a pde for the multi time joint probability of the airy process abstract this paper gives a pde for multi time joint probability of the airy process which generalizes adler and van moerbeke s result on the 2 time case as an intermediate step the pde for the multi time joint probability of the dyson brownian motion is also given

matrices coupled in a chain ii spacing functions for the eigenvalues of p complex hermitian matrices coupled in a chain we give a method of calculating the spacing functions this is a generalization of the one matrix case which has been known for a long time

recursion relations for unitary integrals combinatorics and the toeplitz lattice in a discussion in spring 2001 alexei borodin showed us recursion relations for the toeplitz determinants going with the symbols e t z z 1 and 1 xiz alpha 1 xiz 1 beta borodin obtained these relations using riemann hilbert methods see the recent work of borodin b and baik baik the nature of borodin s recursion relations pointed towards the toeplitz lattice and its virasoro algebra introduced by us in 3 in this paper we take the toeplitz lattice and virasoro algebra approach for a fairly large class of symbols leading to a systematic way of generating recursion relations the latter are very naturally expressed in terms of the l matrices appearing in the toeplitz lattice equations as a surprise we find compared to borodin s a different set of relations except for the 3 step relations associated with the symbol e t z z 1 the painleve analysis of the toeplitz lattice enables us to show the singularity confinement for these recursion relations

asymptotics and integrable structures for biorthogonal polynomials associated to a random two matrix model abstract we give a rigorous construction of complete families of biorthonormal polynomials associated to a planar measure of the form e n v x w y 2 xy d x d y for polynomial v and w we are further able to show that the zeroes of these polynomials are all real and distinct a complex analytical construction of the biorthonormal polynomials is given in terms of a non local riemann hilbert problem which given our prior result provides an avenue for developing uniform asymptotics for the statistical distributions of these zeroes as n becomes large the biorthonormal polynomials considered here play a fundamental role in the analysis of certain random multi matrix models we show that the evolutions of the recursion matrices for the polynomials induced by linear deformations of v and w coincide with a semi infinite generalization of the completely integrable full kostant toda lattice this connection could be relevant for understanding aspects of scaling limits for the multi matrix model

various equations for gap probabilities of coupled gaussian matrices versions of tracy widom tw and adler shiota van moerbeke asvm approaches are applied to derive various partial differential equations pde satisfied by joint eigenvalue spacing probabilities of two coupled gaussian hermitian matrices coupled gue all the lowest third order pde satisfied by the probabilities for the largest eigenvalues of two coupled gue are found and the results of both approaches are compared the tw approach allows us to derive all pde at once while in the asvm one starting with different bilinear identities leads to different subsets of the full set of equations an interesting result is that the joint probability for the largest eigenvalues of coupled gaussian matrices satisfies a number of different pde and the previously known adler van moerbeke equation avm 1999 ann math 149 921 76 is only one of them some of the new equations appear like coupled painlev iv and have the usual painlev iv equation as the one matrix limit i e when the spectral endpoint of one of the matrices goes to infinity this is in contrast to the avm equation which becomes trivial in this limit moreover the new pde which stem from the matrix kernel approach of tracy and widom 2004 commun math phys 252 7 41 do not contain derivatives w r t the strength of coupling unlike the avm equation in other words they contain fewer independent variables and in this sense are simpler

end point behaviour of solutions to hypersingular integral equations we consider one dimensional hypersingular integral equations over finite intervals the integral must be interpreted as a finite part integral such equations arise naturally in various physical situations involving thin rigid bodies or cracks examples are given a method is developed for determining the behaviour of the solution to a hypersingular integral equation near the end points of the interval of integration the method uses the mellin transform several examples are worked out in detail

non singular somigliana stress identities in elasticity the paper presents two fully equivalent and regular forms of the hypersingular somigliana stress identity in elasticity that are appropriate for problems in which the displacement field and resulting stresses is c1 continuous each form is found as the result of a single decomposition process on the kernels of the somigliana stress identity in three dimensions the results show that the use of a simple stress state for regularization arises in a direct manner from the somigliana stress identity just as the use of a constant displacement state regularization arose naturally for the somigliana displacement identity the results also show that the same construction leads naturally to a finite part form of the same identity while various indirect constructions of the equivalents to these findings are published none of the earlier forms address the fundamental issue of the usual discontinuities of boundary data in the hypersingular somigliana stress identity that arise at corners and edges these new findings specifically focus on the corner problem and establish that the previous requirements for continuity on the densities in the hypersingular somigliana stress identity are replaced by a sole requirement on displacement field continuity the resulting regularized and finite part forms of the somigliana stress identity leads to a regularized form of the stress boundary integral equation stress bie the regularized stress bie is shown to properly allow piecewise discontinuity of the boundary data subject only to c1 continuity of the underlying displacement field the importance of the findings is in their application to boundary element modeling of the hypersingular problem the piecewise discontinuity derivation for corners is found to provide a rigorous and non singular basis for collocation of the discontinuous boundary data for both the regularized and finite part forms of the stress bie the boundary stress solution for both forms is found to be an average of the computed stresses at collocation points at the vertices of boundary element meshes collocation at these points is shown to be without any unbounded terms in the formulation thereby eliminating the use of non conforming elements for the hypersingular equations the analytical findings in this paper confirm the correct use of both regularized and finite part forms of the stress bie that have been the basis of boundary element analysis previously published by the first author of the current paper

numerical evaluation of two dimensional singular boundary integrals theory and fortran code in this paper an approach is presented for the numerical evaluation of weakly strongly hyper and supersingular boundary integrals which exist in the cauchy principal value sense in two dimensional problems in this approach the singularities involved in integration kernels are analytically removed by expressing the nonsingular parts of the integration kernels as polynomials of the distance r a self contained fortran code is listed and described for implementation of the proposed approach the attached code is also able to evaluate general regular integrals using gaussian quadrature which enables the code to evaluate any two dimensional boundary integral some examples are provided to verify the correctness of the presented formulations and the included code

average characteristic polynomials in the two matrix model the two matrix model is defined on pairs of hermitian matrices m1 m2 of size n n by the probability measure 1znexp tr v m1 w m2 m1m2 dm1dm2 where v and w are given potential functions and r we study averages of products and ratios of characteristic polynomials in the two matrix model where both matrices m1 and m2 may appear in a combined way in both numerator and denominator we obtain determinantal expressions for such averages the determinants are constructed from several building blocks the biorthogonal polynomials pn x and qn y associated with the two matrix model certain transformed functions pn w and qn v and finally cauchy type transforms of the four eynard mehta kernels k1 1 k1 2 k2 1 and k2 2 in this way we generalize known results for the one matrix model our results also imply a new proof of the eynard mehta theorem for correlation functions in the two matrix model and they lead to a generating function for averages of products of traces

the correspondence between tracy widom and adler shiota van moerbeke approaches in random matrix theory the gaussian case two approaches to the derivation of integrable differential equations for random matrix probabilities are compared both methods are rewritten in such a form that simple and explicit relations between all tw dependent variables and functions of asvm are found for the example of finite size gaussian matrices orthogonal function systems and toda lattice are seen as the core structure of both approaches and their relationship

uniform exponential attractors for non autonomous klein gordon schrodinger lattice systems in weighted spaces abstract in weighted spaces of infinite sequences we study the existence of uniform exponential attractors for non autonomous klein gordon schrodinger lattice dynamical systems with quasi periodic symbols where we show that the solution semigroup associated with such a system is lipschitz continuous contraction and satisfying the squeezing property in such a case following corollary 1 we get the main results namely theorem 1 theorem 2 given by the end of this work

random matrices virasoro algebras and noncommutative kp department of mathematics brandeis university waltham mass 02254 usa email adler math brandeis edu the support of a national science foundation grant dms 9503246 is gratefully acknowledged department of mathematics kyoto university kyoto 606 japan the hospitality of the university of louvain and brandeis university is gratefully acknowledged department of mathematics universite de louvain 1348 louvain la neuve belgium and brandeis university waltham mass 02254 usa e mail vanmoerbeke geom ucl ac be and math brandeis edu the support of a national science foundation grant dms 9503246 a nato a fnrs and a francqui foundation grant is gratefully acknowledged a corrected version has appeared in the duke mathematical journal 94 379 431 1998

the spectrum of coupled random matrices the study of the spectrum of coupled random matrices has received rather little attention to the best of our knowledge coupled random matrices have been studied to some extent by mehta in this work we explain how the integrable technology can be brought to bear to gain insight into the nature of the distribution of the spectrum of coupled hermitean random matrices and the equations the associated probabilities satisfy in particular the two toda lattice its algebra of symmetries and its vertex operators will play a prominent role in this interaction namely the method is to introduce time parameters in an artificial way and to dress up a certain matrix integral with a vertex integral operator for which we find virasoro like differential equations these methods lead to very simple nonlinear third order partial differential equations for the joint statistics of the spectra of two coupled gaussian random matrices comment 56 pages published version abstract added in migration

inertial manifolds for nonlinear evolutionary equations abstract in this paper we introduce the concept of an inertial manifold for nonlinear evolutionary equations in particular for ordinary and partial differential equations these manifolds which are finite dimensional invariant lipschitz manifolds seem to be an appropriate tool for the study of questions related to the long time behavior of solutions of the evolutionary equations the inertial manifolds contain the global attractor they attract exponentially all solutions and they are stable with respect to perturbations furthermore in the infinite dimensional case they allow for the reduction of the dynamics to a finite dimensional ordinary differential equation

differential systems for biorthogonal polynomials appearing in 2 matrix models and the associated riemann hilbert problem we consider biorthogonal polynomials that arise in the study of a generalization of two matrix hermitian models with two polynomial potentials v1 x v2 y of any degree with arbitrary complex coefficients finite consecutive subsequences of biorthogonal polynomials windows of lengths equal to the degrees of the potentials v1 and v2 satisfy systems of ode s with polynomial coefficients as well as pde s deformation equations with respect to the coefficients of the potentials and recursion relations connecting consecutive windows a compatible sequence of fundamental systems of solutions is constructed for these equations the stokes sectorial asymptotics of these fundamental systems are derived through saddle point integration and the riemann hilbert problem characterizing the differential equations is deduced

universality in the two matrix model with a monomial quartic and a general even polynomial potential in this paper we studied the asymptotic eigenvalue statistics of the 2 matrix model with the probability measure z 1 n rm exp left n left rm tr v m 1 w m 2 tau m 1m 2 right right rm d m 1 rm d m 2 in the case where w frac y 4 4 and v is a general even polynomial we studied the correlation kernel for the eigenvalues of the matrix m1 in the limit as n we extended the results of duits and kuijlaars in 14 to the case when the limiting eigenvalue density for m1 is supported on multiple intervals the results are achieved by constructing the parametrix to a riemann hilbert problem obtained in 14 with theta functions and then showing that this parametrix is well defined for all n by studying the theta divisor

lozenge tilings and hurwitz numbers we give a new proof of the fact that near a turning point of the frozen boundary the vertical tiles in a uniformly random lozenge tiling of a large sawtooth domain are distributed like the eigenvalues of a gue random matrix our argument uses none of the standard tools of integrable probability in their place it uses a combinatorial interpretation of the harish chandra itzykson zuber integral as a generating function for desymmetrized hurwitz numbers

orthogonal laurent polynomials on the unit circle extended cmv ordering and 2d toda type integrable hierarchies abstract we connect the theory of orthogonal laurent polynomials on the unit circle and the theory of toda like integrable systems using the gauss borel factorization of a cantero moral velazquez moment matrix that we construct in terms of a complex quasi definite measure supported on the unit circle the factorization of the moment matrix leads to orthogonal laurent polynomials on the unit circle and the corresponding second kind functions we obtain jacobi operators 5 term recursion relations christoffel darboux kernels and corresponding christoffel darboux formulas from this point of view in a completely algebraic way we generalize the cantero moral velazquez sequence of laurent monomials recursion relations christoffel darboux kernels and corresponding christoffel darboux formulas in this extended context we introduce continuous deformations of the moment matrix and we show how they induce a time dependent orthogonality problem related to a toda type integrable system which is connected with the well known toeplitz lattice we obtain the lax and zakharov shabat equations using the classical integrability theory tools we explicitly derive the dynamical system associated with the coefficients of the orthogonal laurent polynomials and we compare it with the classical toeplitz lattice dynamical system for the verblunsky coefficients of szeg polynomials for a positive measure discrete flows are introduced and related to darboux transformations finally we obtain the representation of the orthogonal laurent polynomials and their second kind functions using the formalism of miwa shifts in terms of functions and the subsequent bilinear equations

a lax representation for the vertex operator and the central extension integrable hierarchies viewed as isospectral deformations of an operatorl may admit symmetries they are time dependent vector fields transversal to and commuting with the hierarchy and forming an algebra in this work the commutation relations for the symmetries are shown to be based on a non commutative lie algebra splitting theorem the symmetries viewed as vector fields onl are expressed in terms of a lax pair

matrix orthogonal laurent polynomials on the unit circle and toda type integrable systems matrix orthogonal laurent polynomials in the unit circle and the theory of toda like integrable systems are connected using the gauss borel factorization of two left and a right cantero morales velazquez block moment matrices which are constructed using a quasi definite matrix measure a block gauss borel factorization problem of these moment matrices leads to two sets of biorthogonal matrix orthogonal laurent polynomials and matrix szeg polynomials which can be expressed in terms of schur complements of bordered truncations of the block moment matrix the corresponding block extension of the christoffel darboux theory is derived deformations of the quasidefinite matrix measure leading to integrable systems of toda type are studied the integrable theory is given in this matrix scenario wave and adjoint wave functions lax and zakharov shabat equations bilinear equations and discrete flows connected with darboux transformations we generalize the integrable flows of the cafasso s matrix extension of the toeplitz lattice for the verblunsky coefficients of szeg polynomials an analysis of the miwa shifts allows for the finding of interesting connections between christoffel darboux kernels and miwa shifts of the matrix orthogonal laurent polynomials

janossy densities of coupled random matrices we explicitly calculate the janossy densities for a special class of finite determinantal random point processes with several types of particles introduced by prahofer and spohn and in the full generality by johansson in connection with the analysis of polynuclear growth models the results of this paper generalize the theorem we proved earlier with borodin about the janossy densities in biorthogonal ensembles in particular our results can be applied to ensembles of random matrices coupled in a chain which provide a very important example of determinantal ensembles we study

cpv and hfp integrals and their applications in the boundary element method the subject of this paper is hadamard finite part hfp and cauchy principal value cpv representations of certain singular integrals of primary concern here are hfp interpretations of strongly singular integrals for which cpv interpretations also exist a relationship between the two is stated and proved this matter is then illustrated by two examples and also discussed in the context of singular boundary integral equations bies for linear elasticity

direct computation of cauchy principal value integrals in advanced boundary elements this paper presents a new method for the direct computation of strongly singular integrals existing in the cauchy principal value sense it can be usefully applied in the solution by the direct boundary element method of many different problems initially some considerations are provided in order to summarize the state of the art on this issue then the features of the proposed method are reported the procedure allows the direct calculation of cauchy principal value integrals with first order singularity and it is applicable even in advanced boundary element methods employing high order elements it requires only the use of standard gaussian quadrature formulae plus the computation of a logarithmic term some examples show the effectiveness and efficiency of the procedure

global attractors for the klein gordon schrodinger equation in unbounded domains abstract in this paper we study the long time behavior of solutions for the klein gordon schrodinger equation in the whole space r n with n 3 we first prove the continuity of the solutions on initial data and then establish the asymptotic compactness of solutions finally we show the existence of the global attractor for this model in the space hk r n hk r n for each integer k 1

duality biorthogonal polynomials and multi matrix models the statistical distribution of eigenvalues of pairs of coupled random matrices can be expressed in terms of integral kernels having a generalized christoffel darboux form constructed from sequences of biorthogonal polynomials for measures involving exponentials of a pair of polynomials v1 v2 in two different variables these kernels may be expressed in terms of finite dimensional windows spanned by finite subsequences having length equal to the degree of one or the other of the polynomials v1 v2 the vectors formed by such subsequences satisfy dual pairs of first order systems of linear differential equations with polynomial coefficients having rank equal to one of the degrees of v1 or v2 and degree equal to the other they also satisfy recursion relations connecting the consecutive windows and deformation equations determining how they change under variations in the coefficients of the polynomials v1 and v2 viewed as overdetermined systems of linear difference differential deformation equations these are shown to be compatible and hence to admit simultaneous fundamental systems of solutions the main result is the demonstration of a spectral duality property namely that the spectral curves defined by the characteristic equations of the pair of matrices defining the dual differential systems are equal upon interchange of eigenvalue and polynomial parameters

analysis of shell like structures by the boundary element method based on 3 d elasticity formulation and verification in this paper the boundary element method bem for 3 d elastostatic problems is studied for the analysis of shell or shell like structures it is shown that the conventional boundary integral equation cbie for 3 d elasticity does not degenerate when applied to shell like structures contrary to the case when it is applied to crack like problems where it does degenerate due to the closeness of the two crack surfaces the treatment of the nearly singular integrals which is a crucial step in the applications of bies to thin shapes is presented in detail to verify the theory numerical examples of spherical and ellipsoidal vessels are presented using the bem approach developed in this paper it is found that the system of equations using the cbie is well conditioned for all the thickness studied for the vessels the advantages disadvantages and potential applications of the proposed bem approach to shell like structures as compared with the fem regarding modelling and accuracy are discussed in the last section applications of this bem approach to shell like structures with non uniform thickness stiffeners and layers will be reported in a subsequent paper 1998 john wiley sons ltd

the generalized additional symmetries of the two toda lattice hierarchy the generalized additional symmetries of the two toda lattice hierarchy are investigated in this paper the algebraic structure of this generalized additional symmetry is showed as w w and the actions of the generalized additional symmetries on function are also discussed by restricting the two toda lattice hierarchy to the semi infinite case

discrete polynuclear growth and determinantal processes we consider a discrete polynuclear growth png process and prove a functional limit theorem for its convergence to the airy process this generalizes previous results by prahofer and spohn the result enables us to express the f1 goe tracy widom distribution in terms of the airy process we also show some results and give a conjecture about the transversal fluctuations in a point to line last passage percolation problem furthermore we discuss a rather general class of measures given by products of determinants and show that these measures have determinantal correlation functions

a dual bie approach for large scale modelling of 3 d electrostatic problems with the fast multipole boundary element method a dual boundary integral equation bie formulation is presented for the analysis of general 3 d electrostatic problems especially those involving thin structures this dual bie formulation uses a linear combination of the conventional bie and hypersingular bie on the entire boundary of a problem domain similar to crack problems in elasticity the conventional bie degenerates when the field outside a thin body is investigated such as the electrostatic field around a thin conducting plate the dual bie formulation however does not degenerate in such cases most importantly the dual bie is found to have better conditioning for the equations using the boundary element method bem compared with the conventional bie even for domains with regular shapes thus the dual bie is well suited for implementation with the fast multipole bem the fast multipole bem for the dual bie formulation is developed based on an adaptive fast multiple approach for the conventional bie several examples are studied with the fast multipole bem code including finite and infinite domain problems bulky and thin plate structures and simplified comb drive models having more than 440 thin beams with the total number of equations above 1 45 million and solved on a pc the numerical results clearly demonstrate that the dual bie is very effective in solving general 3 d electrostatic problems as well as special cases involving thin perfect conducting structures and that the adaptive fast multipole bem with the dual bie formulation is very efficient and promising in solving large scale electrostatic problems copyright 2007 john wiley sons ltd

on the squared eigenfunction symmetry of the toda lattice hierarchy the squared eigenfunction symmetry for the toda lattice hierarchy is explicitly constructed in the form of the kronecker product of the vector eigenfunction and the vector adjoint eigenfunction which can be viewed as the generating function for the additional symmetries when the eigenfunction and the adjoint eigenfunction are the wave function and the adjoint wave function respectively then after the fay like identities and some important relations about the wave functions are investigated the action of the squared eigenfunction related to the additional symmetry on the tau function is derived which is equivalent to the adler shiota van moerbeke formulas

generalized orthogonal polynomials discrete kp and riemann hilbert problems classically a single weight on an interval of the real line leads to moments orthogonal polynomials and tridiagonal matrices appropriately deforming this weight with times t t 1 t 2 leads to the standard toda lattice and tau functions expressed as hermitian matrix integrals this paper is concerned with a sequence of t perturbed weights rather than one single weight this sequence leads to moments polynomials and a fuller matrix evolving according to the discrete kp hierarchy the associated tau functions have integral as well as vertex operator representations among the examples considered we mention nested calogero moser systems concatenated solitons and m periodic sequences of weights the latter lead to 2m 1 band matrices and generalized orthogonal polynomials also arising in the context of a riemann hilbert problem we show the riemann hilbert factorization is tantamount to the factorization of the moment matrix into the product of a lower times upper triangular matrix

multiple orthogonal polynomials of mixed type gauss borel factorization and the multi component 2d toda hierarchy multiple orthogonality is considered in the realm of a gauss borel factorization problem for a semi infinite moment matrix perfect combinations of weights and a finite borel measure are constructed in terms of m nikishin systems these perfect combinations ensure that the problem of mixed multiple orthogonality has a unique solution that can be obtained from the solution of a gauss borel factorization problem for a semi infinite matrix which plays the role of a moment matrix this leads to sequences of multiple orthogonal polynomials their duals and second kind functions it also gives the corresponding linear forms that are bi orthogonal to the dual linear forms expressions for these objects in terms of determinants from the moment matrix are given recursion relations are found which imply a multi diagonal jacobi type matrix with snake shape and results like the abc theorem or the christoffel darboux formula are re derived in this context using the factorization problem and the generalized hankel symmetry of the moment matrix the connection between this description of multiple orthogonality and the multi component 2d toda hierarchy which can be also understood and studied through a gauss borel factorization problem is discussed deformations of the weights natural for m nikishin systems are considered and the correspondence with solutions to the integrable hierarchy represented as a collection of lax equations is explored corresponding lax and zakharov shabat matrices as well as wave functions and their adjoints are determined the construction of discrete flows is discussed in terms of miwa transformations which involve darboux transformations for the multiple orthogonality conditions the bilinear equations are derived and the tau function representation of the multiple orthogonality is given

multivariate orthogonal polynomials and integrable systems multivariate orthogonal polynomials in d real dimensions are considered from the perspective of the cholesky factorization of a moment matrix the approach allows for the construction of corresponding multivariate orthogonal polynomials associated second kind functions jacobi type matrices and associated three term relations and also christoffel darboux formulae the multivariate orthogonal polynomials their second kind functions and the corresponding christoffel darboux kernels are shown to be quasi determinants as well as schur complements of bordered truncations of the moment matrix quasi tau functions are introduced it is proven that the second kind functions are multivariate cauchy transforms of the multivariate orthogonal polynomials discrete and continuous deformations of the measure lead to toda type integrable hierarchy being the corresponding flows described through lax and zakharov shabat equations bilinear equations are found varying size matrix nonlinear partial difference and differential equations of the 2d toda lattice type are shown to be solved by matrix coefficients of the multivariate orthogonal polynomials the discrete flows which are shown to be connected with a gauss borel factorization of the jacobi type matrices and its quasi determinants lead to expressions for the multivariate orthogonal polynomials and their second kind functions in terms of shifted quasi tau matrices which generalize to the multidimensional realm those that relate the baker and adjoint baker functions to ratios of miwa shifted tau functions in the 1d scenario in this context the multivariate extension of the elementary darboux transformation is given in terms of quasi determinants of matrices built up by the evaluation at a poised set of nodes lying in an appropriate hyperplane in r d of the multivariate orthogonal polynomials the multivariate christoffel formula for the iteration of m elementary darboux transformations is given as a quasi determinant it is shown using congruences in the space of semi infinite matrices that the discrete and continuous flows are intimately connected and determine nonlinear partial difference differential equations that involve only one site in the integrable lattice behaving as a kadomstev petviashvili type system finally a brief discussion of measures with a particular linear isometry invariance and some of its consequences for the corresponding multivariate polynomials is given in particular it is shown that the toda times that preserve the invariance condition lay in a secant variety of the veronese variety of the fixed point set of the linear isometry

pfaff tau functions consider the two dimensional toda lattice with certain skew symmetric initial condition which is preserved along the locus s t of the space of time variables restricting the solution to s t we obtain another hierarchy called pfaff lattice which has its own tau function being equal to the square root of the restriction of 2d toda tau function we study its bilinear and fay identities w and virasoro symmetries relation to symmetric and symplectic matrix integrals and quasiperiodic solutions

matrices coupled in a chain i eigenvalue correlations the general correlation function for the eigenvalues of p complex hermitian matrices coupled in a chain is given as a single determinant for this we use a slight generalization of a theorem of dyson

coupled map lattices some topological and ergodic properties abstract lattice dynamical systems ldss form the class of extended systems that is the intermediate one between partial differential equations pdes and cellular automata the most popular class of ldss is formed by coupled map lattices cmls while being introduced rather recently ldss allowed already to clarify and to define exactly some notions that form the basis of the modern phenomenological theory of spatio temporal dynamics and to obtain some new and even rigorous results on space time chaos intermittency and pattern formation we discuss the rigorous results that were obtained in this area but eventually give also some applications

the w1 gls symmetries of the s component kp hierarchy adler shiota and van moerbeke obtained for the kp and toda lattice hierarchies a formula which translates the action of the vertex operator on tau functions to an action of a vertex operator of pseudodifferential operators on wave functions this relates the additional symmetries of the kp and toda lattice hierarchy to the w1 respectively w1 w1 algebra symmetries in this paper we generalize the results to the s component kp hierarchy the vertex operators generate the algebra w1 gls the matrix version of w1 since the toda lattice hierarchy is formally equivalent to the 2 component kp hierarchy the results of this article uncover in that particular case a much richer structure than the one obtained by adler shiota and van moerbeke

evaluation of hypersingular integrals in the boundary element method by complex variable techniques strongly singular and hypersingular integrals that occur in many two dimensional applications of the boundary element method e g potential theory and liner elasticity can be treated in a unified manner by complex analysis this paper presents an elegant approach for the evaluation of such hypersingular integrals the limit to the boundary method where a source point is moved from the inside or outside of a body to its boundary is carried out using complex variables the connection of this limit with the hadamard finite part of hypersingular integrals and principal value integrals is established many examples are worked out throughout the paper to illustrate the theory

on the variation in the cohomology of the symplectic form of the reduced phase space is called the momentum mapping of the hamiltonian t action given 1 1 the condition 1 2 just means that t acts along the fibers of j for the basic definitions and properties of non commutative hamiltonian group actions see am the results of this paper can easily be extended to hamiltonian actions of arbitrary compact connected lie groups by applying our results to the action of its maximal toms and using the equivariance of the momentum mapping for some more details see the remarks at the end of sect 2 we will assume throughout this paper that the momentum map is proper that is j i u is compact for each compact subset u of t now let r be a regular value of j that is t j tmm t is surjective for all meye j 1 4 then ye is a smooth submanifold of m compact because

string orthogonal polynomials string equations and 2 toda symmetries 1 the 2 toda lattice and its generic symmetries 2 a larger class of symmetries for special initial conditions 3 borel decomposition of moment matrices tau functions and string orthogonal polynomials 4 from string orthogonal polynomials to the 2 toda lattice and the string equation 5 virasoro constraints on two matrix integrals comment 45 pages latex

pdes for the joint distributions of the dyson airy and sine processes in a celebrated paper dyson shows that the spectrum of an n x n random hermitian matrix diffusing according to an ornstein uhlenbeck process evolves as n noncolliding brownian motions held together by a drift term the universal edge and bulk scalings for hermitian random matrices applied to the dyson process lead to the airy and sine processes in particular the airy process is a continuous stationary process describing the motion of the outermost particle of the dyson brownian motion when the number of particles gets large with space and time appropriately rescaled in this paper we answer a question posed by kurt johansson to find a pde for the joint distribution of the airy process at two different times similarly we find a pde satisfied by the joint distribution of the sine process this hinges on finding a pde for the joint distribution of the dyson process which itself is based on the joint probability of the eigenvalues for coupled gaussian hermitian matrices the pde for the dyson process is then subjected to an asymptotic analysis consistent with the edge and bulk rescalings the pdes enable one to compute the asymptotic behavior of the joint distribution and the correlation for these processes at different times t 1 and t 2 when t 2 t 1 infinity as illustrated in this paper for the airy process this paper also contains a rigorous proof that the extended hermite kernel governing the joint probabilities for the dyson process converges to the extended airy and sine kernels after the appropriate rescalings

pdes for the gaussian ensemble with external source and the pearcey distribution the present paper studies a gaussian hermitian random matrix ensemble with external source given by a fixed diagonal matrix with two eigenvalues a as a first result the probability that the eigenvalues of the ensemble belong to an interval e satisfies a fourth order pde with quartic nonlinearity the variables are the eigenvalue a and the boundary of e this equation enables one to find a pde for the pearcey distribution the latter describes the statistics of the eigenvalues near the closure of a gap i e when the support of the equilibrium measure for large size random matrices has a gap that can be made to close the gaussian hermitian random matrix ensemble with external source described above has this feature the pearcey distribution is shown to satisfy a fourth order pde with cubic nonlinearity this also gives the pde for the transition probability of the pearcey process a limiting process associated with nonintersecting brownian motions on r c 2006 wiley periodicals inc

the partition function of the two matrix model as an isomonodromic τ function we consider the itzykson zuber eynard mehta two matrix model and prove that the partition function is an isomonodromic function in a sense that generalizes that of jimbo et al monodromy preserving deformation of linear ordinary differential equations with rational coefficients physica d 2 306 1981 in order to achieve the generalization we need to define a notion of function for isomonodromic systems where the adregularity of the leading coefficient is not a necessary requirement

universality in the two matrix model a riemann hilbert steepest descent analysis the eigenvalue statistics of a pair m 1 m 2 of n x n hermitian matrices taken randomly with respect to the measure 1 z n exp n tr v m 1 w m 2 tau m 1 m 2 dm 1 dm 2 can be descri

large scale correlations in normal non hermitian matrix ensembles we compute the large scale macroscopic correlations in ensembles of normal random matrices with a general non gaussian measure and in ensembles of general non hermitian matrices with a class of non gaussian measures in both cases the eigenvalues are complex and in the large n limit they occupy a domain in the complex plane for the case when the support of eigenvalues is a connected compact domain we compute two three and four point connected correlation functions in the first non vanishing order in 1 n in a manner that the algorithm of computing higher correlations becomes clear the correlation functions are expressed through the solution of the dirichlet boundary problem in the domain complementary to the support of eigenvalues

noncolliding brownian motion and determinantal processes a system of one dimensional brownian motions bms conditioned never to collide with each other is realized as i dyson s bm model which is a process of eigenvalues of hermitian matrix valued diffusion process in the gaussian unitary ensemble gue and as ii the h transform of absorbing bm in a weyl chamber where the harmonic function h is the product of differences of variables the vandermonde determinant the karlin mcgregor formula gives determinantal expression to the transition probability density of absorbing bm we show from the karlin mcgregor formula if the initial state is in the eigenvalue distribution of gue the noncolliding bm is a determinantal process in the sense that any multitime correlation function is given by a determinant specified by a matrix kernel by taking appropriate scaling limits spatially homogeneous and inhomogeneous infinite determinantal processes are derived we note that the determinantal processes related with noncolliding particle systems have a feature in common such that the matrix kernels are expressed using spectral projections of appropriate effective hamiltonians on the common structure of matrix kernels continuity of processes in time is proved and general property of the determinantal processes is discussed

asymptotic smoothing effect of solutions to weakly dissipative klein gordon schrodinger equations abstract in this paper the authors consider the cauchy problem of weakly dissipative klein gordon schrodinger equations through yukawa coupling in r 3 making use of a strichartz type inequality and a suitable decomposition of the solution semigroup they prove the asymptotic smoothing effect of the solutions

smoothness relaxation strategies for singular and hypersingular integral equations three stages are involved in the formulation of a typical direct boundary element method derivation of an integral representation taking a limit to the boundary ltb so as to obtain an integral equation and discretization we examine the second and third stages focussing on strategies that are intended to permit the relaxation of standard smoothness assumptions two such strategies are indicated therst is the introduction of various apparent or pseudo ltbs the second is relaxed regularization in which a regularized integral equation derived rigorously under certain smoothness assumptions is used when less smoothness is available both strategies are shown to be based on inconsistent reasoning nevertheless reasons are oered for having some condence in numerical results obtained with certain strategies our work is done in two physical contexts namely two dimensionalpotential theory using functions of a complex variable and three dimensionalelastostatics 1998 john wiley sons ltd

on additional symmetries of the kp hierarchy and sato s backlund transformation a short proof is given to the fact that the additional symmetries of the kp hierarchy defined by their action on pseudodifferential operators according to fuchssteiner chen lee lin orlov shulman coincide with those defined by their action on functions as sato s backlund transformations a new simple formula for the generator of additional symmetries is also presented

differential equations for dyson processes we call a dyson process any process on ensembles of matrices in which the entries undergo diffusion we are interested in the distribution of the eigenvalues or singular values of such matrices in the original dyson process it was the ensemble of n n hermitian matrices and the eigenvalues describe n curves given sets x1 xm the probability that for each k no curve passes through xk at time k is given by the fredholm determinant of a certain matrix kernel the extended hermite kernel for this reason we call this dyson process the hermite process similarly when the entries of a complex matrix undergo diffusion we call the evolution of its singular values the laguerre process for which there is a corresponding extended laguerre kernel scaling the hermite process at the edge leads to the airy process which was introduced by prahofer and spohn as the limiting stationary process for a polynuclear growth model and in the bulk to the sine process scaling the laguerre process at the edge leads to the bessel process

a jacobi type christoffel darboux formula for multiple orthogonal polynomials of mixed type abstract an alternative expression for the christoffel darboux formula for multiple orthogonal polynomials of mixed type is derived from the lu factorization of the moment matrix of a given measure and two sets of weights we use the action of the generalized jacobi matrix j also responsible for the recurrence relations on the linear forms and their duals to obtain the result

use of simple solutions for boundary integral methods in elasticity and fracture analysis the simple solutions or indirect method of analysing cauchy and hypersingular integrands in the gradient flux or traction boundary integral equation bie is applied to linear elastic fracture analysis because of the geometric singularity of the crack surface application of the simple solutions formulas on the crack face requires integration over a temporary closure surface rather than the remainder of the body closure surface constructions are exhibited for crack surfaces allowing the gradient bie to be applied as a constraint equation on a crack surface where the primary bie is degenerate computational results are given for two benchmark fracture problems

attractor for dissipative klein gordon schrodinger equations inr3 abstract in this paper the authors consider the cauchy problem of coupled dissipative klein gordon schrodinger equations in r 3 and prove the existence of the maximal attractor

a pde for non intersecting brownian motions and applications abstract consider n n 1 n 2 n p non intersecting brownian motions on the real line starting from the origin at t 0 with n i particles forced to reach p distinct target points i at time t 1 with 1 2 p this can be viewed as a diffusion process in a sector of r n this work shows that the transition probability that is the probability for the particles to pass through windows e k at times t k satisfies in a new set of variables a non linear pde which can be expressed as a near wronskian that is a determinant of a matrix of size p 1 with each row being a derivative of the previous except for the last column it is an interesting open question to understand those equations from a more probabilistic point of view as an application of these equations let the number of particles forced to the extreme points 1 and p tend to infinity keep the number of particles forced to intermediate points fixed inliers but let the target points themselves go to infinity according to a proper scale a new critical process appears at the point of bifurcation where the bulk of the particles forced to n depart from those going to n these statistical fluctuations near that point of bifurcation are specified by a kernel which is a rational perturbation of the pearcey kernel this work also shows that such equations are an essential tool in obtaining certain asymptotic results finally the paper contains a conjecture

two coupled matrices eigenvalue correlations and spacing functions for two n n hermitian matrices a and b with joint probability density proportional to exp tr u a v b 2cab where u and v are polynomials a method is given to calculate all correlation cluster and spacing functions of the eigenvalues of either one or both matrices the method relies on the introduction of two sets of bi orthogonal polynomials with non local weights in a linear chain of coupled matrices if one looks for the statistical properties of the eigenvalues of only one matrix two matrices situated anywhere in the chain then we can proceed as a one matrix two matrices problem

hypersingular and finite part integrals in the boundary element method abstract a new definition of the hadamard finite part hfp of hypersingular integrals is proposed in this paper this definition does not involve a limiting process it is completely general and is valid for one as well as higher dimensional integrals on closed as well as on open surfaces it reduces respectively to the cauchy principal value cpv and riemann integral respectively for the special cases of strongly singular and weakly singular integrands of course suitable symmetric exclusion zones must be chosen to realize cpv integrals starting with this new definition of the hfp of certain hypersingular boundary integral equations hbie that arise in potential theory and in wave scattering a regularization method is carried out in order to express the hypersingular integrals in terms of ones that are at most weakly singular the regularized versions are completely consistent with those available in the recent literature where a different definition of the hfp was employed

the additional symmetries for the btl and ctl hierarchies in this paper we construct the additional symmetries for the toda lattice hierarchies of b type and c type the btl and ctl hierarchies and show their algebraic structures are w b w b and w c w c respectively and also we discuss the generating functions of the additional symmetries