Symmetric Stable Processes in Cones

We study exponents of integrability of the first exit time from generalized cones for conditioned rotation invariant stable Lévy processes. Along the way, we introduce the spherical fractional Laplacian and derive some of its spectral properties.     

Jordan Algebras and Dual Affine Connections on Symmetric Cones

We study dually flat structures on symmetric cones associated with Jordan algebras. We give an interpretation of connections, a geometrical concept, in terms of Jordan algebras and show a relation between doubly autoparallel submanifolds and Jordan subalgebras.     

Riesz Exponential Families on Symmetric Cones

Let E be a simple Euclidean Jordan algebra of rank r and let be its symmetric cone. Given a Jordan frame on E, the generalized power _s (– ^–1) defined on – is the Laplace transform of some positive measure R _s on E if and only if s is in a given subset of R ^r . The aim of this paper is to study the natural exponential families (NEFs) F(R _s ) associated to the measures R _s . We give a condition on s so that R _s generates a NEF, we calculate the variance function of F(R _s ) and we show that a NEF F on E is invariant by the triangular group if and only if there exists s in such that either F=F(R _s ) or F is the image of F(R _s ) under the map x–x.     

Invariant Differential Operators on Symmetric Cones and Hermitian Symmetric Spaces

We give a brief survey on the study of constructions of invariant differential operators on Riemannian symmetric spaces and of combinatorial and analytical properties of their eigenvalues, and pose some open questions.     

局部对称仿射超曲面

本文用活动标架法证明了：仿射空间Ａｎ＋１中维数ｎ大于２的非退化超曲面Ｍｎ，相对于诱导的仿射连络局部对称，当且仅当Ｍ是虚（ｉｍｐｒｏｐｅｒ）仿射球或者是非退化的二次超曲面．     

Affine Rank 3 Groups on Symmetric Designs

In this note we determine the symmetric designs which admit a primitive rank 3 group with a solvable normal subgroup. This completes the investigations of Dempwolff [4].     

Representations in L-Spaces on Infinite-Dimensional Symmetric Cones

In this paper we study representations of the automorphism groups of classical infinite-dimensional tube domains. In particular we construct the L²-realization of all unitary highest weight representations, including the vector-valued case. We also find a projective representation of the full identity component of the affine automorphism group of the Hilbert-Schmidt version of the tube domain with trivial cocycle on the subgroup corresponding to the trace class version, but non-trivial on the large group. Finally we show that the operator-valued measures corresponding to the vector valued highest weight representations have densities of a rather weak type with respect to Wishart distributions which makes it possible to determine their “supports.”     

Self-Scaled Barrier Functions on Symmetric Cones and Their Classification

Self-scaled barrier functions on self-scaled cones were axiomatically introduced by Nesterov and Todd in 1994 as a tool for the construction of primal—dual long-step interior point algorithms. This paper provides firm foundations for these objects by exhibiting their symmetry properties, their close ties with the symmetry groups of their domains of definition, and subsequently their decomposition into irreducible parts and their algebraic classification theory. In the first part we recall the characterization of the family of self-scaled cones as the set of symmetric cones and develop a primal—dual symmetric viewpoint on self-scaled barriers, results that were first discovered by the second author. We then show in a short, simple proof that any pointed, convex cone decomposes into a direct sum of irreducible components in a unique way, a result which can also be of independent interest. We then proceed to showing that any self-scaled barrier function decomposes, in an essentially unique way, into a direct sum of self-scaled barriers defined on the irreducible components of the underlying symmetric cone. Finally, we present a complete algebraic classification of self-scaled barrier functions using the correspondence between symmetric cones and Euclidean—Jordan algebras. December 5, 1999. Final version received: September 6, 2001.     

Geodesics for Dual Connections and Means on Symmetric Cones

Means on self-dual and homogeneous cones (, i.e., symmetric cones) are discussed from a viewpoint of differential geometry with affine connections. We first define means on symmetric cones in an axiomatic way following [8]. Next we consider dualistic differential geometry (, i.e., Riemannian metric and affine connections) [1] naturally introduced on symmetric cones. Elucidating the relation between the geodesics defined by each affine connection, and operator monotone functions that generate means, we show an important class of means are expressed by the (mid)points on geodesics. Related results on the means and submanifolds in a symmetric cone are also presented.     

A Note on Normality of Cones Over Symmetric Varieties

Let G be a semisimple and simply connected algebraic group, and let H ⁰ be the subgroup of points fixed by an involution of G. Let V be an irreducible representation of G with a nonzero vector v fixed by H ⁰. In this article, we prove a property of the normalization of the coordinate ring of the closure of G·[v] in ?(V).     

Restricted Normal Cones and Sparsity Optimization with Affine Constraints

The problem of finding a vector with the fewest nonzero elements that satisfies an underdetermined system of linear equations is an NP-complete problem that is typically solved numerically via convex heuristics or nicely behaved nonconvex relaxations. In this paper we consider the elementary method of alternating projections (MAP) for solving the sparsity optimization problem without employing convex heuristics. In a parallel paper we recently introduced the restricted normal cone which generalizes the classical Mordukhovich normal cone and reconciles some fundamental gaps in the theory of sufficient conditions for local linear convergence of the MAP algorithm. We use the restricted normal cone together with the notion of superregularity, which is inherently satisfied for the affine sparse optimization problem, to obtain local linear convergence results with estimates for the radius of convergence of the MAP algorithm applied to sparsity optimization with an affine constraint.     

On Transformations and Determinants of Wishart Variables on Symmetric Cones

Let x and y be independent Wishart random variables on a simple Jordan algebra V. If c is a given idempotent of V, write for the decomposition of x in where V(c,) equals the set of v such that cv=v. In this paper we compute E(det(ax+by)) and some generalizations of it (Theorems 5 and 6). We give the joint distribution of (x ₁, x ₁₂, y ₀) where and P is the quadratic representation in V. In statistics, if x is a real positive definite matrix divided into the blocks x ₁₁, x ₁₂, x ₂₁, x ₂₂, then y ₀ is equal to . We also compute the joint distribution of the eigenvalues of x (Theorem 9). These results have been known only when V is the algebra of Hermitian matrices with entries in the real or the complex field. To obtain our results, we need to prove several new results on determinants in Jordan algebras. They include in particular extensions of some classical parts of linear algebra like Leibnitz's determinant formula (Proposition 2) or Schur's complement (Eqs. (3.3) and (3.6)).     

Application of Hilbert＇s Projective Metric on Symmetric Cones

Let Ω be a symmetric cone. In this note, we introduce Hilbert＇s projective metric on Ω in terms of Jordan algebras and we apply it to prove that, given a linear invertible transformation g such that g（Ω） = Ω and a real number p, ｜p｜ 〉 1, there exists a unique element x ∈ Ω satisfying g（x） = x^p.     

Jordan-Algebraic Aspects of Nonconvex Optimization over Symmetric Cones

We illustrate the usefulness of Jordan-algebraic techniques for nonconvex optimization by considering a potential-reduction algorithm for a nonconvex quadratic function over the domain obtained as the intersection of a symmetric cone with an affine subspace.     

Equations Defining Symmetric Varieties and Affine Grassmannians

Suppose be a simpleinvolution of a semisimple algebraic group and suppose H is the subgroup of G of pointsfixed by . If the restrictedroot system is of type or and G is simplyconnected, or if the restricted root system is of type and G is of adjoint type, then we describe astandard monomial theory and the equations for the coordinatering using the standardmonomial theory and the Plücker relations of an appropriate(maybe infinite-dimensional) Grassmann variety.     

Affine Locally Symmetric Surfaces in R~4

The nondegenerate affine locally symmetric surfaces in R^4 with the transversal bundle defined by Nomizu and Vrancken have been studied and a complete classification of the locally symmetric surfaces with flat normal bundle has been given.     

Symmetric Stable Processes on Unbounded Domains

Let be a symmetric -stable process in , , . We give necessary and sufficient condition under which the expectation of a very general function of the exit time from horns is finite. These domains include the symmetric domains given by increasing functions studied earlier by various authors. Our methods differ from those in earlier papers in that we obtain our results from estimates on the transition densities instead of harmonic measure. Some of this estimates are of independent interest.Supported in part by NSF grant #9700585-DMS and RTN Harmonic Analysis and Related Problems contract HPRN-CT-2001-00273-HARP.