global attractor for the kirchhoff type equation with a strong dissipation abstract the paper studies the longtime behavior of the kirchhoff type equation with a strong dissipation u t t m u 2 u u t h u t g u f x it proves that the related continuous semigroup s t possesses in the phase space with low regularity a global attractor which is connected and an example is shown

long time dynamics of kirchhoff wave models with strong nonlinear damping we study well posedness and long time dynamics of a class of quasilinear wave equations with a strong damping we accept the kirchhoff hypotheses and assume that the stiffness and damping coefficients are functions of the l2 norm of the gradient of the displacement we prove the existence and uniqueness of weak solutions and study their properties for a wide class of nonlinearities which covers the case of possible degeneration or even negativity of the stiffness coefficient and the case of a supercritical source term our main results deal with global attractors for strictly positive stiffness factors we prove that in the natural energy space endowed with a partially strong topology there exists a global finite dimensional attractor in the non supercritical case this attractor is strong in this case we also establish the existence of a fractal exponential attractor and give conditions that guarantee the existence of a finite number of determining functionals

compact composition operators on weighted hilbert spaces of analytic functions we characterize the compactness of composition operators acting on a large family of hilbert spaces of analytic functions which lie between bergman and dirichlet spaces our characterization is given in terms of generalized nevanlinna counting functions

holonomic functions of several complex variables and singularities of anisotropic ising n fold integrals focusing on examples associated with holonomic functions we try to bring new ideas on how to look at phase transitions for which the critical manifolds are not points but curves depending on a spectral variable or even fill higher dimensional submanifolds lattice statistical mechanics often provides a natural holonomic framework to perform singularity analysis with several complex variables that would in the most general mathematical framework be too complex or simply could not be defined in a learn by example approach considering several picard fuchs systems of two variables above calabi yau odes associated with double hypergeometric series we show that d finite holonomic functions are actually a good framework for finding properly the singular manifolds the singular manifolds are found to be genus zero curves we then analyze the singular algebraic varieties of quite important holonomic functions of lattice statistical mechanics the n fold integrals n corresponding to the n particle decomposition of the magnetic susceptibility of the anisotropic square ising model in this anisotropic case we revisit a set of so called nickelian singularities that turns out to be a two parameter family of elliptic curves we then find the first set of non nickelian singularities for 3 and 4 that also turns out to be rational or elliptic curves we underline the fact that these singular curves depend on the anisotropy of the ising model or equivalently that they depend on the spectral parameter of the model this has important consequences on the physical nature of the anisotropic n s which appear to be highly composite objects we address from a birational viewpoint the emergence of families of elliptic curves and that of calabi yau manifolds on such problems we also address the question of singularities of non holonomic functions with a discussion on the accumulation of these singular curves for the non holonomic anisotropic full susceptibility this article is part of lattice models and integrability a special issue of journal of physics a mathematical and theoretical in honour of f y wu s 80th birthday

some supercongruences occurring in truncated hypergeometric series abstract for the purposes of this paper supercongruences are congruences between terminating hypergeometric series and quotients of p adic gamma functions that are stronger than those one can expect to prove using commutative formal group laws we prove a number of such supercongruences by using classical hypergeometric transformation formulae these formulae most of which are decades or centuries old allow us to write the terminating series as the ratio of products of values at this point sums have become quotients writing these quotients as p quotients we are in a situation that is well suited for proving p adic congruences these p functions can be p adically approximated by their taylor series expansions sometimes there is cancellation of the lower order terms leading to stronger congruences using this technique we prove among other things a conjecture of kibelbek and a strengthened version of a conjecture of van hamme

singular limits in the cauchy problem for the damped extensible beam equation abstract we study the cauchy problem of the ball model for an extensible beam t 2 u t u x 4 u t x 4 u r x u 2 d x r t x u x u d x x 2 u the aim of this paper is to investigate singular limits as 0 for this problem in the authors previous paper 8 decay estimates of solutions u to the equation in the case 0 were shown with the help of the decay estimates we describe the singular limit in the sense of the following uniform in time estimate u u 0 l 0 h 2 r c

picard fuchs equations for relative periods and abel jacobi map for calabi yau hypersurfaces we study the variation of relative cohomology for a pair consisting of a smooth projective hypersurface and an algebraic subvariety in it we construct an inhomogeneous picard fuchs equation by applying a picard fuchs operator to the holomorphic top form on a calabi yau hypersurface in toric variety and deriving a general formula for the d exact form on one side of the equation we also derive a double residue formula giving a purely algebraic way to compute the inhomogeneous picard fuchs equations for the abel jacobi map which has played an important role in the recent study of d branes by morrison and walcher using the variation formalism we prove that the relative periods of toric b branes on calabi yau hypersurface satisfy the enhanced gkz hypergeometric system proposed in physics literature by alim hecht mayr and mertens and discuss the relations between several works in the recent study of open string mirror symmetry we also give the general solutions to the enhanced hypergeometric system

a criterion for the integrality of the taylor coefficients of mirror maps in several variables we give a necessary and sufficient condition for the integrality of the taylor coefficients at the origin of formal power series qi z ziexp gi z f z with z z1 zd and where f z and gi z log zi f z i 1 d are particular solutions of certain a systems of differential equations this criterion is based on the analytical properties of landau s function which is classically associated with sequences of factorial ratios and it generalizes the criterion in the case of one variable presented in e delaygue critere pour l integralite des coefficients de taylor des applications miroir j reine angew math 662 2012 205 252 one of the techniques used to prove this criterion is a generalization of a version of a theorem of dwork on formal congruences between formal series proved by krattenthaler and rivoal in c krattenthaler t rivoal multivariate p adic formal congruences and integrality of taylor coefficients of mirror maps in l di vizio t rivoal eds theories galoisiennes et arithmetiques des equations differentielles in seminaire et congres vol 27 soc math france paris 2011 pp 279 307

longtime behavior of the kirchhoff type equation with strong damping on rn abstract the paper studies the longtime behavior of the kirchhoff type equation with strong damping on r n u t t m u 2 u u t u u t g x u f x it shows that the related continuous semigroup s t possesses a global attractor which is connected and has finite fractal and hausdorff dimension

generalized hypergeometric functions and rational curves on calabi yau complete intersections in toric varieties we formulate general conjectures about the relationship between the a model connection on the cohomology of ad dimensional calabi yau complete intersectionv ofr hypersurfacesv1 vr in a toric varietyp and the system of differential operators annihilating the special generalized hypergeometric series 0 constructed from the fan using this generalized hypergeometric series we propose conjectural mirrorsv ofv and the canonicalq coordinates on the moduli spaces of calabi yau manifolds

hyperbolic parabolic singular perturbation for nondegenerate kirchhoff equations with critical weak dissipation we consider the hyperbolic parabolic singular perturbation problem for a nondegenerate quasilinear equation of kirchhoff type with weak dissipation this means that the dissipative term is multiplied by a coefficient b t which tends to 0 as t the case where b t 1 t p with p 1 but they remain true in the critical case p 1

on the number of complete intersection calabi yau manifolds the intersection numbers and the action of the pontryagin class on the integral cohomology are used to distinguish between the many cicy manifolds that have the same hodge numbers it is shown by examining manifolds embedded in fewer than six projective spaces that at least 2590 of the manifolds are distinct

hyperbolic parabolic singular perturbation for quasilinear equations of kirchhoff type abstract we consider a hyperbolic parabolic singular perturbation problem for a quasilinear equation of kirchhoff type and obtain parameter dependent time decay estimates of the difference between the solutions of a quasilinear dissipative hyperbolic equation of kirchhoff type and the corresponding quasilinear parabolic equation for this purpose we show time decay estimates for hyperbolic parabolic singular perturbation problem for linear equations with a time dependent coefficient

necessary condition on the dissipative term for the asymptotically free property of dissipative kirchhoff equations abstract it is known that the solutions of linear abstract dissipative wave equation with integrable coefficient of the dissipation term are asymptotically free namely the solutions have the same asymptotic behavior as that of the solutions of an abstract free wave equation in this paper we conclude that the integrability is not enough for asymptotic free property of kirchhoff equation by showing the following for the existence of a non trivial asymptotically free solution of abstract kirchhoff equation with initial data in some class an essentially stronger decay condition on the coefficient of dissipation term is necessary

composition operators with univalent symbol in schatten classes abstract we study composition operators induced by a sub domain of the unit disc whose boundary intersects the unit circle at 1 and which has in a neighborhood of 1 a polar equation 1 r see fig 1 we obtain an explicit characterization for the membership in schatten p classes in terms of

differential algebra on lattice green functions and calabi yau operators we revisit miscellaneous linear differential operators mostly associated with lattice green functions in arbitrary dimensions but also calabi yau operators and order 7 operators corresponding to exceptional differential galois groups we show that these irreducible operators are not only globally nilpotent but are such that they are homomorphic to their formal adjoints considering these operators or sometimes equivalent operators we show that they are also such that either their symmetric square or their exterior square have a rational solution this is a general result an irreducible linear differential operator homomorphic to its formal adjoint is necessarily such that either its symmetric square or its exterior square has a rational solution and this situation corresponds to the occurrence of a special differential galois group we thus define the notion of being special geometry for a linear differential operator if it is irreducible globally nilpotent and such that it is homomorphic to its formal adjoint since many derived from geometry n fold integrals periods occurring in physics are seen to be diagonals of rational functions we address several examples of minimal order operators annihilating diagonals of rational functions and remark that they also seem to be systematically associated with irreducible factors homomorphic to their adjoint

composition operators on weighted bergman spaces of dirichlet series abstract we study boundedness and compactness of composition operators on weighted bergman spaces of dirichlet series particularly we obtain in some specific cases upper and lower bounds of the essential norm of these operators and a criterion of compactness on classical weighted bergman spaces moreover a sufficient condition of compactness is obtained using the notion of carleson s measure

mildly degenerate kirchhoff equations with weak dissipation global existence and time decay abstract we consider the hyperbolic parabolic singular perturbation problem for a degenerate quasilinear kirchhoff equation with weak dissipation this means that the coefficient of the dissipative term tends to zero when t we prove that the hyperbolic problem has a unique global solution for suitable values of the parameters we also prove that the solution decays to zero as t with the same rate of the solution of the limit problem of parabolic type

finite dimensional attractors for the kirchhoff equation with a strong dissipation abstract the paper studies the existence of the finite dimensional global attractors and exponential attractors for the dynamical system associated with the kirchhoff type equation with a strong dissipation u t t m u 2 u u t h u t g u f x it proves that the above mentioned dynamical system possesses a global attractor which has finite fractal dimension and an exponential attractor for application the fact shows that for the concerned viscoelastic flow the permanent regime global attractor can be observed when the excitation starts from any bounded set in phase space and the dimension of the attractor that is the number of degree of freedom of the turbulent phenomenon and thus the level of complexity concerning the flow is finite

compact composition operators on bergman orlicz spaces we construct an analytic self map of the unit disk and an orlicz functionfor which the composition operator of symbol is compact on the hardy orlicz space h but not on the bergman orlicz space b for that we first prove a carleson embedding theorem and then characterize the compact ness of composition operators on bergman orlicz spaces in terms of carleson function of order 2 we show that this carleson function is equivalent to the nevanlinna counting function of order 2 mathematics subject classification primary 47b33 secondary 30d50 30d55 46e15

an attractor for the singularly perturbed kirchhoff equation with supercritical nonlinearity abstract we consider a quasilinear wave equation of kirchhoff type u t t 1 u 2 u u t f u g x where 0 is a small parameter without any growth restrictions on the nonlinearity f u we prove the existence of a finite dimensional global attractor on an appropriate bounded phase space the key step is the estimate of the difference between the solutions of a quasilinear dissipative hyperbolic equation of kirchhoff type and the corresponding quasilinear parabolic equation

on the global solvability of kirchhoff equation for non analytic initial data abstract we give sufficient conditions for the global solvability of kirchhoff equation in terms of the spectral resolutions of the initial data u 0 x u t 0 x we assume no smallness conditions and only sobolev type regularity

threshold results for the existence of global and blow up solutions to kirchhoff equations with arbitrary initial energy abstract in this paper we will apply the modified potential well method and variational method to the study of the long time behaviors of solutions to a class of parabolic equation of kirchhoff type global existence and blow up in finite time of solutions will be obtained for arbitrary initial energy to be a little more precise we will give a threshold result for the solutions to exist globally or to blow up in finite time when the initial energy is subcritical and critical respectively the decay rate of the l 2 norm is also obtained for global solutions in these cases moreover some sufficient conditions for the existence of global and blow up solutions are also derived when the initial energy is supercritical

two sphere partition functions and gromov witten invariants many n 2 2 two dimensional nonlinear sigma models with calabi yau target spaces admit ultraviolet descriptions as n 2 2 gauge theories gauged linear sigma models we conjecture that the two sphere partition function of such ultraviolet gauge theories recently computed via localization by benini et al and doroud et al yields the exact kahler potential on the quantum kahler moduli space for calabi yau threefold target spaces in particular this allows one to compute the genus zero gromov witten invariants for any such calabi yau threefold without the use of mirror symmetry more generally when the infrared superconformal fixed point is used to compactify string theory this provides a direct method to compute the spacetime kahler potential of certain moduli e g vector multiplet moduli in type iia exactly in we compute these quantities for the quintic and for rodland s pfaffian calabi yau threefold and find agreement with existing results in the literature we then apply our methods to a codimension four determinantal calabi yau threefold in p 7 recently given a nonabelian gauge theory description by the present authors for which no mirror calabi yau is currently known we derive predictions for its gromov witten invariants and verify that our predictions satisfy nontrivial geometric checks

on the integrality of the taylor coefficients of mirror maps we continue our study begun in on the integrality of the taylor coefficients of mirror maps duke math j to appear of the fine integrality properties of the taylor coefficients of the series q z z exp g z f z where f z and g z log z f z are specific solutions of certain hypergeometric differential equations with maximal unipotent monodromy at z 0 more precisely we address the question of finding the largest integer v such that the taylor coefficients of z 1q z 1 v are still integers in particular we determine the dwork kontsevich sequence un n 1 where un is the largest integer such that qn z n is a series with integer coefficients where qn z exp gn z fn z fn z m 0 nm z m and gn z m 1 hnm hm nm z m with hn denoting the n th harmonic number conditional on the conjecture that there are no prime number p and integer n such that the p adic valuation of hn 1 is strictly greater than 3

local soltutions for a nonlinear degenerate hyperbolic equation on etudie l existence et l unicite des solutions locales pour le probleme mixte associe a l equation hyperbolique degenere du type 2 u 2 m u x t 2 dx u 0 ou est un ouvert borne de r n a frontiere lisse et x est un vecteur de r n

stability of attractors for the kirchhoff wave equation with strong damping and critical nonlinearities abstract the paper investigates longtime dynamics of the kirchhoff wave equation with strong damping and critical nonlinearities u t t 1 u 2 u u t h u t g u f x with 0 1 the well posedness and the existence of global and exponential attractors are established and the stability of the attractors on the perturbation parameter is proved for the ibvp of the equation provided that both nonlinearities h s and g s are of critical growth

hyperbolic parabolic singular perturbation for mildly degenerate kirchhoff equations time decay estimates we consider degenerate kirchhoff equations with a small parameter e in front of the second order time derivative it is well known that these equations admit global solutions when e is small enough and that these solutions decay as t with the same rate of solutions of the limit problem of parabolic type in this paper we prove decay error estimates for the difference between a solution of the hyperbolic problem and the solution of the corresponding parabolic problem these estimates show in the same time that the difference tends to zero both as e 0 and as t concerning the decay rates it turns out that the difference decays faster than the two terms separately as t proofs involve a nonlinear step where we separate fourier components with respect to the lowest frequency followed by a linear step where we exploit weighted versions of classical energies

global solutions for dissipative kirchhoff strings with m r rp p 1 abstract we investigate the evolution problem u u m a 1 2 u 2 h au 0 u 0 u 0 u 0 u 1 where h is a hilbert space a is a self adjoint non negative operator on h with domain d a 0 is a parameter and m r r p with p u 0 u 1 d a i 2 d a i 1 2 satisfy a suitable smallness assumption and the non degeneracy condition m a 1 2 u 0 2 h 0 where p 2 i and i 2 i 1 moreover we prove for this solution decay with a polynomial rate as t these results apply to degenerate hyperbolic pdes with non local non linearities

contact points and schatten composition operators we study composition operators on hardy and dirichlet spaces belonging to schatten classes we give some new examples and analyse the size of contact set of the symbol of such operators

on the better behaved version of the gkz hypergeometric system we consider a version of the generalized hypergeometric system introduced by gelfand kapranov and zelevinsky gkz suited for the case when the underlying lattice is replaced by a finitely generated abelian group in contrast to the usual gkz hypergeometric system the rank of the better behaved gkz hypergeometric system is always the expected one we give largely self contained proofs of many properties of this system the discussion is intimately related to the study of the variations of hodge structures of hypersurfaces in algebraic tori

a uniqueness result for kirchhoff equations with non lipschitz nonlinear term abstract we consider the second order cauchy problem u m a 1 2 u 2 a u 0 u 0 u 0 u 0 u 1 where m 0 0 is a continuous function and a is a self adjoint nonnegative operator with dense domain on a hilbert space it is well known that this problem admits local in time solutions provided that u 0 and u 1 are regular enough depending on the continuity modulus of m it is also well known that the solution is unique when m is locally lipschitz continuous in this paper we prove that if either a u 0 u 1 0 or a 1 2 u 1 2 m a 1 2 u 0 2 a u 0 2 then the local solution is unique even if m is not lipschitz continuous

monodromy of picard fuchs differential equations for calabi yau threefolds abstract in this paper we are concerned with the monodromy of picardfuchs differential equations associated with one parameter families of calabiyau threefolds our results show that in the hypergeometric cases the matrix representations of monodromy relative to the frobenius bases can be expressed in terms of the geometric invariants of the underlying calabi yau threefolds this phenomenon is also verified numerically for other families of calabi yau threefolds in the paper furthermore we discover that under a suitable change of bases the monodromy groups are contained in certain congruence subgroups of sp 4 z of finite index and whose levels are related to the geometric invariants of the calabi yau threefolds

optimal decay error estimates for the hyperbolic parabolic singular perturbation of a degenerate nonlinear equation abstract we consider a degenerate hyperbolic equation of kirchhoff type with a small parameter e in front of the second order time derivative in a recent paper under a suitable assumption on initial data we proved decay error estimates for the difference between solutions of the hyperbolic problem and the corresponding solutions of the limit parabolic problem these estimates show in the same time that the difference tends to zero both as e 0 and as t in particular in that case the difference decays faster than the two terms separately in this paper we consider the complementary assumption on initial data and we show that now the optimal decay error estimates involve a decay rate which is slower than the decay rate of the two terms in both cases the improvement or deterioration of decay rates depends on the smallest frequency represented in the fourier components of initial data

global solutions for dissipative kirchhoff strings with non lipschitz nonlinear term we investigate the evolution problem u u m a 1 2 u 2 au 0 u 0 u0 u 0 u1 where h is a hilbert space a is a self adjoint nonnegative operator on h with domain d a 0i s a parameter and m r is a nonnegative function such that m 0 0a ndm is nonnecessarily lipschitz continuous in a neighborhood of 0 we prove that this problem has a unique global solution for positive times provided that the initial data u0 u 1 d a d a 1 2 satisfy a suitable smallness assumption and the nondegeneracy condition m a 1 2 u0 2 0 moreover we study the decay of the solution as t these results apply to degenerate hyperbolic pdes with nonlocal nonlinearities

an attractor for a nonlinear dissipative wave equation of kirchhoff type abstract in this paper we prove the existence and some absorbing properties of an attractor in a local sense for the initial boundary value problem of a quasilinear wave equation of kirchhoff type with a standard dissipation u t

mirror manifolds in higher dimension we describe mirror manifolds in dimensions different from the familiar case of complex threefolds we isolate certain simplifying features present only in dimension three and supply alternative methods that do not rely on these special characteristics and hence can be generalized to other dimensions although the moduli spaces for calabi yaud folds are not special kahler manifolds whend 3 they still have a restricted geometry and we indicate the new geometrical structures which arise we formulate and apply procedures which allow for the construction of mirror maps and the calculation of order by order instanton corrections to yukawa couplings mathematically these corrections are expected to correspond to calculating chern classes of various parameter spaces hilbert schemes for rational curves on calabi yau manifolds our mirror aided calculations agree with those chern class calculations in the limited number of cases for which the latter can be carried out with current mathematical tools finally we make explicit some striking relations between instanton corrections for various yukawa couplings derived from the associativity of the operator product algebra

mirror symmetry and rational curves on quintic threefolds a guide for mathematicians we give a mathematical account of a recent string theory calcula tion which predicts the number of rational curves on the generic quintic three fold our account involves the interpretation of yukawa couplings in terms of variations of hodge structure a new q expansion principle for functions on the moduli space of calabi yau manifolds and the mirror symmetry phe nomenon recently observed by string theorists department of mathematics duke university durham north carolina 27706 e mail address drm math duke edu this content downloaded from 157 55 39 224 on wed 14 dec 2016 04 59 36 utc all use subject to http about jstor org terms

optimal attractors of the kirchhoff wave model with structural nonlinear damping abstract the paper investigates the well posedness and longtime dynamics of the kirchhoff wave model with structural nonlinear damping u t t u 2 u u 2 u t f u g x with 1 2 1 we find a new critical exponent p n 2 n 2 p n 2 n 2 n 3 and show that when the growth exponent p of the nonlinearity f u is up to the range 1 p p i the ibvp of the equation is well posed and its solution is of additionally global regularity when t 0 ii for each 1 2 1 the related solution semigroup has in natural energy space h an optimal global attractor a whose compactness and attractiveness are in the regularized space h 1 where a lies and an optimal exponential attractor e whose compactness boundedness of the fractional dimension and the exponential attractiveness are in h 1 where e lies respectively iii the family of global attractors a 1 2 1 is upper semi continuous at each point 0 1 2 1 the paper breaks though the longstanding existed growth restriction 1 p p for p had been considered a uniqueness index deepens and extends the results in literature 6 25 27

the a hypergeometric system associated with a monomial curve introduction inthis paper we make a detailed analysis of the hypergeometric system or gkz system associated with a monomial curve and integral hence resonant exponents we describe all rational solutions and show in theorem 1 10 that they are in fact laurent polynomials we also show that for any exponent there are at most two linearly independent laurent solutions and that the upper bound is reached if and only if the curve is not arithmetically cohen macaulay we then construct for all integral parameters a basis of local solutions in terms of the roots of the generic univariate polynomial 0 5 associated with we also determine in theorem 3 7 the holonomic rank r for all z 2 and show that d r d 1 where d is the degree of the curve moreover the value d 1 is attained only for those exponents for which there are two linearly independent rational solutions and therefore r d for all if and only if the curve is arithmetically cohen macaulay inorder to place these results intheir appropriate con text we recall the defin ition

mirror duality and string theoretic hodge numbers we prove in full generality the mirror duality conjecture for string theoretic hodge numbers of calabi yau complete intersections in gorenstein toric fano varieties the proof is based on properties of intersection cohomology

long time existence and singular perturbation results for quasilinear hyperbolic equations with small parameter and dissipation term iii on considere des solutions lisses du probleme aux valeurs initiales hyperbolique quasilineaire eu tt u t ij 1 n a ij i j u 0 u 0 u 0 u t 0 u 1 ou e 0 0 u u x t r x r n on etudie l existence de solutions globales correspondant a des donnees arbitrairement grandes initiales u 0 et u 1 et a leur convergence quand e 0 vers la solution du probleme parabolique correspondant

decay estimates for evolution equations with classical and fractional time derivatives abstract using energy methods we prove some power law and exponential decay estimates for classical and nonlocal evolutionary equations the results obtained are framed into a general setting which comprise among the others equations involving both standard and caputo time derivative complex valued magnetic operators fractional porous media equations and nonlocal kirchhoff operators both local and fractional space diffusion are taken into account possibly in a nonlinear setting the different quantitative behaviours which distinguish polynomial decays from exponential ones depend heavily on the structure of the time derivative involved in the equation

composition operators acting on weighted dirichlet spaces abstract we study composition operators acting on weighted dirichlet spaces we obtain estimates for the essential norm describe the membership in schatten von neumann ideals and characterize the composition operators with closed range

finding the exact decay rate of all solutions to some second order evolution equations with dissipation we consider an abstract second order evolution equation with damping the elastic term is represented by a self adjoint nonnegative operator a with discrete spectrum and the nonlinear term has order greater than one at the origin we investigate the asymptotic behavior of solutions we prove the coexistence of slow solutions and fast solutions slow solutions live close to the kernel of a and decay as negative powers of t as solutions of the first order equation obtained by neglecting the operator a and the second order time derivatives in the original equation fast solutions live close to the range of a and decay exponentially as solutions of the linear homogeneous equation obtained by neglecting the nonlinear terms in the original equation the abstract results apply to semilinear dissipative hyperbolic equations

ising n fold integrals as diagonals of rational functions and integrality of series expansions we show that the n fold integrals n of the magnetic susceptibility of the ising model as well as various other n fold integrals of the ising class or n fold integrals from enumerative combinatorics like lattice green functions correspond to a distinguished class of functions generalizing algebraic functions they are actually diagonals of rational functions as a consequence the power series expansions of the analytic at x 0 solutions of these linear differential equations derived from geometry are globally bounded which means that after just one rescaling of the expansion variable they can be cast into series expansions with integer coefficients we also give several results showing that the unique analytical solution of calabi yau odes and more generally picard fuchs linear odes with solutions of maximal weights are always diagonals of rational functions besides in a more enumerative combinatorics context generating functions whose coefficients are expressed in terms of nested sums of products of binomial terms can also be shown to be diagonals of rational functions we finally address the question of the relations between the notion of integrality series with integer coefficients or more generally globally bounded series and the modularity of odes

mirror symmetry and toric degenerations of partial flag manifolds in this paper we propose and discuss a mirror construction for complete intersections in partial flag manifolds f n1 nl n this construction includes our previous mirror construction for complete intersection in grassmannians and the mirror construction of givental for complete flag manifolds the key idea of our construction is a degeneration of f n1 nl n to a certain gorenstein toric fano variety p n1 nl n which has been investigated by gonciulea and lakshmibai we describe a natural small crepant desingularization of p n1 nl n and prove a generalized version of a conjecture of gonciulea and lakshmibai on the singular locus of p n1 nl n mathematisches institut eberhard karls universitat tubingen d 72076 tubingen germany email address batyrev bastau mathematik uni tuebingen de department of mathematics oklahoma state university stillwater ok 74078 usa email address ciocan math okstate edu department of mathematics university of california davis davis ca 95616 usa email address bumsig math ucdavis edu fb 17 mathematik johannes gutenberg universitat mainz d 55099 mainz germany email address straten mathematik uni mainz de

asymptotic behavior of solutions of quasilinear hyperbolic equations with linear damping we consider the asymptotic behavior of the solution of quasilinear hyperbolic equation with linear dampingvtt a vx x vt 0 x t r 0 subsequent to k nishihara j differential equations131 1996 171 188 in that article the system with dampingvt ux 0 ut p v x u p v 0 was treated and the convergence rates to the diffusion wave by l hsiao and t p liu comm math phys 143 1992 599 605 and l hsiao and t p liu chinese ann math ser b14 1993 465 480 were improved for initial data v0 u0 x satisfying v0 u0 v0 0 and v0 x v0 dx 0 this system is reduced to the quasilinear hyperbolic equation with linear dampingvtt p v0 vx p v0 x vt 0 and the decay rates v t l o t 1 2 vx t l o t 1 vt t l o t 3 2 were obtained provided that v vt t 0 h3 h2 is small and inl1 l1 from these decay rates vis expected to behave as a solution to the parabolic equation the aims of this paper are to seek the asymptotic profile x t satisfying t l o t 1 2 x t l o t 1 t t l o t 3 2 and to show that the solution of satisfies v t l o t 1 v x t l o t 3 2 v t t l o t 2 the proof is based on the elementary energy method and the green function method for the parabolic equation

quantum hyperplane section theorem for homogeneous spaces author s kim bumsig abstract we formulated a mirror free approach to the mirror conjecture namely quantum hyperplane section conjecture and proved it in the case of nonnegative complete intersections in homogeneous manifolds for the proof we followed the scheme of givental s proof of a mirror theorem for toric complete intersections

some nonlinear degenerate wave equations on etudie la resolubilite locale du probleme aux valeurs limites et initiales suivant 2 u t 2 m gradu x t 2 dx u 0 dans 0 u 0 sur w 0 u 0 0 u t 0 dans ou m r est une fonction c 1 non negative sur 0 un domaine borne de r n a frontiere lisse w

nevanlinna counting function and carleson function of analytic maps we show that the maximal nevanlinna counting function and the carleson function of analytic self maps of the unit disk are equivalent up to constants

a spectral radius type formula for approximation numbers of composition operators abstract for approximation numbers a n c of composition operators c on weighted analytic hilbert spaces including the hardy bergman and dirichlet cases with symbol of uniform norm lim n a n c 1 n e 1 cap d where cap d is the green capacity of d in d this formula holds also for h p with 1 p

compact composition operators on the dirichlet space and capacity of sets of contact points we prove several results about composition operators on the dirichlet space d for every compact set k d of logarithmic capacity capk 0 there exists a schur function both in the disk algebra a d and in d such that the composition operator c is in all schatten classes sp d p 0 and for which k eit eit 1 eit eit 1 for every bounded composition operator c on d and every d the logarithmic capacity of eit eit is 0 every compact composition operator c on d is compact on b 2 and on h 2 in particular c is in every schatten class sp p 0 both on h2 and on b2 there exists a schur function such that c is compact on h 2 but which is not even bounded on d there exists a schur function such that c is compact on d but in no schatten class sp d

mirror symmetry mirror map and applications to calabi yau hypersurfaces mirror symmetry picard fuchs equations and instanton corrected yukawa couplings are discussed within the framework of toric geometry it allows to establish mirror symmetry of calabi yau spaces for which the mirror manifold had been unavailable in previous constructions mirror maps and yukawa couplings are explicitly given for several examples with two and three moduli

binomial d modules fil dickenstein alicia marcela universidad de buenos aires facultad de ciencias exactas y naturales departamento de matematica argentina consejo nacional de investigaciones cientificas y tecnicas oficina de coordinacion administrativa ciudad universitaria instituto de investigaciones matematicas luis a santalo argentina

non commutative periods and mirror symmetry in higher dimensions we study an analog for higher dimensional calabi yau manifolds of the standard predictions of mirror symmetry we introduce periods associated with non commutative deformations of calabi yau manifolds these periods define a map on the moduli space of such deformations which is a local isomorphism using these non commutative periods we introduce invariants of variations of semi infinite generalized hodge structures living over the moduli space it is shown that the generating function of such invariants satisfies the system of wdvv equations exactly as in the case of gromov witten invariants we prove that the total collection of rational gromov witten invariants of complete intersection calabi yau manifold can be identified with the collection of invariants of variations of generalized semi infinite hodge structures attached to the mirror variety the basic technical tool utilized is the deformation theory

global solvability for kirchhoff equation in special classes of non analytic functions abstract in this paper we derive the following two properties the first one is a precise representation of wkb solution to the cauchy problem of a linear wave equation with a variable coefficient with respect to time and the second one is the global solvability for kirchhoff equation in some special classes of nonreal analytic functions which is proved by grace of the first property

the ising model from elliptic curves to modular forms and calabi yau equations we show that almost all the linear differential operators factors obtained in the analysis of the n particle contributions s of the susceptibility of the ising model for n 6 are linear differential operators associated with elliptic curves beyond the simplest differential operators factors which are homomorphic to symmetric powers of the second order operator associated with the complete elliptic integral e the second and third order differential operators z2 f2 f3 can actually be interpreted as modular forms of the elliptic curve of the ising model a last order 4 globally nilpotent linear differential operator is not reducible to this elliptic curve modular form scheme this operator is shown to actually correspond to a natural generalization of this elliptic curve modular form scheme with the emergence of a calabi yau equation corresponding to a selected 4f3 hypergeometric function this hypergeometric function can also be seen as a hadamard product of the complete elliptic integral k with a remarkably simple algebraic pull back square root extension the corresponding calabi yau fourth order differential operator having a symplectic differential galois group the mirror maps and higher order schwarzian odes associated with this calabi yau ode present all the nice physical and mathematical ingredients we had with elliptic curves and modular forms in particular an exact isogenies representation of the generators of the renormalization group extending the modular group to a symmetry group