integration of quantum many body problems by affine knizhnik zamolodchikov equations abstract an equivalence of the affine analogues of knizhnik zamolodchikov and dunkl equations is established for arbitrary root systems as a corollary an isomorphism is obtained from the solutions of the above kz equations in the induced representations of hecke algebras to the eigenfunctions for the corresponding matrix quantum calogero sutherland operators which generalizes the matsuo map

yang s system of particles and hecke algebras the graded hecke algebra has a simple realization as a certain algebra of operators acting on a space of smooth functions this operator algebra arises from the study of the root system analogue of yang s system of n particles on the real line with delta function potential it turns out that the spectral problem for this generalization of yang s system is related to the problem of finding the spherical tempered representations of the graded hecke algebra this observation turns out to be very useful for both these problems application of our technique to affine hecke algebras yields a simple formula for the formal degree of the generic iwahori spherical discrete series representations

some conjectures and results concerning the homology of nilpotent lie algebras in this paper we consider the lie algebra homology of for l a complex lie algebra our goal is to express the homology of l in terms of the homology of l this problem comes up in previous work by the author on the macdonald root system conjectures we present a number of conjectures related to this problem the simplest of these conjectures asserts that h l h l l i

affine yangians and deformed double current algebras in type a we study the structure of yangians of affine type and deformed double current algebras which are deformations of the enveloping algebras of matrix w1 8 algebras we prove that they admit a pbw type basis establish a connection limit construction between these two types of algebras and toroidal quantum algebras and we give three equivalent definitions of deformed double current algebras we construct a schur weyl functor between these algebras and rational cherednik algebras

fast analysis of transient acoustic wave scattering from rigid bodies using the multilevel plane wave time domain algorithm the analysis of transient wave scattering from rigid bodies using integral equation based techniques is computationally intensive if carried out using classical schemes the evaluation of the velocity potential on the surface of a three dimensional scatterer represented in terms of ns spatial basis functions for nt time steps requires o ntns2 operations the recently developed plane wave time domain pwtd algorithm permits the rapid evaluation of transient fields that are generated by bandlimited source distributions it has been shown that incorporation of the pwtd algorithm into integral equation based solvers in a two level setting reduces the computational complexity of a transient analysis to o ntns1 5 logns in this paper it is shown that casting the pwtd scheme into a multilevel framework permits the analysis of transient acoustic surface scattering phenomena in o ntnslog2ns operations using o ntns memory numerical examples that demonstrate the efficacy of the multilevel implementation are also presented

a unification of knizhnik zamolodchikov and dunkl operators via affine hecke algebras some generalizations of the lusztig lascoux schutzenberger operators for affine hecke algebras are considered as corollaries we obtain lusztig s isomorphisms from affine hecke algebras to their degenerate versions a natural interpretation of the dunkl operators and a new class of differential difference operators generalizing dunkl s ones and the knizhnik zamolodchikov operators from the two dimensional conformal field theory

toroidal and level 0 uq sln 1 actions on uq gln 1 modules 1 utilizing a braid group action on a completion of uq sln 1 an algebra homomorphism from the toroidal algebra uq sln 1 tor n 2 to a completion of uq gln 1 is obtained 2 the toroidal actions by saito induces a level 0 uq sln 1 action on level 1 integrable highest weight modules of uq sln 1 another level 0 uq sln 1 action was defined by jimbo et al in the case n 1 using the fact that the intertwiners of uq sln 1 modules are intertwiners of toroidal modules for an appropriate comultiplication the relation between these two level 0 uq sln 1 actions is clarified

ruijsenaars commuting difference operators and invariant subspace spanned by theta functions we study a family of mutually commutative difference operators introduced by ruijsenaars the conjugations of these operators with an appropriate function give the hamiltonians of some relativistic quantum systems these operators can be regarded as elliptic analogs of the macdonald operators and their coefficients consist of the jacobi theta functions we show that these operators act on the space of meromorphic functions on the cartan subalgebra of affine lie algebras and that the space spanned by characters of a fixed positive level is invariant under the action of these operators

a finite difference delay modeling approach to the discretization of the time domain integral equations of electromagnetics a new method for solving the time domain integral equations of electromagnetic scattering from conductors is introduced this method called finite difference delay modeling appears to be completely stable and accurate when applied to arbitrary structures the temporal discretization used is based on finite differences specifically based on a mapping from the laplace domain to the z transform domain first and second order unconditionally stable methods are derived spatial convergence is achieved using the higher order divergence conforming vector bases of graglia et al low frequency instability problems are avoided with the loop tree decomposition approach numerical results will illustrate the accuracy and stability of the technique

cyclic homology and the macdonald conjectures leta k denote the ring t t k 1 and letg be a reductive complex lie algebra with exponentsm 1 m n this paper concerns the lie algebra cohomology ofg a k considered as a bigraded algebra here one of the gradings is homological degree and the other which we callweight is inherited from the obvious grading ofg a k we conjecture that this lie algebra cohomology is an exterior algebra withk 1 generators of homological degree 2m s 1 fors 1 2 n of thesek 1 generators of degree 2m s 1 one has weight 0 and the others have weights k 1 m s t fort 1 2 k it is shown that this conjecture about the lie algebra cohomology of a k implies the macdonald root system conjectures next we consider the case thatg is a classical lie algebra with root systema n b n c n ord n it is shown that our conjecture holds in the limit onn asn approaches infinity which amounts to the computation of the cyclic and dihedral cohomologies ofa k lastly we discuss the relevance of this limiting case to the case of finiten in this situation

on the internal resonant modes in marching on in time solution of the time domain electric field integral equation internal resonant modes are always observed in the marching on in time mot solution of the time domain electric field integral equation efie although relaxed initial conditions which are enforced at the beginning of time marching should in theory prevent these spurious modes from appearing it has been conjectured that numerical errors built up during time marching establish the necessary initial conditions and induce the internal resonant modes however this conjecture has never been proved by systematic numerical experiments our numerical results in this communication demonstrate that the internal resonant modes amplitudes are indeed dictated by the numerical errors additionally it is shown that in a few cases the internal resonant modes can be made invisible by significantly suppressing the numerical errors these tests prove the conjecture that the internal resonant modes are induced by numerical errors when the time domain efie is solved by the mot method

bosonic and fermionic eigenstates for generalized sutherland models we construct bosonic and fermionic eigenstates for the generalized sutherland models associated with arbitrary reduced root systems respectively through w symmetrization and w anti symmetrization of heckman opdam s nonsymmetric jacobi polynomials square norms of the nonsymmetric heckman opdam polynomials are evaluated from their rodrigues formulae the w symmetrization and w anti symmetrization of the nonsymmetric polynomials enable us to evaluate square norms of bosonic and fermionic eigenstates for the generalized sutherland models

t equivariant k theory of generalized flag varieties let g be a kac moody group with borel subgroup b and compact maximal torus t analogous to kostant and kumar kostant b kumar s 1986 proc nati acad sci usa 83 1543 1545 we define a certain ring y purely in terms of the weyl group w associated to g and its action on t by dualizing y we get another ring 1 which we prove is canonically isomorphic with the t equivariant ktheory kt g b of gib now kt g b apart from being an algebra over kt pt a t also has a weyl group action and moreover kt g b admits certain operators dw w similar to the demazure operators defined on a t we prove that these structures on kt g b come naturally from the ring y by evaluating the a t module w at 1 we recover k g b together with the above mentioned structures we believe that many of the results of this paper are new in the finite case i e g is a finite dimensional semisimple group over c

theta functions and weil representations of loop symplectic groups we introduce weil representations for loop symplectic groups and prove the convergence and modularity of the related theta functions

stable electric field tdie solvers via quasi exact evaluation of mot matrix elements prior theoretical studies and experience confirm that the stability of marching on in time mot solvers pertinent to the analysis of scattering from free standing three dimensional perfect electrically conducting surfaces hinges on the accurate evaluation of mot matrix elements resulting from a galerkin discretization of the underlying time domain integral equation tdie unfortunately the accurate evaluation of the four dimensional spatial integrals involved in the expressions for these matrix elements is prohibitively expensive when performed by computational means here a method that permits the quasi exact evaluation of mot matrix elements is presented specifically the proposed method permits the analytical evaluation of three out of the four spatial integrations leaving only one integral to be evaluated numerically since the latter has finite range and a piecewise smooth integrand it can be evaluated to very high accuracy using standard quadrature rules as a result the proposed method permits the fast evaluation of mot matrix elements with arbitrary user specified accuracy extensive numerical experiments show that an mot solver for the electric field tdie that uses the proposed quasi exact method is stable for a very wide range of time step sizes and yields solutions that decay exponentially after the excitation vanishes

fast analysis of transient electromagnetic scattering phenomena using the multilevel plane wave time domain algorithm the computational complexity of classical marching on in time mot methods for solving time domain integral equations tdies pertinent to the analysis of transient scattering phenomena involving perfectly conducting targets grows as o n sub t n sub s sup 2 n sub t and n sub s denote the number of temporal and spatial degrees of freedom dof of the electric current on the target this scaling law impedes the application of these schemes to the analysis of large scale scattering phenomena the recently developed plane wave time domain pwtd algorithm permits the rapid evaluation of transient wave fields generated by temporally bandlimited sources and hence the acceleration of marching on in time based tdie solvers previously we described a two level pwtd enhanced tdie solver for analyzing electromagnetic scattering from perfectly conducting targets the computational complexity of this algorithm scales as o n sub t n sub s sup 1 5 logn sub s here a multilevel pwtd scheme for rapidly evaluating electric fields due to temporally bandlimited electric current sources is described in addition a multilevel pwtd enhanced tdie solver for analyzing electromagnetic scattering from perfectly conducting scatterers using o n sub t n sub s log sup 2 n sub s cpu resources is outlined last the accuracy and cpu memory efficiency of this solver are demonstrated by analyzing transient scattering from electrically large bodies

parallel pwtd accelerated explicit solution of the time domain electric field volume integral equation a parallel plane wave time domain pwtd accelerated explicit marching on in time mot scheme for solving the time domain electric field volume integral equation td efvie is presented the proposed scheme leverages pulse functions and lagrange polynomials to spatially and temporally discretize the electric flux density induced throughout the scatterers and a finite difference scheme to compute the electric fields from the hertz electric vector potentials radiated by the flux density the flux density is explicitly updated during time marching by a predictor corrector pc scheme and the vector potentials are efficiently computed by a scalar pwtd scheme the memory requirement and computational complexity of the resulting explicit pwtd pc efvie solver scale as o n s log n s and o n s n t respectively here n s is the number of spatial basis functions and n t is the number of time steps a scalable parallelization of the proposed mot scheme on distributed memory cpu clusters is described the efficiency accuracy and applicability of the resulting parallelized pwtd pc efvie solver are demonstrated via its application to the analysis of transient electromagnetic wave interactions on canonical and real life scatterers represented with up to 25 million spatial discretization elements

a new generalisation of macdonald polynomials we introduce a new family of symmetric multivariate polynomials whose coefficients are meromorphic functions of two parameters q t and polynomial in a further two parameters u v we evaluate these polynomials explicitly as a matrix product at u v 0 they reduce to macdonald polynomials while at q 0 u v s they recover a family of inhomogeneous symmetric functions originally introduced by borodin

longitudinal data analysis using generalized linear models summary this paper proposes an extension of generalized linear models to the analysis of longitudinal data we introduce a class of estimating equations that give consistent estimates of the regression parameters and of their variance under mild assumptions about the time dependence the estimating equations are derived without specifying the joint distribution of a subject s observations yet they reduce to the score equations for multivariate gaussian outcomes asymptotic theory is presented for the general class of estimators specific cases in which we assume independence m dependence and exchangeable correlation structures from each subject are discussed efficiency of the proposed estimators in two simple situations is considered the approach is closely related to quasi likelih ood some key ironh estimating equation generalized linear model longitudinal data quasi likelihood repeated measures

residue construction of hecke algebras abstract hecke algebras are usually defined algebraically via generators and relations we give a new algebra geometric construction of affine and double affine hecke algebras the former is known as the iwahori hecke algebra and the latter was introduced by cherednik in terms of residues more generally to any generalized cartan matrixaand a pointqin a one dimensional complex algebraic group c we associate an associate algebrahq ifais of finite type and c c the algebrahqis theaffinehecke algebra of the corresponding finite root system ifais of affine type and c c thenhqis essentially the cherednik algebra the case c c corresponds to degenerate counterparts of the above objects considered by drinfeld and lusztig finally taking c to be an elliptic curve one gets some new elliptic analogues of the affine hecke algebra

a stabilized marching on in degree scheme for the transient solution of the electric field integral equation time domain electric integral equation methods td efie suffer from the numerical stability problem numerous schemes have been proposed over the years for the purpose of taking care of this instability among them the marching on in degree mod method is one of the newly proposed algorithms for calculating the various transient responses the mod solver employs a set of associated laguerre functions and incorporates exact temporal galerkin testing although the mod scheme is assumed to be more stable when compared with marching on in time mot there are few numerical or theoretical investigations on the stability of the various numerical recursive operations for the mod methodology it has been shown that the time domain magnetic field integral equation td mfie using the mod method can be stabilized by using a filon type quadrature formula to accurately integrate the highly oscillatory laguerre functions which has caused instability for the mod solvers using a higher degree of approximation on the other hand the analytical expression of temporal derivatives used in the mod scheme is strongly affected by an unbounded initial value of the transient current in this paper a novel filon type radial integration method and a sequential constraint term for td efie mod are proposed which make the stabilized mod solver accurate and stable for the arbitrary degree of the laguerre polynomials some numerical results are presented to illustrate the validity of the stabilized mod solution of td efie for transient scattering problems

yangian gelfand zetlin bases gl n jack polynomials and computation of dynamical correlation functions in the spin calogero sutherland model we consider the invariant calogero sutherland models with n 1 2 3 in the framework of symmetric polynomials

implicit runge kutta methods for the discretization of time domain integral equations implicit runge kutta based schemes are proposed for solving the time domain integral equations of electromagnetic theory the proposed technique maps a laplace domain equation to a domain equation using the butcher tableau of the implicit runge kutta scheme a discrete time domain system is recovered by computing the inverse transform numerically the resulting technique is capable of third or fifth order accuracy in time and is stable independent of time step numerical results illustrate the accuracy and stability of the technique

quantum knizhnik zamolodchikov equations and affine root systems quantum difference knizhnik zamolodchikov equations s1 fr are generalized for ther matrices from ch1 with the arguments in arbitrary root systems and their formal counterparts in particular qkz equations with certain boundary conditions are introducted the self consistency of the equations from fr and the cross derivative integrability conditions for ther matrix kz equations from ch2 are obtained as corollaries a difference counterpart of the quantum many body problem connected with macdonald s operators is defined as an application

geometric representations of graded and rational cherednik algebras abstract we provide geometric constructions of modules over the graded cherednik algebra h gr and the rational cherednik algebra h rat attached to a simple algebraic group g together with a pinned automorphism these modules are realized on the cohomology of affine springer fibers of finite type that admit c actions in the rational cherednik algebra case the standard grading on these modules is derived from the perverse filtration on the cohomology of affine springer fibers coming from its global analog hitchin fibers when is trivial we show that our construction gives the irreducible finite dimensional spherical modules l triv of h gr and of h rat we give a formula for the dimension of l triv and give a geometric interpretation of its frobenius algebra structure the rank two cases are studied in further details

on the stability of time domain magnetic field integral equation using laguerre functions the marching on in degree mod method has been proposed for solving time domain integral equations tdies and it uses the associated laguerre functions as temporal basis and testing functions although the mod scheme is assumed to be stable and accurate there exists little investigation into the accuracy of the integral operations involving the associated laguerre functions of high degree it is shown in this paper that the numerical quadrature for integrating the highly oscillatory laguerre functions may not be accurate and this may result in instability for the mod solvers in this paper the highly oscillatory nature of the high degree associated laguerre functions is investigated first based on that the numerical procedures are described which result in the inaccuracy of the numerical quadrature associated with the highly oscillatory nature for the higher order laguerre functions this can produce instability for the mod solvers when a solution requires a high degree of the laguerre functions hence a novel filon type quadrature method and a stabilized mod technique for time domain magnetic field integral equations td mfie smod are presented in this paper some numerical results are presented to illustrate the validity of the proposed filon type quadrature rule resulting in stable solution of the td mfie for transient scattering problems

analysis of transient scattering from composite arbitrarily shaped complex structures a time domain surface integral equation approach based on the electric field formulation is utilized to calculate the transient scattering from both conducting and dielectric bodies consisting of arbitrarily shaped complex structures the solution method is based on the method of moments mom and involves the modeling of an arbitrarily shaped structure in conjunction with the triangular patch basis functions an implicit method is described to solve the coupled integral equations derived utilizing the equivalence principle directly in the time domain the usual late time instabilities associated with the time domain integral equations are avoided by using an implicit scheme detailed mathematical steps are included along with representative numerical results

unramified whittaker functions on p adic loop groups we define unramified whittaker functions on the p adic points of an affine kac moody group and establish an analogue of the casselman shalika formula for these functions

on the reliability and precision of within and between population estimates of relative rate parameters comparisons of individual and aggregate level analyses of data from a multigroup observational study are made using an exponential form relative rate model stratified analytic random effects and aggregate random effects analyses are studied estimating equations are developed to give a consistent estimation procedure across analyses and corresponding information matrices are compared simulation studies provide insight into the efficiency and bias of relative rate parameter estimates with respect to covariate dispersion confounding and covariate measurement error

time domain integral equation analysis of scattering from composite bodies via exact evaluation of radiation fields a novel marching on in time and poggio miller chang harrington wu tsai pmchwt based time domain integral equation solver for analyzing transient scattering from composite multiregion structures comprising homogeneous penetrable volumes and perfectly conducting surfaces is presented to render the marching on in time procedure stable observer and source integrals are approximated via multipoint collocation and evaluated in closed form the proposed scheme avoids the costly evaluation of five dimensional integrals required in full space time galerkin schemes and proves to be stable for all targets investigated including coated perfect electrically conducting surfaces and multiregion penetrable volumes

the homflypt skein algebra of the torus and the elliptic hall algebra we give a generators and relations presentation of the homflypt skein algebra h of the torus t2 and we give an explicit description of the module corresponding to the solid torus using this presentation we show that h is isomorphic to the 1 specialization of the elliptic hall algebra of burban and schiffmann as an application for an iterated cable k of the unknot we use the elliptic hall algebra to construct a 3 variable polynomial that specializes to the colored homflypt polynomial of k we show that this polynomial also specializes to one constructed by cherednik and danilenko using the gln double affine hecke algebra this proves one of the connection conjectures in their recent work

macdonald s constant term conjectures for exceptional root systems we announce proofs of macdonald s constant term conjectures for the affine root systems s f 4 and s f 4 v we also give an algorithm for deciding the conjectures for the remaining root systems s e 6 s e 7 and s e 8 and prove that the constant term in question can indeed be expressed in closed form combined with previous work of zeilberger bressoud kadell and gustafson our results imply that macdonald s conjectures are true in form for any root system

numerical evaluation of singular and near singular potential integrals a simple and efficient numerical procedure using a singularity cancellation scheme is presented for evaluating singular and near singular potential integrals with 1 r singularities the procedure not only has several advantages over singularity subtraction methods but also improves on some aspects of other singularity cancellation methods such as polar and duffy transformations a theoretical analysis is presented for triangles quadrilaterals tetrahedrons bricks and prisms and numerical results are presented for triangles and prisms

the spherical hecke algebra for affine kac moody groups i we dene the spherical hecke algebra for an untwisted ane kacmoody group over a local non archimedian eld we prove a generalization of the satake isomorphism for this algebra relating it to integrable representations of the langlands dual ane kac moody group in the next publication we shall use these results to dene and study the notion of hecke eigenfunction for the group ga

group actions on algebras and the graded lie structure of hochschild cohomology hochschild cohomology governs deformations of algebras and its graded lie structure plays a vital role we study this structure for the hochschild cohomology of the skew group algebra formed by a nite group acting on an al gebra by automorphisms we examine the gerstenhaber bracket with a view toward deformations and developing bracket formulas we then focus on the linear group actions and polynomial algebras that arise in orbifold theory and representation theory deformations in this context include graded hecke alge bras and symplectic reection algebras we give some general results describing when brackets are zero for polynomial skew group algebras which allow us in particular to nd noncommutative poisson structures for abelian groups we express the bracket using inner products of group characters lastly we inter pret results for graded hecke algebras

orthogonality relations and cherednik identities for multivariable baker akhiezer functions abstract we establish orthogonality relations for the baker akhiezer ba eigenfunctions of the macdonald difference operators we also obtain a version of cherednik macdonald mehta integral for these functions as a corollary we give a simple derivation of the norm identity and cherednik macdonald mehta integral for macdonald polynomials in the appendix written by the first author we prove a summation formula for ba functions we also consider more general identities of cherednik type which we use to introduce and construct more general twisted ba functions this leads to a construction of new quantum integrable models of macdonald ruijsenaars type

aggregate data studies of disease risk factors summary statistical methods are proposed for estimating relative rate parameters based on estimated disease rates and covariate data from random samples of individuals from each of several cohorts a random effects model is used to derive mean and variance models for estimated disease rates estimating equations for relative rate parameters are then developed by replacing cohort covariate averages by corresponding sample averages the asymptotic distribution of regression parameter estimates is derived and the asymptotic bias is shown to be small even if covariates are contaminated by classical random measurement errors provided the covariate sample size in each cohort is not small simulation studies motivated by international data on diet and breast cancer provide insights into the properties of the proposed estimators

double affine hecke algebras and 2 dimensional local fields the concept of an n dimensional local field was introduced by a n parshin pal with the aim of generalizing the classical adelic formalism to absolutely ndimensional schemes by definition a 0 dimensional local field is just a finite field and an n dimensional local field n 0 is a complete discrete valued field whose residue field is n 1 dimensional local thus for n 1 we get locally compact fields such as qp fq t and for n 2 we get fields such as qp t fq ti t2 etc in representation theory harmonic analysis on reductive groups over 0and 1dimensional local fields leads in particular to consideration of the finite and affine hecke algebras hq hq associated to aniy finite root system r and anly q e c these algebras can be defined in several ways one being by generators and relations another as the convolution algebra with respect to the haar measure of functions on the group bi invariant with respect to an appropriate subgroup i e as the algebra of double cosets harmonic analysis on groups over 2 dimensional local fields has not been developed the main difficulty being the infinite dimensionality absense of local compactness of such fields however the double affine hecke

stability properties of the time domain electric field integral equation using a separable approximation for the convolution with the retarded potential the state of art of time domain integral equation tdie solvers has grown by leaps and bounds over the past decade during this time advances have been made in i the development of accelerators that can be retrofitted with these solvers and ii understanding the stability properties of the electric field integral equation as is well known time domain electric field integral equation solvers have been notoriously difficult to stabilize research into methods for understanding and prescribing remedies have been on the uptick the most recent of these efforts are i lubich quadrature and ii exact integration in this paper we re examine the solution to this equation using i the undifferentiated form of the time domain electric field integral equation tdefie and ii a separable approximation to the spatio temporal convolution the proposed scheme can be constructed such that the spatial integrand over the source and observer domains is smooth and integrable as several numerical results will demonstrate the proposed scheme yields stable results for long simulation times and a variety of targets both of which have proven extremely challenging in the past

on the accommodation of disease rate correlations in aggregate data studies of disease risk factors prentice and sheppard 1995 biometrika 82 113 125 proposed a method for estimating relative risks associated with poorly measured exposures using disease rates from multiple populations and exposure and confounding factor data from sample surveys of persons in each population the method involved an assumption of independence of disease rates across populations conditional on exposures and confounding factors here this assumption is relaxed by allowing dependencies among the disease rates within a partition of the populations perhaps defined on the basis of geographic proximity or cultural similarity such dependencies could for example derive from unmeasured risk factors that are shared among neighboring populations dependencies within an element of the partition are modeled by allowing a common correlation between the disease rates of contiguous populations and by inducing correlations between the rates for noncontiguous populations using a multivariate lognormal assumption estimating equations are proposed for relative risk parameter estimation and robustness and efficiency properties are assessed through simulations the relaxed estimation procedure is shown to yield useful efficiency gains when moderate or strong disease rate dependencies are present especially when disease rate variances are large relative to binomial variance

a galerkin bem for transient acoustic scattering by an absorbing obstacle this paper deals with the numerical resolution of the so called time domain boundary integral equations the scattering problem by an absorbing obstacle serves as the model for this discussion a new system of retarded potential boundary integral equations rpbie is set up to solve the problem using an energy identity we are able to prove the unconditional stability for standard conforming galerkin approximation of the rpbie the actual space time boundary elements are described in some detail and their implementation has confirmed that stability in a wide range of situations and for large run times never reached before

a bispectral q hypergeometric basis for a class of quantum integrable models for the class of quantum integrable models generated from the q onsager algebra a basis of bispectral multivariable q orthogonal polynomials is exhibited in the first part it is shown that the multivariable askey wilson polynomials with n variables and n 3 parameters introduced by gasper and rahman dev math 13 209 2005 generate a family of infinite dimensional modules for the q onsager algebra whose fundamental generators are realized in terms of the multivariable q difference and difference operators proposed by iliev trans am math soc 363 1577 2011 raising and lowering operators extending those of sahi sigma 3 002 2007 are also constructed in the second part finite dimensional modules are constructed and studied for a certain class of parameters and if the n variables belong to a discrete support in this case the bispectral property finds a natural interpretation within the framework of tridiagonal pairs in the third part eigenfunctions of the q dolan grady hierarchy are considered in the polynomial basis in particular invariant subspaces are identified for certain conditions generalizing nepomechie s relations in the fourth part the analysis is extended to the special case q 1 this framework provides a q hypergeometric formulation of quantum integrable models such as the open xxz spin chain with generic integrable boundary conditions q 1 for the class of quantum integrable models generated from the q onsager algebra a basis of bispectral multivariable q orthogonal polynomials is exhibited in the first part it is shown that the multivariable askey wilson polynomials with n variables and n 3 parameters introduced by gasper and rahman dev math 13 209 2005 generate a family of infinite dimensional modules for the q onsager algebra whose fundamental generators are realized in terms of the multivariable q difference and difference operators proposed by iliev trans am math soc 363 1577 2011 raising and lowering operators extending those of sahi sigma 3 002 2007 are also constructed in the second part finite dimensional modules are constructed and studied for a certain

discrete series characters for affine hecke algebras and their formal degrees we introduce the generic central character of an irreducible discrete series representation of an affine hecke algebra using this invariant we give a new classification of the irreducible discrete series characters for all abstract affine hecke algebras except for the types e n 1 n 6 7 8 with arbitrary positive parameters and we prove an explicit product formula for their formal degrees in all cases

matrix product formula for macdonald polynomials we derive a matrix product formula for symmetric macdonald polynomials our results are obtained by constructing polynomial solutions of deformed knizhnik zamolodchikov equations which arise by considering representations of the zamolodchikov faddeev and yang baxter algebras in terms of t deformed bosonic operators these solutions are generalised probabilities for particle configurations of the multi species asymmetric exclusion process and form a basis of the ring of polynomials in n variables whose elements are indexed by compositions for weakly increasing compositions anti dominant weights these basis elements coincide with non symmetric macdonald polynomials our formulas imply a natural combinatorial interpretation in terms of solvable lattice models they also imply that normalisations of stationary states of multi species exclusion processes are obtained as macdonald polynomials at q 1

commuting difference operators arising from the elliptic c2 1 face model we study a pair of commuting difference operators arising from the elliptic c2 1 face model the operators whose coefficients are expressed in terms of the jacobi s elliptic theta function act on the space of meromorphic functions on the weight space of the c2 type simple lie algebra we show that the space of functions spanned by the level one characters of the affine lie algebra sp 4 c is invariant under the action of the difference operators

the influence of the exact evaluation of radiation fields in finite precision arithmetic on the stability of the time domain integral equation method transient electromagnetic scattering phenomena can effectively be simulated with time domain integral equation methods the stability and accuracy of the marching on in time scheme is usually established with exact evaluation of the radiation fields due to singularities in their analytical expressions straightforward evaluation in finite precision arithmetic can jeopardize the accuracy of the radiation fields computational experiments confirm this and show that it can even lead to late time instability of the numerical model hence a reformulation is necessary to remove the singular behavior and obtain the robustness required for industrial application to this end highly accurate and well behaved expressions for the quasi exact integration method in finite precision arithmetic will be derived in this paper numerical experiments confirm the robustness and stability of the improved marching on in time scheme

irreducible representations of the rational cherednik algebra associated to the coxeter group h3 abstract this paper describes irreducible representations in category o of the rational cherednik algebra h c h 3 h associated to the exceptional coxeter group h 3 and any complex parameter c we compute the characters of all these representations explicitly as a consequence we classify all the finite dimensional irreducible representations of h c h 3 h

a dc stable and large time step well balanced td efie based on quasi helmholtz projectors the marching on in time mot solution of the time domain electric field integral equation td efie has traditionally suffered from a number of issues including the emergence of spurious static currents dc instability and ill conditioning at large time steps low frequencies in this contribution a space time galerkin discretization of the td efie is proposed which separates the loop and star components of both the equation and the unknown judiciously integrating or differentiating these components with respect to time leads to an equation which is free from dc instability by choosing the correct temporal basis and testing functions for each of the components a stable mot system is obtained furthermore the scaling of these basis and testing functions ensure that the system remains well conditioned for large time steps the loop star decomposition is performed using quasi helmholtz projectors to avoid the explicit transformation to the unstable bases of loops and stars or trees and to avoid the search for global loops which is a computationally expensive operation

a stable marching on in time scheme for solving the time domain electric field volume integral equation on high contrast scatterers a time domain electric field volume integral equation td efvie solver is proposed for characterizing transient electromagnetic wave interactions on high contrast dielectric scatterers the td efvie is discretized using the schaubert wilton glisson swg and approximate prolate spherical wave apsw functions in space and time respectively the resulting system of equations cannot be solved by a straightforward application of the marching on in time mot scheme since the two sided apsw interpolation functions require the knowledge of unknown future field samples during time marching causality of the mot scheme is restored using an extrapolation technique that predicts the future samples from known past ones unlike the extrapolation techniques developed for mot schemes that are used in solving time domain surface integral equations this scheme trains the extrapolation coefficients using samples of exponentials with exponents on the complex frequency plane this increases the stability of the mot td efvie solver significantly since the temporal behavior of decaying and oscillating electromagnetic modes induced inside the scatterers is very accurately taken into account by this new extrapolation scheme numerical results demonstrate that the proposed mot solver maintains its stability even when applied to analyzing wave interactions on high contrast scatterers

twisted strong macdonald theorems and adjoint orbits the strong macdonald theorems state that for l reductive and s an odd variable the cohomology algebras h l z z n and h l z s are freely generated and describe the cohomological s and z degrees of the generators the resulting identity for the z weighted euler characteristic is equivalent to macdonald s constant term identity for a finite root system we calculate h mathfrak p z n mathfrak p and h mathfrak p s for mathfrak p a standard parahoric in a twisted loop algebra giving strong macdonald theorems that take into account both a parabolic component and a possible diagram automorphism twist in particular we show that h mathfrak p z n mathfrak p contains a parabolic subalgebra of the coinvariant algebra of the fixed point subgroup of the weyl group of l and thus is no longer free we also prove a strong macdonald theorem for h mathfrak b s mathfrak n and h mathfrak b z n mathfrak n when mathfrak b and mathfrak n are iwahori and nilpotent subalgebras respectively of a twisted loop algebra for each strong macdonald theorem proved taking z weighted euler characteristics gives an identity equivalent to macdonald s constant term identity for the corresponding affine root system as part of the proof we study the regular adjoint orbits for the adjoint action of the twisted arc group associated to l proving an analogue of the kostant slice theorem

scattering theory of discrete pseudo laplacians on a weyl chamber to a crystallographic root system we associate a system of multivariate orthogonal polynomials diagonalizing an integrable system of discrete pseudo laplacians on the weyl chamber we develop the time dependent scattering theory for these discrete pseudo laplacians and determine the corresponding wave operators and scattering operators in closed form as an application we describe the scattering behavior of certain hyperbolic ruijsenaars schneider type lattice calogero moser models associated with the macdonald polynomials

treating the instabilities in marching on in time method from a different perspective electromagnetic scattering a general method is presented to treat the instabilities which are frequently observed in the electromagnetic transient solutions using the marching on in time method the basic idea is to apply an finite impulse response fir filter with a constant group delay during the course of marching in time an electric field integral equation efie formulation for perfectly conducting bodies is used as a vessel to demonstrate the method sample numerical results are presented and discussed the computed results while showing good agreement with the data obtained from other methods present great stability improvement

double affine hecke algebras and macdonald s conjectures in this paper we prove the main results about the structure of double affine hecke algebras announced in c1 c2 the technique is based on the realization of these algebras in terms of demazure lusztig operators bgg d l2 ls c3 and rather standard facts from the theory of affine weyl groups in particular it completes the proof partially published in c2 of the macdonald scalar product conjecture see ml 12 6 including the famous macdonald constant term conjecture the q t case we mainly follow the opdam paper 0 where the macdonald mehta conjectures in the degenerate differential case were deduced from certain properties of the heckman opdam operators ho and the existence of the shift operators heckman s interpretation of these operators via the so called dunkl operators see he and also c5 was important to our approach we note that the ho operators are closely related to the so called quantum many body problem calogero sutherland moser olshanetsky perelomov the conformal field theory knizhnik zamolodchikov equations the harmonic analysis on symmetric spaces harish chandra helgason etc and last but not the least the classic theory of the hypergeometric functions establishing the connection between the difference counterparts of heckman opdam operators introduced in c4 and the macdonald theory ml m2 including the construction of the difference shift operators is the main thrust of this paper once the connection is established it is not very difficult to calculate the scalar squares of the macdonald polynomials and to prove the constant term conjecture from his fundamental paper m3 to simplify the exposition we discuss the reduced root systems only and impose the relation q tk for k e z to avoid infinite products in the definition of macdonald s pairing the purpose of this work is to present a concrete application of the new technique arbitrary q t can be handled in a

iwahori hecke algebras for p adic loop groups this paper is a continuation of braverman and kazhdan ann math 2 174 3 1603 1642 2011 in which the first two authors have introduced the spherical hecke algebra and the satake isomorphism for an untwisted affine kac moody group over a non archimedian local field in this paper we develop the theory of the iwahori hecke algebra associated to these same groups the resulting algebra is shown to be closely related to cherednik s double affine hecke algebra furthermore using these results we give an explicit description of the affine satake isomorphism generalizing macdonald s formula for the spherical function in the finite dimensional case the results of this paper have been previously announced in braverman and kazhdan european congress of mathematics european mathematical society zurich 2014

quasiharmonic polynomials for coxeter groups and representations of cherednik algebras we introduce and study deformations of finite dimensional modules over rational cherednik algebras our main tool is a generalization of usual harmonic polynomials for each coxeter group the so called quasiharmonic polynomials a surprising application of this approach is the construction of canonical elementary symmetric polynomials and their deformations for all coxeter groups

bispectrality for the quantum ruijsenaars model and its integrable deformation an elementary construction of the eigenfunctions for the quantum rational ruijsenaars model with integer coupling parameter is presented as a by product we establish the bispectral duality between this model and the trigonometric calogero moser model in particular this gives a new way for calculating jack polynomials we propose also a certain one parameter deformation of the ruijsenaars model proving its integrability and bispectrality the generalizations related to other root systems and difference operators by macdonald are considered