moments of spectral functions monte carlo evaluation and verification the subject of the present study is the monte carlo path integral evaluation of the moments of spectral functions such moments can be computed by formal differentiation of certain estimating functionals that are infinitely differentiable against time whenever the potential function is arbitrarily smooth here i demonstrate that the numerical differentiation of the estimating functionals can be more successfully implemented by means of pseudospectral methods e g exact differentiation of a chebyshev polynomial interpolant which utilize information from the entire interval ensuremath ensuremath beta ensuremath hbar 2 ensuremath beta ensuremath hbar 2 the algorithmic detail that leads to robust numerical approximations is the fact that the path integral action and not the actual estimating functional are interpolated although the resulting approximation to the estimating functional is nonlinear the derivatives can be computed from it in a fast and stable way by contour integration in the complex plane with the help of the cauchy integral formula e g by lyness method an interesting aspect of the present development is that hamburger s conditions for a finite sequence of numbers to be a moment sequence provide the necessary and sufficient criteria for the computed data to be compatible with the existence of an inversion algorithm finally the issue of appearance of the sign problem in the computation of moments albeit in a milder form than for other quantities is addressed

lower bounds for the condition number of vandermonde matrices we derive lower bounds for the condition number of then n vandermonde matrixvn x in the cases where the node vectorxt x1 x2 xn has positive elements or real elements located symmetrically with respect to the origin the bounds obtained grow exponentially inn witho 2n ando 2n 2 respectively we also compute the optimal spectral condition numbers ofvn x for the two node configurations including the optimal nodes and compare them with the bounds obtained

bezout hankel and loewner matrices abstract relations between the classes of bezout hankel and loewner matrices and of their inverses are investigated all these classes appear as matrix realizations of one operator in suitable bases different bases in the space of all polynomials of degree not exceeding n 1 are considered these bases correspond to different expansions of the polynomial w z p x p z w x z x we introduce a matrix k corresponding to this polynomial this matrix together with its limit forms may be used to represent all the classes of matrices mentioned in particular we obtain a new interpretation of the variant of generalized loewner matrices introduced recently by the second named author

a systolic realization for 2 d digital filters systolic hardware realizations for two dimensional finite impulse response fir and infinite impulse response iir digital filters are presented the structure permits the 2 d input data to be scanned row wise and broadcasted one value at a time to various processing elements shift registers are used to store the required data needed by the 2 d recursive equation a processing element can be implemented in vlsi and as many of these as are required to satisfy the order of the filter can be used to build the total structure

vandermonde systems on gauss lobatto chebyshev nodes abstract this paper deals with vandermonde matrices v n whose nodes are the gauss lobatto chebyshev nodes also called extrema chebyshev nodes we give an analytic factorization and explicit formula for the entries of their inverse and explore its computational issues we also give asymptotic estimates of the frobenius norm of both v n and its inverse and present an explicit formula for the determinant of v n

generalized inversion of finite rank hankel and toeplitz operators with rational matrix symbols abstract the goal of the paper is a generalized inversion of finite rank hankel operators and hankel or toeplitz operators with block matrices having finitely many rows to attain it a left coprime fractional factorization of a strictly proper rational matrix function and the bezout equation are used generalized inverses of these operators and generating functions for the inverses are explicitly constructed in terms of the fractional factorization

on the partial realization problem abstract in this paper we take a unified approach to the partial realization problem in which we seek to incorporate ideas from numerical linear algebra most of which were originally developed in other contexts we approach the partial realization problem from several different angles and explore the connections to such topics as factoriza tion of hankel matrices block tridiagonalization generalizations of the lanczos process for biorthogonalization the euclidean algorithm and the principal part con tinued fractions of arne magnus the pade table and the berlekamp masseyalgo rithm in this way we are able to clarify some previous results by rissanen kalman and others and place them in a broader context this leads to several results and concepts which we think are new our analysis is restricted to the scalar case but some definitions and formulations have been rigged to facilitate an extension to the matrix case

vandermonde matrices on integer nodes the rectangular case we consider a rectangular vandermonde matrix v on integer nodes using combinatorial identities we give an explicit cholesky factorization of the matrix b v rm t v a factorization of the pseudo inverse of v and asymptotic estimates of the extremal eigenvalues of b we also discuss the numerical properties of the proposed formulas

recursive partial realization from the combined sequence of markov parameters and moments abstract in this paper we develop the recursive structure of partial realization from a two sided sequence of markov parameters and moments the specific feature of this recursive structure is that it accommodates sequences growing in two directions like in the two point pade approximation we present a recursive algorithm for the minimal partial realization from a sequence of markov parameters and moments that amounts to recursive lu factorization of a toeplitz behavior matrix leaping over nongeneric sections it turns out that this algorithm reveals a new continued fraction expansion which extends the classical cauer third form as a corollary various results related to toeplitz matrices two point pade approximations continued fractions etc are unified within this partial realization setting

hankel and loewner matrices abstract mutual relations between the hankel toeplitz bezout and loewner matrices as well as further connections to rational interpolation and projective geometry are investigated

existence of positive solutions for second order functional differential equations we afford a existence criterion of positive solutions of a boundary value problem concerning a second order functional differential equation by using the krasnoselskii fixed point theorem on cones in banach spaces moreover we also apply our results to establish several existence theorems of multiple positive solutions for some functional differential equations

minimal condition number for positive definite hankel matrices using semidefinite programming abstract we present a semidefinite programming approach for computing optimally conditioned positive definite hankel matrices of order n unlike previous approaches our method is guaranteed to find an optimally conditioned positive definite hankel matrix within any desired tolerance since the condition number of such matrices grows exponentially with n this is a very good test problem for checking the numerical accuracy of semidefinite programming solvers our tests show that semidefinite programming solvers using fixed double precision arithmetic are not able to solve problems with n 30 moreover the accuracy of the results for 24 n 30 is questionable in order to accurately compute minimal condition number positive definite hankel matrices of higher order we use a mathematica 6 0 implementation of the sdpha solver that performs the numerical calculations in arbitrary precision arithmetic by using this code we have validated the results obtained by standard codes for n 24 and we have found optimally conditioned positive definite hankel matrices up to n 100

inversion des matrices de hankel abstract hankel determinants can be viewed as special schur symmetric functions this provides without computations of determinants or matrices most of the algebra of hankel determinants and explains their links with other classical fields

a decomposition theorem and its implications to the design and realization of two dimensional filters it is shown that an arbitrary rational 2 d transfer function can be expanded in first order terms each one of which is a function of only one of the two variables this method leads naturally to reconfigurable filters with great modularity and parallelism which can realize any rational transfer function up to a given order

convex subclass of harmonic starlike functions complex valued harmonic functions that are univalent and sense preserving in the unit disk u can be written in the form f h g where h and g are analytic in u we define and investigate a convex subclass of harmonic starlike functions of order a 0 a 1 we obtain coefficient conditions extreme points distortion bounds convolution conditions and convex combination for the above class of harmonic functions

discrete probability distributions and moment problem numerical aspects discrete probability distributions with finite range generated by maximum entropy technique and constrained by first moments have been considered numerical aspects as stability of parameter s computation entropy variation spectral properties of involved hankel s matrices have been studied the ill conditioning of hankel s matrices suggests minimization methods like newton s should be unreliable as a high number of moments are kept while methods relying upon relative error estimate between assigned and calculated moments should be possible candidates

how bad are hankel matrices considered are hankel vandermonde and krylov basis matrices it is proved that for any real positive definite hankel matrix of order n its spectral condition number is bounded from below by 3 cdot 2 n 6 also proved is that the spectral condition number of a krylov basis matrix is bounded from below by 3 frac 1 2 cdot 2 frac n 2 3 for v v x 1 ldots x n a vandermonde matrix with arbitrary but pairwise distinct nodes x 1 ldots x n we show that mbox rm cond 2 v geq 2 n 2 n frac 1 2 if either x j leq 1 or x j geq 1 for all j then mbox rm cond 2 v geq 2 n 2

neighborhoods of certain p valently analytic functions with negative coefficients in this investigation by making use of the familiar concept of neighborhoods of p valently analytic functions the authors prove coefficient bounds distortion inequalities associated inclusion relations for the n neighborhoods of a family of p valently analytic functions and its certain derivatives which is defined by means of a certain non homogenous differential equation

on highly modular systolic structures for separable in denominator 2 d digital filters it is shown that the proposed structure in raghuramireddy and unbehaven 1991 for a separable in denominator recursive 2 d digital filter with complex poles structure ii is not valid the modified and correct version of the proposed structure is given in chakrabarti 1993 a similar error correction and ratification of the realization has been independently reported a general systolic realization is also presented with emphasis on parallelism modularity and amenability to vlsi design and implementation

harmonic functions starlike in the unit disk complex valued harmonic functions that are univalent and sense preserving in the unit disk can be written in the form f h where h and g are analytic in we give univalence criteria and sufficient coefficient conditions for normalized harmonic functions that are starlike of order 0 1 these coefficient conditions are also shown to be necessary when h has negative and g has positive coefficients these lead to extreme points and distortion bounds

semi infinite programming using high degree polynomial interpolants and semidefinite programming in a common formulation of semi infinite programs the infinite constraint set is a requirement that a function parametrized by the decision variables is nonnegative over an interval if this function is sufficiently closely approximable by a polynomial or a rational function then the semi infinite program can be reformulated as an equivalent semidefinite program which in turn can be solved with interior point methods very efficiently to high accuracy on the other hand solving this semidefinite program is challenging if the polynomials involved are of high degree due to numerical difficulties and bad scaling arising both from the polynomial approximations and from the fact that the semidefinite programming constraints coming from the sum of squares representation of nonnegative polynomials are badly scaled we combine polynomial function approximation techniques and polynomial programming to overcome these numerical difficulties using sum of squares interpolants specifically it is shown that the c

neighborhoods of a new class of harmonic multivalent functions abstract we introduce and investigate a new subclass of harmonic multivalent functions defined by using a differential operator we obtain coefficient conditions distortion bounds extreme points convex combination for the above class of harmonic multivalent functions we also derive inclusion relationships involving the neighborhoods of harmonic multivalent functions

neighborhood properties of certain classes of multivalently analytic functions associated with the convolution structure abstract in this paper by making use of the familiar concept of neighborhoods of p valently analytic functions we prove coefficient bounds distortion inequalities and associated inclusion relations for the n neighborhoods of a family of p valently analytic functions and their derivatives which is defined by means of a certain general family of non homogenous cauchy euler differential equations

functional models in linear algebra abstract this paper examines the role of functional models in linear algebra and some related areas of application notably linear systems theory this approach adopts an intermidiate level of abstraction somewhere between abstract module theory on the one hand and matrix theory on the other it is a powerful method which simplifies many proofs provides insights and illuminates the less obvious connections between linear algebra and operator theory approximation theory moment problems interpolation and control theory

vandermonde systems on equidistant nodes in 0 1 accurate computation this paper deals with vandermonde matrices v whose nodes are the equidistant points in 0 1 we give an analytic factorization and explicit formula for the entries of their inverse and explore its computational issues we also give asymptotic estimates of the frobenius norm of both v and its inverse and show that a new representation of the floating point number system allows one to build an accurate algorithm for the interpolation problem on equidistant nodes in 0 1

displacement structure approach to polynomial vandermonde and related matrices abstract we introduce a new class of what we call polynomial vandermonde like matrices this class generalizes the polynomial vandermonde matrices studied earlier by various authors who derived explicit inversion formulas and fast algorithms for inversion and for solving the associated linear systems a displacement structure approach allows us to carry over all these results to the wider class of polynomial vandermonde like matrices

injectivity of sections of univalent harmonic mappings abstract if s denotes the class of functions h z z n 2 a n z n which are analytic and univalent in the unit disk z 1 then the classical result of szego shows that every section s n h z k 1 n a k z k of h is univalent in z 1 4 exact largest radius of univalence r n of s n h remains an open problem although the corresponding results for sections of various geometric subclasses of s have been obtained which include those h s for which re h z 0 holds in the unit disk however no attempt has been made to derive harmonic analog of these results the central object in this case is the class s h 0 of sense preserving harmonic univalent mappings f h g defined on the unit disk normalized so that h 0 g 0 h 0 1 g 0 0 our primary objective in this paper is to solve the univalency of every section of a harmonic function in the class p h 0 where p h 0 denotes the class of normalized univalent harmonic mapping f h g in the unit disk z 1 satisfying the condition re h z g z for z 1 where g 0 0 and 0 1 in this paper we first present sharp bounds for the moduli of the coefficients a n b n of f p h 0 and then determine the value of r so that the partial sums of f p h 0 are close to convex in z r

additive preconditioning and aggregation in matrix computations we combine our novel svd free additive preconditioning with aggregation and other relevant techniques to facilitate the solution of a linear system of equations and other fundamental matrix computations our analysis and experiments show the power of our algorithms guide us in selecting most effective policies of preconditioning and aggregation and provide some new insights into these and related subjects compared to the popular svd based multiplicative preconditioners our additive preconditioners are generated more readily and for a much larger class of matrices furthermore they better preserve matrix structure and sparseness and have a wider range of applications e g they facilitate the solution of a consistent singular linear system of equations and of the eigenproblem

a certain subclass of analytic functions involving operators of fractional calculus making use of certain operators of fractional calculus we introduce a new class t n of functions which are analytic and univalent in the open unit disk u and obtain a necessary and sufficient condition for a function to be in the class t n the various other results presented here for the class t n include the radii of close to convexity starlikeness and convexity and some growth and distortion theorems involving fractional integrals and fractional derivatives some interesting consequences and possible further generalizations of these results are also pointed out

regularized nonlinear acceleration we describe a convergence acceleration technique for generic optimization problems our scheme computes estimates of the optimum from a nonlinear average of the iterates produced by any optimization method the weights in this average are computed via a simple linear system whose solution can be updated online this acceleration scheme runs in parallel to the base algorithm providing improved estimates of the solution on the fly while the original optimization method is running numerical experiments are detailed on classical classification problems

group and moore penrose invertability of bezoutians abstract a characterization of the group and moore penrose inverse of bezoutians generalized bezoutians and related matrices is presented the results are obtained by combining specific factorizations in the literature with recent theory on generalized inverses

the lu decomposition theorem and its implications to the realization of two dimensional digital filters a new approach to the realization of two dimensional 2 d fir and iir digital filters is introduced based on the so called lower upper triangular lu decomposition of matrix coefficients of their two dimensional polynomials it is demonstrated that the lu realization scheme enjoys a number of attributes for vlsi implementation including high parallelism modularity and regularity this paper also shows that the computational requirements of the lu realization structure are much smaller than those of the jordan jd singular value sv and canonical direct realizations furthermore it is shown that if a canonical 2 d iir filter is realizable then a realizable lu decomposition form can be always obtained a property that is not shared by the jd and sv decomposition forms finally the extension of the lu decomposition approach to the realization of m d digital filters is discussed

harmonic univalent functions with negative coefficients we give sufficient coefficient conditions for normalized harmonic functions to map onto either starlike or convex regions these conditions are also shown to be necessary when the coefficients are negative this leads to distortion bounds and extreme points

inclusion and neighborhood properties for certain classes of multivalently analytic functions of complex order associated with the convolution structure making use of the familiar convolution structure of analytic functions in this paper we introduce and investigate two new subclasses of multivalently analytic functions of complex order among the various results obtained here for each of these function classes we derive the coefficient inequalities and other interesting properties and characteristics for functions belonging to the class introduced here

explicit expressions for bezoutians abstract the following commutator identity is proved u s v s v 1 s u 1 s here s is the n by n matrix of the truncated shift operator s i i 1 i 0 1 n 1 and u v are two polynomials of degree not exceeding n the reciprocal polynomial f 1 of a polynomial f of degree n is defined by f 1 z z n f 1 z the commutator identity is closely related to some properties of the bezoutian matrix of a pair of polynomials it is used to obtain the bezoutian matrix in the form of a simple expression in terms of s and s to demonstrate the advantage of this expression we show how it can be used to obtain simple proofs of some interesting corollaries

a new realization fo 2 d digital filters a systolic array realization for 2 d digital filters is presented it is based on chan s local state space model this structure has the minimum number of states minimum delays a technique is also presented to reduce the number of multipliers to its minimum conditions that the system eigenvalues have to satisfy to ensure low sensitivity with respect to the variations in the multipliers are developed several examples are included to illustrate the proposed techniques

highly molecular systolic structures for denominator separable 2 d recursive filters two systolic realization structures for the implementation of 2 d denominator separable recursive filters are presented these structures possess a high degree of modularity and parallelism with low roundoff noise the number of delays and multipliers is fewer than for similar realizations in the literature the proposed structures are more suitable for special purpose hardware and amenable to vlsi design

remarks on the inversion of hankel matrices the paper continues the investigation into the links between algebraic system theory more specifically the partial realization problem and the problem of hankel matrix inversion the representation of the inverse of a hankel matrix as a bezoutiant and the bezout equation play a principal role

neighborhoods of certain p valent analytic functions with complex order in this paper the authors prove several coefficient bounds distortion inequalities of the class s g n p l b b of p valent analytic functions of complex order by making use of the familiar concept of neighborhoods of p valent analytic functions they obtained several inclusion relations associated with the n p n d neighborhood of this class and its certain derivatives which is defined by means of a certain non homogeneous cauchy euler differential equations

neighborhoods of a certain family of multivalent functions with negative coefficients abstract by making use of the familiar concept of neighborhoods of analytic and p valent functions the authors prove coefficient bounds and distortion inequalities and associated inclusion relations for the n neighborhoods of a family of multivalent functions with negative coefficients which is defined by means of a certain nonhomogeneous cauchy euler differential equation relevant connections of the various function classes investigated in this paper with those considered by earlier workers on the subject are also mentioned

neighborhoods of certain multivalent functions with negative coefficients by means of a differential operator we introduce and investigate two new subclasses of p valently analytic functions of complex order the various results obtained here for each of these function classes include coefficient inequalities and the consequent inclusion relationships involving the neighborhoods of the p valently analytic functions

neighborhoods for certain subclasses of multivalently analytic functions defined by using a differential operator in the present investigation by making use of the familiar concept of neighborhoods of analytic and multivalent functions we derive coefficient bounds and distortion inequalities associated inclusion relations for the n d neighborhoods of subclasses of analytic and multivalent functions with negative coefficients which are defined by means of a certain nonhomogenous differential equation several special cases of the main results are mentioned

fractional calculus and certain starlike functions with negative coefficients abstract a certain subclass t n p of starlike functions in the unit disk is introduced the object of the present paper is to derive several interesting properties of functions belonging to the class t n p various distortion inequalities for fractional calculus of functions in the class t n p are also given

pluriharmonic mappings in cn and complex banach spaces abstract in this paper we obtain a sufficient condition for pluriharmonic mappings on the euclidean unit ball b n to be univalent sense preserving quasiconformal and bi lipschitz diffeomorphisms on b n and to have linearly connected images also we give a sufficient condition for pluriharmonic mappings on b n to have quasiconformal extensions to c n next we generalize the harmonic schwarz lemma to pluriharmonic mappings of the unit ball b x of a complex banach space x into the unit ball b n of c n with respect to an arbitrary norm further we obtain a generalization of the harmonic schwarz pick lemma to the case of pluriharmonic mappings of the homogeneous unit ball b x of a complex banach space x into the unit ball b n we also obtain a version of the holomorphic schwarz pick lemma for the jacobian determinant on the euclidean unit ball b n to the case of pluriharmonic mappings of the homogeneous unit ball b x into b n in the case that b x is an open subset of c n finally we obtain the landau and the bloch theorems for pluriharmonic or holomorphic mappings on finite dimensional homogeneous unit balls

a new algorithm for computing orthogonal polynomials the aim of this paper is to present an algorithm for computing orthogonal polynomials the fast fourier transform completes the connection given by draux between the three term recurrence relationship and the euclidian algorithm applications to hankel systems and numerical examples illustrate our purpose

on the inversion of the vandermonde matrix abstract the inversion of the vandermonde matrix has received much attention for its role in the solution of some problems of numerical analysis and control theory this work deals with the problem of getting an explicit formula for the generic element of the inverse we derive two algorithms in o n2 and o n3 and compare them with the parker traub and the bjorck pereyra algorithms

connections between various subclasses of planar harmonic mappings involving hypergeometric functions the main purpose of this paper is to establish connections between various subclasses of harmonic mappings in the plane by applying certain convolution operators involving hypergeometric functions to be more precise we investigate such connections with harmonic k uniformly starlike and harmonic k uniformly convex mappings in the plane

on inversion of toeplitz and close to toeplitz matrices abstract it is proved that three columns of an inverse of a scalar toeplitz matrix when properly chosen are always enough to reconstruct it a parametrization of invertible toeplitz matrices in terms of the first and a combination of two consecutive columns of the inverse matrix is also given

rectangular vandermonde matrices on chebyshev nodes abstract we consider a rectangular vandermonde matrix v on chebyshev nodes using combinatorial identities we give an explicit factorization of the matrix v a factorization of thepseudo inverse of v and prove that the condition number 2 v does not depend onthe dimension of nodes set we also discuss the numerical properties of the proposed formulas

bounds on elementary symmetric functions abstract tight bounds on an elementary symmetric function are established under the assumption that the values of the elementary symmetric functions of lower orders are given the explicit form of the inverse hankel moment matrix leads to inequalities for moments and for elementary symmetric polynomials

matrix representations of bezoutians abstract an approach is presented to get matrix representations for classical and more general r bezoutians with entries from a noncommutative ring formulas due to ammar and gader barnett gohberg and semencul gohberg and heinig kravitsky and russakovskii lerer and tismentsky ptak and wimmer as well as some generalizations and modifications are obtained in a unified way

sharp coefficient estimates for certain subclasses of starlike functions of complex order in this paper the authors derive several sharp coefficient estimates for the subclassess l c a b andk l c a b m m of the class of normalized starlike functions of complex order email protected email protected 0 the results presented here would generalize as well as improve the recent work by srivastava et al 2011 20 some closely related open problems are also posed

comments on highly modular systolic structures for denominator separable 2 d recursive filters with reply some errors in the above titled paper see ibid vol 39 no 12 p 2725 8 1991 are pointed out and rectified realizations with a reduced number of multipliers are also proposed the authors reply that the correction is appreciated

duality in polynomial models with some applications to geometric control theory duality is studied in the context of polynomial models for linear systems the output injection group the dual of the feedback group is studied and a polynomial characterization of c a invariant subspaces as well as of the maximal reachability subspaces contained in ker c is given

eigenvalue clustering control energy and logarithmic capacity abstract we prove two bounds showing that if the eigenvalues of a matrix are clustered in a region of the complex plane then the corresponding discrete time linear system requires significant energy to control a curious feature of one of our bounds is that the dependence on the region is via its logarithmic capacity which is a measure of how well a unit of mass may be spread out over the region to minimize a logarithmic potential