the inverses of toeplitz band matrices abstract the elements of the inverse of a toeplitz band matrix are given in terms ofthe solution of a difference equation the expression for these elements is a quotient of determinants whose orders depend the number of nonzero superdiagonals but not on the order of the matrix thus the formulae are particularly simple for lower triangular and lower hessenberg toeplitz matrices when the number of nonzero superdiagonals is small sufficient conditions on the solution of the abovementioned difference equation can be given to ensure that the inverse matrix is positive if the inverse is positive the row sums can be expressed in terms of the solution of the difference equation

a fast elementary algorithm for computing the determinant of toeplitz matrices in recent years a number of fast algorithms for computing the determinant of a toeplitz matrix were developed the fastest algorithm we know so far is of order k 2 log n k 3 where n is the number of rows of the toeplitz matrix and k is the bandwidth size this is possible because such a determinant can be expressed as the determinant of certain parts of the n th power of a related k i k companion matrix in this paper we give a new elementary proof of this fact and provide various examples we give symbolic formulas for the determinants of toeplitz matrices in terms of the eigenvalues of the corresponding companion matrices when k is small

a generalization of the entropy power inequality with applications the authors prove the following generalization of the entropy power inequality h ax or h ax where h denotes joint differential entropy x x sub 1 x sub n is a random vector with independent components x x x sub n is a gaussian vector with independent components such that h x sub i h x sub i i 1 n and a is any matrix this generalization of the entropy power inequality is applied to show that a non gaussian vector with independent components becomes closer to gaussianity after a linear transformation where the distance to gaussianity is measured by the information divergence another application is a lower bound greater than zero for the mutual information between nonoverlapping spectral components of a non gaussian white process they also describe a dual generalization of the fisher information inequality

three results in connection with inverse matrices abstract first we show that the moore penrose solution of an arbitrary system of linear equations is a convex combination of the solutions of all uniquely solvable partial systems the other two results concern the elements of inverse toeplitz band matrices namely the asymptotic behavior of a determinant appearing in a formula of d s meek and a modification of a formula of w d hoskins and p j ponzo for matrices with binomial coefficients in the limit case

spectral and computational properties of band symmetric toeplitz matrices abstract we are investigating spectral properties of band symmetric toeplitz matrices bst matrices by giving a suitable representation of a bst matrix we achieve separation results and multiplicity conditions for the eigenvalues of a 7 r 5 diagonal bst matrix and also structural properties of the eigenvectors we give eigenvalue bounds for a 2k 1 diagonal bst matrix and also necessary and sufficient conditions for positive definiteness which are easy to check the same conditions apply in the case of block bst matrices either with full blocks or with bst blocks we exhibit fast computational methods for the evaluation of the determinant and the characteristic polynomial of a bst matrix either for sequential or for parallel computations two algorithms based on the bisection technique and newton s method are shown to be very fast for computing the eigenvalues of a 7 or 5 diagonal bst matrix

simple deterministic dynamical systems with fractal diffusion coefficients we analyze a simple model of deterministic diffusion the model consists of a one dimensional array of scatterers with moving point particles the particles move from one scatterer to the next according to a piecewise linear expanding deterministic map on unit intervals the microscopic chaotic scattering process of the map can be changed by a control parameter the macroscopic diffusion coefficient for the moving particles is well defined and depends upon the control parameter we calculate the diffusion coefficent and the largest eigenmodes of the system by using markov partitions and by solving the eigenvalue problems of respective topological transition matrices for different boundary conditions we find that the largest eigenmodes of the map match the ones of the simple phenomenological diffusion equation our main result is that the diffusion coefficient exhibits a fractal structure as a function of the control parameter we provide qualitative and quantitative arguments to explain features of this fractal structure

unimodality convexity and applications properties of univariate unimodal distributions concepts of multivariate unimodality some more notions of unimodality unimodality for discrete distributions unimodality of infinitely divisible distributions unimodality and notions of dependence ordering of distributions by peakedness applications of unimodality in statistical inference convexity in reliability theory appendix

an elementary algorithm for computing the determinant of pentadiagonal toeplitz matrices over the last 25 years various fast algorithms for computing the determinant of a pentadiagonal toeplitz matrices were developed in this paper we give a new kind of elementary algorithm requiring email protected email protected email protected 30k o logn operations where k 4 is an integer that needs to be chosen freely at the beginning of the algorithm for example we can compute det t n in n o logn and 82n o logn operations if we choose k as 56 and 2815 n 4 respectively for various applications it will be enough to test if the determinant of a pentadiagonal toeplitz matrix is zero or not as in another result of this paper we used modular arithmetic to give a fast algorithm determining when determinants of such matrices are non zero this second algorithm works only for toeplitz matrices with rational entries

a new entropy power inequality a strengthened version of shannon s entropy power inequality for the case where one of the random vectors involved is gaussian is proved in particular it is shown that if independent gaussian noise is added to an arbitrary multivariate random variable the entropy power of the resulting random variable is a concave function of the variance power of the added noise the strengthened inequality is shown to hold for the class of stable distributions

efficient algorithms for the evaluation of the eigenvalues of block banded toeplitz matrices on presente des algorithmes de calcul des valeurs propres des matrices bandes blocs de toeplitz

spectral analysis of nonsymmetric quasi toeplitz matrices with applications to preconditioned multistep formulas the eigenvalue spectrum of a class of nonsymmetric preconditioned matrices arising in time dependent partial differential equations is analyzed and discussed the matrices generated by the underlying numerical integrators are small rank perturbations of block toeplitz matrices circulant like preconditioners based on the former are considered the eigenvalue distribution of the preconditioned matrix influences often crucially the convergence of krylov iterative accelerators due to several reasons lack of symmetry band structure and coefficients depending on the size the classical approach based on smooth generating functions gives very little insight here therefore to characterize the eigenvalues a difference equation approach exploiting the band toeplitz and circulant patterns generalizing the well known results of trench is proposed

the concavity of renyi entropy power we associate to the pth renyi entropy a definition of entropy power which is the natural extension of shannon s entropy power and exhibits a nice behavior along solutions to the p nonlinear heat equation in rn we show that the renyi entropy power of general probability densities solving such equations is always a concave function of time whereas it has a linear behavior in correspondence to the barenblatt source type solutions this result extends costa s concavity inequality for shannon s entropy power to renyi entropies

exact solution of corner modified banded block toeplitz eigensystems motivated by the challenge of seeking a rigorous foundation for the bulk boundary correspondence for free fermions we introduce an algorithm for determining exactly the spectrum and a generalized eigenvector basis of a class of banded block quasi toeplitz matrices that we call corner modified corner modifications of otherwise arbitrary banded block toeplitz matrices capture the effect of boundary conditions and the associated breakdown of translational invariance our algorithm leverages the interplay between a non standard projector based method of kernel determination physically a bulk boundary separation and families of linear representations of the algebra of matrix laurent polynomials thanks to the fact that these representations act on infinite dimensional carrier spaces in which translation symmetry is restored it becomes possible to determine the eigensystem of an auxiliary projected block laurent matrix this results in an analytic eigenvector ansatz independent of the system size which we prove is guaranteed to contain the full solution of the original finite dimensional problem the actual solution is then obtained by imposing compatibility with a boundary matrix also independent of system size as an application we show analytically that zero energy topological boundary modes of short ranged fermionic tight binding models may display power law corrections to exponential decay and demonstrate the phenomenon for the paradigmatic majorana chain of kitaev

determinant of block toeplitz band matrices some expressions are given for the determinant of an mn mn block toeplitz band matrix l li j with bandwidth p q 1 n mn in terms of the n n generating matrix polynomial l p qj 0 jlp j detl q 0 in the scalar case this yields formulas for the determinant expressed via the zeros of the generating scalar polynomial the approach adopted in this work leans heavily on the recently developed spectral theory of matrix polynomials

a note on asymptotic zero distribution of orthogonal polynomials abstract we strengthen a theorem of kuijlaars and serra capizzano on the distribution of zeros of a sequence of orthogonal polynomials p n n 1 for which the coefficients in the three term recurrence relation are clustered at finite points the proof uses a matrix argument motivated by a theorem of tyrtyshnikov

mutual information and minimum mean square error in gaussian channels this paper deals with arbitrarily distributed finite power input signals observed through an additive gaussian noise channel it shows a new formula that connects the input output mutual information and the minimum mean square error mmse achievable by optimal estimation of the input given the output that is the derivative of the mutual information nats with respect to the signal to noise ratio snr is equal to half the mmse regardless of the input statistics this relationship holds for both scalar and vector signals as well as for discrete time and continuous time noncausal mmse estimation this fundamental information theoretic result has an unexpected consequence in continuous time nonlinear estimation for any input signal with finite power the causal filtering mmse achieved at snr is equal to the average value of the noncausal smoothing mmse achieved with a channel whose snr is chosen uniformly distributed between 0 and snr

numerical solution of the inverse eigenvalue problem for real symmetric toeplitz matrices h j landau has recently given a nonconstructive proof of an existence theorem for the inverse eigenvalue problem for real symmetric toeplitz rst matrices this paper presents a procedure for the numerical solution of this problem the procedure is based on an implementation of newton s method that exploits landau s theorem and other special spectral properties of rst matrices with this version of newton s method together with the strategy proposed for applying it the method appears to be globally convergent however this is not proved

a strengthened central limit theorem for smooth densities abstract we consider the central limit theorem for the probability density function n x of the standardized sum of independent and identically distributed random variables with finite variance and regular probability density function by showing boundedness of different convex functionals along the sequence fn we prove convergence in various norms to the normal density

on the entropy power inequality for the renyi entropy of order 0 1 using a sharp version of the reverse young inequality and a renyi entropy comparison result due to fradelizi madiman and wang 2016 the authors derive renyi entropy power inequalities for log concave random vectors when renyi parameters belong to 0 1 furthermore the estimates are shown to be sharp up to absolute constants

a simple proof of the entropy power inequality this correspondence gives a simple proof of shannon s entropy power inequality epi using the relationship between mutual information and minimum mean square error mmse in gaussian channels

simple proof of the concavity of the entropy power with respect to added gaussian noise a very simple proof of m h costa s result see ibid vol it 31 p 751 60 1985 that the entropy power of x sub t x n o ti is concave in t is derived as an immediate consequence of an inequality concerning fisher information this relationship between fisher information and entropy is found to be useful for proving the central limit theorem thus one who seeks new entropy inequalities should try first to find new equalities about fisher information or at least to exploit the existing ones in new ways

eigenvectors of hermitian toeplitz matrices with smooth simple loop symbols abstract the paper is devoted to the structure and the asymptotics of the eigenvector matrix of hermitian toeplitz matrices with moderately smooth symbols which trace out a simple loop on the real line the results extend existing results on banded toeplitz matrices to full toeplitz matrices with temperate decay of the entries in the first row and column we establish formulas for both the exact and the asymptotic computation of arbitrarily prescribed individual components of arbitrarily prescribed individual eigenvectors these formulas are in terms of the wiener hopf factorization of a function which depends solely on the symbol and the corresponding eigenvalue or even only on an approximation to this eigenvalue the paper does not aim at providing numerical algorithms its main purpose is rather to reveal the structure underneath the eigenvectors which are shown to be the sum of two harmonics and certain edge corrections

a concavity property for the reciprocal of fisher information and its consequences on costa s epi abstract we prove that the reciprocal of fisher information of a log concave probability density x in r n is concave in t with respect to the addition of a gaussian noise z t n 0 t i n as a byproduct of this result we show that the third derivative of the entropy power of a log concave probability density x in r n is nonnegative in t with respect to the addition of a gaussian noise z t for log concave densities this improves the well known costa s concavity property of the entropy power costa 1985

on the eigenvalue problem for toeplitz band matrices abstract a formula is given for the characteristic polynomial of an n th order toeplitz band matrix with bandwidth k n in terms of the zeros of a k th degree polynomial with coefficients independent of n the complexity of the formula depends on the bandwidth k and not on the order n also given is a formula for eigenvectors in terms of the same zeros and k coefficients which can be obtained by solving a k k homogeneous system

is every matrix similar to a toeplitz matrix abstract we show that every n n complex nonderogatory matrix is similar to a unique unit upper hessenberg toeplitz matrix the proof is constructive and can be adapted to nonderogatory matrices with entries in any field of characteristic zero or characteristic greater than n we also prove that every n n complex matrix with n 4 is similar to a toeplitz matrix

a note on computing eigenvalues of banded hermitian toeplitz matrices it is pointed out that the author s o n 2 algorithm for computing individual eigenvalues of an arbitrary n times n hermitian toeplitz matrix t n reduces to an o rn algorithm if t n is banded with bandwidth r

real eigenvalues of certain tridiagonal matrix polynomials with queueing applications many queueing problems lead to tridiagonal lambda matrices containing polynomials that have except for the diagonal non negative coefficients this paper deals with the question addressed in the literature only for special cases whether the eigenvalues corresponding to such lambda matrices are real in most cases they are as the theorems of this paper show but sometimes complex eigenvalues occur our results are derived by using sturm sequences in addition to simplifying the proofs of our theorems sturm sequences are also valuable to verify whether or not a given interval contains eigenvalues

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

toeplitz matrices commuting with tridiagonal matrices abstract we prove that if r is a nonscalar toeplitz matrix ri j r i j which commutes with a tridiagonal matrix with simple spectrum then r k r 1 u k 1 r 2 r 1 cos p u k 1 cos p k 4 5 with uk the chebychev polynomial of the second kind where p is determined from cos p 1 2 r 2 1 r 1 r 3 r 2 2 r 1 r 3

nonlinear diffusions extremal properties of barenblatt profiles best matching and delays abstract in this paper we consider functionals based on moments and nonlinear entropies which have a linear growth in time in case of source type solutions to the fast diffusion or porous medium equations that are also known as barenblatt solutions as functions of time these functionals have convexity properties for generic solutions so that their asymptotic slopes are extremal for barenblatt profiles the method relies on scaling properties of the evolution equations and provides a simple and direct proof of sharp gagliardo nirenberg sobolev inequalities in scale invariant form the method also gives refined estimates of the growth of the second moment and as a consequence establishes the monotonicity of the delay corresponding to the best matching barenblatt solution compared to the barenblatt solution with same initial second moment here the notion of best matching is defined in terms of a relative entropy

positive integer powers of certain tridiagonal matrices abstract in j rimas on computing of arbitrary positive integer powers for one type of symmetric tridiagonal matrices of even order i appl math comput 168 2005 783 787 and j rimas on computing of arbitrary positive integer powers for one type of symmetric tridiagonal matrices of odd order i appl math comput 171 2005 1214 1217 rimas derived a general expression for the entries of the qth power q n of the n n real symmetric tridiagonal matrix tridiag n 1 0 1 for all n n in this paper we present an extension of that interesting work deriving a similar expression for the entries of the qth power q n of the n n hermitian tridiagonal matrix tridiag n a 1 a 0 a 1 for all n n

information theoretic proofs of entropy power inequalities while most useful information theoretic inequalities can be deduced from the basic properties of entropy or mutual information up to now shannon s entropy power inequality epi is an exception existing information theoretic proofs of the epi hinge on representations of differential entropy using either fisher information or minimum mean square error mmse which are derived from de bruijn s identity in this paper we first present an unified view of these proofs showing that they share two essential ingredients 1 a data processing argument applied to a covariance preserving linear transformation 2 an integration over a path of a continuous gaussian perturbation using these ingredients we develop a new and brief proof of the epi through a mutual information inequality which replaces stam and blachman s fisher information inequality fii and an inequality for mmse by guo shamai and verdu used in earlier proofs the result has the advantage of being very simple in that it relies only on the basic properties of mutual information these ideas are then generalized to various extended versions of the epi zamir and feder s generalized epi for linear transformations of the random variables takano and johnson s epi for dependent variables liu and viswanath s covariance constrained epi and costa s concavity inequality for the entropy power