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

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

coulomb gas ensembles and laplacian growth we consider weight functions q that are locally in a suitable sobolev space and impose a logarithmic growth condition from below we use q as a confining potential in the model of one component plasma 2 dimensional coulomb gas and study the configuration of the electron cloud as the number n of electrons tends to infinity while the confining potential is rescaled we use mq in place of q and let m tend to infinity as well we show that if m and n tend to infinity in a proportional fashion with n m t where 0 t is fixed then the electrons accumulate on a compact set st which we call the droplet the set s t can be obtained as the coincidence set of an obstacle problem if we remove a small set the shallow points moreover on the droplet st the density of electrons is asymptotically q the growth of the droplets st as t increases is known as the laplacian growth it is well known that laplacian growth is unstable to analyse this feature we introduce the notion of a local droplet which involves removing part of the obstacle away from the set s t the local droplets are no longer uniquely determined by the time parameter t but at least they may be partially ordered we show that the growth of the local droplets may be terminated in a maximal local droplet or by the droplets growing to infinity in some direction fingering

partition functions and the continuum limit in penner matrix models we present an implementation of the method of orthogonal polynomials which is particularly suitable to study the partition functions of penner random matrix models to obtain their explicit forms in the exactly solvable cases and to determine the coefficients of their perturbative expansions in the continuum limit the method relies on identities satisfied by the resolvent of the jacobi matrix in the three term recursion relation of the associated families of orthogonal polynomials these identities lead to a convenient formulation of the string equations as an application we show that in the continuum limit the free energy of certain exactly solvable models like the linear and double penner models can be written as a sum of gaussian contributions plus linear terms to illustrate the one cut case we discuss the linear double and cubic penner models and for the two cut case we discuss theoretically and numerically the existence of a double branch structure of the free energy for the gaussian penner model

high accuracy wave simulation revised derivation numerical analysis and testing of a nearly analytic integration discrete method for solving acoustic wave equation the nearly analytic integration discrete naid method for solving the two dimensional acoustic wave equation has been fully mathematically revised analyzed and tested the naid method is an alternative numerical modeling method for generating synthetic seismograms the acoustic wave equation is first transformed into a system of first order ordinary differential equations odes with respect to time variable t and then directly integrated at a small time interval of tn tn 1 to obtain semi discrete ordinary differential equations the integral kernel is expanded into a truncated taylor series to which the integration operator is explicitly applied the high order temporal derivatives involved in the integral kernel are replaced by high order spatial derivatives which then are approximately calculated as a weighted linear combination of the displacement the particle velocity and their spatial gradients in this article we investigate the theoretical properties of the revised naid method including the discrete error and the stability criteria numerical results for constant and layered velocity models show that comparing to the lax wendroff correction lwc scheme and the staggered grid finite difference method the naid method can effectively suppress the numerical dispersion and source noises caused by the discretization of the acoustic wave equation with too coarse spatial grids or when models have strong velocity contrasts between adjacent layers the proposed naid method has been applied in computing the acoustic wavefields for two heterogeneous models the corner edge model and the marmousi model promising numerical results illustrate that the naid method provides an encouraging tool for large scale and complex wave simulation and inversion problems based on the acoustic equation

multipeakons and the classical moment problem abstract classical results of stieltjes are used to obtain explicit formulas for the peakon antipeakon solutions of the camassa holm equation the closed form solution is expressed in terms of the orthogonal polynomials of the related classical moment problem it is shown that collisions occur only in peakon antipeakon pairs and the details of the collisions are analyzed using results from the moment problem a sharp result on the steepening of the slope at the time of collision is given asymptotic formulas are given and the scattering shifts are calculated explicitly

density of eigenvalues of random normal matrices the relation between random normal matrices and conformal mappings discovered by wiegmann and zabrodin is made rigorous by restricting normal matrices to have spectrum in a bounded set it is shown that for a suitable class of potentials the asymptotic density of eigenvalues is uniform with support in the interior domain of a simple smooth curve

random matrix model with external source and a constrained vector equilibrium problem we consider the random matrix model with external source in the case where the potential v x is an even polynomial and the external source has two eigenvalues a of equal multiplicity we show that the limiting mean eigenvalue distribution of this model can be characterized as the first component of a pair of measures 1 2 that solve a constrained vector equilibrium problem the proof is based on the steepest descent analysis of the associated riemann hilbert problem for multiple orthogonal polynomials we illustrate our results in detail for the case of a quartic double well potential v x 1 4x 4 t 2x 2 we are able to determine the precise location of the phase transitions in the ta plane where either the constraint becomes active or the two intervals in the support come together or both 2010 wiley periodicals inc

random matrices in 2d laplacian growth and operator theory since it was first applied to the study of nuclear interactions by wigner and dyson almost 60 years ago random matrix theory rmt has developed into a field of its own within applied mathematics and is now essential to many parts of theoretical physics from condensed matter to high energy the fundamental results obtained so far rely mostly on the theory of random matrices in one dimension the dimensionality of the spectrum or equilibrium probability density in the last few years this theory has been extended to the case where the spectrum is two dimensional or even fractal with dimensions between 1 and 2 in this paper we review these recent developments and indicate some physical problems where the theory can be applied

solution of matrix riemann hilbert problems with quasi permutation monodromy matrices in this paper we solve an arbitrary matrix riemann hilbert inverse monodromy problem with irreducible quasi permutation monodromy representation outside of a divisor in the space of monodromy data this divisor is characterized in terms of the theta divisor on the jacobi manifold of an auxiliary compact riemann surface realized as an appropriate branched covering of open image in new windowp1 the solution is given in terms of a generalization of szego kernel on the riemann surface in particular our construction provides a new class of solutions of the schlesinger system the isomonodromy tau function of these solutions is computed up to a nowhere vanishing factor independent of the elements of monodromy matrices results of this work generalize the results of papers 13 and 14 where the 2 2 case was solved

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

growth in gaussian elimination for weighing matrices w n n 1 abstract we consider the values for large minors of a skew hadamard matrix or conference matrix w of order n and find that maximum n n minor equals to n 1 n 2 maximum n 1 n 1 minor equals to n 1 n 2 1 maximum n 2 n 2 minor equals to 2 n 1 n 2 2 and maximum n 3 n 3 minor equals to 4 n 1 n 2 3 this leads us to conjecture that the growth factor for gaussian elimination ge of completely pivoted cp skew hadamard or conference matrices and indeed any cp weighing matrix of order n and weight n 1 is n 1 and that the first and last few pivots are 1 2 2 3 or 4 n 1 or n 1 2 n 1 2 n 1 for n 14 we show the unique w 6 5 has a single pivot pattern and the unique w 8 7 has at least two pivot structures we give two pivot patterns for the unique w 10 9

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

the supercritical regime in the normal matrix model with cubic potential abstract the normal matrix model with a cubic potential is ill defined and it develops a critical behavior in finite time we follow the approach of bleher and kuijlaars to reformulate the model in terms of orthogonal polynomials with respect to a hermitian form this reformulation was shown to capture the essential features of the normal matrix model in the subcritical regime namely that the zeros of the polynomials tend to a number of segments the motherbody inside a domain the droplet that attracts the eigenvalues in the normal matrix model in the present paper we analyze the supercritical regime and we find that the large n behavior is described by the evolution of a spectral curve satisfying the boutroux condition the boutroux condition determines a system of contours 1 consisting of the motherbody and whiskers sticking out of the domain we find a second critical behavior at which the original motherbody shrinks to a point at the origin and only the whiskers remain in the regime before the second criticality we also give strong asymptotics of the orthogonal polynomials by means of a steepest descent analysis of a 3 3 matrix valued riemann hilbert problem it follows that the zeros of the orthogonal polynomials tend to 1 with the exception of at most three spurious zeros

strong asymptotics for cauchy biorthogonal polynomials with application to the cauchy two matrix model we apply the nonlinear steepest descent method to a class of 3 3 riemann hilbert problems introduced in connection with the cauchy two matrix random model the general case of two equilibrium measures supported on an arbitrary number of intervals is considered in this case we solve the riemann hilbert problem for the outer parametrix in terms of sections of a spinorial line bundle on a three sheeted riemann surface of arbitrary genus and establish strong asymptotic results for the cauchy biorthogonal polynomials

critical measures quadratic differentials and weak limits of zeros of stieltjes polynomials we investigate the asymptotic zero distribution of heine stieltjes polynomials polynomial solutions of second order differential equations with complex polynomial coefficients in the case when all zeros of the leading coefficients are all real zeros of the heine stieltjes polynomials were interpreted by stieltjes as discrete distributions minimizing an energy functional in a general complex situation one deals instead with a critical point of the energy we introduce the notion of discrete and continuous critical measures saddle points of the weighted logarithmic energy on the plane and prove that a weak limit of a sequence of discrete critical measures is a continuous critical measure thus the limit zero distributions of the heine stieltjes polynomials are given by continuous critical measures we give a detailed description of such measures showing their connections with quadratic differentials in doing that we obtain some results on the global structure of rational quadratic differentials on the riemann sphere that have an independent interest

an eigenvalue approach evaluating minors for weighing matrices w n n 1 abstract in the present paper we concentrate our study on the evaluation of minors for weighing matrices w n n 1 theoretical proofs concerning their minors up to the order of n 4 n 4 are derived introducing an eigenvalue approach a general theorem specifying the analytical form of any n l n l minor is developed an application to the growth problem for weighing matrices is given

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

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

conformal maps and integrable hierarchies we show that conformal maps of simply connected domains with an analytic boundary to a unit disk have an intimate relation to the dispersionless 2d toda integrable hierarchy the maps are determined by a particular solution to the hierarchy singled out by the conditions known as string equations the same hierarchy locally solves the 2d inverse potential problem i e reconstruction of the domain out of a set of its harmonic moments this is the same solution which is known to describe 2d gravity coupled to c matter we also introduce a concept of the function for analytic curves

orthogonal polynomials in the normal matrix model with a cubic potential abstract we consider the normal matrix model with a cubic potential the model is ill defined and in order to regularize it elbau and felder introduced a model with a cut off and corresponding system of orthogonal polynomials with respect to a varying exponential weight on the cut off region on the complex plane in the present paper we show how to define orthogonal polynomials on a specially chosen system of infinite contours on the complex plane without any cut off which satisfy the same recurrence algebraic identity that is asymptotically valid for the orthogonal polynomials of elbau and felder the main goal of this paper is to develop the riemann hilbert rh approach to the orthogonal polynomials under consideration and to obtain their asymptotic behavior on the complex plane as the degree n of the polynomial goes to infinity as the first step in the rh approach we introduce an auxiliary vector equilibrium problem for a pair of measures 1 2 on the complex plane we then formulate a 3 3 matrix valued rh problem for the orthogonal polynomials in hand and we apply the nonlinear steepest descent method of deift zhou to the asymptotic analysis of the rh problem the central steps in our study are a sequence of transformations of the rh problem based on the equilibrium vector measure 1 2 and the construction of a global parametrix the main result of this paper is a derivation of the large n asymptotics of the orthogonal polynomials on the whole complex plane we prove that the distribution of zeros of the orthogonal polynomials converges to the measure 1 the first component of the equilibrium measure we also obtain analytical results for the measure 1 relating it to the distribution of eigenvalues in the normal matrix model which is uniform in a domain bounded by a simple closed curve

critical measures for vector energy global structure of trajectories of quadratic differentials abstract saddle points of a vector logarithmic energy with a vector polynomial external field on the plane constitute the vector valued critical measures a notion that finds a natural motivation in several branches of analysis we study in depth the case of measures 1 2 3 when the mutual interaction comprises both attracting and repelling forces for arbitrary vector polynomial external fields we establish general structural results about critical measures such as their characterization in terms of an algebraic equation solved by an appropriate combination of their cauchy transforms and the symmetry properties or the s properties exhibited by such measures in consequence we conclude that vector valued critical measures are supported on a finite number of analytic arcs that are trajectories of a quadratic differential globally defined on a three sheeted riemann surface the complete description of the so called critical graph for such a differential is the key to the construction of the critical measures we illustrate these connections studying in depth a one parameter family of critical measures under the action of a cubic external field this choice is motivated by the asymptotic analysis of a family of non hermitian multiple orthogonal polynomials that is subject of a forthcoming paper here we compute explicitly the riemann surface and the corresponding quadratic differential and analyze the dynamics of its critical graph as a function of the parameter giving a detailed description of the occurring phase transitions when projected back to the complex plane this construction gives us the complete family of vector valued critical measures that in this context turn out to be vector valued equilibrium measures

on the growth problem for skew and symmetric conference matrices abstract koukouvinos et al c koukouvinos m mitrouli j seberry growth in gaussian elimination for weighing matrices w n n 1 linear algebra appl 306 2000 189 202 conjectured that the growth factor for gaussian elimination of any completely pivoted weighing matrix of order n and weight n 1 is n 1 and that the first and last few pivots are 1 2 2 3 or 4 n 1 or n 1 2 n 1 2 n 1 for n 14 in the present paper we study the growth problem for skew and symmetric conference matrices an algorithm for extending a k k matrix with elements 0 1 to a skew and symmetric conference matrix of order n is described by using this algorithm we show that the unique w 8 7 has two pivot structures we also prove that the unique w 10 9 has three pivot patterns

the small dispersion limit of the korteweg de vries equation ii in part i the scattering transform method is used to study the weak limit of solutions to the initial value problem for the korteweg devries kdv equation as the dispersion tends to zero in that limit the associated schrodinger operator becomes semiclassical so the exact scattering data is replaced by its corresponding wkb expressions only nonpositive initial data are considered in that case the limiting reflection coefficient vanishes the explicit solution of kay and moses for the reflectionless inverse transform is then analyzed and the weak limit valid for all time is characterized by a quadratic minimum problem with constraints this minimum problem is reduced to a riemann hilbert problem in function theory in parts ii and iii we use function theoretical methods to solve the riemann hilbert problem in terms of solutions to an auxiliary initial value problem the weak limit satisfies the kdv equation with the dispersive term dropped until its derivatives become infinite up to that time the weak limit is a strong l2 limit at later times the weak limit is locally described by whitham s averaged equations or more generally by the equations found by flaschka et al using multiphase averaging for large times behavior of the weak limit is studied directly from the minimum problem

3 d large scale wave propagation modeling by spectral element method on cray t3e multiprocessor highly efficient algorithms and massively parallel computers are needed for acoustic full wave modeling in large scale 3 d realistic unbounded media in this work a spectral element method in conjunction with a new iterative solution technique is presented the very high spatial accuracy of the algorithm allows for increased computational efficiency when compared to standard finite element method in addition the use of an iterative solver based on a local element by element formulation and a matrix vector product which is factored and converted to a faster matrix matrix product allows for a significant reduction both in storage requirements and in the computational complexity the resulting algorithm can easily be implemented on high performance vector and or parallel processors the numerical scheme has been tested on a cray t3e supercomputer our experimental results show that efficiency is very high and that good load balancing can be obtained at run time because relevant blocks of computations are performed at the element level and are totally independent

strong asymptotics of the orthogonal polynomials with respect to a measure supported on the plane we consider the orthogonal polynomials pn z with respect to the measure z a 2nce n z 2da z over the whole complex plane we obtain the strong asymptotic of the orthogonal polynomials in the complex plane and the location of their zeros in a scaling limit where n grows to infinity with n the asymptotics are described in terms of three probability measures associated with the problem the first measure is the limit of the counting measure of zeros of the polynomials which is captured by the g function much in the spirit of ordinary orthogonal polynomials on the real line the second measure is the equilibrium measure that minimizes a certain logarithmic potential energy supported on a region k of the complex plane the third measure is the harmonic measure of kc with a pole at this appears as the limit of the probability measure given up to the normalization constant by the squared modulus of the nth orthogonal polynomial times the orthogonality measure i e pn z 2 z a 2nce n z 2da z the compact region k that is the support of the second measure undergoes a topological transition under the variation of the parameter t n n in a double scaling limit near the critical point given by tc a a 2c we observe the hastings mcleod solution to painleve ii in the asymptotics of the orthogonal polynomials 2014 wiley periodicals inc

solving elastodynamics in a fluid solid heterogeneous sphere a parallel spectral element approximation on non conforming grids we present a spectral element approach for modeling elastic wave propagation in a solid fluid sphere where the local effects of gravity are taken into account the equations are discretized in terms of the displacement in the solid and the velocity potential in a neutrally stratified fluid the spatial approximation is based upon a spherical mesh of hexahedra in which local refinement allows for adapting the discretization to the variation of elastic parameters in both the solid and the fluid regions continuity constraints across the non conforming interfaces are introduced through lagrange multipliers which are further discretized by the mortar element method due to the spherical nature of the non conforming interfaces the mortar method turns out to be functionally conforming and allows for an equal order interpolation of the primal variables and the lagrange multipliers the method is shown to provide an accurate solution when compared to analytical calculations obtained for radial models of elastic parameters its parallel implementation is based upon a simple domain decomposition strategy which makes it efficient to solve large problems as those imposed by planetary scales

the behavior of the weyl function in the zero dispersion kdv limit the moment formulas that globally characterize the zero dispersion limit of the korteweg devries kdv equation are known to be expressed in terms of the solution of a maximization problem here we establish a direct relation between this maximizer and the zero dispersion limit of the logarithm of the jost functions associated with the inverse spectral transform all the kdv conserved densities are encoded in the spatial derivative of these functions known as weyl functions we show the weyl functions are densities of measures that converge in the weak sense to a limiting measure this limiting measure encodes all of the weak limits of the kdv conserved densities moreover we establish the weak limit of spectral measures associated with the dirichlet problem

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

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

dft modal analysis of spectral element methods for the 2d elastic wave equation the dft modal analysis is a dispersion analysis technique that transforms the equations of a numerical scheme to the discrete fourier transform domain sampled in the mesh nodes this technique provides a natural matching of exact and approximate modes of propagation we extend this technique to spectral element methods for the 2d isotropic elastic wave equation by using a rayleigh quotient approximation of the eigenvalue problem that characterizes the dispersion relation taking full advantage of the tensor product representation of the spectral element matrices numerical experiments illustrate the dependence of dispersion errors on the grid resolution polynomial degree and discretization in time we consider spectral element methods with chebyshev and legendre collocation points

two dimensional simulation of rayleigh waves with staggered sine cosine transforms and variable grid spacing simulation of rayleigh waves requires high accuracy and an adequate spatial sampling at the surface discrete cosine and sine transforms are used to compute spatial derivatives along the direction perpendicular to the surface of the earth unlike the standard fourier method these transforms allow nonperiodicboundaryconditionstobesatisfied inparticular the stress free conditions at the surface because simulation of surface waves requires more points per minimum wavelengthatthesurfacethansimulationofbodywaves theequispaced grid is not efficient to overcome this problem a grid compression is performed at the surface to obtain a denser spatial sampling grid size is minimal at the surface and increases with depth until reaching at a relatively shallow depth the grid points per wavelength required by the body waves the stress free boundary conditions are naturally handled by expanding the appropriate stress components in terms of the discrete sine transform the wave equation is solved in the particle velocity and stress formulation using a runge kutta time integration and the convolutional pml cpmlmethodtopreventreflectionsfromthemeshboundaries the simulations are very accurate for shallow sources andreceiversandlargeoffsets

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

planar orthogonal polynomials as type ii multiple orthogonal polynomials we show that the planar orthogonal polynomials with l logarithmic singularities in the potential are the multiple orthogonal polynomials hermite pade polynomials of type ii with l measures we also find the ratio between the determinant of the moment matrix corresponding to the multiple orthogonal polynomials and the determinant of the moment matrix from the original planar measure

non intersecting squared bessel paths and multiple orthogonal polynomials for modified bessel weights we study a model of n non intersecting squared bessel processes in the confluent case all paths start at time t 0 at the same positive value x a remain positive and are conditioned to end at time t t at x 0 in the limit n after appropriate rescaling the paths fill out a region in the tx plane that we describe explicitly in particular the paths initially stay away from the hard edge at x 0 but at a certain critical time t the smallest paths hit the hard edge and from then on are stuck to it for t t we obtain the usual scaling limits from random matrix theory namely the sine airy and bessel kernels a key fact is that the positions of the paths at any time t constitute a multiple orthogonal polynomial ensemble corresponding to a system of two modified bessel type weights as a consequence there is a 3 3 matrix valued riemann hilbert problem characterizing this model that we analyze in the large n limit using the deift zhou steepest descent method there are some novel ingredients in the riemann hilbert analysis that are of independent interest

the cauchy two matrix model we introduce a new class of two multi matrix models of positive hermitian matrices coupled in a chain the coupling is related to the cauchy kernel and differs from the exponential coupling more commonly used in similar models the correlation functions are expressed entirely in terms of certain biorthogonal polynomials and solutions of appropriate riemann hilbert problems thus paving the way to a steepest descent analysis and universality results the interpretation of the formal expansion of the partition function in terms of multicolored ribbon graphs is provided and a connection to the o 1 model a steepest descent analysis of the partition function reveals that the model is related to a trigonal curve three sheeted covering of the plane much in the same way as the hermitian matrix model is related to a hyperelliptic curve

universal hypotrochoidic law for random matrices with cyclic correlations the celebrated elliptic law describes the distribution of eigenvalues of random matrices with correlations between off diagonal pairs of elements having applications to a wide range of physical and biological systems here we investigate the generalization of this law to random matrices exhibiting higher order cyclic correlations between k tuples of matrix entries we show that the eigenvalue spectrum in this ensemble is bounded by a hypotrochoid curve with k fold rotational symmetry this hypotrochoid law applies to full matrices as well as sparse ones and thereby holds with remarkable universality we further extend our analysis to matrices and graphs with competing cycle motifs which are described more generally by polytrochoid spectral boundaries

numerical simulation of interface waves by high order spectral modeling techniques few problems in elastodynamics have a closed form analytical solution the others can be investigated with semianalytical methods but in general one is not sure whether these methods give reliable solutions the same happens with numerical techniques for instance finite difference methods solve in principle any complex problem including those with arbitrary inhomogeneities and boundary conditions however there is no way to verify the quantitative correctness of the solutions the major problems are stability with respect to material properties numerical dispersion and the treatment of boundary conditions in practice these problems may produce inaccurate solutions in this paper the study of complex problems with two different numerical grid techniques in order to cross check the solutions is proposed interface waves in particular are emphasized since they pose the major difficulties due to the need to implement boundary conditions the first method is based on global differential operators where the solution is expanded in terms of the fourier basis and chebyshev polynomials while the second is the spectral element method an extension of the finite element method that uses chebyshev polynomials as interpolating functions both methods have spectral accuracy up to approximately the nyquist wave number of the grid moreover both methods implement the boundary conditions in a natural way particularly the spectral element algorithm we first solve lamb s problem and compare numerical and analytical solutions then the problem of dispersed rayleigh waves and finally the two quarter space problem we show that the modeling algorithms correctly reproduce the analytical solutions and yield a perfect matching when these solutions do not exist the combined modeling techniques provide a powerful tool for solving complex problems in elastodynamics

3 d wave simulation in anelastic media using the kelvin voigt constitutive equation we design a numerical algorithm for wave simulation in anelastic media in the presence of free surface which can be used to model seismic waves at the earth s surface and ultrasonic waves in heterogeneous materials the stress strain relation is based on the kelvin voigt mechanical model which has the advantage of not requiring additional field variables the model requires two anelastic parameters and twice the spatial derivatives of the lossless case the high frequency components of the wave field are more attenuated than the low frequency components with the attenuation factors being approximately proportional to the square of the frequency the modeling simulates 3 d waves by using the fourier and chebyshev methods to compute the spatial derivatives along the horizontal and vertical directions respectively we stretch the mesh in the vertical direction to increase the minimum grid spacing and reduce the computational cost instabilities of the chebyshev differential operator due to the implementation of the free surface boundary conditions are solved with a characteristic approach where the characteristic variables are evaluated at the source central frequency the results of the modeling are verified by comparisons to the analytical solutions for lamb s problem and propagation in unbounded homogeneous media examples illustrating the propagation of rayleigh and love waves are presented

critical measures for vector energy asymptotics of non diagonal multiple orthogonal polynomials for a cubic weight abstract we consider the type i multiple orthogonal polynomials mops a n m b n m deg a n m n 1 deg b n m m 1 and type ii mops p n m deg p n m n m satisfying non hermitian orthogonality with respect to the weight e z 3 on two unbounded contours 1 and 2 on c with in the case of type ii mops n conditions on 1 and m on 2 under the assumption that n m n n m 0 1 we find the detailed rescaled asymptotics of a n m b n m and p n m on c and describe the phase transitions of this limit behavior as a function of this description is given in terms of the vector critical measure 1 2 3 the saddle point of an energy functional comprising both attracting and repelling forces this critical measure is characterized by a cubic equation spectral curve and its components j live on trajectories of a canonical quadratic differential on the riemann surface of this equation the structure of these trajectories and their deformations as functions of was object of study in our previous paper 42 and some of the present results strongly rely on the analysis carried out there we conclude that the asymptotic zero distribution of the polynomials a n m and p n m are given by appropriate combinations of the components j of the vector critical measure however in the case of the zeros of b n m the behavior is totally different and can be described in terms of the balayage of the signed measure 2 3 onto certain curves on the plane these curves are constructed with the aid of the canonical quadratic differential and their topology has three very distinct characters depending on the value of and are obtained from the critical graph of once the trajectories and vector critical measures are studied the main asymptotic technical tool is the deift zhou nonlinear steepest descent analysis of a 3 3 matrix valued riemann hilbert problem characterizing both a n m b n m and p n m which allows us to obtain the limit behavior of both types of mops simultaneously we illustrate our findings with results of several numerical experiments explain the computational methodology and formulate some conjectures and empirical observations based on these experiments

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

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

large n limit of gaussian random matrices with external source part i we continue the study of the hermitian random matrix ensemble with external source open image in new window where a has two distinct eigenvalues a of equal multiplicity this model exhibits a phase transition for the value a 1 since the eigenvalues of m accumulate on two intervals for a 1 and on one interval for 0 1 was treated in part i where it was proved that local eigenvalue correlations have the universal limiting behavior which is known for unitarily invariant random matrices that is limiting eigenvalue correlations are expressed in terms of the sine kernel in the bulk of the spectrum and in terms of the airy kernel at the edge in this paper we establish the same results for the case 0 a 1 as in part i we apply the deift zhou steepest descent analysis to a 3 3 matrix riemann hilbert problem due to the different structure of an underlying riemann surface the analysis includes an additional step involving a global opening of lenses which is a new phenomenon in the steepest descent analysis of riemann hilbert problems

cancellation of spurious arrivals in green s function extraction and the generalized optical theorem the extraction of the green s function by cross correlation of waves recorded at two receivers nowadays finds much application we show that for an arbitrary small scatterer the cross terms of scattered waves give an unphysical wave with an arrival time that is independent of the source position this constitutes an apparent inconsistency because theory predicts that such spurious arrivals do not arise after integration over a complete source aperture this puzzling inconsistency can be resolved for an arbitrary scatterer by integrating the contribution of all sources in the stationary phase approximation to show that the stationary phase contributions to the source integral cancel the spurious arrival by virtue of the generalized optical theorem this work constitutes an alternative derivation of this theorem when the source aperture is incomplete the spurious arrival is not canceled and could be misinterpreted to be part of the green s function we give an example of how spurious arrivals provide information about the medium complementary to that given by the direct and scattered waves the spurious waves can thus potentially be used to better constrain the medium

simulation of anisotropic wave propagation based upon a spectral element method we introduce a numerical approach for modeling elastic wave propagation in 2 d and 3 d fully anisotropic media based upon a spectral element method the technique solves a weak formulation of the wave equation which is discretized using a high order polynomial representation on a finite element mesh for isotropic media the spectral element method is known for its high degree of accuracy its ability to handle complex model geometries and its low computational cost we show that the method can be extended to fully anisotropic media the mass matrix obtained is diagonal by construction which leads to a very efficient fully explicit solver we demonstrate the accuracy of the method by comparing it against a known analytical solution for a 2 d transversely isotropic test case and by comparing its predictions against those based upon a finite difference method for a 2 d heterogeneous anisotropic medium we show its generality and its flexibility by modeling wave propagation in a 3 d transversely isotropic medium with a symmetry axis tilted relative to the axes of the grid

a highly accurate discontinuous galerkin method for complex interfaces between solids and moving fluids we have extended a new highly accurate numerical scheme for unstructured 2d and 3d meshes based on the discontinuous galerkin approach to simulate seismic wave propagation in heterogeneous media containing fluid solid interfaces because of the formulation of the wave equations as a unified first order hyperbolic system in velocity stress the fluid can be in movement along the interface the governing equations within the moving fluid are derived from well known first principles in fluid mechanics the discontinuous galerkin approach allows for jumps of the material parameters and the solution across element interfaces which are handled by riemann solvers or numerical fluxes the use of riemann solvers at the element interfaces makesthe treatment of the fluid particularly simple bysetting the shearmodulus in the fluid region to zero no additional compatibility relations such as vanishing shear stress or continuity of normal stresses are necessary to couple the solid and fluid along an interface the

scaling limits of random normal matrix processes at singular boundary points abstract we introduce a method for taking microscopic limits of normal matrix ensembles and apply it to study the behaviour near certain types of singular points on the boundary of the droplet our investigation includes ensembles without restrictions near the boundary as well as hard edge ensembles where the eigenvalues are confined to the droplet we establish in both cases existence of new types of determinantal point fields which differ from those which can appear at a regular boundary point or in the bulk

universality of the local eigenvalue statistics for a class of unitary invariant random matrix ensembles this paper is devoted to the rigorous proof of the universality conjecture of random matrix theory according to which the limiting eigenvalue statistics ofn n random matrices within spectral intervals ofo n 1 is determined by the type of matrix real symmetric hermitian or quaternion real and by the density of states we prove this conjecture for a certain class of the hermitian matrix ensembles that arise in the quantum field theory and have the unitary invariant distribution defined by a certain function the potential in the quantum field theory satisfying some regularity conditions

optimal blended spectral element operators for acoustic wave modeling spectral element methods based on high order polynomials are among the most commonly used techniques for computing accurate simulations of wave propagation phenomena in complex media however to retain computational efficiency very high order polynomials cannot be used and errors such as numerical dispersion and numerical anisotropy cannot be totally avoided in the present work we devise an approach for reducing such errors by considering modified discrete wave operators we analyze consistent and lumped operators together with blended operators weighted averages of consistent and lumped operators furthermore using the operator blending approach and a novel dispersion analysis method we develop optimal spectral element operators that have increased numerical accuracy without resorting to very high order operators the new operators are faster and computationally more efficient than consistent operators our approach is based on the tensor product decomposition of the element matrices into 1d f

the spectral element method for elastic wave equations application to 2 d and 3 d seismic problems a spectral element method for the approximate solution of linear elastodynamic equations set in a weak form is shown to provide an efficient tool for simulating elastic wave propagation in realistic geological structures in two and three dimensional geometries the computational domain is discretized into quadrangles or hexahedra defined with respect to a reference unit domain by an invertible local mapping inside each reference element the numerical integration is based on the tensor product of a gauss lobatto legendre 1 d quadrature and the solution is expanded onto a discrete polynomial basis using lagrange interpolants as a result the mass matrix is always diagonal which drastically reduces the computational cost and allows an efficient parallel implementation absorbing boundary conditions are introduced in variational form to simulate unbounded physical domains the time discretization is based on an energy momentum conserving scheme that can be put into a classical explicit implicit predictor multicorrector format long term energy conservation and stability properties are illustrated as well as the efficiency of the absorbing conditions the accuracy of the method is shown by comparing the spectral element results to numerical solutions of some classical two dimensional problems obtained by other methods the potentiality of the method is then illustrated by studying a simple three dimensional model very accurate modelling of rayleigh wave propagation and surface diffraction is obtained at a low computational cost the method is shown to provide an efficient tool to study the diffraction of elastic waves and the large amplification of ground motion caused by three dimensional surface topographies copyright 1999 john wiley sons ltd

cubic string boundary value problems and cauchy biorthogonal polynomials cauchy biorthogonal polynomials appear in the study of special solutions to the dispersive nonlinear partial differential equation called the degasperis procesi dp equation as well as in certain two matrix random matrix models another context in which such biorthogonal polynomials play a role is the cubic string a third order ode boundary value problem f zgf which is a generalization of the inhomogeneous string problem studied by krein a general class of such boundary value problems going beyond the original cubic string problem associated with the dp equation is discussed under the assumption that the source of inhomogeneity g is a discrete measure it is shown that by a suitable choice of a generalized fourier transform associated with these boundary value problems one can establish a parseval type identity which aligns cauchy biorthogonal polynomials with certain natural orthogonal systems on l2g

semiclassical asymptotics of orthogonal polynomials riemann hilbert problem and universality in the matrix model we derive semiclassical asymptotics for the orthogonal polynomials pn z on the line with respect to the exponential weight exp inv z where v z is a double well quartic polynomial in the limit when n n 1 we assume that n n cri for some 0 where cr is the critical value which separates orthogonal polynomials with two cuts from the ones with one cut simultaneously we derive semiclassical asymptotics for the recursive coe cients of the orthogonal polynomials and we show that these coe cients form a cycle of period two which drifts slowly with the change of the ratio n n the proof of the semiclassical asymptotics is based on the methods of the theory of integrable systems and on the analysis of the appropriate matrix riemannhilbert problem as an application of the semiclassical asymptotics of the orthogonal polynomials we prove the universality of the local distribution of eigenvalues in the matrix model with the double well quartic interaction in the presence of two cuts contents

solving the 3d acoustic wave equation on generalized structured meshes a finite difference time domain approach abstractthe key computational kernel of most advanced 3d seismic imaging and inversion algorithms used in exploration seismology involves calculating solutions of the 3d acoustic wave equation most commonly with a finite difference time domain fdtd methodology although well suited for regularly sampled rectilinear computational domains fdtd methods seemingly have limited applicability in scenarios involving irregular 3d domain boundary surfaces and mesh interiors best described by non cartesian geometry e g surface topography using coordinate mapping relationships and differential geometry an fdtd approach can be developed for generating solutions to the 3d acoustic wave equation that is applicable to generalized 3d coordinate systems and quadrilateral faced hexahedral structured meshes the developed numerical implementation is similar to the established cartesian approaches save for a necessary introduction of weighted first and mixed second order partial derivative operators that account

values of minors of some infinite families of matrices constructed from supplementary difference sets and their application to the growth problem abstract we obtain explicit formulae for the values of the 2 v j minors j 0 1 2 of some infinite families of matrices of order 2 v constructed from supplementary difference sets of the form 2 v k 1 k 2 this allows us to obtain information on the growth problem for families of matrices with moderate growth several examples specifying the pivot pattern of these matrices are presented and a conjecture about their pivot structure is posed

universality conjecture and results for a model of several coupled positive definite matrices the paper contains two main parts in the first part we analyze the general case of p geq 2 matrices coupled in a chain subject to cauchy interaction similarly to the itzykson zuber interaction model the eigenvalues of the cauchy chain form a multi level determinantal point process we first compute all correlations functions in terms of cauchy biorthogonal polynomials and locate them as specific entries of a p 1 times p 1 matrix valued solution of a riemann hilbert problem in the second part we fix the external potentials as classical laguerre weights we then derive strong asymptotics for the cauchy biorthogonal polynomials when the support of the equilibrium measures contains the origin as a result we obtain a new family of universality classes for multi level random determinantal point fields which include the bessel nu universality for 1 level and the meijer g universality for 2 level our analysis uses the deift zhou nonlinear steepest descent method and the explicit construction of a p 1 times p 1 origin parametrix in terms of meijer g functions the solution of the full riemann hilbert problem is derived rigorously only for p 3 but the general framework of the proof can be extended to the cauchy chain of arbitrary length p

a comparative study of finite element and spectral element methods in seismic wavefield modeling the computational accuracy and efficiency of finite element method and spectral element method sem are investigated thoroughly in time domain elastic wavefield modeling the diagonal mass matrices of the fem and sem free from matrix inversion are compared comprehensively by making full use of the mass lumped technique and quadrature rules we investigate the fem and sem based respectively on quadrilateral with the polynomials of degrees one and two and on triangular grids with the polynomials of degrees one and three generally the numerical solutions based on quadrilateral grids have a higher precision than those computed on triangular grids when the same order of polynomials is used the fem has a comparable accuracy to the sem with the same number of interpolant points in view of the triangular and quadrilateral sems the former suffers from larger computational costs and relatively lower accuracy compared with the latter furthermore the convergence study proves that the triangular sem produces consistently larger errors than the quadrilateral sem for any order and element sizes however the triangular sem can adapt to arbitrary complex geometries effectively in terms of efficiency the fem has an efficiency comparable with the sem on condition that the order of interpolation polynomials is identical in addition a perfectly matched layer pml boundary condition in variational form is deduced by introducing four intermediate variables in frequency domain the pml avoids convolution calculation and obtains an exact solution through inverse fourier transform in time domain the numerical examples verify the validity and effectiveness of the pml

mass lumping techniques in the spectral element method on the equivalence of the row sum nodal quadrature and diagonal scaling methods abstract in the context of wave propagation analysis the spectral element method sem in conjunction with a diagonal mass matrix is often the method of choice therefore it is of high importance to investigate the influence of different mass lumping schemes on the accuracy of the numerical results to this end we compare the performance of three established methods including the row sum method the nodal quadrature method and the diagonal scaling method the theoretical analysis of these methods reveals a close connection between them under certain conditions that are discussed in detail in this article we are able to show a direct equivalence between these three approaches in this regard the attainable accuracy of the numerical integration of the mass matrix plays an important role by means of several dynamic benchmark problems we verify the theoretical results and illustrate the convergence properties of the lumped mass matrix sem in comparison to a formulation based on the consistent mass matrix

unconstrained plastering hexahedral mesh generation via advancing front geometry decomposition the generation of all hexahedral finite element meshes has been an area of ongoing research for the past two decades and remains an open problem unconstrained plastering is a new method for generating all hexahedral finite element meshes on arbitrary volumetric geometries starting from an unmeshed volume boundary unconstrained plastering generates the interior mesh topology without the constraints of a pre defined boundary mesh using advancing fronts unconstrained plastering forms partially defined hexahedral dual sheets by decomposing the geometry into simple shapes each of which can be meshed with simple meshing primitives by breaking from the tradition of previous advancing front algorithms which start from pre meshed boundary surfaces unconstrained plastering demonstrates that for the tested geometries high quality boundary aligned orientation insensitive all hexahedral meshes can be generated automatically without pre meshing the boundary examples are given for meshes from both solid mechanics and geotechnical applications copyright 2009 john wiley sons ltd

fluctuations of eigenvalues of random normal matrices in this article we consider a fairly general potential in the plane and the corresponding boltzmann gibbs distribution of eigenvalues of random normal matrices as the order of the matrices tends to infinity the eigenvalues condensate on a certain compact subset of the plane the droplet we prove that fluctuations of linear statistics of eigenvalues of random normal matrices converge on compact subsets of the interior of the droplet to a gaussian field and we discuss various ramifications of this result

cauchy laguerre two matrix model and the meijer g random point field we apply the general theory of cauchy biorthogonal polynomials developed in bertola et al commun math phys 287 3 983 1014 2009 and bertola et al j approx th 162 4 832 867 2010 to the case associated with laguerre measures in particular we obtain explicit formulae in terms of meijer g functions for all key objects relevant to the study of the corresponding biorthogonal polynomials and the cauchy two matrix model associated with them the central theorem we prove is that a scaling limit of the correlation functions for eigenvalues near the origin exists and is given by a new determinantal two level random point field the meijer g random field we conjecture that this random point field leads to a novel universality class of random fields parametrized by exponents of laguerre weights we express the joint distributions of the smallest eigenvalues in terms of suitable fredholm determinants and evaluate them numerically we also show that in a suitable limit the meijer g random field converges to the bessel random field and hence the behavior of the eigenvalues of one of the two matrices converges to the one of the laguerre ensemble

the ellipse law kirchhoff meets dislocations in this paper we consider a nonlocal energy i whose kernel is obtained by adding to the coulomb potential an anisotropic term weighted by a parameter alpha in mathbb r the case 0 corresponds to purely logarithmic interactions minimised by the circle law 1 corresponds to the energy of interacting dislocations minimised by the semi circle law we show that for alpha in 0 1 the minimiser is the normalised characteristic function of the domain enclosed by the ellipse of semi axes sqrt 1 alpha and sqrt 1 alpha this result is one of the very few examples where the minimiser of a nonlocal anisotropic energy is explicitly computed for the proof we borrow techniques from fluid dynamics in particular those related to kirchhoff s celebrated result that domains enclosed by ellipses are rotating vortex patches called kirchhoff ellipses