Asymptotic properties of type I elliptical random vectors

Let be an elliptical random vector with a non-singular square matrix and a spherical random vector in , and let be a sequence of vectors in such that . We assume in this paper that the associated random radius R _k =(S ₁ + S ₂ +...+S _k )^1/2 is almost surely positive, and it has distribution function in the Gumbel max-domain of attraction. Relying on extreme value theory we obtain an exact asymptotic expansion of the tail probability for converging as to a boundary point. Further we discuss density convergence under a suitable transformation. We apply our results to obtain an asymptotic approximation of the distribution of partial excess above a high threshold, and to derive a conditional limiting result. Further, we investigate the asymptotic behaviour of concomitants of order statistics, and the tail asymptotics of associated random radius for subvectors of .      

Numerically satisfactory solutions of Kummer recurrence relations

Pairs of numerically satisfactory solutions as for the three-term recurrence relations satisfied by the families of functions , , are given. It is proved that minimal solutions always exist, except when and z is in the positive or negative real axis, and that is minimal as whenever . The minimal solution is identified for any recurrence direction, that is, for any integer values of and . When the confluent limit , with fixed, is the main tool for identifying minimal solutions together with a connection formula; for , is the main tool to be considered.     

Periodic automorphisms of takiff algebras, contractions, and θ-groups

Let be an algebraic Lie algebra and a (generalised) Takiff algebra. Any finite-order automorphism θ of induces an automorphism of of the same order, denoted . We study invariant-theoretic properties of representations of the fixed point subalgebra of on other eigenspaces of in . We use the observation that, for special values of m, the fixed point subalgebra, , turns out to be a contraction of a certain Lie algebra associated with and θ. To my teacher Supported in part by R.F.B.R. grant 06-01-72550.     

Regularity in Capacity and the Dirichlet Laplacian

Given an open set in , we prove that every function in is zero everywhere on the boundary if and only if is regular in capacity. If in addition is bounded, then it is regular in capacity if and only if the mapping from into is injective, where denotes the Perron solution of the Dirichlet problem. Let be the set of all open subsets of which are regular in capacity. Then one can define metrics and on only involving the resolvent of the Dirichlet Laplacian. Convergence in those metrics will be defined to be the local/global uniform convergence of the resolvent of the Dirichlet Laplacian applied to the constant function . We prove that the spaces and are complete and contain the set of all open sets which are regular in the sense of Wiener (or Dirichlet regular) as a closed subset.     

Reduction of exterior differential systems with infinite dimensional symmetry groups

A symmetry-based method for constructing solutions to systems of differential equations founded on the reduction of exterior differential systems invariant under the action of an infinite dimensional pseudogroup is proposed. One can associate to any system of differential equations Δ=0 with a symmetry group an exterior differential system invariant under so that solutions of Δ=0 correspond to integral manifolds . The -invariant exterior differential system gives rise to a reduced system specified on a cross section to the pseudogroup orbits, and it is shown that solutions to Δ=0 can be reconstructed from integral manifolds by solving an equation of generalized Lie type for the jets of pseudogroup transformations. In particular, as opposed to the classical method of symmetry reduction, every solution to the system of differential equations can, under some mild regularity assumptions, be constructed by the present algorithm. AMS subject classification (2000)  58A15, 58A20, 58H05, 58J70     

Reduced Minimum Modulus Preserving in Banach Space

Let be the algebra of all bounded linear operators on a complex Banach space X and γ(T) be the reduced minimum modulus of operator . In this work, we prove that if , is a surjective linear map such that is an invertible operator, then , for every , if and only if, either there exist two bijective isometries and such that for every , or there exist two bijective isometries and such that for every . This generalizes for a Banach space the Mbekhta’s theorem [12].      

Slices of Essentially Algebraic Categories

This paper is a contribution to the theory of functor slices of J. Sichler and V. Trnková. For every ordinal α we introduce a basket , prove that every essentially algebraic category of height α is a slice of , characterize small slices of and give a common generalization of known results about slices of the algebraic basket .      

Hilbert Space Representations of Cross Product Algebras II

In this paper, we study and classify Hilbert space representations of cross product -algebras of the quantized enveloping algebra with the coordinate algebras of the quantum motion group and of the complex plane, and of the quantized enveloping algebra with the coordinate algebras of the quantum group and of the quantum disc. Invariant positive functionals and the corresponding Heisenberg representations are explicitly described.Presented by S.L. Woronowicz.     

Ranks of abelian varieties over infinite extensions of the rationals

Let S be an infinite set of rational primes and, for some p ∈ S, let be the compositum of all extensions unramified outside S of the form , for . If , let be the intersection of the fixed fields by , for i = 1, . . , n. We provide a wide family of elliptic curves such that the rank of is infinite for all n ≥ 0 and all , subject to the parity conjecture. Similarly, let be a polarized abelian variety, let K be a quadratic number field fixed by , let S be an infinite set of primes of and let be the maximal abelian p-elementary extension of K unramified outside primes of K lying over S and dihedral over . We show that, under certain hypotheses, the -corank of sel _p ∞(A/F) is unbounded over finite extensions F/K contained in . As a consequence, we prove a strengthened version of a conjecture of M. Larsen in a large number of cases.     

Limit distributions of upper order statistics for families of multivariate distributions

We consider a portfolio of dependent exchangeable random variables , where the dependence structure is generated by a mixture model (Archimedean copulas belong to this class of models). Define the ordered sample . We prove results of the following type: fix and choose appropriately, then converges in distribution to a random vector as , for which we can explicitly give the distribution.     

State space collapse for asymptotically critical multi-class fluid networks

We consider a class of fluid queueing networks with multiple fluid classes and feedback allowed, which are fed by N heavy tailed ON/OFF sources. We study the asymptotic behavior when N→∞ of these queueing systems in a heavy traffic regime (that is, when they are asymptotically critical). As performance processes we consider the workload W ^N (the amount of time needed for each server to complete processing of all the fluid in queue), and the fluid queue Z ^N (the quantity of each fluid class in the system). We show the convergence of and (to and ) in heavy traffic if state space collapse (SSC) holds. (SSC) is a condition that establishes a relationship between those components of that correspond to fluid classes processed by the same server, which implies that for a deterministic lifting matrix Δ. Our main contribution is to prove that assuming that the other hypotheses are true, (SSC) is not only sufficient for this convergence, but necessary. Furthermore, we prove that processes and , conveniently scaled in time, converge to W (a reflected fractional Brownian motion) and Z (=ΔW). We illustrate the application of our results with some examples including a tandem queue. Supported by project MEC-FEDER ref. MTM2006-06427.     

A Relation Between Closure Operators on a Small Category and Its Category of Presheaves

In this paper we show that each factorization structure on a small category , satisfying certain conditions, yields a presheaf on and a morphism of presheaves . We then give connections, and set up one to one correspondences, between subclasses of the following classes: (a) closure operators on (b) subobjects of (c) morphisms from to (d) weak Lawvere–Tierney topologies (e) weak Grothendieck topologies (f) closure operators on .     

Primitive Ideals and Automorphisms of Quantum Matrices

Let be a field and q be a nonzero element of that is not a root of unity. We give a criterion for 〈0〉 to be a primitive ideal of the algebra of quantum matrices. Next, we describe all height one primes of ; these two problems are actually interlinked since it turns out that 〈0〉 is a primitive ideal of whenever has only finitely many height one primes. Finally, we compute the automorphism group of in the case where m ≠ n. In order to do this, we first study the action of this group on the prime spectrum of . Then, by using the preferred basis of and PBW bases, we prove that the automorphism group of is isomorphic to the torus when m ≠ n and (m,n) ≠ (1, 3),(3, 1). This research was supported by a Marie Curie Intra-European Fellowship within the 6th European Community Framework Programme and by Leverhulme Research Interchange Grant F/00158/X.     

Schur–Weyl Reciprocity between the Quantum Superalgebra and the Iwahori–Hecke Algebra

In this paper, we establish Schur–Weyl reciprocity between the quantum general super Lie algebra and the Iwahori–Hecke algebra . We introduce the sign -permutation representation of on the tensor space of dimensional -graded -vector space . This action commutes with that of derived from the vector representation on . Those two subalgebras of satisfy Schur–Weyl reciprocity. As special cases, we obtain the super case (), and the quantum case (). Hence this result includes both the super case and the quantum case, and unifies those two important cases.Presented by A. Verschoren.     

An iterative method for the least squares symmetric solution of matrix equation AXB = C

This paper an iterative method is presented to solve the minimum Frobenius norm residual problem: with unknown symmetric matrix . By the iterative method, for any initial symmetric matrix , a solution can be obtained within finite iteration steps in the absence of roundoff errors, and the solution with least Frobenius norm can be obtained by choosing a special kind of initial symmetric matrix. In addition, in the solution set of the minimum Frobenius norm residual problem, the unique optimal approximation solution to a given matrix in Frobenius norm can be expressed as , where is the least norm symmetric solution of the new minimum residual problem: with . Given numerical examples are show that the iterative method is quite efficient.Research supported by Scientific Research Fund of Hunan Provincial Education Department of China (05C797), by China Postdoctoral Science Foundation (2004035645) and by National Natural Science Foundation of China (10571047).     

Deep Matrix Algebras of Finite Type

Deep matrix algebras based on a set over a ring are introduced and studied by McCrimmon when has infinite cardinality. Here, we construct a new -module that is faithful for of any cardinality. For a field of arbitrary characteristic and , is studied in depth. The algebra is shown to possess a unique proper non-zero ideal, isomorphic to . This leads to a new and interesting simple algebra, , the quotient of by its unique ideal. Both and the quotient algebra are shown to have centers isomorphic to . Via the new faithful representation, all automorphisms of are shown to be inner in the sense of Definition 18.Presented by D. Passman.     

Coherence for Product Monoids and their Actions

Let and be two monoids (algebras) in a monoidal category . Further let be a distributive law in the sense of [J. Beck, Lect. Notes Math., 80:119–140, 1969]; naturally yields a monoid . Consider a word in the symbols , , and . The first coherence theorem proved in this paper asserts that all morphisms coincide in , provided they arise as composites of morphisms which are -products of ’s ‘canonical’ structure morphisms, and of , , , , , , , and . Assume now that an object is endowed with both an -object structure , and an -object structure . Further assume that these two structures are compatible, in the sense that they naturally yield an -object . Let be a word in , , , and , which contains a single instance of , in the rightmost position. The second coherence theorem states that all morphisms coincide in , provided they arise as composites of morphisms which are -products of ’s ‘canonical’ structure morphisms, and of , , , , , , , , , and .     

Compact Differences of Weighted Composition Operators on Weighted Banach Spaces of Analytic Functions

Let be the weighted Banach space of analytic functions with a topology generated by weighted sup-norm. In the present article, we investigate the analytic mappings and which characterize the compactness of differences of two weighted composition operators on the space . As a consequence we characterize the compactness of differences of composition operators on weighted Bloch spaces.      

Stacks of Algebras and Their Homology

For any increasing function which takes only finitely many distinct values, a connected finite dimensional algebra is constructed, with the property that for all ; here is the -generated finitistic dimension of . The stacking technique developed for this construction of homological examples permits strong control over the higher syzygies of -modules in terms of the algebras serving as layers. The first author was supported in part by a fellowship stipend from the National Physical Science Consortium and the National Security Agency. The second author was partially supported by a grant from the National Science Foundation.