , # 2 # 1#2#3#4#4 19#3 phys . fluids , # 1 , # 2 # 1#2#3#4#4 19#3 proc . phys . soc . , # 1 , # 2 # 1#2#3#4#4 19#3 nucl . phys . , # 1 , # 2 # 1#2#3#4#4 19#3 mod . phys . lett . , # 1 , # 2 # 1#2#3#4#4 19#3 ap . j. , # 1 , # 2 # 1#2#3#4#4 19#3 astr . j. , # 1 , # 2 # 1#2#3#4#4 19#3 acta astr . , # 1 , # 2 # 1#2#3#4#4 19#3 rev . mod . phys . , # 1 , # 2 # 1#2#3#4#4 19#3 nuovo cimento c , # 1 , # 2 # 1#2#3#4#4 19#3 sov . phys . jetp , # 1 , # 2 # 1#2#3#4#4 19#3 sov . ast . aj , # 1 , # 2 # 1#2#3#4#4 19#3 pub . ast . soc . japan , # 1 , # 2 # 1#2#3#4#4 19#3 pub . ast . soc . pacific , # 1 , # 2 # 1#2#3#4#4 19#3 ann . phys . ( ny ) , # 1 , # 2 # 1#2#3#4#4 19#3 yad . fiz . , # 1 , # 2 # 1#2#3#4#4 19#3 sov . j. nucl . phys . , # 1 , # 2 # 1#2#3#4#4 19#3 ast . ap . , # 1 , # 2 # 1#2#3#4#4 19#3 ann . rev . astr . ap . , # 1 , # 2 # 1#2#3#4#4 19#3 m.n.r.a.s . , # 1 , # 2 # 1#2#3#4#4 19#3 j. de physique , # 1,#2 # 1#2#3#4#4 19#3 j. quant . spec . rad . transfer , # 1,#2 # 1#2#3#4#4 19#3 j.e.t.p . lett . , # 1,#2 # 1#2#3#4#4 19#3 ap . j. ( letters ) . , # 1,#2 # 1#2#3#4#4 19#3 ap . j. ( supp . ) . , # 1,#2 # 1#2#3#4#4 19#3 ap . lett . , # 1,#2 # 1#2#3#4#4 19#3 ap . sp . sci . , # 1,#2 # 1#2#3#4#4 19#3 nature , # 1,#2 # 1#2#3#4#4 19#3 sp . sci . rev . , # 1,#2 # 1#2#3#4#4 19#3 adv . sp . res . , # 1,#2

astro-ph9609094

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we have calculated the vertical structure and uv spectrum of an accretion disk irradiated by hard x - rays from a corona , as predicted by many models for the x - ray emission in agn ( pounds 1990 ; hm93 ) . consistent with these x - ray production models , we assume there is no viscous dissipation in the accretion disk . the gravitational potential energy of the accreting gas is dissipated in a corona and radiated as hard x - rays with a power law spectrum . photoionization of oxygen and iron by the coronal x - rays heat a geometrically thin skin at the surface of the accretion disk . because there is no dissipation inside the disk proper , its temperature is constant as a function of altitude and the gas density and optical depth are much larger than in a conventional dissipative disk ( sz95 ) . to find the structure of the x - ray heated skin , we have solved the differential equations of radiative transfer , hydrostatic equilibrium and surface mass density subject to the constraints of radiative equilibrium and charge conservation . the inner boundary conditions for this numerical solution were chosen to match an analytical solution for the underlying isothermal accretion disk . we tested our method by computing the structure and spectrum of a conventional accretion disk and comparing to the previous state - of - the - art calculation ( ln89 , sincell & krolik 1996 ) . the inner edge of the x - ray heated skin is 3 - 4 scale heights from the midplane of the isothermal disk . in the skin , but not the body of the disk , the force of the uv flux supports the gas against gravity and the momentum of the absorbed x - rays . the gas pressure in the skin is orders of magnitude smaller than the pressure at the midplane of the disk and is nearly constant throughout the skin . the gas temperature typically increases from the inner edge of the skin to the surface , although the temperature gradient occasionally changes sign . the changes in the temperature gradient correspond to changes in the ionization state and opacity of the gas . the gas above the column density @xmath108 is unable to cool because cooling by thermal radiation processes is no longer able to balance heating by x - ray photoionization . this transition takes place at a characteristic value of the ionization parameter @xmath257 . compton cooling was not included in this model so we could not find a solution for @xmath109 . however , we estimate the gas temperature in the soft x - ray corona to be @xmath258 . we find that @xmath108 increases with disk radius , because the cooler gas at large radii radiates less efficiently , and the average value of @xmath108 for a given disk increases with the accretion rate , because @xmath147 at the surface of the heated skin . for our specific x - ray spectrum and set of approximations , the fraction of the incident hard x - ray flux which is reprocessed into uv photons ranges from @xmath259 at @xmath151 to 0.62 when @xmath141 . the exact value of @xmath143 depends on the shape of the x - ray spectrum , and our method ( see 4.1 ) can be expected to slightly underestimate @xmath143 . nevertheless , the decrease in @xmath143 with accretion rate is a direct consequence of our assumption that all the gravitational potential energy is dissipated in the corona ( see eq . [ eq : skin gas pressure ] ) and should be a general property of irradiated disks . the reprocessed uv continuum depends significantly on the specific parameters of the model ( accretion rate , central mass ) , but certain features are generic : it has strong spectral curvature , in the sense that the spectrum softens with increasing frequency ; and there is a significant lyman edge emission feature . the low frequency continuum ( parameterized by the spectral index at @xmath260 ) hardens with increasing x - ray flux ( @xmath2 ) and ranges from -1.6 when @xmath261 to 0.8 when @xmath190 . the continuum slope in this frequency range is only weakly dependent on viewing angle . the lyman edge is in emission because the photosphere is higher in the atmosphere at frequencies above the edge , and the temperature increases upward . additional emission features at the hei and heii edges may also occur in the spectra of the hotter disks . the amplitude of the lyman edge feature increases with central mass and is insensitive to the accretion rate . this is because both the photoionization and bremsstrahlung opacities are proportional to @xmath192 , so the gas cools less efficiently as @xmath3 increases . this results in larger temperature inversions which increase the strength of the emission feature . doppler boosting spreads the lyman edge over a range @xmath262 . the amplitude of the lyman edge increases with disk inclination because the non - ionizing continuum is limb - darkened and the ionizing flux is ( relatively ) limb - brightened . irradiated disks are self - gravitating , and unstable , unless there is a small amount of internal dissipation . we find that the strictly isothermal disk is gravitationally unstable at @xmath263 , so that strictly isothermal disks with luminosities @xmath264 ergs s@xmath166 are unstable at all radii . if , as is likely , there is a small amount of internal dissipation , and this stabilizes the disk , we expect @xmath143 to decrease , and the amplitude of the lyman edge feature to increase relative to what our models show for the limiting case of @xmath265 . these effects are likely to become stronger for larger luminosity and central mass . some , but not all , of these predictions are consistent with observations . the average optical - uv power - law index for quasars is @xmath266 ( laor 1990 ) but a single power law is a poor description of the continuum spectrum ( francis , 1991 ) . the continuum of the composite quasar spectrum , formed by combining all the quasars in the large bright quasar survey , softens with increasing frequency ( francis , 1991 ) . both the mean optical - uv slope and the spectral curvature are consistent with our predicted uv spectrum of an irradiated disk if @xmath267 . on the other hand , our finding that a lyman edge emission feature is a general property of x - ray - irradiated disks is difficult to reconcile with recent observations of agn . lyman edge features are rarely seen , and then usually as partial absorption edges ( antonucci , kinney & ford 1989 , koratkar , kinney & bohlin 1992 ) . emission features at the lyman edge have been detected only in polarized light , and in these two quasars the total flux spectrum shows the lyman edge in absorption ( koratkar , 1995 ) . the polarization of this emission is quite high and detailed polarization dependent calculations ( blaes & agol 1996 ) will be required to determine if these features could be produced by an irradiated disk . internal dissipation might , in some cases , reduce the amplitude of the lyman edge feature . conventional disks ( sincell & krolik 1996 ) sometimes produce emission features at the lyman edge , and sometimes absorption , depending primarily on @xmath0 . however , both the fraction of the total gravitational energy dissipated in the disk and @xmath0 must be very finely tuned to eliminate lyman edge features from all agn . inverse compton scattering in the disk coronae can smooth out the lyman edge feature only if the hot corona producing the hard x - rays covers most of the disk and has an optical depth @xmath7 . because most recent treatments of x - ray production suggest that either the optical depth is rather smaller than this , or else the hot corona is highly clumped , it is unlikely that compton scattering smooths the lyman edge . the only escape we see from this contradiction makes use of the fact that the evidence for x - ray irradiation comes almost entirely from low luminosity agn ( seyfert galaxies ) , while spectra of the lyman edge region exist only for high luminosity agn ( quasars ) . the reason for this separation is that only the seyfert galaxies , which are relatively nearby , are bright enough for detailed x - ray spectroscopy , while in order for us to see the lyman edge , we must look at objects with redshifts great enough to bring it below the lyman cut - off imposed by our galaxy s interstellar medium . perhaps , then , seyfert galaxies have lyman edges in emission ( if we could only see them ) , while quasar disks are not so thoroughly irradiated by x - rays . x - ray spectroscopy of quasars by future , more sensitive instruments , will be able to test this idea . thus , our calculation of the uv spectrum expected from irradiated disks raises serious questions for accretion disk models . while the x - ray production and x - ray reflection models which motivated this study have had substantial success in describing agn x - ray spectra , we now see that they also predict strong lyman edge emission features , a prediction that is contradicted by the simplest interpretation of the observations . we would like to thank ari laor for helpful conversations and for providing many disk spectra . we also thank omer blaes and eric agol for ongoing conversations . mws also thanks the observatoire de meudon for hospitality during part of this work . mws received support for this work from nasa grants nagw-3129 , 1583 and nag 5 - 2925 , and nsf grant ast 93 - 15133 . jhk was partially supported by nasa grant nagw-3156 . blaes , o. m. & e. agol , 1996 , submitted to ap . j. letters ( hm93 ) ( ln89 ) laor , a. 1992 , in testing the agn paradigm : aip conference proceedings 254 , american institute of physics , new york martocchia , a. and matt , g. 1996 , m.n.r.a.s . in press mihalas , d. m. , 1978 , stellar atmospheres , w. h. freeman & co. , san francisco nandra , k. 1997 , in iau coll . 159 , emission lines in active galactic nuclei : new methods and techniques , eds . b. peterson , f .- z . cheng , and a.s . wilson novikov , i. & k. s. thorne , 1973 in black holes , p. 422 , eds : dewitt , c. & b. dewitt , gordon & breach , new york pounds , k. a. & t. j. turner , 1988 proceedings of the iau symposium 134 active galactic nuclei " , eds . j. s. miller & d. e. osterbrock , kluwer , dordrecht press , w. h. , s. a. teukolsky , w. t. vetterling & b. p. flannery , 1992 , numerical recipes in c , cambridge university press rybicki , g. b. & a. p. lightman , 1979 , radiative processes in astrophysics , john wiley & sons , inc . , new york ( ss73 ) sincell , m. w. & j. h. krolik , 1996 , in preparation ( sz95 ) zheng , w. , kriss , g.a . , telfer , r.c . , grimes , j.p . , and davidsen , a.f . 1996 , ap.j . in press

motivated by recent work indicating that the uv continuum in agn may be produced by reradiation of energy absorbed from x - rays irradiating an accretion disk , we present a calculation of the vertical structures and ultraviolet spectra of x - ray irradiated accretion disks around massive non - rotating black holes . after finding the radial dependence of vertically - integrated quantities for these disks , the slope of the continuum ( ) varies from -1.6 to 0.8 as increases from 0.001 to 1 . here , the lyman edge appears in emission . the amplitude of the lyman edge feature increases with but is relatively independent of . the amplitude of the lyman edge emission feature increases with disk inclination . compton scattering in disk coronae can smooth the lyman edge feature only if , where is the thomson depth of the coronae . while the overall spectral shape predicted by x - ray irradiation may be compatible with observations , the lyman edge emission feature it predicts is not . ast . ast . j. nucl . ap . , # 1 , # 2 # 1#2#3#4#4 19#3 m.n.r.a.s . , # 1 , # 2 # 1#2#3#4#4 19#3 j. de physique , # 1,#2 # 1#2#3#4#4 19#3 j. quant . spec . transfer , # 1,#2 # 1#2#3#4#4 19#3 j.e.t.p . j. ( letters ) .

motivated by recent work indicating that the uv continuum in agn may be produced by reradiation of energy absorbed from x - rays irradiating an accretion disk , we present a calculation of the vertical structures and ultraviolet spectra of x - ray irradiated accretion disks around massive non - rotating black holes . after finding the radial dependence of vertically - integrated quantities for these disks , we solve the equations of hydrostatic equilibrium , energy balance , and frequency - dependent radiation transfer as functions of altitude . to solve the last set of equations , we use a variable eddington factor method . we include electron scattering , free - free , and hi , hei , and heii bound - free opacities and the corresponding continuum cooling processes . the incident x - ray flux heats a thin layer of material 3 - 4 scale heights above the midplane of the disk . this x - ray heated skin has two layers : a radiation pressure supported region in which the uv flux is created , and , immediately above this layer , a warmer zone , optically thin to uv radiation , formed where the x - ray ionization parameter is large . in the lower layer the gas pressure is nearly independent of altitude but the temperature increases upward . the fraction of the incident hard x - ray flux which emerges in the uv falls with increasing ( the accretion rate in eddington units ) . at frequencies below the lyman edge the slope of the continuum ( ) varies from -1.6 to 0.8 as increases from 0.001 to 1 . here is the mass of the central black hole in units of . in all cases examined ( and ) , the lyman edge appears in emission . the amplitude of the lyman edge feature increases with but is relatively independent of . the amplitude of the lyman edge emission feature increases with disk inclination . compton scattering in disk coronae can smooth the lyman edge feature only if , where is the thomson depth of the coronae . while the overall spectral shape predicted by x - ray irradiation may be compatible with observations , the lyman edge emission feature it predicts is not . this finding raises questions for many otherwise plausible models in which x - ray irradiation plays a major role . # 1#2#3#4#4 19#3 phys . rev . d , # 1 , # 2 # 1#2#3#4#4 19#3 phys . lett . , # 1 , # 2 # 1#2#3#4#4 19#3 phys . rev . lett . , # 1 , # 2 # 1#2#3#4#4 19#3 phys . rev . , # 1 , # 2 # 1#2#3#4#4 19#3 phys . rep . , # 1 , # 2 # 1#2#3#4#4 19#3 phys . fluids , # 1 , # 2 # 1#2#3#4#4 19#3 proc . phys . soc . , # 1 , # 2 # 1#2#3#4#4 19#3 nucl . phys . , # 1 , # 2 # 1#2#3#4#4 19#3 mod . phys . lett . , # 1 , # 2 # 1#2#3#4#4 19#3 ap . j. , # 1 , # 2 # 1#2#3#4#4 19#3 astr . j. , # 1 , # 2 # 1#2#3#4#4 19#3 acta astr . , # 1 , # 2 # 1#2#3#4#4 19#3 rev . mod . phys . , # 1 , # 2 # 1#2#3#4#4 19#3 nuovo cimento c , # 1 , # 2 # 1#2#3#4#4 19#3 sov . phys . jetp , # 1 , # 2 # 1#2#3#4#4 19#3 sov . ast . aj , # 1 , # 2 # 1#2#3#4#4 19#3 pub . ast . soc . japan , # 1 , # 2 # 1#2#3#4#4 19#3 pub . ast . soc . pacific , # 1 , # 2 # 1#2#3#4#4 19#3 ann . phys . ( ny ) , # 1 , # 2 # 1#2#3#4#4 19#3 yad . fiz . , # 1 , # 2 # 1#2#3#4#4 19#3 sov . j. nucl . phys . , # 1 , # 2 # 1#2#3#4#4 19#3 ast . ap . , # 1 , # 2 # 1#2#3#4#4 19#3 ann . rev . astr . ap . , # 1 , # 2 # 1#2#3#4#4 19#3 m.n.r.a.s . , # 1 , # 2 # 1#2#3#4#4 19#3 j. de physique , # 1,#2 # 1#2#3#4#4 19#3 j. quant . spec . rad . transfer , # 1,#2 # 1#2#3#4#4 19#3 j.e.t.p . lett . , # 1,#2 # 1#2#3#4#4 19#3 ap . j. ( letters ) . , # 1,#2 # 1#2#3#4#4 19#3 ap . j. ( supp . ) . , # 1,#2 # 1#2#3#4#4 19#3 ap . lett . , # 1,#2 # 1#2#3#4#4 19#3 ap . sp . sci . , # 1,#2 # 1#2#3#4#4 19#3 nature , # 1,#2 # 1#2#3#4#4 19#3 sp . sci . rev . , # 1,#2 # 1#2#3#4#4 19#3 adv . sp . res . , # 1,#2

1310.4677

i

electromagnetic energy transfer between dielectric bodies at different temperatures is commonly described using the fluctuational electrodynamics ( fed ) approach @xcite developed by rytov @xcite and first applied to condensed matter physics by lifshitz @xcite . according to fed , thermal motion of charged particles in a body creates random currents , which induce electromagnetic fields . outside the body , the field is then either radiated to free space or absorbed in the near or far - field regime by another body . proximity effects involving the evanescent waves in the near - field were first observed by hargreaves @xcite , and fed was consequently applied to theoretically predict strong near - field enhancement of heat transfer in various geometries @xcite . the predictions have been explored in more detail also experimentally @xcite . near - field effects are expected to have numerous applications in , e.g. , thermal microscopy @xcite , infrared thermophotovoltaics @xcite and narrow - band infrared antennas @xcite . as a statistical model , fed is closely related to langevin dynamics commonly applied to describe the random thermal motion of non - charged bodies @xcite . in langevin dynamics the particle is assumed to be coupled to a bath of harmonic oscillators , whose effect on the particle can effectively be described by a random force and deterministic friction @xcite . huttner and barnett @xcite essentially applied langevin dynamics to study the quantization of the electromagnetic field in absorbing dielectrics by coupling the polarization field to a bath of harmonic oscillators . the explicit inclusion of the microscopic degrees of freedom responsible for absorption solves the problem of temporally decaying field commutators arising if one blindly applies the standard electromagnetic field quantization methods to dielectrics . relation of the langevin dynamics to fed was recently highlighted by rosa , dalvit and milonni @xcite , who used langevin dynamics to derive the fluctuation - dissipation theorem ( fdt ) @xcite for the fluctuating polarization field from the microscopic motion of the oscillating dipoles . the goal of this paper is to show that the microscopic dipole oscillator model combined with quantum langevin dynamics can be used to transparently and microscopically treat also the problem of electromagnetic heat transfer between dielectric bodies held at different temperatures . the primary motivation for studying heat transfer using the quantum langevin equation instead of fed is that we can derive heat transfer rates in full analogy with phononic @xcite and electronic @xcite heat transfer and , by following the mathematical manipulations presented in ref . @xcite , we are able to arrive at an identical landauer - bttiker - like formula for the energy transmission function . the theory presented here enables , therefore , a unification of phononic , electronic and photonic heat transfer under the common langevin theory . when written in terms of particle polarizabilities and the electromagnetic green s function , the transmission function reduces to the form derived earlier from fed @xcite . in contrast to these works , we ( i ) consistently include the electromagnetic self - interaction produced by the local electromagnetic green s dyadic by following the discrete dipole approximation @xcite , ( ii ) include the inhomogeneous environment enabling , e.g. , accounting for cavity resonance effects , and ( iii ) present an alternative and conceptually simple way to derive electromagnetic energy transfer rates starting from the microscopic equations of motion . for presentational simplicity , we initially assume the particles to be small enough for the dipole approximation to hold . however , overcoming this assumption by following the well - established discrete dipole approximation mentioned above is also discussed . as an application of the formalism , we study the enhancement of heat transfer rates between sic particles placed in a microcavity and close to a polar surface supporting surface phonon polaritons ( spps ) . in a microcavity , the heat transfer rate between particles is shown to oscillate as a function of their distance and the cavity enhancement can be several orders of magnitude as compared to the free space heat current with a similar interparticle distance . enhanced heat current is predicted also for particles close to a sic surface , where the spps transport electromagnetic energy between the particles . the paper is organized as follows . in sec . [ sec : theory ] , we present the formulation to calculate thermal energy transfer between dielectric particles and the environment . we ( i ) represent the polarization fields inside the particles by their total dipole moments , ( ii ) solve the quantum langevin equations of motion for coupled dipole moment dynamics in terms of the dipole displacement green s function , ( iii ) calculate the thermal averages of heat currents using the fluctuation - dissipation theorem for the bath noises and the thermal field of the environment , and ( iv ) express the heat currents in terms of particle polarizabilities and the electromagnetic green s dyadic . in secs . [ sec : results_cavity ] and [ sec : results_surface ] , we investigate heat transfer between sic nanoparticles in a microcavity and above a sic surface , respectively .

the theory is , in a sense , the microscopic generalization of the well - known fluctuational electrodynamics theory and thereby provides an alternative and conceptually simple way to calculate the local emission and absorption rates from the local langevin bath currents . the results show that the heat current between the dipoles placed in a cavity oscillates as a function of their position and distance and can be enhanced by several orders of magnitude as compared to the free space heat current with a similar interparticle distance . similar effects are also observed in the interparticle heat transfer between dipoles located next to a surface of a polar material supporting surface phonon polaritons .

near - field and resonance effects have a strong influence on the nanoscale electromagnetic energy transfer , and detailed understanding of these effects is required for the design of new , optimized nano - optical devices . we provide a comprehensive microscopic view of electromagnetic energy transfer phenomena by introducing quantum langevin heat baths as local noise sources in the equations of motion for the thermally fluctuating electric dipoles forming dielectric bodies . the theory is , in a sense , the microscopic generalization of the well - known fluctuational electrodynamics theory and thereby provides an alternative and conceptually simple way to calculate the local emission and absorption rates from the local langevin bath currents . we apply the model to study energy transfer between silicon carbide nanoparticles located in a microcavity formed of two mirrors and next to a surface supporting propagating surface modes . the results show that the heat current between the dipoles placed in a cavity oscillates as a function of their position and distance and can be enhanced by several orders of magnitude as compared to the free space heat current with a similar interparticle distance . the predicted enhancement can be viewed as a many - body generalization of the well - known cavity purcell effect . similar effects are also observed in the interparticle heat transfer between dipoles located next to a surface of a polar material supporting surface phonon polaritons .

0905.4094

i

in this article , we have analyzed new ccd observations of the eclipsing binary system ar boo obtained during three successive seasons beginning in 2006 february . the light curves show partial eclipses and season - to - season light variability . our light - curve solution indicates that ar boo is a w - subtype contact binary composed of a hotter g9 primary star and a cooler k1 companion , the derived spectral types depend , of course , on the adopted temperature for the secondary component and the observed ( @xmath43 ) color index . the disturbed light curves of the system were best modeled by using a two - spot model with cool and hot spots on the secondary star . we speculate that both spots may be produced by magnetic dynamo - related activity . according to the thermal relaxation oscillation ( tro ) theory ( lucy 1976 ; lucy & wilson 1979 ) , contact binaries must oscillate cyclically between contact and non - contact conditions . because our detailed study of period and light variations show that the orbital period is increasing and that mass is transferring from the less massive primary to the more massive secondary star , ar boo may presently be in a transition state evolving from a contact to a non - contact configuration as suggested by this theory . our new orbital period study with all available timings reveals that the @xmath21@xmath22 residuals have varied in a cyclical oscillation superposed on a secular period increase . the semi - amplitude and period of the cyclical variation are low and short , respectively , and their values command modest confidence because the observed ccd timings cover only about 1.5 cycle of this period . the oscillation may be produced , in principle , either by the ltt effect due to a stellar- or substellar - mass companion of @xmath0=0.081 @xmath1 or by a active magnetic cycle in the more massive secondary component but not by the asymmetries of eclipse light curves due to starspot activity . the circumstance that there has existed even a limited association between phase in the 7.57-year cycle and the spot activity seen in the light curves diminishes confidence in the postulated 3rd body while suggesting that a more searching attempt be made to understand magnetic activity mediating angular momentum exchange between the spin of the stellar mass distribution and the orbit . there s a further message to be drawn here . three of the authors had never even heard of this binary before the loao observations were in train . it is a very nondescript object , yet the assiduous accumulation of the best possible observations has given weight to two ideas : starspots measurably inflect eclipse timings and magnetic cycles in stellar components of close binaries express themselves dynamically . every binary has its individual information encoded for us but , a priori , we have no idea of the richness of that information and we must uncover it and piece it together with information from all the others that we observe . in a perfect world there would be seasonal accumulation of light curves ( and radial velocities ) of ar boo over the next 8 years and then surely understanding of this system and its activity would be advanced . it is even possible that the declination of the binary permits accumulation of two independent light curves per observing season . a justification for such a strenuous effort may be found in looking back at the 2007 and 2008 light curves . whereas those of the earlier year were compiled over the 6 nights from march 11 through march 17 , the data for 2008 were observed over 17 nights from december 23 , 2007 through april 12 , 2008 . we wish to thank professor chun - hwey kim for his help in using the @xmath21@xmath22 database of eclipsing binaries . we also thank the staff of mt . lemmon optical astronomy observatory for assistance with our observations . this research has made use of the simbad database maintained at cds , strasbourg , france . applegate , j. h. 1992 , apj , 385 , 621 baki , v. , dogru , s. s. , baki , h. , dogru , d. , erdem , a. , cick , c. , & demircan , o. 2005 , inf . variable stars , no . 5662 binnendijk , l. 1970 , vistas in astronomy , 12 , 217 blttler , e. 1998 , bbsag bull . 117 blttler , e. 2002 , bbsag bull . 127 brt , l. , zejda , m. , & svoboda , p. 2007 , open european journal on variable stars , 74 , 1 diethelm , r. 1996 , bbsag bull . 112 diethelm , r. 1997 , bbsag bull . 115 diethelm , r. 1998 , bbsag bull . 117 diethelm , r. 2001 , inf . bull . variable stars , no . 5027 diethelm , r. 2003 , inf . bull . variable stars , no . 5438 diethelm , r. 2004 , inf . variable stars , no . 5543 diethelm , r. 2005 , inf . variable stars , no . 5653 diethelm , r. 2006 , inf . bull . variable stars , no . 5713 flower , p. j. 1996 , apj , 469 , 355 han , w. , et al . 2005 , pasj , 57 , 821 harmanec , p. 1988 czechoslovakia , 39 , 329 hilditch , r. w. , king , d. j. , & mcfarlane , t. m. 1988 , mnras , 231 , 341 houck , j. c. , & pollock , j. t. 1986 , pasp , 98 , 461 hbscher , j. , paschke , a. , & walter , f. 2005 , inf . variable stars , no . 5657 hbscher , j. , paschke , a. , & walter , f. 2006 , inf . bull . variable stars , no . 5731 hbscher , j. , steinbach , h .- m . , & walter , f. 2009 , inf . variable stars , no . 5874 irwin , j. b. 1952 , apj , 116 , 211 irwin , j. b. 1959 , aj , 64 , 149 kang , y .- w . , hong , k .- s . , & lee , j. 2007 , in asp conf . 362 , the seventh pacific rim conference on stellar astrophysics , ed . kang et al . ( san francisco : asp ) , 19 kim , c .- h . , jeong , j. h. , demircan , o. , myesserolu , z. , & budding , e. 1997 , aj , 114 , 2753 kim , c .- h . , lee , j. w. , kim , h .- , kyung , j .- m . , & koch , r. h. 2003 , aj , 126 , 1555 krajci , t. 2005 , inf . bull . variable stars , no . 5592 kreiner , j. m. , kim , c .- h . , & nha , i .- s . 2001 , an atlas of @xmath21@xmath22 diagrams of eclipsing binary stars ( krakow : wydawn . pedagogicznej ) kurochkin , n. e. 1960 , perem . zvezdy , 13 , 84 kwee , k. k. , & van woerden , h. 1956 , bull . netherlands , 12 , 327 lanza , a. f. , rodono , m. , & rosner , r. 1998 , mnras , 296 , 893 lee , j. w. , kim , c .- h . , han , w. , kim , h .- i . , & koch , r. h. 2004 , mnras , 352 , 1041 lee , j. w. , kim , h .- i . , & kim , s .- l . 2007 , pasp , 119 , 1099 lee , j. w. , youn , j .- h . , kim , c .- h . , lee , c .- u . , & kim , h .- i . 2008 , aj , 135 , 1523 lucy , l. b. 1976 , apj , 205 , 208 lucy , l. b. , & wilson , r. e. 1979 , apj , 231 , 502 maceroni , c. , & vant veer , f. 1994 , a&a , 289 , 871 nelson , r. h. 2007 , inf . bull . variable stars , no . 5760 nelson , r. h. 2009 , inf . bull . variable stars , no . 5875 press , w. h. , teukolsky , s. a. , vetterling , w. t. , & flannery , b. p. 1992 , numerical recipes ( cambridge : cambridge univ . press ) , chapter 15 pribulla , t. , & rucinski , s. m. 2006 , aj , 131 , 2986 qian , s .- b . , yuan , j .- z . , soonthornthum , b. , zhu , l .- y . , he , j .- j . , & yang , y .- g . 2007 , apj , 671 , 811 richards , m. t. , & albright , g. e. 1999 , apjs , 123 , 537 af , j. , & zejda , m. 2000 , inf . bull . variable stars , no . 4887 af , j. , & zejda , m. 2002 , inf . bull . variable stars , no . 5263 samec , r. g. , lofling , t. s. , & van hamme , w. 2006 , inf . bull . variable stars , no . 5696 schlegel , d. j. , finkbeiner , d. p. , & davis , m. 1998 , apj , 500 , 525 van hamme , w. 1993 , aj , 106 , 2096 wilson , r. e. , & devinney , e. j. 1971 , apj , 166 , 605 wolf , m. , borovika , j. , arounov , l. , af , j. , & afov , e. 1998 , inf . bull . variable stars , no . 4601 yang , y .- g . , dai , j .- m . , yin , x .- , & xiang , f .- y . 2007 , aj , 134 , 179 yang , y .- g . , lu , g .- zhu , c .- h . , & nakajima , k. 2009 , aj , 137 , 236 zejda , m. 2002 , inf . variable stars , no . 5287 zejda , m. 2004 , inf . variable stars , no . 5583 , @xmath2 , and @xmath11 bandpasses . the differences between the two seasons are shown in the middle panel and the magnitude difference between the check and comparison stars in the lower panel . the open circle and plus symbols in the upper and lower panels are the individual measures of the 2007 and 2008 seasons , respectively . the dotted lines in the middle panel refer to values of 0.0 mag . ] @xmath22 diagram of ar boo . in the upper panel the continuous curve represents the final improved ephemeris . cc , pg , vi , and p stand for ccd , photographic , visual , and photographic plate minima , respectively . the middle and lower panels display the ccd residuals from the linear and quadratic terms and from the complete ephemeris , respectively . the 7.57-year cycle is shown propagated back in time only to indicate that it could not have been recognized in the noisy earlier minimum timings . in the two lower panels , plus symbols represent the minimum times obtained by re - analyzing individual eclipse curves with the wd code . ] ( the sum of the residuals squared ) of ar boo as a function of mass ratio @xmath49 . the filled and open circles represent the @xmath49-search results for the 2007 and 2008 data sets , respectively . the arrows indicate minimum values for each data set . ] cccccc 2,454,171.79150 & @xmath660.919 & 2,454,171.79340 & @xmath661.063 & 2,454,171.79482 & @xmath661.173 + 2,454,171.79648 & @xmath660.866 & 2,454,171.79842 & @xmath661.026 & 2,454,171.79984 & @xmath661.118 + 2,454,171.80151 & @xmath660.817 & 2,454,171.80350 & @xmath660.972 & 2,454,171.80501 & @xmath661.059 + 2,454,171.80671 & @xmath660.746 & 2,454,171.80872 & @xmath660.912 & 2,454,171.81020 & @xmath661.000 + 2,454,171.81192 & @xmath660.684 & 2,454,171.81392 & @xmath660.841 & 2,454,171.81541 & @xmath660.938 + 2,454,171.81712 & @xmath660.624 & 2,454,171.81913 & @xmath660.782 & 2,454,171.82061 & @xmath660.899 + 2,454,171.82232 & @xmath660.578 & 2,454,171.82432 & @xmath660.763 & 2,454,171.82582 & @xmath660.879 + 2,454,171.82754 & @xmath660.567 & 2,454,171.82956 & @xmath660.758 & 2,454,171.83105 & @xmath660.893 + 2,454,171.83276 & @xmath660.594 & 2,454,171.83476 & @xmath660.792 & 2,454,171.83626 & @xmath660.926 + 2,454,171.83797 & @xmath660.646 & 2,454,171.83998 & @xmath660.846 & 2,454,171.84147 & @xmath660.988 + lrrrcl 0,180.4108 & @xmath120.0028 & -6.0 & 0.00131 & i & safr & zejda ( 2000 ) + 0,182.4791 & @xmath120.0002 & 0.0 & 0.00036 & i & wolf et al . ( 1998 ) + 0,192.4789 & @xmath120.0030 & 29.0 & -0.00119 & i & diethelm ( 1996 ) + 0,200.4128 & @xmath120.0002 & 52.0 & 0.00060 & i & wolf et al . ( 1998 ) + 0,200.5850 & @xmath120.0005 & 52.5 & 0.00037 & ii & wolf et al . ( 1998 ) + 0,547.3555 & @xmath120.0002 & 1058.0 & -0.00015 & i & wolf et al . ( 1998 ) + 0,551.4936 & @xmath120.0018 & 1070.0 & -0.00054 & i & diethelm ( 1997 ) + 0,607.3635 & @xmath120.0004 & 1232.0 & -0.00025 & i & wolf et al . ( 1998 ) + 0,607.5346 & @xmath120.0003 & 1232.5 & -0.00159 & ii & wolf et al . ( 1998 ) + 0,611.5025 & @xmath120.0005 & 1244.0 & 0.00026 & i & wolf et al . ( 1998 ) + 0,638.4028 & @xmath120.0002 & 1322.0 & 0.00038 & i & wolf et al . ( 1998 ) + 0,888.4365 & @xmath120.0002 & 2047.0 & 0.00037 & i & wolf et al . ( 1998 ) + 0,899.4721 & @xmath120.0003 & 2079.0 & 0.00000 & i & wolf et al . ( 1998 ) + 0,923.4410 & @xmath120.0017 & 2148.5 & 0.00015 & ii & diethelm ( 1998 ) + 0,925.5099 & @xmath120.0014 & 2154.5 & -0.00019 & ii & diethelm ( 1998 ) + 0,926.3730 & @xmath120.0006 & 2157.0 & 0.00072 & i & blttler ( 1998 ) + 0,927.4073 & @xmath120.0003 & 2160.0 & 0.00040 & i & wolf et al . ( 1998 ) + 1,270.5551 & @xmath120.0021 & 3155.0 & -0.00137 & i & safr & zejda ( 2002 ) + 1,277.4539 & @xmath120.0025 & 3175.0 & -0.00005 & i & safr & zejda ( 2002 ) + 1,284.3511 & @xmath120.0015 & 3195.0 & -0.00033 & i & safr & zejda ( 2002 ) + 1,320.7360 & @xmath120.0008 & 3300.5 & 0.00038 & ii & diethelm ( 2001 ) + 1,580.5980 & @xmath120.0033 & 4054.0 & -0.00016 & i & zejda ( 2002 ) + 1,626.4677 & @xmath120.0015 & 4187.0 & 0.00121 & i & zejda ( 2002 ) + 1,641.2961 & @xmath120.0016 & 4230.0 & -0.00002 & i & brt et al . ( 2007 ) + 1,684.4062 & @xmath120.0025 & 4355.0 & 0.00063 & i & brt et al . ( 2007 ) + 1,956.5138 & @xmath120.0015 & 5144.0 & 0.00039 & i & brt et al . ( 2007 ) + 1,965.4811 & @xmath120.0022 & 5170.0 & 0.00088 & i & zejda ( 2004 ) + 1,968.5846 & @xmath120.0025 & 5179.0 & 0.00049 & i & brt et al . ( 2007 ) + 2,014.4530 & @xmath120.0022 & 5312.0 & 0.00025 & i & brt et al . ( 2007 ) + 2,043.4225 & @xmath120.0022 & 5396.0 & 0.00009 & i & brt et al . ( 2007 ) + 2,053.4230 & @xmath120.0015 & 5425.0 & -0.00084 & i & brt et al . ( 2007 ) + 2,287.5953 & @xmath120.0004 & 6104.0 & 0.00015 & i & blttler ( 2002 ) + 2,344.4994 & @xmath120.0004 & 6269.0 & -0.00039 & i & blttler ( 2002 ) + 2,365.5369 & @xmath120.0032 & 6330.0 & -0.00037 & i & zejda ( 2004 ) + 2,723.3454 & @xmath120.0009 & 7367.5 & -0.00120 & ii & diethelm ( 2003 ) + 2,723.5173 & @xmath120.0007 & 7368.0 & -0.00174 & i & diethelm ( 2003 ) + 2,730.4160 & @xmath120.0049 & 7388.0 & -0.00057 & i & zejda ( 2004 ) + 2,808.3591 & @xmath120.0003 & 7614.0 & 0.00048 & i & baki et al . ( 2005 ) + 3,090.4682 & @xmath120.0001 & 8432.0 & 0.00077 & i & krajci ( 2005 ) + 3,094.4343 & @xmath120.0013 & 8443.5 & 0.00079 & ii & diethelm ( 2004 ) + 3,117.5406 & @xmath120.0012 & 8510.5 & 0.00038 & ii & hbscher et al . ( 2005 ) + 3,131.8527 & @xmath120.0003 & 8552.0 & 0.00012 & i & samec et al . ( 2006 ) + 3,135.8183 & @xmath120.0013 & 8563.5 & -0.00036 & ii & samec et al . ( 2006 ) + 3,143.2339 & @xmath120.0003 & 8585.0 & 0.00040 & i & krajci ( 2005 ) + 3,165.8238 & @xmath120.0003 & 8650.5 & 0.00091 & ii & samec et al . ( 2006 ) + 3,203.4157 & @xmath120.0004 & 8759.5 & 0.00130 & ii & hbscher et al . ( 2005 ) + 3,351.7109 & @xmath120.0002 & 9189.5 & -0.00026 & ii & hbscher et al . ( 2006 ) + 3,409.6501 & @xmath120.0044 & 9357.5 & -0.00026 & ii & hbscher et al . ( 2005 ) + 3,463.4508 & @xmath120.0002 & 9513.5 & -0.00023 & ii & hbscher et al . ( 2006 ) + 3,515.5263 & @xmath120.0010 & 9664.5 & -0.00103 & ii & diethelm ( 2005 ) + 3,767.9772 & @xmath120.0005 & 10396.5 & 0.00059 & ii & this article + 3,769.0115 & @xmath120.0003 & 10399.5 & 0.00026 & ii & this article + 3,770.0456 & @xmath120.0004 & 10402.5 & -0.00027 & ii & this article + 3,797.9806 & @xmath120.0002 & 10483.5 & -0.00022 & ii & this article + 3,813.4999 & @xmath120.0018 & 10528.5 & -0.00034 & ii & hbscher et al . ( 2006 ) + 3,821.7770 & @xmath120.0003 & 10552.5 & -0.00027 & ii & nelson ( 2007 ) + 3,859.5407 & @xmath120.0003 & 10662.0 & -0.00049 & i & diethelm ( 2006 ) + 4,171.8267 & @xmath120.0001 & 11567.5 & 0.00038 & ii & this article + 4,172.8614 & @xmath120.0001 & 11570.5 & 0.00045 & ii & this article + 4,173.8959 & @xmath120.0001 & 11573.5 & 0.00032 & ii & this article + 4,174.7577 & @xmath120.0001 & 11576.0 & -0.00007 & i & this article + 4,176.8269 & @xmath120.0001 & 11582.0 & -0.00012 & i & this article + 4,177.8615 & @xmath120.0001 & 11585.0 & -0.00015 & i & this article + 4,521.0145 & @xmath120.0003 & 12580.0 & 0.00035 & i & this article + 4,522.9109 & @xmath120.0002 & 12585.5 & -0.00008 & ii & this article + 4,539.8093 & @xmath120.0002 & 12634.5 & -0.00074 & ii & this article + 4,540.8442 & @xmath120.0002 & 12637.5 & -0.00048 & ii & nelson ( 2009 ) + 4,551.8804 & @xmath120.0003 & 12669.5 & -0.00041 & ii & this article + 4,553.9497 & @xmath120.0003 & 12675.5 & -0.00038 & ii & this article + 4,554.8127 & @xmath120.0002 & 12678.0 & 0.00042 & i & this article + 4,554.9844 & @xmath120.0002 & 12678.5 & -0.00032 & ii & this article + 4,555.5023 & @xmath120.0014 & 12680.0 & 0.00026 & i & hbscher et al . ( 2009 ) + 4,555.8471 & @xmath120.0002 & 12681.0 & 0.00019 & i & this article + 4,556.8819 & @xmath120.0004 & 12684.0 & 0.00035 & i & this article + 4,568.7795 & @xmath120.0002 & 12718.5 & -0.00038 & ii & this article + 4,569.8142 & @xmath120.0002 & 12721.5 & -0.00031 & ii & this article + 4,569.9873 & @xmath120.0003 & 12722.0 & 0.00035 & i & this article + 4,598.4399 & @xmath120.0006 & 12804.5 & 0.00042 & ii & hbscher et al . ( 2009 ) + ccc @xmath65 & 2,450,182.47724(28 ) & hjd + @xmath67 & 0.344874257(32 ) & d + @xmath68 & 1.264(27)@xmath69 & d + @xmath70 & 0.259(67 ) & au + @xmath18 & 280(18 ) & deg + @xmath17 & 0.59(28 ) & + @xmath71 & 0.1301(17 ) & deg d@xmath40 + @xmath20 & 2,446,142(97 ) & hjd + @xmath72 & 7.573(99 ) & yr + @xmath25 & 0.00149(38 ) & d + @xmath73 & 0.000304(79 ) & @xmath1 + @xmath74 ( @xmath75=90 deg)@xmath76 & 0.081 & @xmath1 + @xmath74 ( @xmath75=60 deg)@xmath76 & 0.095 & @xmath1 + @xmath74 ( @xmath75=30 deg)@xmath76 & 0.170 & @xmath1 + @xmath37/@xmath38 & 2.676(58)@xmath77 & d yr@xmath40 + @xmath78/@xmath38 & 1.482@xmath77 & @xmath1 yr@xmath40 + cccc @xmath26 & 0.1069 & 0.1069 & s + @xmath79 & @xmath80 & @xmath80 & + @xmath27 & @xmath81 & @xmath82 & g @xmath83 + @xmath28 & @xmath84 & @xmath85 & g @xmath86 s@xmath40 + @xmath87 & @xmath88 & @xmath89 & g @xmath86 + @xmath30 & @xmath90 & @xmath91 & s@xmath40 + @xmath92 & @xmath93 & @xmath94 & + @xmath31 & @xmath95 & @xmath96 & erg + @xmath32 & @xmath97 & @xmath98 & erg s@xmath40 + & 0.162 & 0.088 & @xmath99 + & 0.506 & 0.144 & @xmath100 + @xmath33 & @xmath120.174 & @xmath120.085 & mag + @xmath4 & 20640 & 14576 & g ccc @xmath65 ( hjd ) & + @xmath67 ( d ) & + @xmath49 & + @xmath101 ( deg ) & + @xmath20 ( k ) & 5398(14 ) & 5100 + @xmath102 & 6.036(6 ) & 6.102 + @xmath68 & 0.5 & 0.5 + @xmath103 & 0.32 & 0.32 + @xmath45 , @xmath46 & 0.645 , 0.186 & 0.643 , 0.169 + @xmath104 , @xmath105 & 0.945(51 ) , 0.070 & 0.866(39 ) , 0.006 + @xmath106 , @xmath107 & 0.770(53 ) , 0.183 & 0.854(44 ) , 0.131 + @xmath108 , @xmath109 & 0.605(55 ) , 0.219 & 0.798(48 ) , 0.186 + @xmath110 , @xmath111 & 0.924(49 ) , 0.070 & 0.808(40 ) , 0.006 + @xmath112 , @xmath113 & 0.737(51 ) , 0.183 & 0.771(44 ) , 0.131 + @xmath114 , @xmath115 & 0.607(52 ) , 0.219 & 0.699(48 ) , 0.186 + @xmath116 & 0.376(2 ) & 0.624 + @xmath117 & 0.371(2 ) & 0.629 + @xmath118 & 0.366(2 ) & 0.634 + @xmath119 & 0.372(2 ) & 0.628 + @xmath120 & 0.366(2 ) & 0.634 + @xmath121 & 0.356(2 ) & 0.644 + @xmath122 ( pole ) & 0.2841(4 ) & 0.4412(4 ) + @xmath122 ( side ) & 0.2968(5 ) & 0.4725(6 ) + @xmath122 ( back ) & 0.3331(9 ) & 0.5007(7 ) + @xmath122 ( volume)@xmath123 & 0.3064 & 0.4729 + @xmath53 & + cccccccc @xmath124@xmath123 & -0.00027(10 ) & -0.00052(9 ) & & & + @xmath49 & 2.601(7 ) & 2.591(4 ) & & & + @xmath101 ( deg ) & 77.4(4 ) & 77.6(4 ) & & & + @xmath125 ( k ) & 5388(16 ) & 5378(14 ) & & & + @xmath126=@xmath127 & 6.007(11 ) & 5.982(9 ) & & & + @xmath128 ( % ) & 12.3 & 14.3 & & & + @xmath129 & 0.942(69 ) & 0.966(57 ) & & & + @xmath130 & 0.756(68 ) & 0.766(56 ) & & & + @xmath131 & 0.575(68 ) & 0.627(58 ) & & & + @xmath132 & 0.857(92 ) & 0.849(67 ) & & & + @xmath133 & 0.793(85 ) & 0.789(68 ) & & & + @xmath134 & 0.700(84 ) & 0.704(71 ) & & & + @xmath135/(@xmath136+@xmath137+@xmath138)@xmath139 & 0.361(4 ) & 0.360(3 ) & & & + @xmath135/(@xmath136+@xmath137+@xmath138)@xmath140 & 0.359(4 ) & 0.359(3 ) & & & + @xmath135/(@xmath136+@xmath137+@xmath138)@xmath141 & 0.357(4 ) & 0.352(4 ) & & & + _ @xmath142_@xmath143 & 0.036(18 ) & 0.025(14 ) & & & + _ @xmath144_@xmath143 & 0.018(17 ) & 0.012(14 ) & & & + _ @xmath145_@xmath143 & 0.006(17 ) & 0.007(14 ) & & & + @xmath146 ( pole ) & 0.2853(10 ) & 0.2865(8 ) & & & + @xmath146 ( side ) & 0.2981(12 ) & 0.2995(9 ) & & & + @xmath146 ( back ) & 0.3350(21 ) & 0.3371(16 ) & & & + @xmath146 ( volume ) & 0.3079 & 0.3095 & & & + @xmath147 ( pole ) & 0.4414(8 ) & 0.4420(7 ) & & & + @xmath147 ( side ) & 0.4728(11 ) & 0.4736(9 ) & & & + @xmath147 ( back ) & 0.5014(15 ) & 0.5026(12 ) & & & + @xmath147 ( volume ) & 0.4734 & 0.4743 & & & + + colatitude@xmath148 ( deg ) & 93.4(2.6 ) & 90.4(0.6 ) & & 91.5(2.8 ) & 103.0(2.4 ) & 83.2(1.0 ) & 103.5(1.5 ) + longitude@xmath148 ( deg ) & 172.5(1.3 ) & 97.4(1.3 ) & & 160.4(0.4 ) & 39.7(2.8 ) & 89.4(0.4 ) & 64.6(1.1 ) + radius@xmath148 ( deg ) & 14.99(46 ) & 16.07(21 ) & & 14.42(35 ) & 17.25(92 ) & 17.71(18 ) & 14.63(73 ) + @xmath20@xmath149/@xmath20@xmath150 & 0.928(3 ) & 0.917(3 ) & & 0.925(2 ) & 1.033(2 ) & 0.918(2 ) & 1.024(3 ) + @xmath53 & 0.0068 & 0.0084 & & & cccccccccc @xmath65 ( hjd ) & & & + @xmath67 ( day ) & & & + @xmath49 & & & + @xmath101 ( deg ) & & & + @xmath20 ( k ) & & & & & + @xmath102 & & & & & + @xmath128 ( % ) & & & + @xmath151 & & & & & + @xmath152 & & & & & + @xmath153 & & & & & + @xmath154 & & & & & + @xmath135/(@xmath136+@xmath137+@xmath138)@xmath139 & & & & & + @xmath135/(@xmath136+@xmath137+@xmath138)@xmath140 & & & & & + @xmath135/(@xmath136+@xmath137+@xmath138)@xmath141 & & & & & + @xmath135/(@xmath136+@xmath137+@xmath138)@xmath155 & & & & & + _ @xmath142_@xmath156 & & & + _ @xmath144_@xmath156 & & & + _ @xmath145_@xmath156 & & & + _ @xmath157_@xmath156 & & & + @xmath122 ( pole ) & & & & & + @xmath122 ( side ) & & & & & + @xmath122 ( back ) & & & & & + @xmath122 ( volume ) & & & & & + colatitude ( deg)@xmath143 & & 90.7(13.0 ) & 115.3(2.9 ) & & & 79.9(0.8 ) & 98.9(9.4 ) + longitude ( deg)@xmath143 & & 150.3(2.9 ) & 35.7(2.5 ) & & & 106.7(0.7 ) & 298(11.0 ) + radius ( deg)@xmath143 & & 11.4(1.4 ) & 19.0(0.8 ) & & & 20.8(0.4 ) & 13.1(2.9 ) + @xmath20@xmath158/@xmath20@xmath159@xmath143 & & 0.931(0.012 ) & 1.086(0.006 ) & & & 0.870(0.006 ) & 1.024(0.010 ) ccc @xmath160 ( @xmath1 ) & 0.35 & 0.90 + @xmath11 ( @xmath161 ) & 0.65 & 1.00 + @xmath162 @xmath103 ( cgs ) & 4.36 & 4.39 + @xmath163 ( @xmath99 ) & 0.32 & 0.61 + @xmath164 ( mag ) & 5.94 & 5.23 + bc ( mag ) & @xmath660.17 & @xmath660.26 + @xmath165 ( mag ) & 6.11 & 5.49 + distance ( pc ) & + ccccccl 0,607.3635 & 0,607.36271 & @xmath120.00021 & @xmath30.00079 & @xmath5 & i & wolf et al . + 0,607.5346 & 0,607.53490 & @xmath120.00022 & @xmath660.00030 & @xmath5 & ii & wolf et al . + 0,611.5025 & 0,611.50142 & @xmath120.00028 & @xmath30.00108 & @xmath11 & i & wolf et al . + 0,638.4028 & 0,638.40188 & @xmath120.00013 & @xmath30.00092 & @xmath11 & i & wolf et al . + 3,131.8527 & 3,131.85333 & @xmath120.00016 & @xmath660.00063 & @xmath6 & i & samec et al . + 3,135.8183 & 3,135.81836 & @xmath120.00016 & @xmath660.00060 & @xmath6 & ii & samec et al . + 3,165.8238 & 3,165.82275 & @xmath120.00020 & @xmath30.00105 & @xmath6 & ii & samec et al . + 4,171.8267 & 4,171.82622 & @xmath120.00006 & @xmath30.00048 & @xmath7 & ii & this article + 4,172.8614 & 4,172.86085 & @xmath120.00009 & @xmath30.00055 & @xmath7 & ii & this article + 4,173.8959 & 4,173.89546 & @xmath120.00005 & @xmath30.00044 & @xmath7 & ii & this article + 4,174.7577 & 4,174.75757 & @xmath120.00007 & @xmath30.00013 & @xmath7 & i & this article + 4,176.8269 & 4,176.82668 & @xmath120.00008 & @xmath30.00022 & @xmath7 & i & this article + 4,177.8615 & 4,177.86136 & @xmath120.00007 & @xmath30.00014 & @xmath7 & i & this article + 4,521.0145 & 4,521.01425 & @xmath120.00011 & @xmath30.00025 & @xmath7 & i & this article + 4,522.9109 & 4,522.91122 & @xmath120.00012 & @xmath660.00032 & @xmath7 & ii & this article + 4,539.8093 & 4,539.80960 & @xmath120.00012 & @xmath660.00030 & @xmath7 & ii & this article + 4,551.8804 & 4,551.88093 & @xmath120.00009 & @xmath660.00053 & @xmath7 & ii & this article + 4,553.9497 & 4,553.95013 & @xmath120.00015 & @xmath660.00043 & @xmath7 & ii & this article + 4,554.8127 & 4,554.81233 & @xmath120.00009 & @xmath30.00037 & @xmath7 & i & this article + 4,554.9844 & 4,554.98474 & @xmath120.00012 & @xmath660.00034 & @xmath7 & ii & this article + 4,555.8471 & 4,555.84711 & @xmath120.00011 & @xmath660.00001 & @xmath7 & i & this article + 4,556.8819 & 4,556.88168 & @xmath120.00015 & @xmath30.00022 & @xmath7 & i & this article + 4,568.7795 & 4,568.77993 & @xmath120.00009 & @xmath660.00043 & @xmath7 & ii & this article + 4,569.8142 & 4,569.81443 & @xmath120.00016 & @xmath660.00023 & @xmath7 & ii & this article + 4,569.9873 & 4,569.98727 & @xmath120.00011 & @xmath30.00003 & @xmath7 & i & this article +

new ccd photometric observations of the eclipsing system ar boo were obtained from february 2006 to april 2008 . mass transfer from the less massive primary to the more massive secondary component is likely responsible for at least a significant part of the secular period change . the cyclical variation with a period of 7.57 yrs and a semi - amplitude of 0.0015 d can be produced either by a light - travel - time effect due to an unseen companion with a scaled mass of=0.081 or by a magnetic period modulation in the secondary star . our solutions confirm that ar boo belongs to the w - subtype contact binary class , consisting of a hotter , less massive primary star with a spectral type of g9 and a companion of spectral type k1 .

new ccd photometric observations of the eclipsing system ar boo were obtained from february 2006 to april 2008 . the star s photometric properties are derived from detailed studies of the period variability and of all available light curves . we find that over about 56 years the orbital period of the system has varied due to a combination of an upward parabola and a sinusoid rather than in a monotonic fashion . mass transfer from the less massive primary to the more massive secondary component is likely responsible for at least a significant part of the secular period change . the cyclical variation with a period of 7.57 yrs and a semi - amplitude of 0.0015 d can be produced either by a light - travel - time effect due to an unseen companion with a scaled mass of=0.081 or by a magnetic period modulation in the secondary star . historical light curves of ar boo , as well as our own , display season - to - season light variability , which are best modeled by including both a cool spot and a hot one on the secondary star . we think that the spots express magnetic dynamo - related activity and offer limited support for preferring the magnetic interpretation of the 7.57-year cycle over the third - body understanding . our solutions confirm that ar boo belongs to the w - subtype contact binary class , consisting of a hotter , less massive primary star with a spectral type of g9 and a companion of spectral type k1 .

1604.07778

c

with our investigations of the one - dimensional voter model , we want to achieve two goals : ( i ) a better understanding of the _ non - linearity _ in the voter dynamics that was mostly studied as a linear model , only , ( ii ) a probabilistic description , and possible approximations , of the dynamics for the fraction of opinion 1 , @xmath119 . the non - linearity can be easily expressed by means of a free parameter @xmath16 . the critical value @xmath220 distinguishes between two different rules , _ majority _ voting ( @xmath221 ) and _ minority _ voting ( @xmath222 ) , whereas @xmath223 refers to the border case of the _ linear _ voter model . it is known from well - mixed populations that majority voting should result in _ consensus _ , i.e. the asymptotic dominance of only one opinion , whereas minority voting should result in _ coexistence _ , i.e. the occurrence of both opinions in different fractions . our main focus was the role of local correlations in determining this outcome . for this we have used one - dimensional cellular automata ( ca ) in which each cell @xmath4 , characterized by its opinion @xmath7 , has a defined neighborhood of two cells with possibly different opinions , denoted as _ triplet_. the transition probability of a cell to change its opinion is then determined by the local _ frequency of opinions _ in its neighborhood ( including the focal cell ) and the _ non - linear response _ to this information , expressed by means of @xmath16 . the values of @xmath16 define a certain probability to switch to the opposite opinion , i.e. we can use them to switch between a deterministic ( @xmath224 ) or a stochastic ( @xmath225 ) dynamics . in the latter case , we have further assumed a very small probability @xmath226 to perturb a state of complete consensus , which allows a non - stationary dynamics of the ca as shown in figure [ fig : ca1dp ] ( bottom right ) . while the minority rule only results in _ random coexistence _ of the two different opinions , the majority rule generates more interesting results . in particular , we observe a _ correlated coexistence _ characterized by the formation of large domains of the same opinion which change continuously . i.e. , we have a non - equilibrium dynamics in which each opinion , for a certain time , can form large clusters of the majority opinion . the question then is how to describe this dynamics mathematically . in this paper , we follow a probabilistic approach , i.e. each cell has a certain probability of a given opinion @xmath227 , which also depends on the probabilities of the nearest neighbors , second - nearest neighbors , and so forth . in order to close the dynamics , we have proposed three different approximations at different levels of the description . the first level is the aggregated description in terms of the global fraction of opinion 1 , @xmath119 , for which we derive a dynamics for the _ expected value _ , @xmath13 . on this level , we discuss two approximations . the simplest one is the _ mean - field approximation _ , in which _ no correlations _ between neighboring states are considered . so , we call this the zero - order approximation . it gives us a prediction for @xmath13 derived from the well - mixed case . in contrast , the _ pair approximation _ considers a correlation between a cell and its neighbor , i.e. the triplet consisting of a cell and its two neighbors is decomposed in two cell - neighbor pairs . correlations between neighbors are not considered , so we call this the 1st - order approximation . hence , we have a prediction for @xmath13 coupled to the dynamics of the pair correlations @xmath14 . the third approximation does not refer to the aggregated level , but to the stochastic dynamics of a triplet , i.e. a cell with its two neighbors , that is determined by the larger neighborhood of a quintuplet , i.e. considers also the second nearest neighbors of the cell . therefore , we call this the 2nd - order approximation . using certain assumptions , we are able to provide a _ closed form dynamics _ for this larger neighborhood in terms of a probabilistic equation . to compare the validity of these mathematical approximations , we use as a _ reference case _ stochastic computer simulations of the one - dimensional ca , which are averaged over a larger number of runs . we have discussed the majority and the minority voting , as well as the deterministic and the stochastic dynamics , separately . in conclusion , we can summarize that the zero - order approximation only predicts the stationary outcome of the minority voting correctly , but fails for the majority voting rule , both with respect to the dynamics and the stationary outcome . the first - order approximation performs comparably well in comparison to the second - order approximation only for the majority voting . the stationary outcome is correctly predicted both for the deterministic and the stochastic case , also the dynamics is covered fairly good . however , the first - order approximation fails to predict the dynamics of the minority voting . this case is only well covered by the second - order approximation that gives not only a correct description of @xmath13 , but also of the pair correlations @xmath218 . commenting specifically on the correlations , we recall again that _ both _ the first and second - order approximations lead to comparably good asymptotic results only for the majority voting . but they clearly predict a faster formation of domains , i.e. a convergence to their stationary value , as compared to the ca simulations . this limits their usability to fully understand the emergence of long - range correlations . in the case of minority voting , our first - order approximation fails , while the second - order approximation for @xmath218 could be even seen as accurate in its computational prediction . this is not so surprising if we recall that minority voting rules , different from majority voting , do _ not _ result in long - range correlations . hence , we can conclude that the quintuplet approximation that covers also the second - order neighborhood , is accurate enough to describe the dynamics of the ca on the macroscopic level . while it is of course understandable , that an approximation that considers more information is usually more accurate , we should also relate this conclusion to the computational effort . here , it turns out that the 2nd - order approximation , although stochastic , i.e. needs to be averaged over a number of runs , performs very fast because of the closed form dynamics . of course , the 1st - order approximation , the closed form of which is given in the appendix , is computationally even simpler . but there is a trade - off with the accuracy of the prediction . still , as long only majority voting is considered , the 1st - order approximation should be preferred , both for simplicity , accuracy and computational effort . our last remark is about the _ coexistence _ of the two opinions , which is the more interesting scenario compared to _ consensus _ , i.e. the existence of only one opinion . here , we are not so much interested in the trivial case of _ random coexistence _ without any structure formation , which is characterized by @xmath141 and @xmath228 . we focus more on the case of _ correlated coexistence _ which has an interesting complexity because of the formation of local structures , nicely shown in fig . [ fig : ca1dp ] ( lower left ) . on the level of our approximations , this state is characterized by @xmath141 and @xmath229 . i.e. , both opinions form at times large domains , indicated by the high pair correlation , but none of the two opinions entirely dominates the dynamics , as the expected value for its fraction is about 0.5 . such insights can be generalized to other cases of frequency dependent processes which e.g. play a role in population ecology ( invasion or extinction of species ) . for most of these applications a two - dimensional ca is more appropriate as it was discussed in @xcite . the one - dimensional ca investigated here , on the other hand , allows a mathematical approximation of the stochastic dynamics in terms of the 2nd - order neighborhood , which gives a much higher predictive power . this formalism can be also used for other frequency dependent processes in one - dimensional ca . the authors acknowledge fruitful discussions with h. mhlenbein on an early version of this paper . 21 natexlab#1#1url # 1`#1`urlprefixselectlanguage # 1 agapie , a. ; hns , r. ; mhlenbein , h. ( 2004 ) . markov chain analysis for one - dimensional asynchronous cellular automata . _ methodology and computing in applied probability _ * 6(2 ) * , 181201 . banisch , s. ( 2014 ) . from microscopic heterogeneity to macroscopic complexity in the contrarian voter model . _ advances in complex systems _ * 17(05 ) * , 1450025 . banisch , s. ; arajo , t. ; lou , j. ( 2010 ) . opinion dynamics and communication networks . _ advances in complex systems _ * 13(01 ) * , 95111 . behera , l. ; schweitzer , f. ( 2003 ) . _ international journal of modern physics c _ * 14(10 ) * , 13311354 . brown , d. t. ( 1959 ) . a note on approximations to discrete probability distributions . _ information and control _ * 2(4 ) * , 386392 . castellano , c. ; fortunato , s. ; loreto , v. ( 2008 ) . statistical physics of social dynamics . _ review of modern physics _ . castellano , c. ; muoz , m. a. ; pastor - satorras , r. ( 2009 ) . nonlinear q - voter model . _ physical review e _ * 80(4 ) * , 041129 . gleeson , j. p. ( 2013 ) . binary - state dynamics on complex networks : pair approximation and beyond . _ physical review x _ * 3(2 ) * , 021004 . grnerup , o. ; jacobi , m. n. ( 2008 ) . a method for inferring hierarchical dynamics in stochastic processes . _ advances in complex systems _ * 11(01 ) * , 116 . keitt , t. h. ; lewis , m. a. ; holt , r. d. ( 2001 ) . allee effects , invasion pinning , and species borders . _ the american naturalist _ * 157(2 ) * , 203216 . krause , s. m. ; bttcher , p. ; bornholdt , s. ( 2012 ) . mean - field - like behavior of the generalized voter - model - class kinetic ising model . _ physical review e _ * 85(3 ) * , 031126 . mhlenbein , h. ; hns , r. ( 2002 ) . stochastic analysis of cellular automata with applications to the voter model . _ advances in complex systems _ * 5(2 ) * , 301337 . l . ; wang , j. ( 2015 ) . entropy and recurrence measures of a financial dynamic system by an interacting voter system . _ entropy _ * 17(5 ) * , 2590 . pfante , o. ; bertschinger , n. ; olbrich , e. ; ay , n. ; jost , j. ( 2014 ) . comparison between different methods of level identification . _ advances in complex systems _ * 17(02 ) * , 1450007 . przybya , p. ; sznajd - weron , k. ; tabiszewski , m. ( 2011 ) . exit probability in a one - dimensional nonlinear q - voter model . _ physical review e _ * 84(3 ) * , 031117 . schweitzer , f. ; behera , l. ( 2009 ) . nonlinear voter models : the transition from invasion to coexistence . _ the european physical journal b _ * 67(3 ) * , 301 318 . stark , h .- u . ; tessone , c. j. ; schweitzer , f. ( 2008 ) . slower is faster : fostering consensus formation by heterogeneous inertia . _ advances in complex systems _ * 11(04 ) * , 551563 . stauffer , d. ( 2002 ) . better be third than second in a search for a majority opinion . _ advances in complex systems _ * 5(1 ) * , 97100 . suchecki , k. ; eguluz , v. m. ; san miguel , m. ( 2005 ) . conservation laws for the voter model in complex networks . _ europhysics letters _ * 69 * , 228234 . suchecki , k. ; eguluz , v. m. ; san miguel , m. ( 2005 ) . voter model dynamics in complex networks : role of dimensionality , disorder , and degree distribution . _ physical review e _ * 72 * , 036132 . xiong , f. ; liu , y. ; zhu , j. ( 2013 ) . competition of dynamic self - confidence and inhomogeneous individual influence in voter models . _ entropy _ * 15(12 ) * , 52925304 .

this allows for voting rules different from majority voting . while the linear voter model is known to reach consensus , non - linear voter models can result in the coexistence of opposite opinions . this is compared with an analytic pair approximation for the expected value of the global fraction of opinions and a mean - field approximation . we further discuss the interesting phenomenon of a correlated coexistence , characterized by the formation of large domains of opinions that dominate for some time , but slowly change . _

non - linear voter models assume that the opinion of an agent depends on the opinions of its neighbors in a non - linear manner . this allows for voting rules different from majority voting . while the linear voter model is known to reach consensus , non - linear voter models can result in the coexistence of opposite opinions . our aim is to derive approximations to correctly predict the time dependent dynamics , or at least the asymptotic outcome , of such local interactions . emphasis is on a probabilistic approach to decompose the opinion distribution in a second - order neighborhood into lower - order probability distributions . this is compared with an analytic pair approximation for the expected value of the global fraction of opinions and a mean - field approximation . our reference case are averaged stochastic simulations of a one - dimensional cellular automaton . we find that the probabilistic second - order approach captures the dynamics of the reference case very well for different non - linearities , i.e for both majority and minority voting rules , which only partly holds for the first - order pair approximation and not at all for the mean - field approximation . we further discuss the interesting phenomenon of a correlated coexistence , characterized by the formation of large domains of opinions that dominate for some time , but slowly change . _ keywords : _ opinion dynamics ; voter model ; pair approximation ; higher - order probability distribution , cellular automata

nucl-th9501036

i

electromagnetic probes provide invaluable information about nuclear structure . the quasifree process @xmath1 has been used extensively to study proton hole states and to determine single particle spectroscopic factors @xcite . this reaction is advantageous because the electromagnetic interaction is known ; the relative weakness of the reaction permits the probe to interact almost uniformly through the entire nucleus , and the first order of perturbation theory should provide an adequate description of the process . coincidence measurements of the @xmath1 reaction can provide detailed information about the single particle structure of the nucleus over a wide range of momentum transfer . the @xmath1 reaction has been widely studied both non - relativistically @xcite and relativistically @xcite , and there are some discrepancies between the results of these investigations . both analyses begin with a lagrangian which allows for the interaction of the photon with both electrons and protons . non - relativistic analyses involve the reduction of the free electron - proton interaction to a form involving two - component spinors for the nucleon . this results in a hamiltonian which is expanded in powers of @xmath2 where @xmath3 is the nucleon mass @xcite . the resulting interaction hamiltonian is sandwiched between schrdinger wave functions describing the nucleons in order to form the nuclear current . relativistic analyses are based on the feynman diagram for one - photon exchange between the projectile electron and a proton which is imbedded in the nucleus . the electrons and nucleons are all described relativistically as spin 1/2 objects via the dirac equation containing appropriate potentials @xcite . a long - standing problem in quasifree electron scattering has been that the spectroscopic factors extracted from non - relativistic analyses are smaller than expected from shell model calculations . spectroscopic factors which are found on the basis of the relativistic approach are generally larger than those found via the non - relativistic approach @xcite . several groups have attempted to understand the underlying differences between these two approaches . this mainly involved looking at the sensitivity of quasifree electron scattering calculations to different optical potentials and renormalizations of the continuum wave function @xcite . this concentration on optical potentials was largely a result of the improvement in the description of proton elastic scattering observables via dirac phenomenology over the standard non - relativistic optical model description . boffi et al . @xcite have multiplied the non - relativistic continuum wave function by a potential - dependent factor @xmath4 / \left ( e + m \right ) \right\}^{1/2}$ ] , where @xmath5 is the dirac scalar potential and @xmath6 is the vector potential . this modification essentially changes the two - component schrdinger wave function into the upper component of the dirac wave function , while no other change is made in the non - relativistic calculation . they find that extracted occupation probabilities are larger than those obtained from the unmodified non - relativistic analysis . the analysis of udias et al . @xcite replaces the non - relativistic bound state wave function with the upper component of a dirac wave function , and the non - relativistic continuum wave function is modified by factors of the same shape as the factor used by boffi et al . the continuum wave function in this case is generated from schrdinger - equivalent potentials @xcite . the nuclear current operators are obtained in the standard way by expansion to order @xmath7 . their `` non - relativistic '' calculations then involve non - relativistic nuclear current operators surrounded by the upper components of dirac wave functions . with these choices little difference is found between the relativistic and `` non - relativistic '' calculations . their conclusion is that differences in observed cross sections are not due to non - relativistic reduction , but to the choice of optical potential . jin and onley @xcite have presented a model which can take either relativistic or non - relativistic optical potentials while keeping other aspects of the calculation the same . they find that different optical potentials can change the results by as much as 14% . these results clearly demonstrate the variability due to final state interactions , however , the issue is clouded by the occasional use of upper components of dirac wave functions in a non - relativistic calculation . we believe that the essential difference between relativistic and non - relativistic approaches are not just in the changes in the optical potentials ; these are usually phenomenological and equivalent potentials can always be found . rather the essential difference is in the appearance of the nuclear potentials in the nuclear current operators when the relativistic amplitude is reduced to a non - relativistic form . such medium effects on the nuclear currents are absent in the standard non - relativistic calculations . in this paper we study the differences between the relativistic and non - relativistic approaches in calculating the amplitude for the @xmath1 reaction . we do this through an effective pauli reduction of the relativistic transition amplitude @xcite . an expansion of the amplitude in powers of @xmath8 allows us to recover a non - relativistic limit , which matches the standard non - relativistic calculations , with the difference that optical potentials used to generate the distorted waves are exactly equivalent to those used in the relativistic calculations . the main difference , as mentioned above is that the nuclear currents are potential dependent . we compare the two approaches and thus explain why they can still give different values for the extracted spectroscopic factors , even when equivalent optical potentials are used . we introduce the relativistic amplitude for quasifree electron scattering in section [ rel - amp ] . section [ pauli ] outlines the pauli reduction of the amplitude and some of its relevant features are discussed in section [ e - results ] . the non - relativistic limit is discussed in section [ nonrel ] . in section [ n - results ] we compare our non - relativistic calculations with and without nuclear potentials in the nuclear current operators , to the results of the fully relativistic calculations . our conclusions are given in section [ concl ] .

an expansion of the amplitude results in a power series in the nuclear potentials . the results can be cast in a form which reproduces the non - relativistic amplitudes in the limit that the potentials are removed from the nuclear current operator . we find that the non - relativistic calculations with potentials included in the nuclear current up to second order give results which are close to those of the fully relativistic calculation .

non - relativistic reduction of the s - matrix for the quasifree electron scattering process is studied in order to understand the source of differences between non - relativistic and relativistic models . we perform an effective pauli reduction on the relativistic expression for the s - matrix in the one - photon exchange approximation . the reduction is applied to the nucleon current only ; the electrons are treated fully relativistically . an expansion of the amplitude results in a power series in the nuclear potentials . the series is found to converge rapidly only if the nuclear potentials are included in the nuclear current operator . the results can be cast in a form which reproduces the non - relativistic amplitudes in the limit that the potentials are removed from the nuclear current operator . large differences can be found between calculations which do and do not include the nuclear potentials in the different orders of the nuclear current operator . in the high missing momentum region we find that the non - relativistic calculations with potentials included in the nuclear current up to second order give results which are close to those of the fully relativistic calculation . this behavior is an indication of the importance of the medium modifications of the nuclear currents in this model , which are naturally built into the relativistic treatment of the reaction .

1612.00136

i

modelling nonparametric time series has received increasing interest among scholars for a few decades , see , for example , @xcite . in classical time series analysis , the stationarity of time series is a fundamental assumption . yet , it may be violated on some occasions in such the fields as finance , sound analysis and neuroscience , especially when the time span of observations tends to infinity . so , it is necessary to generalize the stationary process to the nonstationary process . priestley ( 1965 ) @xcite first introduced a stochastic process with evolutionary spectra , which locally displays an approximately stationary behavior . but in his framework , it is impossible to establish an asymptotic statistical inference . dahlhaus ( 1997 ) proposed a new generalization of stationarity , called locally stationary process , and investigated its statistical inference . more details can refer to @xcite . in essence , the locally stationary process is locally close to a stationary process over short periods of time , but its second order characteristic is gradually changing as time evolves . a formal description of locally stationary process can refer to assumption ( a1 ) in the appendix . in parametric context , the statistical inference of locally stationary process has been studied extensively by @xcite . in nonparametric context , vogt ( 2012 ) @xcite considered the time - varying nonlinear autoregressive ( tvnar ) models including its general form and estimated the time - varying multivariate regression function using the kernel - type method . however , it still suffers the `` curse of dimensionality '' problem when the dimension of covariates is high . in order to solve the aforementioned problem , a familiar way is to adopt the additive nonparametric regression model suggested by @xcite . it not only remedies the `` curse of dimensionality '' , but also has an independent interest in practical applications due to its flexibility and interpretability . there exists abound research findings about the additive regression model in the literature . in the case of iid observations , the additive nonparametric component functions can be estimated by kernel - based methods : the classic backfitting estimators of @xcite , the marginal integration estimators of @xcite , the smoothing backfitting estimators of @xcite , and two - stage estimators of @xcite . in the stationary time series context , there are kernel estimators via marginal integration of @xcite , spline estimators of @xcite , and the spline - backfitted kernel ( sbk ) estimators which borrow the strength of both kernel estimation and spline estimation , see @xcite . vogt @xcite considered the locally stationary additive model and proposed smooth backfitting method to estimate bivariate additive component functions . on the other hand , the varying - coefficient model is a natural extension of linear model which allows the coefficients to change over certain common covariates instead of being invariant . this model succeeds to relax the parameter limitation of linear model and may have practical as well as theoretical significance . for this model , there are three types of estimation methods : local polynomial smoothing method @xcite , polynomial spline estimation method @xcite and smoothing spline method @xcite . zhang and wang @xcite proposed a so - called varying - coefficient additive model to catch the evolutionary nature of time - varying regression function in the analysis of functional data . their model assumes the evolutionary regression function has the form @xmath2 which is more flexible in the sense that it covers both varying - coefficient model and additive model as special cases . specifically speaking , it reduces to an additive model when @xmath3 are all constants , and a varying - coefficient model if @xmath4 are all linear functions . extracting the special meaning of time in functional data analysis , one can generalize time to some other common covariates . in this paper , we model locally stationary time series . to concreteness , let @xmath5 be a length-@xmath6 realization of @xmath7 dimension locally stationary time series , and assume that the data is generated by varying - coefficient additive model as follows @xmath8 where @xmath9 s are i.i.d , @xmath10 is the varying - coefficient component function , @xmath11 is the additive component function and @xmath12 is a bivariate nonparametric function , which allows the heteroscedasticity case . without loss of generality , we require @xmath13^{p}},$ ] where the superscript ` @xmath14 ' means transposition of vector or matrix . in order to identify these multiplied component functions , we require that @xmath15 where @xmath16 is the @xmath1 norm of any function @xmath17 defined on @xmath18 $ ] such that @xmath19 for functional data , zhang and wang @xcite proposed a two - step spline estimation procedure . in the first step , sorting the data within each subject in ascending order of time and averaging the response for each subject using trapezoidal rule to fit an additive model , then , in the second step , fitting a varying - coefficient model by substituting the estimated additive function into varying - coefficient additive model . his estimation methodology works since there are dense observation for every subject and covariates is independent of observation time within subject . however , for some other practical problems , such as longitudinal data with finite observertion time , time series data , such an assumption fails . to circumvent this problem , under mild assumptions , we derive an initial estimation of additive component by employing a segmentation technique . then we can fit a varying - coefficient model and an additive model , respectively , to get spline estimators of varying - coefficient function and additive function . as expected , we show that the proposed estimators of @xmath10 and @xmath11 are consistent and present the corresponding @xmath1 rate of convergence . on the other hand , the product term in may simply reduce to a varying - coefficient term or an additive term in the case of @xmath11 being linear function or @xmath10 being constant . so , in the parsimony sense , identifying additive terms and varying - coefficient terms in are of interest . to this end , we propose a two - stage penalized least squares estimator based on scad penalty function , and , furthermore , show that our model identification strategy is consistent , i.e. , the additive term and the varying - coefficient term are correctly selected with probability approaching to 1 . meantime , @xmath1 rate of convergence of penalized spline estimator of each component function achieves the rate of the spline estimator of univariate nonparametric function . the rest of this paper is organized as follows . we propose a three - step spline estimation method in section 2 and a two - stage model identification procedure in section 3 . section 4 describes the smoothing parameter selection strategies . section 5 establishes the asymptotic properties of the proposed model estimation and identification methods . simulation studies are illustrated in section 6 . the main technical proofs are presented in the appendix . lemmas and other similar proofs are given in the supplementary .

nonparametric regression models with locally stationary covariates have received increasing interest in recent years . as a nice relief of `` curse of dimensionality '' induced by large dimension of covariates , additive regression model is commonly used . however , in locally stationary context , to catch the dynamic nature of regression function , we adopt a flexible varying - coefficient additive model where the regression function has the form for this model , we propose a three - step spline estimation method for each univariate nonparametric function , and show its consistency and rate of convergence . furthermore , based upon the three - step estimators , we develop a two - stage penalty procedure to identify pure additive terms and varying - coefficient terms in varying - coefficient additive model . . locally stationary process , varying - coefficient additive regression model , b - spline , scad , penalized least squares

nonparametric regression models with locally stationary covariates have received increasing interest in recent years . as a nice relief of `` curse of dimensionality '' induced by large dimension of covariates , additive regression model is commonly used . however , in locally stationary context , to catch the dynamic nature of regression function , we adopt a flexible varying - coefficient additive model where the regression function has the form for this model , we propose a three - step spline estimation method for each univariate nonparametric function , and show its consistency and rate of convergence . furthermore , based upon the three - step estimators , we develop a two - stage penalty procedure to identify pure additive terms and varying - coefficient terms in varying - coefficient additive model . as expected , we demonstrate that the proposed identification procedure is consistent , and the penalized estimators achieve the same rate of convergence as the polynomial spline estimators . simulation studies are presented to illustrate the finite sample performance of the proposed three - step spline estimation method and two - stage model selection procedure . locally stationary process , varying - coefficient additive regression model , b - spline , scad , penalized least squares

1401.5799

i

over the last several decades , with new evidence , the objects we call galaxies " have become much larger . extended dark - matter halos " were proposed to produce flat rotation curves at large radii in disk galaxies ( rubin et al . 1980 ) , and a corona " of hot interstellar gas at the galaxy s virial temperature was predicted by spitzer ( 1956 ) to provide pressure confinement of high - latitude clouds . more recently , astronomers have observed galactic kinematic tracers ( blue horizontal branch stars , globular clusters , satellite galaxies ) to distances of 50 - 250 kpc , and x - ray absorption - line ( ) spectroscopy and stacked soft x - ray emission have provided evidence of large reservoirs of hot ionized gas in milky way halo ( miller & bregman 2013 ) and the outskirts of external galaxies ( soltan 2006 ; anderson et al . 2013 ) . in ultraviolet spectroscopy , the cosmic origins spectrograph ( cos ) on the _ hubble space telescope _ ( hst ) has recently detected extended ( 100 - 150 kpc ) reservoirs of highly ionized oxygen ( ) around star - forming galaxies ( tumlinson et al . 2011 , 2013 ; stocke et al . 2013 ) likely created by outflows of metal - enriched gas from star formation . thus , our picture of galaxies has evolved to a system of stars and gas embedded in an extended dark - matter halo , often associated with an even larger gaseous circumgalactic medium ( cgm ) . galaxies must also be viewed in a cosmological context , in which most of the baryonic matter in the universe is unseen ( persic & salucci 1992 ) and likely distributed through the intergalactic medium ( igm ) in a cosmic web " shaped by dark - matter structure and inefficient galaxy formation ( cen & ostriker 1999 ; dav et al . 1999 ; smith et al . 2011 ) . observations and modeling ( shull et al . 2012 ) suggest that 60 - 80% of the cosmological baryons reside in the low - redshift , multi - phase igm , with perhaps 20% in collapsed form ( galaxies , groups , clusters ) . although these baryon fractions have some uncertainty , current research is devoted to understanding the physical conditions , spatial extent , and evolution of the gas at distances of 100 kpc to a few mpc from galaxies . at low redshift ( @xmath13 ) the influence of galactic winds and metal injection out to @xmath14 mpc has been inferred from the association of qso absorption systems of and with nearby galaxies ( penton et al . 2002 ; prochaska et al . 2011 ; stocke et al . 2006 , 2013 ) . galactic outflows have been detected in absorption toward intermediate - redshift galaxies ( steidel et al . 2004 ; martin et al . 2012 ; tripp et al . 2011 ) , while at higher redshifts , @xmath15 , rudie et al . ( 2012 ) found an enhancement of circumgalactic ( @xmath16 kpc ) absorbers in a sample of 886 star - forming galaxies probed by 15 background qsos . thus , the connection between galaxies and extended absorption systems seems secure . perhaps because of the recent nature of these discoveries , a semantic problem has arisen regarding the proposed structures : halo , cgm , igm . when does gas cease to be circumgalactic and become intergalactic ? are the edges of galaxies defined by gravity or gas outflows ? are quasar absorption lines the extended halos of intervening galaxies ( bahcall & spitzer 1969 ) or filaments of intergalactic gas ( sargent et al . similarly , the phrase circumgalactic medium " appears to have replaced the concept of a gaseous halo " or galactic corona " of hot interstellar gas at the galaxy s virial temperature . ionized gas with high covering factor has been detected above the galactic disk in uv absorption - line surveys in metal ions such as ( sembach et al . 2003 ) and ( shull et al . 2009 ; lehner & howk 2011 ) and in soft x - ray absorption lines of or at @xmath17 ( nicastro et al . 2002 ; mckernan et al . 2005 ; wang et al . 2005 ) . the uv absorbers appear to come primarily from gas within 2 - 10 kpc of the disk plane , elevated by supernovae and star formation in the disk . the galactic x - ray absorption suggests hot gas at @xmath18 k , but its radial extent is controversial . it may come from a 50-kpc halo ( anderson & bregman 2010 ; gupta et al . 2012 ) although absorption toward background agn ( fang et al . 2006 ; hagihara et al . 2010 ) and x - ray binaries ( yao & wang 2005 ; hagihara et al . 2011 ) suggests that much of the absorption comes within several kpc of the disk . the disk model is consistent with both x - ray observations and total mass considerations ( collins et al . 2005 ; fang et al . 2006 ) . the intent of this paper is to improve the definition of galaxy halos as regions of strong gravitational influence , using dynamical principles and observational constraints . section 2 discusses physical measures of the spatial extent of large ( @xmath19 ) galaxies including the milky way and andromeda . we discuss the somewhat arbitrary and occasionally misused definition of virial radius " . somewhat better defined are the gravitational radius " , @xmath20 , derived from galactic mass and potential energy , the gravitational sphere of influence , accretion radius , and tidal radius . section 2.3 develops a physically realistic definition of @xmath21 for halos of mass @xmath3 , assembled primarily at redshifts @xmath4 with further mass accretion down to the present epoch . these new virial radii are typically 50 - 60% the sizes used to analyze hot halo gas with hst / cos . section 3 discusses four estimates of galaxy extent : ( 1 ) recent kinematical studies of the milky way and andromeda ( m31 ) ; ( 2 ) the galactopause " where outflow ram pressure balances thermal pressure of the cgm ; ( 3 ) qso absorber cross sections derived from metal absorption - line frequency in redshift ; and ( 4 ) virial radii and halo masses obtained from galaxy abundance - matching . we also discuss recent estimates of the mass and size of the milky way and andromeda halos from kinematics of stars , galactic satellites , and the local group . most of our estimates suggest a smaller spatial extent ( @xmath6 kpc ) for galaxies of mass @xmath22 , comparable to the milky way and m31 . section 4 concludes with applications of the new definition of virial radius to observations of extended gas around galaxies made with hst / cos , and to recent mass and size measurements for the milky way and m31 .

our current view of galaxies considers them as systems of stars and gas embedded in extended halos of dark matter , much of it formed by the infall of smaller systems at earlier times . other physical estimates of the extent of gravitational influence include the gravitational radius , gas accretion radius , and galactopause " arising from outflows that stall at 100 - 200 kpc over a range of outflow parameters and confining gas pressures . physical criteria are proposed to define bound structures , including a more realistic definition of for stellar mass and halo mass , half of which formed at assembly redshifts " ranging from . the new virial radii , with mean kpc , are 40 - 50% smaller than values estimated in recent hst / cos detections of and absorbers around galaxies . in the new formalism this formalism is intended to clarify semantic differences arising from observations of extended gas in galactic halos , circumgalactic medium ( cgm ) , and filaments of the intergalactic medium ( igm ) .

our current view of galaxies considers them as systems of stars and gas embedded in extended halos of dark matter , much of it formed by the infall of smaller systems at earlier times . the true extent of a galaxy remains poorly determined , with the virial radius " ( ) providing a characteristic separation between collapsed structures in dynamical equilibrium and external infalling matter . other physical estimates of the extent of gravitational influence include the gravitational radius , gas accretion radius , and galactopause " arising from outflows that stall at 100 - 200 kpc over a range of outflow parameters and confining gas pressures . physical criteria are proposed to define bound structures , including a more realistic definition of for stellar mass and halo mass , half of which formed at assembly redshifts " ranging from . we estimate the extent of bound gas and dark matter around galaxies to be kpc . the new virial radii , with mean kpc , are 40 - 50% smaller than values estimated in recent hst / cos detections of and absorbers around galaxies . in the new formalism , the milky way stellar mass , , would correspond to kpc for half - mass halo assembly at . the frequency per unit redshift of low - redshift absorption lines in qso spectra suggests absorber sizes kpc when related to intervening galaxies . this formalism is intended to clarify semantic differences arising from observations of extended gas in galactic halos , circumgalactic medium ( cgm ) , and filaments of the intergalactic medium ( igm ) . astronomers should refer to _ bound gas _ in the galactic halo or cgm , and _ unbound _ gas at the cgm - igm interface , on its way into the igm .

1401.5799

c

this paper reviews current observations and inferences about the spatial extent of galaxies , and the extent of their gravitational influence . these included several dynamical measures : the gravitational radius " @xmath38 , the radius of influence " @xmath304 , and the accretion radius " @xmath305 . clearly , these definitions are related for systems near dynamical equilibrium in the gravitational potential of the galactic halo . for example , in section 2.1 and table 1 , we found a ratio @xmath306 for nfw halos with concentrations ranging from @xmath69 to @xmath67 . various estimates of radial extent from rotation curves and injection of fast stars into extended halos suggest 150 - 200 kpc extents for halo masses of @xmath83 . kinematic observations of the milky way and m31 give masses @xmath307 within 60 - 80 kpc . the inferred galactic escape velocity , @xmath163 km s@xmath78 , together with the local circular velocity @xmath308 km s@xmath78 , is consistent with halos extending beyond 80 - 100 kpc . somewhat larger halo masses , @xmath309 and @xmath310 ( all masses in @xmath294 units ) have been inferred for the milky way and m31 out to larger radii , using globular clusters , satellite galaxies , and stellar streams . their mass sum , @xmath311 , is comparable to recent estimates of the timing mass of the local group , @xmath312 ( van der marel 2012 ) , although a recent study ( gonzalez et al . 2013 ) used cosmological simulations of halo pairs and a likelihood correction to find a lower value , @xmath313 . for either of these masses , the milky way and andromeda are likely to have radial extents @xmath6 kpc . the answer to the question posed in the title to this paper depends on how one wishes to define or observe the edge of a galaxy " . does a galaxy end because of gravity or outflows ? this paper suggests that the best definition of @xmath0 is in terms of gravitational binding . dark matter halos have very large extents , but galactic outflows can drive gas out of the stellar disk , into the ( bound ) halo , and often into the ( unbound ) cgm and igm . we have formulated a new definition of virial radius in this spirit , based on galaxy assembly in the past , when the background density was higher . as a result , we find virial radii smaller by factors of 0.5 - 0.6 , with important consequences for assessing whether the extended absorption seen with cos is bound halo gas , or unbound gas on its way to the igm . * a revised formulation of virial radius , @xmath314 , expresses the critical overdensity @xmath88 relative to the background matter density , @xmath315 , evaluated at @xmath96 , the redshift of galaxy assembly when half the halo mass had collapsed . here , @xmath96 depends on @xmath3 and is inferred from stellar mass @xmath2 . for dynamical consistency , the circumgalactic medium of large galaxies is defined by their gravitationally bound regions , estimated from the virial radius or gravitational radius , @xmath316 . * for galaxies with halo mass @xmath317 observed by hst / cos at redshifts @xmath318 , the assembly redshifts are @xmath319 . because virialization was largely determined by early collapse , when the background density was higher , we adopt @xmath320 and @xmath0 is reduced by a factor @xmath321 compared to @xmath110 . * halo masses can be estimated from stellar masses by halo - matching based on a physically motivated model ( ttc13 ) for galaxy assembly consistent with rest - frame uv luminosity functions from @xmath322 . applied to 44 cos - halos galaxies , this formalism yields virial radii smaller by factors of 0.5 - 0.6 . galaxies with @xmath323 ( @xmath324 ) have @xmath325 kpc , consistent with their observed radial extent . * if confined by cgm gas pressure , the wind galactopause is predicted to occur at distances @xmath326 kpc for sfr = 10 - 30 @xmath294 yr@xmath78 , mass - loading factors @xmath327 , and the observed range of cgm gas pressures and wind velocities . stronger galaxy winds could break out into the lower density igm , with radial extents beyond 200 kpc determined by the strength , collimation , and duration of the outflow . * for dynamical masses of the large local group galaxies , we adopt : milky way ( @xmath328 ) and m31 ( @xmath329 ) with ( gravitational ) radial extents of 150 - 200 kpc . in our new formalism , the milky way stellar mass , @xmath8 , would correspond to somewhat higher halo mass @xmath299 and virial radiius @xmath9 kpc for assembly at @xmath10 . the inferred @xmath3 for m31 is anomalously high ( @xmath301 ) , which suggests caution in using halo - matching to obtain @xmath3 and @xmath0 for massive , luminous galaxies ( @xmath291 ) . as noted earlier , the new definition of @xmath0 is important in deciding whether gas in the vicinity of galaxies is gravitationally bound or unbound . whether galaxies are open or closed boxes " helps to determine the metal evolution of galaxies and the extent of metal - pollution of the igm . it also suggests that astronomers should be careful in making interchangeable use of the terms cgm and halo . matter in the galactic halo is _ gravitationally bound _ , while the circumgalactic ( gaseous ) medium may in fact be _ unbound outflows _ that will merge into the igm . what many observers are calling the cgm is probably gas at the cgm igm interface . in observations of large metal - enriched reservoirs around star - forming galaxies ( tumlinson et al . 2013 ; stocke et al . 2013 ) , the radial offset distances @xmath46 of the qso sight lines were normalized to @xmath0 , with much of the metal - enriched gas within @xmath330 . with the revised ( smaller ) virial radii ( table 3 and figure 1 ) some of this metal - enriched gas extends beyond the region of strong gravitational influence , probably on its way out to the igm . this trend is seen in the 28 actively star - forming galaxies in the cos - halos sample , whereas the undetected around 16 passive galaxies all had sight lines with @xmath330 . in future work , both data sets will be examined carefully with the new formalism . i thank crystal martin , michele trenti , massimo ricotti , john stocke , brian keeney , blair savage , and vasily belokurov for helpful discussions , and evan tilton and joshua moloney for comments on the manuscript . this research was supported by the astrophysical theory program at the university of colorado boulder ( grant nnx07-ag77 g from nasa ) . i am grateful to the institute of astronomy at cambridge university for their stimulating scientific environment and support through the sackler visitor program . lccc uniform sphere & @xmath331 & @xmath46 & @xmath332 + plummer model & @xmath333^{-5/2}$ ] & @xmath52 & @xmath334 + jaffe model & @xmath335 & @xmath336 & @xmath337 + hernquist model & @xmath338 & @xmath336 & @xmath339 + nfw model ( @xmath340 ) & @xmath341 & @xmath54 & @xmath342 + nfw model ( @xmath343 ) & @xmath341 & @xmath54 & @xmath344 + nfw model ( @xmath345 ) & @xmath341 & @xmath54 & @xmath346 + nfw model ( @xmath347 ) & @xmath341 & @xmath54 & @xmath348 + 9.00 & 1.709 & & & 12.25 & 1.112 + 9.25 & 1.666 & & & 12.50 & 1.070 + 9.50 & 1.620 & & & 12.75 & 1.026 + 9.75 & 1.572 & & & 13.00 & 0.979 + 10.00 & 1.531 & & & 13.25 & 0.936 + 10.25 & 1.486 & & & 13.50 & 0.891 + 10.50 & 1.439 & & & 13.75 & 0.844 + 10.75 & 1.394 & & & 14.00 & 0.806 + 11.00 & 1.346 & & & 14.25 & 0.765 + 11.25 & 1.302 & & & 14.50 & 0.722 + 11.50 & 1.255 & & & 14.75 & 0.684 + 11.75 & 1.216 & & & 15.00 & 0.645 + 12.00 & 1.155 & & lccccccrrr j0226 + 0015(268,22 ) & 0.22744 & 0.51 & 10.8 & 12.62 & 12.73 & 303 & 178 & 1.03 & 80 + j0401 - 0540 ( 67,24 ) & 0.21969 & 0.37 & 10.2 & 12.08 & 11.86 & 200 & 85 & 1.19 & 86 + j0803 + 4332(306,20 ) & 0.25347 & 1.63 & 11.3 & 13.48 & 13.84 & 581 & 462 & 0.83 & 79 + j0910 + 1014 ( 34,46 ) & 0.14274 & 0.60 & 10.6 & 12.48 & 12.39 & 279 & 133 & 1.09 & 116 + j0910 + 1014(242,34 ) & 0.26412 & 2.16 & 11.5 & 13.76 & 14.40 & 716 & 747 & 0.74 & 139 + j0914 + 2823 ( 41,27 ) & 0.24431 & 0.37 & 9.8 & 11.87 & 11.48 & 169 & 61 & 1.26 & 104 + j0925 + 4004(196,22 ) & 0.24745 & 1.48 & 11.3 & 13.45 & 13.84 & 569 & 462 & 0.83 & 86 + j0928 + 6025(110,35 ) & 0.15400 & 0.52 & 10.8 & 12.65 & 12.73 & 317 & 178 & 1.03 & 94 + j0935 + 0204 ( 15,28 ) & 0.26228 & 0.79 & 11.0 & 12.88 & 13.13 & 365 & 251 & 0.95 & 114 + j0943 + 0531(106,34 ) & 0.22839 & 0.74 & 10.8 & 12.61 & 12.73 & 300 & 178 & 1.03 & 125 + j0943 + 0531(216,61 ) & 0.14311 & 0.72 & 11.0 & 12.80 & 13.13 & 382 & 251 & 0.95 & 154 + j0943 + 0531(227,19 ) & 0.35295 & 0.24 & 9.6 & 11.66 & 11.31 & 141 & 69 & 1.29 & 95 + j0950 + 4831(177,27 ) & 0.21194 & 1.44 & 11.2 & 13.30 & 13.59 & 511 & 371 & 0.88 & 94 + j1009 + 0713(204,17 ) & 0.22784 & 0.28 & 9.9 & 11.90 & 11.56 & 174 & 66 & 1.24 & 62 + j1009 + 0713(170,9 ) & 0.35569 & 0.29 & 10.3 & 12.06 & 11.98 & 189 & 94 & 1.16 & 45 + j1016 + 4706(274,6 ) & 0.25195 & 0.23 & 10.2 & 12.10 & 11.86 & 202 & 85 & 1.19 & 24 + j1016 + 4706(359,16 ) & 0.16614 & 0.44 & 10.5 & 12.35 & 12.24 & 251 & 117 & 1.12 & 46 + j1112 + 3539(236,14 ) & 0.24670 & 0.52 & 10.3 & 12.17 & 11.98 & 214 & 94 & 1.16 & 54 + j1133 + 0327(110,5 ) & 0.23670 & 1.94 & 11.2 & 13.19 & 13.59 & 515 & 371 & 0.88 & 14 + j1133 + 0327(164,21 ) & 0.15449 & 0.19 & 10.1 & 12.09 & 11.76 & 206 & 78 & 1.21 & 56 + j1157 - 0022(230,7 ) & 0.16378 & 0.55 & 10.9 & 12.72 & 12.91 & 334 & 207 & 1.00 & 20 + j1220 + 3853(225,38 ) & 0.27371 & 0.66 & 10.8 & 12.53 & 12.73 & 279 & 178 & 1.03 & 159 + j1233 + 4758(94,38 ) & 0.22210 & 0.70 & 10.8 & 12.58 & 12.73 & 295 & 178 & 1.03 & 137 + j1233 - 0031(168,7 ) & 0.31850 & 0.37 & 10.6 & 12.30 & 12.39 & 230 & 133 & 1.09 & 33 + j1241 + 5721(199,6 ) & 0.20526 & 0.20 & 10.2 & 12.10 & 11.86 & 205 & 85 & 1.19 & 21 + j1241 + 5721(208,27 ) & 0.21780 & 0.20 & 10.1 & 12.02 & 11.76 & 192 & 78 & 1.21 & 92 + j1245 + 3356(236,36 ) & 0.19248 & 0.20 & 9.9 & 11.81 & 11.56 & 178 & 66 & 1.24 & 117 + j1322 + 4645(349,11 ) & 0.21418 & 0.58 & 10.8 & 12.50 & 12.73 & 303 & 178 & 1.03 & 39 + j1330 + 2813(289,28 ) & 0.19236 & 0.24 & 10.3 & 12.22 & 11.98 & 225 & 94 & 1.16 & 91 + j1342 - 0053(157,10 ) & 0.22702 & 1.08 & 11.0 & 12.67 & 13.13 & 345 & 251 & 0.95 & 37 + j1342 - 0053(77,10 ) & 0.20127 & 0.25 & 10.5 & 12.34 & 12.24 & 247 & 117 & 1.12 & 34 + j1419 + 4207(132,30 ) & 0.17925 & 0.55 & 10.6 & 12.46 & 12.39 & 272 & 133 & 1.09 & 92 + j1435 + 3604(126,21 ) & 0.26226 & 0.33 & 10.4 & 12.20 & 12.01 & 218 & 97 & 1.15 & 86 + j1435 + 3604(68,12 ) & 0.20237 & 1.37 & 11.1 & 13.08 & 13.35 & 433 & 302 & 0.92 & 40 + j1437 + 5045(317,38 ) & 0.24600 & 0.46 & 10.2 & 12.06 & 11.86 & 196 & 85 & 1.19 & 149 + j1445 + 3428(232,33 ) & 0.21764 & 0.29 & 10.4 & 12.26 & 12.01 & 230 & 97 & 1.15 & 117 + j1514 + 3619(287,14 ) & 0.21223 & 0.18 & 9.7 & 11.81 & 11.39 & 164 & 57 & 1.28 & 49 + j1550 + 4001(197,23 ) & 0.31247 & 1.78 & 11.4 & 13.50 & 14.10 & 578 & 577 & 0.79 & 106 + j1550 + 4001(97,33 ) & 0.32179 & 0.82 & 10.9 & 12.54 & 12.91 & 311 & 207 & 1.00 & 155 + j1555 + 3628(88,11 ) & 0.18930 & 0.54 & 10.5 & 12.38 & 12.24 & 254 & 117 & 1.12 & 35 + j1617 + 0638(253,39 ) & 0.15258 & 2.65 & 11.5 & 14.03 & 14.40 & 912 & 747 & 0.74 & 104 + j1619 + 3342(113,40 ) & 0.14137 & 0.19 & 10.1 & 12.12 & 11.76 & 211 & 78 & 1.21 & 100 + j2257 + 1340(270,40 ) & 0.17675 & 0.67 & 10.9 & 12.78 & 12.91 & 348 & 207 & 1.00 & 120 + j2345 - 0059(356,12 ) & 0.25389 & 0.80 & 10.9 & 12.63 & 12.91 & 304 & 207 & 1.00 & 48 + all galaxies & 44 & 10.61 & 12.53 & 12.55 & 315 & 200 & 0.563 + @xmath349 & 12 & 10.23 & 12.12 & 11.89 & 207 & 81 & 0.422 + @xmath350 & 10 & 10.84 & 12.62 & 12.80 & 309 & 190 & 0.613 + @xmath351 & 7 & 11.34 & 13.53 & 13.97 & 626 & 534 & 0.845 +

, the milky way stellar mass , , would correspond to kpc for half - mass halo assembly at . astronomers should refer to _ bound gas _ in the galactic halo or cgm , and _ unbound _ gas at the cgm - igm interface , on its way into the igm .

our current view of galaxies considers them as systems of stars and gas embedded in extended halos of dark matter , much of it formed by the infall of smaller systems at earlier times . the true extent of a galaxy remains poorly determined , with the virial radius " ( ) providing a characteristic separation between collapsed structures in dynamical equilibrium and external infalling matter . other physical estimates of the extent of gravitational influence include the gravitational radius , gas accretion radius , and galactopause " arising from outflows that stall at 100 - 200 kpc over a range of outflow parameters and confining gas pressures . physical criteria are proposed to define bound structures , including a more realistic definition of for stellar mass and halo mass , half of which formed at assembly redshifts " ranging from . we estimate the extent of bound gas and dark matter around galaxies to be kpc . the new virial radii , with mean kpc , are 40 - 50% smaller than values estimated in recent hst / cos detections of and absorbers around galaxies . in the new formalism , the milky way stellar mass , , would correspond to kpc for half - mass halo assembly at . the frequency per unit redshift of low - redshift absorption lines in qso spectra suggests absorber sizes kpc when related to intervening galaxies . this formalism is intended to clarify semantic differences arising from observations of extended gas in galactic halos , circumgalactic medium ( cgm ) , and filaments of the intergalactic medium ( igm ) . astronomers should refer to _ bound gas _ in the galactic halo or cgm , and _ unbound _ gas at the cgm - igm interface , on its way into the igm .

0903.0616

m

the kepler mission will soon yield precise high - cadence time - series photometry of hundreds of pulsating stars every few months for at least 3.5 years @xcite . we will then face the challenge of determining the fundamental properties of these stars from the data , by attempting to match them with the output of computer models . the traditional approach to this task is to make informed guesses for each of the model parameters , and then to adjust them iteratively until an adequate match is found . the volume of asteroseismic data that will emerge from the kepler mission calls for a more automated approach to modeling that initially explores a broad range of model parameters in an objective manner . the cornerstone of our model - fitting approach is a global optimization method using a parallel genetic algorithm . the result of the global search provides the starting point for a local analysis using a levenberg - marquardt algorithm with singular value decomposition , which also allows us to explore the information content of the observables and the impact of including additional observational constraints @xcite . we have recently adapted the aarhus stellar evolution code ( astec ; * ? ? ? * ) and adiabatic pulsation code ( adipls ; * ? ? ? * ) to interface with the parallel genetic algorithm . these are essentially the same models that were developed for the analysis of helioseismic data , and are the source of model s of @xcite , which has been used extensively as a reference model for solar inversions . using these models for the analysis of pulsations in solar - type stars will provide some internal consistency in our understanding of solar - like oscillations . briefly , these stellar models use the opal 2005 equation of state ( see * ? ? ? * ) and the most recent opal opacities ( see * ? ? ? * ) , supplemented by kurucz opacities at low temperatures . the nuclear reaction rates come from @xcite , convection is described by the mixing - length theory of @xcite , and we included the effects of helium settling as described by @xcite . each model evaluation involves the computation of a stellar evolution track from the zero - age main sequence through a mass - dependent number of internal time steps , terminating prior to the beginning of the red - giant stage . rather than calculate the pulsation frequencies for each of the 200300 models along the track , we exploit the fact that the average frequency spacing of consecutive radial overtones @xmath6 in most cases is a monotonically decreasing function of age @xcite . once the evolution track is complete , we start with a pulsation analysis of the model at the middle time step and then use a binary decision tree comparing the observed and calculated values of @xmath6to select older or younger models along the track . in practice , this recipe allows us to interpolate the age between the two nearest time steps by running the pulsation code on just 8 models from each stellar evolution track . since we are interested in developing a general - purpose modeling tool for asteroseismic data from the kepler mission , we need to select a global method for optimizing the match between our model output and the available observations of any given star . using only observations and the constitutive physics of the model to restrict the range of possible values for each parameter , a genetic algorithm ( ga ; * ; * ? ? ? * ) can provide a relatively efficient means of searching globally for the optimal model . although it is more difficult for a ga to find _ precise _ values for the optimal set of parameters efficiently , it is well suited to search for the _ region _ of parameter space that contains the global minimum . in this sense , the ga is an objective means of obtaining a good first guess for a more traditional local analysis method , which can narrow in on the precise values and uncertainties of the optimal model parameters . @xcite developed a fully parallel and distributed implementation of the pikaia genetic algorithm that was originally written by @xcite . @xcite used this modeling tool in the context of white dwarf asteroseismology , which ultimately led to a number of interesting physical results , including : ( 1 ) a precise estimate of the astrophysically important ( @xmath7c + @xmath8he @xmath9 @xmath10o ) nuclear reaction rate @xcite , ( 2 ) the first unambiguous detection of a crystallized core in a massive pulsating white dwarf @xcite , and ( 3 ) asteroseismic confirmation of a key prediction of diffusion theory in white dwarf envelopes @xcite . the impact of this method on the analysis of pulsating white dwarfs suggests that seismological modeling of other types of stars could also benefit from this approach . our implementation of the ga optimizes four adjustable model parameters ; these are the stellar mass ( @xmath11 ) from 0.75 to 1.75 @xmath12 , the metallicity ( @xmath13 ) from 0.002 to 0.05 ( equally spaced in @xmath14 ) , the initial helium mass fraction ( @xmath15 ) from 0.22 to 0.32 , and the mixing - length parameter ( @xmath16 ) from 1 to 3 . the stellar age ( @xmath17 ) is optimized internally during each model evaluation by matching the observed value of @xmath6 ( see [ sec2.1 ] ) . the ga uses two - digit decimal encoding , so there are 100 possible values for each parameter within the ranges specified above . each run of the ga evolves a population of 128 models through 200 generations to find the optimal set of parameters , and we execute 4 independent runs with different random initialization to ensure that the best model identified is truly the global solution . this method requires about @xmath18 model evaluations , compared to @xmath19 models for a complete grid at the same sampling density , making the ga nearly 1000 times more efficient than a complete grid ( currently 1 week of computing time , compared to many years for a grid ) . of course , a grid could in principle be applied to hundreds of observational data sets without calculating additional models but the ga approach also gives us the flexibility to improve the physical ingredients in the future , while the physics of a grid would be fixed . once the ga brings us close enough to the global solution , we can switch to a local optimization method . we implement a modified levenberg - marquardt ( lm ) algorithm that uses singular value decomposition ( svd ) on the calculated design matrices to filter the least important information from the observables ( some of which may be dominated by noise ) . this then provides an effective local inversion technique . lm is relatively fast and stable , and convergence typically occurs within 34 iterations . we treat the local analysis as a @xmath20-minimization problem , where one seeks to find the set of parameters * p * that minimizes @xmath21 here @xmath22 and @xmath23 are the @xmath24 measurements and errors , while the @xmath25 are the calculated model observables resulting from * p*. the lm+svd method requires an initial guess of * p * , and these are taken to be the results from the global search by the ga . the lm+svd analysis subsequently uses derivative information from the model at the current parameter values to calculate suggested parameter changes @xmath26 that will bring the model observables closer to the observations . we have three main motivations for implementing a local optimization method at the end of the global search . first , the ga has a limited resolution for each parameter , and the values that match the observations best are most likely between the fixed sample points . for example , the parameter @xmath13 near solar values is tuned by the ga along sample points spaced 0.0006 apart , corresponding to a precision of roughly 3% . considering that the search resolution is limited for _ all _ of the model parameters , this significantly limits the precision of the global optimization . the resolution of the local analysis is limited only by the precision of the stellar evolution and pulsation codes , so we use it to adjust the models below the resolution of the ga search . our second motivation for the local analysis is to quantify the final parameter uncertainties and correlations , and to probe the information content of the observables . we do this by calculating the derivatives of the model observables with respect to each of the fitted parameters , and then dividing each vector by the corresponding measurement error . this matrix can be referred to simply as the design matrix * d*. we subsequently calculate the singular value decomposition of * d * for some of its very useful properties ( see appendix [ appb ] ) . the final motivation for implementing a local analysis is to explore the effects of using different physical descriptions of the stellar interior @xcite . when the changes to the underlying physics are relatively subtle , we can assume that the global search by the ga using one set of assumptions will also provide a good starting point for a local analysis under slightly perturbed conditions . by applying the techniques explained in @xcite , such an analysis could demonstrate that the observational data contain enough information to distinguish clearly between different choices for the equation of state , for example . ultimately , this technique could reveal discrepancies in the observables that indicate which physical description is most suitable for the star under investigation . we apply the local analysis to each of the sets of optimal parameters identified by the four independent runs of the ga . the local analysis involves the gradual refinement of the optimal parameters through the iterative application of several steps : 1 . scan the @xmath13 parameter within @xmath270.002 of its original value , and perform a new minimization for each starting value of @xmath13 . we retain the set of parameters * p@xmath28 * that result in the lowest value of the reduced @xmath20 . 2 . starting with * p@xmath28 * , scan the @xmath15 parameter within @xmath270.01 of its initial value , and perform a new minimization for each starting value of @xmath15 . again we retain the * p@xmath28 * that yields the lowest reduced @xmath20 . 3 . rescan the @xmath13 parameter beginning with * p@xmath28*. we adopt as the optimal model the set of parameters that yield the lowest reduced @xmath20 from this final iteration . the reason for scanning @xmath15 and @xmath13 becomes clear when we begin to understand the intrinsic parameter correlations , which are enhanced by the limited set of observations ( see [ sec3.2 ] ) . in most cases the best model found by the global search leads to the final best model after the local optimization . however , sometimes a model that appears to be marginally worse at the end of the global search is improved more substantially during the local analysis . we take our final solution to be the best match from the four independent analyses , and for clarity we report the results of the corresponding global search even when it is not the best of the four models identified by the ga ( cf . table [ tab1 ] ) . the final uncertainties in the parameter values ( @xmath29 ) and in the model observables ( @xmath30 ) are calculated using svd ( see appendix [ appb ] ) .

solar - like oscillations have already been detected from the ground in several stars , and nasa s kepler mission is poised to unleash a flood of stellar pulsation data . deriving reliable asteroseismic information from these observations demands a significant improvement in our analysis methods . in this paper the cornerstone of our automated approach is an optimization method using a parallel genetic algorithm . # 1]*#1 ] *

over the past two decades , helioseismology has revolutionized our understanding of the interior structure and dynamics of the sun . asteroseismology will soon place this knowledge into a broader context by providing structural data for hundreds of sun - like stars . solar - like oscillations have already been detected from the ground in several stars , and nasa s kepler mission is poised to unleash a flood of stellar pulsation data . deriving reliable asteroseismic information from these observations demands a significant improvement in our analysis methods . in this paper we report the initial results of our efforts to develop an objective stellar model - fitting pipeline for asteroseismic data . the cornerstone of our automated approach is an optimization method using a parallel genetic algorithm . we describe the details of the pipeline and we present the initial application to sun - as - a - star data , yielding an optimal model that accurately reproduces the known solar properties . # 1]*#1 ] *

0903.0616

r

the overall goal of our model - fitting pipeline is to take a range of oscillation frequencies and other constraints as input , to identify and refine the model that best matches these observations , and to produce the optimal values of several parameters and other characteristics of the model as output . to ensure that our pipeline yields reliable results , we must begin by applying it to data where the basic stellar properties are already known . the most obvious choice is the sun , where high - quality sun - as - a - star observations are available from multiple experiments . if we feed the pipeline a stellar - like set of solar data , we can judge the experiment a success if the pipeline returns the known solar properties within acceptable tolerances . to reach this goal , we must first optimize the efficiency of the search method by passing synthetic data through the model - fitting procedure ( [ sec3.1 ] ) , calibrate the differences between our models and real observations due to near - surface effects and ensure that the resulting empirical correction does not introduce any large systematic errors in the final model parameters ( [ sec3.2 ] ) , and finally quantify any differences in the derived model parameters due to the source and error properties of the solar data ( [ sec3.3 ] ) . the results of these experiments are listed in table [ tab1 ] , and described in the following subsections . we will eventually want to validate this model - fitting pipeline using other stars that differ from the sun , where the physical properties are known with lower precision . however , because of the computation - intensive nature of our method ( which currently requires about 1 week on 512 processors ) , we will consider here only the initial validation using solar data . although genetic algorithms are often more efficient than other comparably global optimization methods , they are still quite demanding computationally . fortunately , the procedure is inherently parallelizable ; we need to calculate many models , and each one of them is independent of the others . so the number of available processors determines the number of models that can be calculated in parallel . also , there is very little communication overhead ; parameter values are sent to each processor , and they return either a list of observables or just a goodness - of - fit measure if the predictions have already been compared to the observations . the parallel version of the pikaia genetic algorithm is perfectly general , and did not require any structural modifications to interface with our stellar evolution and pulsation codes . the efficiency of genetic - algorithm - based optimization can be defined as the number of model evaluations required to yield the global solution , relative to the number of models that would be required for a complete grid at the same sampling density . in practice , a ga is usually hundreds or even thousands of times more efficient than a complete grid , and its performance is fairly insensitive to the few internal parameters that control its operation . we initially set these internal parameters ( e.g. population size , run length , crossover and mutation rates ) based on our experience with white dwarf models , but we also ran synthetic data through the optimization procedure in a series of `` hare & hound '' ( h&h ) exercises to ensure that the input parameters were recovered faithfully . the basic procedure for an h&h exercise is for one team member to calculate the theoretical oscillation frequencies and other observables for a specific set of model parameters ( @xmath29 ) . a subset of the predictions , typical of whatever is available from actual observations , is then given to another team member who does not know the source parameters . the data are passed through the complete optimization method in an attempt to recover the source parameters without any additional information . the results of such exercises are used to quantify the success rate of the optimization method ( the fraction of independent runs that lead to the known source parameters ) , and to improve its efficiency ( minimize the number of model evaluations ) if possible . for our first h&h exercise ( with source parameters listed in the row labeled `` h&h1 '' in table [ tab1 ] ) , we assume that typical asteroseismic data from the kepler mission will include twelve frequencies for each of the radial ( @xmath31 ) , dipole ( @xmath32 ) , and quadrupole ( @xmath33 ) modes , with consecutive radial orders in the range @xmath34 is necessary for the operation of our pipeline . information about the value of @xmath3 is only provided for completeness . ] . thus , we allowed the ga to fit a total of 36 oscillation frequencies . we assigned statistical uncertainties to each frequency by scaling up the errors on the corresponding modes in bison data @xcite by a factor of 10 , which is roughly what we expect from kepler data ( @xmath35hz ) . we complemented this synthetic asteroseismic information with artificial data on the effective temperature and luminosity , with errors comparable to what is expected for stars in the kepler input catalog ( * ? ? ? * @xmath36 k , @xmath37 ) . this is the simplest conceivable test of the pipeline if we generate a set of model data and then use the optimization method to match those data using exactly the same models , how quickly will the ga find the true values of the model parameters ? the results of such a test verify the basic functionality of the algorithm , and tell us the approximate number of iterations ( or `` generations '' of the ga ) that we need to execute before stopping the search . of the best solution in the population as a function of generation number for four independent runs of the genetic algorithm on the h&h1 data set . all four runs converge to the global solution within 150 generations.[fig1 ] ] the results of h&h1 for four independent runs of the ga are illustrated in figure [ fig1 ] . each convergence curve shows the reduced @xmath20 ( hereafter @xmath38 where @xmath39 is the number of constraints and @xmath40 is the number of parameters ) for the best model in the ga population as a function of generation number . all of the runs converge in less than @xmath41150 generations , in each case leading to the same mass as the source model and values for the other parameters offset by less than @xmath411% from the source values ( the best model from the ga is listed in the row labeled `` global '' under h&h1 in table [ tab1 ] ) . since we expect the algorithm to converge more quickly in this highly idealized case , we decided to continue running the ga for 200 generations for the more difficult tests that follow . the local analysis using lm+svd brings the final value of the metallicity ( @xmath13 ) closer to the source model , while retaining comparable accuracy for the other parameters ( see the row labeled `` local '' under h&h1 in table [ tab1 ] ) . note that the unusually small values of @xmath42 in table [ tab1 ] are a consequence of adopting scaled - up bison errors for the model frequencies , which vastly overestimate the true theoretical uncertainties in this case . the biggest challenge to comparing the oscillation frequencies from theoretical models with those actually observed in solar - type stars are the systematic errors due to _ surface effects_. the mixing - length parameterization of convection that is used in most stellar models is insufficient to describe the near - surface layers , and this leads to a systematic difference of several @xmath43hz ( up to about 0.3% for a solar model ) between the observed and calculated frequencies ( see figure [ fig2 ] ) . the offset is nearly independent of the spherical degree ( @xmath4 ) of the mode and grows larger towards the acoustic cutoff frequency . the 3d simulations of convection that might in principle reduce this discrepancy for individual stars are far too computationally expensive for the model - fitting approach that we are developing . instead , we adopt the method for empirical correction of surface effects described by @xcite , which uses the discrepancies between model s and golf data for the sun @xcite to calibrate the empirical surface correction . the models that we have adopted for the pipeline include a slightly different set of physical ingredients than what was used to produce model s from @xcite . to characterize the surface effects , we need a model that uses the same physics as the other models in the present investigation , while matching the model s frequencies as closely as possible . to find such a model ( hereafter model ) , we used our pipeline to fit the computed frequencies of model s. since this involves the comparison of two sets of model frequencies both of which include a mixing - length parameterization of convection there are no surface effects to consider . again we allowed the ga to fit 36 frequencies for modes with @xmath34 and @xmath44 and we used the same scaled bison errors and the same constraints on the effective temperature and luminosity ( see [ sec3.1 ] ) . the resulting optimal models from the global search and the local analysis had the parameter values listed in the rows below `` model '' in table [ tab1 ] . following @xcite , we fit a power law to the differences between the frequencies of the radial modes of this model and the corresponding frequencies from bison data to characterize the surface effects ( see figure [ fig2 ] and appendix [ appa ] ) . we found a power law exponent @xmath45 , slightly lower than the value ( @xmath46 ) derived by @xcite using data from the golf experiment . with this exponent fixed , the recipe of @xcite describes how to predict the surface effect for any other set of calculated oscillation data , allowing us to apply this empirical correction to each of our models before comparing them to observations . ] if our strategy of making this empirical correction to each of our models is to succeed , it must not only work _ well _ for models in a certain region of the search space it must work _ best _ for the model that simultaneously matches all of the independent observational constraints within their uncertainties . our second h&h exercise ( with parameters listed in the row labeled `` h&h2 '' in table [ tab1 ] ) was designed to test the behavior of the pipeline with the surface correction included . the source model and other constraints were identical to those used for h&h1 , but the surface effect was first calculated from eq.([eq.fobs - fref ] ) and then applied to each model frequency prior to fitting so that the h&h2 data would mimic real observations . we fixed the value of @xmath47 to 4.82 ( as determined above ) and we calculated the value of @xmath48 using eq.([eq.fobs - fref ] ) with the scaling factor @xmath49 determined using bison data as the reference set of observations . the fitting procedure then included the empirical surface correction attempting to remove the systematic frequency errors that we artificially introduced as detailed in appendix [ appa ] . again this test reveals how long we must run the ga for it to converge to the global solution , but it also quantifies any systematic errors on the derived parameter values that arise from our implementation of the empirical surface correction . as expected , the ga converges to the global solution more slowly with the inclusion of the surface correction . within 150200 generations , two of the four runs converged to the parameters listed in the row labeled `` global '' under h&h2 in table [ tab1 ] , while the other two remained in a nearby local minimum . if we had continued the search , these two runs would eventually also have found the global solution . the largest systematic error appears in the value of @xmath13 , which is about 6% ( two sample points ) higher in the global solution than in the source model . there are also smaller errors ( @xmath411% ) on the derived values of @xmath15 and @xmath17 . the results of the local analysis ( listed in the row labeled `` local '' under h&h2 in table [ tab1 ] ) bring both @xmath13 and @xmath17 closer to the source values while preserving the accuracy of the other parameters . the remaining errors are not a deficiency in the ability of the method to find the true solution the identified model actually results in a lower @xmath42 than the source parameters . the reason for this counter - intuitive result is that we can only generate a surface correction when there is a reference set of observed oscillation frequencies ( see appendix [ appa ] ) . our source model used the bison data for reference , and these data have a slightly different value of the large frequency separation ( @xmath50hz ) than the resulting source model ( @xmath51hz ) . since this quantity is used by the binary decision tree to fit the age of the model , the small difference leads to a slightly offset derived age which subsequently modifies the optimal composition through intrinsic parameter correlations . in other words , when generating a model with the surface correction included from a given set of parameters , those same parameters will generally not provide the best fit to the resulting observables . this actually highlights our inability to generate realistic data with surface effects , rather than an inherent limitation in our fitting method . in any case , this exercise demonstrates that the systematic errors resulting from our surface correction are small , and that running the ga for 200 generations should be sufficient for real observations . ultimately , our model - fitting pipeline can only be judged a success if it leads to accurate estimates of the stellar properties for the star that we know best : the sun . up to this point , we have essentially been fitting models to synthetic data using the solar data from bison only to calibrate the empirical surface correction and to provide realistic errors . there are many other ingredients in our models that could in principle be insufficient descriptions of the actual conditions inside of real stars deficiencies that could easily lead to systematic errors in our determinations of the optimal model parameters for a given set of oscillation data . for example , we initially tried to use models that employed the simpler eff equation of state @xcite for computational expediency , but this led to estimates of the stellar mass about 10% too high for the sun , and unacceptably large systematic errors on many of the other stellar properties . even attempting to ignore the effects of helium settling proved to be too coarse an approximation , leading to 5% errors on the mass . the only potential ingredient that we omitted without serious consequences was heavy element settling . this is not to say that simpler stellar models can not be used in the analysis of asteroseismic data , but rather that some of the more sophisticated ingredients are required to obtain accurate results from a _ global _ search of the parameter space . having demonstrated the effectiveness of the method by fitting our models to synthetic data , and after calibrating the empirical surface correction using the differences between model and the bison data , we finally applied our model - fitting pipeline to solar data from the bison and golf experiments . the oscillation frequencies from these two sources are identical to each other within the observational uncertainties , but their noise properties are slightly different allowing us to quantify any systematic errors that might arise from subtle effects in the data acquisition and analysis methods . in both cases we used the same set of modes referenced in the earlier experiments ( @xmath34 , @xmath44 ) with the respective errors again scaled up by a factor of 10 , and the same constraints on the effective temperature and luminosity ( see [ sec3.1 ] ) . the two sets of input data differed only in the absolute values of the oscillation frequencies ( yielding distinct values of @xmath6 for fitting the stellar age ) , and in the statistical uncertainties assigned to each mode ( leading to subtle differences in the weighting of the fit ) . in general , the four independent runs of the ga for each data set led to slightly different results after 200 generations , so we list only the solution leading to the best model in the rows below `` bison '' and `` golf '' in table [ tab1 ] . we found that if we continued running the ga for up to 300 generations , the same optimal solution was identified in 23 of the independent runs . to reduce the total computing time required for future experiments , we decided to stop the ga earlier when 12 of the runs would still reliably identify the global solution . both data sets lead to identical values of the mass and metallicity from the global search , with slight variations in the values of the other parameters . these minor differences largely disappear after the local analysis . note that because we multiplied the true observational errors by a factor of 10 for the fitting , the resulting values of @xmath42 are @xmath410.1 . although the fits used a limited range of frequencies and did not include @xmath52 modes , the optimal models also match the modes with lower frequencies and higher degree ( see the bison fit in figure [ fig3 ] ) and accurately reproduce the known solar properties . and plot them against the oscillation frequency , along with the optimal model from our asteroseismic modeling pipeline ( open points ) . note that the pipeline only used the @xmath44 frequencies between the dashed lines for the fit , but the resulting optimal model also matches the @xmath4=3 modes and frequencies outside of the fitting range.[fig3 ] ] rccccccccc h&h1 & 1.000 & 0.0191 & 0.271 & 2.06 & 4.54 & 5780 & 1.001 & 1.000 & @xmath53 + global : & 1.000 & 0.0197 & 0.273 & 2.04 & 4.51 & 5763 & 0.990 & 1.000 & 0.052 + local : & 1.001 & 0.0193 & 0.269 & 2.03 & 4.58 & 5755 & 0.986 & 1.000 & 0.004 + error : & 0.009 & 0.0014 & 0.012 & 0.07 & 0.16 & 61 & 0.040 & 0.004 & @xmath53 + model & @xmath53&@xmath53&@xmath53&@xmath53&@xmath53&@xmath53&@xmath53&@xmath53&@xmath53 + global : & 0.990 & 0.0191 & 0.277 & 2.04 & 4.56 & 5778 & 0.993 & 0.996 & 0.062 + local : & 0.993 & 0.0188 & 0.275 & 2.05 & 4.52 & 5787 & 1.001 & 0.997 & 0.039 + error : & 0.009 & 0.0013 & 0.010 & 0.07 & 0.16 & 60 & 0.042 & 0.003 & @xmath53 + h&h2 & 1.000 & 0.0191 & 0.271 & 2.06 & 4.54 & 5780 & 1.001 & 1.000 & @xmath53 + global : & 1.010 & 0.0210 & 0.272 & 2.12 & 4.67 & 5773 & 1.005 & 1.004 & 0.042 + local : & 1.010 & 0.0197 & 0.268 & 2.09 & 4.55 & 5776 & 1.006 & 1.003 & 0.007 + error : & 0.010 & 0.0016 & 0.012 & 0.07 & 0.16 & 61 & 0.041 & 0.004 & @xmath53 + bison & @xmath53&@xmath53&@xmath53&@xmath53&@xmath53&@xmath53&@xmath53&@xmath53&@xmath53 + global : & 1.010 & 0.0217 & 0.278 & 2.14 & 4.52 & 5793 & 1.018 & 1.004 & 0.197 + local : & 1.012 & 0.0200 & 0.267 & 2.12 & 4.65 & 5779 & 1.010 & 1.004 & 0.146 + error : & 0.008 & 0.0014 & 0.010 & 0.06 & 0.17 & 55 & 0.039 & 0.003 & @xmath53 + golf & @xmath53&@xmath53&@xmath53&@xmath53&@xmath53&@xmath53&@xmath53&@xmath53&@xmath53 + global : & 1.010 & 0.0217 & 0.273 & 2.10 & 4.71 & 5752 & 0.990 & 1.004 & 0.183 + local : & 1.011 & 0.0199 & 0.267 & 2.11 & 4.66 & 5779 & 1.008 & 1.004 & 0.124 + error : & 0.014 & 0.0017 & 0.017 & 0.07 & 0.21 & 67 & 0.043 & 0.005 & @xmath53

over the past two decades , helioseismology has revolutionized our understanding of the interior structure and dynamics of the sun . we describe the details of the pipeline and we present the initial application to sun - as - a - star data , yielding an optimal model that accurately reproduces the known solar properties .

over the past two decades , helioseismology has revolutionized our understanding of the interior structure and dynamics of the sun . asteroseismology will soon place this knowledge into a broader context by providing structural data for hundreds of sun - like stars . solar - like oscillations have already been detected from the ground in several stars , and nasa s kepler mission is poised to unleash a flood of stellar pulsation data . deriving reliable asteroseismic information from these observations demands a significant improvement in our analysis methods . in this paper we report the initial results of our efforts to develop an objective stellar model - fitting pipeline for asteroseismic data . the cornerstone of our automated approach is an optimization method using a parallel genetic algorithm . we describe the details of the pipeline and we present the initial application to sun - as - a - star data , yielding an optimal model that accurately reproduces the known solar properties . # 1]*#1 ] *

0804.3215

i

0.275 in optical packet - switched ring wavelength division multiplexing ( wdm ) networks have emerged as a promising solution to alleviate the capacity shortage in the metropolitan area , which is commonly referred to as metro gap . packet - switched ring networks , such as the resilient packet ring ( rpr ) @xcite , overcome many of the shortcomings of circuit - switched ring networks , such as low provisioning flexibility for packet data traffic @xcite . in addition , the use of multiple wavelength channels in wdm ring networks , see e.g. , @xcite , overcomes a key limitation of rpr , which was originally designed for a single - wavelength channel in each ring direction . in optical packet - switched ring networks , the destination nodes typically remove ( strip ) the packets destined to them from the ring . destination stripping _ allows the destination node as well as other nodes downstream to utilize the wavelength channel for their own transmissions . with this so - called _ spatial wavelength reuse _ , multiple simultaneous transmissions can take place on any given wavelength channel . spatial wavelength reuse is maximized through shortest path routing , whereby the source node sends a packet in the ring direction that reaches the destination with the smallest hop distance , i.e. , traversing the smallest number of intermediate network nodes . multicast traffic is widely expected to account for a large portion of the metro area traffic due to multi - party communication applications , such as tele - conferences @xcite , virtual private network interconnections , interactive distance learning , distributed games , and content distribution . these multi - party applications are expected to demand substantial bandwidths due to the trend to deliver the video component of multimedia content in the high - definition television ( hdtv ) format or in video formats with even higher resolutions , e.g. , for digital cinema and tele - immersion applications . while there is at present scant quantitative information about the multicast traffic volume , there is ample anecdotal evidence of the emerging significance of this traffic type @xcite . as a result , multicasting has been identified as an important service in optical networks @xcite and has begun to attract significant attention in optical networking research as outlined in section [ lit : sec ] . metropolitan area networks consist typically of edge rings that interconnect several access networks ( e.g. , ethernet passive optical networks ) and connect to a metro core ring @xcite . the metro core ring interconnects several metro edge rings and connects to the wide area network . the node connecting a metro edge ring to the metro core ring is typically a traffic hotspot as it collects / distributes traffic destined to / originating from other metro edge rings or the wide area network . similarly , the node connecting the metro core ring to the wide area network is typically a traffic hotspot . examining the capacity of optical packet - switched ring networks for hotspot traffic is therefore very important . in this paper we examine the multicast capacity ( maximum achievable long run average multicast packet throughput ) of bidirectional wdm optical ring networks with a single hotspot for a general fanout traffic model comprising unicast , multicast , and broadcast traffic . we consider an arbitrary traffic mix composed of uniform traffic , hotspot destination traffic ( from regular nodes to the hotspot ) , and hotspot source traffic ( from the hotspot to regular nodes ) . we study the widely considered node architecture that allows nodes to transmit on all wavelength channels , but to receive only on one channel . we initially examine shortest path routing by deriving bounds and approximations for the ring segment utilization probabilities due to uniform , hotspot destination , and hotspot source packet traffic . we prove that there are three ring segments ( in a given ring direction ) that govern the maximum segment utilization probability . for the clockwise direction in a network with nodes @xmath0 and wavelengths @xmath1 ( with @xmath2 ) , whereby node 1 receives on wavelength 1 , node 2 on wavelength 2 , @xmath3 , node @xmath4 on wavelength @xmath4 , node @xmath5 on wavelength 1 , and so on , and with node @xmath6 denoting the index of the hotspot node , the three critical segments are identified as : * the segment connecting the hotspot , node @xmath6 , to node 1 on wavelength 1 , * the segment connecting node @xmath7 to node @xmath4 on wavelength @xmath4 , and * the segment connecting node @xmath8 to node @xmath6 on wavelength @xmath4 . the utilization on these three segments limits the maximum achievable multicast packet throughput . we observe from the derived utilization probability expressions that the utilizations of the first two identified segments exceed 1/2 ( and approach 1 ) for large fractions of hotspot source multi- and broadcast traffic , whereas the utilization of the third identified segment is always less than or equal to 1/2 . thus , shortest path routing achieves a long run average multicast throughput of less than two simultaneous packet transmissions ( and approaching one simultaneous packet transmission ) for large portions of hotspot source multi- and broadcast traffic . we specify one - copy routing which sends only one packet copy for hotspot source traffic , while uniform and hotspot destination packet traffic is still served using shortest path routing . one - copy routing ensures a capacity of at least two simultaneous packet transmissions for arbitrary hotspot source traffic , and at least approximately two simultaneous packet transmissions for arbitrary overall traffic . we verify the accuracy of our bounds and approximations for the segment utilization probabilities , which are exact in the limit @xmath9 , through comparisons with utilization probabilities obtained from discrete event simulations . we also quantify the gains in maximum achievable multicast throughput achieved by the one - copy routing strategy over shortest path routing through simulations . this paper is structured as follows . in the following subsection , we review related work . in section [ model : sec ] , we introduce the detailed network and traffic models and formally define the multicast capacity . in section [ gen_prop : sec ] , we establish fundamental properties of the ring segment utilization in wdm packet rings with shortest path routing . in section [ util_laneqla : sec ] , we derive bounds and approximations for the ring segment utilization due to uniform , hotspot destination , and hotspot source packet traffic on the wavelengths that the hotspot is not receiving on , i.e. , wavelengths @xmath10 in the model outlined above . in section [ util_laeqla : sec ] , we derive similar utilization probability bounds and approximations for wavelength @xmath4 that the hotspot receives on . in section [ max_segment : sec ] , we prove that the three specific segments identified above govern the maximum segment utilization and multicast capacity in the network , and discuss implications for packet routing . in section [ num : sec ] , we present numerical results obtained with the derived utilization bounds and approximations and compare with verifying simulations . we conclude in section [ concl : sec ] . there has been increasing research interest in recent years for the wide range of aspects of multicast in general mesh circuit - switched wdm networks , including lightpath design , see for instance @xcite , traffic grooming , see e.g. , @xcite , routing and wavelength assignment , see e.g. , @xcite , and connection carrying capacity @xcite . similarly , multicasting in packet - switched single - hop star wdm networks has been intensely investigated , see for instance @xcite . in contrast to these studies , we focus on packet - switched wdm ring networks in this paper . multicasting in circuit - switched wdm rings , which are fundamentally different from the packet - switched networks considered in this paper , has been extensively examined in the literature . the scheduling of connections and cost - effective design of bidirectional wdm rings was addressed , for instance in @xcite . cost - effective traffic grooming approaches in wdm rings have been studied for instance in @xcite . the routing and wavelength assignment in reconfigurable bidirectional wdm rings with wavelength converters was examined in @xcite . the wavelength assignment for multicasting in circuit - switched wdm ring networks has been studied in @xcite . for unicast traffic , the throughputs achieved by different circuit - switched and packet - switched optical ring network architectures are compared in @xcite . optical _ packet - switched _ wdm ring networks have been experimentally demonstrated , see for instance @xcite , and studied for unicast traffic , see for instance @xcite . multicasting in packet - switched wdm ring networks has received increasing interest in recent years @xcite . the photonics level issues involved in multicasting over ring wdm networks are explored in @xcite , while a node architecture suitable for multicasting is studied in @xcite . the general network architecture and mac protocol issues arising from multicasting in packet - switched wdm ring networks are addressed in @xcite . the fairness issues arising when transmitting a mix of unicast and multicast traffic in a ring wdm network are examined in @xcite . the multicast capacity of packet - switched wdm ring networks has been examined for uniform packet traffic in @xcite . in contrast , we consider non - uniform traffic with a hotspot node , as it commonly arises in metro edge rings @xcite . studies of non - uniform traffic in optical networks have generally focused on issues arising in circuit - switched optical networks , see for instance @xcite . a comparison of circuit - switching to optical burst switching network technologies , including a brief comparison for non - uniform traffic , was conducted in @xcite . the throughput characteristics of a mesh network interconnecting routers on an optical ring through fiber shortcuts for non - uniform unicast traffic were examined in @xcite . the study @xcite considered the throughput characteristics of a ring network with uniform unicast traffic , where the nodes may adjust their send probabilities in a non - uniform manner . the multicast capacity of a single - wavelength packet - switched ring with non - uniform traffic was examined in @xcite . in contrast to these works , we consider non - uniform traffic with an arbitrary fanout , which accommodates a wide range of unicast , multicast , and broadcast traffic mixes , in a wdm ring network .

, we analyze the maximum achievable long - run average packet throughput , which we refer to as _ multicast capacity _ , of bi - directional shortest - path routed wdm rings . we characterize the segment utilization probabilities through bounds and approximations , which we verify through simulations . we discover that shortest - path routing can lead to utilization probabilities above one half for moderate to large portions of hotspot source multi- and broadcast traffic , and consequently multicast capacities of less than two simultaneous packet transmissions . we outline a one - copy routing strategy that guarantees a multicast capacity of at least two simultaneous packet transmissions for arbitrary hotspot source traffic . keywords : hotspot traffic , multicast , packet throughput , shortest path routing , spatial reuse , wavelength division multiplexing ( wdm ) .

packet - switching wdm ring networks with a hotspot transporting unicast , multicast , and broadcast traffic are important components of high - speed metropolitan area networks . for an arbitrary multicast fanout traffic model with uniform , hotspot destination , and hotspot source packet traffic , we analyze the maximum achievable long - run average packet throughput , which we refer to as _ multicast capacity _ , of bi - directional shortest - path routed wdm rings . we identify three segments that can experience the maximum utilization , and thus , limit the multicast capacity . we characterize the segment utilization probabilities through bounds and approximations , which we verify through simulations . we discover that shortest - path routing can lead to utilization probabilities above one half for moderate to large portions of hotspot source multi- and broadcast traffic , and consequently multicast capacities of less than two simultaneous packet transmissions . we outline a one - copy routing strategy that guarantees a multicast capacity of at least two simultaneous packet transmissions for arbitrary hotspot source traffic . keywords : hotspot traffic , multicast , packet throughput , shortest path routing , spatial reuse , wavelength division multiplexing ( wdm ) .

0804.3215

c

we have analytically characterized the segment utilization probabilities in a bi - directional wdm packet ring network with a single hotspot . we have considered arbitrary mixes of unicast , multicast , and broadcast traffic in combination with an arbitrary mix of uniform , hotspot destination , and hotspot source traffic . for shortest - path routing , we found that there are three segments that can attain the maximum utilization , which in turn limits the maximum achievable long - run average multicast packet throughput ( multicast capacity ) . through verifying simulations , we found that our bounds and approximations of the segment utilization probabilities , which are exact in the limit for many nodes in a network with a fixed number of wavelength channels , are fairly accurate for networks with on the order of ten nodes receiving on a wavelength . importantly , we observed from our segment utilization analysis that shortest - path routing does _ not _ maximize the achievable multicast packet throughput when there is a significant portion of multi- or broadcast traffic emanating from the hotspot , as arises with multimedia distribution , such as ip tv networks . we proposed a one - copy routing strategy with an achievable long run average multicast packet throughout of about two simultaneous packet transmissions for such distribution scenarios . this study focused on the maximum achievable multicast packet throughput , but did not consider packet delay . a thorough study of the packet delay in wdm ring networks with a hotspot transporting multicast traffic is an important direction for future research .

packet - switching wdm ring networks with a hotspot transporting unicast , multicast , and broadcast traffic are important components of high - speed metropolitan area networks . for an arbitrary multicast fanout traffic model with uniform , hotspot destination , and hotspot source packet traffic we identify three segments that can experience the maximum utilization , and thus , limit the multicast capacity .

packet - switching wdm ring networks with a hotspot transporting unicast , multicast , and broadcast traffic are important components of high - speed metropolitan area networks . for an arbitrary multicast fanout traffic model with uniform , hotspot destination , and hotspot source packet traffic , we analyze the maximum achievable long - run average packet throughput , which we refer to as _ multicast capacity _ , of bi - directional shortest - path routed wdm rings . we identify three segments that can experience the maximum utilization , and thus , limit the multicast capacity . we characterize the segment utilization probabilities through bounds and approximations , which we verify through simulations . we discover that shortest - path routing can lead to utilization probabilities above one half for moderate to large portions of hotspot source multi- and broadcast traffic , and consequently multicast capacities of less than two simultaneous packet transmissions . we outline a one - copy routing strategy that guarantees a multicast capacity of at least two simultaneous packet transmissions for arbitrary hotspot source traffic . keywords : hotspot traffic , multicast , packet throughput , shortest path routing , spatial reuse , wavelength division multiplexing ( wdm ) .

1010.3017

i

high - redshift star - forming galaxies are becoming an important probe of galaxy formation , reionization , and cosmology . ionizing photons of young stars in star - forming galaxies ionize neutral hydrogen atoms in the interstellar medium ( ism ) , and each subsequent recombination has a probability of @xmath2 of ending up as a photon @xcite . after escaping the ism , these photons can be scattered by neutral hydrogen atoms in the circumgalactic and intergalactic media ( igm ) , which tends to make the emission extended . in this paper , we present predictions for the extended emission associated with high - redshift star - forming galaxies from a radiative transfer model of emission applied to a hydrodynamic cosmological simulation @xcite . as a result of reprocessed ionizing photons , prominent emission can be a characteristic of star - forming galaxies , which can be used to detect high - redshift galaxies . galaxies detected through the strong emission associated with them ( e.g. , from narrowband photometry ) are dubbed lyman - alpha emitters ( laes ) and an increasing number of such galaxies have been discovered ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? high - redshift star - forming galaxies can also be detected from broadband photometry with lyman break technique , which are termed lyman break galaxies ( lbgs ; e.g. , @xcite ) . as long as photons reprocessed from ionizing photons can escape from the ism , emission is also expected to be associated with lbgs . emission encodes useful information about star - forming galaxies and their environments , such as star formation rate ( e.g. , @xcite ) , kinematics of ism gas ( e.g. , @xcite ) , and neutral fraction of igm gas ( e.g. , @xcite ) . photons usually experience complex radiative transfer processing ( resonant scatterings ) in media with neutral hydrogen atoms , which complicates the interpretation of observed emission properties . @xcite ( hereafter paper i ) present a physical model of emission from laes by solving radiative transfer in the circumgalactic and intergalactic media . the radiative transfer calculation is performed in a cosmological volume ( @xmath3 on a side ) from a state - of - the - art radiation - hydrodynamic reionization simulation @xcite . the calculation uses a @xmath4 grid to sample the density , velocity , and temperature of the neutral hydrogen gas in the simulation and is applied to all the sources residing in halos above @xmath5 . resonant scatterings enable photons to probe the circumgalactic and intergalactic environments ( density and velocity structures ) around star - forming galaxies . this leads to a coupling between the observed emission properties and the environments . this simple physical model is able to explain an array of observed properties of @xmath65.7 laes the subaru/_xmm - newton _ deep survey ( sxds ; @xcite ) , including spectra , morphology , apparent luminosity function ( lf ) , shape of the ultraviolet ( uv ) lf , and the distribution of equivalent width . the selection imposed by the environment dependent radiative transfer also introduces interesting new features in the clustering of laes ( @xcite , hereafter paper ii ) . in the above radiative transfer model , while the number of photons is conserved after they escape the ism , the scatterings in the circumgalactic and intergalactic media cause the emission to spread spatially . therefore , one generic prediction of the model is an extended emission halo around a star - forming galaxy . observationally , only a fraction of photons can be detected for an individual source , those included in the central part of the extended emission with high enough surface brightness ( tip of the iceberg ) . the outskirts of the halo with low surface brightness is typically buried in the sky noise . in this paper , we show that it is possible to detect the bottom of the iceberg by stacking the narrowband images of a large number of sources to suppress the sky noise . we present the predictions of the extended emission in the stacked image from our radiative transfer model and discuss what we can learn from it . our radiative transfer modeling is performed for sources at @xmath7 . we study properties of surface brightness profile from stacked images in section 2 . in section 3 , we discuss the observational prospects . we summarize our results in section 4 . throughout the paper , we adopt a spatially flat @xmath8cdm cosmological model for our calculations , with a matter density parameter @xmath9 and a hubble constant @xmath10 in units of @xmath11 . distances are expressed in comoving units unless mentioned clearly otherwise .

photons that escape the interstellar medium of star - forming galaxies may be resonantly scattered by neutral hydrogen atoms in the circumgalactic and intergalactic media , thereby increasing the angular extent of the galaxy s emission . we present predictions of this extended , low surface brightness emission based on radiative transfer modeling in a cosmological reionization simulation .

photons that escape the interstellar medium of star - forming galaxies may be resonantly scattered by neutral hydrogen atoms in the circumgalactic and intergalactic media , thereby increasing the angular extent of the galaxy s emission . we present predictions of this extended , low surface brightness emission based on radiative transfer modeling in a cosmological reionization simulation . the extended emission can be detected from stacked narrowband images of emitters ( laes ) or of lyman break galaxies ( lbgs ) . its average surface brightness profile has a central cusp , then flattens to an approximate plateau beginning at an inner characteristic scale below.2 mpc ( comoving ) , then steepens again beyond an outer characteristic scale of mpc . the inner scale marks the transition from scattered light of the central source to emission from clustered sources , while the outer scale marks the spatial extent of scattered emission from these clustered sources . both scales tend to increase with halo mass , uv luminosity , and observed luminosity . the extended emission predicted by our simulation is already within reach of deep narrowband photometry using large ground - based telescopes . such observations would test radiative transfer models of emission from laes and lbgs , and they would open a new window on the circumgalactic environment of high - redshift star - forming galaxies .

1010.3017

c

after escaping from the ism , photons converted from ionizing photons in star - forming galaxies experience scatterings in the circumgalactic and intergalactic media . such a radiative transfer process makes emission from these galaxies spatially extended . in this paper , built on the radiative transfer modeling of laes in @xcite and @xcite , we investigate the predicted spatial distribution of emission that can be measured from the stacked narrowband image of these galaxies . in general , the predicted surface brightness profile measured from the stacked image has two characteristic scales , an inner one at tenths of mpc and an outer one at about 1 mpc . the profile shows a central cusp inside the inner scale , an approximate plateau between the two scales , and an extended tail beyond the outer scale . the inner scale may possibly be an upper limit , as an effect of the grid resolution in the simulation ( see [ sec : resolution ] ) . the stacked surface brightness profile can be understood as a superposition of the brightness distribution from the stacked sources themselves ( one - halo term ) and that from neighboring clustered sources ( two - halo term ) . the two - halo term is the profile of the ( angular ) two - point correlation function ( plus one ) smoothed by the extended emission profile ( psf ) of individual sources . the smoothing makes the profile flattened on scales smaller than the spatial extent of emission of clustered sources . the outer characteristic scale in the stacked surface brightness profile marks this spatial extent , and the plateau in the profile is a consequence of the smoothing effect . the transition from one - halo term domination to two - halo term domination leads to the inner characteristic scale seen in the stacked surface brightness profile . for continuum selected galaxies ( lbgs ) , the amplitude of the stacked surface brightness profile increases with the source uv luminosity ( or halo mass ) on both small and large scales . the two characteristic scales also increase with the uv luminosity . the central cusp becomes steeper for higher uv luminosity . for line selected galaxies ( laes ) , the amplitude of the central cuspy profile increases with observed luminosity and the slope is steeper than that of lbgs . beyond the inner characteristic scale , the amplitude of the surface brightness profile only has a weak dependence on observed luminosity . the inner and outer characteristic scales do not show strong dependence on luminosity , either . because of the contribution from source clustering to the stacked surface brightness profile , the cumulative luminosity from the stacked image does not converge to the intrinsic luminosity of the stacked sources . it is therefore not straightforward to estimate the total emission from the stacked image . in detail , the surface brightness profile depends on the initial line profile of emission ( after photons escape the ism ) . the source would appear to be more compact in the narrowband image as the initial line shift toward red increases . if this shift is caused by galactic wind , the wind velocity needs to be comparable to a few times the virial velocity of the host halo and the wind needs to be largely isotropic to make emission point - like . conversely , from the measured surface brightness profile in the narrowband image , it may be possible to constrain the effect of galactic winds . it is worth further study along this line . our particular prediction here is for sources after reionization is complete ( @xmath7 ) . it is interesting to investigate how the prediction changes for star - forming galaxies at lower redshifts , because of the large amount of efforts in observing star - forming galaxies around redshift 24 ( e.g. , @xcite ) and in the local universe ( e.g. , @xcite ) . it is also necessary to study the extended emission for sources at higher redshifts , before reionization is complete , given the increasing observational efforts ( e.g. , @xcite ) . at the time of our initial submission , we conclude that deep narrowband photometry from large ground - based telescopes is on the verge of detecting the extended emission around star - forming galaxies , including lbgs and laes ( e.g. , * ? ? ? * ; * ? ? ? now the latest observation by @xcite indeed revealed the extended emission to large radii ( @xmath99 ) around lbgs and laes , starting to verify our prediction and provide stringent test to the theoretical model . the detection of the predicted emission supports the generic picture that radiative transfer in the circumgalactic and intergalactic environments produces extended emission . the extended emission opens a new window to study the circumgalactic and intergalactic environment of high - redshift star - forming galaxies . the surface brightness profile encodes information about cold baryons around galaxies , including their density , temperature , and velocity . all of these properties could be modified by galactic wind , so the extended emission in principle can put constraints on the galactic wind . the stacked image also includes contributions from faint galaxies that can not be detected in a single exposure , which provides an opportunity to study low luminosity star - forming galaxies . the extended emission gives us a better idea on the total amount of emission from galaxies . when compared with the uv emission or other optically - thin line emission ( e.g. , h@xmath75 emission ) , we may infer the dust distribution and its effect on photons . with integral - field - units observations of high - redshift star - forming galaxies , we would have stacked spectra for the extended emission and expect to learn more about galaxy environments . details on how to extract all the information encoded in the extended emission need to be investigated , and radiative transfer calculation , in combination with sophisticated models of star - forming galaxies and their environments , will play an irreplaceable role . zz thanks masami ouchi , tomoki saito , and chuck steidel for helpful discussions and the hospitality of the institute for the physics and mathematics of the universe . we thank andy gould for useful comments and the referee for constructive suggestions . zz gratefully acknowledges support from yale center for astronomy and astrophysics through a ycaa fellowship . zz and dhw thank the institute for advanced study for their hospitality during part of the work . rc acknowledges support from nasa grants nng06gi09 g and nnx08ah31 g . jm is supported by the spanish grant aya2009 - 09745 . atek , h. , kunth , d. , hayes , m. , stlin , g. , & mas - hesse , j. m. 2008 , , 488 , 491 atek , h. , schaerer , d. , & kunth , d. 2009 , , 502 , 791 berlind , a. a. & weinberg , d. h. 2002 , , 575 , 587 bland , j. , & tully , b. 1988 , , 334 , 43 bond , n. a. , feldmeier , j. j. , matkovi , a. , gronwall , c. , ciardullo , r. , & gawiser , e. 2010 , , 716 , l200 bouwens , r. j. , et al . 2010 , , 708 , l69 , cooray , a. , & sheth , r. 2002 , , 372 , 1 cowie , l. l. , & hu , e. m. 1998 , , 115 , 1319 dijkstra , m. , lidz , a. , & wyithe , j. s. b. 2007 , , 377 , 1175 dijkstra , m. , & wyithe , j. s. b. 2010 , , 408 , 352 faucher - gigure , c .- a . , kere , d. , dijkstra , m. , hernquist , l. , & zaldarriaga , m. 2010 , , 725 , 633 finkelstein , s. l. , et al . 2011 , , 735 , 5 fynbo , j. u. , mller , p. , & thomsen , b. 2001 , , 374 , 443 fynbo , j. p. u. , ledoux , c. , mller , p. , thomsen , b. , & burud , i. 2003 , , 407 , 147 gawiser , e. , et al . 2007 , , 671 , 278 giavalisco , m. , koratkar , a. , & calzetti , d. 1996 , , 466 , 831 gould , a. , & weinberg , d. h. 1996 , , 468 , 462 guaita , l. , et al . 2010 , , 714 , 255 hansen , m. , & oh , s. p. 2006 , , 367 , 979 hayashino , t. , et al . 2004 , , 128 , 2073 hayes , m. , stlin , g. , atek , h. , kunth , d. , mas - hesse , j. m. , leitherer , c. , jimnez - bailn , e. , & adamo , a. 2007 , , 382 , 1465 hu , e. , & mcmahon , r. g. 1996 , , 382 , 281 kashikawa , n. , et al . 2011 , , 734 , 119 keel , w. c. 2005 , , 129 , 1863 kollmeier , j. a. , zheng , z. , dav , r. , gould , a. , katz , n. , miralda - escud , j. , & weinberg , d. h. 2010 , , 708 , 1048 kravtsov , a. v. , berlind , a. a. , wechsler , r. h. , klypin , a. a. , gottlber , s. , allgood , b. , & primack , j. r. 2004 , , 609 , 35 kunth , d. , leitherer , c. , mas - hesse , j. m. , stlin , g. , & petrosian , a. 2003 , , 597 , 263 laursen , p. , & sommer - larsen , j. 2007 , , 657 , l69 loeb , a. , & rybicki , g. b. 1999 , , 524 , 527 madau , p. , pozzetti , l. , & dickinson , m. 1998 , , 498 , 106 malhotra , s. , & rhoads , j. 2004 , , 617 , l5 moller , p. , & warren , s. j. 1998 , , 299 , 661 neufeld , d. a. 1991 , , 370 , l85 ono , y. , ouchi , m. , shimasaku , k. , dunlop , j. , farrah , d. , mclure , r. , & okamura , s. 2010 , , 724 , 1524 stlin , g. , hayes , m. , kunth , d. , mas - hesse , j. m. , leitherer , c. , petrosian , a. , & atek , h. 2009 , , 138 , 923 ouchi , m. , et al . 2008 , , 176 , 301 ouchi , m. , et al . 2010 , , 723 , 869 partridge , r.b . , & peebles , p.j.e . 1967 , , 147 , 868 pieri , m. m. , frank , s. , weinberg , d. h. , mathur , s. , & york , d. g. 2010 , , 724 , l69 rauch , m. , et al . 2008 , , 681 , 856 rhoads , j. e. , et al . 2003 , , 125 , 1006 seljak , u. 2000 , , 318 , 203 shopbell , p. l. , & bland - hawthorn , j. 1998 , , 493 , 129 steidel , c. c. , adelberger , k. l. , shapley , a. e. , pettini , m. , dickinson , m. , & giavalisco , m. 2003 , , 592 , 728 steidel , c. c. , erb , d. k. , shapley , a. e. , pettini , m. , reddy , n. , bogosavljevi , m. , rudie , g. c. , & rakic , o. 2010 , , 717 , 289 steidel , c. c. , bogosavljevi , m. , shapley , a. e. , kollmeier , j. a. , reddy , n. a. , erb , d. k. , & pettini , m. 2011 , in press , arxiv:1101.2204 taniguchi , y. , et al . 2009 , , 701 , 915 trac , h. , & cen , r. 2007 , , 671 , 1 trac , h. , cen , r. , & loeb , a. 2008 , , 689 , l81 veilleux , s. , & rupke , d. s. 2002 , , 565 , l63 verhamme , a. , schaerer , d. , & maselli , a. 2006 , , 460 , 397 wang , j .- x . , malhotra , s. , rhoads , j. e. , zhang , h .- t . , & finkelstein , s. l. 2009 , , 706 , 762 zheng , z. , cen , r. , trac , h. , & miralda - escude , j. 2010 , , 716 , 574 zheng , z. , cen , r. , trac , h. , & miralda - escude , j. 2011 , , 726 , 38

the extended emission can be detected from stacked narrowband images of emitters ( laes ) or of lyman break galaxies ( lbgs ) . its average surface brightness profile has a central cusp , then flattens to an approximate plateau beginning at an inner characteristic scale below.2 mpc ( comoving ) , then steepens again beyond an outer characteristic scale of mpc . the extended emission predicted by our simulation is already within reach of deep narrowband photometry using large ground - based telescopes . such observations would test radiative transfer models of emission from laes and lbgs , and they would open a new window on the circumgalactic environment of high - redshift star - forming galaxies .

photons that escape the interstellar medium of star - forming galaxies may be resonantly scattered by neutral hydrogen atoms in the circumgalactic and intergalactic media , thereby increasing the angular extent of the galaxy s emission . we present predictions of this extended , low surface brightness emission based on radiative transfer modeling in a cosmological reionization simulation . the extended emission can be detected from stacked narrowband images of emitters ( laes ) or of lyman break galaxies ( lbgs ) . its average surface brightness profile has a central cusp , then flattens to an approximate plateau beginning at an inner characteristic scale below.2 mpc ( comoving ) , then steepens again beyond an outer characteristic scale of mpc . the inner scale marks the transition from scattered light of the central source to emission from clustered sources , while the outer scale marks the spatial extent of scattered emission from these clustered sources . both scales tend to increase with halo mass , uv luminosity , and observed luminosity . the extended emission predicted by our simulation is already within reach of deep narrowband photometry using large ground - based telescopes . such observations would test radiative transfer models of emission from laes and lbgs , and they would open a new window on the circumgalactic environment of high - redshift star - forming galaxies .

physics0409071

i

[ [ spthist ] ] historical background of study p- and t - parity nonconservation effects in heavy - atom molecules . + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + after discovery of the combined charge ( c ) and space ( p ) parity violation , or cp - violation , in @xmath0-meson decay @xcite , the search for the electric dipole moments ( edms ) of elementary particles has become one of the most fundamental problems in physics @xcite . permanent edm is induced by the weak interaction that breaks both the space symmetry inversion and time - reversal invariance ( t ) @xcite . considerable experimental effort has been invested in measuring atomic edms induced by the edms of proton , neutron and electron and by p , t - odd interactions between them . the best available restriction for the electron edm , @xmath1 , was obtained in the atomic tl experiment @xcite , which established an upper limit of @xmath2 cm , where @xmath3 is the charge of the electron . the benchmark upper limit on a nucleus edm is obtained in atomic experiment on @xmath4hg @xcite , @xmath5 cm , from which the best restriction on the proton edm , @xmath6 cm , was also recently obtained by dmitriev & senkov @xcite . since 1967 , when sandars suggested to use polar heavy - atom molecules in the experimental search for the proton edm @xcite , the molecules are considered as the most promising objects for such experiments . sandars also noticed earlier @xcite that the p- and p , t - parity nonconservation ( pnc ) effects are strongly enhanced in heavy - atom systems due to relativistic and other effects . for example , in paramagnetic atoms the enhancement factor for an electron edm , @xmath7 , is roughly proportional to @xmath8 , where @xmath9 is the fine structure constant , @xmath10 is the nuclear charge and @xmath11 is the atomic polarisability . it can be of order 100 or greater for highly polarizable heavy atoms ( @xmath12 ) . furthermore , the effective inner molecular electric field @xmath13 acting on electrons in polar molecules can be a few orders of magnitude higher than the maximal field @xmath14 accessible in a laboratory , @xmath15 . the first molecular edm experiment ( on search for the proton edm and other nuclear p , t - odd effects ) was performed on tlf by sandars _ _ @xcite ( oxford , uk ) . in 1991 , in the last series of the @xmath16tlf experiments by hinds _ et al . _ @xcite ( yale , usa ) , the restriction @xmath17 was obtained ( that was recalculated in 2002 by petrov _ et al . _ @xcite as @xmath18 ) . in 1978 the experimental investigation of the electron edm and other pnc effects was further stimulated by labzowsky _ _ @xcite and sushkov & flambaum @xcite who clarified the possibilities of additional enhancement of these effects in diatomic radicals like bis and pbf due to the closeness of levels of opposite parity in @xmath19-doublets having the @xmath20 ground state . then sushkov _ et al . _ @xcite and flambaum & khriplovich @xcite have suggested to use @xmath19-doubling in diatomic radicals with the @xmath21 ground state for such experiments and the hgf , hgh and baf molecules were first studied semiempirically by kozlov @xcite . at the same time , the first two - step _ ab initio _ calculation of pnc effects in pbf initiated by labzowsky was finished in st .- petersburg ( russia ) @xcite . a few years later , hinds started experimental search for the electron edm on the ybf molecule , on which the first result was obtained by his group in 2002 ( sussex , uk ) @xcite , @xmath22 . though that restriction is worse than the best available now @xmath1 datum calculated from the tl experiment ( see above ) , nevertheless , it is limited by only counting statistics , as hinds _ _ pointed out in @xcite . new series of the electron edm experiments on ybf by hinds group ( london , uk ) are in progress and new generation of the electron edm experiments using a vapor cell , on the metastable @xmath23 state of pbo , is prepared by group of demille ( yale , usa ) . the unique suitability of pbo for searching the elusive @xmath1 is demonstrated by the very high statistical sensitivity of the yale experiment to the electron edm . in prospect , it allows one to detect @xmath1 of order of @xmath24 cm @xcite , three four orders of magnitude lower than the current limit quoted above . some other candidates for the edm experiments , hgh , hgf , and teo@xmath25 , are yet discussed and the experiment on pbf is planned in oklahoma , usa . in order to interpret the data measured in such molecular experiments in terms of the electron edm or other fundamental constants of p- and p , t - odd interactions , high - precision calculation of the electronic wave function near a heavy nucleus is required . moreover , _ ab initio _ calculations of some molecular properties are usually required already prior to the stage of preparation of the experimental setup . [ [ shamcalc ] ] heavy - atom molecules , computational strategies . + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + the most straightforward way for calculation of electronic structure of molecules containing heavy atoms is solution of the eigenvalue problem using the dirac - coulomb or dirac - coulomb - breit hamiltonian when some approximation for the four - component wave function is chosen . however , even applying the relativistic scf approximation to heavy - atom molecules when solving the dirac - fock or dirac - fock - breit equations followed by the transformation of the two - electron integrals is not always an easy task because much larger sets of primitive basis functions are required @xcite for such all - electron four - component calculations as compared to the nonrelativistic case . starting from the pauli approximation and foldy & wouthuysen transformation , many different two - component approaches were developed in which only large components are treated explicitly ( e.g. , see and references ) . the most computational savings are yet achieved when the two - component relativistic effective core potential ( recp ) approximation proposed by lee _ et al . _ @xcite is used . there are the following reasons for it . the recp approach allows one to exclude large number of chemically inactive electrons from molecular calculations and to treat explicitly the valence electrons only . then , the oscillations of the valence spinors are usually smoothed in heavy - atom cores . as a result , the number of primitive basis functions can be reduced dramatically . this approach is based on a well - developed nonrelativistic technique of calculations , however , an effective spin - orbit interaction and other scalar - relativistic effects are taken into account usually by means of radially - local or separable potentials @xcite . many complications of the dirac - coulomb(-breit ) molecular calculations @xcite are avoided when employing recps . the radially - local recp approaches for `` shape - consistent '' ( or `` norm - conserving '' ) pseudoorbitals are the most widely applied in calculations of molecules with heavy elements though `` energy - adjusted / consistent '' pseudopotentials @xcite by stuttgart team and huzinaga - type `` _ ab initio _ model potentials '' @xcite are also actively used . ( in plane wave calculations of many - atomic systems and in molecular dynamics , the separable pseudopotentials are more popular now because they provide linear scaling of computational effort on the basis set size . ) the nonrelativistic shape - consistent effective core potential has been first proposed by durand & barthelat @xcite and then a modified scheme of the pseudoorbital constructing was suggested by christiansen _ _ @xcite and by hamann _ et al . _ @xcite . however , inaccuracy of the conventional radially - local recp approaches reaches 10003000 @xmath26 for the transition and dissociation energies that may be insufficient in practice . in deep reorganization of electronic configuration structure of molecules containing , in particular , transition metals , lanthanides , actinides , and superheavy elements the conventional radially - local recp approaches can not provide reliable results for a wide variety of properties ( as it was shown both in many calculations and theoretically , see @xcite and references ) though otherwise is sometime stated ( e.g. , see @xcite ) such problems can be efficiently overcomed by applying the generalized recp ( grecp ) approach , that involves both radially - local , separable and huzinaga - type potentials as its components and as particular cases . in the grecp concept , the inner core , outer core and valence electrons are first treated employing different approximations for each . nevertheless , calculation of such properties as electronic densities near nuclei , hyperfine structure constants , p , t - parity nonconservation effects , chemical and isotopic shifts etc . with the help of the two - component pseudospinors smoothed in cores is impossible . for evaluation the matrix elements of the operators singular on nuclei , proper shapes of the valence molecular four - component spinors must be restored in atomic core regions after the recp calculation of that molecule performed . in 1959 , a nonrelativistic procedure of restoration of the orbitals from smoothed phillips kleinman pseudoorbitals was proposed @xcite based on the orthogonalization of the latters to the original atomic core orbitals . in 1985 , pacios & christiansen @xcite suggested a modified orthogonalization scheme in the case of shape - consistent pseudospinors . at the same time , a simple procedure of one - center restoration employing the idea of generation of equivalent basis sets in four - component dirac - fock and two - component recp / scf calculations was proposed in @xcite ( i.e. nocr procedure , see below ) and first applied to evaluation of the p , t - odd spin - rotational hamiltonian parameters in the pbf molecule . in 1994 , similar procedure was used by blchl inside the augmentation regions @xcite in solids to construct the transformation operator between pseudoorbitals ( `` ps '' ) and original orbitals ( `` ae '' ) in his projector augmented - wave method . all the above restoration schemes can be called by `` nonvariational '' as compared to the `` variational '' one - center restoration ( vocr , see below ) procedure proposed in @xcite . proper behavior of the molecular orbitals ( four - component spinors in the relativistic case ) in atomic cores of molecules can be restored in the scope of a variational procedure if the molecular pseudoorbitals ( two - component pseudospinors ) match correctly the original orbitals ( large components of bispinors ) in the valence region after the molecular recp calculation . this condition is rather correct when the shape - consistent recp is involved to the molecular calculation with explicitly treated outermost core orbitals and , especially , when the grecp operator is used as is demonstrated in @xcite . at the restoration stage , a one - center expansion on the spherical harmonics with numerical radial parts is most appropriate both for orbitals ( spinors ) and for the description of `` external '' interactions with respect to the core regions of a considered molecule . in the scope of the discussed two - step methods of the electronic structure calculation of a molecule , finite nucleus models and quantum electrodynamic terms including , in particular , two - electron breit interaction @xcite may be used without problems . one - center expansion had been applied first to whole molecules by desclaux & pyykk in relativistic and nonrelativistic hartree - fock calculations for the series ch@xmath27 to pbh@xmath27 @xcite and then in the dirac - fock calculations of cuh , agh and auh @xcite and other molecules @xcite . a large bond length contraction due to the relativistic effects has been estimated . however , the accuracy of such calculations is limited in practice because the orbitals of the hydrogen atom are reexpanded on a heavy nucleus in all the coordinate space . it is worth to note that the recp and one - center expansion approaches were considered earlier as alternatives to each other @xcite . the applicability of the proposed two - step algorithms for calculation of wave functions of molecules with heavy atoms is a consequence of the fact that the valence and core electrons may be considered as two subsystems , interaction between which is described mainly by some integrated and not by detailed properties of these subsystems . the methods for consequent calculation of the valence and core parts of electronic structure of molecules give us a way to combine relative simplicity and accessibility of both molecular recp calculations in gaussian basis set and relativistic finite - difference one - center calculations inside a sphere with the atomic core radius . the first two - step calculations of the p , t - odd spin - rotational hamiltonian parameters were performed for the pbf radical about 20 years ago @xcite with the semiempirical accounting for the spin - orbit interaction . before , only nonrelativistic scf calculation of the tlf molecule using the relativistic scaling was carried out @xcite , in which the p , t - odd values were almost three times underestimated as compared to the relativistic df calculations . the latter were first performed only in 1997 by laerdahl _ et al . _ @xcite and by parpia @xcite . the next two - step calculation , for pbf and hgf molecules @xcite , was carried out with the spin - orbit recp part taken into account using the method suggested in @xcite . later we performed correlation grecp / nocr calculations of the core properties in ybf @xcite , baf @xcite , again in ybf @xcite and in tlf @xcite . in 1998 , first all - electron dirac - fock calculations of the ybf molecule were also performed by quiney _ _ @xcite and by parpia @xcite . very recently we finished extensive two - step calculations of the p , t - odd properties and hyperfine structure of the excited states of the pbo molecule @xcite . in the paper , the main features of the used two - step method are presented and only the last series of the two - step calculations are discussed , in which electron correlations are taken into account by a combined method of the second - order perturbation theory ( pt2 ) and configuration interaction ( ci ) , or `` pt2/ci '' @xcite ( for baf and ybf ) , by the relativistic coupled cluster ( rcc ) method @xcite ( for tlf and pbo ) , and by the spin - orbit direct - ci method ( for pbo ) . in the discussed _ ab initio _ calculations the best up to - date accuracy was attained for the hyperfine constants and p , t - odd parameters regarding the molecules containing heavy atoms .

precise calculations of core properties in heavy - atom systems which are described by the operators heavily concentrated in atomic cores , like to hyperfine structure and p , t - parity nonconservation effects , usually require accounting for relativistic effects . the valence molecular spinors are usually smoothed in atomic cores and , as a result , direct calculation of electronic densities near heavy nuclei is impossible . in the paper , the methods of nonvariational and variational one - center restoration of correct shapes of four - component spinors in atomic cores after a two - component recp calculation of a molecule are discussed . their efficiency is illustrated in correlation calculations of hyperfine structure and parity nonconservation effects in heavy - atom molecules ybf , baf , tlf , and pbo . * short name : * studying core properties in molecules * keywords for indexing : * electronic structure , atom in a molecule , molecules with heavy atoms , method of ab initio molecular calculation , relativistic effective core potential , one - center restoration .

precise calculations of core properties in heavy - atom systems which are described by the operators heavily concentrated in atomic cores , like to hyperfine structure and p , t - parity nonconservation effects , usually require accounting for relativistic effects . unfortunately , completely relativistic treatment of molecules containing heavy elements is very consuming already at the stages of calculation and transformation of two - electron integrals with a basis set of four - component spinors . in turn , the relativistic effective core potential ( recp ) calculations of valence ( spectroscopic , chemical etc . ) properties of molecules are very popular because the recp method allows one to treat quite satisfactory the correlation and relativistic effects for the valence electrons of a molecule and to reduce significantly the computational efforts . the valence molecular spinors are usually smoothed in atomic cores and , as a result , direct calculation of electronic densities near heavy nuclei is impossible . in the paper , the methods of nonvariational and variational one - center restoration of correct shapes of four - component spinors in atomic cores after a two - component recp calculation of a molecule are discussed . their efficiency is illustrated in correlation calculations of hyperfine structure and parity nonconservation effects in heavy - atom molecules ybf , baf , tlf , and pbo . * short name : * studying core properties in molecules * keywords for indexing : * electronic structure , atom in a molecule , molecules with heavy atoms , method of ab initio molecular calculation , relativistic effective core potential , one - center restoration .

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the global ocean dominates the iconic image of earth viewed from space , leading to the now famous `` blue marble '' descriptor for our planet . the ocean covers more than 70% of the planetary surface and ocean processes are critical to life - sustaining ( and life - challenging ) events and processes occurring across broad ranges of temporal and spatial scales . understanding issues of ocean resource consumption ( e.g. , fisheries , coastal pollution , etc . ) lead to foci on ocean ecosystem dynamics and their coupling to physical processes ( e.g. , mixing , transports , upwelling , ice dynamics , etc . ) . understanding the ocean role in climate dynamics ( e.g. , sequestration of atmospheric @xmath0 , absorption of atmospheric heat , impacts on the hydrologic cycle , teleconnections , etc . ) lead to foci on massive and complex simulations and forecasts based on equations of geophysical fluid dynamics . in all instances , the broad range of scales , the energetic exchanges across them , and the associated uncertainties drive innovations that involve methods of modern statistical modeling.=1 oceanography has historically been a `` data poor '' science and the need to use advanced statistical methodology to perform inference and prediction has been paramount throughout its development . although it is the case that there are too few in situ observations of the ocean to characterize its evolution and its interaction with marine ecosystems , in an ironic twist , the discipline also suffers from having an abundance of particular data types when one factors in the satellite observations that have become available in the last couple of decades . the need for statistical collaboration in oceanography results from both the situation of not having enough observations in some parts of the system and having huge amounts of data in other parts of the system . the physical ocean is governed by basic laws of physics ( see , e.g. , @xcite ) . the primitive equations consist of the following : three equations corresponding to the conservation of momentum ( for the two horizontal and one vertical components of velocity ) , a continuity equation representing the conservation of mass , an equation of state ( relating density , pressure , temperature and salinity ) , and equations corresponding to the conservation of temperature and salinity . there are seven state variables ( three velocity components , density , pressure , temperature and salinity ) . this system of equations is nonlinear and exhibits a huge range of spatial and temporal scales of variability . given that many of these scales of variability are not resolved in data or in deterministic or `` forward '' ocean models , the equations are typically simplified by scale analysis arguments and the small scale ( turbulent ) structures are parameterized . these parameterizations involve relationships between the mean of the state variables and their gradients . in this way the eddy viscosity and diffusivity terms serve as so - called `` sub - grid scale '' parameterizations , representing the unresolved processes that are sinks of momentum and heat at the grid scale of a given model , for example , @xmath1 km in global ocean models and @xmath2 km in regional ocean models . the ocean system is nonlinearly coupled to the atmosphere across a broad range of scales . at the largest scales , air sea fluxes of heat and fresh water drive mostly vertical or `` thermohaline '' circulations while the surface shear stress and wind stress curl drive mostly horizontal or wind - driven gyre circulations ( e.g. , @xcite ) . at smaller scales , the vertical - horizontal separation breaks down and the ocean response to external forcing and internal instabilities results in a broadband spectrum of vigorous eddy circulations ( e.g. , @xcite ) . on all scales , the circulation provides the context for ocean processes affecting other components of the ocean system such as those related to ocean biology and chemistry . there are significant nonlinear interactions between the ocean chemistry , biology and its physical state . ocean biogeochemistry is concerned with the interaction of the biology , chemistry and geology of the ocean ( e.g. , @xcite ) . this is a very complex system that contains many interactions across a variety of scales . the system can be simply illustrated by thinking about the interactions of broad classes of its components . for example , the presence of nutrients near the ocean surface , where there is light , allows for the growth of phytoplankton , which deplete the nutrients as their population expands . the increased abundance of phytoplankton then provides a food source for zooplankton , which leads to growth in the zooplankton population . the consumption of phytoplankton leads to waste products from the zooplankton that settles as detritus to the ocean floor . as the zooplankton deplete the phytoplankton , the zooplankton population decreases due to the lack of a sufficient food source . eventually , the detritus at depth is transferred to the surface through upwelling and mixing , providing the nutrients that lead to another bloom in phytoplankton , etc . this simple four component system is a vast oversimplification , as there are many different species interacting at any one time . more critically , this lower trophic ecosystem is also coupled to higher levels of the food web , for example , with foraging fish predating the zooplankton , which are in turn predated by higher trophic fish , marine mammals , commercial fishing , etc . the chemical component of the cycle is critical in several respects . it provides a way for carbon to be removed from the atmosphere , as the phytoplankton remove carbon from the ocean water and are consumed by the zooplankton . some of that carbon is contained in the detritus that sinks to the ocean floor , becoming buried in the sediment and leading to a ( temporary ) carbon sink in the global carbon cycle , which is very important in the context of sequestration of carbon relative to potential climate change from greenhouse gases . in addition , the biological cycle in the ocean is closely tied to the distribution of dissolved oxygen in the water and also influences the distribution of other chemicals , such as silicon , nitrates and phosphates , for example , in the shells of diatoms . given the complexities in the ocean system , it is not surprising that there are numerous sources of uncertainty . first , although the large - scale equations of motion are in some sense deterministic , the scale issues that lead to eddy viscosity / diffusivity parameterizations are inherently uncertain . furthermore , the forms of the linkages between system components ( e.g. , wind stress , heat and moisture fluxes between atmosphere and ocean ) are not known with certainty . the components of traditional biogeochemical models are even more uncertain , both in terms of the functional forms and parameters . the process and parameter uncertainty is compounded by the inherent data issues in the ocean system . in situ observations of the ocean are quite limited in terms of spatial and temporal resolution , and in terms of the variables measured . for example , it is a painstaking process to measure zooplankton abundance , often requiring a scientist or technician to literally count critters through a microscope . fortunately , many surface variables can be observed remotely , particularly through satellite proxies . in some cases , for example , near surface winds from scatterometers , sea surface height from altimeters and sea surface temperature ( sst ) from radiometers , the satellite observations are typically quite precise , albeit with gaps corresponding to orbital geometries , swath widths and fields of view . in other cases , for example , ocean color as a proxy for phytoplankton , the observational representation of the process is more uncertain , at least on fairly short time scales . thus , a key issue in state prediction , parameter estimation and inference is to deal with incomplete observations that vary in precision and in spatial and temporal support . this is particularly important when one considers ocean `` data assimilation , '' that is , the blending of prior information ( e.g. , a numerical solution of the deterministic representation of the ocean state ) with observations . another traditionally important component of uncertainty in ocean process modeling corresponds to the selection of reduced - dimensional representations of the process . given the assumption that much of the larger scale processes in the ocean can be represented in a lower dimensional manifold , with smaller scales corresponding to turbulent scales ( that certainly may interact with the larger scale modes , or at least suggest the form of parameterizations or stochastic noise terms ) , there has been considerable attention given to different approaches to obtain the reduced - dimension basis functions . the choices vary depending on the part of the system being considered as well as whether one is looking at the system diagnostically or predictively . in the context of statistical models used to describe or predict portions of the ocean system , the nature of the error structures is important . given the nonlinearity that is inherent in the system , many process distributions are not well represented bygaussian errors . in addition , in some cases ( e.g. , biological abundance variables ) the distributions can only have positive support . the ocean and atmospheric sciences have benefitted from a strong tradition in applying fairly complex statistical methods to deal with many of the uncertainty issues described above . in particular , general monographs such as @xcite , @xcite , and @xcite provide comprehensive descriptions of traditional methods used to analyze such data . in addition to overviews of basic statistical concepts , these books describe multivariate methods ( e.g. , principal components empirical orthogonal functions , canonical correlation analysis , discriminant analysis , etc . ) , spectral methods ( e.g. , cross - spectral analysis ) and dynamically - based reduction methods ( e.g. , principal oscillation patterns ) to facilitate analysis of high - dimensional data that has inherent dependence in time and space . this is in addition to more focused monographs such as @xcite(@xcite ) , which gives a comprehensive overview ofeigen - decomposition methods , and several books and review papers devoted to various aspects of data assimilation ( see section [ sec3.1 ] ) . statistical presentations of many of these methods can be found in @xcite and @xcite . recognizing the challenges related to uncertainty in the ocean system and the need to foster more collaborative research between oceanographers and statisticians , the u.s . national research council(nrc ) commissioned a panel to write a report on `` statistics and oceanography '' @xcite ; see also the accompanying article by @xcite and published comments . this report contains a very nice review of physical oceanography for nonoceanographers and outlined the need for research in several key areas , including the change of support problem and the indirect nature of satellite observations , non - gaussian random fields , the incorporation of lagrangian and eulerian data , data assimilation , inverse modeling , model / data comparison and feature identification , to name some of the most prominent . the report focused on the physical component of the ocean and did not address biogeochemistry nor many issues of current interest , such as climate change and reduced - dimensional representations . our goal with this review is to provide an overview of some of the advancements that have occurred at the interface of statistics and oceanography since the @xcite report . in particular , we believe strongly that the hierarchical statistical perspective has played a significant role in this development and will focus our review from that perspective . section [ sec2 ] presents a brief discussion of hierarchical modeling , both empirical and bayesian , with some discussion of the need for computational tools . we note that although this paper is in a statistics journal , we hope that it will generate interest from both statisticians and oceanographers . for the same reason that we gave a brief and general overview of oceanography above , we will also give a brief and general overview of hierarchical modeling for those readers with little exposure to these ideas . in section [ sec3 ] we focus in more depth on examples related to data assimilation and inverse modeling , long lead forecasting and uncertainty quantification in biogeochemical models . we will follow this review with a brief discussion of current and future challenges in section [ sec4 ] .

processes in ocean physics , air sea interaction and ocean biogeochemistry span enormous ranges in spatial and temporal scales , that is , from molecular to planetary and from seconds to millennia . identifying and implementing sustainable human practices depend critically on our understandings of key aspects of ocean physics and ecology within these scale ranges . the set of all ocean data is distorted such that three- and four - dimensional ( i.e. , time - dependent ) in situ data are very sparse , while observations of surface and upper ocean properties from space - borne platforms have become abundant in the past few decades . precisions in observations of all types vary as well . in the face of these challenges , the interface between statistics and oceanography has proven to be a fruitful area for research and the development of useful models . with the recognition of the key importance of identifying , quantifying and managing uncertainty in data and models of ocean processes , a hierarchical perspective has become increasingly productive . as examples , we review a heterogeneous mix of studies from our own work demonstrating bayesian hierarchical model applications in ocean physics , air sea interaction , ocean forecasting and ocean ecosystem models . this review is by no means exhaustive and we have endeavored to identify hierarchical modeling work reported by others across the broad range of ocean - related topics reported in the statistical literature . we conclude by noting relevant ocean - statistics problems on the immediate research horizon , and some technical challenges they pose , for example , in terms of nonlinearity , dimensionality and computing . , ,

processes in ocean physics , air sea interaction and ocean biogeochemistry span enormous ranges in spatial and temporal scales , that is , from molecular to planetary and from seconds to millennia . identifying and implementing sustainable human practices depend critically on our understandings of key aspects of ocean physics and ecology within these scale ranges . the set of all ocean data is distorted such that three- and four - dimensional ( i.e. , time - dependent ) in situ data are very sparse , while observations of surface and upper ocean properties from space - borne platforms have become abundant in the past few decades . precisions in observations of all types vary as well . in the face of these challenges , the interface between statistics and oceanography has proven to be a fruitful area for research and the development of useful models . with the recognition of the key importance of identifying , quantifying and managing uncertainty in data and models of ocean processes , a hierarchical perspective has become increasingly productive . as examples , we review a heterogeneous mix of studies from our own work demonstrating bayesian hierarchical model applications in ocean physics , air sea interaction , ocean forecasting and ocean ecosystem models . this review is by no means exhaustive and we have endeavored to identify hierarchical modeling work reported by others across the broad range of ocean - related topics reported in the statistical literature . we conclude by noting relevant ocean - statistics problems on the immediate research horizon , and some technical challenges they pose , for example , in terms of nonlinearity , dimensionality and computing . , ,

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polymers exposed to an unfavorable environment can collapse or change shape in order to minimize surface energy @xcite . examples of unfavorable environments include a poor solvent or a hydrophilic - hydrophobic interface like the one between water and either air or oil . examples of conformations driven by such energy minimization are rings , loops , coils , spools , tori / toroids , hairpins or tennis rackets @xcite . in filaments comprising aggregated proteins or peptides , ring formation falls into two main classes : fully annealed rings occasionally observed as intermediate states during protein fibrillation , like in apolipoprotein c - ii @xcite and a@xmath1 @xcite ; or ring formation in actively driven systems , where the energy required for filament bending is provided by gtp or atp @xcite . insulin has been shown to form open - ring shaped fibrils when pressure was applied during fibrillation @xcite , which was explained by an anisotropic distribution of void volumes in fibrils and therefore aggregation into bent fibrils . we study amyloid fibrils , which are linear supramolecular assemblies of proteins / peptides that , despite a large diversity in possible peptide sequences , show remarkable structural homogeneity . peptides form @xmath0-sheets that stack , often with chiral registry , to form a filament whose main axis is perpendicular to the @xmath0-strands @xcite . fully formed fibrils can consist of one or , more commonly , multiple filaments , assembled into twisted ribbons with a twist pitch determined by the number of filaments in the fibrils @xcite . their high aspect ratio ( diameter usually less than @xmath2 nm , total contour length up to several @xmath3 m ) leads to liquid crystalline phases in both three ( 3d ) @xcite and two dimensions ( 2d ) @xcite . amyloid fibrils were initially studied due to their involvement in many different degenerative diseases such as diabetes ii or parkinson s disease @xcite . however , protein fibrils have recently experienced a surge of interest in potential applications in materials @xcite , and functional roles have been identified in biological processes such as hormone storage @xcite , emphasizing the importance of understanding their structure and properties in 2d . here , we present experimental evidence for the development of _ curved _ fibrils at interfaces . semiflexible @xmath0-lactoglobulin fibrils are found to undergo a shape change and passively form open rings upon adsorption to an interface ( liquid - liquid or liquid - air ) . we show that this can not be described by a simple bending modulus ; this bending can instead be understood in terms of a _ spontaneous curvature _ induced on symmetry grounds by the chiral and polar nature of the fibril , when interacting with the heterogeneous environment provided by an interface . a comparison of different fibril batches of the same protein shows that the probability of forming rings depends on the average fibril thickness , with batches of thicker fibrils not forming loops . these results imply that flexible non - symmetric bodies embedded in heterogeneous media such as the physiological environment can be expected to deform , bend , and twist , depending on the specific surface interaction with the environment . for example , concentration gradients of ions or ph could enhance shape changes necessary for locomotion in flexible nanoswimmers @xcite , or be used to promote or control self - assembly through shape changes . one could even envision high surface to volume materials such as bicontinuous phases with large length scales being used to process large amounts of flexible shape changers .

protein fibril accumulation at interfaces is an important step in many physiological processes and neurodegenerative diseases as well as in designing materials . here we reason that this spontaneous curvature is governed by structural characteristics on the molecular level and is to be expected when a chiral and polar fibril is placed in an inhomogeneous environment such as an interface . by testing-lactoglobulin fibrils with varying average thicknesses ,

protein fibril accumulation at interfaces is an important step in many physiological processes and neurodegenerative diseases as well as in designing materials . here we show , using-lactoglobulin fibrils as a model , that semiflexible fibrils exposed to a surface do not possess the gaussian distribution of curvatures characteristic for wormlike chains , but instead exhibit a spontaneous curvature , which can even lead to ring - like conformations . the long - lived presence of such rings is confirmed by atomic force microscopy , cryogenic scanning electron microscopy and passive probe particle tracking at air- and oil - water interfaces . we reason that this spontaneous curvature is governed by structural characteristics on the molecular level and is to be expected when a chiral and polar fibril is placed in an inhomogeneous environment such as an interface . by testing-lactoglobulin fibrils with varying average thicknesses , we conclude that fibril thickness plays a determining role in the propensity to form rings . this document is the unedited author s version of a submitted work that was subsequently accepted for publication in acs american chemical society after peer review . to access the final edited and published work see http://pubs.acs.org/doi/abs/10.1021/nn504249x .

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when imaging the air - water interfacial fibril layer by afm using a modified langmuir - schaefer horizontal transfer technique ( see materials and methods ) to resolve 2d liquid crystallinity , we found that , in addition to nematic and isotropic fibril domains @xcite , some @xmath0-lactoglobulin fibrils were present in circular conformations . these rings appear at the lowest interfacial density investigated , where fibril alignment is still negligible @xcite , and persist in the presence of nematic fibril domains up to high densities [ see supplementary note 1 , supplementary fig . s1 and s2 ] . ring diameters range from @xmath4 @xmath5 ( fig . [ fig : afm ] and [ fig : ptsem ] ) , and are consistent whether observed _ via afm _ at the air - water interface , cryogenic scanning electron microscopy ( cryo - sem ) or passive probe particle tracking at the oil - water interface , confirming that fibrils have a similar tendency to bend at air - liquid and liquid - liquid interfaces . a small selection of the vast variety of ring morphologies is presented in fig . [ fig : afm ] . highly complex structures involving several fibrils are quite common ( fig . [ fig : afm]a , b , s1 and s2 ) , whereas relatively few distinct rings or tennis rackets comprise a single fibril and can rather be thought to be intermediate assembly states @xmath6 to final ring structures ( fig . [ fig : afm]c and d ) @xcite . short fibrils , which could be the result of fracture due to the bending strain , exposure to air or inhomogeneous strong surface tension , also assemble into rings ( fig . s3 ) . alternatively , short fibrils frequently accumulate within an outer ring and align either along the circumference of this ring or parallel to each other in the center , with minimal contact with the ring itself ( fig . [ fig : afm]b and e ) . the long - lived presence , and hence inferred stability , of these self - organized conformations was confirmed by passive probe particle tracking experiments performed at the oil - water interface , where fluorescently - labelled spherical tracer particles ( diameter @xmath7 nm ) were observed to move in near - perfect circles or sickle - shaped trajectories over the course of three to four minutes . a simple pathway for ring formation could be the presence of nano- or microbubbles at the liquid surface , which give fibrils the opportunity to bend around their circumference @xcite . this would , however , also lead to a distortion of the peptide layer ( see materials and methods ) at the interface ; once the sample has dried , the bubble would have disappeared but still be visible in afm images as a height discontinuity through the ` bubble ' . the absence of such observations in afm ( fig . s4 ) , or of bubbles ( cavities ) in the cryo - sem images ( fig . [ fig : ptsem ] ) , indicates that there is an inherent predisposition of the fibrils to bend , which then leads to circle formation upon interaction with a liquid surface . understanding these data requires a study of how surface effects influence the shape of fibrils ( or indeed filaments ) . we consider an inextensible fibril of length @xmath8 , represented as a twisted ribbon with chiral wavelength @xmath9 and pitch angle @xmath10 , where @xmath11 is the inscribing radius of the twisted ribbon ( see supplementary note 2 ) . we parametrize the shape by @xmath12 , the direction parallel to the central axis of the ribbon , or equivalently the tangent vector of the fibril . the ribbon twists around its axis @xmath12 by the angle @xmath13 . we will parametrize the bending in terms of the angular rate of deflection @xmath14 , where @xmath15 is the local curvature . hence , @xmath16 , where @xmath17 is the axis about which the tangent vector is deflected during a bend . for a fibril confined to bend on a surface , we take @xmath17 to be outward surface normal vector ( pointing _ into _ the liquid ) , so that @xmath18 can be either positive or negative . the free energy is given by @xcite @xmath19 the first term penalizes bending , and @xmath20 is the bending modulus . the second term penalizes twist relative to the native helical twist , which is parametrized by the chiral wavenumber @xmath21 . here , @xmath22 is the twist modulus . the vector @xmath23 represents the twist - bend couplings allowed by a polar fibril with a non - symmetric local cross section @xcite . in this work we will focus on the bend degrees of freedom , since in filaments with free ends , such as those considered here , the twist degrees of freedom will relax to accomodate any imposed bend . a polar twisted fibril has an anisotropy that distinguishes ` head ' from ` tail ' directions along the fibril axis ; in f - actin this ` polarization ' arises from the orientations required of g - actin monomers to effect self - assembly @xcite ; in an @xmath24-helix the n - c polymerisation breaks the polar symmetry and in cross-@xmath0 amyloid fibrils such as those studied here the polarity is due to the molecular packing of @xmath0-sheets @xcite . the polarity is reflected in variations in molecular structure along the exposed surface of the twisted ribbon . when this structure is placed in a heterogenous environment , as occurs near a solid surface or when immersed within a meniscus between two fluids ( or fluid and gas ) , the inhomogeneity of the environment generally leads to unbalanced torques on the body ( see supplementary note 2 , fig . s5 and s6 for details ) , even when local forces have balanced to place the fibril at the interface . a non - symmetric body , such as a chiral and polar fibril , can thus experience an effective spontaneous curvature @xcite . to demonstrate this effect , we consider a fibril adsorbed _ onto _ a planar surface with which it interacts , rather than immersed _ within _ a meniscus . the effects are qualitatively the same , but the details are easier to understand in the adsorbed case . the surface and the adsorbed ribbon interact _ via _ numerous molecular interactions @xcite . although in principle _ all _ atoms in the fibril interact with every point on the surface due to coulomb interactions , screening limits the interaction to only the adsorbing surface . long - range dispersion interactions are also irrelevant for fibrils that are induced to bend or twist within the plane , since the change in this energy will be negligible . hence , we consider the following surface free energy @xmath25d^2r , \end{aligned}\ ] ] where @xmath9 is the twist pitch or wavelength , the average surface energy @xmath26 controls adsorption , and @xmath27 is the contact area of a the ribbon , which occurs every half wavelength . the asymmetry @xmath28 reflects the polar nature of the interaction and can vary from repulsive to attractive along the repeat patch . a polar moment ( with dimensions of energy ) of the interaction can be defined by @xmath29 where @xmath27 is the area of the patch where the fibril contacts the surface . the polar moment @xmath30 is determined by the nature of the interaction with the surface , and is thus not an intrinsic property of the fibril alone . [ fig : surfacecartoon ] shows an example in which the surface patch is a parallelogram with length @xmath31 and width @xmath32 . for a simple surface potential @xmath33 , where the coordinate @xmath34 is parallel to the fibril axis coordinate @xmath35 , the polar moment ( see supplementary note 2 ) has magnitude @xmath36 . here , @xmath37 is a geometric prefactor whose sign depends on the polarization and chirality , and parametrizes the degree to which the symmetric parellelogram is deformed into a non - symmetric shape to favor one sign of surface ` charge ' . when the twisted ribbon is bent the ribbon - surface contact area changes shape , so that either the repulsive or attractive part of the polar interaction has more contact with the surface , depending on the sign of the bend ( fig . [ fig : surfacecartoon]c ) . this leads to a spontaneous curvature . the contribution of bending to the overall interaction energy can then be written as a chiral coupling between the bending rate @xmath38 and the polar moment @xmath30 : @xmath39 the vector product is the simplest term which has no mirror symmetry , and is thus appropriate for a chiral filament . moreover , under @xmath40 both @xmath18 and @xmath41 change sign , whereas @xmath30 does not , so that the free energy is also reparametrization - invariant . the dimensionless geometric factor @xmath42 and the moment @xmath30 depend on the details of the surface free energy @xmath43 interaction potential @xmath44 , the contact area shape , and its deformation under bending . the polar moment @xmath30 depends on the surface normal vector through its vector nature and the details of the surface - fibril interaction . the ellipses indicate other terms induced by the surface , such as contributions to the bend - twist or curvature moduli , or a spontaneous twist . we choose the convention that the surface normal vector @xmath17 points _ away _ from the surface and thus into the fibril . an example free energy @xmath45 is calculated in the supplementary information for a simple model contact potential . the curvature in eq . [ eq : surf2 ] carries a sign : for @xmath46 the fibril bends in a right - handed sense around the surface normal vector @xmath17 , while for @xmath47 the fibril bends in a left - handed sense . the process of transferring the surface layer for afm observation orients the surface normal towards the afm observer , so that observation is from the liquid side towards the air side ( fig . [ fig : surfacecartoon ] ) . consider a polarization such that @xmath48 , where @xmath49 , and choose @xmath50 ( as observed in the afm image ; see fig . [ fig : surfacecartoon ] ) , where @xmath51 . this implies @xmath52 . consider a bend as shown in fig . [ fig : surfacecartoon ] , in which @xmath53 , where @xmath47 . in supplementary note 2 we find @xmath54 , so that this bend ( @xmath47 ) increases the energy , and thus @xmath46 is favored . similarly , for the opposite sign of @xmath55 a negative curvature @xmath47 is favored . the competition between the surface energy ( eq . [ eq : surf2 ] ) and the ordinary fibril bending energy ( eq . [ fullbulk ] ) leads , by minimization , to a spontaneous curvature @xmath56 given by ( see si ) @xmath57 this is equal to @xmath58 for the simple surface potential @xmath33 . isambert and maggs @xcite articulated how a surface can induce spontaneous curvature in a polar and chiral filament . they proposed a phenomenological free energy with an explicit spontaneous curvature that depends on the twist angle , and a surface interaction that breaks polar symmetry . hence , they have actually introduced a spontaneous curvature ` by hand ' . conversely , we present a model in which a polar surface interaction is itself chiral by virtue of the local chirality of the filament , and this gives rise to an effective spontaneous curvature as a result of total energy minimization . therefore , the functional form of the resulting spontaneous curvature differs from that proposed in ref . [ 30 ] . enhanced curvature is expected for amyloid fibrils with fewer filaments ( as confirmed in fig . [ fig : thickness ] ) , which will have smaller bending moduli @xmath20 , or for fibrils with larger polar moments @xmath59 and thus stronger surface interactions . in addition , the specific details of the surface deformation encapsulated in the function @xmath37 play an important role : fibrils for which the deformation leads to a more symmetric contact area will have a stronger geometric factor and thus a greater expected spontaneous curvature . consider a segment of arc length @xmath60 of a wormlike chain ( wlc ) . the probability @xmath61 of finding this segment curved with curvature @xmath62 , where @xmath63 is the radius of curvature , is governed by the bending modulus and should be gaussianly distributed , @xmath64 , where @xmath65 is the persistence length . deviations from the wlc model can be quite common , as with toroidal dna @xcite , in which the nucleic acids have a smaller persistence length at short length scales @xcite . the presence of rings in our system suggests a characteristic intrinsic curvature or length scale , in addition to the usual @xmath66 . for quantitative analysis , we have extracted the @xmath67 coordinates of fibrils from images acquired at low interfacial fibril densities after short adsorption times , where interactions and contact between fibrils are still minimal , and calculated @xmath68 ( fig . [ fig : curvature ] ; see materials and methods ) . any rings present on the image were excluded from the analysis , since their closed topology would introduce an additional constraint . to benchmark this approach , we first generate conformations based on the discrete wlc model with the @xmath66 obtained from the 2d mean squared end - to - end distance of fibrils at the air - water interface @xcite . these conformations are used to create artificial images of wlc polymers with the same resolution as the afm images and then subjected to the same tracking algorithm used for analyzing the real fibril image . [ fig : curvature ] shows the normalized probability distribution of curvatures @xmath61/@xmath69 for both the original wlcs and the corresponding tracked conformations ( see methods ) . in the tracked conformations the distribution shifts towards lower curvatures : this change is due to finite image resolution ( fig . [ fig : curvature]a ) . importantly , however , both distributions are gaussian . in contrast , and as expected from the theoretical considerations put forth above , the normalized @xmath68 for real fibrils adsorbed at the air - water interface can indeed not be fitted with a single gaussian distribution function but has a pronounced fat tail instead . -lactoglobulin fibrils ( * a * ) at the air - water interface after @xmath70 minutes of adsorption from a @xmath71 w / w fibril suspension and ( * b * ) deposited onto mica for @xmath72 minutes from the bulk with @xmath73 w / w . lower panel : probability distributions of normalized absolute local curvatures @xmath18 extracted from the full ( * a * ) @xmath74 @xmath5 ( see appendix for full image ) and ( * b * ) @xmath75 @xmath5 images ( green diamonds ) with a @xmath60 of @xmath76 and @xmath77 nm , respectively . the curvature distribution of simulated wlcs generated using all relevant parameters from the corresponding afm image ( see methods ) is shown as purple crosses and is successfully fitted with a gaussian probability distribution function ( purple line ) . tracking these wlcs results in a change in the probability densities ( blue crosses ) but the values are still gaussianly distributed ( blue line ) . plotting the normalized probabilities in logarithmic scale as a function of @xmath78 clearly shows fat tails and thus the presence of spontaneous curvature only in real fibrils ( insets in the lower panel).,title="fig:",scaledwidth=65.0% ] it has been argued that differences in @xmath56 are to be expected depending on the strength of adsorption to the surface @xcite and on whether the polymer is in 3d or 2d @xcite . to test this , we compare the curvature distributions from fibrils adsorbed to the air - water interface and transferred horizontally to mica ( fig . [ fig : curvature]a ) to fibrils deposited onto mica from a drop of the bulk solution ( fig . [ fig : curvature]b ) . the modified langmuir - schaefer afm sample preparation is a 2d to 2d transfer from a liquid onto a solid surface , which is much faster ( milliseconds ) than the slower ( seconds ) 3d to 2d equilibration obtained by depositing onto a solid substrate from bulk @xcite . the bending probability of fibrils adsorbed from the bulk to mica , where no rings are observed , was also found to deviate from a typical gaussian distribution ( fig . [ fig : curvature]b ) . fibrils hence bend as a result of their exposure to the inhomogeneous environment of solid - liquid , liquid - liquid , and gas - liquid interfaces , independently of how they initially adsorbed at these phase boundaries . as noted above , we predict a larger fibrillar diameter to imply a larger bending modulus , and hence a smaller likelihood of bending spontaneously ( according to eq.[eq : c0 ] ) . this was confirmed by studying fibrils from different batches of preparation , as well as from different suppliers . [ fig : thickness ] shows ratios of double- to triple - stranded fibrils for @xmath0-lactoglobulin fibrils produced from native protein obtained from three different suppliers . non - identical distributions can be expected due to different fibril processing conditions ( sample volumes , shearing and stirring histories ) between batches , and/or genetic variants between suppliers @xcite . this then affects the individual filament thickness , and number of filaments per fibril , due to subtle differences in proteolysis . thicker filaments , with larger bending moduli , should have much smaller spontaneous curvatures , and not be visibly curved if thick enough . fig . [ fig : thickness ] shows the distribution of number of strands per fibril , which is proportional to thickness , as determined from the afm images . the batch with the highest number of rings ( fig . [ fig : afm ] ) contains the largest amount of double - stranded fibrils ( fig . [ fig : thickness]a ) . by contrast , for batches of fibrils formed with the same protocol but from protein obtained from a different supplier , primarily three - stranded fibrils were found , which did not assemble into rings ( fig . [ fig : thickness]d ) . both a second batch of fibrils from the first source as well as a batch from a third supplier containing a more even mix of double- and triple - stranded fibrils yielded curved conformations ( fig . [ fig : thickness]b and c ) . by separating the data used to calculate the normalized distribution of @xmath68 presented in fig . [ fig : curvature]a into double- and triple - stranded fibrils ( fig . [ fig : thickness]e ) , we confirm that the normalized @xmath68 distribution of thick fibrils has a less pronounced fat tail and these fibrils thus bend less than their thinner counterparts . a similar trend is observed for the different batches in fig . [ fig : thickness]a - d , where a higher fraction of thicker fibrils in the sample results in less curved structures at the air - water interface and less spontaneous curvature ( fig . [ fig : thickness]f ) .

we show , using-lactoglobulin fibrils as a model , that semiflexible fibrils exposed to a surface do not possess the gaussian distribution of curvatures characteristic for wormlike chains , but instead exhibit a spontaneous curvature , which can even lead to ring - like conformations . the long - lived presence of such rings is confirmed by atomic force microscopy , cryogenic scanning electron microscopy and passive probe particle tracking at air- and oil - water interfaces . this document is the unedited author s version of a submitted work that was subsequently accepted for publication in acs american chemical society after peer review . to access the final edited and published work see http://pubs.acs.org/doi/abs/10.1021/nn504249x .

protein fibril accumulation at interfaces is an important step in many physiological processes and neurodegenerative diseases as well as in designing materials . here we show , using-lactoglobulin fibrils as a model , that semiflexible fibrils exposed to a surface do not possess the gaussian distribution of curvatures characteristic for wormlike chains , but instead exhibit a spontaneous curvature , which can even lead to ring - like conformations . the long - lived presence of such rings is confirmed by atomic force microscopy , cryogenic scanning electron microscopy and passive probe particle tracking at air- and oil - water interfaces . we reason that this spontaneous curvature is governed by structural characteristics on the molecular level and is to be expected when a chiral and polar fibril is placed in an inhomogeneous environment such as an interface . by testing-lactoglobulin fibrils with varying average thicknesses , we conclude that fibril thickness plays a determining role in the propensity to form rings . this document is the unedited author s version of a submitted work that was subsequently accepted for publication in acs american chemical society after peer review . to access the final edited and published work see http://pubs.acs.org/doi/abs/10.1021/nn504249x .

1007.1920

i

a good understanding of quantum chromodynamics ( qcd ) at finite temperature and baryon density is crucial for us to understand a wide range of physical phenomena . for instance , to understand the evolution of the universe in the first few seconds , one needs the knowledge of qcd phase transition at temperature @xmath3mev and very small baryon density . on the other hand , understanding the physics of neutron stars requires the knowledge of qcd at high baryon density and very low temperature @xcite . lattice simulation of qcd at finite temperature has been successfully performed in the past few decades ; however , no successful lattice simulation at high baryon density has been done due to the sign problem @xcite : the fermion determinant is not positively definite in presence of a nonzero baryon chemical potential @xmath0 . we thus look for some special cases which have a positively definite fermion determinant . one case is qcd at finite isospin chemical potential @xmath4 @xcite , where the ground state changes from a pion condensate to a bcs superfluid with quark - antiquark condensation with increasing isospin density . another case is the qcd - like theories @xcite where quarks are in a real or pseudoreal representation of the gauge group , including two - color qcd with quarks in the fundamental representation and qcd with quarks in the adjoint representation . while these cases do not correspond to the real world , they can be simulated on the lattice and may give us some information of real qcd at finite baryon density . for all these special cases , chiral perturbation theories predict a continuous quantum phase transition from the vacuum to the matter phase at baryon or isospin chemical potential equal to the pion mass , in contrast to real qcd where the phase transition takes place at @xmath0 approximately equal to the nucleon mass . the resulting matter near the quantum phase transition is a dilute bose condensate of diquarks or pions with weakly repulsive interactions @xcite . the equations of state and elementary excitations in such matter have been investigated many years ago by bogoliubov @xcite and lee , huang , and yang @xcite . bose - einstein condensation ( bec ) phenomenon is believed to widely exist in dense matter , such as pions and kaons can condense in neutron star matter if the electron chemical potential exceeds the effective mass for pions and kaons @xcite . however , the condensation of pions and kaons in neutron star matter is rather complicated due to the meson - nucleon interactions in dense nuclear medium . on the other hand , at asymptotically high density , perturbative qcd calculations show that the ground state is a weakly coupled bcs superfluid with the condensation of overlapping cooper pairs @xcite . it is interesting that the dense bcs superfluid and the dilute bose condensate have the same symmetry breaking pattern and thus are continued with one another . in condensed matter physics , this phenomenon was first discussed by eagles @xcite and leggett @xcite and is now called bec - bcs crossover . it has been successfully realized using ultracold fermionic atoms in the past few years @xcite . while the lattice simulations of two - color qcd at finite baryon chemical potential @xcite and qcd at finite isospin chemical potential @xcite have been successfully performed , we still ask for some effective models to link the physics of bose condensate and the bcs superfluidity . the chiral perturbation theories @xcite as well as the linear sigma models @xcite , which describe only the physics of bose condensate , do not meet our purpose . the nambu jona - lasinio ( njl ) model @xcite with quarks as elementary blocks , which describes well the mechanism of chiral symmetry breaking and low energy phenomenology of the qcd vacuum , is generally believed to work at low and moderate temperatures and densities @xcite . recently , this model has been used to describe the superfluid transition at finite chemical potentials @xcite for the special cases we are interested in this paper . one finds that the critical chemical potential for the superfluid transition predicted by the njl model is indeed equal to the pion mass @xcite , and the chiral and diquark condensates obtained from the mean - field calculation agree with the results from lattice simulations and chiral perturbation theories near the quantum phase transition @xcite . the njl model also predicts a bec - bcs crossover when the chemical potential increases @xcite . a natural problem arises : how can the fermionic njl model describe the weakly interacting bose condensate near the quantum phase transition ? in fact , we do not know how the repulsive interactions among diquarks or mesons enter in the pure mean - field calculations @xcite . in this paper , we will focus on this problem and show that the repulsive interaction is indeed properly included even in the mean - field calculations . this phenomenon is in fact analogous to the bcs description of the molecular condensation in strongly interacting fermi gases studied by leggett many years ago @xcite . fermionic models have been used to describe the bec - bcs crossover in cold fermi gases by the cold atom community . recently , it has been shown that we can recover the equation of state of the dilute bose condensate with correct boson - boson scattering length in the strong coupling limit , including the lee - huang - yang correction by considering the beyond - mean - field corrections @xcite . in appendix [ app1 ] , we give a summary of the many - body theoretical approach in cold atoms , which is useful for us to understand the theoretical approach and the results of this paper . in this paper , using two - color two - flavor qcd as an example and following the theoretical approach of the bec - bcs crossover in cold fermi gases @xcite , we examine how the njl model describes the weakly interacting bose condensate and the bec - bcs crossover . near the quantum phase transition point @xmath1 , we perform a ginzburg - landau expansion of the effective potential at the mean - field level , and show that the ginzburg - landau free energy is essentially the gross - pitaevskii free energy describing weakly interacting bose condensates via a proper redefinition of the condensate wave function . as a by - product , we obtain a diquark - diquark scattering length @xmath5 ( @xmath6 is the pion decay constant ) characterizing the repulsive interaction between the diquarks , which recovers the tree - level result predicted by chiral lagrangian @xcite . we also show analytically that the goldstone mode takes the same dispersion as the bogoliubov excitation in weakly interacting bose condensates , which gives a diquark - diquark scattering length identical to that in the gross - pitaevskii free energy . the mixing between the sigma meson and diquarks plays an important role in recovering the bogoliubov excitation . the results of in - medium chiral and diquark condensates predicted by chiral perturbation theory are analytically recovered . at high density , we find the superfluid matter undergoes a bec - bcs crossover at @xmath7 with @xmath8 being the mass of the sigma meson . at @xmath9 , we find that the chiral symmetry is approximated restored and the spectra of pions and sigma meson become nearly degenerated . well above the chemical potential of chiral symmetry restoration , the degenerate pions and sigma meson undergo a mott transition , where they become unstable resonances . because of the spontaneous breaking of baryon number symmetry , mesons can decay into quark pairs in the superfluid medium at nonzero momentum . the beyond - mean - field corrections are studied . the thermodynamic potential including the gaussian fluctuations is derived . it is shown that the vacuum state @xmath10 is thermodynamically consistent in the gaussian approximation , i.e. , all thermodynamic quantities keep vanishing in the regime @xmath10 even though the beyond - mean - field corrections are included . near the quantum phase transition point , we expand the fluctuation contribution to the thermodynamic potential in powers of the superfluid order parameter . to leading order , the beyond - mean - field correction is quartic and its effect is to renormalize the diquark - diquark scattering length . the correction to the mean - field result is shown to be proportional to @xmath11 . thus , our theoretical approach provides a new way to calculate the diquark - diquark or meson - meson scattering lengths in the njl model beyond the mean - field approximation . we also find that we can obtain a correct transition temperature of bose condensation in the dilute limit , including the beyond - mean - field corrections . the paper is organized as follows : in sec . [ s2 ] , we derive the general effective action of the two - color njl model at finite temperature and density , and determine the model parameters via the vacuum phenomenology . in sec . [ s3 ] , we investigate the properties of dilute bose condensate near the quantum phase transition at the mean - field level . in sec . [ s5 ] , the properties of matter at high density are discussed . beyond - mean - field corrections are studied in sec . we summarize in sec . natural units are used throughout .

qcd - like theories possess a positively definite fermion determinant at finite baryon chemical potential and the lattice simulation can be successfully performed . while the chiral perturbation theories are sufficient to describe the bose condensate at low density , to describe the crossover from bose - einstein condensation ( bec ) to bcs superfluidity at moderate density we should use some fermionic effective model of qcd , such as the nambu jona - lasinio model . in this paper , using two - color two - flavor qcd as an example , we examine how the nambu jona - lasinio model describes the weakly interacting bose condensate at low density and the bec - bcs crossover at moderate density . near the quantum phase transition point ( is the mass of pion / diquark multiplet ) , the ginzburg - landau free energy at the mean - field level can be reduced to the gross - pitaevskii free energy describing a weakly repulsive bose condensate with a diquark - diquark scattering length identical to that predicted by the chiral perturbation theories . the goldstone mode recovers the bogoliubov excitation in weakly interacting bose condensates . the results of in - medium chiral and diquark condensates predicted by chiral perturbation theories are analytically recovered .

qcd - like theories possess a positively definite fermion determinant at finite baryon chemical potential and the lattice simulation can be successfully performed . while the chiral perturbation theories are sufficient to describe the bose condensate at low density , to describe the crossover from bose - einstein condensation ( bec ) to bcs superfluidity at moderate density we should use some fermionic effective model of qcd , such as the nambu jona - lasinio model . in this paper , using two - color two - flavor qcd as an example , we examine how the nambu jona - lasinio model describes the weakly interacting bose condensate at low density and the bec - bcs crossover at moderate density . near the quantum phase transition point ( is the mass of pion / diquark multiplet ) , the ginzburg - landau free energy at the mean - field level can be reduced to the gross - pitaevskii free energy describing a weakly repulsive bose condensate with a diquark - diquark scattering length identical to that predicted by the chiral perturbation theories . the goldstone mode recovers the bogoliubov excitation in weakly interacting bose condensates . the results of in - medium chiral and diquark condensates predicted by chiral perturbation theories are analytically recovered . the bec - bcs crossover and meson mott transition at moderate baryon chemical potential as well as the beyond - mean - field corrections are studied . part of our results can also be applied to real qcd at finite baryon or isospin chemical potential .

1609.07581

i

from the famous einstein - podolsky - rosen ( epr ) paradox @xcite to bell s seminal discovery @xcite , quantum theory has never failed to surprise us with its plethora of intriguing phenomena and mind - boggling applications @xcite . among those who made the bizarre nature of quantum theory evident was schrdinger , who not only coined the term entanglement " , but also pointed out that quantum theory allows for _ steering _ @xcite : through the act of local measurements on one - half of an entangled state , a party can _ remotely _ steer the set of ( conditional ) quantum states accessible by the other party . taking a quantum information perspective , the demonstration of steering can be viewed as the verification of entanglement involving an untrusted party @xcite . imagine that two parties alice and bob share some quantum state and alice wants to convince bob that the shared state is entangled , but bob does not trust her . if alice can convince bob that the shared state indeed exhibits epr steering , then bob would be convinced that they share entanglement , as the latter is a prerequisite for steering . note , however , that shared entanglement is generally insufficient to guarantee steerability . interestingly , steerability is actually a necessary but generally insufficient condition for the demonstration of bell - nonlocality @xcite . hence , steering represents a form of quantum inseparability in between entanglement and bell - nonlocality . apart from entanglement verification in a partially - trusted scenario , steering has also found applications in the distribution of secret keys in partially trusted scenario @xcite . from a resource perspective , the steerability of a quantum state @xmath0 , i.e. , whether @xmath0 is steerable and the extent to which it can exhibit steering turns out to provide also an indication for the usefulness of @xmath0 in other quantum information processing tasks . for instance , steerability as quantified by steering robustness @xcite is monotonously related to the probability of success in the problem of subchannel discrimination when one is restricted to local measurements aided by one - way communications . the characterization of quantum states that are capable of exhibiting steering and the quantification of steerability are thus of relevance not just from a fundamental viewpoint , but also in quantum information . surprisingly , very little is known in terms of which quantum state is ( un)steerable ( see , however , @xcite ) . here , we derive some generic sufficient conditions for steerability that can be applied to quantum state of arbitrary hilbert space dimensions . importantly , in contrast to conventional approach of steering inequalities @xcite where an optimization over the many measurements that can be performed by each party is needed , our criteria only requires the relatively straightforward computation of the fully entangled fraction @xcite . given that some entangled quantum state @xmath0 can not exhibit steering @xcite , a natural question that arises is whether the steerability of such a state can be _ superactivated _ by allowing joint measurements on multiple copies of @xmath0 . in other words , is it possible that some @xmath0 that is not steerable becomes steerable if local measurements are performed instead on @xmath1 for some large enough @xmath2 ? building on some recent results established for bell - nonlocality @xcite , we provide here an affirmative answer to the above question . note that even for a quantum state @xmath0 that is steerable , it is interesting to investigate how their steerability scales with the number of copies . for instance , is it possible to amplify the amount of steering - inequality violation by an _ arbitrarily large _ amount if only a small number of copies is available ( see @xcite for analogous works in the context of bell - nonlocality ) ? again , we provide a positive answer to this question , showing that an unbounded amount of amplification can be obtained by allowing joint measurements on as few as three copies of a quantum state that is barely steerable , or even unsteerable under projective measurements . the rest of this paper is structured as follows . in section [ sec : prelim ] , we give a brief overview on some of the basic notions in bell - nonlocality and epr steering that we will need in subsequent discussions . there , inspired by the work of cavalcanti _ et al . _ @xcite , we also introduce the notion of _ steering fraction _ and _ largest ( steering - inequality ) violation _ , which are crucial quantities that lead to many of the findings mentioned above . for instance , in section [ sec : characterization ] , we use these quantities to derive ( 1 ) a general sufficient condition for an arbitrary quantum state @xmath0 to be steerable and ( 2 ) upper bounds on the largest steering - inequality violation of an arbitrary finite - dimensional maximally entangled state as a function of its hilbert space dimension @xmath3 . quantification of steerability using a strengthened version of steering fraction is discussed in section [ sec : quantifysteering ] there , we also demonstrate how this novel steering monotone @xcite is related to the others , such as the steerable weight @xcite and steering robustness @xcite . in section [ sec : superampli ] , we show the superactivation of steerablity , provide a procedure to construct a steering inequality for this purpose , and demonstrate unbounded amplification of steerability . we conclude in section [ sec : conclude ] with a discussion and some opened problems for future research .

quantum steering , also called einstein - podolsky - rosen steering , is the intriguing phenomenon associated with the ability of spatially separated observers to _ steer_by means of local measurements the set of conditional quantum states accessible by a distant party . in the light of quantum information , _ all _ steerable quantum states are known to be resources for quantum information processing tasks . here , via a quantity dubbed _ steering fraction _ , we derive a simple , but general criterion that allows one to identify quantum states that can exhibit quantum steering ( without having to optimize over the measurements performed by each party ) , thus making an important step towards the characterization of steerable quantum states . the criterion , in turn , also provides upper bounds on the largest steering - inequality violation achievable by arbitrary finite - dimensional maximally entangled states . for the quantification of steerability , we prove that a strengthened version of the steering fraction is a _ in particular , our approach allows one to explicitly construct a steering inequality to manifest this phenomenon .

quantum steering , also called einstein - podolsky - rosen steering , is the intriguing phenomenon associated with the ability of spatially separated observers to _ steer_by means of local measurements the set of conditional quantum states accessible by a distant party . in the light of quantum information , _ all _ steerable quantum states are known to be resources for quantum information processing tasks . here , via a quantity dubbed _ steering fraction _ , we derive a simple , but general criterion that allows one to identify quantum states that can exhibit quantum steering ( without having to optimize over the measurements performed by each party ) , thus making an important step towards the characterization of steerable quantum states . the criterion , in turn , also provides upper bounds on the largest steering - inequality violation achievable by arbitrary finite - dimensional maximally entangled states . for the quantification of steerability , we prove that a strengthened version of the steering fraction is a _ convex steering monotone _ and demonstrate how it is related to two other steering monotones , namely , steerable weight and steering robustness . using these tools , we further demonstrate the _ superactivation _ of steerability for a well - known family of entangled quantum states , i.e. , we show how the steerability of certain entangled , but unsteerable quantum states can be recovered by allowing joint measurements on multiple copies of the same state . in particular , our approach allows one to explicitly construct a steering inequality to manifest this phenomenon . finally , we prove that there exist examples of quantum states ( including some which are unsteerable under projective measurements ) whose steering - inequality violation can be arbitrarily amplified by allowing joint measurements on as little as three copies of the same state . for completeness , we also demonstrate how the largest steering - inequality violation can be used to bound the largest bell - inequality violation and derive , analogously , a simple sufficient condition for bell - nonlocality from the latter .

1609.07581

c

in this paper , we have introduced the tool of steering fraction @xmath280 and used it to establish novel results spanning across various dimensions of the phenomenon of quantum steering . below , we briefly summarize these results and comment on some possibilities for future research . firstly , we have derived a general sufficient condition for _ any _ bipartite quantum state @xmath0 to be steerable ( bell - nonlocal ) in terms of its fully entangled fraction , a quantity closely related to the usefulness of @xmath0 for teleportation @xcite . as we briefly discussed in section [ sec : characterization ] , we do not expect these sufficient conditions to detect all steerable ( bell - nonlocal ) states . nonetheless , let us stress that to determine if a quantum state is steerable ( as with determining if a quantum state can exhibit bell - nonlocality , see , e.g. , @xcite ) is a notoriously difficult problem , which often requires the optimization over the many parameters used to describe the measurements involved in a steering experiment ( and/or the consideration of potentially infinitely many different steering inequalities ) . in contrast , the general criterion that we have presented in theorem [ thm : sufficentsteerability ] for steerability ( and theorem [ theorem : suff_condi_nonlocality ] for bell - nonlocality ) only requires a relatively straightforward computation of the fully entangled fraction of the state of interest . given that these sufficient conditions are likely to be suboptimal , an obvious question that follows is whether one can find an explicit threshold @xmath281 that is smaller than that given by theorem [ thm : sufficentsteerability ] ( theorem [ theorem : suff_condi_nonlocality ] ) such that @xmath282 still guarantees steerability ( bell - nonlocality ) . while this may seem like a difficult problem , recent advances @xcite in the algorithmic construction of local hidden - variable ( -state ) models may shed some light on this . more generally , it will be interesting to know if our sufficient condition can be strengthened while preserving its computability . in particular , it will be very useful to find analogous steerability ( bell - nonlocality ) criteria that are tight . on the other hand , the aforementioned sufficient condition has also enabled us to derive upper bounds as functions of @xmath3on the largest steering - inequality violation @xmath102 ( @xmath283 ) achievable by the maximally entangled state @xmath76 under general povms ( projective measurements ) . in particular , using the general connection between @xmath102 and the largest bell - inequality violation , @xmath284 , established in appendix [ app : bell ] , our upper bounds on @xmath102 and @xmath283 imply upper bounds on @xmath284 and @xmath285 by @xmath76 ( for non - negative * b * ) , respectively . notably , our upper bound on @xmath285 is somewhat tighter than that due to palazuelos @xcite . if any strengthened sufficient conditions for steerability , as discussed above , are found , it would also be interesting to see if they could lead to tighter ( non - asymptotic ) upper bound(s ) on the largest steering - inequality ( and/or bell - inequality ) violation attainable by @xmath76 . the tool of steering fraction @xmath280 , in addition , can be used to quantify steerability . in particular , we showed that if @xmath280 is optimized over all ( non - negative ) * f * , the resulting quantity can be cast as a _ convex steering monotone _ @xcite which we referred to as the _ optimal steering fraction _ @xmath158 . we further demonstrated how this monotone is quantitatively related to two other convex steering monotones steerable weight @xcite and steering robustness @xcite . in the light of quantum information , it would be desirable to determine an operational meaning of @xmath158 , e.g. , in the context of some quantum information tasks ( cf . steering robustness @xcite ) . establishment of quantitative relations between @xmath158 and other convex steering monotones , such as the relative entropy of steering @xcite , would certainly be very welcome . in particular , it would be highly desirable to establish quantitative relations that allow one to estimate @xmath158 from other easily - computable steering monotones , such as the steerable weight , or the steering robustness . using the established sufficient condition for steerability , we have also demonstrated the superactivation of steerability , i.e. , the phenomenon that certain unsteerable quantum state @xmath0 becomes , for sufficiently large @xmath2 , steerable when joint _ local _ measurements on @xmath1 are allowed . a drawback of the examples that we have presented here is that they inherit directly from the superactivation of bell - nonlocality due to palazuelos @xcite and cavalcanti _ et al . _ @xcite . an obvious question that follows is whether one can construct explicit examples for the superactivation of steerability using quantum states whose bell - nonlocality _ can not _ be superactivated via joint measurements . one the other hand , with joint local measurements , we showed that the steering - inequality ( bell - inequality ) violation of certain barely steerable ( bell - nonlocal ) @xmath0 [ or even unsteerable ( bell - local ) with projective measurements ] can be arbitrarily amplified , in particular , giving an arbitrarily large steering - inequality ( bell - inequality ) violation with @xmath286 . could such unbounded amplification be achieved using joint measurements on two copies of the same state ? our proof technique , see eq . , clearly requires a minimum of three copies for unbounded amplification to take place but it is conceivable that a smaller number of copies suffices if some other steering ( bell ) inequality is invoked , a problem that we shall leave for future research . the authors acknowledge useful discussions with nicolas brunner , daniel cavalcanti , flavien hirsch , marco tlio quintino and helpful suggestions from an anonymous referee of aqis2016 . this work is supported by the ministry of education , taiwan , r.o.c . , through aiming for the top university project " granted to the national cheng kung university ( ncku ) , and the ministry of science and technology , taiwan ( grant no.104 - 2112-m-006 - 021-my3 ) . _ note added. _ while completing this manuscript , we became aware of the work of @xcite who independently ( 1 ) derived a sufficient condition of steerability in terms of the fully entangled fraction and ( 2 ) demonstrated the superactivation of steering of the isotropic states .

convex steering monotone _ and demonstrate how it is related to two other steering monotones , namely , steerable weight and steering robustness . using these tools , we further demonstrate the _ superactivation _ of steerability for a well - known family of entangled quantum states , i.e. , we show how the steerability of certain entangled , but unsteerable quantum states can be recovered by allowing joint measurements on multiple copies of the same state .

quantum steering , also called einstein - podolsky - rosen steering , is the intriguing phenomenon associated with the ability of spatially separated observers to _ steer_by means of local measurements the set of conditional quantum states accessible by a distant party . in the light of quantum information , _ all _ steerable quantum states are known to be resources for quantum information processing tasks . here , via a quantity dubbed _ steering fraction _ , we derive a simple , but general criterion that allows one to identify quantum states that can exhibit quantum steering ( without having to optimize over the measurements performed by each party ) , thus making an important step towards the characterization of steerable quantum states . the criterion , in turn , also provides upper bounds on the largest steering - inequality violation achievable by arbitrary finite - dimensional maximally entangled states . for the quantification of steerability , we prove that a strengthened version of the steering fraction is a _ convex steering monotone _ and demonstrate how it is related to two other steering monotones , namely , steerable weight and steering robustness . using these tools , we further demonstrate the _ superactivation _ of steerability for a well - known family of entangled quantum states , i.e. , we show how the steerability of certain entangled , but unsteerable quantum states can be recovered by allowing joint measurements on multiple copies of the same state . in particular , our approach allows one to explicitly construct a steering inequality to manifest this phenomenon . finally , we prove that there exist examples of quantum states ( including some which are unsteerable under projective measurements ) whose steering - inequality violation can be arbitrarily amplified by allowing joint measurements on as little as three copies of the same state . for completeness , we also demonstrate how the largest steering - inequality violation can be used to bound the largest bell - inequality violation and derive , analogously , a simple sufficient condition for bell - nonlocality from the latter .

astro-ph0610088

r

in this section the dependence of stripping on the masses of the large and satellite halos and on the model s parameters is explicitly determined . the maximum ram pressure , the ram pressure at the pericenter of the galaxy s orbit , is given by @xmath58 where @xmath59 is the distance of closest approach , @xmath60 is the orbital energy per unit mass , @xmath61 , @xmath62 , and @xmath63 is the orbital angular momentum per unit mass . both @xmath64 and @xmath65 are expected to be independent of the masses of the satellite and the cluster . at any point in the orbit , @xmath66 , the ram pressure is @xmath67 where dimensionless pressure , @xmath68 is defined . @xmath69\left(1 + \frac{s^2}{(r_c / r_{v , gr})^2 } \right)^{(-3/2)\beta}\ ] ] equations for the orbital speeds are derived in appendix [ ap orbits ] . assuming that @xmath6 , @xmath11 , @xmath70 , and @xmath71 do not depend on the mass of the halo , the @xmath1 dependence of both @xmath72 and @xmath73 is @xmath74 . for a potential of the form in eq . [ eqn . full restoring pressure ] , the force per unit gas mass in the @xmath75 direction is found as follows . @xmath76 @xmath77 assuming that the @xmath78 and @xmath79 introduced in section [ sec . restoring pressure ] and @xmath57 are constant with mass , the mass dependence of the restoring force per unit gas mass is given by @xmath80 . for the gas disk , if the fractional mass of the gas , @xmath81 , and the scaled size of the disk , @xmath82 , are both constant , then @xmath83 combining the above , @xmath84 for any radius along the disk , @xmath85 , there is a maximum restoring pressure . if the maximum restoring pressure is greater than the ram pressure , the satellite holds the gas at this radius . the condition for the satellite holding its gas can be written as @xmath86 @xmath87^{3/2}\ ] ] the restoring pressure of the galactic hot halo is @xmath88 the condition for retaining this gas within @xmath89 is @xmath90^{3/2}\ ] ] if the model s parameters : @xmath71 , @xmath57 , @xmath91 , @xmath92 , @xmath11 , @xmath70 , @xmath6 , the @xmath78 , and the @xmath79 ; are all independent of the mass of the satellite and cluster , then the fraction of gas that is stripped from a satellite depends on the ratio @xmath93 . in this case , @xmath94 and for both the gas disk and hot galactic halo @xmath95 [ model ind ] assuming that this set of parameters is constant is equivalent to assuming that the physical parameters scale with mass in the most obvious way . it assumes that component masses scale as @xmath96 , lengths scale as @xmath97 , and the central densities of the icm and the satellite s hot galactic halo are constant . for any model in which these scalings hold , the result that the extent of stripping depends on @xmath93 will hold . for any cluster mass , the scaled radius will set the icm density and the orbital velocity will be proportional to @xmath98 . the restoring pressure depends on the depth of the satellite galaxy s gravitational potential well and the density of the gas disk . for a generic potential , @xmath99 and the restoring force per unit gas mass is proportional to @xmath100 . if the mass of the gas disk scales with @xmath0 and the disk length with @xmath101 , then the density of the gas disk scales as @xmath100 and the restoring pressure as @xmath102

whether ram pressure stripping of the outer disk or hot galactic halo occurs is found to depend primarily on the ratio of the satellite galaxy mass to the mass of the host group or cluster . how the effectiveness of ram pressure stripping depends on the density of the inter - group gas , the dark matter halo concentrations , and the scale lengths and masses of the satellite components is explored . the predictions of the model are shown to be well matched to h i observations of spirals in a sample of nearby clusters .

ram pressure stripping is an important process in the evolution of both dwarf galaxies and large spirals . large spirals are severely stripped in rich clusters and may be mildly stripped in groups . dwarf galaxies can be severely stripped in both clusters and groups . a model is developed that describes the stripping of a satellite galaxy s outer h i disk and hot galactic halo . the model can be applied to a wide range of environments and satellite galaxy masses . whether ram pressure stripping of the outer disk or hot galactic halo occurs is found to depend primarily on the ratio of the satellite galaxy mass to the mass of the host group or cluster . how the effectiveness of ram pressure stripping depends on the density of the inter - group gas , the dark matter halo concentrations , and the scale lengths and masses of the satellite components is explored . the predictions of the model are shown to be well matched to h i observations of spirals in a sample of nearby clusters . the model is used to predict the range of h i gas fractions a satellite of mass can lose orbiting in a cluster of mass .

astro-ph0610088

c

the model developed here relates the degree of ram pressure stripping a satellite galaxy experiences to the galaxy and cluster masses and can be used to quickly determine the extent to which a galaxy is likely to be stripped . all galaxies lose most of their hot galactic halo . in clusters galaxies at moderate mass ratios , @xmath252 , are moderately stripped , @xmath253 , at intermediate distances , @xmath254 , from the cluster center and severely stripped closer to the cluster center . satellites with lower @xmath93 are severely stripped even at intermediate distances . stripping also occurs in groups . however , the same degree of stripping occurs for lower @xmath93 in groups than in clusters . dwarf galaxies are moderately to severely stripped at intermediate distances in groups , and large spiral galaxies can be moderately stripped if they travel to small @xmath66 . the model is simple and motivated by observations . dark matter profiles are well matched to nfw profiles outside a possible core , observations of x - ray gas in clusters can be fit using @xmath11 profiles , and the average gravitational potential of most galaxies should be well matched by the model potential . the model has a large number of parameters , but the values for many can be taken from observations . however , the model assumes that clusters are static . from an evolutionary standpoint , it is only valid after the group or cluster has acquired an icm . on the other hand , in dynamic clusters bulk icm motions may cause more stripping to occur than is predicted by the model . this is observed in the virgo cluster @xcite . ram pressure stripping is not the only way to remove gas from galaxies . in groups and clusters tidal stripping of both gas and stars can occur ( e.g. , @xcite ) . for dwarf galaxies it has been proposed that supernova winds associated with bursts of star formation may expel gas @xcite , and the efficiency of this mechanism may depend on the environment @xcite . however , ram pressure stripping is capable of removing gas from galaxies across a large range of galaxy masses and environments . in particular ram pressure stripping can act when supernova - driven ejection is likely to be inefficient and can remove gas from galaxies that are either not tidally interacting or that are experiencing only weak tidal interactions . the tidal radius , @xmath255 , at the physical orbital radius , @xmath256 , as estimated using @xmath257 , can be compared to the stripping radius , @xmath85 , as determined using the model . here @xmath258 is the cluster mass enclosed within @xmath256 . in scaled coordinates , @xmath259^{-1/3}$ ] , where @xmath260 . the h i fraction lost to tidal stripping depends only on the scaled size of the gas disk and @xmath66 and not on either mass . because the gas disk resides in the center of the satellite s dark halo , tidal stripping rarely effects the h i disk and practically never competes with the effect rps . for example , in order for the h i disk to be tidally stripped to @xmath261 , a satellite must travel to @xmath262 . by this @xmath66 the entire outer h i disk has been ram pressure stripped for almost all satellites . the effect of varying the model s parameters was studied both to identify the parameters that most effect the extent to which an individual galaxy is stripped and to determine the range of satellite and cluster masses that result in the same @xmath85 . the extent to which an individual galaxy is stripped depends most strongly on the galaxy s @xmath66 or @xmath59 , inclination , @xmath162 , and disk scale length , @xmath82 , and on the cluster s icm profile . galaxies that reach smaller @xmath66 , are face - on to the wind , have denser disks , or orbit through a denser icm are more effectively stripped . galaxies in the same cluster with @xmath0 that differ by as much as a factor of 25 can be stripped to the same @xmath85 . in different clusters , galaxies on identical orbits with identical morphologies , identical @xmath78 , @xmath79 , @xmath17 , can be stripped to the same @xmath85 with @xmath0 values that differ by as much as a factor of 30 . ram pressure stripping of satellite galaxies outer gas disks and hot galactic halos is occurring frequently in both groups and clusters . in general , removing gas from a galaxy reduces star formation . the gas in the outer disk and hot galactic halo is not currently involved in star formation , but may feed future star formation if it is not stripped . therefore , it would be interesting to study in more detail how the removal of this gas affects galaxy evolution and to determine if , to what extent , and how quickly star formation declines after a galaxy is stripped .

ram pressure stripping is an important process in the evolution of both dwarf galaxies and large spirals . a model is developed that describes the stripping of a satellite galaxy s outer h i disk and hot galactic halo . the model can be applied to a wide range of environments and satellite galaxy masses .

ram pressure stripping is an important process in the evolution of both dwarf galaxies and large spirals . large spirals are severely stripped in rich clusters and may be mildly stripped in groups . dwarf galaxies can be severely stripped in both clusters and groups . a model is developed that describes the stripping of a satellite galaxy s outer h i disk and hot galactic halo . the model can be applied to a wide range of environments and satellite galaxy masses . whether ram pressure stripping of the outer disk or hot galactic halo occurs is found to depend primarily on the ratio of the satellite galaxy mass to the mass of the host group or cluster . how the effectiveness of ram pressure stripping depends on the density of the inter - group gas , the dark matter halo concentrations , and the scale lengths and masses of the satellite components is explored . the predictions of the model are shown to be well matched to h i observations of spirals in a sample of nearby clusters . the model is used to predict the range of h i gas fractions a satellite of mass can lose orbiting in a cluster of mass .

1405.4246

i

ultracold atomic gases are among the most successful implementations of a quantum simulator @xcite . some paradigms in condensed matter physics have analogues in ultracold atomic gases and can be studied in an ideal setting with full control of the hamiltonian parameters . for instance , the microcanonical approach to quantum transport @xcite that has been used to test certain assumptions of the scattering approach to conduction in nanoscale systems , can now be fully realized in cold - atom systems with relative ease , thus providing a direct test of several predictions that are difficult to verify in the solid state @xcite . recent advances in experiments have demonstrated artificial electric and magnetic fields from artificial gauge fields for cold atoms @xcite , which offer the opportunity to study a great variety of problems relevant to conventional condensed matter systems . when cold atoms are confined in optical lattices , the hopping coefficients can acquire a phase via peierls substitution @xcite using artificial gauge fields @xcite or lattice modulations @xcite . for example , charge and spin transport in strongly correlated systems @xcite are among the most interesting problems that can now be addressed from a different perspective using ultracold atomic gases driven by artificial gauge fields . in this regard , the superfluid / mott - insulator transition @xcite in the bose - hubbard model has been realized in cold - atom systems @xcite and subsequently studied in a large number of papers ( see @xcite for a review ) . here , we investigate transport properties of the bose - hubbard model by means of a sudden change of the hopping phase that delivers a finite momentum to the gas ( see fig . [ fig : one_1]*a * ) . for low filling or weak interactions the system is close to the continuum limit and the atoms are delocalized . for integer filling and strong enough interactions the atoms localize and the system becomes a mott insulator . it is important to address the issue of how the lattice - induced correlations affect the transport in between these two limits , as the system is tuned from the weakly - interacting regime to the strongly interacting one . in order to achieve this goal , we employ the quasi - exact density matrix renormalization group ( dmrg ) method using a matrix product state ( mps ) ansatz for the wavefunction . recently the static dmrg method has been applied to the study of the bose - hubbard model phase diagram under the influence of artificial gauge fields @xcite . as a step further we study the evolution in time using the time - dependent dmrg algorithm ( tdmrg ) @xcite within the microcanonical picture of transport @xcite , which is ideal to study transport phenomena in closed finite systems as the present ones . we concentrate on the case of ultracold bosons in a one - dimensional optical lattice with a superimposed external confining potential which is constant throughout the system and rises sharply at the boundary . hard - wall confining potentials of this kind are not common in ultracold atoms where harmonic confining potentials are the norm , but have been recently realized @xcite , and the ground state properties of uniform condensates have been measured @xcite . they offer the advantage that in a uniform system one can focus on the intrinsic transport properties without spurious effects due to the external confinement . moreover , boundary effects such as the density waves studied here may not be observable in a harmonic confinement where the whole cloud can move together in a sloshing fashion . after a sudden quench of the hopping phase to a non - zero value , the gas in the superfluid phase is quickly driven into a quasi - steady state with constant current that does not decay with time . we use the quasi - exact tdmrg to extract the quasi - steady state current as a function of filling and interaction strength and we argue that in fact this corresponds to measuring the drude weight @xcite . ultimately this technique can be used to infer the drude weight or , possibly , the superfluid fraction in real experiments in a simple way . near the superfluid - mott insulator transition the current decays quickly from the finite value attained immediately after the quench , as expected from the vanishing drude weight . * a ) * schematic representation of the bose - hubbard hamiltonian with a complex time - dependent hopping term @xmath0 and interaction term @xmath1 . the system is confined by hard - wall ( h.w . ) boundaries at the edges of the lattice @xcite . * b ) * density profiles observed after the quench of the hopping phase @xmath2 for @xmath3 . the light grey profile corresponds to @xmath4 ( @xmath5 ) , the grey to @xmath6 and the black to @xmath7 . the bulk current @xmath8 is driven to the left of the system by the phase quench ( @xmath9 is the bulk density in the initial state and @xmath10 the particle velocity ) . a shock wave with height @xmath11 and front speed @xmath12 forms at the left boundary , while a rarefaction wave with the same height forms at the right boundary . ] another important quantity characterizing the quasi - steady state is the entanglement entropy between two parts of the system . recently it has been found that the rate at which entanglement entropy is generated between two fermi seas is related to the noise statistics of the current through a quantum point contact that couples the two reservoirs @xcite . since fluctuations can be accessed in the ultracold gases counterpart of solid state transport experiments @xcite , we report results for the entanglement entropy production rate in the bose - hubbard model driven by a quench in the hopping phase and we find that the entropy production rate is non - zero only around the superfluid - mott insulator transition . this can serve as an alternative way to detect the transition in experiments . the quasi - steady state is not expected to persist indefinitely in a system with hard - wall boundary conditions . in our case we observe that shock and rarefaction waves form at the boundaries and propagate toward the center of the chain as illustrated in fig . [ fig : one_1]*b*. we assume that the continuously shrinking middle region in between the two waves is a good approximation of a true steady state , since the current reaches its stationary value on a short time scale and is independent of the position within the same region . moreover , in recent experiments @xcite it has been observed that perturbations ( in our case the boundary effects ) propagate at a finite speed , in a light cone manner . it is natural to assume in our case that the speed of propagation of the perturbations is in fact the shock front speed @xmath13 ( see fig . [ fig : one_1]*b * ) . the quench that we consider here is in essence the _ piston problem _ of shock wave theory for a fluid in a lattice rather than in the continuum . in the piston problem a piston is moved at constant speed in a fluid initially at rest in a long cylinder . the piston problem for dispersive shock waves has been considered recently for a fluid described by the nonlinear schrdinger equation @xcite . we find that , regardless of the presence of the lattice , dispersive shock and rarefaction waves form whenever the system is a superfluid , but not in the mott insulator . from our simulations we extract the height of the well defined density step @xmath14 that precedes the shock wave or , equivalently , the wave speed @xmath13 ( see fig . [ fig : one_1]*b * ) . we conclude that dispersive waves are a general signature of the superfluid . we emphasize that the formation of such waves is a consequence of the hard wall potentials at the boundaries . if the system is confined in a harmonic trap , the same kind of quench would simply trigger a periodic oscillation of the center of mass , eventually damped by lattice effects @xcite , but waves do not form . note that recent works have focused on the dynamics of fermi- and bose - hubbard models under a constant force @xcite . the quench protocol considered here is distinct and leads to a quite different dynamics . the difference with respect to the case of an applied constant force for @xmath15 is that the quench considered here corresponds to a large force applied for a short time ( impulsive force ) , after which the system evolves freely . a constant force applied to a lattice system generally leads to bloch oscillations of the gas center of mass @xcite , but no such oscillations have been observed in our case . on the other hand , a large tilted lattice potential applied for a small time can be used to approximate the phase quench considered here . this offers an alternative way to experimentally verify our results . the paper is organized as follows . section [ sec : bh ] briefly reviews the bose - hubbard model and provides details on the tdmrg method used to study it . in section [ sec : noninteracting ] the limiting cases of noninteracting bosons and fermions are discussed , as well as the mapping of the bose - hubbard model into the xxz spin chain . in section [ sec : dynamics ] the bulk properties ( away from the boundaries ) of the quasi - steady state are considered , namely current ( sec . [ sec : steady ] ) , drude weight ( sec . [ sec : drude ] ) and entanglement entropy ( sec . [ sec : entanglement ] ) . in section [ sec : shock ] is focused on the density profile dynamics at the boundaries which is interpreted as the formation and propagation of shock and rarefaction waves . finally , section [ sec : conclusion ] summarizes the main results .

recent experimental realizations of artificial gauge fields for cold atoms are promising for generating steady states carrying a mass current in strongly correlated systems , such as the bose - hubbard model . , we analyze the effect of the lattice and interaction strength on the current generated by a quench in the artificial vector potential when the density varies from low values ( continuum limit ) up to integer filling in the mott - insulator regime . other observable quantities used to characterize the quasi - steady state in the bulk of the system are the drude weight and entanglement entropy production rate . the latter in particular provides a striking signature of the superfluid - mott insulator transition . our results should be verifiable with current experimental capabilities .

recent experimental realizations of artificial gauge fields for cold atoms are promising for generating steady states carrying a mass current in strongly correlated systems , such as the bose - hubbard model . moreover , a homogeneous condensate confined by hard - wall potentials from laser sheets has been demonstrated , which provides opportunities for probing the intrinsic transport properties of isolated quantum systems . using the time - dependent density matrix renormalization group ( tdmrg ) , we analyze the effect of the lattice and interaction strength on the current generated by a quench in the artificial vector potential when the density varies from low values ( continuum limit ) up to integer filling in the mott - insulator regime . there is no observable mass current deep in the mott - insulator state as one may expect . other observable quantities used to characterize the quasi - steady state in the bulk of the system are the drude weight and entanglement entropy production rate . the latter in particular provides a striking signature of the superfluid - mott insulator transition . furthermore , an interesting property of the superfluid state is the formation of shock and rarefaction waves at the boundaries due to the hard - wall confining potentials . we provide results for the height and the speed of the shock front that propagates from the boundary toward the center of the lattice . our results should be verifiable with current experimental capabilities .

1106.2898

i

the advent of the @xmath2 x - ray observatory , with its unprecedented sub - arcsecond resolution , allowed to study better than ever before the main contributors to the total x - ray emission of early - type galaxies ( hereafter etgs ) : the low - mass x - ray binaries ( lmxbs ; fabbiano 2006 ) , a population of weak sources as late type stellar coronae , cataclismic variables , and coronally active binaries ( pellegrini & fabbiano 1994 , revnivtsev et al . 2008 ) , the nuclear emission due to a supermassive black hole ( mbh ; e.g. , gallo et al . 2010 , pellegrini 2010 ) , and a hot interstellar medium ( ism ) with a temperature of a few million degrees . after careful subtraction of the stellar ( resolved and unresolved ) and nuclear emissions , the properties of the hot ism could be characterized with unprecedented accuracy . recently , this has been done for a sample of 30 normal ( non - cd ) etgs observed with @xmath2 to a depth ensuring the detection of bright lmxbs ( boroson et al . 2011 , hereafter bkf ) . this is the first x - ray sample of etgs covering a wide range of galactic luminosity , central velocity dispersion @xmath3 , and hot gas emission @xmath1 , and with the x - ray properties of the hot gas ( e.g. , luminosity @xmath1 and average temperature @xmath0 ) derived in a homogeneous way , using a complete and accurate procedure to subtract all kinds of non - gaseous emission ( nucleus , detected and undetected lmxbs , and unresolved weak stellar sources ) . this approach resulted in a larger fraction of hot gas - poor galaxies than in previous samples , with @xmath1 extending down to much lower values than before ( @xmath12 erg s@xmath5 ) , and showing a variation of up to @xmath13 orders of magnitude at the same galactic luminosity ( see also david et al . 2006 , diehl & statler 2007 , memola et al . such a wide variation , even larger than previously found , had been linked to the origin and evolution of the hot ism , and had provided evidence for the effectiveness of an internal heating mechanism ( as from type ia supernovae , hereafter snia s ) to regulate the gas evolution and produce its very different content in etgs at the present epoch ( loewenstein & mathews 1987 , david et al . 1990 , ciotti et al . 1991 ) ; the action of external agents ( gas stripping , confinement , accretion ) to reduce or enhance the gas content was also invoked ( e.g. , white & sarazin 1991 , brown & bregman 2000 , sun et al . 2007 ) . with this new characterization of the hot gas , bkf revisited the relationships between fundamental properties of the hot gas and of the host galaxy , as the @xmath14 , @xmath15 and @xmath16 relations , where @xmath3 is a representative measure of the depth of the galactic potential well ( eskridge et al . 1995 , osullivan et al . 2001 , 2003 ) . @xmath1 correlates positively with @xmath0 and @xmath3 , though with a wide variation at fixed @xmath8 and @xmath0 . interestingly , the best fit relation @xmath17 , close to what already known for x - ray luminous etgs ( osullivan et al . 2003 ) , is still moderately strong among etgs with low @xmath0 and @xmath1 ; also , in the @xmath15 relation , etgs with @xmath18 kev are the x - ray brightest ( with one exception ) , while those with @xmath19 kev are the x - ray faintest . the least gas rich etgs are then the coolest ones , which seemed contrary to expectation , if low @xmath1 etgs loose their ism in an outflow ( e.g. , david et al . 1990 , ciotti et al . 1991 ) , and the hotter the gas , the stronger is the outflow ( bkf ) . on average @xmath0 increases with @xmath3 , and most etgs lie above a rough estimate of the gas virial temperature ( @xmath20 ) , suggesting the presence of additional heating . etgs with a moderate to high gas content ( @xmath21 erg s@xmath5 ) follow a trend roughly parallel to that of @xmath22 ; instead , etgs with little hot gas ( @xmath23 erg s@xmath5 ) have a similar temperature for @xmath3 ranging from 160 to 250 km s@xmath5 . this lack of correlation was attributed to a different dynamical state of the hot ism in gas - poor with respect to gas - rich etgs , though a full explanation of this aspect remained to be found ( bkf ) . this work takes advantage of the new accurate measurements of the hot gas properties , and of the fundamental relations @xmath15 and @xmath16 , derived down to galaxy masses and x - ray luminosities smaller than ever before ( bkf ) , to investigate the relationship between @xmath0 , the galaxy structure , the internal gas heating mechanisms ( snia s , and those linked to the gravitational potential ) , and the dynamical status of the gas flow . to this purpose , a few characteristic temperatures are introduced , depending on the nature of the gas heating sources and the galaxy structure , and relevant for the various gas flow phases ; these characteristic temperatures are then compared with the observed @xmath0 values . in doing so , galaxy mass models are built according to the most recent understanding of the etgs structure , such as their stellar mass profile and their dark matter content and distribution , as indicated by detailed modeling of optical observations and by the main scaling laws ( e.g. , cappellari et al . 2006 , weijmans et al . 2009 , auger et al . 2010 , napolitano et al . 2010 , shen & gebhardt 2010 ) . the aims are to address the following questions : can the gas heating sources above account for the observed @xmath0 s ? how are the various input energy sources for the gas used in the different flow phases ? is there any relation between @xmath0 and the flow phase ? we present in sect . [ heat ] the sources of mass and heating for the hot ism , in sect . [ esc ] the conditions for the gas to escape from the galaxy , in sect . [ mass ] the galaxy mass models , in sect . [ disc ] the comparison between observed and predicted temperatures , in sect . [ temp ] the relation between gas temperature and flow status , and in sect . [ concl ] the conclusions .

this work investigates the origin of the observed temperatures , by examining the relationship between them and the galaxy structure , the gas heating due to type ia supernovae ( snia s ) and the gravitational potential , and the dynamical status of the gas flow . in galaxies with km s ,

recently , the temperature and luminosity of the hot gas halos of early type galaxies have been derived with unprecedented accuracy from data , for a sample of 30 galaxies covering a wider range of galactic luminosity ( and central velocity dispersion ) than before . this work investigates the origin of the observed temperatures , by examining the relationship between them and the galaxy structure , the gas heating due to type ia supernovae ( snia s ) and the gravitational potential , and the dynamical status of the gas flow . in galaxies with km s , the s are close to a fiducial average temperature for the gas when in outflow ; at 200(km s , the s are generally lower than this , and unrelated with , which requires a more complex gas flow status ; at larger , the s may increase as , as expected for infall heating , though heating from snia s , independent of , should be dominant . all observed s are larger than the virial temperature , by up to kev . this additional heating can be provided in the x - ray brightest galaxies by snia s and infall heating , with a snia s energy input even lower than in standard assumptions ; in the x - ray fainter ones it can be provided by snia s , whose energy input would be required close to the full standard value at the largest . this same energy input , though , would produce temperatures larger than observed at low , if entirely thermalized . the values of the observed s increase from outflows to inflows ; the gas is relatively hotter in outflows , though , if the s are rescaled by the virial temperature . for km s , lower values tend to correspond to lower s , which deserves further investigation .