zappa szep products of semigroups and their c algebras abstract zappa szep products of semigroups provide a rich class of examples of semigroups that include the self similar group actions of nekrashevych we use li s construction of semigroup c algebras to associate a c algebra to zappa szep products and give an explicit presentation of the algebra we then define a quotient c algebra that generalises the cuntz pimsner algebras for self similar actions we indicate how known examples previously viewed as distinct classes fit into our unifying framework we specifically discuss the baumslag solitar groups the binary adding machine the semigroup n n and the a x b semigroup z z

tensor algebras overc correspondences representations dilations andc envelopes tensor algebras overc correspondences are noncommutative generalizations of the disk algebra they contain as special cases analytic crossed products semi crossed products and popescu s noncommutative disk algebras in this paper a dilation theorem and a commutant lifting theorem for representations of tensor algebras is proved these are used to show that for certainc correspondences thec envelopes in the sense of arveson of the tensor algebras are the cuntz pimsner algebras of the correspondences

on the computation of sets of points with low lebesgue constant on the unit disk abstract in this paper of numerical nature we test the lebesgue constant of several available point sets on the disk and propose new ones that enjoy low lebesgue constant furthermore we extend some results in cuyt 2012 analyzing the case of bos arrays whose radii are nonnegative gauss gegenbauer lobatto nodes with exponent noticing that the optimal still allows to achieve point sets on with low lebesgue constant n for degrees n 30 next we introduce an algorithm that through optimization determines point sets with the best known lebesgue constants for n 25 finally we determine theoretically a point set with the best lebesgue constant for the case n 1

kms states on c algebras associated to a family of commuting local homeomorphisms abstract we consider a family of commuting local homeomorphisms on a compact space and build a compactly aligned product system of hilbert bimodules the nica toeplitz algebra of this system carries a gauge action of a higher dimensional torus and there are many possible dynamics obtained by composing with different embeddings of the real line in this torus we study the kms states of these dynamics for large inverse temperatures including we describe the simplex of kms states on the nica toeplitz algebra we illustrate our main theorem by considering backward shifts on the infinite path spaces of a class of k graphs whose shift maps commute

scrambled net variance for integrals of smooth functions hybrids of equidistribution and monte carlo methods of integration can achieve the superior accuracy of the former while allowing the simple error estimation methods of the latter one version randomized t m s nets has the property that the integral estimates are unbiased and that the variance is o 1 n for any square integrable integrand stronger assumptions on the integrand allow one to find rates of convergence this paper shows that for smooth integrands over s dimensions the variance is of order n 3 log n 3 1 compared to n 1 for ordinary monte carlo thus the integration errors are of order n 3 2 log n s 1 2 in probability this compares favorably with the rate n 1 log n s 1 for unrandomized t m s nets

orthogonal polynomial eigenfunctions of second order partial differerential equations in this paper we show that for several second order partial differential equations l u a x y uxx 2b x y uxy c x y uyy d x y ux e x y uy atu which have orthogonal polynomial eigenfunctions these polynomials can be expressed as a product of two classical orthogonal polynomials in one variable this is important since otherwise it is very difficult to explicitly find formulas for these polynomial solutions from this observation and characterization we are able to produce additional examples of such orthogonal polynomials together with their orthogonality that widens the class found by h l krall and sheffer in their seminal work in 1967 moreover from our approach we can answer some open questions raised by krall and sheffer

on bivariate classical orthogonal polynomials abstract we deduce new characterizations of bivariate classical orthogonal polynomials associated with a quasi definite moment functional and we revise old properties for these polynomials more precisely new characterizations of classical bivariate orthogonal polynomials satisfying a diagonal pearson type equation are proved they are solutions of two separate partial differential equations one for every partial derivative their partial derivatives are again orthogonal and every vector polynomial can be expressed in terms of its partial derivatives by means of a linear relation involving only three terms of consecutive total degree moreover we study general solutions of the matrix second order partial differential equation satisfied by classical orthogonal polynomials and we deduce the explicit expressions for the matrix coefficients of the structure relation finally some illustrative examples are given

nica toeplitz algebras associated with product systems over right lcm semigroups abstract we prove uniqueness of representations of nica toeplitz algebras associated to product systems of c correspondences over right lcm semigroups by applying our previous abstract uniqueness results developed for c precategories our results provide an interpretation of conditions identified in work of fowler and fowler raeburn and apply also to their crossed product twisted by a product system in the new context of right lcm semigroups as well as to a new doplicher roberts type c algebra associated to the nica toeplitz algebra as a derived construction we develop nica toeplitz crossed products by actions with completely positive maps this provides a unified framework for nica toeplitz semigroup crossed products by endomorphisms and by transfer operators we illustrate these two classes of examples with semigroup c algebras of right and left semidirect products

discrete product systems and twisted crossed products by semigroups a product systemeover a semigrouppis a family of hilbert spaces es s p together with multiplicationses et est we vieweas a unitary valued cocycle onp and consider twisted crossed productsa e pinvolvingeand an action ofpby endomorphisms of ac algebraa whenpis quasi lattice ordered in the sense of nica we isolate a class of covariant representations ofe and consider a twisted crossed productbp e pwhich is universal for covariant representations ofewhenehas finite dimensional fibres and in general is slightly larger in particular whenp nand dim e1 our algebrabn e n is a new infinite analogue of the toeplitz cuntz algebras ton our main theorem is a characterisation of the faithful representations ofbp e p

product system models for twisted c algebras of topological higher rank graphs abstract we use product systems of c correspondences to introduce twisted c algebras of topological higher rank graphs we define the notion of a continuous t valued 2 cocycle on a topological higher rank graph and present examples of such cocycles on large classes of topological higher rank graphs to every proper source free topological higher rank graph and continuous t valued 2 cocycle c on we associate a product system x of c 0 0 correspondences built from finite paths in we define the twisted cuntz krieger algebra c c to be the cuntz pimsner algebra o x and we define the twisted toeplitz algebra t c c to be the nica toeplitz algebra nt x we also associate to and c a product system y of c 0 correspondences built from infinite paths we prove that there is an embedding of t c c into nt y and an isomorphism between c c and o y

partial differential equations having orthogonal polynomial solutions abstract we show that if a second order partial differential equation l u auxx 2buxy cuyy dux euy nu has orthogonal polynomial solutions then the differential operator l must be symmetrizable and can not be parabolic in any nonempty open subset of the plane we also find rodrigues type formula for orthogonal polynomial solutions of such differential equations

the c algebras of finitely aligned higher rank graphs we generalise the theory of cuntz krieger families and graph algebras to the class of finitely aligned k graphs this class contains in particular all row finite k graphs the cuntz krieger relations for non row finite k graphs look significantly different from the usual ones and this substantially complicates the analysis of the graph algebra we prove a gauge invariant uniqueness theorem and a cuntz krieger uniqueness theorem for the c algebras of finitely aligned k graphs

semigroup c algebras and amenability of semigroups we construct reduced and full semigroup c algebras for left cancellative semigroups our new construction covers particular cases already considered by a nica and also toeplitz algebras attached to rings of integers in number fields due to j cuntz moreover we show how left amenability of semigroups can be expressed in terms of these semigroup c algebras in analogy to the group case

when is the cuntz krieger algebra of a higher rank graph approximately finite dimensional evans d g and sims a when is the cuntz krieger algebra of a higher rank graph approximately finite dimensional preprint arxiv 1112 4549v1

product systems over right angled artin semigroups we build upon mac lane s definition of a tensor category to introduce the concept of a product system that takes values in a tensor groupoid g we show that the existing notions of product systems fit into our categorical framework as do the k graphs of kumjian and pask we then specialize to product systems over right angled artin semigroups these are semigroups that interpolate between free semigroups and free abelian semigroups for such a semigroup we characterize all product systems which take values in a given tensor groupoid g in particular we obtain necessary and sufficient conditions under which a collection of k 1 graphs form the coordinate graphs of a k graph

higher dimensional generalizations of the thompson groups abstract motivated by the goal of generalizing the cuntz krieger algebras and making heavy use of the pioneering work of robertson and steger kumjian and pask generalized the notion of a directed graph to what they termed a higher rank graph the c algebras constructed from such higher rank graphs have proved to be highly interesting higher rank graphs are in fact a class of cancellative categories and so the one vertex higher rank graphs are therefore a class of cancellative monoids in this paper we call them k monoids finite direct products of free monoids belong to this family but there are many examples not of this form it is well known that the classical thompson groups arise from the free monoids and it is easy to show that brin s higher dimensional thompson groups arise from finite direct products of free monoids in this paper we generalize these constructions and show how to construct a family of new groups with simple commutator subgroups from arbitrary aperiodic k monoids thereby subsuming all the above examples although we give simple direct constructions of our groups a key feature of our paper is that we also show that they arise as topological full groups of the usual etale groupoids associated with k monoids in this way our groups are naturally associated with c algebras the question of the mutual isomorphisms amongst this family of groups appears to be a delicate one and is not handled here

structure relations for the bivariate big q jacobi polynomials we give structure relations for the orthogonal polynomials in two variables defined by lewanowicz and wozny s lewanowicz p wozny j comput appl math 233 2010 1554 1561 p n k x y a b c d q p n k y a bcq 2 k 1 dq k q y kdq y q kp kx y c b d y q n 0 k 0 1 n where email protected 0 1 0

affine buildings tiling systems and higher rank cuntz krieger algebras to an r dimensional subshift of finite type satisfying certain special properties we associate a c algebra a this algebra is a higher rank version of a cuntz krieger algebra in particular it is simple purely infinite and nuclear we study an example if is a group acting freely on the vertices of an a2 building with finitely many orbits and if is the boundary of that building then c o is the algebra associated to a certain two dimensional subshift

nuclearity of semigroup c algebras and the connection to amenability abstract we study c algebras associated with subsemigroups of groups for a large class of such semigroups including positive cones in quasi lattice ordered groups and left ore semigroups we describe the corresponding semigroup c algebras as c algebras of inverse semigroups groupoid c algebras and full corners in associated group crossed products these descriptions allow us to characterize nuclearity of semigroup c algebras in terms of faithfulness of left regular representations and amenability of group actions moreover we also determine when boundary quotients of semigroup c algebras are uct kirchberg algebras this leads to a unified approach to cuntz algebras and ring c algebras

linear partial q difference equations on q linear lattices and their bivariate q orthogonal polynomial solutions orthogonal polynomial solutions of an admissible potentially self adjoint linear second order partial q difference equation of the hypergeometric type in two variables on q linear lattices are analyzed a q pearson s system for the orthogonality weight function as well as for the difference derivatives of the solutions are presented giving rise to a solution of the q difference equation under study in terms of a rodrigues type formula the monic orthogonal polynomial solutions are treated in detail giving explicit formulae for the matrices in the corresponding recurrence relations they satisfy lewanowicz and wozny s lewanowicz p wozny j comput appl math 233 2010 1554 1561 have recently introduced a non monic bivariate extension of big q jacobi polynomials together with a partial q difference equation of the hypergeometric type that governs them this equation is analyzed in the last section we provide two more orthogonal polynomial solutions namely a second non monic solution from the rodrigues representation and the monic solution both from the recurrence relation that govern them and also explicitly given in terms of generalized bivariate basic hypergeometric series limit relations as email protected for the partial q difference equation and for the all three q orthogonal polynomial solutions are also presented

semigroup crossed products and the toeplitz algebras of nonabelian groups abstract we consider the quasi lattice ordered groups g p recently introduced by nica we realise their universal toeplitz algebra as a crossed product b p p by a semigroup of endomorphisms and show that the toeplitz representation is faithful precisely when an amenability condition is satisfied we then show directly that many interesting free products are amenable the final result contains theorems of cuntz and dinh on the uniqueness of toeplitz cuntz algebras as special cases

the interplay between k graphs and the yang baxter equation in this paper we initiate the study of the interplay between k graphs and the yang baxter equation for this we provide two very different perspectives on one hand we show that the set of all set theoretic solutions of the yang baxter equation is a special class of single vertex k graphs as a consequence we construct an infinite family of large solutions of the yang baxter equation from an arbitrarily given one on the other hand we prove that all single vertex k graphs are yb semigroups of square free involutive solutions of the yang baxter equation other various connections are also investigated

high order divergence conforming constrained bases for triangular cells this paper presents an algebraic technique for generating arbitrary order divergence conforming bases for curvilinear triangular cells the bases are constructed by enforcing appropriate constraints on a linear combination of general functions and then extracting the desired bases using singular value decompositions koornwinder dubiner polynomials are chosen as the general function set basic constraints are presented to obtain divergence conforming bases and additional constraints are presented to further enforce the bases to be nedelec results from a variety of problems are given to show that the bases exhibit high order convergence and also produce systems that are relatively well conditioned compared to other basis sets

a nearly conservative high order lagrange galerkin method for the resolution of scalar convection dominated equations in non divergence free velocity fields abstract in this work we present a novel lagrange galerkin method for the resolution of the scalar pure convection and convection dominated diffusion equations in non divergence free velocity fields the scheme has been formulated so that i time integration is carried out by means of an arbitrary order backward differentiation formula ii finite element space discretizations of any order can be employed and iii it satisfies a discrete mass balance equation at each simulation instant and preserves mass in those cases in which no fluid enters nor leaves the control domain as long as the integrals involved are computed exactly these properties are achieved by posing a conservation law for a weighted mass from which the standard mass can be seen as a particular case that can be easily discretized in time and in space with any order of accuracy numerical experiments which include a three dimensional test show that the method is as accurate in time as the employed time marching formula as accurate in space as the finite element discretization and generally more accurate especially in terms of mass conservation errors and efficient than the classic non conservative lagrange galerkin method to the best of our knowledge this is the first conservative lagrange galerkin method for convection dominated equations that considers both non divergence free velocity fields and time and space discretizations of any order

pure infiniteness and ideal structure of c algebras associated to fell bundles abstract we investigate structural properties of the reduced cross sectional algebra c r b of a fell bundle b over a discrete group g conditions allowing one to determine the ideal structure of c r b are studied notions of aperiodicity paradoxicality and b infiniteness for the fell bundle b are introduced and investigated by themselves and in relation to the partial dynamical system dual to b several criteria of pure infiniteness of c r b are given it is shown that they generalize and unify corresponding results obtained in the context of crossed products by the following duos laca spielberg 34 jolissaint robertson 21 sierakowski rordam 47 giordano sierakowski 18 and ortega pardo 39 for exact separable fell bundles satisfying the residual intersection property primitive ideal space of c r b is determined the results of the paper are shown to be optimal when applied to graph c algebras applications to a class of exel larsen crossed products are presented

multivariate orthogonal polynomials and integrable systems multivariate orthogonal polynomials in d real dimensions are considered from the perspective of the cholesky factorization of a moment matrix the approach allows for the construction of corresponding multivariate orthogonal polynomials associated second kind functions jacobi type matrices and associated three term relations and also christoffel darboux formulae the multivariate orthogonal polynomials their second kind functions and the corresponding christoffel darboux kernels are shown to be quasi determinants as well as schur complements of bordered truncations of the moment matrix quasi tau functions are introduced it is proven that the second kind functions are multivariate cauchy transforms of the multivariate orthogonal polynomials discrete and continuous deformations of the measure lead to toda type integrable hierarchy being the corresponding flows described through lax and zakharov shabat equations bilinear equations are found varying size matrix nonlinear partial difference and differential equations of the 2d toda lattice type are shown to be solved by matrix coefficients of the multivariate orthogonal polynomials the discrete flows which are shown to be connected with a gauss borel factorization of the jacobi type matrices and its quasi determinants lead to expressions for the multivariate orthogonal polynomials and their second kind functions in terms of shifted quasi tau matrices which generalize to the multidimensional realm those that relate the baker and adjoint baker functions to ratios of miwa shifted tau functions in the 1d scenario in this context the multivariate extension of the elementary darboux transformation is given in terms of quasi determinants of matrices built up by the evaluation at a poised set of nodes lying in an appropriate hyperplane in r d of the multivariate orthogonal polynomials the multivariate christoffel formula for the iteration of m elementary darboux transformations is given as a quasi determinant it is shown using congruences in the space of semi infinite matrices that the discrete and continuous flows are intimately connected and determine nonlinear partial difference differential equations that involve only one site in the integrable lattice behaving as a kadomstev petviashvili type system finally a brief discussion of measures with a particular linear isometry invariance and some of its consequences for the corresponding multivariate polynomials is given in particular it is shown that the toda times that preserve the invariance condition lay in a secant variety of the veronese variety of the fixed point set of the linear isometry

two variable orthogonal polynomials of big q jacobi type a four parameter family of orthogonal polynomials in two discrete variables is defined for a weight function of basic hypergeometric type the polynomials which are expressed in terms of univariate big q jacobi polynomials form an extension of dunkl s bivariate little q jacobi polynomials c f dunkl orthogonal polynomials in two variables of q hahn and q jacobi type siam j algebr discrete methods 1 1980 137 151 we prove orthogonality property of the new polynomials and show that they satisfy a three term relation in a vector matrix notation as well as a second order partial q difference equation

bivariate orthogonal polynomials in the lyskova class classical orthogonal polynomials in two variables can be characterized as the polynomial solutions of a matrix second order partial differential equation involving matrix polynomial coefficients in this work we study classical orthogonal polynomials in two variables whose partial derivatives satisfy again a second order partial differential equation of the same type

monte carlo variance of scrambled net quadrature hybrids of equidistribution and monte carlo methods of integration can achieve the superior accuracy of the former while allowing the simple error estimation methods of the latter this paper studies the variance of one such hybrid scrambled nets by applying a multidimensional multiresolution wavelet analysis to the integrand the integrand is assumed to be measurable and square integrable but not necessarily of bounded variation in simple monte carlo every nonconstant term of the multiresolution contributes to the variance of the estimated integral for scrambled nets certain low dimensional and coarse terms do not contribute to the variance for any integrand in l2 the sampling variance tends to zero faster under scrambled net quadrature than under monte carlo sampling as the number of function evaluations n tends to infinity some finite n results bound the variance under scrambled net quadrature by a small constant multiple of the monte carlo variance uniformly over all integrands f latin hypercube sampling is a special case of scrambled net quadrature

orthogonal polynomials in two variables and second order partial differential equations abstract we study the second order partial differential equations l u au xx 22 bu xy cu yy du x eu y n u which have orthogonal polynomials in two variables as solutions by using formal functional calculus on moment functionals we first give new simpler proofs and improvements of the results by krall and sheffer and littlejohn we then give a two variable version of al salam and chihara s characterization of classical orthogonal polynomials in one variable we also study in detail the case when l belongs to the basic class that is a y c x 0 in particular we characterize all such differential equations which have a product of two classical orthogonal polynomials in one variable as solutions

the algebra of two dimensional generalized chebyshev koornwinder oscillator in the previous works of borzov and damaskinsky chebyshev koornwinder oscillator theor math phys 175 3 765 772 2013 and ladder operators for chebyshev koornwinder oscillator in proceedings of the days on diffraction 2013 the authors have defined the oscillator like system that is associated with the two variable chebyshev koornwinder polynomials we call this system the generalized chebyshev koornwinder oscillator in this paper we study the properties of infinite dimensional lie algebra that is analogous to the heisenberg algebra for the chebyshev koornwinder oscillator we construct the exact irreducible representation of this algebra in a hilbert space of functions that are defined on a region which is bounded by the steiner hypocycloid the functions are square integrable with respect to the orthogonality measure for the chebyshev koornwinder polynomials and these polynomials form an orthonormalized basis in the space the generalized oscillator which is studied in the work can be considered as the simplest nontrivial example of multiboson quantum system that is composed of three interacting oscillators h h

weak classical orthogonal polynomials in two variables orthogonal polynomials in two variables constitute a very old subject in approximation theory usually they are studied as solutions of second order partial differential equations in this work we study two variable orthogonal polynomials associated with a moment functional satisfying the two variable analogue of the pearson differential equation from this approach we derive the extension of some of the usual characterizations of the classical orthogonal polynomials in one variable

algorithms and geometry for graph products of groups recent work of gromov epstein cannon thurston and many others has generated strong interest in the geometric and algorithmic structure of finitely generated infinite groups see 16 17 and 14 many of these structures are preserved by taking graph products the graph product of groups not to be confused with the fundamental group of a graph of groups is a product mixing direct and free products whether the product between two groups in the graph product is free or direct is determined by a simplicial graph given a simplicial graph we say that two vertices are adjacent if they are joined by a single edge

bivariate second order linear partial differential equations and orthogonal polynomial solutions in this paper we construct the main algebraic and differential properties and the weight functions of orthogonal polynomial solutions of bivariate second order linear partial differential equations which are admissible potentially self adjoint and of hypergeometric type general formulae for all these properties are obtained explicitly in terms of the polynomial coefficients of the partial differential equation using vector matrix notation moreover rodrigues representations for the polynomial eigensolutions and for their partial derivatives of any order are given as illustration these results are applied to a two parameter monic appell polynomials finally the non monic case is briefly discussed 2011 elsevier inc

second order partial differential equations for gradients of orthogonal polynomials in two variables in this work we introduce the classical orthogonal polynomials in two variables as the solutions of a matrix second order partial differential equation involving matrix polynomial coefficients the usual gradient operator and the divergence operator here we show that the successive gradients of these polynomials also satisfy a matrix second order partial differential equation closely related to the first one

a distributional study of discrete classical orthogonal polynomials abstract for the sequences of discrete classical orthogonal polynomials charlier meixner hahn we can find a functional u which satisfies the difference distributional equation u u where and are polynomials of degrees 2 and 1 respectively from this it follows that these polynomials are solutions of a second order difference equation also they can be represented by a rodrigues type formula the sequence of difference polynomials derived from them constitutes an orthogonal polynomial sequence their weight functions satisfy a pearson type difference equation a structure relation pn 1 anpn 2 bnpn 1 cnpn also holds

a semiclassical perspective on multivariate orthogonal polynomials differential properties for orthogonal polynomials in several variables are studied we consider multivariate orthogonal polynomials whose gradients satisfy some quasi orthogonality conditions we obtain several characterizations for these polynomials including the analogues of the semiclassical pearson differential equation the structure relation and a differential difference equation

on the q polynomials a distributed study in this paper we present a uni1ed distributional study of the classical discrete q polynomials in the hahn s sense from the distributional q pearson equation we will deduce many of their properties such as the three term recurrence relations structure relations etc also several characterizations of such q polynomials are presented c

topological aperiodicity for product systems over semigroups of ore type we prove a version of uniqueness theorem for cuntz pimsner algebras of discrete product systems over semigroups of ore type to this end we introduce doplicher roberts picture of cuntz pimsner algebras and the semigroup dual to a product system of regular c correspondences under a certain aperiodicity condition on the latter we obtain the uniqueness theorem and a simplicity criterion for the algebras in question these results generalize the corresponding ones for crossed products by discrete groups due to archbold and spielberg and for exel s crossed products due to exel and vershik they also give interesting conditions for topological higher rank graphs and p graphs and apply to the new cuntz c algebra qn arising from the ax b semigroup over n

on koornwinder classical orthogonal polynomials in two variables in 1975 tom koornwinder studied examples of two variable analogues of the jacobi polynomials in two variables those orthogonal polynomials are eigenfunctions of two commuting and algebraically independent partial differential operators some of these examples are well known classical orthogonal polynomials in two variables such as orthogonal polynomials on the unit ball on the simplex or the tensor product of jacobi polynomials in one variable but the remaining cases are not considered classical by other authors the definition of classical orthogonal polynomials considered in this work provides a different perspective on the subject we analyze in detail koornwinder polynomials and using the koornwinder tools new examples of orthogonal polynomials in two variables are given

a construction of real weight functions for certain orthogonal polynomials in two variables abstract h l krall and i m sheffer considered the problem of classifying certain second order partial differential equations having an algebraically complete weak orthogonal bivariate polynomial system of solutions two of the equations that they considered are x 2 y u x x 2 x y u x y y 2 u y y g x u x g y 1 u y u and x 2 u x x 2 x y u x y y 2 y u y y g x 1 u x g y u y u even though they showed that these equations have a sequence of weak orthogonal polynomial solutions they were unable to show that these polynomials were in fact orthogonal the orthogonality of these two polynomial sequences was recently established by kwon littlejohn and lee solving an open problem from 1967 in this paper we construct explicit weight functions for these two orthogonal polynomial sequences using a method first developed by littlejohn and then further developed by han kim and kwon moreover two additional partial differential equations were found by kwon littlejohn and lee that have sequences of orthogonal polynomial solutions these equations are given by x 2 x u x x 2 x y u x y y 2 u y y d x e u x d y h u y u x u x x 2 y u x y d x e u x d y h u y u in each of these examples we also produce explicit orthogonalizing weight functions

the structure relation for askey wilson polynomials an explicit structure relation for askey wilson polynomials is given this involves a divided q difference operator which is skew symmetric with respect to the askey wilson inner product and which sends polynomials of degree n to polynomials of degree n 1 by specialization of parameters and by taking limits similar structure relations as well as lowering and raising relations can be obtained for other families in the q askey scheme and the askey scheme this is explicitly discussed for jacobi polynomials continuous q jacobi polynomials continuous q ultraspherical polynomials and for big q jacobi polynomials an already known structure relation for this last family can be obtained from the new structure relation by using the three term recurrence relation and the second order q difference formula the results are also put in the framework of a more general theory their relationship with earlier work by zhedanov and bangerezako is discussed there is also a connection with the string equation in discrete matrix models and with the sklyanin algebra

sobolev orthogonal polynomials in two variables and second order partial differential equations abstract we consider polynomials in two variables which satisfy an admissible second order partial differential equation of the form a u x x 2 b u x y c u y y d u x e u y u and are orthogonal relative to a symmetric bilinear form defined by p q p q p x q x where a e are polynomials in x and y is an eigenvalue parameter and are linear functionals on polynomials we find a condition for the partial differential equation to have polynomial solutions which are orthogonal relative to a symmetric bilinear form also examples are provided

a new duality theory for compact groups the tannaka krein duality theory characterizes the category g of finite dimensional continuous unitary representations of a compact group as a subcategory of the category of hilbert spaces we prove a more powerful result characterizing g as an abstract category every strict symmetric monoidalc category with conjugates which has subobjects and direct sums and for which thec algebra of endomorphisms of the monoidal unit reduces to the complex numbers is isomorphic to a category g for a compact groupg unique up to isomorphism

semiclassical orthogonal polynomials in two variables in this work semiclassical orthogonal polynomials in two variables are defined as the orthogonal polynomials associated with a quasi definite linear functional satisfying a matrix pearson type differential equation semiclassical functionals are characterized by means of the analogue of the structure relation in one variable moreover non trivial examples of semiclassical orthogonal polynomials in two variables are given

two dimensional krall sheffer polynomials and integrable systems two dimensional krall sheffer polynomials are analogues of the classical orthogonal polynomial they are eigenfunctions of second order linear partial differential operators and moreover satisfy orthogonality conditions we show that all krall sheffer polynomials are connected with two dimensional superintegrable systems on spaces with constant curvature

the sparse structure of high dimensional integrands abstract we consider a discretization of a functional of white noise over a finite time interval we explore the possible interest of representing the white noise in the orthonormal bases of orthogonal polynomials or wavelets for the numerical evaluation of the expected value of this functional using the wiener ito decomposition of the functional the sparsity is studied of the representation of the functional in these bases an approximation scheme is proposed that uses existing low dimensional quasi monte carlo rules and takes profit of the sparse structure of the quadratic part of the functional

topological freeness for c correspondences abstract we study conditions that ensure uniqueness theorems of cuntz krieger type for relative cuntz pimsner algebras o j x associated to a c correspondence x over a c algebra a we give general sufficient conditions phrased in terms of a multivalued map x acting on the spectrum a of a when x j is of type i we construct a directed graph dual to x and prove a uniqueness theorem using this graph when x j is liminal we show that topological freeness of this graph is equivalent to the uniqueness property for o j x as well as to an algebraic condition which we call j acyclicity of x as an application we improve the fowler raeburn uniqueness theorem for the toeplitz algebra t x we give new simplicity criteria for o x we generalize and enhance uniqueness results for relative quiver c algebras of muhly and tomforde we also discuss applications to crossed products by endomorphisms

new steps on sobolev orthogonality in two variables sobolev orthogonal polynomials in two variables are defined via inner products involving gradients such a kind of inner product appears in connection with several physical and technical problems matrix second order partial differential equations satisfied by sobolev orthogonal polynomials are studied in particular we explore the connection between the coefficients of the second order partial differential operator and the moment functionals defining the sobolev inner product finally some old and new examples are given

symmetrizability of differential equations having orthogonal polynomial solutions abstract we show that if a linear differential equation of spectral type with polynomial coefficients l n y x i 0 n l i x n y x has an orthogonal polynomial system of solutions then the differential operator ln must be symmetrizable we also give a few applications of this result

structure relations for monic orthogonal polynomials in two discrete variables in this paper extensions of several relations linking differences of bivariate discrete orthogonal polynomials and polynomials themselves are given by using an appropriate vector matrix notation three term recurrence relations are presented for the partial differences of the monic polynomial solutions of admissible second order partial difference equation of hypergeometric type structure relations difference representations as well as lowering and raising operators are obtained finally expressions for all matrix coefficients appearing in these finite type relations are explicitly presented for a finite set of hahn and kravchuk orthogonal polynomials

discrete product systems and their c algebras continuous product systems or product systems over r were recently introduced by arveson to develope an index theory for continuous semigroups of endomorphisms of b h in this paper we continue his work by studying discrete product systems or product systems over g where g is a countable dense subgroup of r we give examples of discrete product systems and compute their automorphism groups to each discrete product system e over g we associate a separable c algebra oe g whose non degenerate representations correspond bijectively to representations of e this family of c algebras may be regarded as analogues of cuntz algebras or car algebras in some degenerate cases these c algebras are the generalized toeplitz algebras we show that oe g is simple and study its gauge automorphisms quasi free automorphisms and fock state