hypergeometric functions on complex matrix space the authors had presented the general foundations for a theory of hypergeometric functions of matrix argument over real division algebras here they further develop the fine structure of these functions over the complex field including series expansions integral representations asymptotics differential equations addition formulas multiplication formulas summation theorems transformation properties etc especially important are the operator valued hypergeometric functions required for nonspherical expansions such as addition formulas by the noncommutativity of matrix multiplication

a gauss newton approach to solving generalized inequalities generalized inequalities are systems of the form gx k 0 where g maps between normed linear spaces and k denotes the partial order induced by the closed convex cone k e g k r m1 0 rm2 in this paper a gauss newton type algorithm is presented for minimizing the distance function rhox mbox dist gx k inf vert gx k vert k in k the technique globalizes the well known newton methods for solving generalized inequalities and overcomes the difficulties associated with subgradient methods for the global minimization of

spherical functions on symmetric cones in this note we obtain a recursive formula for the spherical functions associated with the symmetric cone of a formally real jordan algebra we use this formula as an inspiration for a similar recursive formula involving the jack polynomials

subdifferential regularities of perturbed distance functions outside the target set in banach spaces abstract this paper is devoted to study the frechet and proximal regularities at points outside the target set of perturbed distance functions d s j determined by a closed subset s and a lipschitz function j also we provide some important results on the frechet proximal and clarke subdifferentials of d s j at those points in banach spaces

certain hypergeometric series related to the root system bc we show that the generalized hypergeometric function 2 f 1 of matrix argument is the series expansion at the origin of a special case of the hypergeometric function associated with the root system of type bc in addition we prove that the jacobi polynomials of matrix argument correspond to the jacobi polynomials associated with the root system of type bc we also give a precise relation between jack polynomials and the jacobi polynomials associated with the root system of type a as a side result one obtains generalized hook length formulas which are related to harish chandra s c function and one can prove a conjecture due to macdonald relating two inner products on a space of symmetric functions

complex bimatrix variate generalised beta distributions in this paper the study of bivariate generalised beta types i and ii distributions is extended to the complex matrix variate case for which the corresponding density functions are found in addition for complex bimatrix variate beta type i distributions several basic properties including the joint eigenvalue density and the maximum eigenvalue distribution are studied

total positivity properties of generalized hypergeometric functions of matrix argument in multivariate statistical analysis several authors have studied the total positivity properties of the generalized 0f1 hypergeometric function of two real symmetric matrix arguments in this paper we make use of zonal polynomial expansions to obtain a new proof of a result that these 0f1 functions fail to satisfy certain pairwise total positivity properties this proof extends both to arbitrary generalized rfs functions of two matrix arguments and to the generalized hypergeometric functions of hermitian matrix arguments in the case of the generalized hypergeometric functions of two hermitian matrix arguments we prove that these functions satisfy certain modified pairwise tp2 properties the proofs of these results are based on sylvester s formula for compound determinants and the condensation formula of c l dodgson lewis carroll 1866

generating functions for weighted hurwitz numbers double hurwitz numbers enumerating weighted n sheeted branched coverings of the riemann sphere or equivalently weighted paths in the cayley graph of sn generated by transpositions are determined by an associated weight generating function a uniquely determined 1 parameter family of 2d toda functions of hypergeometric type is shown to consist of generating functions for such weighted hurwitz numbers four classical cases are detailed in which the weighting is uniform okounkov s double hurwitz numbers for which the ramification is simple at all but two specified branch points the case of belyi curves with three branch points two with specified profiles the general case with a specified number of branch points two with fixed profiles the rest constrained only by the genus and the signed enumeration case with sign determined by the parity of the number of branch points using the exponentiated quantum dilogarithm function as a weight generator three new types of weighted enumerations are intr

bessel systems for jordan algebras of rank 2 and 3 special functions of matrix argument arise in a diverse range of applications in harmonic analysis number theory multivariate statistics quantum physics and molecular chemistry this paper presents series expansions for the bessel functions associated with jordan algebras of rank 2 or 3 detailed information on the domains of convergence of these series are given where horn s theorems for double and triple hypergeometric series are used

special functions of matrix argument i algebraic induction zonal polynomials and hypergeometric functions hypergeometric functions of matrix argument arise in a diverse range of applications in harmonic analysis multivariate statistics quantum physics molecular chemistry and number theory this paper presents a general theory of such functions for real division algebras these functions which generalize the classical hypergeometric functions are defined by infinite series on the space s s n f of all n x n hermitian matrices over the division algebra f the theory depends intrinsically upon the representation theory of the general linear group g gl n f of invertible n x n matrices over f and the theme of this work is the full exploitation of the inherent group theory the main technique is the use of the method of algebraic induction to realize explicitly the appropriate representations of g to decompose the space of polynomial functions on s and to describe the zonal polynomials from which the hypergeometric functions are constructed detailed descriptions of the convergence properties of the series expansions are given and integral representations are provided future papers in this series will develop the fine structure of these functions 0 introduction we begin a series of articles in which we develop the fine structure of generalized hypergeometric functions of matrix argument by fine structure we allude to the analogues of such classical results as series expansions integral formulas asymptotics differential equations summation formulas addition theorems composition formulas and recurrence relations this first paper in which we simultaneously treat real complex and quaternionic analysis is the result of our desire to present a complete theory of hypergeometric functions of matrix argument over real division algebras not only as a framework for the body of detailed results to follow in later papers but also to clarify the representationtheoretic foundation for such a theory although these hypergeometric functions are of interest on purely analytic grounds they arise in a wide range of applications indeed various classes of received by the editors may 19 1986 1980 malhema1zics s ject claszficain 1985 ren primary 22e30 22e45 33a75 43a85 43a90 62h10 secondary 20g20 32a07 32m15 44a10 62e15 key w and phrases generalized hypergeometric functions zonal polynomials representation theory algebraic induction multivariate statistics general linear group generalized gamma functions pochhammer symbols laplace transforms maximal compact subgroup invariant polynomials positive cones symmetric spaces schur functions special functions of matrix argument the first author is on leave from the university of wyoming the second author was partially supported by the national science foundation under grant mcs 8403381 and by the research council of the university of north carolina b1987 american mathematical society 0002 9947 87 1 00 25 per page

a sequential quadratic programming method for potentially infeasible mathematical programs abstract the paper describes a modification to the wilson han powell sequential quadratic programming technique that is applicable to potentially infeasible nonlinear programs the method is designed to locate stationary points of a nonlinear program in a generalized sense definition 2 8 these stationary points include the familiar kuhn tucker and fritz john points but also include what we call external and strong external stationary points the external and strong external stationary points are not feasible thus the problem may be infeasible and yet have stationary points loosely speaking an external stationary point is one that satisfies a certain first order necessary optimality condition for being as close to feasibility as possible the proposed technique is similar to those described by j v burke and s p han a modified sequential quadratic programming method manuscript and m sahba j optim theory appl 52 1987 291 309 however it is shown to possess substantially enhanced stability properties the global convergence properties of the method are described along with some continuity results for the quadratic programming subproblems

on the largest principal angle between random subspaces formulas are derived for the probability density function and the probability distribution function of the largest canonical angle between two p dirnensional subspaces of r n chosen from the uniform distribution on the grassmann manifold which is the unique distribution invariant by orthogonal transformations of r n the formulas involve the gamma function and the hypergeometric function of a matrix argument c 2005 elsevier inc all rights reserved

generic solutions for some perturbed optimization problem in non reflexive banach spaces abstract let z be a closed boundedly relatively weakly compact nonempty subset of a banach space x and j z r a lower semicontinuous function bounded from below if x 0 is a convex subset in x and x 0 has approximatively z property k then the set of all points x in x 0 z for which there exists z 0 z such that j z 0 x z 0 x and every sequence z n z satisfying lim n j z n x z n x for x contains a subsequence strongly convergent to an element of z is a dense g subset of x 0 z moreover under the assumption that x 0 is approximatively z strictly convex we show more namely that the set of all points x in x 0 z for which there exists a unique point z 0 z such that j z 0 x z 0 x and every sequence z n z satisfying lim n j z n x z n x for x converges strongly to z 0 is a dense g subset of x 0 z here x inf j z x z z z these extend s cobzas s result j math anal appl 243 2000 344 356

divergence free bases for finite element schemes in hydrodynamics in a number of finite element schemes satisfying a divergence free condition a difficulty has been the lack of explicit bases or even their dimension in the present paper we employ graph theory to calculate the dimensions of these vector spaces and to obtain bases for them

a class of iterative methods for solving saddle point problems we consider the numerical solution of indefinite systems of linear equations arising in the calculation of saddle points we are mainly concerned with sparse systems of this type resulting from certain discretizations of partial differential equations we present an iterative method involving two levels of iteration similar in some respects to the uzawa algorithm we relate the rates of convergence of the outer and inner iterations proving that under natural hypotheses the outer iteration achieves the rate of convergence of the inner iteration the technique is applied to finite element approximations of the stokes equations

the largest eigenvalue of real symmetric hermitian and hermitian self dual random matrix models with rank one external source part i we consider the limiting location and limiting distribution of the largest eigenvalue in real symmetric 1 hermitian 2 and hermitian self dual 4 random matrix models with rank 1 external source they are analyzed in a uniform way by a contour integral representation of the joint probability density function of eigenvalues assuming the one band condition and certain regularities of the potential function we obtain the limiting location of the largest eigenvalue when the nonzero eigenvalue of the external source matrix is not the critical value and further obtain the limiting distribution of the largest eigenvalue when the nonzero eigenvalue of the external source matrix is greater than the critical value when the nonzero eigenvalue of the external source matrix is less than or equal to the critical value the limiting distribution of the largest eigenvalue will be analyzed in a subsequent paper in this paper we also give a definition of the external source model for all 0

the iwasawa main conjectures for gl2 we prove the one two and three variable iwasawa greenberg main conjectures for a large class of modular forms that are ordinary with respect to an odd prime p the method of proof involves an analysis of an eisenstein ideal for ordinary hida families for gu 2 2

well posedness of a class of perturbed optimization problems in banach spaces abstract let x be a banach space and z a nonempty subset of x let j z r be a lower semicontinuous function bounded from below and p 1 this paper is concerned with the perturbed optimization problem of finding z 0 z such that x z 0 p j z 0 inf z z x z p j z which is denoted by min j x z the notions of the j strictly convex with respect to z and of the kadec with respect to z are introduced and used in the present paper it is proved that if x is a kadec banach space with respect to z and z is a closed relatively boundedly weakly compact subset then the set of all x x for which every minimizing sequence of the problem min j x z has a converging subsequence is a dense g subset of x z 0 where z 0 is the set of all points z z such that z is a solution of the problem min j z z if additionally p 1 and x is j strictly convex with respect to z then the set of all x x for which the problem min j x z is well posed is a dense g subset of x z 0

weighted laplace transforms and bessel functions on hermitian symmetric spaces this paper defines ir weighted laplace transforms on the spaces of ir covariant functions by the inverse laplace transform we define operator valued bessel functions we also study the holomorphic discrete series of the automorphism group of a siegel domain of type ii

limiting subdifferentials of perturbed distance functions in banach spaces abstract we explore in arbitrary banach spaces the frechet type e subdifferentials and the limiting subdifferentials for the perturbed distance function d s j determined by a closed subset s and a lower semicontinuous function j defined on s in particular upper and lower estimates for the frechet type e subdifferentials and for the limiting subdifferentials are provided in terms of the corresponding subdifferentials of the sum of the associated functions j and s

proximal analysis and the minimal time function under general hypotheses on the target set s and the dynamics of the system we show that the minimal time function ts cdot is a proximal solution to the hamilton jacobi equation uniqueness results are obtained with two different kinds of boundary conditions a new propagation result is proven and as an application we give necessary and sufficient conditions for ts cdot to be lipschitz continuous near s a petrov type modulus condition is also shown to be sufficient for continuity of ts cdot near s

on bessel integrals for reducible degenerate principal series representations abstract we discuss generalized bessel integrals with nondegenerate characters which are assigned to irreducible submodules of a reducible degenerate principal series representation of sp n r then we give sufficient conditions for their vanishings which are based on the signatures of the nondegenerate characters this consequently suggests a reasonable correspondence between open gl n r orbits in the set of real symmetric matrices of size n and irreducible submodules of the reducible principal series representations

a triple product identity for schur functions abstract in this paper we expand the product of n different classical theta functions into a laurent sum of schur symmetric functions of their arguments this expansion is a natural extension of the jacobi triple product identity we derive our result from a q binomial theorem for i g macdonald s schur function multiple basic hypergeometric series of matrix argument

porosity of perturbed optimization problems in banach spaces let x be a banach space and z a nonempty closed subset of x l etj z r be a lower semicontinuous function bounded from below this paper is concerned with the perturbed optimization problem infz z j z x z denoted by x j inf for x x in the case when x is compactly fully 2 convex it is proved in the present paper that the set of all points x in x for which there does not exist z0 z such that j z0 x z0 infz z j z x z is a porous set in x f urthermore ifx is assumed additionally to be compactly locally uniformly convex we verify that the set of all points x x z 0 such that the problem x j inf fails to be approximately compact is a porous set in x z 0 w herez 0 denotes the set of all z z such that z pz z moreover a counterexample to which some results of ni r x ni generic solutions for some perturbed optimization problem in nonreflexive banach space j math anal appl 302 2005 417 424 fail is provided

parallel algorithm for solving linear systems arising from pde and integral equations in this paper we consider solving matrix systems arising from the linear integral equations and pde by two recent parallel techniques the systems are indefinite due to linear constraints imposed on the fluid velocity the first approach known as the multilevel algorithm employs a hierarchical technique to compute the constrained linear space for the unknowns followed by the iterative solution of a positive definite reduced problem the second approach exploits the banded structure of sparse matrices to obtain a different reduced system which is determined by the unknowns common to adjacent block rows although the reduced system in this approach may still be indefinite the algorithm converges to the solution at an accelerated rate these methods have two desirable characteristics namely robust numerical convergence and efficient parallelizability

on differential operators attached to certain representations of classical groups classification et construction de sous espaces irreductibles de sr integrales contenant les applications polynomiales l operateur differentiel d operateurs differentiels de type arithmetique applications aux series de eisenstein

hypergeometric functions associated to cones of convex type in a euclidean jordan algebra abstract in this paper we study a family of hypergeometric functions associated to cones of convex type in a euclidean jordan algebra we obtain upper and lower bounds and give information about their singularities the family under consideration was introduced by sahi invent math 110 1992 409 418 and it plays an important role in the construction of some small representations

confluent hypergeometric functions on tube domains our problem specialized to the case of the siegel upper half space h of degree m concerns the fourier expansion of a series s z fl det z a det a a l here z is a variable on h l is a lattice in the space v of all real symmetric matrices of size m and fl ec it can be shown that this is convergent if re f l m and also that if re m 2 and r e f l m 2 it has a fourier expansion of the form

on wishart distribution some extensions this paper proposes a unified approach that enables the wishart distribution to be studied simultaneously in the real complex quaternion and octonion cases under elliptical models in particular the matrix multivariate elliptical distribution the noncentral generalised wishart distribution the joint density of the eigenvalues and the distribution of the maximum eigenvalue are obtained for real normed division algebras

operator valued bessel functions on schrodinger fock spaces and siegel domains of type ii in this paper a general theory of operator valued bessel functions on mixed schrodinger fock spaces and siegel domains of type ii is presented in general bessel functions j are associated to the partial fourier transform on a mixed schrodinger fock space j can be realized as a reduced bessel function k which is a generalized inverse laplace transform on a siegel domain of type ii the application of this theory to representation theory will be studied in the next paper

on the evaluation of some selberg like integrals abstract several methods of evaluation are presented for a family i n d p of selberg like integrals that arise in the computation of the algebraic geometric degrees of a family of spherical nilpotent orbits associated to the symmetric space of a simple real lie group adapting the technique of nishiyama ochiai and zhu we present an explicit evaluation in terms of certain iterated sums over permutation groups the resulting formula however is only valid when the integrand involves an even power of the vandermonde determinant we then apply to the general case the theory of symmetric functions and obtain an evaluation of the integral i n d p as a product of polynomial of fixed degree times a particular product of gamma factors thereby identifying the asymptotics of the integrals with respect to their parameters lastly we derive a recursive formula for evaluation of another general class of selberg like integrals by applying some of the technology of generalized hypergeometric functions

totally positive kernels polya frequency functions and generalized hypergeometric series abstract recently k i gross and the author j approx theory 59 224 246 1989 analyzed the total positivity of kernels of the form k x y p f q xy x y r wherep f q denotes a classical generalized hypergeometric series there we related the determinants which define the total positivity of k to the hypergeometric functions of matrix argument defined on the space of n n hermitian matrices in the first part of this paper we apply these methods to derive the total positivity properties of the kernels k for more general choices of the parameters in these hypergeometric series in particular we prove that if ai 0 and ki is a positive integer i 1 p then k x y p f q a 1 a p a 1 k 1 a p k p xy is strictly totally positive on r2 in the second part we use the theory of entire functions to derive some polya frequency function properties of the hypergeometric series p f q x some of these latter results suggest a curious duality with the total positivity properties described above for instance we prove that if p 1 and the ai and ki are as above then there exists a probability density function f on r such that f is a strict polya frequency function and 1 p f p a 1 k 1 a p k p a 1 a p z l f z the laplace transform of f

generic existence of solutions for some perturbed optimization problems abstract let x be a real banach space z a closed nonvoid subset of x and j z r a lower semicontinuous function bounded from below if x is reflexive and has the kadets property then the set of all x x for which there exists z0 z such that j z0 x z0 inf j z x z z z is g and dense in x this extends a result of k s lau 1978 indiana univ math j 27 791 795 on nearest points the case j 0