rational kernels theory and algorithms many classification algorithms were originally designed for fixed size vectors recent applications in text and speech processing and computational biology require however the analysis of variable length sequences and more generally weighted automata an approach widely used in statistical learning techniques such as support vector machines svms is that of kernel methods due to their computational efficiency in high dimensional feature spaces we introduce a general family of kernels based on weighted transducers or rational relations rational kernels that extend kernel methods to the analysis of variable length sequences or more generally weighted automata we show that rational kernels can be computed efficiently using a general algorithm of composition of weighted transducers and a general single source shortest distance algorithm not all rational kernels are positive definite and symmetric pds or equivalently verify the mercer condition a condition that guarantees the convergence of training for discriminant classification algorithms such as svms we present several theoretical results related to pds rational kernels we show that under some general conditions these kernels are closed under sum product or kleene closure and give a general method for constructing a pds rational kernel from an arbitrary transducer defined on some non idempotent semirings we give the proof of several characterization results that can be used to guide the design of pds rational kernels we also show that some commonly used string kernels or similarity measures such as the edit distance the convolution kernels of haussler and some string kernels used in the context of computational biology are specific instances of rational kernels our results include the proof that the edit distance over a non trivial alphabet is not negative definite which to the best of our knowledge was never stated or proved before rational kernels can be combined with svms to form efficient and powerful techniques for a variety of classification tasks in text and speech processing or computational biology we describe examples of general families of pds rational kernels that are useful in many of these applications and report the result of our experiments illustrating the use of rational kernels in several difficult large vocabulary spoken dialog classification tasks based on deployed spoken dialog systems our results show that rational kernels are easy to design and implement and lead to substantial improvements of the classification accuracy

theta lifting of nilpotent orbits for symmetric pairs we consider a reductive dual pair g g in the stable range with g the smaller member and of hermitian symmetric type we study the theta lifting of nilpotent k c orbits where k is a maximal compact subgroup of g and we describe the precise k c module structure of the regular function ring of the closure of the lifted nilpotent orbit of the symmetric pair g k as an application we prove sphericality and normality of the closure of certain nilpotent k c orbits obtained in this way we also give integral formulas for their degrees

theta lifting of unitary lowest weight modules and their associated cycles we consider a reductive dual pair g g in the stable range with g the smaller member and of hermitian symmetric type we study theta lifting of holomorphic nilpotent k c orbits in relation to theta lifting of unitary lowest weight modules of g we determine the associated cycles of all such representations in particular we prove that the multiplicity in the associated cycle is preserved under theta lifting

multivariate hermite interpolation by algebraic polynomials a survey abstract this is a survey of that theory of multivariate lagrange and hermite interpolation by algebraic polynomials which has been developed in the past 20 years its purpose is not to be encyclopedic but to present the basic concepts and techniques which have been developed in that period of time and to illustrate them with examples it takes classical hermite interpolation as a starting point but then successively broadens the assumptions so that finally interpolation of arbitrary functionals and the theory of singularities from algebraic geometry is discussed

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

nilpotent orbits and complex dual pairs w x x w w w w x g end w and similarly for our main theorem describes the behaviour of closures of nilpotent orbits under the action of moment maps it is easy to see that for a nilpotent coadjoint orbit o g the set 1 o is the union of nilpotent coadjoint orbits in g it turns out that it is a closure of a single orbit theorem 1 1 let o g be a nilpotent coadjoint orbit there exists a unique nilpotent coadjoint orbit o g such that 1 o o

transcending classical invariant theory let sp2 r sp2 sp be the real symplectic group of rank n let sp denote the two fold cover of sp there is a unitary representation constructed by shale sh and weil wa of sp that is of considerable interest in the theory of automorphic forms we denote this representation by co and call it the oscillator representation the purpose of this paper is to establish about c a fact that should be useful for clarifying the structure of spaces of automorphic forms constructed using co we begin by formulating the result let e be a reductive subgroup of sp and let e denote the inverse image of e in sp denote by r e the set of infinitesimal equivalence classes of continuous irreducible admissible representations of e on locally convex spaces let co be the smooth representation of sp associated to co let co be realized on a hilbert space and let the subspace of smooth vectors on which space co is defined be written denote by r9 e co the set of elements of e which are realized as quotients by co e invariant closed subspaces of t g consider a reductive dual pair g g c sp h2 it is not hard to show that g and g commute with one another hence we may regard co i g g as a representation of gx g it is well known f that r gx g s9 g xr9 g the identification associates to p e r g and p e i g the tensor product p x p we note that p x p is not defined as a topological vector space nevertheless the infinitesimal equivalence class of p op is well defined select p x p e g x g suppose that in fact p p e 9 g g c then clearly p e r g c and p e 9 g c hence g co defines the graph of a correspondence between certain subsets of r9 g cv and j g co in fact the situation is quite precise

learning state machine based string edit kernels during the past few years several works have been done to derive string kernels from probability distributions for instance the fisher kernel uses a generative model m e g a hidden markov model and compares two strings according to how they are generated by m on the other hand the marginalized kernels allow the computation of the joint similarity between two instances by summing conditional probabilities in this paper we adapt this approach to edit distance based conditional distributions and we present a way to learn a new string edit kernel we show that the practical computation of such a kernel between two strings x and x built from an alphabet s requires i to learn edit probabilities in the form of the parameters of a stochastic state machine and ii to calculate an infinite sum over s by resorting to the intersection of probabilistic automata as done for rational kernels we show on a handwritten character recognition task that our new kernel outperforms not only the state of the art string kernels and string edit kernels but also the standard edit distance used by a neighborhood based classifier

singular unitary representations of classical groups on presente une description de la partie relativement singuliere du dual unitaire d un groupe classique de type i sur un corps local f de caracteristique autre que 2

the selberg jack symmetric functions abstract k aomoto has recently given a simple proof of an extension of a selberg s integral we prove the following generalization of aomoto s theorem for eachk 0 there exists a family sk t of homogeneous symmetric functions such that if the integrand of selberg s integral is multiplied bysk t1 tn then the integral has a certain closed form sincesk t is a renormalized jack symmetric function with 1 k we call sk t the selberg jack symmetric functions r p stanley and i g macdonald have extended many of the basic properties of the i j schur functions to the jack symmetric functions stanley s extensions of the pieri formula in a form obtained by macdonald s extension of the duality property and the combinatorial representation arise naturally in our proof we give proofs of these and other results of stanley and macdonald we give two constant term identities which are equivalent to the integration formula these results extend w g morris theorem foranand the macdonald morris constant term identity forbcn by exploring selberg s proof of his integral we give a recurrence relation outline a method for calculating the selberg jack symmetric functions and show that a constant term orthogonality relation recently proved by macdonald arises naturally we give an expansion which allows us to evaluate the normalization of this orthogonality thus giving an alternative proof of the integration formula

multiplicity free actions and the geometry of nilpotent orbits let g be a reductive lie group take a maximal compact subgroup k of g and denote their lie algebras by mathfrak g 0 and mathfrak k 0 respectively we get a cartan decomposition mathfrak g 0 mathfrak k 0 oplus mathfrak s 0 let mathfrak g be the complexification of mathfrak g 0 and mathfrak g mathfrak k oplus mathfrak s the complexified decomposition the adjoint action restricted to k preserves the space mathfrak s 0 hence k mathbb c acts on mathfrak s where k mathbb c denotes the complexification of k in this paper we consider a series of small nilpotent k mathbb c orbits in mathfrak s which are obtained from the dual pair g g o p q sp 2 n r r howe transcending classical invariant theory j amer math soc 2 1989 no 3 535 552 we explain astonishing simple structures of these nilpotent orbits using generalized null cones for example these orbits have a linear ordering with respect to the closure relation and k mathbb c acts on them in multiplicity free manner we clarify the k mathbb c module structure of the regular function ring of the closure of these nilpotent orbits in detail and prove the normality all these results naturally comes from the analysis on the null cone mathfrak n in a matrix spacew and the double fibration of nilpotent orbits in mathfrak s and mathfrak s the classical invariant theory assures that the regular functions on our nilpotent orbits are coming from harmonic polynomials on w with repspect to k mathfrak c or k mathfrak c we also provide many interesting examples of multiplicity free actions on conic algebraic varieties

multivariate delta goncarov and abel polynomials abstract classical goncarov polynomials are polynomials which interpolate derivatives delta goncarov polynomials are polynomials which interpolate delta operators e g forward and backward difference operators we extend fundamental aspects of the theory of classical bivariate goncarov polynomials and univariate delta goncarov polynomials to the multivariate setting using umbral calculus after introducing systems of delta operators we define multivariate delta goncarov polynomials show that the associated interpolation problem is always solvable and derive a generating function an appell relation for them we show that systems of delta goncarov polynomials on an interpolation grid z r d are of binomial type if and only if z a n d for some d d matrix a this motivates our definition of delta abel polynomials to be exactly those delta goncarov polynomials which are based on such a grid finally compact formulas for delta abel polynomials in all dimensions are given for separable systems of delta operators this recovers a former result for classical bivariate abel polynomials and extends previous partial results for classical trivariate abel polynomials to all dimensions

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

asymptotic cone of semisimple orbits for symmetric pairs abstract let g be a reductive algebraic group over c and denote its lie algebra by g let o h be a closed g orbit through a semisimple element h g by a result of borho and kraft 1979 4 it is known that the asymptotic cone of the orbit o h is the closure of a richardson nilpotent orbit corresponding to a parabolic subgroup whose levi component is the centralizer z g h in g in this paper we prove an analogue on a semisimple orbit for a symmetric pair more precisely let be an involution of g and k g a fixed point subgroup of then we have a cartan decomposition g k s of the lie algebra g lie g which is the eigenspace decomposition of on g let x h y be a normal sl 2 triple where x y s are nilpotent and h k semisimple in addition we assume x y where x denotes the complex conjugation which commutes with then a 1 x y is a semisimple element in s and we can consider a semisimple orbit ad k a in s which is closed our main result asserts that the asymptotic cone of ad k a in s coincides with ad g x s if x is even nilpotent

unipotent representations attached to spherical nilpotent orbits a coadjoint nilpotent orbit inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i of a complex semisimple lie group g is said to be spherical if it contains an open orbit of a borel subgroup we determine when and how to attach unitary representations to such an orbit for the real orthogonal and symplectic groups our results actually extend to a larger class of nilpotent coadjoint orbits

edit distance based kernel functions for structural pattern classification a common approach in structural pattern classification is to define a dissimilarity measure on patterns and apply a distance based nearest neighbor classifier in this paper we introduce an alternative method for classification using kernel functions based on edit distance the proposed approach is applicable to both string and graph representations of patterns by means of the kernel functions introduced in this paper string and graph classification can be performed in an implicit vector space using powerful statistical algorithms the validity of the kernel method cannot be established for edit distance in general however by evaluating theoretical criteria we show that the kernel functions are nevertheless suitable for classification and experiments on various string and graph datasets clearly demonstrate that nearest neighbor classifiers can be outperformed by support vector machines using the proposed kernel functions

remarks on classical invariant theory a uniform formulation applying to all classical groups simultaneously of the first fundamental theory of classical invariant theory is given in terms of the weyl algebra the formulation also allows skew symmetric as well as symmetric variables examples illustrate the scope of this formulation