Rates of Convergence for Double Sequences

In 1900 Pringsheim presented the following definition for convergence of a double sequence (i.e. ordinary infinite matrices). A double sequence [x] has limit L if the terms of the double sequence approaches L as both the column and row indices increases. Using this notion for convergence I will present definitions for asymptotically equivalent double sequences, rate preserving four dimensional matrix transformation, and these definitions shall be used to present two natural invariance theorems.AMS Subject Classification (2000): Primary 42B15, Secondary 40C05     

数列的极限及其应用分析

根据数列的极限及级数的收敛的定义和准则,详尽地分析了两个具体的物理问题的极限存在性(即物理解的存在性),据此便捷地求解了这两个问题,体现了数列极限思想在物理问题中的重要性.     

Comparison Theorems for Four Dimensional Regular Matrices

A four dimensional summability matrix is stronger than convergence in the Pringsheim sense if it sums a divergent double sequence. Presented in this paper are sufficient conditions on a four dimensional matrix transformation which ensure that the summability matrix is stronger than convergence in the Pringsheim sense. Also other sufficient conditions will be presented to ensure the failure of inclusion between two summability matrices.AMS Subject Classification (2000): Primary 40B05.     

Rough convergence of double sequences of fuzzy numbers

In this paper, we define the concepts of rough convergence and rough Cauchy sequence of double sequences of fuzzy numbers. Then, we investigate some relations between rough limit set and extreme limit points of such sequences.     

Matrix characterization of oscillation for double sequences

The notion of oscillation for ordinary sequences was presented by Hurwitz in 1930. Using this notion Agnew and Hurwitz presented regular matrix characterization of the resulting sequence space. The primary goal of this article is to extend this definition to double sequences, which grants us the following definition: the double oscillation of a double sequence of real or complex number is given P-lim sup_{(m,n)→∞;(α,β)→∞}|S _m,n -S _α,β |. Using this concept a matrix characterization of double oscillation sequence space is presented. Other implication and variation shall also be presented.      

关于二重极限的某种定义的一个注记

本文考虑二重极限的某种定义的缺陷.     

Banach and Statistical Cores of Bounded Sequences

In this paper, we are mainly concerned with characterizing matrices that map every bounded sequence into one whose Banach core is a subset of the statistical core of the original sequence.     

Martingale-coboundary representation for a class of random fields

It is known that under some conditions, a stationary random sequence admits a representation as a sum of two sequences: one of them is a martingale difference sequence, and another one is a so-called coboundary. Such a representation can be used for proving some limit theorems by means of the martingale approximation. A multivariate version of such a decomposition is presented in the paper for a class of random fields generated by several commuting, noninvertible, probability preserving transformations In this representation, summands of mixed type appear, which behave with respect to some group of directions of the parameter space as reversed rnultiparameter martingale differences (in the sense of one of several known definitions), while they look as coboundaries relative to other directions. Applications to limit theorems will be published elsewhere. Bibliography: 14 titles.     

On statistical limit points of double sequences

We investigate the structure of the set of all statistical limit points of a double sequence and prove certain results, mainly showing that this set can be characterized as a F_σ-set.     

Core theorems for double subsequences and rearrangements

The concepts of subsequence and rearrangement of double sequence are used to present multidimensional analogues of the following core questions. If x is a bounded real sequence and A is a matrix summability method, under what conditions does there exist y, a subsequence (rearrangement) of x such that each number t in the core of x is a limit point of Ay?     

Almost convergence and a core theorem for double sequences

The idea of almost convergence was introduced by Moricz and Rhoades [Math. Proc. Cambridge Philos. Soc. 104 (1988) 283-294] and they also characterized the four dimensional strong regular matrices. In this paper we define the M-core for double sequences and determine those four dimensional matrices which transform every bounded double sequence x=[x_jk] into one whose core is a subset of the M-core of x.     

关于任意随机序列的强收敛性

该文的目的是要研究任意随机序列的强极限定理。作为推论，得到了一类鞅差序列的强大数定律，一类随机序列公平比的强极限定理，以及任意随机序列部分和估计定理。      

任意随机变量序列泛函及其随机变换的一类极限定理

采用鞅方法研究对任意随机变量序列普遍成立的强极限定理.并作为推论得到了m阶马氏过程,鞅序列,鞅差序列,独立随机变量序列的一类强极限定理.并把赌博系统的随机变换概念推广到任意随机变量序列的情况,得到任意随机变量序列随机选择与公平比的若干极限定理.     

关于条件期望的注记

利用条件期望的定义和性质给出了两个例子说明随机变量序列的条件期望唯一存在的情况.     

一道数列极限的多种解法

利用数学分析中关于数列极限的定义、收敛数列的性质及数列极限存在的条件,介绍一道数列极限问题的多种解法.     

基于“以学为中心”的数列极限教学设计研究

本文选取数列极限的定义这一部分内容,基于“以学为中心”教学理念介绍如何设计数列极限定义的教学过程,从九个环节进行设计旨在使学生更好的理解掌握数列极限的本质和内涵,达到以学为中心的教学目标.     

On different geometric formulations of Lagrangian formalism

We consider two geometric formulations of Lagrangian formalism on fibred manifolds: Krupka's theory of finite order variational sequences, and Vinogradov's infinite order variational sequence associated with the -spectral sequence. On one hand, we show that the direct limit of Krupka's variational bicomplex is a new infinite order variational bicomplex which yields a new infinite order variational sequence. On the other hand, by means of Vinogradov's -spectral sequence, we provide a new finite order variational sequence whose direct limit turns out to be the Vinogradov's infinite order variational sequence. Finally, we provide an equivalence of the two finite order and infinite order variational sequences modulo the space of Euler-Lagrange morphisms.     

Endpoint theory for set-valued nonlinear asymptotic contractions with respect to generalized pseudodistances in uniform spaces

In uniform spaces, inspired by ideas of Banach, Tarafdar and Yuan, we introduce the concepts of generalized pseudodistances and generalized gauge maps, for set-valued dynamic systems we define various nonlinear asymptotic contractions and contractions with respect to these pseudodistances and gauges, provide conditions on the iterates of these set-valued dynamic systems and present a method which is useful for establishing conditions guaranteeing the existence and uniqueness of endpoints (stationary points) of these set-valued dynamic systems and conditions that each generalized sequence of iterations (in particular, each dynamic process) converges and the limit of a generalized sequence of iterations is an endpoint. The definitions, the results and the method are new for set-valued dynamic systems in uniform, locally convex and metric spaces and even for single-valued maps. The paper includes a number of various examples which show a fundamental difference between our results and those existing in the literature.     

Affine Yangians and deformed double current algebras in type A

We study the structure of Yangians of affine type and deformed double current algebras, which are deformations of the enveloping algebras of matrix W1+∞-algebras. We prove that they admit a PBW-type basis, establish a connection (limit construction) between these two types of algebras and toroidal quantum algebras, and we give three equivalent definitions of deformed double current algebras. We construct a Schur-Weyl functor between these algebras and rational Cherednik algebras.     

两类统计收敛的表示定理

本文证明了Banach空间X中的序列Δ(Δ~m)-统计收敛表示,以及Banach空间X中的双序列(双序列双-Lacunary)统计收敛的表示.