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

mesh refinement processes based on the generalized bisection of simplices mesh refinement processes based on the generalized bisection of simplices are discussed and characterized in the context of the general theory of rheinboldt siam j numer anal 17 1980 pp 766 778 then it is proved that such processes allow the construction of sequences of naturally smooth conforming and nested nonuniform meshes in fact it is shown that there exists a generalized mesh refinement operator that for any conforming triangulation delta and for any v subset delta produces a nested smooth conforming triangulation delta containing successors of all elements of v and such that the minimum cell size of delta is bounded from below by half the minimum cell size of delta two conforming mesh refinement algorithms that allow the selective refinement of computational triangulations are explicitly given

vicious walks with a wall noncolliding meanders and chiral and bogoliubov de gennes random matrices spatially and temporally inhomogeneous evolution of one dimensional vicious walkers with wall restriction is studied we show that its continuum version is equivalent with a noncolliding system of stochastic processes called brownian meanders here the brownian meander is a temporally inhomogeneous process introduced by yor as a transform of the bessel process that is the motion of radial coordinate of the three dimensional brownian motion represented in spherical coordinates it is proved that the spatial distribution of vicious walkers with a wall at the origin can be described by the eigenvalue statistics of gaussian ensembles of bogoliubov de gennes hamiltonians of the mean field theory of superconductivity which have a particle hole symmetry we report that a time evolution of the present stochastic process is fully characterized by the change of symmetry classes from type c to type ci in the nonstandard classes of random matrix theory of altland and zirnbauer the relation between the noncolliding systems of the generalized meanders of yor which are associated with the even dimensional bessel processes and the chiral random matrix theory is also clarified

topology optimization using polytopes abstract meshing complex engineering domains is a challenging task arbitrary polyhedral meshes can provide the much needed flexibility in automated discretization of such domains the geometric property of polyhedral meshes such as its unstructured nature and the connectivity of faces between elements makes them specially attractive for topology optimization applications numerical anomalies in designs such as the single node connections and checkerboard pattern can be naturally circumvented with polyhedrons in the current work we solve the governing three dimensional elasticity state equation using the virtual element method vem approach the main characteristic difference between vem and standard finite element methods fem is that in vem the canonical basis functions are not constructed explicitly rather the stiffness matrix is computed directly utilizing a projection map which extracts the linear component of the deformation such a construction guarantees the satisfaction of the patch test used by engineers as an indicator of optimal convergence of numerical solutions under mesh refinement finally the computations reduce to the evaluation of matrices which contain purely geometric surface facet quantities the present work focuses on the first order vem in which the degrees of freedom are associated with the vertices the features of the current optimization approach are demonstrated using numerical examples for compliance minimization and compliant mechanism problems

noncolliding brownian motion with drift and time dependent stieltjes wigert determinantal point process using the determinantal formula of biane bougerol and o connell we give multitime joint probability densities to the noncolliding brownian motion with drift where the number of particles is finite we study a special case such that the initial positions of particles are equidistant with a period a and the values of drift coefficients are well ordered with a scale we show that at each time t 0 the single time probability density of particle system is exactly transformed to the biorthogonal stieltjes wigert matrix model in the chern simons theory introduced by dolivet and tierz here one parameter extensions extensions of the stieltjes wigert polynomials which are themselves q extensions of the hermite polynomials play an essential role the two parameters a and of the process combined with time t are mapped to the parameters q and of the biorthogonal polynomials by the transformation of normalization factor of our probability density the partition function of the chern simons matrix model is readily obtained we study the determinantal structure of the matrix model and prove that at each time t 0 the present noncolliding brownian motion with drift is a determinantal point process in the sense that any correlation function is given by a determinant governed by a single integral kernel called the correlation kernel using the obtained correlation kernel we study time evolution of the noncolliding brownian motion with drift

a superconvergent ldg hybridizable galerkin method for second order elliptic problems we identify and study an ldg hybridizable galerkin method which is not an ldg method for second order elliptic problems in several space dimensions with remarkable convergence properties unlike all other known discontinuous galerkin methods using polynomials of degree k 0 for both the potential as well as the flux the order of convergence in l 2 of both unknowns is k 1 moreover both the approximate potential as well as its numerical trace superconverge in l 2 like norms to suitably chosen projections of the potential with order k 2 this allows the application of element by element postprocessing of the approximate solution which provides an approximation of the potential converging with order k 2 in l 2 the method can be thought to be in between the hybridized version of the raviart thomas and that of the brezzi douglas marini mixed methods

correlation functions cluster functions and spacing distributions for random matrices the usual formulas for the correlation functions in orthogonal and symplectic matrix models express them as quaternion determinants from this representation one can deduce formulas for spacing probabilities in terms of fredholm determinants of matrix valued kernels the derivations of the various formulas are somewhat involved in this article we present a direct approach which leads immediately to scalar kernels for the unitary ensembles and matrix kernels for the orthogonal and symplectic ensembles and the representations of the correlation functions cluster functions and spacing distributions in terms of them

conformal welding problem flow line problem and multiple schramm loewner evolution a quantum surface qs is an equivalence class of pairs d h of simply connected domains d c and random distributions h on d induced by the conformal equivalence for random metric spaces this distribution valued random field is extended to a qs with n 1 marked boundary points mbps with n z 0 we propose the conformal welding problem for it in the case of n z 1 if n 1 it is reduced to the problem introduced by sheffield who solved it by coupling the qs with the schramm loewner evolution sle when n 3 there naturally appears room for making the configuration of mbps random and hence a new problem arises how to determine the probability law of the configuration we report that the multiple sle in h driven by the dyson model on r helps us to fix the problems and makes them solvable for any n 3 we also propose the flow line problem for an imaginary surface with boundary condition changing points bccps in the case when the number of bccps is two this problem was solved by miller and sheffield we address the general case with an arbitrary number of bccps in a similar manner to the conformal welding problem we again find that the multiple sle driven by the dyson model plays a key role to solve the flow line problem

unified hybridization of discontinuous galerkin mixed and continuous galerkin methods for second order elliptic problems we introduce a unifying framework for hybridization of finite element methods for second order elliptic problems the methods fitting in the framework are a general class of mixed dual finite element methods including hybridized mixed continuous galerkin nonconforming and a new wide class of hybridizable discontinuous galerkin methods the distinctive feature of the methods in this framework is that the only globally coupled degrees of freedom are those of an approximation of the solution defined only on the boundaries of the elements since the associated matrix is sparse symmetric and positive definite these methods can be efficiently implemented moreover the framework allows in a single implementation the use of different methods in different elements or subdomains of the computational domain which are then automatically coupled finally the framework brings about a new point of view thanks to which it is possible to see how to devise novel methods displaying very localized and simple mortaring techniques as well as methods permitting an even further reduction of the number of globally coupled degrees of freedom

riemannian symmetric superspaces and their origin in random matrix theory gaussian random matrix ensembles defined over the tangent spaces of the large families of cartan s symmetric spaces are considered such ensembles play a central role in mesoscopic physics as they describe the universal ergodic limit of disordered and chaotic single particle systems the generating function for the spectral correlations of each ensemble is reduced to an integral over a riemannian symmetric superspace in the limit of large matrix dimension such a space is defined as a pair g h m r where g h is a complex analytic graded manifold homogeneous with respect to the action of a complex lie supergroup g and m r is a maximal riemannian submanifold of the support of g h

convergence analysis of the high order mimetic finite difference method we prove second order convergence of the conservative variable and its flux in the high order mfd method the convergence results are proved for unstructured polyhedral meshes and full tensor diffusion coefficients for the case of non constant coefficients we also develop a new family of high order mfd methods theoretical result are confirmed through numerical experiments

a locking free weak galerkin finite element method for elasticity problems in the primal formulation this paper presents an arbitrary order locking free numerical scheme for linear elasticity on general polygonal polyhedral partitions by using weak galerkin wg finite element methods like other wg methods the key idea for the linear elasticity is to introduce discrete weak strain and stress tensors which are defined and computed by solving inexpensive local problems on each element such local problems are derived from weak formulations of the corresponding differential operators through integration by parts locking free error estimates of optimal order are derived in a discrete h 1 norm and the usual l 2 norm for the approximate displacement when the exact solution is smooth numerical results are presented to demonstrate the efficiency accuracy and the locking free property of the weak galerkin finite element method

strong asymptotics of orthogonal polynomials with respect to exponential weights we consider asymptotics of orthogonal polynomials with respect to weights w x dx e q x dx on the real line where q x 2m k 0 qkx k q2m 0 denotes a polynomial of even order with positive leading coefficient the orthogonal polynomial problem is formulated as a riemann hilbert problem following 22 23 we employ the steepest descent type method introduced in 18 and further developed in 17 19 in order to obtain uniform plancherel rotach type asymptotics in the entire complex plane as well as asymptotic formulae for the zeros the leading coefficients and the recurrence coefficients of the orthogonal polynomials c 1999 john wiley sons inc

continuous and discrete painleve equations arising from the gap probability distribution of the finite n gaussian unitary ensembles in this paper we study the gap probability problem in the gaussian unitary ensembles of n by n matrices the probability that the interval j a a is free of eigenvalues in the works of tracy and widom adler and van moerbeke and forrester and witte on this subject it has been shown that two painleve type differential equations arise in this context the first is the jimbo miwa okomoto sigma form and the second is a particular painleve iv using the ladder operator technique of orthogonal polynomials we derive three quantities associated with the gap probability denoted as sigma n a r n a and r n a we show that each one satisfies a second order painleve type differential equation as well as a discrete painleve type equation in particular in addition to providing an elementary derivation of the aforementioned sigma form and painleve iv we are able to show that the quantity r n a satisfies a particular case of chazy s second degree second order differential equation for the discrete equations we show that the quantity r n a satisfies a particular form of the modified discrete painleve ii equation obtained by grammaticos and ramani in the context of backlund transformations we also derive second order second degree difference equations for the quantities r n a and sigma n a

brownian motions of ellipsoids on donne un traitement elementaire des mouvements browniens d ellipsoides et en particulier des resultats remarquables obtenus par dynkin

phase transitions and equilibrium measures in random matrix models the paper is devoted to a study of phase transitions in the hermitian random matrix models with a polynomial potential in an alternative equivalent language we study families of equilibrium measures on the real line in a polynomial external field the total mass of the measure is considered as the main parameter which may be interpreted also either as temperature or time our main tools are differentiation formulas with respect to the parameters of the problem and a representation of the equilibrium potential in terms of a hyperelliptic integral using this combination we introduce and investigate a dynamical system system of odes describing the evolution of families of equilibrium measures on this basis we are able to systematically derive a number of new results on phase transitions such as the local behavior of the system at all kinds of phase transitions as well as to review a number of known ones

universal shocks in the wishart random matrix ensemble we show that the derivative of the logarithm of the average characteristic polynomial of a diffusing wishart matrix obeys an exact partial differential equation valid for an arbitrary value of n the size of the matrix in the large n limit this equation generalizes the simple inviscid burgers equation that has been obtained earlier for hermitian or unitary matrices the solution through the method of characteristics presents singularities that we relate to the precursors of shock formation in the burgers equation the finite n effects appear as a viscosity term in the burgers equation using a scaling analysis of the complete equation for the characteristic polynomial in the vicinity of the shocks we recover in a simple way the universal bessel oscillations so called hard edge singularities familiar in random matrix theory

a new weak galerkin finite element method for elliptic interface problems a new weak galerkin wg finite element method is introduced and analyzed in this paper for solving second order elliptic equations with discontinuous coefficients and interfaces comparing with the existing wg algorithm for solving the same type problems the present wg method has a simpler variational formulation and fewer unknowns moreover the new wg algorithm allows the use of finite element partitions consisting of general polytopal meshes and can be easily generalized to high orders optimal order error estimates in both h 1 and l 2 norms are established for the present wg finite element solutions extensive numerical experiments have been conducted to examine the accuracy flexibility and robustness of the proposed wg interface approach in solving regular elliptic interface problems high order convergences are numerically confirmed by using piecewise polynomial basis functions of high degrees moreover the wg method is shown to be able to accommodate very complicated interfaces due to its flexibility in choosing finite element partitions finally in dealing with challenging problems with low regularities the piecewise linear wg method is capable of delivering a second order of accuracy in l norm for both c 1 and h 2 continuous solutions

residual based error estimate and quasi interpolation on polygonal meshes for high order bem based fem abstract only a few numerical methods can treat boundary value problems on polygonal and polyhedral meshes the bem based finite element method is one of the new discretization strategies which make use of and benefits from the flexibility of these general meshes that incorporate hanging nodes naturally the article in hand addresses quasi interpolation operators for the approximation space over polygonal meshes to prove interpolation estimates the poincare constant is bounded uniformly for patches of star shaped elements these results give rise to the residual based error estimate for high order bem based fem and its reliability as well as its efficiency are proven such a posteriori error estimates can be used to gauge the approximation quality and to implement adaptive fem strategies numerical experiments show optimal rates of convergence for meshes with non convex elements on uniformly as well as on adaptively refined meshes

discontinuous galerkin finite element methods for interface problems a priori and a posteriori error estimations discontinuous galerkin dg finite element methods were studied by many researchers for second order elliptic partial differential equations and a priori error estimates were established when the solution of the underlying problem is piecewise h 3 2 epsilon smooth with epsilon 0 however elliptic interface problems with intersecting interfaces do not possess such a smoothness in this paper we establish a quasi optimal a priori error estimate for interface problems whose solutions are only h 1 alpha smooth with alpha in 0 1 and hence fill a theoretical gap of the dg method for elliptic problems with low regularity the second part of the paper deals with the design and analysis of robust residual and recovery based a posteriori error estimators theoretically we show that the residual and recovery estimators studied in this paper are robust with respect to the dg norm i e their reliability and efficiency bounds do not depend on the jump provided that the distribution of coefficients is locally quasi monotone

a posteriori error estimates stopping criteria and adaptivity for multiphase compositional darcy flows in porous media in this paper we derive a posteriori error estimates for the compositional model of multiphase darcy flow in porous media consisting of a system of strongly coupled nonlinear unsteady partial differential and algebraic equations we show how to control the dual norm of the residual augmented by a nonconformity evaluation term by fully computable estimators we then decompose the estimators into the space time linearization and algebraic error components this allows to formulate criteria for stopping the iterative algebraic solver and the iterative linearization solver when the corresponding error components do not affect significantly the overall error moreover the spatial and temporal error components can be balanced by time step and space mesh adaptation our analysis applies to a broad class of standard numerical methods and is independent of the linearization and of the iterative algebraic solvers employed we exemplify it for the two point finite volume method with fully implicit euler time stepping the newton linearization and the gmres algebraic solver numerical results on two real life reservoir engineering examples confirm that significant computational gains can be achieved thanks to our adaptive stopping criteria already on fixed meshes without any noticeable loss of precision

spatial correlations of the 1d kpz surface on a flat substrate we study the spatial correlations of the one dimensional kpz surface for the flat initial condition it is shown that the multi point joint distribution for the height is given by a fredholm determinant with its kernel in the scaling limit explicitly obtained this may also describe the dynamics of the largest eigenvalue in the goe dyson s brownian motion model our analysis is based on a reformulation of the determinantal green s function for the totally asep in terms of a vicious walk problem

handbook of brownian motion facts and formulae i theory i stochastic processes in general ii linear diffusions iii stochastic calculus iv brownian motion v local time as a markov process vi differential systems associated to brownian motion appendix 1 briefly on some diffusions ii tables of distributions of functionals of brownian motion and related processes 1 brownian motion 2 brownian motion with drift 3 reflecting brownian motion 4 bessel process of order 5 bessel process of order 1 2 6 bessel process of order zero 7 ornstein uhlenbeck process 8 radial ornstein uhlenbeck process 9 geometric brownian motion appendix 2 special functions appendix 3 inverse laplace transforms appendix 4 differential equations and their solutions appendix 5 formulae for n fold differentiation

a priori and a posterior error estimate of new weak galerkin finite element methods for second order elliptic interface problems on polygonal meshes abstract in this paper we present a posteriori error estimate of a new weak galerkin wg finite element method for the second order elliptic interface problems instead of introducing the local weak derivatives on each element the bilinear form is computed through the inner product of traditional defined derivatives the proposed numerical scheme is symmetric and positive definite and can be applied to polygonal meshes in the a posteriori error estimate we construct a error indicator that can be applied to polygonal meshes or meshes with hanging nodes the reliability and efficiency of the designed error estimator have been proved in this work extensive numerical tests are performed to validate our algorithm these results demonstrate the effectiveness of the adaptive mesh refinement guided by the proposed error estimator

discontinuous hp finite element methods for advection diffusion reaction problems we consider the hp version of the discontinuous galerkin finite element method dgfem for second order partial differential equations with nonnegative characteristic form this class of equations includes second order elliptic and parabolic equations advection reaction equations as well as problems of mixed hyperbolic elliptic parabolic type our main concern is the error analysis of the method in the absence of streamline diffusion stabilization in the hyperbolic case an hp optimal error bound is derived here we consider only advection reaction problems which satisfy a certain standard positivity condition in the self adjoint elliptic case an error bound that is h optimal and p suboptimal by frac 1 2 a power of p is obtained these estimates are then combined to deduce an error bound in the general case for elementwise analytic solutions the method exhibits exponential rates of convergence under p refinement the theoretical results are illustrated by numerical experiments

symmetric polynomials generalized jacobi trudi identities and τ functions an element grnh f of the grassmannian of n dimensional subspaces of the hardy space h h2 extended over the field f c x1 xn may be associated to any polynomial basis 0 1 for c x the plucker coordinates s n x1 xn of labeled by partitions provide an analog of jacobi s bi alternant formula defining a generalization of schur polynomials applying the recursion relations satisfied by the polynomial system to the analog hi 0 of the complete symmetric functions generates a doubly infinite matrix hi j of symmetric polynomials that determine an element h grn h f this is shown to coincide with implying a set of generalized jacobi identities extending a result obtained by sergeev and veselov moscow math j 14 161 168 2014 for the case of orthogonal polynomials the symmetric polynomials s n x1 xn are shown to be kp kadomtsev petviashvili functions in terms of the power sums x of the xa s viewed as kp flow variables a fermionic operator representation is derived for these as well as for the infinite sums s n x s n t associated to any pair of polynomial bases which are shown to be 2d toda lattice functions a number of applications are given including classical group character expansions matrix model partition functions and generators for random processes an element grnh f of the grassmannian of n dimensional subspaces of the hardy space h h2 extended over the field f c x1 xn may be associated to any polynomial basis 0 1 for c x the plucker coordinates s n x1 xn of labeled by partitions provide an analog of jacobi s bi alternant formula defining a generalization of schur polynomials applying the recursion relations satisfied by the polynomial system to the analog hi 0 of the complete symmetric functions generates a doubly infinite matrix hi j of symmetric polynomials that determine an element h grn h f this is shown to coincide with implying a set of generalized jacobi identities extending a result obtained by sergeev and veselov moscow math j 14 161 168 2014 for the case of orthogonal polynomials the symmetric polynomials s n x1 xn are shown to be kp kadomtsev petviashvili functions in terms of the power sums x of the xa s viewed as kp flow variables a fermionic operator re

rank 1 real wishart spiked model in this paper we consider n dimensional real wishart matrices y in the class input amssym w bbb r sigma m in which all but one eigenvalue of is 1 let the nontrivial eigenvalue of be 1 then as n m with m n 2 finite and nonzero the eigenvalue distribution of y will converge into the marchenko pastur distribution inside a bulk region when increases from 0 one starts to see a stray eigenvalue of y outside of the support of the marchenko pastur density as this stray eigenvalue leaves the bulk region a phase transition will occur in the largest eigenvalue distribution of the wishart matrix in this paper we will compute the asymptotics of the largest eigenvalue distribution when the phase transition occurs we will first establish the results that are valid for all n and m and will use them to carry out the asymptotic analysis in particular we have derived a contour integral formula for the harish chandra itzykson zuber integral when x and y are real symmetric and y is a rank 1 matrix this allows us to write down a fredholm determinant formula for the largest eigenvalue distribution and analyze it using orthogonal polynomial techniques as a result we obtain an integral formula for the largest eigenvalue distribution in the large n limit characterized by painleve transcendents the approach used in this paper is very different from a recent paper by bloemenal and virag in which the largest eigenvalue distribution was obtained using a stochastic operator method in particular the painleve formula for the largest eigenvalue distribution obtained in this paper is new 2012 wiley periodicals inc

janossy densities i determinantal ensembles we derive an elementary formula for janossy densities for determinantal point processes with a finite rank projection type kernel in particular for 2 polynomial ensembles of random matrices we show that the janossy densities on an interval i can be expressed in terms of the christoffel darboux kernel for the orthogonal polynomials on the complement of i

a weak galerkin finite element scheme for solving the stationary stokes equations a weak galerkin wg finite element method for solving the stationary stokes equations in two or three dimensional spaces by using discontinuous piecewise polynomials is developed and analyzed the variational form we considered is based on two gradient operators which is different from the usual gradient divergence operators the wg method is highly flexible by allowing the use of discontinuous functions on arbitrary polygons or polyhedra with certain shape regularity optimal order error estimates are established for the corresponding wg finite element solutions in various norms numerical results are presented to illustrate the theoretical analysis of the new wg finite element scheme for stokes problems

maximum distributions of bridges of noncolliding brownian paths one dimensional brownian motion starting from the origin at time t 0 conditioned to return to the origin at time t 1 and to stay positive during time interval 0 t 1 is called the bessel bridge with duration 1 we consider an n particle system of such bessel bridges conditioned never to collide with each other in 0 t 1 which is the continuum limit of the vicious walk model in watermelon configuration with a wall distributions of maximum values of paths attained in the time interval t epsilon 0 1 are studied to characterize the statistics of random patterns of the repulsive paths on the spatiotemporal plane for the outermost path the distribution function of maximum value is exactly determined for general n we show that the present n path system of noncolliding bessel bridges is realized as the positive eigenvalue process of the 2nx2n matrix valued brownian bridge in the symmetry class c using this fact computer simulations are performed and numerical results on the n dependence of the maximum value distributions of the inner paths are reported the present work demonstrates that the extreme value problems of noncolliding paths are related to random matrix theory the representation theory of symmetry and number theory

on semiclassical linear functionals integral representations abstract if is a functional of moments and satisfies the equation d o where x and x are arbitrary polynomials but with the only condition deg 1 then is said to be a semiclassical functional when is semiclassical and regular its corresponding orthogonal polynomial sequence ops pn x n 0 is called a semiclassical ops in this paper integral representations are given for semiclassical functionals regular or not in cases a deg deg and b deg deg but in the latter case when deg 0 the fundamental result is the following if n with n 0 are the moments of an a functional then n cm n this allows us to find integral representations for a functionals using standard techniques taken from pollaczek 1951

the nonconforming virtual element method for the stokes equations we present the nonconforming virtual element method vem for the numerical approximation of velocity and pressure in the steady stokes problem the pressure is approximated using discontinuous piecewise polynomials while each component of the velocity is approximated using the nonconforming virtual element space on each mesh element the local virtual space contains the space of polynomials of up to a given degree plus suitable nonpolynomial functions the virtual element functions are implicitly defined as the solution of local poisson problems with polynomial neumann boundary conditions as typical in vem approaches the explicit evaluation of the non polynomial functions is not required this approach makes it possible to construct nonconforming virtual spaces for any polynomial degree regardless of the parity for two and three dimensional problems and for meshes with very general polygonal and polyhedral elements we show that the nonconforming vem is inf sup stable and establish optimal a pri

fluctuations of the one dimensional polynuclear growth model in half space we consider the multi point equal time height fluctuations of the one dimensional polynuclear growth model in half space for special values of the nucleation rate at the origin the multi layer version of the model is reduced to a process with a determinantal weight for which the asymptotics can be analyzed in the scaling limit the fluctuations near the origin are shown to be equivalent to those of the largest eigenvalue of the orthogonal symplectic to unitary transition ensemble at soft edge in random matrix theory

interacting particles on the line and dunkl intertwining operator of type a application to the freezing regime we consider a one dimensional system of brownian particles that repel each other through a logarithmic potential we study two formulations for the system and the relation between them the first dyson s brownian motion model has an interaction coupling constant determined by the parameter 0 when 1 2 and 4 this model can be regarded as a stochastic realization of the eigenvalue statistics of gaussian random matrices the second system comes from dunkl processes which are defined using differential difference operators dunkl operators associated with finite abstract vector sets called root systems when the type a root system is specified dunkl processes constitute a one parameter system similar to dyson s model with the difference that its particles interchange positions spontaneously we prove that the type a dunkl processes with parameter k 0 starting from any symmetric initial configuration are equivalent to dyson s model with the parameter 2k we focus on the intertwining operators since they play a central role in the mathematical theory of dunkl operators but their general closed form is not yet known using the equivalence between symmetric dunkl processes and dyson s model we extract the effect of the intertwining operator of type a on symmetric polynomials from these processes transition probability densities in the strong coupling limit the intertwining operator maps all symmetric polynomials onto a function of the sum of their variables in this limit dyson s model freezes and it becomes a deterministic process with a final configuration proportional to the roots of the hermite polynomials multiplied by the square root of the process time while being independent of the initial configuration

a discontinuous skeletal method for advection diffusion reaction on general meshes we design and analyze an approximation method for advection diffusion reaction equa tions where the generalized degrees of freedom are polynomials of order k 0 at mesh faces the method hinges on local discrete reconstruction operators for the diffusive and advective derivatives and a weak enforcement of boundary conditions fairly general meshes with poly topal and nonmatching cells are supported arbitrary polynomial orders can be considered including the case k 0 which is closely related to mimetic finite difference mixed hybrid finite volume methods the error analysis covers the full range of peclet numbers including the delicate case of local degeneracy where diffusion vanishes on a strict subset of the domain computational costs remain moderate since the use of face unknowns leads to a compact stencil with reduced communications numerical results are presented

asymptotics for the partition function in two cut random matrix models we obtain large n asymptotics for the random matrix partition function z n v int mathbb r n prod i j x i x j 2 prod j 1 ne nv x j dx j in the case where v is a polynomial such that the random matrix eigenvalues accumulate on two disjoint intervals the two cut case we compute leading and sub leading terms in the asymptotic expansion for log zn v up to terms that are small as n to infty our approach is based on the explicit computation of the first terms in the asymptotic expansion for a quartic symmetric potential v afterwards we use deformation theory of the partition function and of the associated equilibrium measure to generalize our results to general two cut potentials v the asymptotic expansion of log zn v as n to infty contains terms that depend analytically on the potential v and that have already appeared in the literature in addition our method allows us to compute the v independent terms of the asymptotic expansion of log zn v which to the best of our knowledge had not appeared before in the literature we use rigorous orthogonal polynomial and riemann hilbert techniques which had to this point only been successful to compute asymptotics for the partition function in the one cut case

partition functions for matrix models and isomonodromic tau functions for one matrix models with polynomial potentials the explicit relationship between the partition function and the isomonodromic tau function for the 2 2 polynomial differential systems satisfied by the associated orthogonal polynomials is derived

large n limit for the product of two coupled random matrices for a pair of coupled rectangular random matrices we consider the squared singular values of their product which form a determinantal point process we show that the limiting mean distribution of these squared singular values is described by the second component of the solution to a vector equilibrium problem this vector equilibrium problem is defined for three measures with an upper constraint on the first measure and an external field on the second measure we carry out the steepest descent analysis for a 4 times 4 matrix valued riemann hilbert problem which characterizes the correlation kernel and is related to mixed type multiple orthogonal polynomials associated with the modified bessel functions a careful study of the vector equilibrium problem combined with this asymptotic analysis ultimately leads to the aforementioned convergence result for the limiting mean distribution an explicit form of the associated spectral curve as well as local sine meijer g and airy universality results for the squared singular values considered

symmetry of matrix valued stochastic processes and noncolliding diffusion particle systems as an extension of the theory of dyson s brownian motion models for the standard gaussian random matrix ensembles we report a systematic study of hermitian matrix valued processes and their eigenvalue processes associated with the chiral and nonstandard random matrix ensembles in addition to the noncolliding brownian motions we introduce a one parameter family of temporally homogeneous noncolliding systems of the bessel processes and a two parameter family of temporally inhomogeneous noncolliding systems of yor s generalized meanders and show that all of the ten classes of eigenvalue statistics in the altland zirnbauer classification are realized as particle distributions in the special cases of these diffusion particle systems as a corollary of each equivalence in distribution of a temporally inhomogeneous eigenvalue process and a noncolliding diffusion process a stochastic calculus proof of a version of the harish chandra itzykson zuber formula of integral over unitary group is established

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

maximum of n independent brownian walkers till the first exit from the half space we consider the one dimensional target search process that involves an immobile target located at the origin and n searchers performing independent brownian motions starting at the initial positions all on the positive half space the process stops when the target is first found by one of the searchers we compute the probability distribution of the maximum distance m visited by the searchers till the stopping time and show that it has a power law tail for large m thus all moments of m up to the order n 1 are finite while the higher moments diverge the prefactor bn increases with n faster than exponentially our solution gives the exit probability of a set of n particles from a box 0 l through the left boundary incidentally it also provides an exact solution of laplace s equation in an n dimensional hypercube with some prescribed boundary conditions the analytical results are in excellent agreement with monte carlo simulations

character expansions for the orthogonal and symplectic groups formulas for the expansion of arbitrary invariant group functions in terms of the characters for the sp 2n so 2n 1 and so 2n groups are derived using a combinatorial method the method is similar to one used by balantekin to expand group functions over the characters of the u n group all three expansions have been checked for all n by using them to calculate the known expansions of the generating function of the homogeneous symmetric functions an expansion of the exponential of the traces of group elements appearing in the finite volume gauge field partition functions is worked out for the orthogonal and symplectic groups

a hybrid high order locking free method for linear elasticity on general meshes abstract we devise an arbitrary order locking free method for linear elasticity the method relies on a pure displacement primal formulation and leads to a symmetric positive definite system matrix with compact stencil the degrees of freedom are vector valued polynomials of arbitrary order k 1 on the mesh faces so that in three space dimensions the lowest order scheme only requires 9 degrees of freedom per mesh face the method can be deployed on general polyhedral meshes the key idea is to reconstruct the symmetric gradient and divergence inside each mesh cell in terms of the degrees of freedom by solving inexpensive local problems the discrete problem is assembled cell wise using these operators and a high order stabilization bilinear form locking free error estimates are derived for the energy norm and for the l 2 norm of the displacement with optimal convergence rates of order k 1 and k 2 respectively for smooth solutions on general meshes the theoretical results are confirmed numerically and the cpu cost is evaluated on both standard and polygonal meshes

a posteriori error estimates for a discontinuous galerkin approximation of second order elliptic problems several a posteriori error estimators are introduced and analyzed for a discontinuous galerkin formulation of a model second order elliptic problem in addition to residual type estimators we introduce some estimators that are couched in the ideas and techniques of domain decomposition results of numerical experiments are presented

two bessel bridges conditioned never to collide double dirichlet series and jacobi theta function it is known that the moments of the maximum value of a one dimensional conditional brownian motion the three dimensional bessel bridge with duration 1 started from the origin are expressed using the riemann zeta function we consider a system of two bessel bridges in which noncolliding condition is imposed we show that the moments of the maximum value is then expressed using the double dirichlet series or using the integrals of products of the jacobi theta functions and its derivatives since the present system will be provided as a diffusion scaling limit of a version of vicious walker model the ensemble of 2 watermelons with a wall the dominant terms in long time asymptotics of moments of height of 2 watermelons are completely determined for the height of 2 watermelons with a wall the average value was recently studied by fulmek by a method of enumerative combinatorics

the dependence on the monodromy data of the isomonodromic tau function the isomonodromic tau function defined by jimbo miwa ueno vanishes on the malgrange s divisor of generalized monodromy data for which a vector bundle is nontrivial or which is the same a certain riemann hilbert problem has no solution in their original work jimbo miwa ueno provided an algebraic construction of its derivatives with respect to isomonodromic times however the dependence on the generalized monodromy data i e monodromy representation and stokes parameters was not derived we fill the gap by providing a simpler and more general description in which all the parameters of the problem monodromy changing and monodromy preserving are dealt with at the same level we thus provide variational formulae for the isomonodromic tau function with respect to the generalized monodromy data the construction applies more generally given any sufficiently well behaved family of riemann hilbert problems rhp where the jump matrices depend arbitrarily on deformation parameters we can construct a one form not necessarily closed on the deformation space malgrange s differential defined off malgrange s divisor we then introduce the notion of discrete schlesinger transformation it means that we allow the solution of the rhp to have poles or zeros at prescribed point s even if is not closed its difference evaluated along the original solution and the transformed one is shown to be the logarithmic differential on the deformation space of a function as a function of the position of the points of the schlesinger transformation it yields a natural generalization of the sato formula for the baker akhiezer vector even in the absence of a tau function and it realizes the solution of the rhp as such ba vector some exemplifications in the setting of the painleve ii equation and finite toplitz hankel determinants are provided

a weak galerkin mixed finite element method for second order elliptic problems abstract a new weak galerkin wg method is introduced and analyzed for the second order elliptic equation formulated as a system of two first order linear equations this method called wg mfem is designed by using discontinuous piecewise polynomials on finite element partitions with arbitrary shape of polygons polyhedra the wg mfem is capable of providing very accurate numerical approximations for both the primary and flux variables allowing the use of discontinuous approximating functions on arbitrary shape of polygons polyhedra makes the method highly flexible in practical computation optimal order error estimates in both discrete h and l norms are established for the corresponding weak galerkin mixed finite element solutions

multidimensional yamada watanabe theorem and its applications to particle systems we prove a multidimensional version of the yamada watanabe theorem i e a theorem giving conditions on coefficients of a stochastic differential equation for existence and pathwise uniqueness of strong solutions it implies an existence and uniqueness theorem for the eigenvalue and eigenvector processes of matrix valued stochastic processes called a spectral matrix yamada watanabe theorem the multidimensional yamada watanabe theorem is also applied to particle systems of squared bessel processes corresponding to matrix analogues of squared bessel processes wishart and jacobi matrix processes the versions of these particle systems are also considered

on some gaussian ensembles of hermitian matrices a gaussian ensemble of hermitian matrices with an arbitrary ratio of their symmetric and anti symmetric parts is studied similarly a gaussian ensemble of hermitian quaternion matrices with an arbitrary ratio of their self dual and anti self dual parts is studied analytical expressions for the correlation and cluster functions as well as their limits when the order of the matrices is large are derived for both ensembles

vicious walkers friendly walkers and young tableaux ii with a wall we derive new results for the number of star and watermelon configurations of vicious walkers in the presence of an impenetrable wall by showing that these follow from standard results in the theory of young tableaux and combinatorial descriptions of symmetric functions for the problem of n friendly walkers we derive exact asymptotics for the number of stars and watermelons both in the absence of a wall and in the presence of a wall

nonintersecting brownian motions on the half line and discrete gaussian orthogonal polynomials we study the distribution of the maximal height of the outermost path in the model of n nonintersecting brownian motions on the half line as n showing that it converges in the proper scaling to the tracy widom distribution for the largest eigenvalue of the gaussian orthogonal ensemble this is as expected from the viewpoint that the maximal height of the outermost path converges to the maximum of the airy2 process minus a parabola our proof is based on riemann hilbert analysis of a system of discrete orthogonal polynomials with a gaussian weight in the double scaling limit as this system approaches saturation we consequently compute the asymptotics of the free energy and the reproducing kernel of the corresponding discrete orthogonal polynomial ensemble in the critical scaling in which the density of particles approaches saturation both of these results can be viewed as dual to the case in which the mean density of eigenvalues in a random matrix model is vanishing at one point

2 matrix versus complex matrix model integrals over the unitary group as triangular integrals we prove that the 2 hermitian matrix model and the complex matrix model obey the same loop equations and as a byproduct we find a formula for itzykzon zuber type integrals over the unitary group integrals over u n are rewritten as gaussian integrals over triangular matrices and then computed explicitly that formula is an efficient alternative to the former shatashvili s formula

two step asymptotics of scaled dunkl processes dunkl processes are generalizations of brownian motion obtained by using the differential difference operators known as dunkl operators as a replacement of spatial partial derivatives in the heat equation special cases of these processes include dyson s brownian motion model and the wishart laguerre eigenvalue processes which are well known in random matrix theory it is known that the dynamics of dunkl processes is obtained by transforming the heat kernel using dunkl s intertwining operator it is also known that under an appropriate scaling their distribution function converges to a steady state distribution which depends only on the coupling parameter as the process time t tends to infinity we study scaled dunkl processes starting from an arbitrary initial distribution and we derive expressions for the intertwining operator in order to calculate the asymptotics of the distribution function in two limiting situations in the first one is fixed and t tends to infinity approach to the steady s

commuting difference operators spinor bundles and the asymptotics of orthogonal polynomials with respect to varying complex weights abstract the paper has three parts in the first part we apply the theory of commuting pairs of pseudo difference operators to the formal asymptotics of orthogonal polynomials using purely geometrical arguments we show heuristically that the asymptotics for large degrees of orthogonal polynomial with respect to varying weights is intimately related to certain spinor bundles on a hyperelliptic algebraic curve reproducing formulae appearing in the works of deift et al on the subject in the second part we show that given an arbitrary nodal hyperelliptic curve satisfying certain conditions of admissibility we can reconstruct a sequence of polynomials orthogonal with respect to semiclassical complex varying weights supported on several curves in the complex plane the strong asymptotics of these polynomials will be shown to be given by the spinors introduced in the first part using a riemann hilbert analysis in the third part we use strebel theory of quadratic differentials and the procedure of welding to reconstruct arbitrary admissible hyperelliptic curves as a result we can obtain orthogonal polynomials whose zeroes may become dense on a collection of jordan arcs forming an arbitrary forest of trivalent loop free trees

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

a weak galerkin finite element method for second order elliptic problems this paper introduces a finite element method by using a weakly defined gradient operator over generalized functions the use of weak gradients and their approximations results in a new concept called discrete weak gradients which is expected to play an important role in numerical methods for partial differential equations this article intends to provide a general framework for managing differential operators on generalized functions as a demonstrative example the discrete weak gradient operator is employed as a building block in the design of numerical schemes for a second order elliptic problem in which the classical gradient operator is replaced by the discrete weak gradient the resulting numerical scheme is called a weak galerkin wg finite element method it can be seen that the weak galerkin method allows the use of totally discontinuous functions in the finite element procedure for the second order elliptic problem an optimal order error estimate in both a discrete h 1 and l 2 norms are established for the corresponding weak galerkin finite element solutions a superconvergence is also observed for the weak galerkin approximation

a posteriori error estimates for the virtual element method an posteriori error analysis for the virtual element method vem applied to general elliptic problems is presented the resulting error estimator is of residual type and applies on very general polygonal polyhedral meshes the estimator is fully computable as it relies only on quantities available from the vem solution namely its degrees of freedom and element wise polynomial projection upper and lower bounds of the error estimator with respect to the vem approximation error are proven the error estimator is used to drive adaptive mesh refinement in a number of test problems mesh adaptation is particularly simple to implement since elements with consecutive co planar edges faces are allowed and therefore locally adapted meshes do not require any local mesh post processing

on tau functions for orthogonal polynomials and matrix models let v be a real polynomial of even degree and let rho be the equilibrium probability measure for v with support s to that v x greeater than or equal to int 2log x y rho dy c for some constant c with equality on s then s is the union of finitely many boundd intervals with endpoints delta j and rho is given by an algebraic weight w x on s then the system of orthogonal polynomials for w gives rise to a system of differential equations known as the schlesinger equations this paper identifies the tau function of this system with the hankel determinant det int x j k rho dx the solutions of the magnus schlesinger equation are realised by a linear system which is used to compute the tau functions in terms of a gelfand levitan equation the tau function is associated with a potential q and a scattering problem for the schrodinger equation with potential q the paper describes cases where this is integrable in terms of the nonlinear fourier transform

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

a posteriori error estimation for discontinuous galerkin finite element approximation it is shown that the interelement discontinuities in a discontinuous galerkin finite element approximation are subordinate to the error measured in the broken h 1 seminorm one consequence is that the dg norm of the error is equivalent to the broken energy seminorm computable a posteriori error bounds are obtained for the error measured in both the dg norm and the broken energy seminorm and are shown to be efficient in the sense that they also provide lower bounds up to a constant and higher order data oscillation terms the estimators are completely free of unknown constants and provide guaranteed numerical bounds for the error

determinantal process starting from an orthogonal symmetry is a pfaffian process when the number of particles n is finite the noncolliding brownian motion bm and the noncolliding squared bessel process with index 1 besq are determinantal processes for arbitrary fixed initial configurations in the present paper we prove that if initial configurations are distributed with orthogonal symmetry they are pfaffian processes in the sense that any multitime correlation functions are expressed by pfaffians the 2 2 skew symmetric matrix valued correlation kernels of the pfaffians processes are explicitly obtained by the equivalence between the noncolliding bm and an appropriate dilatation of a time reversal of the temporally inhomogeneous version of noncolliding bm with finite duration in which all particles start from the origin n 0 and by the equivalence between the noncolliding besq and that of the noncolliding squared generalized meander starting from n 0

some basic formulations of the virtual element method vem for finite deformations abstract we present a general virtual element method vem framework for finite elasticity which emphasizes two issues element level volume change volume average of the determinant of the deformation gradient and stabilization to address the former issue we provide exact evaluation of the average volume change in both 2d and 3d on properly constructed local displacement spaces for the later issue we provide a new stabilization scheme that is based on the trace of the material tangent modulus tensor which captures highly heterogeneous and localized deformations two vem formulations are presented a two field mixed and an equivalent displacement based which is free of volumetric locking convergence and accuracy of the vem formulations are verified by means of numerical examples and engineering applications are demonstrated

gaussian ensembles of random hermitian matrices intermediate between orthogonal and unitary ones a gaussian ensemble of hermitian matrices depending on a parameter is considered when 0 the ensemble is gaussian orthogonal and when 1 it is gaussian unitary an analytic expression for then level correlation and cluster functions is given for anyn and 0 1 this ensemble is of relevance in the study of time reversal symmetry breaking of nuclear interactions

monodromy preserving deformation of linear ordinary differential equations with rational coefficients iii abstract a unified treatment of monodromy and spectrum preserving deformations is presented in particular a general procedure is described to reduce the latter into the former consistently the concept of the function previously introduced for the former 2 is extended to the isospectral context it is shown that the general monodromy and spectrum preserving deformation equations can be written as hirota s bilinear differential equations by using the functions as dependent variables

scaling limit of vicious walks and two matrix model we consider the diffusion scaling limit of the one dimensional vicious walker model of fisher and derive a system of nonintersecting brownian motions the spatial distribution of n particles is studied and it is described by use of the probability density function of eigenvalues of nxn gaussian random matrices the particle distribution depends on the ratio of the observation time t and the time interval t in which the nonintersecting condition is imposed as t t is going on from 0 to 1 there occurs a transition of distribution which is identified with the transition observed in the two matrix model of pandey and mehta despite of the absence of matrix structure in the original vicious walker model in the diffusion scaling limit accumulation of contact repulsive interactions realizes the correlated distribution of eigenvalues in the multimatrix model as the particle distribution

semiclassical orthogonal polynomials matrix models and isomonodromic tau functions the differential systems satisfied by orthogonal polynomials with arbitrary semiclassical measures supported on contours in the complex plane are derived as well as the compatible systems of deformation equations obtained from varying such measures these are shown to preserve the generalized monodromy of the associated rank 2 rational covariant derivative operators the corresponding matrix models consisting of unitarily diagonalizable matrices with spectra supported on these contours are analyzed and it is shown that all coefficients of the associated spectral curves are given by logarithmic derivatives of the partition function or more generally the gap probabilities the associated isomonodromic tau functions are shown to coincide within an explicitly computed factor with these partition functions

polynuclear growth model with external source and random matrix model with deterministic source we present a random matrix interpretation of the distribution functions which have appeared in the study of the one dimensional polynuclear growth png model with external sources it is shown that the distribution goe2 which is defined as the square of the gaussian orthogonal ensemble goe tracy widom distribution can be obtained as the scaled largest eigenvalue distribution of a special case of a random matrix model with a deterministic source which have been studied in a different context previously compared to the original interpretation of the goe2 as the square of goe ours has an advantage in that it can also describe the transition from the gaussian unitary ensemble gue tracy widom distribution to the goe2 we further demonstrate that our random matrix interpretation can be obtained naturally by noting the similarity of the topology between a certain noncolliding brownian motion model and the multilayer png model with an external source this provides us with a multimatrix model interpretation of the multipoint height distributions of the png model with an external source

universal structure and universal equations pde for unitary ensembles random matrix ensembles with unitary invariance of measure ue are described in a unified way using a combination of tracy widom tw and adler shiota van moerbeke approaches to the derivation of partial differential equations pdes for spectral gap probabilities first general three term recurrence relations for ue restricted to subsets of real line or in other words for functions in the resolvent kernel are obtained using them simple universal relations between all tw dependent variables and one dimensional toda lattice functions are found a universal system of pde for ue is derived from previous relations which leads also to a single independent pde for spectral gap probability of various ue thus orthogonal function bases and toda lattice are seen at the core of correspondence of different approaches moreover toda akns system provides a common structure of pde for unitary ensembles interestingly this structure can be seen in two very different forms one arises from orthogonal functi