new design and stability analysis of fuzzy proportional derivative control systems this paper describes the design principle tracking performance and stability analysis of a fuzzy proportional derivative pd controller first the fuzzy pd controller is derived from the conventional continuous time linear pd controller then the fuzzification control rule base and defuzzification in the design of the fuzzy pd controller are discussed in detail the resulting controller is a discrete time fuzzy version of the conventional pd controller which has the same linear structure in the proportional and the derivative parts but has nonconstant gains both the proportional and derivative gains are nonlinear functions of the input signals the new fuzzy pd controller thus preserves the simple linear structure of the conventional pd controller yet enhances its self tuning control capability computer simulation results have demonstrated this advantage of the fuzzy pd controller particularly when the process to be controlled is nonlinear after a detailed stability analysis where a simple and realistic sufficient condition for the bounded input bounded output stability of the overall feedback control system was derived several computer simulation results are compared with the conventional pd controller although the conventional and fuzzy pd controllers are not exactly comparable the authors compare them in order to have a sense of how well the fuzzy pd controller performs for this reason in the simulations several first order and second order linear systems with or without time delays are first used to test the performance of the fuzzy pd controller for step reference inputs the fuzzy pd control systems show remarkable performance as well as if not better than the conventional pd control systems moreover the fuzzy pd controller is compared to the conventional pd controller for a particular second order linear system showing the advantage of the fuzzy pd controller over the conventional one in the sense that in order to obtain the same control performance the conventional pd controller has to employ an extremely large gain while the fuzzy controller uses a reasonably small gain finally in the case of nonlinear systems the authors provide some examples to show that the fuzzy pd controller can track the set points satisfactorily but the conventional pd controller cannot

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

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

on the hypergeometric matrix function abstract this paper deals with the study of the hypergeometric function with matrix arguments f a b c z conditions for matrices a b c so that the series representation of the hypergeometric function be convergent for z 1 and satisfies a matrix differential equation are given after the study of beta and gamma matrix functions an integral representation of f a b c z is obtained for the case where b c and c b are positive stable matrices with bc cb

typical takagi sugeno pi and pd fuzzy controllers analytical structures and stability analysis in the first part of this paper we investigate explicit structure of the typical takagi sugeno ts fuzzy pi and pd controllers by typical we mean the use of at least three trapezoidal or triangular input fuzzy sets for each input variable fuzzy rules with linear consequent zadeh fuzzy logic and operator and the centroid defuzzifier this configuration covers many ts fuzzy controllers in the literature we mathematically prove these fuzzy controllers to be nonlinear pi or pd controllers with proportional gain and integral gain or derivative gain changing with output of the controlled system we reveal these fuzzy controllers to be inherently nonlinear gain scheduling pi or pd controllers with different variable gains in different regions of input space the gains constantly change even in the same region and switch continuously and smoothly between adjacent regions the explicit structural expressions of the fuzzy controllers are derived and their characteristics including the bounds and geometrical shape of the gain variation are studied in the second part of the paper we apply the small gain theorem to the derived structures of the fuzzy controllers and establish a sufficient global bibo stability criterion for nonlinear systems controlled by the fuzzy controllers local stability is also examined

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

general analytical structure of typical fuzzy controllers and their limiting structure theorems abstract the fuzzy controllers studied in this paper have r input variables error rate change of error of process output etc n members of input fuzzy sets are employed to fuzzify the inputs nr nonlinear control rules represented by an arbitrary function f are used which are evaluated by any type of fuzzy logic in conjunction with any inference method the center of gravity algorithm is utilized for defuzzification the author proves that the general analytical structure of the fuzzy controllers is the sum of a global f dependent nonlinear controller and a local nonlinear controller which locally adjusts control action of the global controller as n the global nonlinear controller approaches a global f dependent nonlinear controller while the local nonlinear controller disappears a necessary and sufficient condition is obtained for judging convergence of limiting structure of the fuzzy controllers if linear control rules are used the global controller is a global r dimensional multilevel relay which approaches a global linear controller as n two illustrative examples which deal with linear product maximum and minimum control rules are given

a nonlinear fuzzy controller with linear control rules is the sum of a global two dimensional multilevel relay and a local nonlinear proportional integral controller abstract the author analytically proves that a nonlinear fuzzy controller with linear control rules and n members for input fuzzy sets is the sum of a global two dimensional multilevel relay and a local nonlinear proportional integral pi controller which adjusts the control action generated by the global multilevel relay as n increases the resolution of the global multilevel relay is enhanced but the role of the local nonlinear pi controller in total control action is decreased as n approaches the global multilevel relay approaches a regular linear pi controller while the control action from the local nonlinear pi controller approaches zero the role of the global multilevel relay and the local nonlinear pi controller in total control action is quantitatively described as is the degree of nonlinearity of the fuzzy controllers with different n

trace formulas and borg type theorems for matrix valued jacobi and dirac finite difference operators abstract borg type uniqueness theorems for matrix valued jacobi operators h and supersymmetric dirac difference operators d are proved more precisely assuming reflectionless matrix coefficients a b in the self adjoint jacobi operator h as a s b with s the right left shift operators on the lattice z and the spectrum of h to be a compact interval e e e e we prove that a and b are certain multiples of the identity matrix an analogous result which however displays a certain novel nonuniqueness feature is proved for supersymmetric self adjoint dirac difference operators d with spectrum given by e 1 2 e 1 2 e 1 2 e 1 2 0 e e our approach is based on trace formulas and matrix valued exponential herglotz representation theorems as a by product of our techniques we obtain the extension of flaschka s borg type result for periodic scalar jacobi operators to the class of reflectionless matrix valued jacobi operators

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

stability analysis and design of fuzzy control systems abstract the stability analysis and the design technique of fuzzy control systems using fuzzy block diagrams are discussed first we show the concept of fuzzy blocks and consider the connection problems of fuzzy blocks diagrams we derive some theorems and corollaries with respect to two basic types of connections of fuzzy blocks in order to preserve some properties in a connection of fuzzy blocks continuous piecewise polynomial membership functions are defined secondly a sufficient condition which guarantees the stability of fuzzy systems is obtained in terms of lyapunov s direct method we give an important fact based on this condition thirdly we propose a new design technique of a fuzzy controller the fuzzy block diagrams and the stability analysis are applied to the design problems of a model based fuzzy controller

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

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

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

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

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

towards a paradigm for fuzzy logic control the purpose of this paper is to lay a foundation for fuzzy logic control flc design constructing a class of fuzzy logic controllers that is suitable for a broad range of controls applications some scalar definitions in flc are extended to the n dimensional case including vector fuzzy number and membership vector a rigorous mathematical expression is given for the function g x called the reasoning surface manufactured by a fuzzy associative memory fam for the existence and uniqueness of solutions in any closed loop system with a fuzzy logic controller it is shown that the fam function g x must be lipschitz this places restrictions on the allowed rules in the flc it is argued that for effective control design the membership functions should not be uniformly spaced the triangular membership functions generally suffice and product inferencing should be used under these conditions the flc manufactures a piecewise multilinear lipschitz fam function g x that is a generalized proportional plus derivative plus offset controller the scalar and two dimensional cases are studied in detail

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

matrix differential equations and scalar polynomials satisfying higher order recursions abstract we show that any scalar differential operator with a family of polynomials as its common eigenfunctions leads canonically to a matrix differential operator with the same property the construction of the corresponding family of matrix valued polynomials has been studied in a duran a generalization of favard s theorem for polynomials satisfying a recurrence relation j approx theory 74 1993 83 109 a duran on orthogonal polynomials with respect to a positive definite matrix of measures canad j math 47 1995 88 112 a duran w van assche orthogonal matrix polynomials and higher order recurrence relations linear algebra appl 219 1995 261 280 but the existence of a differential operator having them as common eigenfunctions had not been considered this correspondence goes only one way and most matrix valued situations do not arise in this fashion we illustrate this general construction with a few examples in the case of some families of scalar valued polynomials introduced in f a grunbaum l haine bispectral darboux transformations an extension of the krall polynomials int math res not 8 1997 359 392 we take a first look at the algebra of all matrix differential operators that share these common eigenfunctions and uncover a number of phenomena that are new to the matrix valued case

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

fi modules and stability for representations of symmetric groups in this paper we introduce and develop the theory of fi modules we apply this theory to obtain new theorems about the cohomology of the configuration space of n distinct ordered points on an arbitrary connected oriented manifold the diagonal coinvariant algebra on r sets of n variables the cohomology and tautological ring of the moduli space of n pointed curves the space of polynomials on rank varieties of n n matrices the subalgebra of the cohomology of the genus n torelli group generated by h1 and more the symmetric group sn acts on each of these vector spaces in most cases almost nothing is known about the characters of these representations or even their dimensions we prove that in each fixed degree the character is given for n large enough by a polynomial in the cycle counting functions that is independent of n in particular the dimension is eventually a polynomial in n in this framework representation stability in the sense of church farb for a sequence of sn representations is converted to a finite generation property for a single fi module

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

schur nevanlinna potapov sequences of rational matrix functions we study particular sequences of rational matrix functions with poles outside the unit circle these schur nevanlinna potapov sequences are recursively constructed based on some complex numbers with norm less than one and some strictly contractive matrices the main theme of this paper is a thorough analysis of the matrix functions belonging to the sequences in question essentially such sequences are closely related to the theory of orthogonal rational matrix functions on the unit circle as a further crosslink we explain that the functions belonging to schur nevanlinna potapov sequences can be used to describe the solution set of an interpolation problem of nevanlinna pick type for matricial schur functions

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

the simplest fuzzy controllers using different inference methods are different nonlinear proportional integral controllers with variable gains abstract the author analytically proves that the simplest fuzzy controllers using different inference methods are different nonlinear proportional integral pi controllers with proportional gains and integral gains changing with inputs of the controllers the inference methods involved are mamdani s minimum inference method larsen s product inference method the drastic product inference method and the bounded product inference method configuration of the fuzzy controllers is minimal which includes two input fuzzy sets three output fuzzy sets four control rules zadeh fuzzy logic and lukasiewicz fuzzy logic or and a center of gravity defuzzification algorithm after analytically investigating properties of the nonlinear pi controllers the author reveals that the bounded product inference method is inappropriate for the control purpose while the other three inference methods are appropriate dynamic and static control behaviors of the fuzzy controllers with the appropriate inference methods are analytically compared with each other and are also compared with those of the linear pi controller finally it is analytically proven that the fuzzy control systems have the same local stability at the equilibrium point as the corresponding linear pi control system does

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

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

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

differential coefficients of orthogonal matrix polynomials we find explicit formulas for raising and lowering first order differential operators for orthogonal matrix polynomials we derive recurrence relations for the coefficients in the raising and lowering operators some examples are given

structural analysis of fuzzy controllers with nonlinear input fuzzy sets in relation to nonlinear pid control with variable gains the popular linear pid controller is mostly effective for linear or nearly linear control problems nonlinear pid controllers however are needed in order to satisfactorily control highly nonlinear plants time varying plants or plants with significant time delay this paper extends our previous papers in which we show rigorously that some fuzzy controllers are actually nonlinear pi pd and pid controllers with variable gains that can outperform their linear counterparts in the present paper we study the analytical structure of an important class of two and three dimensional fuzzy controllers we link the entire class as opposed to one controller at a time to nonlinear pi pd and pid controllers with variable gains by establishing the conditions for the former to structurally become the latter unlike the results in the literature which are exclusively for the fuzzy controllers using linear fuzzy sets for the input variables this class of fuzzy controllers employs nonlinear input fuzzy sets of arbitrary types our structural results are thus more general and contain the existing ones as special cases two concrete examples are provided to illustrate the usefulness of the new results

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

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

pid type fuzzy controller and parameters adaptive method abstract the authors of this paper try to analyze the dynamic behavior of the product sum crisp type fuzzy controller revealing that this type of fuzzy controller behaves approximately like a pd controller that may yield steady state error for the control system by relating to the conventional pid control theory we propose a new fuzzy controller structure namely pid type fuzzy controller which retains the characteristics similar to the conventional pid controller in order to improve further the performance of the fuzzy controller we work out a method to tune the parameters of the pid type fuzzy controller on line producing a parameter adaptive fuzzy controller simulation experiments are made to demonstrate the fine performance of these novel fuzzy controller structures

cyclic tridiagonal pairs higher order onsager algebras and orthogonal polynomials abstract the concept of cyclic tridiagonal pairs is introduced and explicit examples are given for a fairly general class of cyclic tridiagonal pairs with cyclicity n we associate a pair of divided polynomials the properties of this pair generalize the ones of tridiagonal pairs of racah type the algebra generated by the pair of divided polynomials is identified as a higher order generalization of the onsager algebra it can be viewed as a subalgebra of the q onsager algebra for a proper specialization at q the primitive 2 n th root of unity orthogonal polynomials beyond the leonard duality are revisited in light of this framework in particular certain second order dunkl shift operators provide a realization of the divided polynomials at n 2 or q i

on weyl titchmarsh theory for singular finite difference hamiltonian systems we develop the basic theory of matrix valued weyl titchmarsh m functions and the associated green s matrices for whole line and half line self adjoint hamiltonian finite difference systems with separated boundary conditions

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

on a multivariable extension of jacobi matrix polynomials the classical jacobi matrix polynomials only for commutative matrices were first studied by defez et al e defez l jodar a law jacobi matrix differential equation polynomial solutions and their properties comput math appl 48 2004 789 803 the main aim of this paper is to construct a multivariable extension with the help of the classical jacobi matrix polynomials jmps generating matrix functions and recurrence relations satisfied by these multivariable matrix polynomials are derived furthermore general families of multilinear and multilateral generating matrix functions are obtained and their applications are presented

central stability homology we give a new categorical way to construct the central stability homology of putman and sam and explain how it can be used in the context of representation stability and homological stability in contrast to them we cover categories with infinite automorphism groups we also connect central stability homology to randal williams and wahl s work on homological stability we also develop a criterion that implies that functors that are polynomial in the sense of randal williams and wahl are centrally stable in the sense of putman

a class of orthogonal polynomials on the radial rays in the complex plane abstract we introduce a class of polynomials orthogonal on some radial rays in the complex plane and investigate their existence and uniqueness a recurrence relation for these polynomials a representation and the connection with standard polynomials orthogonal on 0 1 are derived it is shown that their zeros are simple and distributed symmetrically on the radial rays with the possible exception of a multiple zero at the origin an analogue of the jacobi polynomials and the corresponding problem with the generalized laguerre polynomials are also treated

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

a general technique for deriving analytical structure of fuzzy controllers using arbitrary trapezoidal input fuzzy sets and zadeh and operator deriving the analytical structure of fuzzy controllers is very important as it creates a solid foundation for better understanding insightful analysis and more effective design of fuzzy control systems we previously developed a technique for deriving the analytical structure of the fuzzy controllers that use zadeh fuzzy and operator and the symmetric identical trapezoidal or triangular input fuzzy sets many fuzzy controllers use arbitrary trapezoidal triangular input fuzzy sets that are asymmetric at present there exists no technique capable of deriving the analytical structure of these fuzzy controllers extending our original technique we now present a novel method that can accomplish rigorously the structure derivation for any fuzzy controller mamdani type or ts type that employs the arbitrary trapezoidal input fuzzy sets and zadeh fuzzy and operator the new technique contains our original technique as a special case given the importance of pid control we focus on mamdani fuzzy pi and pd controllers in this paper and show in detail how to use the new technique for different configurations of the fuzzy pi pd controllers the controllers use two arbitrary trapezoidal fuzzy sets for each input variable four arbitrary singleton output fuzzy sets four fuzzy rules zadeh fuzzy and operator and the centroid defuzzifier this configuration is more general and complicated than the mamdani fuzzy pi pd controllers in the current literature it actually contains them as special cases we call this configuration the generalized fuzzy pi pd controller

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

pi fuzzy controllers for integral plants to ensure robust stability new pi fuzzy controllers comprising takagi sugeno and mamdani fuzzy systems to control a class of integral plants specific to servo systems are proposed in this paper linear pi controllers are designed in the first phase they are tuned using the extended symmetrical optimum method to ensure the desired overshoot and settling time with respect to step setpoint modifications and three types of load disturbance inputs this type of pi controller design guarantees robust stability of the closed loop system in response to parametric variations in a controlled plant following this based on the results from the linear case and the modal equivalence principle an original development method for pi fuzzy controllers is proposed some experimental results are included to illustrate the effectiveness of the design process

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

spectral decomposition and matrix valued orthogonal polynomials the relation between the spectral decomposition of a self adjoint operator which is realizable as a higher order recurrence operator and matrix valued orthogonal polynomials is investigated a general construction of such operators from scalar valued orthogonal polynomials is presented two examples of matrix valued orthogonal poly nomials with explicit orthogonality relations and three term recurrence relation are pre sented which both can be considered as 2 2 matrix valued analogues of subfamilies of askey wilson polynomials

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

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

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

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

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

stability in the homology of congruence subgroups the homology groups of many natural sequences of groups g n n 1 infty e g general linear groups mapping class groups etc stabilize as n rightarrow infty indeed there is a well known machine for proving such results that goes back to early work of quillen church and farb discovered that many sequences of groups whose homology groups do not stabilize in the classical sense actually stabilize in some sense as representations they called this phenomena representation stability we prove that the homology groups of congruence subgroups of mathrm gl n r for almost any reasonable ring r satisfy a strong version of representation stability that we call central stability the definition of central stability is very different from church farb s definition of representation stability it is defined via a universal property but we prove that it implies representation stability our main tool is a new machine for proving central stability that is analogous to the classical homological stability machine

studies on the output of fuzzy controller with multiple inputs abstract in this paper we discuss the output of a fuzzy controller in which an arbitrary number of fuzzy control inputs and an arbitrary number of fuzzy sets employed to fuzzify every linguistic variable are considered in the fuzzy control rules a rule base with a rule mapping concept is considered a defuzzifier with the general t norm and t conorm operators is formulated a triangular family is adopted so that the defuzzifier of the complete rule mapping is represented by a formula consisting of only some rules using a specific logic pair to evaluate values we derive that the output of the fuzzy controller can be simply represented by a linear parametric function of the inputs to the fuzzy controller

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

on some n 1 symmetric linear functionals abstract many authors have dealt with problems related to the symmetrization of sequences of orthogonal polynomials on a real line or on a unit circle particular aspects are treated for quadratic or cubic decompositions in this paper we present a technique that unifies the treatment of these topics for an arbitrary positive order of decomposition n by considering a more general definition of orthogonality this technique is based on the decomposition with respect to the cyclic group of order n the main theorem is applied to the orthogonality on n 1 symmetric curves some particular cases are singled out these results may be useful in studying hermite pade approximations vector continued fractions and dynamical systems

theorectical analysis of a fuzzy logic controller with unequally spaced triangular membership functions abstract the mathematical analysis concerning the explicit input output relation of the fuzzy logic controller flc is addressed in this article under the assumption of simple rule mapping and the use of arbitrarily located triangular fuzzy partitions as the membership functions the inferred output of the flc can be decomposed into two terms the global multilevel relay and the local nonlinear compensator the ultimate control behavior of this flc and the equivalence to nonlinear proportional integral pi controller as the number of linguistic terms are approaching to infinity are also investigated the local stability criteria for pi type flcs are derived and the global stability properties for pd type flcs are established this analysis is a generalization to previous related research works where the flc is characterized by equally spaced triangular fuzzy partitions on the universal sets

on a weyl titchmarsh theory for discrete symplectic systems on a half line recently bohner and sun introduced basic elements of a weyl titchmarsh theory into the study of discrete symplectic systems we extend this development through the introduction of weyl titchmarsh functions together with a preliminary study of their properties a limit point criterion is described and characterized green s function for the half line is introduced as a limit of such functions in the regular case and half line solutions obtained are seen to satisfy lambda dependent boundary conditions at infinity

representations of the schrodinger group and matrix orthogonal polynomials the representations of the schrodinger group in one space dimension are explicitly constructed in the basis of the harmonic oscillator states these representations are seen to involve matrix orthogonal polynomials in a discrete variable that have charlier and meixner polynomials as building blocks the underlying lie theoretic framework allows for a systematic derivation of the structural formulas recurrence relations difference equations rodrigues formula etc that these matrix orthogonal polynomials satisfy

homological stability for automorphism groups abstract given a family of groups admitting a braided monoidal structure satisfying mild assumptions we construct a family of spaces on which the groups act and whose connectivity yields via a classical argument of quillen homological stability for the family of groups we show that stability also holds with both polynomial and abelian twisted coefficients with no further assumptions this new construction of a family of spaces from a family of groups recovers known spaces in the classical examples of stable families of groups such as the symmetric groups general linear groups and mapping class groups by making systematic the proofs of classical stability results we show that they all hold with the same type of coefficient systems obtaining in particular without any further work new stability theorems with twisted coefficients for the symmetric groups braid groups automorphisms of free groups unitary groups mapping class groups of non orientable surfaces and mapping class groups of 3 manifolds our construction can also be applied to families of groups not considered before in the context of homological stability as a byproduct of our work we construct the braided analogue of the category fi of finite sets and injections relevant to the present context and define polynomiality for functors in the context of pre braided monoidal categories

open quantum random walks on the half line the karlin mcgregor formula path counting and foster s theorem in this work we consider open quantum random walks on the non negative integers by considering orthogonal matrix polynomials we are able to describe transition probability expressions for classes of walks via a matrix version of the karlin mcgregor formula we focus on absorbing boundary conditions and for simpler classes of examples we consider path counting and the corresponding combinatorial tools a non commutative version of the gambler s ruin is studied by obtaining the probability of reaching a certain fortune and the mean time to reach a fortune or ruin in terms of generating functions in the case of the hadamard coin a counting technique for boundary restricted paths in a lattice is also presented we discuss an open quantum version of foster s theorem for the expected return time together with applications

on polynomials orthogonal with respect to an inner product involving differences abstract in this paper we consider the inner product r x g x d x c g c where c r and is a distribution function with infinite spectrum such that has no points of increase in the interval c c 1 furthermore 0 and g are functions on r and c c 1 c let qn x be the sequence of monic orthogonal polynomials with respect to this inner product and pn x pnc x the sequences of monic standard orthogonal polynomials 0 with respect to d x and x c x c 1 d x respectively we derive an explicit representation for qn x in terms of pn x and pnc x and we present some results on the distribution of the zeros of qn x in relation to the zeros of pn x finally we treat the special case where pn x are charlier polynomials and c 0

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 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

exceptional laguerre and jacobi polynomials and the corresponding potentials through darboux crum transformations a simple derivation is presented of the four families of infinitely many shapeinvariant hamiltonians corresponding to the exceptional laguerre and jacobi polynomials the darboux crum transformations are applied to connect the well known shape invariant hamiltonians of the radial oscillator and the darboux p oschl teller potential to the shape invariant potentials of odake sasaki dutta and roy derived the two lowest members of the exceptional laguerre polynomials by this method the method is expanded to its full generality and many other ramifications including the aspects of the generalized bochner problem and the bispectral property of the exceptional orthogonal polynomials are discussed

an approach to fuzzy control of nonlinear systems stability and design issues presents a design methodology for stabilization of a class of nonlinear systems first the authors represent a nonlinear plant with a takagi sugeno fuzzy model then a model based fuzzy controller design utilizing the concept of the so called parallel distributed compensation is employed the main idea of the controller design is to derive each control rule so as to compensate each rule of a fuzzy system the design procedure is conceptually simple and natural moreover the stability analysis and control design problems can be reduced to linear matrix inequality lmi problems therefore they can be solved efficiently in practice by convex programming techniques for lmis the design methodology is illustrated by application to the problem of balancing and swing up of an inverted pendulum on a cart

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

matrix valued spherical functions associated to the complex projective plane abstract the main purpose of this paper is to compute all irreducible spherical functions on g su 3 of arbitrary type k where k s u 2 u 1 u 2 this is accomplished by associating to a spherical function on g a matrix valued function h on the complex projective plane p2 c g k it is well known that there is a fruitful connection between the hypergeometric function of euler and gauss and the spherical functions of trivial type associated to a rank one symmetric pair g k but the relation of spherical functions of types of dimension bigger than one with classical analysis has not been worked out even in the case of an example of a rank one pair the entries of h can be described by solutions of two systems of ordinary differential equations there is no ready made approach to such a pair of systems or even to a single system of this kind in our case the situation is very favorable and the solution to this pair of systems can be exhibited explicitly in terms of a special class of generalized hypergeometric functions p 1fp

orthogonal polynomials on sobolev spaces old and new directions abstract during the last years orthogonal polynomials on sobolev spaces have attracted considerable attention algebraic properties distribution of their zeros and fourier expansions as well as their relevance in the analysis of spectral methods for partial differential equations provide a very large field to explore and to compare with the standard case in this paper we present an introductory survey about the subject the origin of the problems and their development show the interest and the promising future of this field

orthogonal polynomials and analytic functions associated to positive definite matrices abstract for a positive definite infinite matrix a we study the relationship between its associated sequence of orthonormal polynomials and the asymptotic behaviour of the smallest eigenvalue of its truncation a n of size n n for the particular case of a being a hankel or a hankel block matrix our results lead to a characterization of positive measures with finite index of determinacy and of completely indeterminate matrix moment problems respectively

a method for deriving the analytical structure of a broad class of typical interval type 2 mamdani fuzzy controllers type 2 t2 fuzzy controllers are emerging as the related t2 fuzzy logic and algorithms have recently been advancing rapidly at present a t2 fuzzy controller is viewed and used as a black box function generator that produces a desired nonlinear mapping between the input and output of the controller we call the mapping analytical structure the mathematical expression of the analytical structure however is not explicitly known to the controller designer this is in sharp contrast with the analytical structure of a conventional controller which is not only always explicitly known but serves as a starting point for system analysis and design obviously the knowledge of a t2 fuzzy controller s analytical structure can have significant benefits they include 1 understanding more precisely how the controller works in the same sense as we understand how a conventional controller e g the pid controller functions 2 making t2 fuzzy control more acceptable to safety critical fields such as biomedicine 3 taking advantage of the well developed nonlinear control theory to develop better analysis and design methods for t2 control systems e g less conservative system stability criteria and 4 permitting rigorous comparative exploration on differences between the t2 and type 1 t1 fuzzy controllers and their relative merits and pitfalls e g performance and structural complexity in this paper we develop an innovative technique which is capable of deriving the analytical structure for a wide class of interval t2 mamdani fuzzy controllers the configuration of the controllers is typical and is substantially more general than the related efforts in the literature it uses any number and types of interval t2 input fuzzy sets any number and types of general or interval t2 output fuzzy sets arbitrary fuzzy rules zadeh and operator karnik mendel center of sets type reducer and the centroid defuzzifier we show in detail how the derivation method works in a general setting and provide the analytical structure of an example t2 controller as well in addition we utilize the method to prove that a subset of the t2 fuzzy controllers are the sum of two nonlinear pi or pd controllers each of which has a variable proportional gain and a variable integral gain or derivative gain plus a variable offset if and only if the input fuzzy sets are piecewise linear e g triangular and or trapezoidal the sum of the two nonlinear pi or pd controllers is a new discovery relative to the current literature

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

deriving analytical input output relationship for fuzzy controllers using arbitrary input fuzzy sets and zadeh fuzzy and operator a fuzzy controller uses either zadeh or product fuzzy and operator with the former being more frequently used than the latter we have recently published a novel technique for deriving analytical input output relation for the fuzzy controllers that use zadeh and operator and arbitrary trapezoidal input fuzzy sets including triangular ones as special cases in this paper we have developed a general technique based on that technique to cover arbitrary types of input fuzzy sets moreover we have established some necessary and sufficient conditions to characterize general relationship between shape of input fuzzy sets and shape of input space divisions an important and integral issue because analytical relationship differs in different regions of input space the new technique and the shape relations are applicable to any type of fuzzy controllers e g mamdani type or takagi sugeno type the analytical structures that we have derived provide an unprecedented opportunity to insightfully and rigorously examine the advantages and shortcomings of different design choices available for various components of the fuzzy controllers we have focused on type selection for input fuzzy sets of mamdani fuzzy controllers our preliminary analysis indicates that the fuzzy controllers using trapezoidal fuzzy sets may be understood and possibly analyzed and designed more sensibly and easily in the context of conventional control theory than the fuzzy controllers using any other types of fuzzy sets our proposition is that trapezoidal fuzzy sets should be the first choice and used most of time possible implication for automatic learning of input fuzzy sets via neural networks or genetic algorithms is briefly discussed

a systematic study of fuzzy pid controllers function based evaluation approach a function based evaluation approach is proposed for a systematic study of fuzzy proportional integral derivative pid like controllers this approach is applied for deriving process independent design guidelines from addressing two issues simplicity and nonlinearity to examine the simplicity of fuzzy pid controllers we conclude that direct action controllers exhibit simpler design properties than gain scheduling controllers then we evaluate the inference structures of direct action controllers in five criteria control action composition input coupling gain dependency gain role change and rule parameter growth three types of fuzzy pid controllers using one two and three input inference structures are analyzed the results according to the criteria demonstrate some shortcomings in mamdani s two input controllers for keeping the simplicity feature like a linear pid controller a one input fuzzy pid controller with one to three mapping inference engine is recommended we discuss three evaluation approaches in a nonlinear approximation study function estimation based generalization capability based and nonlinearity variation based approximations the study focuses on the last approach a nonlinearity evaluation is then performed for several one input fuzzy pid controllers based on two measures nonlinearity variation index and linearity approximation index using these quantitative indices one can make a reasonable selection of fuzzy reasoning mechanisms and membership functions without requiring any process information from the study we observed that the zadeh mamdani s max min gravity scheme produces the highest score in terms of nonlinearity variations which is superior to other schemes such as mizumoto s product sum gravity and takagi sugeno kang schemes

theoretical analysis of crisp type fuzzy logic controllers using various t norm sum gravity inference methods the input output parametric relationship of a class of crisp type fuzzy logic controllers flcs using various t norm sum gravity inference methods is studied four most important t norms are used to calculate the matching level of each control rule and the explicit mathematical forms of reasoning surfaces obtained by using these four t norms are addressed reasoning surfaces of these crisp type flcs are proved to be composed of a two dimensional multilevel relay no matter which t norm is used and a local position dependent nonlinear compensator with output pattern influenced by the t norms is selected by analyzing the intrinsic operation of the four t norms the authors find that both standard intersection and algebraic product are suitable operators to perform the inference of the flc however bounded difference and drastic intersection are disqualified because they cannot satisfy some important criteria a measure of relative degree of nonlinearity is defined to examine the output figures of these crisp type flcs the ultimate behavior of these crisp type flcs as the number of linguistic terms approaches infinity is also explored the local stability criteria for the proportional integral pi type fuzzy control systems and the natural global stability characteristic for the proportional derivative pd type fuzzy control systems are also examined

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

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

representation theory and homological stability abstract we introduce the idea of representation stability and several variations for a sequence of representations v n of groups g n a central application of the new viewpoint we introduce here is the importation of representation theory into the study of homological stability this makes it possible to extend classical theorems of homological stability to a much broader variety of examples representation stability also provides a framework in which to find and to predict patterns from classical representation theory littlewood richardson and murnaghan rules stability of schur functors to cohomology of groups pure braid torelli and congruence groups to lie algebras and their homology to the equivariant cohomology of flag and schubert varieties to combinatorics the n 1 n 1 conjecture the majority of this paper is devoted to exposing this phenomenon through examples in doing this we obtain applications theorems and conjectures beyond the discovery of new phenomena the viewpoint of representation stability can be useful in solving problems outside the theory in addition to the applications given in this paper it is applied by church ellenberg farb in preparation 20 to counting problems in number theory and finite group theory representation stability is also used by church 2012 19 to give broad generalizations and new proofs of classical homological stability theorems for configuration spaces on oriented manifolds

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