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

multistability of second order competitive neural networks with nondecreasing saturated activation functions in this paper second order interactions are introduced into competitive neural networks nns and the multistability is discussed for second order competitive nns socnns with nondecreasing saturated activation functions firstly based on decomposition of state space cauchy convergence principle and inequality technique some sufficient conditions ensuring the local exponential stability of 2n equilibrium points are derived secondly some conditions are obtained for ascertaining equilibrium points to be locally exponentially stable and to be located in any designated region thirdly the theory is extended to more general saturated activation functions with 2r corner points and a sufficient criterion is given under which the socnns can have r 1 n locally exponentially stable equilibrium points even if there is no second order interactions the obtained results are less restrictive than those in some recent works finally three examples with their simulations are presented to verify the theoretical analysis

singular perturbation analysis of competitive neural networks with different time scales the dynamics of complex neural networks must include the aspects of long and short term memory the behavior of the network is characterized by an equation of neural activity as a fast phenomenon and an equation of synaptic modification as a slow part of the neural system the main idea of this paper is to apply a stability analysis method of fixed points of the combined activity and weight dynamics for a special class of competitive neural networks we present a quadratic type lyapunov function for the flow of a competitive neural system with fast and slow dynamic variables as a global stability method and a modality of detecting the local stability behavior around individual equilibrium points

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

global exponential stability of bidirectional associative memory neural networks with time delays in this paper we consider delayed bidirectional associative memory bam neural networks nns with lipschitz continuous activation functions by applying young s inequality and holder s inequality techniques together with the properties of monotonic continuous functions global exponential stability criteria are established for bam nns with time delays this is done through the use of a new lyapunov functional and an m matrix the results obtained in this paper extend and improve previous results

double products and hypersymplectic structures on ℝ4n in this paper we give a procedure to construct hypersymplectic structures on 4n beginning with affine symplectic data on 2n these structures are shown to be invariant by a 3 step nilpotent double lie group and the resulting metrics are complete and not necessarily flat explicit examples of this construction are exhibited

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

exponential stability of impulsive high order hopfield type neural networks with time varying delays this paper considers the problems of global exponential stability and exponential convergence rate for impulsive high order hopfield type neural networks with time varying delays by using the method of lyapunov functions some sufficient conditions for ensuring global exponential stability of these networks are derived and the estimated exponential convergence rate is also obtained as an illustration an numerical example is worked out using the results obtained

hadamard s theorem for locally lipschitzian maps if the derivativef p of a continuously differentiable mapffrom r into itself is an invertible linear transformation for each p then by the classical inverse function theorem f is a local homeomorphism every p has a neighborhood u mapped homeomorphically onto its image f u a natural and intriguing question under what further conditions will f be a homeomorphism of r onto itself was answered in one way by hadamard 8 in 1906 using a condition on the norms b max bh h i of the inverses b f p

toric hypersymplectic quotients we study the hypersymplectic spaces obtained as quotients of flat hypersymplectic space r 4d by the action of a compact abelian group these 4n dimensional quotients carry a multi hamilitonian action of an n torus the image of the hypersymplectic moment map for this torus action may be described by a configuration of solid cones in r 3n we give precise conditions for smoothness and non degeneracy of such quotients and show how some properties of the quotient geometry and topology are constrained by the combinatorics of the cone configurations examples are studied including non trivial structures on r 4n and metrics on complements of hypersurfaces in compact manifolds

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

adaptive lag synchronization for competitive neural networks with mixed delays and uncertain hybrid perturbations this paper investigates the problem of adaptive lag synchronization for a kind of competitive neural network with discrete and distributed delays mixed delays as well as uncertain nonlinear external and stochastic perturbations hybrid perturbations a simple but robust adaptive controller is designed such that the response system can lag synchronize with a drive system based on the lyapunov stability theory and some suitable lyapunov krasovskii functionals several sufficient conditions ensuring the lag synchronization are developed our synchronization criteria are easily verified and do not need to solve any linear matrix inequality some existing results are improved and extended moreover the designed adaptive controller has better anti interference capacity and is more practical than the usual adaptive controller numerical simulations are exploited to show the effectiveness of the theoretical results

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

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

zeros of 3 f 2 hypergeometric polynomials abstract we investigate the behaviour of the zeros of three distinct types of 3 f 2 hypergeometric polynomials each type of polynomial considered has degree 2 n and contains a free parameter b we describe the location of the zeros as b varies continuously through real values numerical tables of the zeros generated by mathematica are presented in confirmation of our stated results

a comment on on equilibria stability and instability of hopfield neural networks and reply it is pointed out that the main analysis results about the existence uniqueness and global asymptotic stability of the equilibrium of a continuous time hopfield type neural network given in the paper by zhi hong guan et al 2000 are special cases of relevant ones previously obtained in the literature in reply the original authors consider the reasoning of xue bin liang s comments and state that their analysis method is in fact different to existing ones

polya frequency sequences and real zeros of some f23 polynomials we prove that the zeros of some families of f23 hypergeometric polynomials are all real and negative this result has a connection with the theory of polya frequency sequences and functions as a consequence we establish the asymptotic distribution of these zeros when the degree of the polynomials tends to infinity

the matrix valued hypergeometric equation the hypergeometric differential equation was found by euler euler l 1769 opera omnia ser 1 11 13 and was extensively studied by gauss gauss c f 1812 comm soc reg sci ii 3 123 162 kummer kummer e j 1836 riene ang math 15 39 83 kummer e j 1836 riene ang math 15 127 172 and riemann riemann b 1857 k gess wiss 7 1 24 the hypergeometric function known also as gauss function is the unique solution of the hypergeometric equation analytic at z 0 and with value 1 at z 0 this function because of its remarkable properties has been used for centuries in the whole subject of special functions in this article we give a matrix valued analog of the hypergeometric differential equation and of gauss function one can only speculate that many of the connections that made gauss function a vital part of mathematics at the end of the 20th century will be shared by its matrix valued version discussed here

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

existence and global exponential stability of almost periodic solution for cellular neural networks with variable coefficients and time varying delays in this paper we study cellular neural networks with almost periodic variable coefficients and time varying delays by using the existence theorem of almost periodic solution for general functional differential equations introducing many real parameters and applying the lyapunov functional method and the technique of young inequality we obtain some sufficient conditions to ensure the existence uniqueness and global exponential stability of almost periodic solution the results obtained in this paper are new useful and extend and improve the existing ones in previous literature

deriving sufficient conditions for global asymptotic stability of delayed neural networks via nonsmooth analysis ii in this paper we obtain new sufficient conditions ensuring existence uniqueness and global asymptotic stability gas of the equilibrium point for a general class of delayed neural networks dnns via nonsmooth analysis which makes full use of the lipschitz property of functions defining dnns based on this new tool of nonsmooth analysis we first obtain a couple of general results concerning the existence and uniqueness of the equilibrium point then those results are applied to show that existence assumptions on the equilibrium point in some existing sufficient conditions ensuring gas are actually unnecessary and some strong assumptions such as the boundedness of activation functions in some other existing sufficient conditions can be actually dropped finally we derive some new sufficient conditions which are easy to check comparison with some related existing results is conducted and advantages are illustrated with examples throughout our paper spectral properties of the matrix a a sup spl tau play an important role which is a distinguished feature from previous studies here a and a sup spl tau are respectively the feedback and the delayed feedback matrix defining the neural network under consideration

the vortex filament in the minkowski 3 space and generalized bi schrodinger maps by extending the concept of generalized bi schrodinger maps to the case that the target manifold is a para kahler manifold we show that the third order timelike and spacelike correction models of the vortex filament in the minkowski 3 space are equivalent to the generalized bi schrodinger maps from to the hyperbolic 2 space and the de sitter 2 space respectively as a consequence all three typical second to fourth order integrable systems of the akns hierarchy are interpreted in a unified way in terms of generalized bi schrodinger maps based on this exploitation and a general discussion of the generalized bi schodinger maps from into an arbitrary riemannian surface we reveal a property of generalized bi schrodinger maps that is not admitted for schrodinger maps

global exponential stability and global convergence in finite time of delayed neural networks with infinite gain this paper introduces a general class of neural networks with arbitrary constant delays in the neuron interconnections and neuron activations belonging to the set of discontinuous monotone increasing and possibly unbounded functions the discontinuities in the activations are an ideal model of the situation where the gain of the neuron amplifiers is very high and tends to infinity while the delay accounts for the finite switching speed of the neuron amplifiers or the finite signal propagation speed it is known that the delay in combination with high gain nonlinearities is a particularly harmful source of potential instability the goal of this paper is to single out a subclass of the considered discontinuous neural networks for which stability is instead insensitive to the presence of a delay more precisely conditions are given under which there is a unique equilibrium point of the neural network which is globally exponentially stable for the states with a known convergence rate the conditions are easily testable and independent of the delay moreover global convergence in finite time of the state and output is investigated in doing so new interesting dynamical phenomena are highlighted with respect to the case without delay which make the study of convergence in finite time significantly more difficult the obtained results extend previous work on global stability of delayed neural networks with lipschitz continuous neuron activations and neural networks with discontinuous neuron activations but without delays

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

robust local stability of multilayer recurrent neural networks we derive a condition for robust local stability of multilayer recurrent neural networks with two hidden layers the stability condition follows from linking theories about linearization robustness analysis of linear systems under nonlinear perturbation and matrix inequalities a characterization of the basin of attraction of the origin is given in terms of the level set of a quadratic lyapunov function similar to the nl sub q theory the local stability is imposed around the origin and the apparent basin of attraction is made large by applying the criterion while the proven basin of attraction is relatively small due to conservatism of the criterion modification of the dynamic backpropagation by the new stability condition is discussed and illustrated by simulation examples

global exponential stability of multitime scale competitive neural networks with nonsmooth functions in this paper we study the global exponential stability of a multitime scale competitive neural network model with nonsmooth functions which models a literally inhibited neural network with unsupervised hebbian learning the network has two types of state variables one corresponds to the fast neural activity and another to the slow unsupervised modification of connection weights based on the nonsmooth analysis techniques we prove the existence and uniqueness of equilibrium for the system and establish some new theoretical conditions ensuring global exponential stability of the unique equilibrium of the neural network numerical simulations are conducted to illustrate the effectiveness of the derived conditions in characterizing stability regions of the neural network

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

on the zeros of meixner polynomials we investigate the zeros of a family of hypergeometric polynomials m n x beta c beta n 2f 1 n x beta 1 frac 1 c n in mathbb n known as meixner polynomials that are orthogonal on 0 infty with respect to a discrete measure for beta 0 and 0 1 as well as the polynomials k n x p n with non hermitian orthogonality for 0 p 1 and n n 1 n 2 ldots we also show that the polynomials m n x beta c beta in mathbb r are real rooted when c rightarrow 0

neutral calabi yau structures on kodaira manifolds we construct neutral calabi yau metrics and hypersymplectic structures on some kodaira manifolds our structures are symmetric with respect to the central tori

the geometry of recursion operators we study the fields of endomorphisms intertwining pairs of symplectic structures using these endomorphisms we prove an analogue of moser s theorem for simultaneous isotopies of two families of symplectic forms we also consider the geometric structures defined by pairs and triples of symplectic forms for which the squares of the intertwining endomorphisms are plus or minus the identity for pairs of forms we recover the notions of symplectic pairs and of holomorphic symplectic structures for triples we recover the notion of a hypersymplectic structure and we also find three new structures that have not been considered before one of these is the symplectic formulation of hyper kahler geometry which turns out to be a strict generalization of the usual definition in terms of differential or kahler geometry

multiperiodicity and attractivity of delayed recurrent neural networks with unsaturating piecewise linear transfer functions this paper studies multiperiodicity and attractivity for a class of recurrent neural networks rnns with unsaturating piecewise linear transfer functions and variable delays using local inhibition conditions for boundedness and global attractivity are established these conditions allow coexistence of stable and unstable trajectories moreover multiperiodicity of the network is investigated by using local invariant sets it shows that under some interesting conditions there exists one periodic trajectory in each invariant set which exponentially attracts all trajectories in that region correspondingly simulations are carried out to illustrate the theories

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

stability analysis for the generalized cohen grossberg neural networks with inverse lipschitz neuron activations in this paper by using nonsmooth analysis approach linear matrix inequality lmi technique topological degree theory and lyapunov krasovskii function method the issue of global exponential stability is investigated for a class of generalized cohen grossberg neural networks possessing inverse lipschitz neuron activations and nonsmooth behaved functions several novel delay dependent sufficient conditions are established towards the existence uniqueness and global exponential stability of the equilibrium point which are shown in terms of lmis it is noted that the results above require neither the lipschitz continuity of the activation functions nor the smoothness of the behaved functions also for the case of the activation function that satisfies not only the inverse lipschitz conditions but also the lipschitz conditions some conditions are derived which generalize the previous results finally two examples with their simulations are given to show the effectiveness of the theoretical results

learning continuous trajectories in recurrent neural networks with time dependent weights the paper is concerned with a general learning with arbitrary criterion and state dependent constraints of continuous trajectories by means of recurrent neural networks with time varying weights the learning process is transformed into an optimal control framework where the weights to be found are treated as controls a learning algorithm based on a variational formulation of pontryagin s maximum principle is proposed this algorithm is shown to converge under reasonable conditions to an optimal solution the neural networks with time dependent weights make it possible to efficiently find an admissible solution i e initial weights satisfying state constraints which then serves as an initial guess to carry out a proper minimization of a given criterion the proposed methodology may be directly applicable to both classification of temporal sequences and to optimal tracking of nonlinear dynamic systems numerical examples are also given which demonstrate the efficiency of the approach presented

global exponential stability of competitive neural networks with different time scales the dynamics of cortical cognitive maps developed by self organization must include the aspects of long and short term memory the behavior of such a neural network is characterized by an equation of neural activity as a fast phenomenon and an equation of synaptic modification as a slow part of the neural system we present a new method of analyzing the dynamics of a biological relevant system with different time scales based on the theory of flow invariance we are able to show the conditions under which the solutions of such a system are bounded being less restrictive than with the k monotone theory singular perturbation theory or those based on supervised synaptic learning we prove the existence and the uniqueness of the equilibrium a strict lyapunov function for the flow of a competitive neural system with different time scales is given and based on it we are able to prove the global exponential stability of the equilibrium point

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

on para kahler and hyper para kahler lie algebras abstract in this study we consider lie algebras that admit para kahler and hyper para kahler structures we provide new characterizations of these lie algebras and develop many methods for building large classes of examples previously bai considered para kahler lie algebras as left symmetric bialgebras we reconsider this viewpoint and make improvements in order to obtain some new results the study of para kahler and hyper para kahler is intimately linked to the study of left symmetric algebras particularly those that admit invariant symplectic forms in this study we provide many new classes of left symmetric algebras and complete descriptions of all the associative algebras that admit an invariant symplectic form we also describe all four dimensional hyper para kahler lie algebras