universal microscopic correlation functions for products of independent ginibre matrices we consider the product of n complex non hermitian independent random matrices each of size n n with independent identically distributed gaussian entries ginibre matrices the joint probability distribution of the complex eigenvalues of the product matrix is found to be given by a determinantal point process as in the case of a single ginibre matrix but with a more complicated weight given by a meijer g function depending on n using the method of orthogonal polynomials we compute all eigenvalue density correlation functions exactly for finite n and fixed n they are given by the determinant of the corresponding kernel which we construct explicitly in the large n limit at fixed n we first determine the microscopic correlation functions in the bulk and at the edge of the spectrum after unfolding they are identical to that of the ginibre ensemble with n 1 and thus universal in contrast the microscopic correlations we find at the origin differ for each n 1 and generalize the known bessel law in the complex plane for n 2 to a new hypergeometric kernel 0fn 1

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

level spacing distributions and the bessel kernel scaling models of randomn n hermitian matrices and passing to the limitn leads to integral operators whose fredholm determinants describe the statistics of the spacing of the eigenvalues of hermitian matrices of large order for the gaussian unitary ensemble and for many others as well the kernel one obtains by scaling in the bulk of the spectrum is the sine kernel frac sin pi x y pi x y rescaling the gue at the edge of the spectrum leads to the kernel frac ai x ai y ai x ai y x y where ai is the airy function in previous work we found several analogies between properties of this airy kernel and known properties of the sine kernel a system of partial differential equations associated with the logarithmic differential of the fredholm determinant when the underlying domain is a union of intervals a representation of the fredholm determinant in terms of a painleve transcendent in the case of a single interval and also in this case asymptotic expansions for these determinants and related quantities achieved with the help of a differential operator which commutes with the integral operator in this paper we show that there are completely analogous properties for a class of kernels which arise when one rescales the laguerre or jacobi ensembles at the edge of the spectrum namely frac j alpha sqrt x sqrt y j alpha sqrt y sqrt x j alpha sqrt x j alpha sqrt y 2 x y wherej z is the bessel function of order in the cases 1 2 these become after a variable change the kernels which arise when taking scaling limits in the bulk of the spectrum for the gaussian orthogonal and symplectic ensembles in particular an asymptotic expansion we derive will generalize ones found by dyson for the fredholm determinants of these kernels

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

a simple step size selection algorithm for ode codes a very simple way of selecting the step size when solving an initial problem for a system of ordinary differential equations odes is examined it is shown to control the true error and the stability of the integration in a gross measure it is shown to be as efficient as control of the local error numerical results illustrate the analysis and explore the robustness of the approach

singular value correlation functions for products of wishart random matrices we consider the product of m quadratic random matrices with complex elements and no further symmetry where all matrix elements of each factor have a gaussian distribution this generalizes the classical wishart laguerre gaussian unitary ensemble with m 1 in this paper we first compute the joint probability distribution for the singular values of the product matrix when the matrix size n and the number m are fixed but arbitrary this leads to a determinantal point process which can be realized in two different ways first it can be written as a one matrix singular value model with a nonstandard jacobian or second for m 2 as a two matrix singular value model with a set of auxiliary singular values and a weight proportional to the meijer g function for both formulations we determine all singular value correlation functions in terms of the kernels of biorthogonal polynomials which we explicitly construct they are given in terms of the hypergeometric and meijer g functions generalizing the laguerre polynomials for m 1 our investigation was motivated from applications in telecommunication of multi layered scattering multiple input and multiple output channels we present the ergodic mutual information for finite n for such a channel model with m 1 layers of scatterers as an example

products of rectangular random matrices singular values and progressive scattering we discuss the product of m rectangular random matrices with independent gaussian entries which have several applications including wireless telecommunication and econophysics for complex matrices an explicit expression for the joint probability density function is obtained using the harish chandra itzykson zuber integration formula explicit expressions for all correlation functions and moments for finite matrix sizes are obtained using a two matrix model and the method of biorthogonal polynomials this generalizes the classical result for the so called wishart laguerre gaussian unitary ensemble or chiral unitary ensemble at m 1 and previous results for the product of square matrices the correlation functions are given by a determinantal point process where the kernel can be expressed in terms of meijer g functions we compare the results with numerical simulations and known results for the macroscopic level density in the limit of large matrices the location of the end points of support for the latter are analyzed in detail for general m finally we consider the so called ergodic mutual information which gives an upper bound for the spectral efficiency of a mimo communication channel with multifold scattering

smoothness of bounded solutions of nonlinear evolution equations abstract it is shown that in many cases globally defined bounded solutions of evolution equations are as smooth in time as the corresponding operator even if a general solution of the initial value problem is much less smooth i e initial values for bounded solutions are selected in such a way that optimal smoothness is attained in particular solutions which bifurcate from certain steady states such as periodic orbits almost periodic orbits and also homo and heteroclinic orbits have this property as examples a neutral functional differential equation a slightly damped non linear wave equation and a heat equation are considered in the latter case the space variable is included into the discussion of smoothness finally generalized hopf bifurcation in infinite dimensions is considered here smoothness of the bifurcation function is discussed and known results on the order of a focus are generalized

on the plane problem of the flow around a submerged beam abstract we consider the problem of the steady flow of an ideal heavy fluid around a submerged beam the problem is obtained from the free boundary problem of the flow past a submerged obstacle in the limit of bodies of vanishing thickness we introduce a special sobolev space formulation of the problem in term of a perturbed stream function and prove its unique solvability for every value of the unperturbed flow velocity with the possible exception of a discrete set depending on the geometry of the domain the asymptotic properties of the solutions are discussed

universal microscopic correlation functions for products of truncated unitary matrices we investigate the spectral properties of the product of m complex non hermitian random matrices that are obtained by removing l rows and columns of larger unitary random matrices uniformly distributed on the group u n l such matrices are called truncated unitary matrices or random contractions we first derive the joint probability distribution for the complex eigenvalues of the product matrix for fixed n l and m given by a standard determinantal point process in the complex plane the weight however is non standard and can be expressed in terms of the meijer g function the explicit knowledge of all eigenvalue correlation functions and the corresponding kernel allows us to take various large n and l limits at fixed m at strong non unitarity with l n finite the eigenvalues condense on a domain inside the unit circle at the edge and in the bulk we find the same universal microscopic kernel as for a single complex non hermitian matrix from the ginibre ensemble at the origin we find the same new universality classes labeled by m as for the product of m matrices from the ginibre ensemble keeping a fixed size of truncation l when n goes to infinity leads to weak non unitarity with most eigenvalues on the unit circle as for unitary matrices here we find a new microscopic edge kernel that generalizes the known results for m 1 we briefly comment on the case when each product matrix results from a truncation of different size lj

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

stable and unstable manifolds for hyperbolic bi semigroups we show the existence of local lipschitzian stable and unstable manifolds for the ill posed problem of perturbations of hyperbolic bi semigroups we do not assume backward nor forward uniqueness of solutions we do not use cut off functions because we do not assume global smallness conditions on the nonlinearities we introduce what we call dichotomous flows which recovers the symmetry between the past and the future thus we need to prove only a stable manifold theorem we modify the conley mcgehee moeckel approach to avoid appealing to wazewski principle and brouwer degree theory hence we allow both the stable and unstable directions to be infinite dimensional we apply our theorem to the elliptic system u u g u u in an infinite cylinder r

a note on the limiting mean distribution of singular values for products of two wishart random matrices the product of m complex random gaussian matrices of size n has recently been studied by akemann kieburg and wei they showed that for fixed m and n the joint probability distribution for the squared singular values of the product matrix forms a determinantal point process with a correlation kernel determined by certain biorthogonal polynomials that can be explicitly constructed we find that in the case m 2 the relevant biorthogonal polynomials are actually special cases of multiple orthogonal polynomials associated with macdonald functions modified bessel functions of the second kind which was first introduced by van assche and yakubovich with known results on asymptotic zero distribution of these polynomials and general theory on multiple orthogonal polynomial ensembles it is then easy to obtain an explicit expression for the distribution of squared singular values for the product of two complex random gaussian matrices in the limit of large matrix dimensions

small amplitude steady water waves with critical layers non symmetric waves abstract the problem for two dimensional steady water waves with vorticity is considered using methods of spatial dynamics we reduce the problem to a finite dimensional hamiltonian system the reduced system describes all small amplitude solutions of the problem and as an application we give a proof of the existence of non symmetric steady water waves whenever the number of roots of the dispersion equation is greater than one

universal distribution of lyapunov exponents for products of ginibre matrices starting from exact analytical results on singular values and complex eigenvalues of products of independent gaussian complex random n n matrices also called the ginibre ensemble we rederive the lyapunov exponents for an infinite product we show that for a large number t of product matrices the distribution of each lyapunov exponent is normal and we compute its t dependent variance as well as corrections in a large t expansion originally lyapunov exponents are defined for the singular values of the product matrix that represents a linear time evolution surprisingly a similar construction for the moduli of the complex eigenvalues yields the very same exponents and normal distributions to leading order we discuss a general mechanism for 2 2 matrices showing why the singular values and the radii of complex eigenvalues collapse onto the same value in the large t limit we thereby rederive newman s triangular law which has a simple interpretation as the radial density of complex eigenvalues in the circular law and study the commutativity of the two limits and on the global scale and on the local scale we show as a mathematical byproduct that a particular asymptotic expansion of a meijer g function with large index leads to a gaussian

singular values of products of ginibre random matrices multiple orthogonal polynomials and hard edge scaling limits akemann ipsen and kieburg recently showed that the squared singular values of products of m rectangular random matrices with independent complex gaussian entries are distributed according to a determinantal point process with a correlation kernel that can be expressed in terms of meijer g functions we show that this point process can be interpreted as a multiple orthogonal polynomial ensemble we give integral representations for the relevant multiple orthogonal polynomials and a new double contour integral for the correlation kernel which allows us to find its scaling limits at the origin hard edge the limiting kernels generalize the classical bessel kernels for m 2 they coincide with the scaling limits found by bertola gekhtman and szmigielski in the cauchy laguerre two matrix model which indicates that these kernels represent a new universality class in random matrix theory

center manifolds under nonuniform hyperbolicity maximal regularity abstract we establish the existence of invariant center manifolds with maximal c r regularity for a nonautonomous dynamics with discrete time we consider the general case of perturbations of a nonuniform exponential trichotomy our proof uses the fiber contraction principle and allows linear perturbations without any further effort

smooth center manifolds for nonuniformly partially hyperbolic trajectories abstract we establish the existence of unique smooth center manifolds for ordinary differential equations v a t v f t v in banach spaces assuming that v a t v admits a nonuniform exponential trichotomy this allows us to show the existence of unique smooth center manifolds for the nonuniformly partially hyperbolic trajectories in addition we prove that the center manifolds are as regular as the vector field our proof of the c k smoothness of the manifolds uses a single fixed point problem in an appropriate complete metric space to the best of our knowledge we establish in this paper the first smooth center manifold theorem in the nonuniform setting

comparison of some recent numerical methods for initial value problems for stiff ordinary differential equations we consider the combustion equation as one of the candidates from the class of stiff ordinary differential equations a solution over a length of time that is inversely proportional to d 0 where d 0 is a small disturbance of the pre ignition state is sought this problem has a transient at the midpoint of the integration interval the solution changes from being non stiff to stiff and afterwards becomes non stiff again we provide its asymptotic and numerical solution obtained via a variety of methods comparisons are made for the numerical results which we obtain with the matlab ode solvers ode45 ode15s and ode23s and some nonstandard finite difference methods results corresponding to standard finite difference method are also presented furthermore the discussion on these approaches along with the others provides several open problems for new and young researchers

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

the dichotomy theorem for evolution bi families we prove that the operator g the closure of the rst order dierential operator d dt d t on l2 r x is fredholm if and only if the not well posed equation u 0 t d t u t t 2 r has exponential dichotomies on r and r and the ranges of the dichotomy projections form a fredholm pair moreover the index of this pair is equal to the fredholm index of g here x is a hilbert space d t a b t a is the generator of a bi semigroup b is a bounded piecewise strongly continuous operator valued function also we prove some perturbations results and consider various examples of not well posed problems

steady flows of inviscid fluids under localized perturbations on essaie d obtenir tous les ecoulements stationnaires dans un canal avec un petit obstacle pour des nombres de froude proches de la valeur critique fo

abundance of entire solutions to nonlinear elliptic equations by the variational method abstract we study entire bounded solutions to the equation u u u 3 0 in r 2 our approach is purely variational and is based on concentration arguments and symmetry considerations this method allows us to construct in an unified way several types of solutions with various symmetries radial breather type rectangular triangular hexagonal etc both positive and sign changing it is also applicable for more general equations in any dimension

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

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

robustness of nonuniform exponential trichotomies in banach spaces abstract we establish the robustness under sufficiently small linear perturbations of nonuniform exponential trichotomies defined by linear equations x a t x in banach spaces we also establish the continuous dependence on the perturbation of the constants in the notion of trichotomy we consider both trichotomies in semi infinite intervals and trichotomies in r

a center stable manifold theorem for differential equations in banach spaces we prove a center stable manifold theorem for a class of differential equations in infinite dimensional banach spaces

normal hyperbolicity of center manifolds and saint venant s principle the concept of normal hyperbolicity of center manifolds is generalized to infinite dimensional differential equations in particular to elliptic problems in cylindrical domains it is shown that all solutions u staying close to the center manifold for t l l satisfy an estimate of the form left u t tilde u t right leqslant ce alpha l t where c and are independent of l and is a solution on the center manifold these results are applied to saint venant s principle for the static deformation of nonlinearly elastic prismatic bodies the use of the center manifold permits the effective treatment of the general case of non zero resultant forces and moments acting on each cross section

polynomial contractions and lyapunov regularity abstract the purpose of this note is twofold to introduce the notion of polynomial contraction for a linear nonautonomous dynamics with discrete time and to show that it persists under sufficiently small linear and nonlinear perturbations the notion of polynomial contraction mimics the notion of exponential contraction but with the exponential decay replaced by a polynomial decay we show that this behavior is exhibited by a large class of dynamics by giving necessary conditions in terms of polynomial lyapunov exponents finally we establish the persistence of the asymptotic stability of a polynomial contraction under sufficiently small linear and nonlinear perturbations we also consider the case of nonuniform polynomial contractions for which the lyapunov stability is not uniform

continuous numerical methods for odes with defect control abstract over the last decade several general purpose numerical methods for ordinary differential equations odes have been developed which generate a continuous piecewise polynomial approximation that is defined for all values of the independent variable in the range of interest for such methods it is possible to introduce measures of the quality of the approximate solution based on how well the piecewise polynomial satisfies the ode this leads naturally to the notion of defect control numerical methods that adopt error estimation and stepsize selection strategies in order to control the magnitude of the associated defect can be very effective and such methods are now being widely used in this paper we will review the advantages of this class of numerical methods and present examples of how they can be effectively applied we will focus on numerical methods for initial value problems ivps and boundary value problems bvps where most of the developments have been introduced but we will also discuss the implications and related developments for other classes of odes such as delay differential equations ddes and differential algebraic equations daes

moving mesh methods for one dimensional hyperbolic problems using clawpack we develop a one dimensional moving mesh method for hyperbolic systems of conser vation laws this method is based on the high resolution finite volume wave propagation method implemented in the clawpack software package a modified system of conservation laws is solved on a fixed uniform computational grid with a grid mapping function computed simultaneously in such a way that in physical space certain features are tracked by cell interfaces the method is tested on a shock tube problem with multiple reflections where the contact discontinuity is tracked and also on two multifluid problems where the interface between two distinct gases is tracked one is a standard test problem and the other also involves a moving piston whose motion is also tracked by the moving mesh 2003 elsevier science ltd all rights reserved

nonconvergent bounded solutions of semilinear heat equations on arbitrary domains abstract we consider the dirichlet problem for the semilinear heat equation 0 1 u t u g x u x where is an arbitrary bounded domain in r n n 2 with c 2 boundary we find a c function g x u such that 0 1 has a bounded solution whose limit set is a continuum of equilibria this extends and improves an earlier result of the first author with rybakowski in which is a disk in r 2 and g is of finite differentiability class we also show that 0 1 can have an infinite dimensional manifold of nonconvergent bounded trajectories

exact theory of secondary supercritical solutions for steady surface waves over a bump abstract two dimensional steady surface waves on an ideal fluid over a small and symmetric bump are studied when the froude number f of the uniform flow far upstream is greater than unity recent numerical results for a symmetric bump indicate that corresponding to f 1 there may exist two solutions for the problem one approaches the uniform flow and the other a solitary wave as the bump size goes to zero in this paper we use a two parameter asymptotic expansion to derive an explicit approximate expression for the later solution and show that the expansion is indeed an asymptotic approximation to the solution of the exact nonlinear equations

smooth conjugacy of centre manifolds in this paper we prove that for a system of ordinary differential equations of class c r 1 1 r 0 and two arbitrary c r 1 1 local centre manifolds of a given equilibrium point the equations when restricted to the centre manifolds are c r conjugate the same result is proved for similinear parabolic equations the method is based on the geometric theory of invariant foliations for centre stable and centre unstable manifolds

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

mixed type hermite pade approximation inspired by the degasperis procesi equation abstract in this work we present new results on the convergence of diagonal sequences of certain mixed type hermite pade approximants of a nikishin system the study is motivated by a mixed hermite pade approximation scheme used in the construction of solutions of a degasperis procesi peakon problem and germane to the analysis of the inverse spectral problem for the discrete cubic string

exponential dichotomies for solitary wave solutions of semilinear elliptic equations on infinite cylinders in applications solitary wave solutions of semilinear elliptic equations u g u u 0 x y r in infinite cylinders frequently arise as travelling waves of parabolic equations as such their bifurcations are an interesting issue interpreting elliptic equations on infinite cylinders as dynamical systems inxhas proved very useful still there are major obstacles in obtaining for instance bifurcation results similar to those for ordinary differential equations in this article persistence and continuation of exponential dichotomies for linear elliptic equations is proved with this technique at hands lyapunov schmidt reduction near solitary waves can be applied as an example existence of shift dynamics near solitary waves is shown if a perturbation h x u u periodic inxis added

adaptive stiff solvers at low accuracy and complexity this paper is concerned with adaptive stiff solvers at low accuracy and complexity for systems of ordinary differential equations the considered stiff solvers are two second order rosenbrock methods with low complexity and the bdf method of the same order for the adaptive algorithm we propose to use a monitor function defined by comparing a measure of the local variability of the solution times the used step size and the order of magnitude of the solution instead of the classical approach based on some local error estimation this simple step size selection procedure is implemented in order to control the behavior of the numerical solution it is easily used to automatically adjust the step size as the calculation progresses until user specified tolerance bounds for the introduced monitor function are fulfilled this leads to important advantages in accuracy efficiency and general easier of use at the end of the paper we present two numerical tests which show the performance of the implementation of the stiff solvers with the proposed adaptive procedure

dropping the independence singular values for products of two coupled random matrices we study the singular values of the product of two coupled rectangular random matrices as a determinantal point process each of the two factors is given by a parameter dependent linear combination of two independent complex gaussian random matrices which is equivalent to a coupling of the two factors via an itzykson zuber term we prove that the squared singular values of such a product form a biorthogonal ensemble and establish its exact solvability the parameter dependence allows us to interpolate between the singular value statistics of the laguerre ensemble and that of the product of two independent complex ginibre ensembles which are both known we give exact formulae for the correlation kernel in terms of a complex double contour integral suitable for the subsequent asymptotic analysis in particular we derive a christoffel darboux type formula for the correlation kernel based on a five term recurrence relation for our biorthogonal functions it enables us to find its scaling limit at the origin representing a hard edge the resulting limiting kernel coincides with the universal meijer g kernel found by several authors in different ensembles we show that the central limit theorem holds for the linear statistics of the singular values and give the limiting variance explicitly

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

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

eigenvalues and singular values of products of rectangular gaussian random matrices we derive exact analytic expressions for the distributions of eigenvalues and singular values for the product of an arbitrary number of independent rectangular gaussian random matrices in the limit of large matrix dimensions we show that they both have power law behavior at zero and determine the corresponding powers we also propose a heuristic form of finite size corrections to these expressions which very well approximates the distributions for matrices of finite dimensions

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

mathematical and numerical modeling for a bio chemical aquarium based on bio chemical ground we derive an aquarium mathematical model useful for predicting dangerous situations as well as for the startup cycle this model is a basic step toward a more complex advection diffusion reaction model in 3d space variables it defines the reaction part of the more complex partial differential equations model for the numerical solution of our aquarium model we apply a low complexity second order method combined with a simple adaptive step size selection procedure the low accuracy and complexity of the resulting numerical algorithm are motivated because of the high complexity of the final 3d model the reported numerical results and comparisons with the know how available in literature show the validity of the proposed model