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．     

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.     

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]).     

一个因式相通定理及其应用

证明了一个因式相通定理,它是奇点理论中等价性定理的一个推广.这个定理表明单纯高阶项的开折对普适开折因式相通中的参数映射没有贡献.据此本文提出了一种因式相通中参数映射的简洁有效的近似计算方法     

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。     

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.     

概率区间空间中的新型KKM定理及应用*

本文建立了概率区间空间的概念,并在此框架下建立了一个新型的KKM定理。作为应用我们得到了概率区间空间中的一个新的极大极小定理和截口定理,匹配定理及一些重合点定理。所得结果均是全新的,它们不仅包含了Vom Neumann[7]中的主要结果,而且将[1][3～4][6],[8]中的相应结果推广到概率区间空间。     

The equivariant family index theorem in odd dimensions

In this paper, we prove a local odd dimensional equivariant family index theorem which generalizes Freed's odd dimensional index formula. Then we extend this theorem to the noncommutative geometry framework. As a corollary, we get the odd family Lichnerowicz vanishing theorem and the odd family Atiyah-Hirzebruch vanishing theorem.     

Implicit Multifunction Theorems

We prove a general implicit function theorem for multifunctions with a metric estimate on the implicit multifunction and a characterization of its coderivative. Traditional open covering theorems, stability results, and sufficient conditions for a multifunction to be metrically regular or pseudo-Lipschitzian can be deduced from this implicit function theorem. We prove this implicit multifunction theorem by reducing it to an implicit function/solvability theorem for functions. This approach can also be used to prove the Robinson–Ursescu open mapping theorem. As a tool for this alternative proof of the Robinson–Ursescu theorem, we also establish a refined version of the multidirectional mean value inequality which is of independent interest.     

A Note on Nonconvex Minimax Theorem with Separable Homogeneous Polynomials

The minimax theorem for a convex-concave bifunction is a fundamental theorem in optimization and convex analysis, and has a lot of applications in economics. In the last two decades, a nonconvex extension of this minimax theorem has been well studied under various generalized convexity assumptions. In this note, by exploiting the hidden convexity (joint range convexity) of separable homogeneous polynomials, we establish a nonconvex minimax theorem involving separable homogeneous polynomials. Our result complements the existing study of nonconvex minimax theorem by obtaining easily verifiable conditions for the nonconvex minimax theorem to hold.     

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.     

Notes on Embedding Theorems for Local Hardy spaces

In this note we establish an embedding theorem (Theorem 2.4) for local Hardy spaces in the sense of GOLDBERG [G]. This result is a non-homogeneous version of the theorem of BAERNSTEIN and SAWYER (Theorem BS). Also applying this theorem we establish embedding theorem and Fourier embedding theorem (Theorem 4.2, Theorem 4.3 and Corollary 4.4) for local Hardy spaces.     

A KY FAN MATCHING THEOREM FOR TRANSFER COMPACTLY OPEN COVERS AND THE APPLICATION TO THE FIXED POINT

In this article, a Ky Fan matching theorem for transfer compactly open covers is established. As applications, a Fan-Browder coincidence theorem, a Ky Fan best approximation theorem and a Brouwer-Schauder-Rothe type fixed point theorem are obtained.     

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.     

L-凸空间中的一个新的GLKKM定理及其对不动点的应用

In this paper a new GLKKM theorem in L-convex spaces is established. As applications, a new variational inequality, section theorem, maximal element theorem and fixed point theorem are obtained in L-convex spaces.     

非紧L-凸度量空间中的GLsKKM定理及其对不动点问题的应用

建立了非紧完备L-凸度量空间中的GLsKKM定理.作为应用,获得了Ky Fan 截口定理,极大元定理和Fan-Browder型不动点定理.     

Whitney定理的一个推广形式

Whitney's theorem is a famous theorem in the local singularity theory. In this paper, as an application of Malgrange preparation theorem, a generalized form of Whitney's theorem will be derived.     

A KY FAN MATCHING THEOREM FOR TRANSFER COMPACTLY OPEN COVERS AND THE APPLICATION TO THE FIXED POINT

In this article, a Ky Fan matching theorem for transfer compactly open covers is established. As applications, a Fan-Browder coincidence theorem, a Ky Fan best approximation theorem and a Brouwer-Schauder-Rothe type fixed point theorem are obtained.