moment matrices and multi component kp with applications to random matrix theory questions on random matrices and non intersecting brownian motions have led to the study of moment matrices with regard to several weights the main result of this paper is to show that the determinants of such moment matrices satisfy upon adding one set of time deformations for each weight the multi component kp hierarchy these determinants are thus tau functions for these integrable hierarchies the tau functions so obtained with appropriate shifts of the time parameters forward and backwards will be expressed in terms of multiple orthogonal polynomials for these weights and their cauchy transforms the main result is a vast generalization of a known fact about infinitesimal deformations of orthogonal polynomials it concerns an identity between the orthogonality of polynomials on the real line the bilinear identity in kp theory and a generating functional for the full kp theory an additional fact not discussed in this paper is that these functions satisfy virasoro constraints with respect to these time parameters

uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory we consider asymptotics for orthogonal polynomials with respect to varying exponential weights wn x dx e nv x dx on the line as n the potentials v are assumed to be real analytic with sufficient growth at infinity the principle results concern plancherel rotach type asymptotics for the orthogonal polynomials down to the axis using these asymptotics we then prove universality for a variety of statistical quantities arising in the theory of random matrix models some of which have been considered recently in 31 and also in 4 an additional application concerns the asymptotics of the recurrence coefficients and leading coefficients for the orthonormal polynomials see also 4 the orthogonal polynomial problem is formulated as a riemann hilbert problem following 19 20 the riemann hilbert problem is analyzed in turn using the steepest descent method introduced in 12 and further developed in 11 13 a critical role in our method is played by the equilibrium measure d v for v as analyzed in 8 1999 john wiley sons inc

a comparison of numerical algorithms for fourier extension of the first second and third kinds the range of fourier methods can be significantly increased by extending a nonperiodic function f x to a periodic function f on a larger interval when f x is analytically known on the extended interval the extension is straightforward when f x is unknown outside the physical interval there is no standard recipe worse still like a radarless aircraft groping through fog the algorithm may wreck on the mountain in fog problem a function f x which is perfectly well behaved on the physical interval may very well have singularities in the extended domain in this article we compare several algorithms for successfully extending a function f x into the fog even when the analytic extension is singular the best third kind extension requires singular value decomposition with iterative refinement but achieves accuracy close to machine precision

a critical phenomenon in the two matrix model in the quartic quadratic case we study a critical behavior for the eigenvalue statistics in the two matrix model in the quartic quadratic case for certain parameters the eigenvalue distribution for one of the matrices has a limit that vanishes like a square root in the interior of the support the main result of the paper is a new kernel that describes the local eigenvalue correlations near that critical point the kernel is expressed in terms of a 4 4 riemann hilbert problem related to the hastings mcleod solution of the painleve ii equation we then compare the new kernel with two other critical phenomena that appeared in the literature before first we show that the critical kernel that appears in case of quadratic vanishing of the limiting eigenvalue distribution can be retrieved from the new kernel by means of a double scaling limit second we briefly discuss the relation with the tacnode singularity in noncolliding brownian motions that was recently analyzed although the limiting density in that model also vanishes like a square root at a certain interior point the process at the local scale is different from the process that we obtain in the two matrix model

random hermitian matrices and gaussian multiplicative chaos n berestycki s work is supported by epsrc grants ep l018896 1 and ep i03372x 1 m d wong is a phd student at the cambridge centre for analysis supported by epsrc grant ep l016516 1 some of this work was carried out while the first and third authors visited the university of helsinki funded in part by epsrc grant ep l018896 1 they also wish to thank the university of helsinki for its hospitality during this visit c webb wishes to thank the isaac newton institute for mathematical sciences for its hospitality during the random geometry program during which this project was initiated c webb was supported by the eemil aaltonen foundation grant stochastic dynamics on large random graphs and academy of finland grants 288318 and 308123

painleve equations nonlinear special functions the six painleve equations pi pvi were first discovered about a hundred years ago by painleve and his colleagues in an investigation of nonlinear second order ordinary differential equations recently there has been considerable interest in the painleve equations primarily due to the fact that they arise as reductions of the soliton equations which are solvable by inverse scattering consequently the painleve equations can be regarded as completely integrable equations and possess solutions which can be expressed in terms of solutions of linear integral equations despite being nonlinear equations although first discovered from strictly mathematical considerations the painleve equations have arisen in a variety of important physical applications including statistical mechanics plasma physics nonlinear waves quantum gravity quantum field theory general relativity nonlinear optics and fibre optics the painleve equations may be thought of a nonlinear analogues of the classical special functions they possess hierarchies of rational solutions and one parameter families of solutions expressible in terms of the classical special functions for special values of the parameters further the painleve equations admit symmetries under affine weyl groups which are related to the associated backlund transformations in this paper i discuss some of the remarkable properties which the painleve equations possess including connection formulae backlund transformations associated discrete equations and hierarchies of exact solutions in particular the second painleve equation pii is used to illustrate these properties and some of the applications of pii are also discussed

the isomonodromy approach to matric models in 2d quantum gravity we consider the double scaling limit in the hermitian matrix model for 2d quantum gravity associated with the measure exp sum limits j 1 n t j z 2j n geqq 3 we show that after the appropriate modification of the contour of integration the cross migdal douglas shenker limit to the painleve i equation in the generic case of the pure gravity is valid and calculate the nonperturbative parameters of the corresponding painleve function our approach is based on the wkb analysis of the l a pair corresponding to the discrete string equation in the framework of the inverse monodromy method here we extend our results which were obtained before for the particular casesn 2 3 our analysis complements the isomonodromy approach proposed by g moore to the general string equations that come from the matrix model in the continuous limit and differ in that we apply the isomonodromy technique to investigate the double scaling limit itself

elliptic problems with lack of compactness via a new fixed point theorem abstract this paper provides a new fixed point theorem for increasing self mappings g b b of a closed ball b x where x is a banach semilattice which is reflexive or has a weakly fully regular order cone x by means of this fixed point theorem we are able to establish existence results of elliptic problems with lack of compactness

discrete painleve equations and their appearance in quantum gravity we discuss an algorithmic approach for both deriving discrete analogues of painleve equations as well as using such equations to characterize similarity reductions of spatially discrete integrable evolution equations as a concrete example we show that a discrete analogue of painleve i can be used to characterize similarity solutions of the kac moerbeke equation it turns out that these similarity solutions also satisfy a special case of painleve iv equation in addition we discuss a methodology for obtaining the relevant continuous limits not only at the level of equations but also at the level of solutions as an example we use the wkb method in the presence of two turning points of the third order to parametrize at the continuous limit the solution of painleve i in terms of the solution of discrete painleve i finally we show that these results are useful for investigating the partition function of the matrix model in 2d quantum gravity associated with the measure exp t1z2 t2z4 t3z6

boundary value problems with discontinuities and monotone discretization in this paper a monotone iteration technique for generating uniform bounds of the solution of a two point boundary value problem with discontinuous nonlinearities is presented moreover the method is combined with a monotone discretization which is implementable

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

on operator and integral equations with discontinuous right hand side on considere des equations operateurs dans c a e ou a est un espace topologique et e est un espace topologique partiellement ordonne ou un groupe

on extremal solutions of inclusion problems with applications to game theory abstract a recursion principle generalized iteration methods and the axiom of choice are applied to prove the existence of extremal fixed points of set valued mappings in posets extremal solutions of an inclusion problem and extremal nash equilibria for a normal form game

fixed point results in ordered normed spaces with applications to abstract and differential equations in s fixed point results are derived for an increasing operator a u u u 01 where u u is a conical segment in an ordered banach space with normal order cone p as the main results an existence a fixed point of a is proved by assuming that a u u is weakly relatively compact and separable if p is also minihedral then a is shown to have least and greatest fixed points no continuity or linearity hypotheses are imposed on a in this paper we shall show for instance that if u u is a conical segment in an ordered normed space e and if a u u u u is increasing and weakly relatively order compact then a has the least and the greatest fixed point provided that the order cone of e is normal or a u u is separable moreover these fixed points are constructed by countable iteration processes the so obtained fixed point results are then applied to derive results on existence of extremal or unique solutions of the operator equation

parameter selection and numerical approximation properties of fourier extensions from fixed data fourier extensions have been shown to be an effective means for the approximation of smooth nonperiodic functions on bounded intervals given their values on an equispaced or in general scattered grid related to this method are two parameters these are the extension parameter t the ratio of the size of the extended domain to the physical domain and the oversampling ratio h the number of sampling nodes per fourier mode the purpose of this paper is to investigate how the choice of these parameters affects the accuracy and stability of the approximation our main contribution is to document the following interesting phenomenon namely if the desired condition number of the algorithm is fixed in advance then the particular choice of such parameters makes little difference to the algorithm s accuracy as a result one is free to choose t without concern that it is suboptimal in particular one may use the value t 2 which corresponds to the case where the extended domain is precisely twice the size of the physical domain for which there is known to be a fast algorithm for computing the approximation in addition we determine the resolution power points per wavelength of the approximation to be equal to email protected and address the trade off between resolution power and stability

on the nonlinear wave equation utt b t u 2 ux 2 uxx f x t u ux ut u 2 ux 2 associated with the mixed homogeneous conditions abstract in this paper we consider the following nonlinear wave equation 1 u tt b t u x 2 u xx f x t u u x u t u x 2 x 0 1 0 2 u x 0 t h 0 u 0 t g 0 t u 1 t g 1 t 3 u x 0 u 0 x u t x 0 u 1 x where b f g 0 g 1 u 0 u 1 are given functions in eq 1 the nonlinear terms b t u x 2 f x t u u x u t u x 2 depending on an integral u x 2 0 1 u x x t 2 dx in this paper we associate with problem 1 3 a linear recursive scheme for which the existence of a local and unique solution is proved by using standard compactness argument in case of b c 3 r 2 b b 0 0 b 1 c 2 r 2 b 1 0 f c 3 0 1 r r 3 r and f 1 c 2 0 1 r r 3 r we obtain from the equation u tt b t u x 2 eb 1 t u x 2 u xx f x t u u x u t u x 2 ef 1 x t u u x u t u x 2 associated to 2 3 a weak solution u e x t having an asymptotic expansion of order 3 in e for e sufficiently small

nonlinear equations for the recurrence coefficients of discrete orthogonal polynomials we study polynomials orthogonal on a uniform grid we show that each weight function gives two potentials and each potential leads to a structure relation lowering operator these results are applied to derive second order difference equations satisfied by the orthogonal polynomials and nonlinear difference equations satisfied by the recursion coefficients in the three term recurrence relations

semiclassical expansions in the toda hierarchy and the hermitian matrix model an iterative algorithm for determining a class of solutions of the dispersionful 2 toda hierarchy characterized by string equations is developed this class includes the solution which underlies the large n limit of the hermitian matrix model in the one cut case it is also shown how the double scaling limit can be naturally formulated in this scheme

on the nonlinear wave equation utt b ux 2 uxx f x t u ux ut ux 2 associated with the mixed homogeneous conditions in this paper we consider the following nonlinear wave equation 1 utt b ux 2 uxx f x t u ux ut ux 2 x 0 1 0 t t 2 ux 0 t h0u 0 t u 1 t 0 3 u x 0 u0 x ut x 0 u1 x where b f u0 u1 are given functions in eq 1 the nonlinear terms b ux 2 f x t u ux ut ux 2 depending on an integral ux 2 01 ux x t 2dx in this paper we associate with problem 1 3 a linear recursive scheme for which the existence of a local and unique solution is proved by using standard compactness argument in case of b cn 1 r b b0 0 b1 cn r b1 0 f cn 1 0 1 r r3 r and f1 cn 0 1 r r3 r we obtain from the following equation utt b ux 2 eb1 ux 2 uxx f x t u ux ut ux 2 ef1 x t u ux ut ux 2 associated to 2 3 a weak solution ue x t having an asymptotic expansion of order n 1 in e for e sufficiently small

algorithms for the fixed point property this survey exhibits various algorithms to decide the question if a given ordered set p has the fixed point property resp if p has a fixed point free order preserving self map while a depth first search algorithm for a fixed point free map is easily written it is also quite inefficient we discuss a reduction algorithm by xia which can be used to speed up the search for a fixed point free self map the ideas used in creating this algorithm show close connections to two problems the decision whether an ordered set has a fixed point free automorphism and the decision whether a given r partite graph has an r clique the latter two problems are shown to be np complete using the work of goddard lubiw and williamson the problem to decide whether a given finite ordered set has a fixed point free order preserving self map has recently been shown to be np complete thus showing that the above close connection is not by accident retraction theorems leading to dismantling algorithms are another approach to the problem we present the classical dismantling procedure by rival and extensions by fofanova li milner rutkowski and the author these theorems give a polynomial algorithm to decide if an ordered set has the fixed point property for some nice classes of ordered sets height 1 width 2 and structural insights for other classes chain complete ordered sets with no infinite antichains sets of interval dimension 2 the related issue of uniqueness of cores gives an insight into birkhoff s problem regarding cancellation of exponents walker s relational fixed point property for which the analogous problem has a very satisfying solution also is discussed another variation on the retraction theme is the use of algebraic topology in deriving fixed point theorems initiated by baclawski and bjorner and continued for example by constantin and fournier after a primer on the basic concepts of integer homology we present their retraction dismantling procedures which always prove acyclicity of the associated simplicial complex and then the fixed point property via a lefschetz type fixed point theorem differences and similarities with the above combinatorial procedures are pointed out the main problem in making these results accessible to entirely combinatorial proofs is the lack of a combinatorial discrete analogue of the continuous concept of a weak resp deformation retract we also present a class of ordered sets without the fixed point property such that all proper retracts have the fixed point property this class is induced via triangulations of n spheres finally we include an indication how methods developed for work on the fixed point property can be used in other areas for example the arguments by abian brown and pelczar to prove that in a chain complete ordered set existence of a point p with p f p implies existence of a fixed point have recently found application to analysis in the work of carl heikkila lakhshmikantham and others

multi critical unitary random matrix ensembles and the general painleve ii equation we study unitary random matrix ensembles of the form z n detm 2 e ntrvwdm where a 1 2 and v is such that the limiting mean eigenvalue density for n n oc and n n 1 vanishes quadratically at the origin in order to compute the double scaling limits of the eigenvalue correlation kernel near the origin we use the deift zhou steepest descent method applied to the riemann hilbert problem for orthogonal polynomials on the real line with respect to the weight x 2ae nv x here the main focus is on the construction of a local parametrix near the origin with functions associated with a special solution qa of the painleve ii equation q sq 2q3 a we show that qa has no real poles for a 1 2 by proving the solvability of the corresponding riemann hilbert problem we also show that the asymptotics of the recurrence coefficients of the orthogonal polynomials can be expressed in terms of qa in the double scaling limit

the fourier extension method and discrete orthogonal polynomials on an arc of the circle abstract the fourier extension method also known as the fourier continuation method is a method for approximating non periodic functions on an interval using truncated fourier series with period larger than the interval on which the function is defined when the function being approximated is known at only finitely many points the approximation is constructed as a projection based on this discrete set of points in this paper we address the issue of estimating the absolute error in the approximation the error can be expressed in terms of a system of discrete orthogonal polynomials on an arc of the unit circle and these polynomials are then evaluated asymptotically using riemann hilbert methods

impossibility of fast stable approximation of analytic functions from equispaced samples it is shown that no stable procedure for approximating functions from equally spaced samples can converge exponentially for analytic functions to avoid instability one must settle for root exponential convergence the proof combines a bernstein inequality of 1912 with an estimate due to coppersmith and rivlin in 1992

a riemann hilbert problem for biorthogonal polynomials we characterize the biorthogonal polynomials that appear in the theory of coupled random matrices via a riemann hilbert problem our riemann hilbert problem is different from the ones that were proposed recently by ercolani and mclaughlin kapaev and bertola et al we believe that our formulation may be tractable to asymptotic analysis

on the nonlinear wave equation utt b t u 2 ux 2 ut 2 uxx f x t u ux ut u 2 ux 2 ut 2 associated with the mixed homogeneous conditions abstract in this paper we consider the following nonlinear wave equation 1 u tt b t u 2 u x 2 u t 2 u xx f x t u u x u t u 2 u x 2 u t 2 x 0 1 0 2 u x 0 t h 0 u 0 t u x 1 t h 1 u 1 t 0 3 u x 0 u 0 x u t x 0 u 1 x where h0 0 h1 0 are given constants and b f u 0 u 1 are given functions in eq 1 the nonlinear terms b t u 2 u x 2 u t 2 f x t u u x u t u 2 u x 2 u t 2 depending on the integrals u 2 u x t 2 d x u x 2 0 1 u x x t 2 d x and u t 2 u t x t 2 d x in this paper we associate with problem 1 3 a linear recursive scheme for which the existence of a local and unique solution is proved by using standard compactness argument in case of b c n 1 r 4 b b 0 0 b 1 c n r 4 b 1 0 f c n 1 0 1 r r 3 r 3 and f1 cn 0 1 r r3 r 3 we obtain from the following equation utt b t u 2 ux 2 ut 2 eb1 t u 2 ux 2 ut 2 uxx f x t u ux ut u 2 ux 2 ut 2 ef1 x t u ux ut u 2 ux 2 ut 2 associated to 2 3 a weak solution ue x t having an asymptotic expansion of order n 1 in e for e sufficiently small

on the origin of integrability in matrix models the matrix integrals involved in 2d lattice gravity are studied at finiten the integrable systems that arise in the continuum theory are shown to result directly from the formulation of the matrix integrals in terms of orthogonal polynomials the partition function proves to be a tau function of the toda lattice hierarchy the associated linear problem is equivalent to finding the polynomial basis which diagonalizes the partition function the cases of one hermitian matrix one unitary matrix and hermitian matrix chains all fall within the toda framework

hermite pade approximants for a pair of cauchy transforms with overlapping symmetric supports hermite pade approximants of type ii are vectors of rational functions with a common denominator that interpolate a given vector of power series at infinity with maximal order we are interested in the situation when the approximated vector is given by a pair of cauchy transforms of smooth complex measures supported on the real line the convergence properties of the approximants are rather well understood when the supports consist of two disjoint intervals angelesco systems or two intervals that coincide under the condition that the ratio of the measures is a restriction of the cauchy transform of a third measure nikishin systems in this work we consider the case where the supports form two overlapping intervals in a symmetric way and the ratio of the measures extends to a holomorphic function in a region that depends on the size of the overlap we derive szeg type formulae for the asymptotics of the approximants identify the convergence and divergence domains the divergence domains appear for angelesco systems but are not present for nikishin systems and show the presence of overinterpolation a feature peculiar for nikishin systems but not for angelesco systems our analysis is based on a riemann hilbert problem for multiple orthogonal polynomials the common denominator 2016 wiley periodicals inc

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

convergence analysis of krylov subspace iterations with methods from potential theory krylov subspace iterations are among the best known and most widely used numerical methods for solving linear systems of equations and for computing eigenvalues of large matrices these methods are polynomial methods whose convergence behavior is related to the behavior of polynomials on the spectrum of the matrix this leads to an extremal problem in polynomial approximation theory how small can a monic polynomial of a given degree be on the spectrum this survey gives an introduction to a recently developed technique to analyze this extremal problem in the case of symmetric matrices it is based on global information on the spectrum in the sense that the eigenvalues are assumed to be distributed according to a certain measure then depending on the number of iterations the lanczos method for the calculation of eigenvalues finds those eigenvalues that lie in a certain region which is characterized by means of a constrained equilibrium problem from potential theory the same constrained equilibrium problem also describes the superlinear convergence of conjugate gradients and other iterative methods for solving linear systems

accurate high order representation of complex three dimensional surfaces via fourier continuation analysis we present a new method for construction of high order parametrizations of surfaces starting from point clouds the method we propose can be used to produce full surface parametrizations by sets of local charts each one representing a large surface patch which typically contains thousands of the points in the original point cloud for complex surfaces of scientific and engineering relevance the proposed approach accurately renders both smooth and non smooth portions of a surface it yields super algebraically convergent fourier series approximations to a given surface up to and including all points of geometric singularity such as corners edges conical points etc in view of their c smoothness except at true geometric singularities and their properties of high order approximation the surfaces produced by this method are suitable for use in conjunction with high order numerical methods for boundary value problems in domains with complex boundaries including pde solvers integral equation solvers etc our approach is based on a very simple concept use of fourier analysis to continue smooth portions of a piecewise smooth function into new functions which defined on larger domains are both smooth and periodic the continuation functions arising from a function f converge super algebraically to f in its domain of definition as discretizations are refined we demonstrate the capabilities of the proposed approach for a number of surfaces of engineering relevance

universality of general beta ensembles we prove the universality of the ensembles with convex analytic potentials and for any 0 i e we show that the spacing distributions of log gases at any inverse temperature coincide with those of the gaussian ensembles

fractional brownian motion with hurst index h 0 and the gaussian unitary ensemble the goal of this paper is to establish a relation between characteristic polynomials of n n gue random matrices h as n and gaussian processes with logarithmic correlations we introduce a regularized version of fractional brownian motion with zero hurst index which is a gaussian process with stationary increments and logarithmic increment structure then we prove that this process appears as a limit of d n z log det h zi on mesoscopic scales as n by employing a fourier integral representation we use this to prove a continuous analogue of a result by diaconis and shahshahani j appl probab 31a 1994 49 62 on the macroscopic scale d n x gives rise to yet another type of gaussian process with logarithmic correlations we give an explicit construction of the latter in terms of a chebyshev fourier random series

random words toeplitz determinants and integrable systems ii this paper connects the analysis of the length of the longest weakly increasing subsequence of inhomogeneous random words to a riemnn hilbert problem and an associated system of integrable partial differential equations in particular we show that the poissonization of the distribution function of this length can be identified as the jimbo miwa ueno tau function

on the resolution power of fourier extensions for oscillatory functions functions that are smooth but non periodic on a certain interval possess fourier series that lack uniform convergence and suffer from the gibbs phenomenon however they can be represented accurately by a fourier series that is periodic on a larger interval this is commonly called a fourier extension when constructed in a particular manner fourier extensions share many of the same features of a standard fourier series in particular one can compute fourier extensions which converge spectrally fast whenever the function is smooth and geometrically fast if the function is analytic much the same as the fourier series of a smooth analytic and periodic function with this in mind the purpose of this paper is to describe analyse and explain the observation that fourier extensions much like classical fourier series also have excellent resolution properties for representing oscillatory functions the resolution power or required number of degrees of freedom per wavelength depends on a user controlled parameter and as we show it varies between 2 and p the former value is optimal and is achieved by classical fourier series for periodic functions for example the latter value is the resolution power of algebraic polynomial approximations thus fourier extensions with an appropriate choice of parameter are eminently suitable for problems with moderate to high degrees of oscillation

critical edge behavior in the modified jacobi ensemble and the painleve v transcendents we study the jacobi unitary ensemble defined by the perturbed chebyshev weight w x 1 x2 12 tn2 x2 x 1 1 tn 1 the algebraic singularity coalesces with the end point of the support of the weight function namely tn 1 the hard edge as n we use the riemann hilbert approach the method of deift and zhou to obtain the asymptotics of the polynomials orthogonal with respect to w x we show that the universality property is preserved the asymptotic behavior of the eigenvalue correlations in the bulk of the spectrum is described in terms of the sine kernel the main result is on the local behavior at the edge of the spectrum the limit kernel at the edge when tn 1 o n 2 is described by the function for a specific solution of the painleve v equation we also show that when tn varies to a fixed t 1 the painleve v kernel transits to the bessel kernel j 12 on the other hand when tn approaches 1 the painleve v kernel can be approximated by another bessel kernel j 12

on the nonlinear wave equation with the mixed nonhomogeneous conditions linear approximation and asymptotic expansion of solutions in this paper we consider the following nonlinear wave equation equation 1 utt x u ux f x t u ux ut 0 x 1 0 t t ux 0 t g t u 1 t 0 u x 0 u 0 x ut x 0 u 1 x where u 0 u 1 ff gg are given functions to problem 1 we associate a linear recursive scheme for which the existence of a local and unique weak solution is proved by applying the faedo galerkin method and the weak compact method in the case of cn 2 r cn 2 r 1 cn 1 r 1 cn 1 r z 0 0 z 0 0 1 z 0 1 z 0 for all z rz r and g c3 r g c3 r f cn 1 0 1 r r3 f cn 1 0 1 r r3 f1 cn 0 1 r r3 f1 cn 0 1 r r3 a weak solution ue1 e2 x t ue1 e2 x t having an asymptotic expansion of order n 1n 1 in two small parameters e1e1 e2e2 is established for the following equation associated to 1 2 3 equation 2 utt x u e1 1 u ux f x t u ux ut e2f1 x t u ux ut

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

existence and comparison results for operator and differential equations in abstract spaces in this paper we apply the chain methods developed in heikkila and lakshmikantham 1994 to obtain new fixed point theorems and new existence and comparison results for operator equations in partially ordered sets these results are then applied to discontinuous implicit functional differential equations in ordered banach spaces

existence results for operator and differential equations and inclusions abstract this paper contains a brief summary of existence results for operator and differential equations and inclusions derived recently via a recursion method and generalized iteration methods introduced in 11 12 23

bounds for polynomials with a unit discrete norm let e be the set of n equidistant points in 1 1 and fn e be the set of all polynomials p of degree n with max p c c e 1 we prove that 7t kn n x max p x clog p e arctan ky s x r v 1 n2 jv2 where n n and c is an absolute constant the result is essentially sharp bounds for kn z z g c uniform for n n are also obtained the method of proof of those results is a general one it allows one to obtain sharp or sharp up to a log n factor bounds for kn under rather general assumptions on e j5 n a model result is announced for a class of sets e main components of the method are discussed in some detail in the process of investigating the case of equally spaced points

on the solvability of painleve ii and iv we introduce a rigorous methodology for studying the riemann hilbert problems associated with certain integrable nonlinear ordinary differential equations for concreteness we investigate the painleve ii and painleve iv equations we show that the cauchy problems for these equations admit in general global meromorphic in t solutions furthermore for special relations among the monodromy data and for t on stokes lines these solutions are bounded for finite t

asymptotics for toeplitz determinants perturbation of symbols with a gap we study the determinants of toeplitz matrices as the size of the matrices tends to infinity in the particular case where the symbol has two jump discontinuities and tends to zero on an arc of the unit circle at a sufficiently fast rate we generalize an asymptotic expansion by widom indiana univ math j 21 277 283 1971 which was known for symbols supported on an arc we highlight applications of our results in the circular unitary ensemble and in the study of fredholm determinants associated to the sine kernel

painleve ii asymptotics near the leading edge of the oscillatory zone for the korteweg de vries equation in the small dispersion limit in the small dispersion limit solutions to the korteweg de vries equation develop an interval of fast oscillations after a certain time we obtain a universal asymptotic expansion for the korteweg de vries solution near the leading edge of the oscillatory zone up to second order corrections this expansion involves the hastings mcleod solution of the painleve ii equation we prove our results using the riemann hilbert approach c 2009 wiley periodicals inc

the fourth painleve equation and associated special polynomials in this article rational solutions and associated polynomials for the fourth painleve equation are studied these rational solutions of the fourth painleve equation are expressible as the logarithmic derivative of special polynomials the okamoto polynomials the structure of the roots of these okamoto polynomials is studied and it is shown that these have a highly regular structure the properties of the okamoto polynomials are compared and contrasted with those of classical orthogonal polynomials further representations are given of the associated rational solutions in the form of determinants through schur functions

on fixed points through iteratively generated chains with applications to differential equations on utilise les iterations pour construire des chaines iterativement generees en utilisant ces chaines on peut demontrer des resultats d existence pour des points fixes maximaux et minimaux d operateurs croissant sur des ensembles ordonnes

emergence of a singularity for toeplitz determinants and painleve v we obtain asymptotic expansions for toeplitz determinants corresponding to a family of symbols depending on a parameter t for t positive the symbols are regular so that the determinants obey szeg s strong limit theorem if t 0 the symbol possesses a fisher hartwig singularity letting t 0 we analyze the emergence of a fisher hartwig singularity and a transition between the two different types of asymptotic behavior for toeplitz determinants this transition is described by a special painleve v transcendent a particular case of our result complements the classical description of wu mccoy tracy and barouch of the behavior of a 2 spin correlation function for a large distance between spins in the two dimensional ising model as the phase transition occurs

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

on the fourier extension of nonperiodic functions we obtain exponentially accurate fourier series for nonperiodic functions on the interval 1 1 by extending these functions to periodic functions on a larger domain the series may be evaluated but not constructed by means of the fft a complete convergence theory is given based on orthogonal polynomials that resemble chebyshev polynomials of the first and second kinds we analyze a previously proposed numerical method which is unstable in theory but stable in practice we propose a new numerical method that is stable both in theory and in practice

on generalized methods of successive approximations abstract there are several generalized methods of monotone successive approximations in ordered sets a which generalize the usual sequences of the iterative of a point a of a under a selfmapping f on a to some well ordered subsets of a these are smallest admissible set containing a by buber and kirk the set w a by abian and brown well ordered chain of f iterations of a by heikkila and smallest complete f chain containing a by fuchssteiner the well ordered chains of f iterations were compared by heikkila in 1999 with the admissible sets we compare here all of the above four methods our study is based on a notion of a generalized orbit which is a generalization of the one which was introduced by the author in 1988 as an answer to a problem posed by banaschewski in 1992 we use in our proofs a minimum of tools in particular we assume that the relation is only asymmetric we do not use the transitivity postulated for orderings and use only extensionality pairing union and separation axioms of zermelo s set theory