new block quadrature rules for the approximation of matrix functions abstract golub and meurant have shown how to use the symmetric block lanczos algorithm to compute block gauss quadrature rules for the approximation of certain matrix functions we describe new block quadrature rules that can be computed by the symmetric or nonsymmetric block lanczos algorithms and yield higher accuracy than standard block gauss rules after the same number of steps of the symmetric or nonsymmetric block lanczos algorithms the new rules are block generalizations of the generalized averaged gauss rules introduced by spalevic applications to network analysis are presented

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

extensions of simple modules for finite chevalley groups let g be an almost simple algebraic group over an algebraically closed field k of positive characteristic p we assume that g is defined and split over the prime field 5 for it 3 1 we denote by g n the finite group consisting of the points of g over the field with p elements our first result states that for a certain class of simple modules for g n the only extensions that can occur are those which are restrictions of g extensions see theorem 2 8 the full story about g extensions is not known yet except for a few rather small groups however much more is known about extensions for the algebraic group than about those for the finite group for instance the linkage principle l gives an upper bound for which g extensions may occur moreover a proof of the lusztig conjecture on the irreducible characters for g also implies strong statements about g extensions see 3 proposition 2 81 it is an open question whether all extensions between simple g n modules can be found via the extension theory for g we prove that in generic cases this is in fact so see theorem 3 2 moreover for the groups x 3 p and sp 4 p we are able to determine completely all extensions between simple modules from the corresponding known results for the algebraic groups x 3 k and sp 4 k our results extend previous results by cline parshall and scott s 91 who consider extensions of the trivial module by a minimal module and smith ls who considers extensions between certain minimal modules the method we use is a generalization of the one used by humphreys 131 who proves the existence of certain selfextensions for sp 4 p it should be pointed out that our methods do not work for p small on the other hand everything we do generalizes easily to the twisted chevalley groups see theorem 3 4 we demonstrate this by working out all extensions for su 3 p see section 4 3

matrix orthogonal polynomials on the unit circle the properties of matrix orthogonal polynomials on the unit circle are investigated beginning with their recurrence formulas the techniques of scattering theory and banach algebras are used in the investigation a matrix generalization of a theorem of baxter is proved

generalizations of orthogonal polynomials we give a survey of recent generalizations of orthogonal polynomials that includes multidimensional matrix and vector orthogonal polynomials and multivariate versions multipole orthogonal rational functions variants and extensions of the orthogonality conditions multiple orthogonality most of these generalizations are inspired by the applications in which they are applied we also give a glimpse of these applications which are usually generalizations of applications where classical orthogonal polynomials also play a fundamental role moment problems numerical quadrature rational approximation linear algebra recurrence relations and random matrices

critical behaviour of the two dimensional potts model with a continuous number of states a finite size scaling analysis we investigate the critical behaviour of the two dimensional q state potts model using finite size scaling and transfer matrix methods for the continuous transition range 0 these results for continuous q were obtained from a transfer matrix constructed for a generalized whitney polynomial representing the potts models an effective algorithm to compute the dominant eigenvalues of this essentially nonsymmetric transfer matrix is developed

matrix polynomial solutions of tangential lagrange sylvester interpolation conditions of low mcmillan degree abstract a matrix polynomial of mcmillan degree n 1 which satisfies n tangential lagrange interpolation conditions is constructed explicitly in realization form analogous results are given for more general bitangential lagrange sylvester interpolation conditions a corollary is a type of euclidean algorithm for matrix polynomials and mcmillan degree the main tool is the recent parametrization of all matrix polynomial solutions of a set of lagrange sylvester interpolation conditions due to ball gohberg and rodman

a divide and conquer method for unitary and orthogonal eigenproblems leth n xn be a unitary upper hessenberg matrix whose eigenvalues and possibly also eigenvectors are to be determined we describe how this eigenproblem can be solved by a divide and conquer method in which the matrixh is split into two smaller unitary upper hessenberg matricesh 1 andh 2 by a rank one modification ofh the eigenproblems forh 1 andh 2 can be solved independently and the solutions of these smaller eigenproblems define a rational function whose zeros on the unit circle are the eigenvalues ofh the eigenvector ofh can be determined from the eigenvalues ofh and the eigenvectors ofh 1 andh 2 the outlined splitting of unitary upper hessenberg matrices into smaller such matrices is carried out recursively this gives rise to a divide and conquer method that is suitable for implementation on a parallel computer whenh n xn is orthogonal the divide and conquer scheme simplifies and is described separately our interest in the orthogonal eigenproblem stems from applications in signal processing numerical examples for the orthogonal eigenproblem conclude the paper

the 2 1 d ising model on a triangular lattice the 2 1 d ising model on a triangular lattice is explored for lattices of up to 5 5 sites finite size scaling techniques are used to estimate the critical parameters xc 0 2096 2 nu 0 627 4 beta 0 322 6 gamma 1 236 8 these agree with previous estimates within errors and with the hypotheses of universality and hyperscaling the behaviour at low temperatures is also studied and theoretical predictions of finite size scaling at a first order transition are confirmed

finite lattice extrapolations for z3 and zs models a method is presented to accelerate the convergence of finite lattice sequences to their bulk limit the calculation of highly accurate estimates of the critical parameters of the bulk system is then possible applied to the hamiltonian version of the z3 model three state potts model in 1 1 dimensions these techniques yield estimates for the exponents gamma 1 444 or 0 0001 nu 0 8333 or 0 0003 and alpha 0 33 or 0 01 for the z5 model the presence of a kosterlitz thouless transition is confirmed

cmv block matrices for symmetric matrix measures on the unit circle abstract in this contribution we study the relationship between the cmv block matrices associated with two symmetric positive definite matrix measures supported on the unit circle we also consider certain transformations of an orthogonality matrix measure supported on the unit circle and deduce connection formulas for the corresponding orthogonal families and their verblunsky matrix coefficients

szego polynomials associated with wiener levinson filters abstract szego polynomials are studied in connection with wiener levinson filters formed from discrete signals xn xn k n 1k 0 our main interest is in the frequency analysis problem of finding the unknown frequencies j when the signal is a trigonometric polynomial x n k j i i j e i j k associated with this signal is the sequence of monic szego polynomials n n z n 0 orthogonal on the unit circle with respect to a distribution function n explicit expressions for the weight function n and associated szego function dn z are given in terms of the z transform xn z of the signal xn several theorems are given to support the following conjecture which was suggested by numerical experiments as n and n increase the 2i 1 zeros of n n z of largest modulus approach the points ei j we conclude by showing that the reciprocal polynomials n n z z n n n 1 z are pade numerators for pade approximants of fixed denominator degree to a meromorphic function related to dn z

on the 1 cohomology of finite groups of lie type for each simply connected semisimple algebraic groupgdefined and split over the prime field fp we establish a uniform bound onnabove which all of the first cohomology groups with values in the simple modules for the finite groupg n are determined by those for the algebraic groupg fp

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

gaussian quadrature formulae for matrix weights we study gaussian quadrature formulae for a matrix weight we firstly show how to generate gaussian quadrature formulae such that the sum of the rank of the quadrature matrix coefficients is a given nonnegative integer l the nodes are the eigenvalues of certain perturbation of the truncated n jacobi matrix of size l or equivalently the zeros of certain matrix combination of two consecutive orthonormal polynomials and the quadrature coefficients are the coefficients in the partial fraction decomposition of the ratio between the inverse of this orthonormal matrix polynomial and the associated polynomial of the second kind we secondly prove that any gaussian quadrature formula for a matrix weight must be of this form

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

generic cohomology for twisted groups let g be a simple algebraic group defined and split over ko fp and let a be a surjective endomorphism of g with finite fixed point set go we give conditions under which cohomology groups of g are isomorphic to cohomology groups of go let g be a simple algebraic group defined and split over k fp and for q pm let g q be the subgroup of fq rational points for a finite dimensional rational g module v and a nonnegative integer e let v e be the g module obtained by twisting the original g action on v by the frobenius endomorphism x x1p of g cline parshall scott and van der kallen proved in 2 that for sufficiently large q and e depending on v and n the restriction map induces an isomorphism from the rational cohomology group h g v e to h g q v e this implies that as q increases the groups h g q v e have a stable or generic value h en g v in this paper we prove an analogous theorem with g q replaced by go for a surjective endomorphism a of g having finite fixed point set the first section of the paper summarizes the basic results on endomorphisms of algebraic groups required for the proof some arithmetic facts are established in the second section and the main theorem is proved in the third section the author is grateful to leonard l scott for several helpful conversations and to the referee for comments on an earlier version of this paper 1 endomorphisms of algebraic groups we briefly review here some results on endomorphisms of algebraic groups which will be needed later we refer the reader to 4 and 5 for more details let k be the algebraic closure of the prime field ko fp and let g be a simple algebraic group defined and split over ko if a is a surjective rational endomorphism of g having finite fixed point set g0 a stabilizes a borel subgroup b and a maximal torus t ua is a t equivariant isomorphism then axa u x cauq a for some received by the editors may 5 1980 and in revised form november 12 1980 1980 mathematics subject classification primary 20g10

asymptotics for szego polynomials associated with wiener levinson filters abstract pan k asymptotics for szego polynomials associated with wiener levinson filters journal of computational and applied mathematics 46 1993 387 394 both weak and strong convergence theorems are given to support a jones njastad and saff conjecture that unknown frequencies in a periodic discrete time signal can be determined by zeros of szego polynomials with respect to some distribution function

finite size scaling and universality in the 2 1 d ising model the energy and magnetisation density of the 2 1 d ising model are investigated for the thermal exponent the author finds xepsilon 1 42 or 0 02 in agreement with hyperscaling the privman fisher universality hypothesis is confirmed by two methods first the author presents at the critical point numerical data for the finite size scaling amplitudes of the singular free energy magnetisation susceptibility and surface tension and checks them for universality second using the explicitly computed eigenvectors of the hamiltonian it is shown that the ratios of matrix elements of the energy and magnetisation density are universal

extensions of simple modules for sl3 2n and su3 2n the group of extensions between any two irreducible 2 modular representations of the groups sl 3 2 m and su 3 2 m is determined

szego polynomials applied to frequency analysis abstract this paper is concerned with the problem of determining unknown frequencies 1 i using an observed discrete time signal xn xn m arising from a continuous waveform that is the superposition of a finite number of sinusoidal waves with well defined frequencies j periods p j 2 n j n z j 1 i we investigate the wiener levinson method formulated here in terms of szeg polynomials n n z orthogonal on the unit circle with respect to a distribution function n defined by the n observed values of the signal we prove that if n0 denotes the number of critical points eiwj then for every n n0 and n 1 the zeros z j n n of n n z can be arranged so that limn z j n n eiwj for each of the frequencies wj this result confirms one of the main parts of the conjecture given by jones njastad and saff this journal 1990 on the convergence of zeros of the szeg polynomials a related result on the convergence of corresponding two point pade approximants is also given

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

stochastic truncation for the 2 1 d ising model the method of symmetrized stochastic truncation is applied to the 2 1 dimensional ising model to calculate the ground state energy mass gap magnetization and susceptibility on lattices of sizes up to 6 6 sites the method is shown to be a powerful and accurate monte carlo technique provided the lattice size is not too large and the model is suitably convergent finite size scaling estimates of the critical indices are nu 0 627 2 alpha 0 12 2 beta 0 324 3 and delta 1 23 1 an estimate of the combination gamma nu 2 beta nu 3 000 1 is obtained in outstanding agreement with the expected hyperscaling relation

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

a class of matrix orthogonal polynomials on the unit circle abstract the study of matrix orthogonal polynomials on the unit circle is connected with problems of applied mathematics such as filtering theory prediction theory and scattering theory in this paper some families of matrix orthogonal polynomials on the unit circle are given explicitly the decomposition a z z a z z h 1 a z where is the linear space of complex algebraic polynomials is the main idea for this approach

jacobi matrix differential equation polynomial solutions and their properties in this paper jacobi matrix polynomials are introduced starting from the hypergeometric matrix function the differential equation satisfied by them is presented a rodrigues formula orthogonality and a three terms matrix recurrence relationship are then developed for jacobi matrix polynomials

fast qr iterations for unitary plus low rank matrices some fast algorithms for computing the eigenvalues of a block companion matrix have recently appeared in the literature in this paper we generalize the approach to encompass unitary plus low rank matrices of the form a u xy h where u is a general unitary matrix three important cases for applications are u unitary diagonal u unitary block hessenberg and u unitary in block cmv form our extension exploits the properties of a larger matrix hat a obtained by a certain embedding of the hessenberg reduction of a suitable to maintain its structural properties we show that hat a can be factored as product of lower and upper unitary hessenberg matrices possibly perturbed in the first k rows and moreover such a data sparse representation is well suited for the design of fast eigensolvers based on the qr iteration the resulting eigenvalue algorithm is fast and backward stable

asymptotic approximations for second order linear difference equations in banach algebras ii abstract a liouville green wkb asymptotic approximation theory is developed for some classes of linear second order difference equations in banach algebras the special case of linear matrix difference equations or equivalently of second order systems is emphasized rigorous and explicitly computable bounds for the error terms are obtained and this when both the sequence index and some parameter that may enter the coefficients go to infinity a simple application is made to orthogonal matrix polynomials in the nevai class

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

magnetisations from finite size scaling a method of obtaining the spontaneous magnetisation from finite lattice matrix elements of the magnetic field operator due to yang 1952 and uzelac 1980 is discussed the method is demonstrated for the case of the ising model in 1 1 dimensions and is shown to provide smooth and rapidly convergent finite lattice sequences applied to the case of the three state potts z3 model in 1 1 dimensions the method yields an estimate beta 0 111 09 or 0 000 05 for the critical exponent this confirms alexander s conjecture of universality with the hard hexagonal model

finite lattice extrapolation algorithms two algorithms for sequence extrapolation due to van den broeck and schwartz 1979 and bulirsch and stoer 1964 are reviewed and critically compared applications to three state and six state quantum chains and to the 2 1 d ising model show that the algorithm of bulirsch and stoer is superior in particular if only very few finite lattice data are available

cluster expansions for the ising and heisenberg spin models in hamiltonian lattice field theory cluster expansion methods are applied to calculate high temperature series for the vacuum energy the susceptibility and the mass gap for the ising model and the o 2 and o 3 heisenberg models in 1 1 dimensions and 2 1 dimensions critical points and critical indices are estimated for the line the square and the triangular lattices the results demonstrate universality with the normal euclidean versions of these models within errors

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

series expansions for the zero temperature transverse ising model series expansions are presented for the magnetisation susceptibility magnetic field derivatives of the susceptibility and specific heat of the zero temperature transverse ising model coefficients in these series have been calculated to tenth order in lambda 2 where lambda is the transverse field for the linear chain to eighth order for the square lattice and to seventh order for the triangular lattice these series yield estimates of the low temperature critical exponents alpha beta gamma and delta of the two and three dimensional ising models they provide good evidence for the symmetry of exponents above and below the critical point e g gamma gamma

asymptotic degeneracy of the transfer matrix spectrum for systems with interfaces relation to surface stiffness and step free energy two and three dimensional ising type systems are considered in the finite cross section cylindrical geometry an interface is forced along the cylinder strip in 2d by the antiperiodic or boundary conditions detailed predictions are presented for the largest asymptotically degenerate set of the transfer matrix eigenvalues for rough interfaces i e for 0 t tc in 2d tr t tc in 3d the eigenvalues are split algebraically and the spectral gaps are governed by thesurface stiffness coefficient for rigid interfaces i e 0 t tr in 3d the eigenvalues are split exponentially with the gaps determined by thestep free energy

gaussian quadrature for matrix valued functions on the real line abstract parallel versions of the block chebyshev algorithm to generate the recursion coefficients of orthonormal matrix polynomials and the gaussian quadrature algorithm to approximate matrix integrals on the real line are implemented on an sp1

perturbations in the nevai matrix class of orthogonal matrix polynomials in this paper we study a jacobi block matrix and the behavior of the limit of its entries when a perturbation of its spectral matrix measure by the addition of a dirac delta matrix measure is introduced 2001 elsevier science inc all rights reserved

on the multiple nevanlinna pick matrix interpolation in the class cp and the caratheodory matrix coefficient problem abstract in the present paper an intrinsic one to one correspondence between the finite multiple nevanlinna pick matrix interpolation np problem in the caratheodory class and the caratheodory matrix coefficient cc problem or equivalently the truncated trigonometric matrix moment tm problem is established thanks to this correspondence the np problem is reduced to what amounts to solving the cc problem or the tm problem associated with the so called toeplitz block vector of the former in both the nondegenerate and degenerate cases simultaneously

computation of matrix valued formally orthogonal polynomials and applications we present a computational procedure for generating formally orthogonal polynomials associated with a given bilinear hankel form with rectangular matrix valued moments our approach covers the most general case of moments of any size and is not restricted to square moments moreover our algorithm has a built in deflation procedure to handle linearly dependent or almost linearly dependent columns and rows of the block hankel matrix associated with the bilinear form possible singular or close to singular leading principal submatrices of the deflated block hankel matrix are avoided by means of look ahead techniques applications of the computational procedure to eigenvalue computations reduced order modeling the solution of multiple linear systems and the fast solution of block hankel systems are also briefly described

finite size scaling in the 2 1 d ising model hamiltonian eigenvalues are calculated for the 2 1 d ising model on lattices up to 5 5 sites finite size scaling techniques are used to estimate the critical coupling xc 0 329 or 0 001 and index v 0 635 or 0 005 the prospects are discussed for further applications of these methods in three and four dimensions

cluster expansions in the 2 1 d ising model the cluster expansion methods of nickel are applied to calculate high temperature series for the vacuum energy and specific heat the susceptibility and the mass gap in the 2 1 d ising model critical points and critical exponents are estimated for the square and triangular lattices the results demonstrate universality with the 3d ising model within errors exact linked cluster expansions are formulated for the quantities above and their convergence and scaling properties are investigated

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

on matrix integration of matrix polynomials this paper is the result of having read a series of recent papers on the quadrature formula for matrix integrals which caused a strong want of clarifying the circumstances for this purpose we have had to revise orthogonality for matrix polynomials being supported by a desire of using means adequate to needs and at the same time of trying to simplify the set up

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

orthogonal matrix polynomials and quadrature formulas we prove that the nodes of a quadrature formula for a matrix weight with the highest degree of precision must necessarily be the zeros of a certain orthonormal matrix polynomial with respect to the matrix weight and the quadrature coefficients are then the coefficients in the partial fraction decomposition of the ratio between the inverse of this orthonormal matrix polynomial and the associated polynomial of the second kind we also extend this result for quadrature formulas with degree of precision one unit smaller than the highest possible

matrix biorthogonal polynomials on the unit circle and non abelian ablowitz ladik hierarchy adler and van moerbeke 2001 commun pure appl math 54 153 205 described a reduction of the 2d toda hierarchy called the toeplitz lattice this hierarchy turns out to be equivalent to the one originally described by ablowitz and ladik 1975 j math phys 16 598 603 using semidiscrete zero curvature equations in this paper we obtain the original semidiscrete zero curvature equations starting directly from the toeplitz lattice and we generalize these computations to the matrix case this generalization leads us to the semidiscrete zero curvature equations for the non abelian or multicomponent version of the ablowitz ladik equations gerdzhikov and ivanov 1982 theor math phys 52 676 85 in this way we extend the link between biorthogonal polynomials on the unit circle and the ablowitz ladik hierarchy to the matrix case

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

critical temperatures of the d 3 s 1 2 ising model the effect of confluent corrections to scaling much attention has been paid to the d 3 s 1 2 ising model on the bcc lattice in recent years because the best high temperature series expansions were available for this lattice however in order to compare series estimates of critical exponents and temperatures with independent evaluations via the monte carlo method lattices of lower coordination number must be considered since monte carlo studies are usually made on these a study of extant susceptibility series on the fcc sc diamond and tetrahedron lattices is reported on and some comments are made on aspects of recent monte carlo analyses of the d 3 s 1 2 ising model on the sc and diamond lattices

a connection between orthogonal polynomials on the unit circle and matrix orthogonal polynomials on the real line szego s procedure to connect orthogonal polynomials on the unit circle and orthogonal polynomials on 1 1 is generalized to nonsymmetric measures it generates the so called semi orthogonal functions on the linear space of laurent polynomials and leads to a new orthogonality structure in the module this structure can be interpreted in terms of a 2 2 matrix measure on 1 1 and semi orthogonal functions provide the corresponding sequence of orthogonal matrix polynomials this gives a connection between orthogonal polynomials on the unit circle and certain classes of matrix orthogonal polynomials on 1 1 as an application the strong asymptotics of these matrix orthogonal polynomials is derived obtaining an explicit expression for the corresponding szego s matrix function

ritz and pseudo ritz values using matrix polynomials abstract in this paper we present a unified analysis of two block krylov subspace methods based on the block arnoldi procedure on a real nonsymmetric matrix a we show that matrix polynomials can be the appropriate tool for analyzing the convergence of jordan ritz pairs when a is not diagonalizable we also prove that analogous arguments can be used for studying the convergence properties of a linear system solver that minimizes the residual norm on the generated krylov subspace bgmres known results for single krylov subspace methods can be naturally generalized to the block setting explicitly demonstrating how the presence of blocks can influence the convergence of the method at hand

high temperature series expansions for the 2 1 dimensional ising model using efficient cluster expansion methods the known high temperature series for the vacuum energy specific heat susceptibility and mass gap of the 2 1 dimensional ising model on the square and triangular lattices have been extended by several terms estimates of the critical indices demonstrate very convincing universality with the euclidean version of the model

finite size scaling in the transverse ising model on a square lattice energy eigenvalues and order parameters are calculated by exact diagonalization for the transverse ising model on square lattices of up to 6 6 sites finite size scaling is used to estimate the critical parameters of the model confirming universality with the three dimensional classical ising model critical amplitudes are also estimated for both the energy gap and the ground state energy

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

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

the 1 1 do 2 model a finite lattice analysis finite size methods and the theory of conformal invariance are applied to the 1 1 do 2 or x y planar model the critical point is found to be xc approximately 2 0 the value sigma 0 501 or 0 005 is obtained for the index governing the critical divergence of the correlation length and the conformal anomaly is confirmed to be c 1 the kosterlitz thouless features of a low temperature phase with zero spontaneous magnetisation but algebraic decay of correlations together with a standard high temperature phase are observed the critical exponent eta is determined as a function of coupling constant and compared with theoretical expectations