Cantor集的性质及应用

Cantor集是实函数论中一类重要的集合.本文从定义、性质及应用三方面研究了Cantor集.目的是帮助初学者对Cantor集有一个较全面的认识.     

与一拓扑紧性反例相关的结果

有反例表明一个紧致拓扑空间不一定是序列紧致的拓扑空间 .我们给出了一个与此反例密切相关的分析结果 ,表明由任一非常值的单边连续周期函数 ,都可构造一个这样的反例 .     

A NOTE OF THE SPECTRUM OF HOMOGENEOUS COMPACT OPERATORS

In this paper, we present some counterexamples which show that there is no theory on the spectrum of homogeneous compact operators which parallels the Riesz-Schauder theory on the spectrum of linear compact operators. These counterexamples also illustrate that it is impossible to study in a unified setting the Fucik spectrum of the Laplacian: -△w = au+ - bu- inΩand u = 0 on (?)Ω, as well as the spectrum of the p-Laplacian: -div(|(?)u| p-2(?)u) = λ|u|p-2u and u = 0 on (?)Ω.     

Rethinking the discovery function of proof within the context of proofs and refutations

Proof and proving are important components of school mathematics and have multiple functions in mathematical practice. Among these functions of proof, this paper focuses on the discovery function that refers to invention of a new statement or conjecture by reflecting on or utilizing a constructed proof. Based on two cases in which eighth and ninth graders engaged in proofs and refutations, we demonstrate that facing a counterexample of a primitive statement can become a starting point of students’ activity for discovery, and that a proof of the primitive statement can function as a useful tool for inventing a new conjecture that holds for the counterexample. An implication for developing tasks by which students can experience this discovery function is mentioned.     

对n-colour有序分拆的一些注记(英文)

本文研究了A.K.Agawarl在文献[1]中给出的n-colour有序分拆的组合性质.利用反例说明其中一个性质的不完全性,并纠正了此性质.此外,还给出了n-colour有序分拆组合性质的两个双射.     

概率论中一类矩不等式的证明

运用证明单边车贝雪夫不等式的方法证明了几个矩不等式,并通过反例说明文献[4]中的一结论是错误的.     

复变函数教学法探讨

复变函数是高等院校工科专业一门重要的基础理论课,它是高等数学的扩展.本文主要探讨在复变函数的教学过程中,教师可以利用"类比"、"转化"和"反例"等教学方法,使学生达到巩固旧知识、探索新课程的目的.     

Complements of Intersections in Constructive Mathematics

We examine, from a constructive perspective, the relation between the complements of S, T, and S ∩ T in X, where X is either a metric space or a normed linear space. The fundamental question addressed is: If x is distinct from each element of S ∩ T, if s ? S, and if t ? T, is x distinct from s or from t? Although the classical answer to this question is trivially affirmative, constructive answers involve Markov's principle and the completeness of metric spaces. Mathematics Subject Classification: 03F65, 46S30.     

On the proximinality of the unit ball of proximinal subspaces in Banach spaces: A counterexample

A known, and easy to establish, fact in Best Approximation Theory is that, if the unit ball of a subspace of a Banach space is proximinal in , then itself is proximinal in . We are concerned in this article with the reverse implication, as the knowledge of whether the unit ball is proximinal or not is useful in obtaining information about other problems. We show, by constructing a counterexample, that the answer is negative in general.