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

on barnes multiple zeta and gamma functions abstract we show how various known results concerning the barnes multiple zeta and gamma functions can be obtained as specializations of simple features shared by a quite extensive class of functions the pertinent functions involve laplace transforms and their asymptotics is obtained by exploiting this we also demonstrate how barnes multiple zeta and gamma functions fit into a recently developed theory of minimal solutions to first order analytic difference equations both of these new approaches to the barnes functions give rise to novel integral representations

properties of generalized univariate hypergeometric functions based on spiridonov s analysis of elliptic generalizations of the gauss hypergeometric function we develop a common framework for 7 parameter families of generalized elliptic hyperbolic and trigonometric univariate hypergeometric functions in each case we derive the symmetries of the generalized hypergeometric function under the weyl group of type e 7 elliptic hyperbolic and of type e 6 trigonometric using the appropriate versions of the nassrallah rahman beta integral and we derive contiguous relations using fundamental addition formulas for theta and sine functions the top level degenerations of the hyperbolic and trigonometric hypergeometric functions are identified with ruijsenaars relativistic hypergeometric function and the askey wilson function respectively we show that the degeneration process yields various new and known identities for hyperbolic and trigonometric special functions we also describe an intimate connection between the hyperbolic and trigonometric theory which yields an expression of the hyperbolic hypergeometric function as an explicit bilinear sum in trigonometric hypergeometric functions

parameter shifts d4 symmetry and joint eigenfunctions for commuting askey wilson type difference operators in previous papers we studied a generalized hypergeometric function that is a joint eigenfunction of four hyperbolic askey wilson difference operators using parameter shifts related to the d4 weight lattice we show that this function is the even linear combination of two elementary joint eigenfunctions for a dense subset of the parameter space more specifically the latter eigenfunctions are equal to a plane wave multiplied by hyperbolic functions

random walks in cones we study the asymptotic behavior of a multidimensional random walk in a general cone we find the tail asymptotics for the exit time and prove integral and local limit theorems for a random walk conditioned to stay in a cone the main step in the proof consists in constructing a positive harmonic function for our random walk under minimal moment restrictions on the increments for the proof of tail asymptotics and integral limit theorems we use a strong approximation of random walks by brownian motion for the proof of local limit theorems we suggest a rather simple approach which combines integral theorems for random walks in cones with classical local theorems for unrestricted random walks we also discuss some possible applications of our results to ordered random walks and lattice path enumeration

a generalized hypergeometric function satisfying four analytic difference equations of askey wilson type the hypergeometric function 2 f 1 can be written in terms of a contour integral involving gamma functions we generalize this barnes representation by using a certain generalized gamma function as a building block in this way we obtain a new 2 f 1 generalization with various symmetry features we determine the analyticity properties of the r function in all of its eight arguments and show that it is a joint eigenfunction of four distinct askey wilson type difference operators two acting on v and two on the askey wilson polynomials can be obtained by a suitable discretization of v or

a generalized hypergeometric function ii asymptotics and d 4 symmetry in previous work we introduced and studied a function r a a bf c v hat v that generalizes the hypergeometric function in this paper we focus on a similarity transformed function mathcal e a a gamma v hat v with parameters 4 related to the couplings c 4 by a shift depending on a a we show that the function is invariant under all maps w with w in the weyl group of type d 4 choosing a a positive and bf gamma hat v real we obtain detailed information on the re v asymptotics of the function in particular we explicitly determine the leading asymptotics in terms of plane waves and the c function that implements the similarity r

kernel functions and backlund transformations for relativistic calogero moser and toda systems we obtain kernel functions associated with the quantum relativistic toda systems both for the periodic version and for the nonperiodic version with its dual this involves taking limits of previously known results concerning kernel functions for the elliptic and hyperbolic relativistic calogero moser systems we show that the special kernel functions at issue admit a limit that yields generating functions of backlund transformations for the classical relativistic calogero moser and toda systems we also obtain the nonrelativistic counterparts of our results which tie in with previous results in the literature

hyperbolic beta integrals abstract hyperbolic beta integrals are analogues of euler s beta integral in which the role of euler s gamma function is taken over by ruijsenaars hyperbolic gamma function they may be viewed as q q bibasic analogues of the beta integral in which the two bases q and q are interrelated by modular inversion and they entail q analogues of the beta integral for q 1 the integrals under consideration are the hyperbolic analogues of the ramanujan integral the askey wilson integral and the nassrallah rahman integral we show that the hyperbolic nassrallah rahman integral is a formal limit case of spiridonov s elliptic nassrallah rahman integral

first order analytic difference equations and integrable quantum systems we present a new solution method for a class of first order analytic difference equations the method yields explicit minimal solutions that are essentially unique special difference equations give rise to minimal solutions that may be viewed as generalized gamma functions of hyperbolic trigonometric and elliptic type euler s gamma function being of rational type we study these generalized gamma functions in considerable detail the scattering and weight functions u and w functions associated to various integrable quantum systems can be expressed in terms of our generalized gamma functions we obtain detailed information on these u and w functions exploiting the difference equations they satisfy

gambler s ruin and the first exit position of random walk from large spheres let t r be the first time a sum s n of nondegenerate i i d random vectors in r d leaves the sphere of radius r in some given norm we characterize in terms of the distribution of the individual summands the following probabilistic behavior s tr s tr has no subsequential weak limit supported on a closed half space in one dimension this result solves a very general form of the gambler s ruin problem we also characterize the existence of degenerate limits and obtain analogous results for triangular arrays along any subsequence r k finally we compute the limiting joint distribution of s tr r s tr s tr

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

a combined octree delaunay method for fully automatic 3 d mesh generation fully automatic three dimensional mesh generation is a fundamental requirement for automating the numerical solution of partial differential equations two techniques in particular the octree and delaunay approaches have been used towards this end a method that combines both approaches to fully automatic mesh generation is presented here the resulting algorithm provides the linear growth rate and divide and conquer approach of the octree method with the simplicity and optimal properties of the delaunay triangulation

advanced general purpose computational mechanics system for large scale analysis and design abstract we have been developing an advanced general purpose computational mechanics system named adventure which is designed to be able to analyze a model of arbitrary shape with a 10 100 million degrees of freedom dofs mesh and additionally to enable parametric and nonparametric shape optimization domain decomposition based parallel algorithms are implemented in pre processes domain decomposition main processes system matrix assembling and solutions and post process visualization respectively the hierarchical domain decomposition method with a pre conditioned iterative solver hddm is adopted in the main processes as one of the major solution techniques module based architecture of the system with standardized i o format and libraries are also developed and employed to attain flexibility portability extensibility and maintainability of the whole system this paper describes some key technologies employed in the system and shows some latest results including elastic stress analysis of a precise three dimensional 3d model of a nuclear reactor vessel with a 60 million dof mesh on hitachi sr2201 1024 pes and nonparametric shape optimization of a support structure for an express way with a one million dof mesh on a pc cluster 10 pes

a generalized hypergeometric function iii associated hilbert space transform for generic parameters a a c 0 2 4 we associate a hilbert space transform to the relativistic hypergeometric function r a a bf c v hat v studied in previous papers restricting the couplings c to a certain polytope we show that the renormalized r function kernel gives rise to an isometry from the even subspace of l 2 mathbb r hat w hat v d hat v to the even subspace of l2 w v dv where hat w hat v and w v are positive and even weight functions we prove that the orthogonal complement of the range of this isometry is spanned by n pairwise orthogonal functions the latter are in essence askey wilson polynomials arising from the r function by choosing hat v i kappa n with kappa 0 ldots kappa n 1 distinct negative numbers the two commuting analytic difference operators acting on the variable v for which r is a joint eigenfunction give rise to two commuting self adjoint hamiltonians on the even subspace of l2 w v dv we explicitly determine the relation of the time dependent scattering theory for these dynamics to their joint spectral transform

some functions that generalize the askey wilson polynomials we determine all biinfinite tridiagonal matrices for which some family of eigenfunctions are also eigenfunctions of a second order q difference operator the solution is described in terms of an arbitrary solution of a q analogue of gauss hypergeometric equation depending on five free parameters and extends the four dimensional family of solutions given by the askey wilson polynomials there is some evidence that this bispectral problem for an arbitrary order q difference operator is intimately related with some q deformation of the toda lattice hierarchy and its virasoro symmetries when tridiagonal matrices are replaced by the schroedinger operator and q 1 this statement holds with toda replaced by kdv in this context this paper determines the analogs of the bessel and airy potentials

ruijsenaars hypergeometric function and the modular double of u q sl 2 c abstract simultaneous eigenfunctions of two askey wilson second order difference operators are constructed as formal matrix coefficients of the principal series representation of the modular double of the quantized universal enveloping algebra u q sl 2 c these eigenfunctions are shown to be equal to ruijsenaars hypergeometric function under a proper parameter correspondence

the compaction of aggregates of non spherical linear viscous particles an approximate discrete element model is used to study the compaction of aggregates of non spherical linear viscous particles the numerical model is developed in which each particle is treated individually with the appropriate constitutive relations kinematic conditions contact constraints and elimination of overlap satisfied for every particle in the aggregate mesh update algorithms are generated based on mixed triangular and quadrilateral element system the algorithms are used to detect contact generate mesh modify concave elements and prevent locking the method could be used in a variety of problems that can be represented using granular media such as asphalt polymers aluminum snow food product and others a series of numerical simulations are shown to demonstrate the efficiency and accuracy of the model

automatic mesh generation of quadrilateral elements using intelligent local approach this paper describes a new automated mesh generation method named intelligent local approach ila which well controls both size and aspect ratio of quadrilateral element in a two dimensional plane the ila can be straightforwardly extended to hexahedral elements in a three dimensional solid here elements are created sequentially considering local information on geometrical constraints and user s demand on quality of elements a user can specify the following two numerical fields a a field of element size and b that of element aspect ratio a field of priority of element creation is also specified within the ila to efficiently deal with various complicated geometrical constraints for high quality quadrilateral elements a fuzzy knowledge processing technique is effectively utilized the ila is implemented using an object oriented technique fundamental performances of the ila are demonstrated in detail through generating several quadrilateral meshes

an element free smoothed radial point interpolation method efs rpim for 2d and 3d solid mechanics problems abstract this paper presents a novel element free smoothed radial point interpolation method efs rpim for solving 2d and 3d solid mechanics problems the idea of the present technique is that field nodes and smoothing cells scs used for smoothing operations are created independently and without using a background grid which saves tedious mesh generation efforts and makes the pre process more flexible in the formulation we use the generalized smoothed galerkin gs galerkin weak form that requires only discrete values of shape functions that can be created using the rpim by varying the amount of nodes and scs as well as their ratio the accuracy can be improved and upper bound or lower bound solutions can be obtained by design the scs can be of regular or irregular polygons in this work we tested triangular quadrangle n sided polygon and tetrahedron as examples stability condition is examined and some criteria are found to avoid the presence of spurious zero energy modes this paper is the first time to create gs galerkin weak form models without using a background mesh that tied with nodes and hence the efs rpim is a true meshfree approach the proposed efs rpim is so far the only technique that can offer both upper and lower bound solutions numerical results show that the efs rpim gives accurate results and desirable convergence rate when comparing with the standard finite element method fem and the cell based smoothed fem cs fem

exit times of n dimensional random walks we study the exit time t of a sum of independent identically distributed random vectors x1 x2 from a subset r of n dimensional euclidean space when n 2 we assume that r is invariant under positive dilations and that the boundary of r satisfies certain regularity conditions the random vector x1 is to have mean zero and a nonsingular covariance matrix we show that there is a critical exponent e independent of x1 x2 such that 0 e implies etp 2