stable algorithm for the stress field around a multiply branched crack we present an algorithm for the computation of the stress field around a branched crack the algorithm is based on an integral equation with good numerical properties our equation is obtained through a left regularization of an integral equation of fredholm s first kind complex valued functions involving repeated products of square roots appear in the regularization a new and effective scheme for correct evaluation of these functions is described for validation mode i and ii stress intensity factors are computed for simple branched geometries the relative errors in the stress intensity factors are typically as low as 10 12 a large scale example is also presented where a crack with 176 branching points is studied

some remarks on the brylinski radon and the fourier transforms abstract we will study a relationship between the brylinski radon and the fourier transforms of a monodromic perverse sheaf in fact we will show that they are equivalent modulo a constant perverse sheaf as an application we will discuss a relation between the sklyanin transform and the brylinski radon transform

langlands program trace formulas and their geometrization the langlands program relates galois representations and auto morphic representations of reductive algebraic groups the trace formula is a powerful tool in the study of this connection and the langlands functorial ity conjecture after giving an introduction to the langlands program and its geometric version which applies to curves over finite fields and over the complex field i give a survey of my recent joint work with robert langlands and ngo bao chau fln fn on a new approach to proving the functoriality conjecture using the trace formulas and on the geometrization of the trace formulas in particular i discuss the connection of the latter to the categorifi cation of the langlands correspondence

dynamic fracture of a rectangular limited permeable crack in magneto electro elastic media under a time harmonic elastic p wave abstract the dynamic behavior of a rectangular limited permeable crack deeply embedded in transversely isotropic magneto electro elastic media was investigated under an incident harmonic stress wave using the generalized almansi s theorem and the schmidt method with the help of the fourier transform this problem was formulated into three pairs of dual integral equations with the jumps in displacement across the crack surfaces as the unknown variables by directly expanding the jumps in displacement across the crack surfaces into a series of jacobi polynomials the solution to the dual integral equations was derived finally the dynamic response of a rectangular crack under a harmonic wave was analyzed and the effects of the electric permittivity and the magnetic permeability of air inside the crack the geometric shape of the rectangular crack and the characteristics of the harmonic wave on the stress the electric displacement and the magnetic flux intensity factors in magneto electro elastic media were concluded

on a vanishing conjecture appearing in the geometric langlands correspondence 0 1 this paper should be regarded as a sequel to 7 there it was shown that the geometric langlands conjecture for gln follows from a certain vanishing conjecture the goal of the present paper is to prove this vanishing conjecture let x be a smooth projective curve over a ground field k let e be an m dimensional local system on x and let bunm be the moduli stack of rank m vector bundles on x the geometric langlands conjecture says that to e we can associate a perverse sheaf fe on bunm which is a hecke eigensheaf with respect to e the vanishing conjecture of 7 says that for all integers n m a certain functor ave depending on e and a parameter d z which maps the category d bunn to itself vanishes identically when d is large enough the fact that the vanishing conjecture implies the geometric langlands conjecture may be regarded as a geometric version of the converse theorem moreover as will be explained in the sequel the vanishing of the functor ave is analogous to the condition that the rankin selberg convolution of e viewed as an m dimensional galois representation and an automorphic form on gln with n m is well behaved both the geometric langlands conjecture and the vanishing conjecture can be formulated in any of the sheaf theoretic situations e g q adic sheaves when char k d modules when char k 0 and sheaves with coefficients in a finite field f again when char k when the ground field is the finite field fq and we are working with adic coefficients it was shown in 7 that the vanishing conjecture can be deduced from lafforgue s theorem that establishes the full langlands correspondence for global fields of positive characteristic cf 9

a survey of the theory of source coding with a fidelity criterion the purpose of this paper is threefold 1 to acquaint the reader with the types of problems that have been considered in the area of source coding with a fidelity criterion 2 to survey results that have been obtained on these problems and 3 to outline future research trends in the area

modular elliptic curves and fermat s last theorem when andrew john wiles was 10 years old he read eric temple bell s the last problem and was so impressed by it that he decided that he would be the first person to prove fermat s last theorem this theorem states that there are no nonzero integers a b c n with n 2 such that an bn cn the object of this paper is to prove that all semistable elliptic curves over the set of rational numbers are modular fermat s last theorem follows as a corollary by virtue of previous work by frey serre and ribet

mod l representations of arithmetic fundamental groups i an analog of serre s conjecture for function fields there is a well known conjecture of serre that any continuous irreducible representation gq gl2 f that is odd arises from a newform here gq is the absolute galois group of q and f is an algebraic closure of the finite field f of elements we formulate such a conjecture for n dimensional mod representations of 1 x for any positive integer n and where x is a geometrically irreducible smooth curve over a finite field k of characteristic p p and we prove this conjecture in a large number of cases in fact a proof of all cases of the conjecture for 2 follows from a result announced conditionally by gaitsgory in g the methods are different

empirical processes typical sequences and coordinated actions in standard borel spaces this paper proposes a new notion of typical sequences on a wide class of abstract alphabets so called standard borel spaces which is based on approximations of memoryless sources by empirical distributions uniformly over a class of measurable test functions in the finite alphabet case we can take all uniformly bounded functions and recover the usual notion of strong typicality or typicality under the total variation distance for a general alphabet however this function class turns out to be too large and must be restricted with this in mind we define typicality with respect to any glivenko cantelli function class i e a function class that admits a uniform law of large numbers and demonstrate its power by giving simple derivations of the fundamental limits on the achievable rates in several source coding scenarios in which the relevant operational criteria pertain to reproducing empirical averages of a general alphabet stationary memoryless source with respect to a suitable function class

automorphic forms for elliptic function fields let f be the function field of an elliptic curve x over mathbb f q in this paper we calculate explicit formulas for unramified hecke operators acting on automorphic forms over f we determine these formulas in the language of the graph of an hecke operator for which we use its interpretation in terms of mathbb p 1 bundles on x this allows a purely geometric approach which involves amongst others a classification of the mathbb p 1 bundles on x we apply the computed formulas to calculate the dimension of the space of unramified cusp forms and the support of a cusp form we show that a cuspidal hecke eigenform does not vanish in the trivial mathbb p 1 bundle further we determine the space of unramified f toroidal automorphic forms where f is the quadratic constant field extension of f it does not contain non trivial cusp forms an investigation of zeros of certain hecke l series leads to the conclusion that the space of unramified toroidal automorphic forms is spanned by the eisenstein series e s where s 1 2 is a zero of the zeta function of x with one possible exception in the case that q is even and the class number h equals q 1

integral equation methods for elliptic problems with boundary conditions of mixed type laplace s equation with mixed boundary conditions that is dirichlet conditions on parts of the boundary and neumann conditions on the remaining contiguous parts is solved on an interior planar domain using an integral equation method rapid execution and high accuracy is obtained by combining equations which are of fredholm s second kind with compact operators on almost the entire boundary with a recursive compressed inverse preconditioning technique then an elastic problem with mixed boundary conditions is formulated and solved in an analogous manner and with similar results this opens up for the rapid and accurate solution of several elliptic problems of mixed type

leray s quantization of projective duality 2 review on correspondences for sheaves and d modules 4 2 1 notations 4 2 1 1 geometry 4 2 1 2 sheaves 4 2 1 3 d modules 5 2 1 4 e modules 6 2 2 correspondences for sheaves and d modules 6

on the geometric langlands conjecture let x be a smooth projective curve and g be a reductive group over a finite field fq the field kx of rational functions on x is what number theorists call a global field and we can do the theory of automorphic functions over it namely we consider the quotient g kx g a here a is the ring of adeles corresponding to kx and study the space of functions on it as a representation of the adele group g a it was essentially an observation of a weil that the quotient g kx g a is closely related to the set of isomorphism classes of principal g bundles on our curve x however g bundles on x possess a richer structure one can view the above set as a set of fq points of an algebraic variety or rather a stack denoted bung moreover instead of studying the vector space of functions on the discrete set of points of bung we will consider the category of adic sheaves on it basically what people call the geometric langlands program is an attempt to understand a spectral decomposition of the above category under the action of the so called hecke functors the latter will be defined in the talk the answer predicted by the geometric langlands conjecture links this spectral decomposition to the moduli stack of local systems on x with respect to the langlands dual group

a universal scheme for wyner ziv coding of discrete sources we consider the wyner ziv wz problem of lossy compression where the decompressor observes a noisy version of the source whose statistics are unknown a new family of wz coding algorithms is proposed and their universal optimality is proven compression consists of sliding window processing followed by lempel ziv lz compression while the decompressor is based on a modification of the discrete universal denoiser dude algorithm to take advantage of side information the new algorithms not only universally attain the fundamental limits but also suggest a paradigm for practical wz coding the effectiveness of our approach is illustrated with experiments on binary images and english text using a low complexity algorithm motivated by our class of universally optimal wz codes

stress intensity factors of a rectangular crack meeting a bimaterial interface using the hypersingular integral equation method based on body force method a planar crack meeting the interface in a three dimensional dissimilar materials is analyzed the singularity of the singular stress field around the crack front terminating at the interface is analyzed by the main part analytical method of hypersingular integral equations then the numerical method of the hypersingular integral equation for a rectangular crack subjected to normal load is proposed by the body force method which the crack opening dislocation is approximated by the product of basic density functions and polynomials numerical solutions of the stress intensity factors of some examples are given

controlling parameter of the stress intensity factors for a planar interfacial crack in three dimensional bimaterials numerical solutions of singular integral equations are discussed in the analysis of a planar rectangular interfacial crack in three dimensional bimaterials subjected to tension the problem is formulated as a system of singular integral equations on the basis of the body force method in the numerical analysis unknown body force densities are approximated by the products of the fundamental density functions and power series where the fundamental density functions are chosen to express singular behavior along the crack front of the interface crack exactly the calculation shows that the present method gives smooth variations of stress intensity factors along the crack front for various aspect ratios the present method gives rapidly converging numerical results and highly satisfied boundary conditions throughout the crack boundary the stress intensity factors are given with varying the material combination and aspect ratio of the crack it is found that the stress intensity factors ki and kii are determined by the bimaterial constant e alone independent of elastic modulus ratio and poisson s ratio

a hypergeometric modules and gauss manin systems abstract let a be a d n integer matrix gel fand et al proved that most a hypergeometric systems have an interpretation as a fourier laplace transform of a direct image the set of parameters for which this happens was later identified by schulze and walther as the set of not strongly resonant parameters of a a similar statement relating a hypergeometric systems to exceptional direct images was proved by reichelt in this article we consider a hybrid approach involving neighborhoods u of the torus of a and consider compositions of direct and exceptional direct images our main results characterize for which parameters the associated a hypergeometric system is the inverse fourier laplace transform of such a mixed gauss manin system in order to describe which u work for such a parameter we introduce the notions of fiber support and cofiber support of a d module if the semigroup ring c n a is normal we show that every a hypergeometric system is mixed gauss manin we also give an explicit description of the neighborhoods u which work for each parameter in terms of primitive integral support functions

entropy points and applications first notions of entropy point and uniform entropy point are introduced using bowen s definition of topological entropy some basic properties of the notions are discussed as an application it is shown that for any topological dynamical system there is a countable closed subset whose bowen entropy is equal to the entropy of the original system then notions of c entropy point are introduced along the line of entropy tuple both in topological and measure theoretical settings it is shown that each c entropy point is an entropy point and the set of c entropy points is the union of sets of c entropy points for all invariant measures

a method for proving multiterminal source coding theorems the two step method used by wyner and ziv to prove the wyner ziv theorem is extended to prove sliding block source coding theorems for coding a general finite alphabet ergodic multiterminal source with respect to a single letter fidelity criterion the first step replaces every stochastic encoding in the network by a deterministic sliding block encoding the second step involves using the slepian wolf theorem to adjust the sliding block encodings so that they will have the desired rate while introducing little additional distortion the method is applied to give quick proofs of sliding block versions of theorems of berger kaspi and tung since the method obtains sliding block coders directly without first obtaining block coders the block coding results can be obtained as an easy corollary the direct methods for proving results on block coding are more difficult and do not imply the corresponding results on sliding block coding indeed in the multiterminal case it is in general not known how to construct good sliding block coders from good block coders

integral transforms with exponential kernels and laplace transform let x z y be a correspondence of complex manifolds f g we study integral transforms associated to kernels exp o with p meromor phic on z acting on formal or moderate cohomologies our main application is the laplace transform in this case x is the projective compactification of the vector space v c y is its dual space z x x y and o z w z w we obtain the isomorphisms f 0 ov f n f ov thom f v thom far n ov this content downloaded from 207 46 13 59 on mon 13 jun 2016 05 54 06 utc all use subject to http about jstor org terms 972 masaki kashiwara and pierre schapira where f is a conic and rll constructible sheaf on v and fa is its fourier sato transform some applications are discussed rims kyoto university kyoto 606 01 japan institut de mathematiques universit9 paris vi case 82 4 pl jussieu 75252 paris france e mail address schapirafmath jussieu fr this content downloaded from 207 46 13 59 on mon 13 jun 2016 05 54 06 utc all use subject to http about jstor org terms

recent advances in the langlands program these are the notes for the lecture given by the author at the mathematical current events special session of the ams meeting in baltimore on january 17 2003 topics reviewed include the langlands correspondence for gl n in the function field case and its proof by v drinfeld and l lafforgue the geometric langlands correspondence for gl n and its proof by d gaitsgory k vilonen and the author and the work of a beilinson and v drinfeld on the quantization of the hitchin system and the langlands correspondence for an arbitrary semisimple algebraic group

the entropy theory of symbolic extensions fix a topological system x t with its space k x t of t invariant borel probabilities if y s is a symbolic system subshift and y s x t is a topological extension factor map then the function h ext on k x t which assigns to each the maximal entropy of a measure on y mapping to is called the extension entropy function of the infimum of such functions over all symbolic extensions is called the symbolic extension entropy function and is denoted by h sex in this paper we completely characterize these functions in terms of functional analytic properties of an entropy structure on x t the entropy structure is a sequence of entropy functions h k defined with respect to a refining sequence of partitions of x or of x z for some auxiliary system z r with simple dynamics whose boundaries have measure zero for all the invariant borel probabilities we develop the functional analysis and computational techniques to produce many dynamical examples for instance we resolve in the negative the question of whether the infimum of the topological entropies of symbolic extensions of x t must always be attained and we show that the maximum value of h sex need not be achieved at an ergodic measure we exhibit several characterizations of the asymptotically h expansive systems of misiurewicz which emerge as a fundamental natural class in the context of the entropy structure the results of this paper are required for the downarowicz newhouse results dn on smooth dynamical systems

application of hypersingular integral equation method to three dimensional crack in electromagnetothermoelastic multiphase composites a three dimensional crack problem in electromagnetothermoelastic multiphase composites emte mcs under extended loads is investigated in this paper using green s functions the extended general displacement solutions are obtained by the boundary element method this crack problem is reduced to solving a set of hypersingular integral equations coupled with boundary integral equations in which the unknown functions are the extended displacement discontinuities then the behavior of the extended displacement discontinuities around the crack front terminating at the interface is analyzed by the main part analysis method of hypersingular integral equations analytical solutions for the extended singular stresses the extended stress intensity factors sifs and the extended energy release rate near the crack front in emte mcs are provided also a numerical method of the hypersingular integral equations for a rectangular crack subjected to extended loads is put forward with the extended displacement discontinuities approximated by the product of basic density functions and polynomials in addition distributions of extended sifs varying with the shape of the crack are presented the results show that the present method accurately yields smooth variations of extended sifs along the crack front

radon and fourier transforms for d modules the fourier and radon hyperplane transforms are closely related and one such relation was established by brylinski 4 in the framework of holonomic d modules the integral kernel of the radon hyperplane transform is associated with the hypersurface scp p of pairs x y where x is a point in the n dimensional complex projective space p belonging to the hyperplane yap as it turns out a useful variant is obtained by considering the integral transform associated with the open complement u of s in p p in the first part of this paper we generalize brylinski s result in order to encompass this variant of the radon transform and also to treat arbitrary quasi coherent d modules as well as twisted abelian sheaves our proof is entirely geometrical and consists in a reduction to the onedimensional case by the use of homogeneous coordinates the second part of this paper applies the above result to the quantization of the radon transform in the sense of 7 first we deal with line bundles more precisely let p 1 4 p v be the projective space of lines in the vector spacev denote by 3 d r

analysis of 3d fluid driven crack propagation problem in co seismic slip under p and s waves by hybrid hypersingular integral method this work reports a new and accurate way of theoretical and numerical description of the extended 3d fluid electromagnetic and flow driven crack progression in co seismic slip under p and s waves first based on the viscous fluid flow reciprocal work theorem the hybrid hypersingular integral equation hie method proposed by the author was defined by combined with the coupled extended wave time domain hie and the extended diffused interface phase field method the general extended 3d fluid flow velocity wave solutions are obtained by the extended wave time domains green s function method the 3d extended dynamic fluid driven crack modeling under fully coupled electromagnetothermoelastic p and s wave and flow field was established then the problem is reduced to solving a set of extended hybrid hies coupled with nonlinear boundary domain integral equations in which the unknown functions are the general extended flow velocity discontinuity waves the behavior of the general extended singular stress indices around the crack front terminating is analyzed by hybrid time domain main part analysis the general extended singular pore stress waves spsws and the extended dynamic stress intensity factors dsifs on the fluid driven crack surface are obtained from closed form solutions in addition a numerical method for the problem is proposed in which the extended velocity discontinuity waves are approximated by the product of time domain density functions and polynomials the extended dsifs and general extended spsws are calculated and the results are presented toward demonstrating the applicability of the proposed method

double affine hecke algebras and 2 dimensional local fields the concept of an n dimensional local field was introduced by a n parshin pal with the aim of generalizing the classical adelic formalism to absolutely ndimensional schemes by definition a 0 dimensional local field is just a finite field and an n dimensional local field n 0 is a complete discrete valued field whose residue field is n 1 dimensional local thus for n 1 we get locally compact fields such as qp fq t and for n 2 we get fields such as qp t fq ti t2 etc in representation theory harmonic analysis on reductive groups over 0and 1dimensional local fields leads in particular to consideration of the finite and affine hecke algebras hq hq associated to aniy finite root system r and anly q e c these algebras can be defined in several ways one being by generators and relations another as the convolution algebra with respect to the haar measure of functions on the group bi invariant with respect to an appropriate subgroup i e as the algebra of double cosets harmonic analysis on groups over 2 dimensional local fields has not been developed the main difficulty being the infinite dimensionality absense of local compactness of such fields however the double affine hecke

rate distortion for correlated sources with partially separated encoders we resolve some open problems concerning the encoding of a pair of correlated sources with respect to a fidelity criterion two encoders are used one encoder observes only the output of the first source while the other is supplied both with the second source output and with partial information about the first source at some predetermined rate a general coding theorem is proved which established that cal s ast a certain region defined in terms of single letter information theoretic quantities is an inner bound to the region of all attainable vectors of rates and distortions for certain cases of interest the converse is proved too thereby establishing the rate distortion region for these cases

a physical origin for singular support conditions in geometric langlands theory we explain how the nilpotent singular support condition introduced into the geometric langlands conjecture by arinkin and gaitsgory arises naturally from the point of view of 4 dimensional mathcal n 4 supersymmetric gauge theory we define what it means in topological quantum field theory to restrict a category of boundary conditions to the full subcategory of objects compatible with a fixed choice of vacuum both in functorial field theory and in the language of factorization algebras for b twisted mathcal n 4 gauge theory with gauge group g the moduli space of vacua is equivalent to mathfrak h w and the nilpotent singular support condition arises by restricting to the vacuum 0 in mathfrak h w we then investigate the categories obtained by restricting to points in larger strata and conjecture that these categories are equivalent to the geometric langlands categories with gauge symmetry broken to a levi subgroup and furthermore that by assembling such for the groups mathrm gl n with n ge 1 one finds a hidden factorization algebra structure for the geometric langlands theory