A theorem of the closed graph type

The author proves a theorem about systems of sets which generalizes the closed graph theorem. To illustrate its application proofs, are given of several generalizations of the closed graph theorem, of selection theorems, a transitivity theorem for C^...-algebras and factorization theorems in Banach algebras.     

Cayley-Hamilton定理的两种简单证法和两个表现定理

本文对Cayley-Hamilton定理提出两种简单证法并应用该定理和罗必塔法则给出两个有关二阶张量函数的表现定理.     

高等代数中两个定理的新证明

本文用新方法证明了关于行列式的 Laplace展开定理和柯西——别内公式 .     

完全覆盖和加标分割在分析中的应用

通过对数学分析中众多定理的证法的分析,说明利用文中的两个引理可使一些定理的证法既直接又易于接受,使黎曼可积的一个充要条件的证明简捷,文章最后给出了罗尔中值定理的一种简单的新证法.     

Intersections of spherically convex sets

New theorems of Helly type are proved concerning the intersection of convex cones with a common vertex or, equivalently, the intersection of sets on a sphere which are convex in the sense of Robinson. The proofs of these theorems are based on a lemma which is a spherical analog and generalization of Radon's theorem.     

Teaching proof in the context of physics

This paper describes an application of statics to geometrical proofs in the classroom. The aim of the study was to find out whether the use of concepts and arguments from statics can help students understand and produce proofs of geometrical theorems. The two theorems studied were (1) that the medians in a triangle meet at a single point which is the centre of gravity of the triangle, and (2) the Varignon theorem, that the lines joining the midpoints of successive sides of a quadrilateral form a parallelogram. The classroom experiment showed that most students were successful in using arguments from statics in their proofs, and that they gained a better understanding of the theorems. These findings lend support to the claim that the introduction of statics helps students produce proofs and grasp their meaning.     

New short proofs to some stability theorems

We present new short proofs to both the exact and the stability result of two extremal problems. The first result is about the extension of Turán’s theorem to hypergraphs, and the second result is about cancellative hypergraphs. Our proofs are concise and straightforward, but give a sharper version of stability theorems to both problems.     

A gentle guide to the basics of two projections theory

This paper is a survey of the basics of the theory of two projections. It contains in particular the theorem by Halmos on two orthogonal projections and Roch, Silbermann, Gohberg, and Krupnik’s theorem on two idempotents in Banach algebras. These two theorems, which deliver the desired results usually very quickly and comfortably, are missing or wrongly cited in many recent publications on the topic, The paper is intended as a gentle guide to the field. The basic theorems are precisely stated, some of them are accompanied by full proofs, others not, but precise references are given in each case, and many examples illustrate how to work with the theorems.     

几何定理机器证明三十年

由于传统的兴趣和多种原因,几何定理的机器证明在自动推理的研究中占有重要的地位.自吴法发表至今30年,几何定理机器证明的研究和实践有了很大的进展.对无序几何命题而言,代数方法、数值方法均能有效地判定其真假,面积法(消点法)、搜索法更能生成其可读的证明.几何不等式机器证明的研究,由于多项式完全判别系统的建立,也有了突破.研究领域已由机器证明扩展为包括几何作图在内的一般几何问题的机器求解,并有了实际的应用.     

Matrix methods in decompositions of modules

Matrix methods of elementary linear algebra are extended to general direct-sum decompositions of modules. These methods are then shown to yield simple proofs of some well-known theorems, notably the Beck-Warfield “mutual exchange property” and the Krull-Schmidt theorem.     

Elementary submodels in infinite combinatorics

The use of elementary submodels is a simple but powerful method to prove theorems, or to simplify proofs in infinite combinatorics. First we introduce all the necessary concepts of logic, then we prove classical theorems using elementary submodels. We also present a new proof of Nash-Williams’s theorem on cycle decomposition of graphs, and finally we improve a decomposition theorem of Laviolette concerning bond-faithful decompositions of graphs.     

Using dynamic geometry to explore non-traditional theorems

The purpose of this article is to provide examples of ‘non-traditional’ theorems that can be explored in a dynamic geometry environment by university and high school students. These theorems were encountered in the dynamic geometry environment. The author believes that teachers can ask their students to construct proofs for these theorems. The Appendix contains proof outlines for each theorem.     

Compactness and semi-continuity

The purpose of this note is to point out that a recent result of Ceder yields easily converses to well known theorems of Wallace and Birkhoff and thus provides two new characterazations of compactness as well as specifying the class of spaces for which the theorems are true. A very slight extension of Ceder’s theorem is also obtained, as well as new and simple proofs.     

Top formal local cohomology module

Periodica Mathematica Hungarica - I give fully detailed proofs of two important theorems—the exact solution of the weak clique game and the compactness theorem—in the theory of...     

Characterization of weak convergence of Birkhoff sums for Gibbs-Markov maps

We investigate limit theorems for Birkhoff sums of locally Hölder functions under the iteration of Gibbs-Markov maps. Aaronson and Denker have given sufficient conditions to have limit theorems in this setting. We show that these conditions are also necessary: there is no exotic limit theorem for Gibbs-Markov maps. Our proofs, valid under very weak regularity assumptions, involve weak perturbation theory and interpolation spaces. For L ² observables, we also obtain necessary and sufficient conditions to control the speed of convergence in the central limit theorem.     

Quelques theoremes de dualite combinatoire

The purpose of this paper is to prevent some new and unified proofs of a number of known results in combinatorial programming: generalized versions of Minty's lemma, of Ford and Fulkerson's and Hoffman's theorems the length-width inequality and the strong complementary slackness theorem. All these results are derived from a “main duality theorem” which can be deduced from the duality theorem of linear programming or from more general results of convex analysis.     

Spine Decompositions and Limit Theorems for a Class of Critical Superprocesses

In this paper we first establish a decomposition theorem for size-biased Poisson random measures. As consequences of this decomposition theorem, we get a spine decomposition theorem and a 2-spine decomposition theorem for some critical superprocesses. Then we use these spine decomposition theorems to give probabilistic proofs of the asymptotic behavior of the survival probability and Yaglom’s exponential limit law for critical superprocesses.

     

不完全资产与现货市场的两期交换经济的非套利均衡的存在性

本文在第一章给出了经济模型及相关的基本概念,第二章中利用Grassmanian流形上的集值映射的不动点理论及非线性泛函分析中一类方程解的有限性定理,给出了不完全资产与现货市场的两期交换经济的非套利均衡存在的简捷证明.     

Stability of Functional Equations in Several Variables

We prove a generalization of Hyers＇ theorem on the stability of approximately additive mapping and a generalization of Badora＇s theorem on an approximate ring homomorphism. We also obtain a more general stability theorem, which gives the stability theorems on Jordan and Lie homomorphisms. The proofs of the theorems given in this paper follow essentially the D. H. Hyers-Th. M. Rassias approach to the stability of functional equations connected with S. M. Ulam＇s problem.     

Converse duality in Banach space

Two theorems on converse duality are obtained for mathematical programs in Banach spaces. The proofs are based on a Banach-space generalization of the F. John necessary condition for a constrained minimum. No use is made of Kuhn-Tucker constraint qualifications. In the second theorem, the primal program contains a nonlinear equality constraint, and a converse duality theorem is obtained, using a modified concept of convexity; this result appears new, even for finite-dimensional programs. The results are applied to a problem in optimal control.