Topological regular variation: I. Slow variation

Motivated by the Category Embedding Theorem, as applied to convergent automorphisms (Bingham and Ostaszewski (in press) [11]), we unify and extend the multivariate regular variation literature by a reformulation in the language of topological dynamics. Here the natural setting are metric groups, seen as normed groups (mimicking normed vector spaces). We briefly study their properties as a preliminary to establishing that the Uniform Convergence Theorem (UCT) for Baire, group-valued slowly-varying functions has two natural metric generalizations linked by the natural duality between a homogenous space and its group of homeomorphisms. Each is derivable from the other by duality. One of these explicitly extends the (topological) group version of UCT due to Bajšanski and Karamata (1969) [4] from groups to flows on a group. A multiplicative representation of the flow derived in Ostaszewski (2010) [45] demonstrates equivalence of the flow with the earlier group formulation. In companion papers we extend the theory to regularly varying functions: we establish the calculus of regular variation in Bingham and Ostaszewski (2010) [13] and we extend to locally compact, σ-compact groups the fundamental theorems on characterization and representation (Bingham and Ostaszewski (2010) [14]). In Bingham and Ostaszewski (2009) [15], working with topological flows on homogeneous spaces, we identify an index of regular variation, which in a normed-vector space context may be specified using the Riesz representation theorem, and in a locally compact group setting may be connected with Haar measure.     

Convex Komlós sets in Banach function spaces

In 1967 Komlós proved that for any sequence n{fn} in L₁(μ), with ‖fn‖?M<∞ (where μ is a probability measure), there exists a subsequence n{gn} of n{fn} and a function g∈L₁(μ) such that for any further subsequence n{hn} of n{gn},

     

实Banach空间中锥上的随机逼近和随机不动点定理

本文在实Ｂａｎａｃｈ空间的锥上证明了集值映射的随机逼近定理．作为应用，讨论了几个随机的不动点定理．我们的工作推广了Ｌｉｎ，Ｓｅｈｇａｌ和Ｓｉｎｇｈ的结果．     

An operator-valued Lyapunov theorem

We generalize Lyapunov's convexity theorem for classical (scalar-valued) measures to quantum (operator-valued) measures. In particular, we show that the range of a nonatomic quantum probability measure is a weak^?-closed convex set of quantum effects (positive operators bounded above by the identity operator) under a sufficient condition on the non-injectivity of integration. To prove the operator-valued version of Lyapunov's theorem, we must first define the notions of essentially bounded, essential support, and essential range for quantum random variables (Borel measurable functions from a set to the bounded linear operators acting on a Hilbert space).     

Directional derivatives and generalized gradients on manifolds

For nonsmooth functions and differential forms on manifolds generalized directional derivatives, subgradients and Lie derivatives are introduced. Some rules for subgradients are given. Cartan’s formula and Stokes’ theorem are formulated for generalized subgradients and Lie derivatives     

A Reflection Principle and an Orthogonal Decomposition in Clifford Algebras

In this article we investigate spaces of functions defined in a domain Ω ? R with values in the Clifford algebra R _n. According to an inner product an orthogonal decomposition is proved. By this decomposition, we obtain a subspace A ₂(Ω) of regular functions with respect to the Dirac operator. In the orthogonal complement the Dirac equation with homogeneous boundary values is solvable. The decomposition can be proved in two ways: by a reflection principle and by Sobolev's regularity theorem. It will turn out, that the existence of the orthogonal decomposition and Sobolev's theorem is equivalent. So also a reflection principle will be proved, which describes the jump behavior of a Cauchy type integral. By the reflection principle, a countable dense subset of A ₂(Ω) can be obtained. Further considerations lead to a minimal generating system, by which the Bergman kernel function can be obtained. As a conclusion we also obtain Runge's theorem.     

Nonhomogeneous multiparameter problems in Orlicz–Sobolev spaces

This paper is concerned with the existence and multiplicity of solutions for a class of problems involving the Φ-Laplacian operator with general assumptions on the nonlinearities, which include both semipositone cases and critical concave convex problems. The research is based on the subsupersolution technique combined with a truncation argument and an application of the Mountain Pass Theorem. The results in this paper improve and complement some recent contributions to this field.     

Stability and other limit laws for exit times of random walks from a strip or a halfplane

We show that the passage time, T^*(r), of a random walk S_n above a horizontal boundary at r (r≥0) is stable (in probability) in the sense that as r→∞ for a deterministic function C(r)>0, if and only if the random walk is relatively stable in the sense that as n→∞ for a deterministic sequence B_n>0. The stability of a passage time is an important ingredient in some proofs in sequential analysis, where it arises during applications of Anscombe's Theorem. We also prove a counterpart for the almost sure stability of T^*(r), which we show is equivalent to E|X|<∞, EX>0. Similarly, counterparts for the exit of the random walk from the strip {|y|≤r} are proved. The conditions arefurther related to the relative stability of the maximal sum and the maximum modulus of the sums. Another result shows that the exit position of the random walk outside the boundaries at ±r drifts to ∞ as r→∞ if and only if the random walk drifts to ∞.