algebraic decoding of cyclic codes using partial syndrome matrices cyclic codes have been widely used in many applications of communication systems and data storage systems this paper proposes a new procedure for decoding cyclic codes up to actual minimum distance the decoding procedure consists of two steps 1 computation of known syndromes and 2 computation of error positions and error values simultaneously to do so a matrix whose all entries are syndromes is called syndrome matrix a matrix whose entries are either syndromes or the elements of a finite field is said to be partial syndrome matrix in this paper two novel methods are presented to determine error positions and error values simultaneously and directly the first method uses a new partial syndrome matrix along with gaussian elimination the partial syndrome matrices for binary respectively ternary cyclic codes of lengths from 69 to 99 respectively 16 to 37 are tabulated for some cyclic codes the partial syndrome matrices contain unknown syndromes the second method constructs a matrix from a system of equations which is generated by the determinants of different partial syndrome matrices and makes use of gaussian elimination to determine its row rank many more cyclic codes beyond the bose chaudhuri hocquenghem bound can be decoded with these methods

solution of population balance equations in applications with fine particles the accurate description and robust simulation at relatively low cost of global quantities e g number density or volume fraction as well as the size distribution of a population of fine particles in a carrier fluid is still a major challenge for many applications for this purpose two types of methods are investigated for solving the population balance equation with aggregation continuous particle size change growth and size reduction and nucleation the extended quadrature method of moments eqmom based on the work of yuan et al 52 and a hybrid method tsm between the sectional and moment methods considering two moments per section based on the work of laurent et al 30 for both methods the closure employs a continuous reconstruction of the number density function of the particles from its moments thus allowing evaluation of all the unclosed terms in the moment equations including the negative flux due to the disappearance of particles here new robust and efficient algorithms are developed for this reconstruction step and two kinds of reconstruction are tested for each method moreover robust and accurate numerical methods are developed ensuring the realizability of the moments the robustness is ensured with efficient and tractable algorithms despite the numerous couplings and various algebraic constraints thanks to a tailored overall strategy eqmom and tsm are compared to a sectional method for various simple but relevant test cases showing their ability to describe accurately the fine particle population with a much lower number of variables these results demonstrate the efficiency of the modeling and numerical choices and their potential for the simulation of real world applications

matrix chebyshev polynomials and continued fractions in the first part we expose the notion of continued fractions in the matrix case in this paper we are interested in their connection with matrix orthogonal polynomials in the second part matrix continued fractions are used to develop the notion of matrix chebyshev polynomials in the case of hermitian coefficients in the recurrence formula we give the explicit formula for the stieltjes transform the support of the orthogonality measure and its density as a corollary we get the extension of the matrix version of the blumenthal theorem proved in j approx theory 84 1 1996 96 the third part contains examples of matrix orthogonal polynomials

a highly stable deferred correction scheme with interpolant for systems of nonlinear two point boundary value problems iterated deferred correction is a very popular approach to the numerical solution of general first order systems of nonlinear two point boundary value problems typically the order of accuracy of the solution is increased by 2 each time that a deferred correction is applied and this allows inexpensive local error estimates to be computed using embedding recent research has shown how deferred correction schemes having the very desirable property of superconvergence can be derived this marks a significant step forward in the design of deferred correction algorithms although the price to be paid for superconvergence is that it is difficult to derive realistic embedded local error estimates in the present paper we show that richardson extrapolation is a viable alternative for error estimation and we consider in some detail the relative merits of embedding and extrapolation guided by this analysis we are able to derive for the first time a lobatto deferred correction code which is very efficient for the solution of stiff problems particularly in cases where the mirk code twpbvp f is unstable furthermore we are able to derive efficient interpolants for this lobatto code which are also applicable to mirk formulae and we consider the problem of estimating the error in these interpolants this completely solves the interpolation problem for our deferred correction codes apart perhaps for deriving interpolants for extremely stiff problems this new lobatto code is available on the web page of one of the present authors and fills a large gap in currently available deferred correction software

a generalized euclidean algorithm for multisequence shift register synthesis the problem of finding a linear feedback shift register of shortest length capable of generating prescribed multiple sequences is considered a generalized euclidean algorithm which is based on a generalized polynomial division algorithm is presented a necessary and sufficient condition for the uniqueness of the solution is given when the solution is not unique the set of all possible solutions is also derived it is shown that the algorithm can be applied to the decoding of many cyclic codes for which multiple syndrome sequences are available when it is applied to the case of a single sequence the algorithm reduces to that introduced by y sugiyama et al inf control vol 27 p 87 9 feb 1975 in the decoding of bch codes

a note on the maximization of matrix valued hankel determinants with applications in this note we consider the problem of maximizing the determinant of moment matrices of matrix measures the maximizing matrix measure can be characterized explicitly by having equal matrix valued weights at the zeros of classical one dimensional orthogonal polynomials the results generalize classical work of schoenberg indag math 62 1959 282 to the case of matrix measures as a statistical application we consider several optimal design problems in linear models which generalize the classical weighing design problems

the role of conditioning in mesh selection algorithms for first order systems of linear two point boundary value problems codes for the numerical solution of two point boundary value problems can now handle quite general problems in a fairly routine and reliable manner when faced with particularly challenging equations such as singular perturbation problems the most efficient codes use a highly non uniform grid in order to resolve the non smooth parts of the solution trajectory this grid is usually constructed using either a pointwise local error estimate defined at the grid points or else by using a local residual control similar error estimates are used to decide whether or not to accept a solution such an approach is very effective in general providing that the problem to be solved is well conditioned however if the problem is ill conditioned then such grid refinement algorithms may be inefficient because many iterations may be required to reach a suitable mesh on which to compute the solution even worse for ill conditioned problems an inaccurate solution may be accepted even though the local error estimates may be perfectly satisfactory in that they are less than a prescribed tolerance the primary reason for this is of course that for ill conditioned problems a small local error at each grid point may not produce a correspondingly small global error in the solution in view of this it could be argued that when solving a two point boundary value problem in cases where we have no idea of its conditioning we should provide an estimate of the condition number of the problem as well as the numerical solution in this paper we consider some algorithms for estimating the condition number of boundary value problems and show how this estimate can be used in the grid refinement algorithm

strict positivstellensatze for matrix polynomials with scalar constraints abstract we extend krivine s strict positivstellensatz for usual real multivariate polynomials to symmetric matrix polynomials with scalar constraints the proof is an elementary computation with schur complements analogous extensions of schmudgen s and putinar s strict positivstellensatz were recently proved by hol and scherer using methods from optimization theory

the general rational interpolation problem in the scalar case and its hankel vector abstract this paper deals with the general rational interpolation problem grip in the scalar case in the recent work of antoulas ball kang and willems the general solution to the grip has been derived in the framework of linear fractions using the so called generating matrix as the main tool within this framework the contribution here consists in a new approach of computing a particular 2 2 polynomial generating matrix based on the deep connection between loewner and hankel matrices it turns out that given a fixed grip we can readily obtain a unique vector called the hankel vector such that both of the allowable mcmillan degrees of interpolants to this grip and the corresponding polynomial generating matrix are completely determined by this hankel vector combined with its characteristic degrees and characteristic polynomials

a finite differences matlab code for the numerical solution of second order singular perturbation problems we show the main features of the matlab code hofid up for solving second order singular perturbation problems the code is based on high order finite differences in particular on the generalized upwind method within its simplicity it uses order variation and continuation for solving any difficult nonlinear scalar problem several numerical tests on linear and nonlinear problems are considered the best performances are reported on problems with perturbation parameters near the machine precision where most of the codes for two point bvps fail

a unified treatment for the matrix stieltjes moment problem abstract this paper presents a unified approach to the solution of the truncated matrix stieltjes moment problem s k 0 u k d u k 0 m in both the cases m 2 n and m 2 n 1 based on the use of the iterative schur algorithm

hamburger moment problem parseval equality anda singularity abstract in one of the author s previous papers the parseval equality was established for any a regular scattering operator function i e for one associated with the generalized bitangential schur nevanlinna pick problem the infinite hamburger moment problem gives an example of an a singular scattering operator function with the parseval equality property this leads to a negative answer to a question due to v m adamjan d z arov and m g krein and what is the same produces a counter example to d sarason s conjecture concerning exposed points in h 1

on invertibility of nonsquare generalized bezoutians abstract conditions for inverting h bezoutians of nonsquare matrix polynomials are studied necessary and sufficient conditions for invertibility of the anderson jury bezoutian are obtained an application to the inversion of block hankel matrices is given

a new mesh selection algorithm based on conditioning for two point boundary value codes we present a hybrid mesh selection strategy for use in codes for the numerical solution of two point boundary value problems this new mesh strategy is based on the estimation of two parameters which characterise the conditioning of the continuous problem as well as on a standard estimate of the local discretisation error we have implemented this algorithm in the well known code twpbvp and have found that the modified code is often considerably more efficient than the original another strong advantage of using the new mesh selection algorithm is that it automatically provides an estimate of the conditioning of the discrete problem this is very valuable arguably indispensible for use either in an a posteriori error estimate or in situations where the conditioning constants are large as a warning that the accuracy obtained in the solution may be worse than anticipated

unbounded subnormal composition operators in l2 spaces abstract a novel criterion for subnormality of unbounded composition operators in l 2 spaces written in terms of measurable families of probability measures satisfying the so called consistency condition is established it becomes a new characterization of subnormality in the case of bounded composition operators pseudo moments of a measurable family of probability measures that satisfies the consistency condition are proved to be given by the radon nikodym derivatives which appear in lambert s characterization of bounded composition operators a criterion for subnormality of composition operators induced by matrices is provided the question of subnormality of composition operators over discrete measure spaces is studied two new classes of subnormal composition operators over discrete measure spaces are introduced a recent criterion for subnormality of weighted shifts on directed trees by the present authors is essentially improved in the case of rootless directed trees and nonzero weights by dropping the assumption of density of c vectors in the underlying l 2 space

on a class of matrix orthogonal polynomials on the real line abstract it is shown how to construct matrix orthogonal polynomials on the real line provided we have a sequence of scalar orthogonal polynomials over an algebraic harmonic curve we consider the moment matrix associated with a distribution on and related to a basis of the linear space of polynomials obtained through the polynomial which defines the curve this particular moment matrix can be regarded as a hankel block matrix and thus matrix orthogonal polynomials appear in a natural way

a survey on orthogonal matrix polynomials satisfying second order differential equations the subject of orthogonal polynomials cuts across a large piece of mathematics and its applications two notable examples are mathematical physics in the 19th and 20th centuries as well as the theory of spherical functions for symmetric spaces it is also clear that many areas of mathematics grew out of the consideration of problems like the moment problem that are intimately associated to the study of scalar valued orthogonal polynomials matrix orthogonality on the real line has been sporadically studied during the last half century since krein devoted some papers to the subject in 1949 see ams translations series 2 vol 97 providence rhode island 1971 pp 75 143 dokl akad nauk sssr 69 2 1949 125 in the last decade this study has been made more systematic with the consequence that many basic results of scalar orthogonality have been extended to the matrix case the most recent of these results is the discovery of important examples of orthogonal matrix polynomials many families of orthogonal matrix polynomials have been found that as the classical families of hermite laguerre and jacobi in the scalar case satisfy second order differential equations with coefficients independent of n the aim of this paper is to give an overview of the techniques that have led to these examples a small sample of the examples themselves and a small step in the challenging direction of finding applications of these new examples

power law subordinacy and singular spectra i half line operators we study hausdorff dimensional spectral properties of certain whole line quasiperiodic discrete schrodinger operators by using the extension of the gilbert pearson subordinacy theory that we previously developed in 19

transformations of matricial α stieltjes non negative definite sequences abstract we investigate the inner structure of power moment sequences of matrix measures on the right semiaxis where is a given real number to a given matrix sequence we associate in a bijective way a new sequence of matrices which we call the right stieltjes parametrization thereby one to one correspondences between power moment sequences on the right semiaxis with additional properties and particular sequences of non negative hermitian matrices are established we consider distinguished transformations of matrix sequences the study of which was suggested by considering some natural transformations of matrix measures on an interval a main theme is to describe the right stieltjes parametrization of the transformed sequence in terms of the right stieltjes parametrization of the original sequence

orthogonal matrix polynomials and higher order recurrence relations abstract it is well known that orthogonal polynomials on the real line satisfy a three term recurrence relation and conversely every system of polynomials satisfying a three term recurrence relation is orthogonal with respect to some positive borel measure on the real line we extend this result and show that every system of polynomials satisfying some 2 n 1 term recurrence relation can be expressed in terms of orthonormal matrix polynomials for which the coefficients are n n matrices we apply this result to polynomials orthogonal with respect to a discrete sobolev inner product and other inner products in the linear space of polynomials as an application we give a short proof of krein s characterization of orthogonal polynomials with a spectrum having a finite number of accumulation points

on double byte error correcting codes this correspondence shows that there is a flaw in the results presented by g l feng et al see ibid vol 44 no 3 p 1152 63 may 1998 a large family of the codes constructed are not double byte error correcting codes as originally claimed

schur analysis of matricial hausdorff moment sequences abstract we develop the algebraic instance of an algorithmic approach to the matricial hausdorff moment problem on a compact interval of the real axis our considerations are along the lines of the classical schur algorithm and the treatment of the hamburger moment problem on the real axis by nevanlinna more precisely a transformation of matrix sequences is constructed which transforms hausdorff moment sequences into hausdorff moment sequences reduced by 1 in length it is shown that this transformation corresponds essentially to the left shift of the associated sequences of canonical moments as an application we show that a matricial version of the arcsine distribution can be used to characterize a certain centrality property of non negative hermitian measures on

a generalization of the berlekamp massey algorithm for multisequence shift register synthesis with applications to decoding cyclic codes a generalization of the berlekamp massey algorithm is presented for synthesizing minimum length linear feedback shift registers for generating prescribed multiple sequences a more general problem is first considered that of finding the smallest initial set of linearly dependent columns in a matrix over an arbitrary field which includes the multisequence problem as a special case a simple iterative algorithm the fundamental iterative algorithm fia is presented for solving this problem the generalized algorithm is then derived through a refinement of the fia application of this generalized algorithm to decoding cyclic codes up to the hartmann tzeng ht bound and roos bound making use of multiple syndrome sequences is considered conditions for guaranteeing that the connection polynomial of the shortest linear feedback shift register obtained by the algorithm will be the error locator polynomial are determined with respect to decoding up to the ht bound and special cases of the roos bound

stieltjes continued fraction and qd algorithm scalar vector and matrix cases the definition in previous studies of vector stieltjes continued fractions in connection with spectral properties of band operators with intermediate zero diagonals left unsolved the question of a direct definition of their coefficients in terms of the original data a vector of stieltjes series the subject was more undefined in the matrix case a new version of the qd algorithm for matrix problem allows to extend to the vector and matrix cases the result of stieltjes expansion of a scalar function in terms of a stieltjes continued fraction beside this connection it solves the inverse miura transform and gives interesting identities between general band matrix and sparse band matrix finally as a consequence we extend to some dynamical systems a method known for toda lattices

a new lower bound for the minimum distance of a cyclic code a new lower bound for the minimum distance of a linear code is derived when applied to cyclic codes both the bose chaudhuri hocquenghem bch bound and the hartmann tzeng ht bound are obtained as corollaries examples for which the new bound is superior to these two bounds as well as to the carlitz uchiyama bound are given

the asep and determinantal point processes we introduce a family of discrete determinantal point processes related to orthogonal polynomials on the real line with correlation kernels defined via spectral projections for the associated jacobi matrices for classical weights we show how such ensembles arise as limits of various hypergeometric orthogonal polynomial ensembles we then prove that the q laplace transform of the height function of the asep with step initial condition is equal to the expectation of a simple multiplicative functional on a discrete laguerre ensemble a member of the new family this allows us to obtain the large time asymptotics of the asep in three limit regimes a for finitely many rightmost particles b gue tracy widom asymptotics of the height function c kpz asymptotics of the height function for the asep with weak asymmetry we also give similar results for two instances of the stochastic six vertex model in a quadrant the proofs are based on limit transitions for the corresponding determinantal point processes

matrix valued orthogonal polynomials on the real line some extensions of the classical theory in the work presented below the classical subject of orthogonal polynomials on the real line is discussed in the matrix setting an analogue of the determinant definition of orthogonal polynomials is presented the classical properties such as the recurrence relation the kernel polynomials and the christoffel darboux formula are discussed a function for the system of matrix valued orthogonal polynomials on the real line is presented some properties of the functions are investigated

a convex optimization model for finding non negative polynomials this paper presents a convex optimization model for the problem of finding some polynomials for which certain linear combinations are non negative polynomials this model is then applied to solve several filter design problems we first reformulate some low pass filter design problems with finite or infinite impulse response as optimization problems over non negative real or complex polynomials whose feasibility problems can be solved by applying our model the whole optimization problems are then solved by using a combination of a bisection search procedure on an appropriate parameter and our convex optimization model to solve the feasibility problems some numerical examples illustrate the method

matrix measures moment spaces and favard s theorem for the interval 0 1 and 0 abstract in this paper we study the moment spaces corresponding to matrix measures on compact intervals and on the nonnegative line 0 a representation for nonnegative definite matrix polynomials is obtained which is used to characterize moment points by properties of generalized hankel matrices we also derive an explicit representation of the orthogonal polynomials with respect to a given matrix measure which generalizes the classical determinant representation of the one dimensional case moreover the coefficients in the recurrence relations can be expressed explicitly in terms of the moments of the matrix measure these results are finally used to prove a refinement of the well known favard theorem for matrix measures which characterizes the domain of the underlying measure of orthogonality by properties of the coefficients in the recurrence relationships

on the minimum distance of cyclic codes the main result is a new lower bound for the minimum distance of cyclic codes that includes earlier bounds i e bch bound ht bound roos bound this bound is related to a second method for bounding the minimum distance of a cyclic code which we call shifting this method can be even stronger than the first one for all binary cyclic codes of length with two exceptions we show that our methods yield the true minimum distance the two exceptions at the end of our list are a code and its even weight subcode we treat several examples of cyclic codes of length geq 63

a code decomposition approach for decoding cyclic and algebraic geometric codes an approach based on code decomposition and partial transform for constructing algorithms for decoding cyclic codes and algebraic geometric ag codes is introduced a general decoding procedure applicable to arbitrary linear codes and their subfield subcodes is then developed in particular we have developed algorithms for error and erasure decoding of cyclic codes up to their actual minimum distance error and erasure decoding of ag codes up to their designed minimum distance and decoding of subfield subcodes of ag codes geometric bch codes moreover a generalization of geometric bch codes is introduced it is shown by an example that for a simple class of ag codes the so called one point codes this generalization brings a better estimate of their minimum distance it is also shown that the algorithms developed in this paper can be applied to decode these codes up to their estimated minimum distance

a real nullstellensatz for free modules abstract let a be the algebra of all n n matrices with entries from r x 1 x d and let g 1 g m f a we will show that f a v 0 for every a r d and v r n such that g i a v 0 for all i if and only if f belongs to the smallest real left ideal of a which contains g 1 g m here a left ideal j of a is real if for every h 1 h k a such that h 1 t h 1 h k t h k j j t we have that h 1 h k j we call this result the one sided real nullstellensatz for matrix polynomials we first prove by induction on n that it holds when g 1 g m f have zeros everywhere except in the first row this auxiliary result can be formulated as a real nullstellensatz for the free module r x 1 x d n

matrix measures on the unit circle moment spaces orthogonal polynomials and the geronimus relations abstract we study the moment space corresponding to matrix measures on the unit circle moment points are characterized by non negative definiteness of block toeplitz matrices this characterization is used to derive an explicit representation of orthogonal polynomials with respect to matrix measures on the unit circle and to present a geometric definition of canonical moments it is demonstrated that these geometrically defined quantities coincide with the verblunsky coefficients which appear in the szego recursions for the matrix orthogonal polynomials finally we provide an alternative proof of the geronimus relations which is based on a simple relation between canonical moments of matrix measures on the interval 1 1 and the verblunsky coefficients corresponding to matrix measures on the unit circle

the general rational interpolation problem and its connection with the nevanlinna pick interpolation and power moment problem abstract the hankel vector approach in a recent work of the author with zhao and zhang on the general rational interpolation problem grip allows the problem to be reduced to what amounts to a limiting case i e a high order rational interpolation at only a single node infinity in particular it gives rise to a new method to find the coefficient matrix of the linear fractional parametrization of interpolants to the grip this approach is extended in the present paper to the nevanlinna pick np interpolation problem with multiple nodes in the class of nevanlinna functions which can be in essence considered as a grip with certain symmetries the hankel vector suitably adapted to the present situation is in fact a finite nonnegative sequence relative to an axis if the problem is solvable in particular a finite positive sequence in the indeterminate case this development leads to an intrinsic connection between the np problem and problems of the truncated power moment and reconstructing generators of the hankel vector in question and therefore opens up certain ways for the discussion and solution of the np problem and its relatives on the basis of the theories of the rational interpolation and algebraic moment problems

moment problems for operator polynomials abstract haviland s theorem states that given a closed subset k in r n each functional l r x r positive on pos k p r x p k 0 admits an integral representation by a positive borel measure schmudgen proved that in the case of compact semialgebraic set k it suffices to check positivity of l on a preordering t having k as the non negativity set further he showed that the compactness of k is equivalent to the archimedianity of t the aim of this paper is to extend these results from functionals on the usual real polynomials to operators mapping from the real matrix or operator polynomials into r m n r or b k

algebraic decoding of some quadratic residue codes with weak locators in this paper an explicit expression of the weak locator polynomial for p ary quadratic residue codes is presented by a modification of the feng tzeng matrix method the differences between the modified version and the original feng tzeng matrix are that in the new matrix not every entry is a syndrome and every syndrome entry is a known syndrome by utilizing this technique an algebraic decoding of the ternary 61 30 12 quadratic residue code is proposed this new result has never been seen in the literature to our knowledge an advantage of the proposed decoding algorithm is that in general the obtained weak locator polynomials can decode efficiently not only all the error patterns of weights four and five but also some error patterns of weight six

on an interpolation problem for j potapov functions abstract the main goal of this paper is the study of an analogue of the matricial schur problem for the potapov class p j d of j contractive meromorphic functions in the open unit disk d using an approach which is based on an analysis of j central j potapov functions we will solve this problem in the most general case moreover in the nondegenerate case we will give a description of the corresponding weyl matrix balls furthermore we will investigate the limit behaviour of the weyl matrix balls associated with the functions belonging to some particular subclass of p j d

the nevanlinna pick interpolation problems and power moment problems for matrix valued functions abstract this paper is concerned with the study of several types of matrix interpolation problems including the matrix nevanlinna pick interpolation problem in the nevanlinna class n p the matrix nevanlinna pick interpolation problem in the stieltjes class l p the truncated matrix power moment problem on the real axis the stieltjes matrix power moment problem we use the so called block hankel vector approach for the treatment of these problems and make explicit the correspondence between solutions to the nevanlinna pick interpolation problem and solutions to an associated power moment problem with the block hankel vector under consideration this enables one not only to find existence criteria for the solutions of the nevanlinna pick interpolation problem but also to describe the solutions based on the results of power moment problem and vice versa

bounded rank one perturbations in sampling theory sampling theory concerns the problem of reconstruction of functions from the knowledge of their values at some discrete set of points in this paper we derive an orthogonal sampling theory and associated lagrange interpolation formulae from a family of bounded rank one perturbations of a self adjoint operator that has only discrete spectrum of multiplicity one

algebraic decoding beyond e sub bch of some binary cyclic codes when e e sub bch for a number of binary cyclic codes with e e sub bch algebraic algorithms are given to find the error locator polynomial thus for these codes more errors can be corrected algebraically than by the berlekamp massey algorithm in some cases all error patterns of weight up to e can be decoded in other cases only error patterns of weight up to e with e sub bch

algebraic decoding of cyclic codes without error locator polynomials the algebraic decoding of a p ary cyclic code consists of four steps 1 computation of the known syndromes using the received word 2 computation of the unknown syndromes from the known syndromes 3 computation of the error positions by a use of the berlekamp massey bm algorithm and chien s search and 4 computation of the error values by solving a linear system this paper addresses two problems of determining the error positions and computing the unknown syndromes to solve the first problem a new matrix together with gaussian elimination instead of the bm algorithm and chien s search is proposed in this new simplified decoder finding an error locator polynomial is completely avoided a main advantage of the presented decoding method is when the bose chaudhuri hocquenghem bound is unequal to the minimum distance of the code some cyclic codes which do not have 2t consecutive known syndromes can be decoded up to their actual minimum distance by using the presented matrix once to solve the second problem two algorithms for different square matrices reported recently by lee et al are also provided to calculate the value of an unknown syndrome finally an algebraic decoding of the 47 23 15 ternary cyclic code is given

on maximum mass measures in truncated matricial hamburger moment problems and related interpolation problems abstract a significant extremal question is considered within the solution sets of two different nondegenerate truncated matricial hamburger moment problems taking an arbitrary r whether there exists one and only one solution of each of those two moment problems which has a concentrated matrix mass at the point equal to the maximum mass the main concern of this paper is to indicate that for the first moment problem problem thm 2 n the answer is yes however for the second moment problem problem thm m the existence is not generally fulfilled and it holds if and only if r has to satisfy some additional condition these observations further serve as starting point to look for the corresponding extremal feature for three nondegenerate matrix interpolation problems of nevanlinna pick type based on the so called block hankel vector approach

on a simultaneous approach to the even and odd truncated matricial stieltjes moment problem ii an α schur stieltjes type algorithm for sequences of holomorphic matrix valued functions abstract the main goal of this paper is to achieve a simultaneous treatment of the even and odd truncated matricial stieltjes moment problems in the most general case these results are generalizations of results of chen and hu 5 19 which considered the particular case 0 our approach is based on schur analysis methods more precisely we use two interrelated versions of schur type algorithms namely an algebraic one and a function theoretic one the algebraic version was worked out in a former paper of the authors it is an algorithm which is applied to finite or infinite sequences of complex matrices the construction and investigation of the function theoretic version of our schur type algorithm is a central theme of this paper this algorithm will be applied to relevant subclasses of holomorphic matrix valued functions of the stieltjes class using recent results on the holomorphicity of the moore penrose inverse of matrix valued stieltjes functions we obtain a complete description of the solution set of the moment problem under consideration in the most general situation

perturbations of orthogonal polynomials with periodic recursion coefficients the results of denisov rakhmanov szeg shohat nevai and killip simon are extended from asymptotically constant orthogonal polynomials on the real line oprl and unit circle opuc to asymptotically periodic oprl and opuc the key tool is a characterization of the isospectral torus that is well adapted to the study of perturbations

error locating pairs for cyclic codes a general decoding method for linear codes is investigated for cyclic codes the decoding consists of solving two systems of linear equations all but four binary cyclic codes of length less than 63 can so be decoded up to their actual distance a new family of codes is given for which the decoding needs only o n sup 2 operations

simplified understanding and efficient decoding of a class of algebraic geometric codes an efficient decoding algorithm for algebraic geometric codes is presented for codes from a large class of irreducible plane curves including hermitian curves it can correct up to d 1 2 errors where d is the designed minimum distance with it we also obtain a proof of d sub min spl ges d without directly using the riemann roch theorem the algorithm consists of gaussian elimination on a specially arranged syndrome matrix followed by a novel majority voting scheme a fast implementation incorporating block hankel matrix techniques is obtained whose worst case running time is o mn sup 2 where m is the degree of the curve applications of our techniques to decoding other algebraic geometric codes to decoding bch codes to actual minimum distance and to two dimensional shift register synthesis are also presented

orthogonal matrix polynomials and applications orthogonal matrix polynomials on the real line or on the unit circle have properties which are natural generalizations of properties of scalar orthogonal polynomials appropriately modified for the matrix calculus we show that orthogonal matrix polynomials both on the real line and on the unit circle appear at various places and we describe some of them the spectral theory of doubly infinite jacobi matrices can be described using orthogonal 2 x 2 matrix polynomials on the real line scalar orthogonal polynomials with a sobolev inner product containing a finite number of derivatives can be studied using matrix orthogonal polynomials on the real line orthogonal matrix polynomials on the unit circle are related to unitary block hessenberg matrices and are very useful in multivariate time series analysis and multichannel signal processing finally we show how orthogonal matrix polynomials can be used for gaussian quadrature of matrix valued functions parallel algorithms for this purpose have been implemented using pvm and some examples are worked out

on dyukarev s resolvent matrix for a truncated stieltjes matrix moment problem under the view of orthogonal matrix polynomials abstract the main result of this paper is a new representation of yu m dyukarev s resolvent matrix for the non degenerate truncated matricial version of the classical stieltjes moment tmcsm problem this new representation is based on the use of a quadruple of q q matrix polynomials which are characterized by certain orthogonality properties in this way useful new interrelations between the moment problems of stieltjes and hamburger which are associated with a stieltjes positive definite sequence s j j 0 2 n of complex q q matrices are found this includes an explicit formula connecting the resolvent matrices used by yu m dyukarev and i v kovalishina respectively furthermore the coefficients in the recurrence formulas for the orthogonal polynomials with respect to a stieltjes positive definite sequence are expressed in terms of the dyukarev stieltjes parameters additionally we obtain two different representations for each extremal solution of the tmcsm problem via matrix continued fraction expansions

high order convergent deferred correction schemes based on parametrerized runge kutta nystrom methods for second order boundary value problems abstract iterated deferred correction is a widely used approach to the numerical solution of first order systems of nonlinear two point boundary value problems normally the orders of accuracy of the various methods used in a deferred correction scheme differ by 2 and as a direct result each time deferred correction is used the order of the overall scheme is increased by a maximum of 2 in 16 however it has been shown that there exist schemes based on parameterized runge kutta methods which allow a higher increase of the overall order a first example of such a high order convergent scheme which allows an increase of 4 orders per deferred correction was based on two mono implicit runge kutta methods in the present paper we will investigate the possibility for high order convergence of schemes for the numerical solution of second order nonlinear two point boundary value problems not containing the first derivative two examples of such high order convergent schemes based on parameterized runge kutta nystrom methods of orders 4 and 8 are analysed and discussed

sum rules for jacobi matrices and their applications to spectral theory we discuss the proof of and systematic application of case s sum rules for jacobi matrices of special interest is a linear combination of two of his sum rules which has strictly positive terms among our results are a complete classification of the spectral measures of all jacobi matrices j for which j j 0 is hilbert schmidt and a proof of nevai s conjecture that the szego condition holds if j j 0 is trace class

n extremal matrices of measures for an indeterminate matrix moment problem in this paper we study the n extremal matrices of measures associated to a completely indeterminate matrix moment problem i e those matrices of measure w solutions of a completely indeterminate matrix moment problem for which the linear space of matrix polynomials is dense in the corresponding l2 w

on the truncated matricial stieltjes moment problem m α sj j 0m abstract this paper gives via stieltjes transform a complete description of the solution set of a matricial truncated stieltjes type power moment problem in the non degenerate and degenerate cases the approach is based on the schur type algorithm which was worked out in the papers 9 10 furthermore the subset of parameters is determined which corresponds to another truncated matricial stieltjes type moment problem

an automatic continuation strategy for the solution of singularly perturbed linear two point boundary value problems an automatic continuation alogorithm for the solution of linear singular perturbation problems is developed and incorporated into two codes which implement global methods for solving two point boundary value problems specifically the algorithm will be based upon error estimates formulated in the collocation code colsys and the deferred correction code hagron the benefits of using continuation are clearly demonstrated for both codes for a large class of problems

polynomial interpolation and gaussian quadrature for matrix valued functions abstract the techniques for polynomial interpolation and gaussian quadrature are generalized to matrix valued functions it is shown how the zeros and rootvectors of matrix orthonormal polynomials can be used to get a quadrature formula with the highest degree of precision

superconvergent deferred correction methods for first order systems of nonlinear two point boundary value problems iterated deferred correction is a widely used approach to the numerical solution of first order systems of nonlinear two point boundary value problems normally the orders of accuracy of the various methods used in a deferred correction scheme differ by 2 and as a direct result each time a deferred correction is applied the order of the overall scheme is increased by a maximum of 2 in this paper we consider the construction of mono implicit runge kutta mirk methods where an increase of four orders of accuracy is obtained for each deferred correction we develop a very powerful yet rather straightforward theory which allows us to identify the appropriate runge kutta formulae for inclusion in such schemes in particular we will focus on the construction of pairs of mirk formulae of order 4 and 8 which will allow this superconvergence to be realized we will further show that it is possible to derive formulae of this type for which high order interpolants and accurate error estimates are readily available

a survey of numerical techniques for solving singularly perturbed ordinary differential equations this survey paper contains a surprisingly large amount of material and indeed can serve as an introduction to some of the ideas and methods of singular perturbation theory starting from prandtl s work a large amount of work has been done in the area of singular perturbations this paper limits its coverage to some standard singular perturbation models considered by various workers and the numerical methods developed by numerous researchers after 1984 2000 the work done in this area during the period 1905 1984 has already been surveyed by the first author of this paper see appl math comput 30 1989 223 for details due to the space constraints we have covered only singularly perturbed one dimensional problems

a unified treatment for the matrix stieltjes moment problem in both nondegenerate and degenerate cases abstract this paper presents a unified method for the solution of the matrix stieltjes moment problem in both the nondegenerate and degenerate cases based on the use of our recent investigation of the matrix hamburger moment problem as a by product a complete solution of the matrix nevanlinna pick interpolation problem in the class of the matrix valued stieltjes functions is derived as well

generalized chebyshev polynomials and discrete schrodinger operators we investigate a solution of the difference equation with the boundary conditions u0a b i u 1a b 0 where a b are hermitian elements of a c algebra una b are usually called generalized chebyshev polynomials of the second kind the equation cannot be easily simplified as in the scalar case because a and b do not need to commute however we are able to compute the spectrum of the corresponding orthogonality measure which is very important for the investigation of the discrete schrodinger operator related to una b

a note on the matrix valued q d algorithm and matrix orthogonal polynomials on 0 1 and 0 in this note a matrix version of the q d algorithm is introduced it is shown that the algorithm may be used to obtain the coefficients of the recurrence relations for matrix orthogonal polynomials on the interval 0 and 0 1 from its moment generating functional the algorithm is illustrated by several examples which generalize classical orthogonal polynomials on the real line

on matrix hurwitz type polynomials and their interrelations to stieltjes positive definite sequences and orthogonal matrix polynomials abstract this paper is a direct continuation of the author s recent investigations 4 on the non degenerate truncated matricial stieltjes problem inspired by a characterization of scalar hurwitz polynomials via continued fractions see e g gantmacher 15 a class of matrix hurwitz type polynomials is introduced it is shown that this class is intimately related with stieltjes positive definite sequences of matrices and can be expressed in terms of the associated stieltjes quadruple of sequences of left orthogonal matrix polynomials it is shown that a monic matrix polynomial is a matrix hurwitz type polynomial if and only if the sequence of its markov parameters is a stieltjes positive definite sequence

extension of the bch decoding algorithm to decode binary cyclic codes up to their maximum error correction capacities the bch algorithm is extended to correct more errors than indicated by the bch bound in the first step of the decoding procedure a number of errors corresponding to a particular case of the hartmann tzeng bound are corrected in the second step full error correction is the goal a measure for the worst case number of field elements of an extension field gf 2 sup m that must be tested for this purpose is given for binary cyclic linear unequal error protection codes as well as for conventional binary cyclic codes

decoding cyclic and bch codes up to actual minimum distance using nonrecurrent syndrome dependence relations the decoding capabilities of algebraic algorithms mainly the berlekamp massey algorithm the euclidean algorithm and the authors 1989 generalizations of these algorithms are basically constrained by the minimum distance bounds of the codes the authors introduce a more general procedure which breaks away from this restriction and which can determine the error locations from nonrecurrent dependence relations among the syndromes it can decode many cyclic and bch codes up to their actual minimum distance and is seen to be a generalization of the procedure introduced by w w peterson and e j weldon 1972

the nevanlinna pick interpolation problems and power moment problems for matrix valued functions iii the infinitely many data case abstract the objective of this paper is to establish an intrinsic and simple correspondence between a multiple nevanlinna pick matrix interpolation np problem in the nevanlinna class n p with a denumerable set of nodes and a certain full hamburger matrix moment problem with the help of the indicated correspondence the solution of an infinite np problem in n p can be reduced to what amounts to the study of the associated full hamburger matrix moment problem with the so called block hankel vector of the former and vice versa these investigations can be regarded as a natural extension of our previous work on both the np problems with a finite set of interpolation data and the truncated power matrix moment problems

use of grobner bases to decode binary cyclic codes up to the true minimum distance a general algebraic method for decoding all types of binary cyclic codes is presented it is shown that such a method can correct t d 1 2 errors where d is the true minimum distance of the given cyclic code the key idea behind this decoding technique is a systematic application of the algorithmic procedures of grobner bases to obtain the error locator polynomial l z the discussion begins from a set of syndrome polynomials f and the ideal t f generated by f it is proved here that the process of transforming f to the normalized reduced grobner basis of i f with respect to the purely lexicographical ordering automatically converges to l z furthermore it is shown that l z can be derived from any normalized grobner basis of i f with respect to any admissible total ordering to illustrate this new approach the procedures for decoding certain bch codes and quadratic residue codes are demonstrated

relative asymptotics for orthogonal matrix polynomials in this paper we study sequences of matrix polynomials that satisfy a non symmetric recurrence relation to study this kind of sequences we use a vector interpretation of the matrix orthogonality in the context of these sequences of matrix polynomials we introduce the concept of the generalized matrix nevai class and we give the ratio asymptotics between two consecutive polynomials belonging to this class we study the generalized matrix chebyshev polynomials and we deduce its explicit expression as well as we show some illustrative examples the concept of a dirac delta functional is introduced we show how the vector model that includes a dirac delta functional is a representation of a discrete sobolev inner product it also allows to reinterpret such perturbations in the usual matrix nevai class finally the relative asymptotics between a polynomial in the generalized matrix nevai class and a polynomial that is orthogonal to a modification of the corresponding matrix measure by the addition of a dirac delta functional is deduced

the classical moment problem as a self adjoint finite difference operator abstract this is a comprehensive exposition of the classical moment problem using methods from the theory of finite difference operators among the advantages of this approach is that the nevanlinna functions appear as elements of a transfer matrix and convergence of pade approximants appears as the strong resolvent convergence of finite matrix approximations to a jacobi matrix as a bonus of this we obtain new results on the convergence of certain pade approximants for series of hamburger

a point interaction for the discrete schrodinger operator and generalized chebyshev polynomials we consider semi infinite jacobi matrices corresponding to a point interaction for the discrete schrodinger operator our goal is to find explicit expressions for the spectral measure the resolvent and other spectral characteristics of such jacobi matrices it turns out that the spectral analysis of this toy problem leads to a new class of orthogonal polynomials generalizing the classical chebyshev polynomials

inversion of a generalized block loewner matrix the minimal partial realization and matrix rational interpolation problem abstract inversion theorems for generalized block loewner matrices with non square blocks are presented connections are made to the general i e hermite matrix rational interpolation theory of partial realization and its extensions theoretical and computational aspects of many other special types of matrices such as block hankel block toeplitz vandermonde block bezout etc

the truncated hamburger matrix moment problems in the nondegenerate and degenerate cases and matrix continued fractions abstract the present paper deals simultaneously with the nondegenerate and degenerate truncated hamburger matrix moment problems in a unified way based on the use of the schur algorithm involving matrix continued fractions a full analysis of them together with a relative matrix moment problem on the real axis is given with the help of the correspondence between the moment problem on the real axis and the nevanlinna pick np interpolation the solutions of the nontangential np interpolation in the nevanlinna class are derived as an application

quadrature formulas for matrix measures a geometric approach a geometric approach to quadrature formulas for matrix measures is presented using the relations between the representations of the boundary points of the moment space generated by all matrix measures and quadrature formulas simple proofs of existence and uniqueness of quadrature formulas of maximal degree of precision are given several new quadrature formulas for matrix measures supported on a compact interval are presented and several examples are discussed additionally a special construction of degenerated quadrature formulas is discussed and some results regarding the location of the zeros of polynomials orthogonal with respect to a matrix measure on a compact interval are obtained as a by product

principal representations and canonical moment sequences for distributions on an interval abstract an initial expository introduction to moment space geometry and canonical moment sequences for distributions on the unit interval leads to the exposure of previously unknown but striking symmetries in stieltjes transforms for these distributions when the transforms are viewed as moment functions the n th and the n 1 st canonical moments corresponding to a moment point interior to the space of the first n moments are shown to depend in a very simple way on the canonical intervals determined by the two principal representations of the moment point for example the n th canonical moment is just the sum of the lengths of the upper canonical intervals

operator positivstellensatze for noncommutative polynomials positive on matrix convex sets abstract this article studies algebraic certificates of positivity for noncommutative nc operator valued polynomials on matrix convex sets such as the solution set d l called a free hilbert spectrahedron of the linear operator inequality loi l x a 0 i j 1 g a j x j 0 where a j are self adjoint linear operators on a separable hilbert space x j matrices and i is an identity matrix if a j are matrices then l x 0 is called a linear matrix inequality lmi and d l a free spectrahedron for monic lmis i e a 0 i and nc matrix valued polynomials the certificates of positivity were established by helton klep and mccullough in a series of articles with the use of the theory of complete positivity from operator algebras and classical separation arguments from real algebraic geometry since the full strength of the theory of complete positivity is not restricted to finite dimensions but works well also in the infinite dimensional setting we use it to tackle our problems first we extend the characterization of the inclusion d l 1 d l 2 from monic lmis to monic lois l 1 and l 2 as a corollary one immediately obtains the description of a polar dual of a free hilbert spectrahedron d l and its projection called a free hilbert spectrahedrop further on using this characterization in a separation argument we obtain a certificate for multivariate matrix valued nc polynomials f positive semidefinite on a free hilbert spectrahedron defined by a monic loi replacing the separation argument by an operator fejer riesz theorem enables us to extend this certificate in the univariate case to operator valued polynomials f finally focusing on the algebraic description of the equality d l 1 d l 2 we remove the assumption of boundedness from the description in the lmis case by an extended analysis however the description does not extend to lois case by counterexamples