iterative methods for a fourth order boundary value problem abstract we consider a fourth order nonlinear ordinary differential equation together with two point boundary conditions and provide a priori error estimates on the length of the interval b a so that the picard s iterative method the approximate picard s iterative method and the quasilinear iterative method convergence to the solution of the problem

products of independent quaternion ginibre matrices and their correlation functions we discuss the product of independent induced quaternion 4 ginibre matrices and the eigenvalue correlations of this product matrix the joint probability density function for the eigenvalues of the product matrix is shown to be identical to that of a single ginibre matrix but with a more complicated weight function we find the skew orthogonal polynomials corresponding to the weight function of the product matrix and use the method of skew orthogonal polynomials to compute the eigenvalue correlation functions for product matrices of finite size the radial behavior of the density of states is studied in the limit of large matrices and the macroscopic density is discussed the microscopic limit at the origin at the edge s and in the bulk is also discussed for the radial behavior of the density of states

iterative methods for solving right focal point boundary value problems abstract we provide necessary and sufficient conditions for the existence of a solution of n th order differential equation satisfying right focal point boundary conditions for the linear problems we propose forward forward and backward backward one pass practical shooting methods which convert the boundary value problems to its equivalent initial value problems for the nonlinear problems iterative methods are discussed and a priori conditions for the convergence are obtained several examples are also illustrated

asymptotic mean density of sub unitary ensembles the asymptotic behaviour of the mean spectral density for the product of n n random unitary matrices distributed according to the haar measure and a fixed diagonal matrix is investigated the large n limit of the exact expression derived by wei and fyodorov 2008 j phys a math theor 41 502001 is calculated by a modification of the saddle point method it is shown that in the bulk the result coincides with the one obtained within the free probability theory by haagerup and larsen 2000 j funct anal 176 331 close to the edge points of the bulk asymptotics the mean density for large n is described by a universal function depending on a certain scaling variable the large deviation formulae valid outside the bulk are also derived obtained formulae agree well with the results of direct numerical calculations

limit laws for random matrices and free products in earlier articles we studied a kind of probability theory in the framework of operator algebras with the tensor product replaced by the free product we prove here that free random variables naturally arise as limits of random matrices and that wigner s semicircle law is a consequence of the central limit theorem for free random variables in this way we obtain a non commutative limit distribution of a general gaussian random matrix as an operator in a certain operator algebra wigner s law being given by the trace of the spectral measure of the selfadjoint component of this operator

products of independent elliptic random matrices for fixed m 1 we study the product of m independent n times n elliptic random matrices as n tends to infinity our main result shows that the empirical spectral distribution of the product converges with probability 1 to the m th power of the circular law regardless of the joint distribution of the mirror entries in each matrix this leads to a new kind of universality phenomenon the limit law for the product of independent random matrices is independent of the limit laws for the individual matrices themselves our result also generalizes earlier results of gotze tikhomirov on the asymptotic spectrum of products of independent random matrices available at http arxiv org abs 1012 2710 and o rourke soshnikov j probab 16 81 2219 2245 2011 concerning the product of independent iid random matrices

spectral theory of sparse non hermitian random matrices sparse non hermitian random matrices arise in the study of disordereda 13 physical systems with asymmetric local interactions and have applications ranginga 13 from neural networks to ecosystem dynamics the spectral characteristics of thesea 13 matrices provide crucial information on system stability and susceptibility however a 13 their study is greatly complicated by the twin challenges of a lack of symmetry anda 13 a sparse interaction structure in this review we provide a concise and systematica 13 introduction to the main tools and results in this field we show how the spectraa 13 of sparse non hermitian matrices can be computed via an analogy with infinitea 13 dimensional operators obeying certain recursion relations with reference to threea 13 illustrative examples adjacency matrices of regular oriented graphs adjacencya 13 matrices of oriented erd s renyi graphs and adjacency matrices of weighted orienteda 13 erd s renyi graphs we demonstrate the use of these methods to obtain both analytica 13 and numerical results for the spectrum the spectral distribution the location of outliera 13 eigenvalues and the statistical properties of eigenvectors a 13

boundary value problems for higher order integro differential equations where r t x t x t x j t and a is a continuous operator map c l al a c ar a the function f is continuous in all of its arguments the problem l l 1 2 includes several particular cases for example the boundary value problems for the differential equations if a 0 considered in 7 8 and the references therein for the integro differential equations of the volterra type if

boundary value problems for second order differential equations of sobolev type abstract in this paper we provide necessary and sufficient conditions for the existence and uniqueness of solutions of second order differential equations of sobolev type satisfying some boundary conditions we also give sufficient conditions so that the picard and approximate picard method converge to the unique solution of these boundary value problems

outlier eigenvalues for deformed i i d random matrices we consider a square random matrix of size n of the form a y where a is deterministic and y has iid entries with variance 1 n under mild assumptions as n grows the empirical distribution of the eigenvalues of a y converges weakly to a limit probability measure beta on the complex plane this work is devoted to the study of the outlier eigenvalues i e eigenvalues in the complement of the support of beta even in the simplest cases a variety of interesting phenomena can occur as in earlier works we give a sufficient condition to guarantee that outliers are stable and provide examples where their fluctuations vary with the particular distribution of the entries of y or the jordan decomposition of a we also exhibit concrete examples where the outlier eigenvalues converge in distribution to the zeros of a gaussian analytic function

travelling wave solutions to the korteweg de vries burgers equation the existence and certain qualitative properties of travelling wave solutions to the korteweg de vries burgers equation are established the limiting behaviour of these waves when e tends to zero and when tends to zero is examined together with a singular limit wherein both e and tend to zero

numerical solution of kdv equation using modified bernstein polynomials abstract here we present an algorithm for approximating numerical solution of korteweg de vries kdv equation in a modified b polynomial basis a set of continuous polynomials over the spatial domain is used to expand the desired solution requiring discretization with only the time variable galerkin method is used to determine the expansion coefficients to construct initial trial functions for the time variable the system of equations is solved using fourth order runge kutta method the accuracy of the solutions is dependent on the size of the b polynomial basis set we have presented our numerical result with an exact analytical result excellent agreement is found between exact and approximate solutions this procedure has a potential to be used in more complex system of differential equations where no exact solution is available

quasilinearization and approximate quasilinearization for multipoint boundary value problems quasilinearization technique has been applied to a general nonlinear lidstone boundary value problem for the construction of a sequence of its approximate solutions xn t sufficient conditions

resolvents of r diagonal operators we consider the resolvent a 1 of any diagonal operator a in a ii 1 factor our main theorem theorem 1 1 gives a universal asymptotic formula for the norm of such a resolvent en route to its proof we calculate the transform of the operator c 2 where c is voiculescu s circular operator and we give an asymptotic formula for the negative moments of a 2 for any diagonal a we use a mixture of complex analytic and combinatorial techniques each giving finer information where the other can give only coarse detail in particular we introduce partition structure diagrams in section 4 a new combinatorial structure arising in free probability

memory free dynamics for the thouless anderson palmer equations of ising models with arbitrary rotation invariant ensembles of random coupling matrices we propose an iterative algorithm for solving the thouless anderson palmer equations of ising models with arbitrary rotation invariant random coupling matrices in the thermodynamic limit we prove by means of the dynamical functional method that the proposed algorithm converges when the so called de almeida thouless criterion is fulfilled moreover we give exact analytical expressions for the rate of the convergence

commutators of free random variables let a be a unital c algebra given together with a specified state a c consider two selfadjoint elements a b of a which are free with respect to in the sense of the free probability theory of voiculescu let us denote c i ab ba where the i in front of the commutator is introduced to make c selfadjoint in this paper we show how the spectral distribution of c can be calculated from the spectral distributions of a and b some properties of the corresponding operation on probability measures are also discussed the methods we use are combinatorial based on the description of freeness in terms of non crossing partitions an important ingredient is the notion of r diagonal pair introduced and studied in our previous paper 12

classical and free infinitely divisible distributions and random matrices we construct a random matrix model for the bijection between classical and free infinitely divisible distributions for every d 1 we associate in a quite natural way to each infinitely divisible distribution a distribution p d on the space of d d hermitian matrices such that p d p d p d the spectral distribution of a random matrix with distribution p d converges in probability to when d tends to it gives among other things a new proof of the almost sure convergence of the spectral distribution of a matrix of the gue and a projection model for the marchenko pastur distribution in an analogous way for every d 1 we associate to each infinitely divisible distribution a distribution l d on the space of complex non hermitian d x d random matrices if is symmetric the symmetrization of the spectral distribution of m d when m d is l d distributed converges in probability to

solutions of differential equations in a bernstein polynomial basis an algorithm for approximating solutions to differential equations in a modified new bernstein polynomial basis is introduced the algorithm expands the desired solution in terms of a set of continuous polynomials over a closed interval and then makes use of the galerkin method to determine the expansion coefficients to construct a solution matrix formulation is used throughout the entire procedure however accuracy and efficiency are dependent on the size of the set of bernstein polynomials and the procedure is much simpler compared to the piecewise b spline method for solving differential equations a recursive definition of the bernstein polynomials and their derivatives are also presented the current procedure is implemented to solve three linear equations and one nonlinear equation and excellent agreement is found between the exact and approximate solutions in addition the algorithm improves the accuracy and efficiency of the traditional methods for solving differential equations that rely on much more complicated numerical techniques this procedure has great potential to be implemented in more complex systems where there are no exact solutions available except approximations

discrete polynomial interpolation green s functions maximum principles error bounds and boundary value problems abstract we construct discrete interpolating polynomials provide explicit representations of discrete green s functions give several identities and inequalities for these green s functions use the explicit forms of the interpolating polynomials and that of green s functions to establish several maximum principles further we obtain error bounds in discrete polynomial interpolation and use them to study existence and uniqueness of the discrete boundary value problems these bounds are also used to provide sufficient conditions for the convergence of the picard s method the approximate picard s method quasilinearization and the approximate quasilinearization the monotone convergence of the picard s iterative method is also analysed

circular law theorem for random markov matrices let xjk jk 1 be i i d nonnegative random variables with bounded density mean m and finite positive variance 2 let m be the n n random markov matrix with i i d rows defined by m jk x jk x j1 cdots x jn in particular when x11 follows an exponential law the random matrix m belongs to the dirichlet markov ensemble of random stochastic matrices let 1 n be the eigenvalues of sqrt n m i e the roots in mathbb c of its characteristic polynomial our main result states that with probability one the counting probability measure frac 1 n delta lambda 1 cdots frac 1 n delta lambda n converges weakly as n to the uniform law on the disk z in mathbb c z leq m 1 sigma the bounded density assumption is purely technical and comes from the way we control the operator norm of the resolvent

r transforms of free joint distributions and non crossing partitions abstract we introduce an extension of the notion of r transform defined by d voiculescu to free joint distributions i e normalized linear functionals on algebras of non commutative polynomials in several indeterminates we point out that the r transform has good behavior with respect to the operations of free product and free convolution of joint distributions we prove that the explicit computation of the inverse r transform of a joint distribution is done via a formula of summation over the lattice of non crossing partitions which shows that the r transform is the operator theoretic counterpart of the free cumulants considered in the combinatorial approach of r speicher moreover in connection to the equivariance of the multidimensional r transform under rotations we show that a natural free analogue of a classical result about rotations of independent random variables is holding

mixed moments of voiculescu s gaussian random matrices abstract we prove an explicit combinatorial formula for the mixed moments of a family of independent selfadjoint gaussian random matrices as a corollary we show that the convergence of such mixed moments proved by voiculescu is as fast as o 1 n 2 and we use this fact to give a short proof of the fact recently proved by hiai and petz that the convergence in voiculescu s result actually holds almost surely this latter result is then used to give an alternative proof of the result by s wassermann that free group factors embed into ultra products of matrix algebras moreover the almost sure convergence result is used to obtain asymptotic lower respectively upper bounds on the maximum respectively minimum of the spectrum of s s for gaussian random matrices s with operator entries in particular those studied by u haagerup and the author 1999 documenta math 4 341 450

lyapunov exponents of free operators abstract lyapunov exponents of a dynamical system are a useful tool to gauge the stability and complexity of the system this paper offers a definition of lyapunov exponents for a sequence of free linear operators the definition is based on the concept of the extended fuglede kadison determinant we establish the existence of lyapunov exponents derive formulas for their calculation and show that lyapunov exponents of free variables are additive with respect to operator product we illustrate these results using an example of free operators whose singular values are distributed by the marchenko pastur law and relate this example to c m newman s triangle law for the distribution of lyapunov exponents of large random matrices with independent gaussian entries as an interesting by product of our results we derive a relation between the extended fuglede kadison determinant and voiculescu s s transform

dt operators and decomposability of voiculescu s circular operator the dt operators are introduced one for every pair c consisting of a compactly supported borel probability measure on the complex plane and a constant c 0 these are operators on hilbert space that are defined as limits in moments of certain upper triangular random matrices the dt operators include voiculescu s circular operator and elliptic deformations of it as well as the circular free poisson operators we show that every dt operator is strongly decomposable we also show that a dt operator generates a ii1 factor whose isomorphism class depends only on the number and sizes of atoms of those dt operators that are also r diagonal are identified for a quasi nilpotent dt operator t we find the distribution of t t and a recursion formula for general moments of t

on the right focal point boundary value problems for integro differential equations abstract for the nth order nonlinear integro differential equations subject to right focal point boundary conditions we provide necessary and sufficient conditions for the existence uniqueness and convergence of an approximate iterative method

the wigner semi circle law and eigenvalues of matrix valued diffusions a system of stochastic differential equations for the eigenvalues of a symmetric matrix whose components are independent ornstein uhlenbeck processes is derived this corresponds to a diffusion model of an interacting particles system with linear drift towards the origin and electrostatic inter particle repulsion the associated empirical distribution of particles is shown to converge weakly as the number of particles tends to infinity to a limiting measure valued process which may be characterized as the weak solution of a deterministic ode the wigner semi circle density is found to be one of the equilibrium points of this limiting equation

on a class of free levy laws related to a regression problem abstract the free meixner laws arise as the distributions of orthogonal polynomials with constant coefficient recursions we show that these are the laws of the free pairs of random variables which have linear regressions and quadratic conditional variances when conditioned with respect to their sum we apply this result to describe free levy processes with quadratic conditional variances and to prove a converse implication related to asymptotic freeness of random wishart matrices

large deviation for the empirical eigenvalue density of truncated haar unitary matrices let u m be an m m haar unitary matrix and u m n be its n n truncation in this paper the large deviation is proven for the empirical eigenvalue density of u m n as m n and n the rate function and the limit distribution are given explicitly u m n is the random matrix model of quq where u is a haar unitary in a finite von neumann algebra q is a certain projection and they are free the limit distribution coincides with the brown measure of the operator quq

continuous family of invariant subspaces for r diagonal operators we show that every r diagonal operator x has a continuous family of invariant subspaces relative to the von neumann algebra generated by x this allows us to find the brown measure of x and to find a new conceptual proof that voiculescu s s transform is multiplicative our considerations base on a new concept of r diagonality with amalgamation for which we give several equivalent characterizations

use of modified bernstein polynomials to solve kdv burgers equation numerically abstract numerical solution of korteweg de veries burgers kdvb equation is presented using modified bernstein polynomials b polynomials over the spatial domain b polynomials are used to expand the desired solution requiring discretization with only the time variable galerkin method is used to determine the expansion coefficients to construct initial trial functions we use fourth order runge kutta method to solve the system of equations for the time variable the accuracy of the solutions is dependent on the size of the b polynomial basis set numerical results obtained using this method are compared with existing analytical results excellent agreement is found between the exact solution and approximate solution obtained by this method

commutative law for products of infinitely large isotropic random matrices ensembles of isotropic random matrices are defined by the invariance of the probability measure under the left and right multiplication by an arbitrary unitary matrix we show that the multiplication of large isotropic random matrices is spectrally commutative and self averaging in the limit of infinite matrix size n the notion of spectral commutativity means that the eigenvalue density of a product abc of such matrices is independent of the order of matrix multiplication for example the matrix abcd has the same eigenvalue density as adcb in turn the notion of self averaging means that the product of n independent but identically distributed random matrices which we symbolically denote by aaa has the same eigenvalue density as the corresponding power a n of a single matrix drawn from the underlying matrix ensemble for example the eigenvalue density of abccabc is the same as that of a 2 b 2 c 3 we also discuss the singular behavior of the eigenvalue and singular value densities of isotropic matrices and their products for small eigenvalues 0 we show that the singularities at the origin of the eigenvalue density and of the singular value density are in one to one correspondence in the limit n the eigenvalue density of an isotropic random matrix has a power law singularity at the origin s with a power se 0 2 when and only when the density of its singular values has a power law singularity with a power s 4 s these results are obtained analytically in the limit n we supplement these results with numerical simulations for large but finite n and discuss finite size effects for the most common ensembles of isotropic random matrices

the analogues of entropy and of fisher s information measure in free probability theory i analogues of the entropy and fisher information measure for random variables in the context of free probability theory are introduced monotonicity properties and an analogue of the cramer rao inequality are proved

brown s spectral distribution measure for r diagonal elements in finite von neumann algebras abstract in 1983 l g brown introduced a spectral distribution measure for non normal elements in a finite von neumann algebra m with respect to a fixed normal faithful tracial state in this paper we compute brown s spectral distribution measure in case t has a polar decomposition t uh where u is a haar unitary and u and h are free when ker t 0 this is equivalent to that t t is an r diagonal pair in the sense of nica and speicher the measure t is expressed explicitly in terms of the s transform of the distribution t t of the positive operator t t in case t is a circular element i e t x1 ix2 2 where x1 x2 is a free semicircular system then sp t d the closed unit disk and t has constant density 1 on d

random regularization of brown spectral measure abstract we generalize a recent result of haagerup namely we show that a convolution with a standard gaussian random matrix regularizes the behavior of fuglede kadison determinant and brown spectral distribution measure in this way it is possible to establish a connection between the limit eigenvalues distributions of a wide class of random matrices and the brown measure of the corresponding limits

eigenvalues of non hermitian random matrices and brown measure of non normal operators hermitian reduction and linearization method abstract we study the brown measure of certain non hermitian operators arising from voiculescu s free probability theory usually those operators appear as the limit in moments of certain ensembles of non hermitian random matrices and the brown measure gives then a canonical candidate for the limit eigenvalue distribution of the random matrices a prominent class for our operators is given by polynomials in free variables other explicit examples include r diagonal elements and elliptic elements for which the brown measure was already known and a new class of triangular elliptic elements our method for the calculation of the brown measure is based on a rigorous mathematical treatment of the hermitian reduction method as considered in the physical literature combined with subordination and the linearization trick

squared eigenvalue condition numbers and eigenvector correlations from the single ring theorem we extend the so called single ring theorem feinberg and zee 1997 nucl phys b 504 579 also known as the haagerup larsen theorem haagerup and larsen 2000 j funct anal 176 331 we do this by showing that in the limit when the size of the matrix goes to infinity a particular correlator between left and right eigenvectors of the relevant non hermitian matrix x being the spectral density weighted by the squared eigenvalue condition number is given by a simple formula involving only the radial spectral cumulative distribution function of x we show that this object allows the calculation of the conditional expectation of the squared eigenvalue condition number we give examples and provide a cross check of the analytic prediction by the large scale numerics

outliers of random perturbations of toeplitz matrices with finite symbols consider an n times n toeplitz matrix t n with symbol varvec a lambda sum ell d 2 d 1 a ell lambda ell perturbed by an additive noise matrix n gamma e n where the entries of e n are centered i i d random variables of unit variance and gamma 1 2 it is known that the empirical measure of eigenvalues of the perturbed matrix converges weakly as n rightarrow infty to the law of varvec a u where u is distributed uniformly on mathbb s 1 in this paper we consider the outliers i e eigenvalues that are at a positive n independent distance from varvec a mathbb s 1 we prove that there are no outliers outside mathrm spec t varvec a the spectrum of the limiting toeplitz operator with probability approaching one as n rightarrow infty in contrast in mathrm spec t varvec a setminus varvec a mathbb s 1 the process of outliers converges to the point process described by the zero set of certain random analytic functions the limiting random analytic functions can be expressed as linear combinations of the determinants of finite sub matrices of an infinite dimensional matrix whose entries are i i d having the same law as that of e n the coefficients in the linear combination depend on the roots of the polynomial p z varvec a lambda varvec a lambda z lambda d 2 and semi standard young tableaux with shapes determined by the number of roots of p z varvec a lambda 0 that are greater than one in moduli

boundary value problems occurring in plate deflection theory abstract the paper deals with several iterative methods for solving some boundary value problems occurring in plate deflection theory making it possible to avoid the lengthy problem preparation which is required to solve these problems by finite difference methods

dual lukacs regressions for non commutative variables abstract dual lukacs type characterizations of random variables in free probability are studied here first we develop a freeness property satisfied by lukacs type transformations of free poisson and free binomial non commutative variables which are free second we give a characterization of non commutative free poisson and free binomial variables by properties of first two conditional moments which mimic lukacs type assumptions known from classical probability more precisely our result is a non commutative version of the following result known in classical probability if u v are independent real random variables such that e v 1 u u v and e v 2 1 u 2 u v are non random then v has a gamma distribution and u has a beta distribution

method of chasing for multipoint boundary value problems the method of chasing developed by gel fand and lokutsiyevskii is completely formulated here for nth order ordinary linear differential equations subject to separated multipoint boundary conditions

sample variance in free probability abstract let x 1 x 2 x n denote i i d centered standard normal random variables then the law of the sample variance q n i 1 n x i x 2 is the 2 distribution with n 1 degrees of freedom it is an open problem in classical probability to characterize all distributions with this property and in particular whether it characterizes the normal law in this paper we present a solution of the free analogue of this question and show that the only distributions whose free sample variance is distributed according to a free 2 distribution are the semicircle law and more generally so called odd laws by which we mean laws with vanishing higher order even cumulants in the way of proof we derive an explicit formula for the free cumulants of q n which shows that indeed the odd cumulants do not contribute and which exhibits an interesting connection to the concept of r cyclicity

eigenvector statistics of the product of ginibre matrices we develop a method to calculate left right eigenvector correlations of the product of m independent n n complex ginibre matrices for illustration we present explicit analytical results for the vector overlap for a couple of examples for small m and n we conjecture that the integrated overlap between left and right eigenvectors is given by the formula o 1 m 2 n 1 and support this conjecture by analytical and numerical calculations we derive an analytical expression for the limiting correlation density as n for the product of ginibre matrices as well as for the product of elliptic matrices in the latter case we find that the correlation function is independent of the eccentricities of the elliptic laws

the single ring theorem we study the empirical measure lan of the eigenvalues of non normal square matrices of the form an untnvn with un vn independent haar distributed on the unitary group and tn real diagonal we show that when the empirical measure of the eigenvalues of tn converges and tn satisfies some technical conditions lan converges towards a rotationally invariant measure on the complex plane whose support is a single ring in particular we provide a complete proof of feinberg zee single ring theorem 6 we also consider the case where un vn are independent haar distributed on the orthogonal group