The Kakutani's precompactness lemma

In this paper we establish a theorem that extends and sharpens an old precompactness lemma due to Kakutani. We use this theorem to derive the classical Arzelà-Ascoli theorem and a theorem of Defant and Floret for families of linear operators. We also use this theorem to derive a theorem for composition operators which yields as immediate corollaries a theorem of Geue and a locally convex version of a theorem of Aron and Schottenloher.     

Generalized KKM theorems and common fixed point theorems

In this paper, we first prove a generalized KKM theorem, and then use this generalized KKM theorem to establish the generalized equi-KKM theorem, common fixed point theorems for a family of multivalued maps, and the Kakutani-Fan-Glicksberg fixed point theorem. We also show that an existence theorem of the common fixed point theorem is equivalent to the Kakutani-Fan-Glicksberg fixed point theorem.     

一个新的Ky Fan匹配定理及其在L-凸度量空间中的应用

建立了转移开覆盖的一个新的Ky Fan型匹配定理．作为应用，获得了Fan-Browder重合定理，不动点定理、极大元定理，相交定理和截口定理．     

几个随机算子定理

得到Banach空间中随机隐函数存在定理、随机反函数定理和随机Hahn-Banach定理，它们是著名隐函数定理、反函数定理和Hahn-Banach控制延拓定理的随机化推广，这些定理在随机算子理论中将起重要作用。     

一类新的KKM定理及其应用*

本文得到一类新的KKM定理,统一和改进了[2,3,0,7,11]中的结果。作为应用,我们得到了几个匹配定理、重合定理、不动点定理、极大极小不等式定理及截口定理。     

向量在几何中的应用

近期将欧氏平面E2上的正弦定理和余弦定理推广到三维欧氏空间E3中,建立了E3中四面体空间角正弦定理、二面角正弦定理和四面体余弦定理,利用向量给出了三维余弦定理和三维正弦定理的简单证明.     

关于Michael选择问题

本文得到一个新的逼近选择定理及一个新的连续选择定理.作为这个新的连续选择定理的应用,我们给出一个新的不动点定理和一个新的重合定理(见文[1-27]).     

播挪运算及其性质

在现有的播挪定义基础上,引入了对偶播挪的概念和任意论域子集族的内并、内交两种新运算,提出了十个定理.它们是：平凡播挪空间定理,最小播挪定理,播挪内运算封闭性定理,播挪任意交定理,播挪上确界定理,播挪运算分配性定理,播挪基判定定理,播挪基闭包定理,播挪基约简定理,播挪基内运算封闭性定理.这些定理对于播挪空间的研究具有一定的理论价值.     

A relative interpolation theorem for infinitary universal Horn logic and its applications

In this paper we deal with infinitary universal Horn logic both with and without equality. First, we obtain a relative Lyndon-style interpolation theorem. Using this result, we prove a non-standard preservation theorem which contains, as a particular case, a Lyndon-style theorem on surjective homomorphisms in its Makkai-style formulation. Another consequence of the preservation theorem is a theorem on bimorphisms, which, in particular, provides a tool for immediate obtaining characterizations of infinitary universal Horn classes without equality from those with equality. From the theorem on surjective homomorphisms we also derive a non-standard Beth-style preservation theorem that yields a non-standard Beth-style definability theorem, according to which implicit definability of a relation symbol in an infinitary universal Horn theory implies its explicit definability by a conjunction of atomic formulas. We also apply our theorem on surjective homomorphisms, theorem on bimorphisms and definability theorem to algebraic logic for general propositional logic.     

连续型概率度量的特征及其应用

讨论了连续型概率度量空间,给出了连续型概率度量的特征定理.利用这个特征定理,我们获得了连续型概率度量空间的闭球套定理和M enger空间完备性特征定理.作为本文的应用获得了一个局部概率压缩映象不动点定理.     

关于“四元数自共轭矩阵与行列式的几个定理”的注记

本刊1985年第4期发表郝稚传的“四元数自共轭矩与行列式的几个定理”(称为〔1〕)的文章的主要工作分为两部分。〔1〕在英文摘要中写到:“(i)This essay has improvedthe conclusion of theorem 8 and theorem 9 in 〔2〕”即〔1〕之第一部分工作在于改进了〔2〕的定理8、9的结果,其根据就是〔1〕的定理1、2.〔1〕称定理1—若     

Minimax Problems for Set-Valued Mappings

In this article, by virtue of the Fan-Browder fixed point theorem, we first obtain a minimax theorem and establish an equivalent relationship between the minimax theorem and a cone saddle point theorem for scalar set-valued mappings. Then, by using scalarization functions, we investigate an existence theorem of the cone saddle point for set-valued mappings. Simultaneously, we also establish a minimax theorem for set-valued mappings.     

An n-dimensional version of Steinhaus' chessboard theorem

The main result of this paper is an n-dimensional version of the Steinhaus' chessboard theorem. Our theorem implies the Poincaré theorem as well as its parametric extension. But it is known that the Poincaré theorem is equivalent to the Brouwer Fixed-Point theorem.     

有关中值定理的探究性教学

通过对中值定理教学思路的设计,给出探究性教学方法的一个实例,即通过导数概念的物理意义导出Lagrange中值定理,经特殊化后推出Rolle定理,再经化归思想给出Lagrange定理的证明,最后推广得到Cauchy中值定理,并借助类比或化归思想分别给出Cauchy定理的证明．     

On the Equivalence of Minimax Theorems

In this note we show that Ky Fan's minimax theorem and its several generalizations such as König's minimax theorem [6], M. Neumann's minimax theorem [8] and Fuchssteiner-König's minimax theorem [3] are equivalent. We also give a direct proof for Fuchssteiner-König's minimax theorem on the basis of Eidelheit's well-known seperation theorem.     

Leray-Schauder不动点定理的一个新证明

给出Leray-Schauder不动点定理的一个新证明.我们首先给出集值映射的焊接引理,利用集值映射的焊接引理和Kakutani不动点定理证明Leray-Schauder不动点定理,并证明Leray-Schauder不动点定理与Brouwer不动点定理等价.     

一类双调和方程弱解的存在性研究

利用上下解方法、嵌入定理和Leray-Schauder不动点定理证明了一类双调和方程弱解的存在性定理.做为定理的应用,给出了一个实例.     

矩阵秩分解定理的一个证明及其应用

矩阵的秩分解定理是矩阵论中的一个基础性定理.本文给出了秩分解定理的一个证明,描述了其中可逆矩阵P,Q的构造,讨论了秩分解定理及其P,Q的构造在解线性方程组中的应用,以及在判别Sylvester不等式等号成立中的应用.     

Existence of the optimal measurable coupling and ergodicity for Markov processes

The existence theorem of the optimal measurable coupling of two probability kernels on a complete separable metric measurable space is proved. Then by this theorem, a general ergodicity theorem for Markov processes is obtained. And as an immediate application to particle systems the uniqueness theorem of the stationary distribution is supplemented, i.e. the uniqueness theorem also implies the existence of the stationary distribution.