a relativistic hypergeometric function we survey our work on a function generalizing 2f1 this function is a joint eigenfunction of four askey wilson type hyperbolic difference operators reducing to the askey wilson polynomials for certain discrete values of the variables it is defined by a contour integral generalizing the barnes representation of 2f1 it has various symmetries including a hidden d4 symmetry in the parameters by means of the associated hilbert space transform the difference operators can be promoted to self adjoint operators provided the parameters vary over a certain polytope in the parameter space for a dense subset of parameter shifts give rise to an explicit evaluation in terms of rational functions of exponentials hyperbolic functions and plane waves

general β jacobi corners process and the gaussian free field we prove that the two dimensional gaussian free field describes the asymptotics of global fluctuations of a multilevel extension of the general jacobi random matrix ensembles our approach is based on the connection of the jacobi ensembles to a degeneration of the macdonald processes that parallels the degeneration of the macdonald polynomials to the heckman opdam hypergeometric functions of type a we also discuss the limit 2015 wiley periodicals inc

duality between the trigonometric bcn sutherland system and a completed rational ruijsenaars schneider van diejen system we present a new case of duality between integrable many body systems where two systems live on the action angle phase spaces of each other in such a way that the action variables of each system serve as the particle positions of the other one our investigation utilizes an idea that was exploited previously to provide group theoretic interpretation for several dualities discovered originally by ruijsenaars in the group theoretic framework one applies hamiltonian reduction to two abelian poisson algebras of invariants on a higher dimensional phase space and identifies their reductions as action and position variables of two integrable systems living on two different models of the single reduced phase space taking the cotangent bundle of u 2n as the upstairs space we demonstrate how this mechanism leads to a new dual pair involving the bcn trigonometric sutherland system thereby we generalize earlier results pertaining to the an trigonometric sutherland system as well as a recent work by pusztai on the hyperbolic bcn sutherland system

reproducing kernel particle method for free and forced vibration analysis abstract a reproducing kernel particle method rkpm is presented to analyze the natural frequencies of euler bernoulli beams as well as kirchhoff plates in addition rkpm is also used to predict the forced vibration responses of buried pipelines due to longitudinal travelling waves two different approaches lagrange multipliers as well as transformation method are employed to enforce essential boundary conditions based on the reproducing kernel approximation the domain of interest is discretized by a set of particles without the employment of a structured mesh which constitutes an advantage over the finite element method meanwhile rkpm also exhibits advantages over the classical rayleigh ritz method and its counterparts numerical results presented here demonstrate the effectiveness of this novel approach for both free and forced vibration analysis

the finite toda lattices connection is established between one dimensional toda lattices constructed on the basis of the systems of simple roots of classical and affine lie algebras and other integrable systems of interacting particles that connection allows us to find new lattices differing from the known ones by the interaction of particles near the ends some of the new lattices admit non abelian generalizations

reproducing kernel particle methods for large deformation analysis of non linear structures abstract large deformation analysis of non linear elastic and inelastic structures based on reproducing kernel particle methods rkpm is presented the method requires no explicit mesh in computation and therefore avoids mesh distortion difficulties in large deformation analysis the current formulation considers hyperelastic and elasto plastic materials since they represent path independent and path dependent material behaviors respectively in this paper a material kernel function and an rkpm material shape function are introduced for large deformation analysis the support of the rkpm material shape function covers the same set of particles during material deformation and hence no tension instability is encountered in the large deformation computation the essential boundary conditions are introduced by the use of a transformation method the transformation matrix is formed only once at the initial stage if the rkpm material shape functions are employed the appropriate integration procedures for the moment matrix and its derivative are studied from the standpoint of reproducing conditions in transient problems with an explicit time integration method the lumped mass matrices are constructed at nodal coordinate so that masses are lumped at the particles several hyperelasticity and elasto plasticity problems are studied to demonstrate the effectiveness of the method the numerical results indicated that rkpm handles large material distortion more effectively than finite elements due to its smoother shape functions and consequently provides a higher solution accuracy under large deformation unlike the conventional finite element approach the nodal spacing irregularity in rkpm does not lead to irregular mesh shape that significantly deteriorates solution accuracy no volumetric locking is observed when applying non linear rkpm to nearly incompressible hyperelasticity and perfect plasticity problems further model adaptivity in rkpm can be accomplished simply by adding more points in the highly deformed areas without remeshing

differential equations in the spectral parameter we determine all the potentialsv x for the schrodinger equation 2 v x k2 such that some family of eigenfunctions satisfies a differential equation in the spectral parameterk of the formb k k o x o for each suchv x we determine the algebra of all possible operatorsb and the corresponding functions x

integral representations for the eigenfunctions of quantum open and periodic toda chains from the qism formalism the integral representations for the eigenfunctions of n particle quantum open and periodic toda chains are constructed within the framework of the quantum inverse scattering method both periodic and open n particle solutions have essentially the same structure being written as a generalized fourier transform over the eigenfunctions of the n 1 particle open toda chain with the kernels satisfying the baxter equations of second and first order respectively in the latter case this leads to recurrent relations which result in a representation of mellin burnes type solutions of an open chain as a byproduct we obtain the gindikin karpelevich formula for the harish chandra function in the case of the gl n group

comultiplication for shifted yangians and quantum open toda lattice abstract we study a coproduct in type a quantum open toda lattice in terms of a coproduct in the shifted yangian of sl 2 at the classical level this corresponds to the multiplication of scattering matrices of euclidean s u 2 monopoles we also study coproducts for shifted yangians for any simply laced lie algebra

a new approach for numerical modal analysis using the element free method a novel approach in analysing the vibration of elastic body systems is presented and discussed in this paper this method uses a meshless spatial approximation based only on nodes which constitutes an advantage over the finite element method as it is applicable to an elastic body with arbitrary shape it also has advantages over the classical rayleigh ritz method and its extensions the paper is organized as follows in section 2 hamilton s principle is used for obtaining the equations of motion for a three dimensonal simply connected elastic body in section 3 the development of weighted base functions for the moving least squares interpolant is described in section 4 the mass and stiffness matrices are derived using the previous results in section 5 a system synthesis description is given in which we also develop the approach used for integrating the boundary conditions finally in section 6 the theoretical results are validated by case studies chosen to highlight various features of the approach including optimization of the shape function parameters the particle method has already been used for structural dynamics liu et al international journal for numerical methods in engineering 1995 38 1655 1679 but for free vibration of beams and plates to the authors knowledge this paper is the first application of the element free method to modal analysis copyright c 1999 john wiley sons ltd

integrability of difference calogero moser systems a general class of n particle difference calogero moser systems with elliptic potentials is introduced besides the step size and two periods the hamiltonian depends on nine coupling constants we prove the quantum integrability of the model for n 2 and present partial results for n 3 in degenerate cases rational hyperbolic or trigonometric limit the integrability follows for arbitrary particle number from previous work connected with the multivariable q polynomials of koornwinder and macdonald liouville integrability of the corresponding classical systems follows as a corollary limit transitions lead to various well known models such as the nonrelativistic calogero moser systems associated with classical root systems and the relativistic calogero moser system

difference calogero moser systems and finite toda chains limits of a recently introduced n particle difference calogero moser system with elliptic potentials are studied we obtain hyperbolic and rational difference calogero moser systems with an eight parameter external field and finite difference toda chains with four parameter potentials acting on the boundary particles hamiltonians for a number of known integrable n particle systems such as ruijsenaars relativistic calogero moser and toda models and their generalizations associated with classical root systems can be seen as special cases of the hamiltonians considered in this paper

action angle maps and scattering theory for some finite dimensional integrable systems i the pure soliton case we construct an action angle transformation for the calogero moser systems with repulsive potentials and for relativistic generalizations thereof this map is shown to be closely related to the wave transformations for a large classl of hamiltonians and is shown to have remarkable duality properties all dynamics inl lead to the same scattering transformation which is obtained explicitly and exhibits a soliton structure an auxiliary result concerns the spectral asymptotics of matrices of the formm exp td ast it pertains to diagonal matricesd whose diagonal elements have pairwise different real parts and to matricesm for which certain principal minors are non zero

complete integrability of relativistic calogero moser systems and elliptic function identities poincare invariant generalizations of the galilei invariant calogero moser a particle systems are studied a quantization of the classical integrals s1 sn is presented such that the operators si9 sn mutually commute as a corollary it follows that si9 9sn poisson commute these results hinge on functional equations satisfied by the weierstrass and 0 functions a generalized cauchy identity involving the function leads to an n x n matrix l whose symmetric functions are proportional to sl5 9sn

wave functions for quantum integrable particle systems via partial confluences of multivariate hypergeometric functions abstract starting from the hyperoctahedral multivariate hypergeometric function of heckman and opdam associated with the b c n root system we arrive via partial confluent limits in the sense of oshima and shimeno at solutions of the eigenvalue equations for the quantum toda chain with one sided boundary perturbations of poschl teller type and for the hyperbolic quantum calogero sutherland system in a morse potential with the aid of corresponding degenerations of the bispectral dual difference equations for the heckman opdam hyperoctahedral hypergeometric function it is deduced that the eigensolutions in question are holomorphic in the spectral variable

on the duality between the hyperbolic sutherland and the rational ruijsenaars schneider models we consider two families of commuting hamiltonians on the cotangent bundle of the group and show that upon an appropriate single symplectic reduction they descend to the spectral invariants of the hyperbolic sutherland and of the rational ruijsenaars schneider lax matrices respectively the duality symplectomorphism between these two integrable models that was constructed by ruijsenaars using direct methods can then be interpreted geometrically simply as a gauge transformation connecting two cross sections of the orbits of the reduction group

aspects of the inverse problem for the toda chain we generalize babelon s approach to equations in dual variables so as to be able to treat new types of operators which we build out of the sub constituents of the model s monodromy matrix further we also apply sklyanin s recent monodromy matrix identities so as to obtain equations in dual variables for yet other operators the schemes discussed in this paper appear to be universal and thus in principle applicable to many models solvable through the quantum separation of variables

difference equation for the heckman opdam hypergeometric function and its confluent whittaker limit abstract we present an explicit difference equation for the heckman opdam hypergeometric function associated with root systems via a confluent hypergeometric limit an analogous difference equation is obtained for the class one whittaker function diagonalizing the open quantum toda chain

properties of some families of hypergeometric orthogonal polynomials in several variables limiting cases are studied of the koornwinder macdonald multivariable generalization of the askey wilson polynomials we recover recently and not so recently introduced families of hypergeometric orthogonal polynomials in several variables consisting of multivariable wilson continuous hahn and jacobi type polynomials respectively for each class of polynomials we provide systems of difference or differential equations recurrence relations and expressions for the norms of the polynomials in terms of the norm of the constant polynomial

on the accuracy and stability of several widely used fdtd approaches for modeling lorentz dielectrics a rigorous and comparative study on the approximation accuracy and stability limits of several widely used finite difference time domain fdtd approaches namely the auxiliary differential equation ade approach the bilinear transform bt approach the z transform approach zt and the piecewise linear recursive convolution plrc approach for modeling dispersive lorentz dielectrics is presented following the given updating equations between the electric flux density and the electric field intensity we find the zt approach with modified material parameters is much more accurate than the original zt approach and the other three approaches for modeling lorentz dielectrics the stability limits of the ade zt and plrc approaches in simulating lorentz dielectrics are also shown to be a bit more stringent than that of bt approach which preserves the courant stability limit as previously reported

positive convolution structure for a class of heckman opdam hypergeometric functions of type bc abstract in this paper we derive explicit product formulas and positive convolution structures for three continuous classes of heckman opdam hypergeometric functions of type bc for specific discrete series of multiplicities these hypergeometric functions occur as the spherical functions of non compact grassmann manifolds g k over one of the skew fields f r c h we write the product formula of these spherical functions in an explicit form which allows analytic continuation with respect to the parameters in each of the three cases we obtain a series of hypergroup algebras which include the commutative convolution algebras of k biinvariant functions on g as special cases the characters are given by the associated hypergeometric functions

hypercomplex fock states for discrete electromagnetic schrodinger operators a bayesian probability perspective abstract we present and study a new class of fock states underlying to discrete electromagnetic schrodinger operators from a multivector calculus perspective this naturally lead to hypercomplex versions of poisson charlier polynomials meixner polynomials among other ones the foundations of this work are based on the exploitation of the quantum probability formulation a la dirac to the setting of bayesian probabilities on which the fock states arise as discrete quasi probability distributions carrying a set of independent and identically distributed i i d random variables by employing mellin barnes integrals in the complex plane we obtain counterparts for the well known multidimensional poisson and hypergeometric distributions as well as quasi probability distributions that may take negative or complex values on the lattice h z n

relativistic toda systems we present and study poincare invariant generalizations of the galilei invariant toda systems the classical nonperiodic systems are solved by means of an explicit action angle transformation

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