有界闭区间上连续函数列一致收敛的充要条件

判别函数列一致收敛的方法有函数列一致收敛定义、Cauchy一致收敛准则、limn→∞supx∈D|fn(x)-f(x)|=0及Dini定理,本文由函数列的等度连续性,可得出几个有界闭区间上连续函数列一致收敛的充要条件,推广了Dini定理.     

度量投影的收敛性（英文）

讨论了赋范空间中度量投影的收敛性．得到了在局部紧集控制下，Chebyshev凸集序列的度量投影的收敛性与K－M收敛，Wijsman收敛和Kuratowski收敛都等价．本文的结论完善了M．Tsukada在[1]和[2]结果．     

On the convergence of the iteration sequence in primal-dual interior-point methods

Recently, numerous research efforts, most of them concerned with superlinear convergence of the duality gap sequence to zero in the Kojima—Mizuno—Yoshise primal-dual interior-point method for linear programming, have as a primary assumption the convergence of the iteration sequence. Yet, except for the case of nondegeneracy (uniqueness of solution), the convergence of the iteration sequence has been an important open question now for some time. In this work we demonstrate that for general problems, under slightly stronger assumptions than those needed for superlinear convergence of the duality gap sequence (except of course the assumption that the iteration sequence converges), the iteration sequence converges. Hence, we have not only established convergence of the iteration sequence for an important class of problems, but have demonstrated that the assumption that the iteration sequence converges is redundant in many of the above mentioned works.This research was supported in part by NSF Coop. Agr. No. CCR-8809615. A part of this research was performed in June, 1991 while the second and the third authors were at Rice University as visiting members of the Center for Research in Parallel Computation.Corresponding author. Research supported in part by AFOSR 89-0363, DOE DEFG05-86ER25017 and ARO 9DAAL03-90-G-0093.Research supported in part by NSF DMS-9102761 and DOE DE-FG05-91ER25100.Research supported in part by NSF DDM-8922636.     

On the Convergence Acceleration of Composite Sequence Transformation

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Convergence near a point and the Arzela-Ascoli-type theorems

Theorems are proved giving necessary and sufficient conditions for the convergence of a sequence of continuous (differentiable) functions to a continuous (differentiable) function. The concepts of convergence near a point and equipotential convergence near a point are introduced. These concepts are introduced locally; on a segment, they are equivalent to the quasiuniform convergence and to the uniform convergence of a sequence of functions, respectively.Translated from Ukrainskii Matemahcheskii Zhurnal, Vol. 45, No. 8, pp. 1090–1095, August, 1993.     

Metaphor as a possible pathway to more formal understanding of the definition of sequence convergence

This study presents how the introduction of a metaphor for sequence convergence constituted an experientially real context in which an undergraduate real analysis student developed a property-based definition of sequence convergence. I use elements from Zandieh and Rasmussen's (2010) Defining as a Mathematical Activity framework to trace the transformation of the student's conception from a non-standard, personal concept definition rooted in the metaphor to a concept definition for sequence convergence compatible with the standard definition. This account of the development of the definition of sequence convergence differs from prior research in the sense that it began neither with examples or visual notions, nor with the statement of the formal definition. This study contributes to the Realistic Mathematics Education literature as it documents a student's progression through the definition-of and definition-for stages of mathematical activity in an interactive lecture classroom context.     

关于模糊值函数序列的C-I平均收敛

在一般模糊测度空间上,利用模糊值Choquet积分定义首次给出了模糊值函数列的C-I平均收敛、C-I平均基本等概念,并针对μ-可积模糊值函数列进一步研究了它的C-I平均收敛与依模糊测度收敛、C-I平均基本与依模糊测度基本之间的蕴涵关系.     

Convergence of the back-and-forth shooting method for solving two-point boundary-value problems

The back-and-forth shooting method of Orava and Lautala (Ref. 1) is considered. The method transforms a given boundary-value problem to a sequence of initial-value problems. The present paper studies the convergence properties of this sequence. A local convergence theorem is given, and the rate of convergence is found to be quadratic in sufficiently smooth cases. The necessary tools for this analysis concerning the Fréchet differentiability of certain mappings are given in the Appendix.     

数列x_(n_m)~r_(n_m+p)审敛原理

“数列xnm~rnm+p审敛原理”,是数列柯西审敛原理的等价命题.采用“数列xnm~rnm+p审敛原理”判别数列(或数项级数)的敛散性比采用柯西审敛原理更便捷;“数列xnm~rnm+p审敛原理”推广了已有的判别数列(或数项级数)敛散性法则,扩大了已有的判别数列(或数项级数)敛散性法则的应用范围.     

NA序列部分和的完全收敛性

本文讨论了非平稳NA随机变量序列部分和的完全收敛性,获得了一般形式的完全收敛速度与矩条件之间的等价关系,其结果与独立情形一致,从而证实了NA序列与独立序列有着极为类似的完全收敛性．     

A sequential iteration algorithm with non-monotoneous behaviour in the method of projections onto convex sets

The method of projections onto convex sets to find a point in the intersection of a finite number of closed convex sets in a Euclidean space, may lead to slow convergence of the constructed sequence when that sequence enters some narrow “corridor” between two or more convex sets. A way to leave such corridor consists in taking a big step at different moments during the iteration, because in that way the monotoneous behaviour that is responsible for the slow convergence may be interrupted. In this paper we present a technique that may introduce interruption of the monotony for a sequential algorithm, but that at the same time guarantees convergence of the constructed sequence to a point in the intersection of the sets. We compare experimentally the behaviour concerning the speed of convergence of the new algorithm with that of an existing monotoneous algorithm.     

Convergence of the Simulated Annealing Algorithm for Continuous Global Optimization

A class of simulated annealing algorithms for continuous global optimization is considered in this paper. The global convergence property is analyzed with respect to the objective value sequence and the minimum objective value sequence induced by simulated annealing algorithms. The convergence analysis provides the appropriate conditions on both the generation probability density function and the temperature updating function. Different forms of temperature updating functions are obtained with respect to different kinds of generation probability density functions, leading to different types of simulated annealing algorithms which all guarantee the convergence to the global optimum.     

不同分布ρ混合序列乘积和的强收敛性

研究了ρ混合序列的收敛性质,利用得到的结果和ρ混合序列的矩不等式讨论了ρ混合序列乘积和的强收敛性质.     

On the convergence of higher-order orthogonal iteration

The higher-order orthogonal iteration (HOOI) has been popularly used for finding a best low-multilinear rank approximation of a tensor. However, its convergence is still an open question. In this paper, we first analyse a greedy HOOI, which updates each factor matrix by selecting from the best candidates one that is closest to the current iterate. Assuming the existence of a block-nondegenerate cluster point, we establish its global iterate sequence convergence through the so-called Kurdyka–?ojasiewicz property. In addition, we show that if the starting point is sufficiently close to any block-nondegenerate globally optimal solution, the greedy HOOI produces an iterate sequence convergent to a globally optimal solution. Relating the iterate sequence by the original HOOI to that by the greedy HOOI, we then show that the original HOOI has global convergence on the multilinear subspace sequence and thus positively address the open question.     

Normalized convergence in stochastic optimization

A new concept of (normalized) convergence of random variables is introduced. This convergence is preserved under Lipschitz transformations, follows from convergence in mean and itself implies convergence in probability. If a sequence of random variables satisfies a limit theorem then it is a normalized convergent sequence. The introduced concept is applied to the convergence rate study of a statistical approach in stochastic optimization.     

HOMOCENTRIC CONVERGENCE BALL OF THE SECANT METHOD

A local convergence theorem and five semi-local convergence theorems of the secant method are listed in this paper.For every convergence theorem,a convergence ball is respectively introduced,where the hypothesis conditions of the corresponding theorem can be satisfied.Since all of these convergence balls have the same center x~*,they can be viewed as a homocentric ball. Convergence theorems are sorted by the different sizes of various radii of this homocentric ball, and the sorted sequence represents the degree of weakness on the conditions of convergence theorems.     

Convergence of the gradient projection method in optimal control problems

The article examines the convergence of the popular gradient projection method for optimal control problems with fixed time and a free right-hand end point. General conditions are derived that substantiate weak convergence of the sequence of control iterations to the set of extremum controls satisfying Pontryagin’s maximum principle. Under stronger assumptions we prove strong convergence of the sequence in the Banach L ₁-norm. __________ Translated from Nelineinaya Dinamika i Upravlenie, No. 3, pp. 139–148, 2003.     

关于广义Φ-压缩映射几种迭代序列收敛的等价性

在广义Φ-压缩映射条件下,分别得到了Picard迭代序列与Krasnoselskii迭代序列以及Mann迭代序列与Ishikawa迭代序列收敛的等价性.     

Addition to the Aitken method for the extrapolation of the limit of slowly convergent sequences

The class of sequences and series in which the Aitken process accelerates the convergence is considerably extended. It is proved that a proper subsequence of a slowly convergent sequence satisfies the sufficient condition for accelerating the convergence of the Aitken transformation. Two numerical examples illustrate the highly accurate limit extrapolation.     

Overlapping approximation of the exponential function on the positive axis

We investigate properties of the exponential function concerning the overlapping approximation which was introduced in former papers. We give bounds for the rate of convergence of the sequence of least deviations and give an exact formula for the convergence speed in the case of only one knot on the positive axis.