Glivenko-Cantelli type theorems: An application of the convergence theory of stochastic suprema

The uniform convergence of empirical processes on certain classes of sets follows from the convergence theory for random lower semicontinuous functions studies in the context of stochastic optimization. In the process, a richer class of sets for which one can prove this type of result is exhibited.Research supported in part by grants from Ministero Publica Istruzione and the National Science Foundation.     

A note on bootstrapping the variance of sample quantile

Summary Letm _{n, p} denote thep-th quantile based onn observations and let λ _p denote the population quantile. In this paper consistency of the bootstrap estimate of variance of is established.     

非紧非单调算子的不动点定理

本文利用和选择公理等价的Zorn引理,得到了一般非单调算子不动点存在性的一些结果.     

Summation formulae for twisted cubic q‐series

A new class of twisted cubic q‐series is investigated by means of the modified Abel lemma on summation by parts. Several remarkable summation and transformation formulae are established for both terminating and nonterminating series.     

Flat functions in Carleman ultraholomorphic classes via proximate orders

Whenever the defining sequence of a Carleman ultraholomorphic class (in the sense of H. Komatsu) is strongly regular and associated with a proximate order, flat functions are constructed in the class on sectors of optimal opening. As consequences, we obtain analogues of both Borel–Ritt–Gevrey theorem and Watson's lemma in this general situation.     

Uniqueness and stability result for Cauchy’s equation of motion for a certain class of hyperelastic materials

We consider Cauchy’s equation of motion for hyperelastic materials. The solution of this nonlinear initial-boundary value problem is the vector field which discribes the displacement which a particle of this material perceives when exposed to stress and external forces. This equation is of greatest relevance when investigating the behavior of elastic, anisotropic composites and for the detection of defects in such materials from boundary measurements. This is why results on unique solvability and continuous dependence from the initial values are of large interest in materials’ research and structural health monitoring. In this article we present such a result, provided that reasonable smoothness assumptions for the displacement field and the boundary of the domain are satisfied for a certain class of hyperelastic materials where the first Piola–Kirchhoff tensor is written as a conic combination of finitely many, given tensors.     

Nontrivial solutions of superlinear p-Laplacian equations

We consider p-Laplacian equations on a bounded domain, where the nonlinearity is superlinear but dose not satisfy the usual Ambrosetti-Rabinowitz condition near infinity, or its dual version near zero. Nontrivial solutions are obtained by computing the critical groups and Morse theory.