关于n维欧氏空间R^n中闭集的教学思考

结合教材《工科数学分析基础》,对n维欧氏空间R^n中闭集的教学进行探讨和设计,同时分别给出了导集、闭包这两类特殊闭集的特征和性质．     

Locating and paired-dominating sets in graphs

In this paper, we continue the study of paired-domination in graphs introduced by Haynes and Slater [T.W. Haynes, P.J. Slater, Paired-domination in graphs, Networks 32 (1998), 199–206]. A paired-dominating set of a graph G with no isolated vertex is a dominating set S of vertices whose induced subgraph has a perfect matching. We consider paired-dominating sets which are also locating sets, that is distinct vertices of G are dominated by distinct subsets of the paired-dominating set. We consider three variations of sets which are paired-dominating and locating sets and investigate their properties.     

拟多项式的迭代

讨论了更广泛的拟多项式映射,研究了拟多项式的迭代,证明了关于逃逸集,充满 Julia集和Julia集的几个定理.推广了多项式动力系统的相关结果.     

An extension of building sets and relative difference sets

Building sets are a successful tool for constructing semi‐regular divisible difference sets and, in particular, semi‐regular relative difference sets. In this paper, we present an extension theorem for building sets under simple conditions. Some of the semi‐regular relative difference sets obtained using the extension theorem are new in the sense that their ambient groups have smaller ranks than previously known. © 2000 John Wiley & Sons, Inc. J Combin Designs 8: 50–57, 2000     

Fuzzy sets-the approximation of semisets

This paper deals with alternative set theory which substantially departs from classical set theory. The main notion of it -a semiset-is proposed to model the same intuitive notion as fuzzy set but it is more general. The reasonable tool for ‘grasping’ semisets could be provided by fuzzy sets whose applicability is beyond discussion. In the paper reasons for such an approximation are given and two ways how to provide it are proposed. The fact that the membership function is modeled within the theory should be stressed. At the end, some problems and an indication for further investigation are discussed.     

On Pseudocomplemented and Stone Ordered Sets

The aim of this paper is to characterize both the pseudocomplemented and Stone ordered sets in a manner similar to that used previously for Boolean and distributive ordered sets. The sublattice G(A) of the Dedekind–Mac Neille completion DM(A) of an ordered set A generated by A is said to be the characteristic lattice of A. We will show that there are distributive pseudocomplemented ordered sets whose characteristic lattices are not pseudocomplemented. We can define a stronger notion of pseudocomplementedness by demanding that both A and G(A) be pseudocomplemented. It turns out that the two concepts are the same for finite and Stone ordered sets.     

Strongly Amorphous Sets and Dual Dedekind Infinity

1. If A is strongly amorphous (i.e., all relations on A are definable), then its power set P(A) is dually Dedekind infinite, i. e., every function from P(A) onto P(A) is injective. 2. The class of “inexhaustible” sets is not closed under supersets unless AC holds.     

Complexity properties of recursively enumerable sets andsQ-completeness

The notions of effectively subcreative set and strongly effectively acceleratable set are introduced. It is proved that the notions of effectively subcreative set, strongly effectively acceleratable set, andsQ-complete recursively enumerable set are equivalent. Translated fromMatematicheskie Zametki, Vol. 62, No. 3, pp. 425–429, September, 1997. Translated by V. N. Dubrovsky     

物元可拓集集合性质研究

在可拓集合概念基础上,提出了向量可拓集和区间可拓集概念,研究了物元可拓集合的交、并、求补等集合运算及其性质.     

On πgs-closed sets in topological spaces

Summary A new class of sets called πgs-closed sets is introduced and its properties are studied. Moreover the notions of πgs-T_1/2 spaces and πgs-continuity are introduced.     

Classification of non‐well‐founded sets and an application

A complete list of Finsler, Scott and Boffa sets whose transitive closures contain 1, 2 and 3 elements is given. An algorithm for deciding the identity of hereditarily finite Scott sets is presented. Anti‐well‐founded (awf) sets, i. e., non‐well‐founded sets whose all maximal ∈‐paths are circular, are studied. For example they form transitive inner models of ZFC minus foundation and empty set, and they include uncountably many hereditarily finite awf sets. A complete list of Finsler and Boffa awf sets with 2 and 3 elements in their transitive closure is given. Next the existence of infinite descending ∈‐sequences in Aczel universes is shown. Finally a theorem of Ballard and Hrbá?ek concerning nonstandard Boffa universes of sets is considerably extended.     

Topology vs generalized rough sets

This paper investigates the relationship between topology and generalized rough sets induced by binary relations. Some known results regarding the relation based rough sets are reviewed, and some new results are given. Particularly, the relationship between different topologies corresponding to the same rough set model is examined. These generalized rough sets are induced by inverse serial relations, reflexive relations and pre-order relations, respectively. We point that inverse serial relations are weakest relations which can induce topological spaces, and that different relation based generalized rough set models will induce different topological spaces. We proved that two known topologies corresponding to reflexive relation based rough set model given recently are different, and gave a condition under which the both are the same topology.     

Solution of the Cameron-Erdös problem for groups of prime order

A subset A of a group G is sum-free if a + b does not belong to A for any a, b ∈ A. Asymptotics of the number of sum-free sets in groups of prime order are proved.     

一类非线性压缩生成的不变集的局部维数

本文以有界变差原理和有界畸变原理为工具,讨论了满足开集条件,且二次可微的一族递增的非线性压缩映射所产生的不变集的局部维数。     

Almost semirecursive sets

LetA be a subset of , and leta∉A. The setA is said to be almost semirecursive, if there is a two-place general recursive functionf such thatf(x, y)ε{x, y, a}∧({x, y}⊆A⇌f(x, y)εA) for all . Among other facts, it is proved that ifA and are almost semirecursive sets, thenA is a semirecursive set, and that there exists a wsr^*-set that is neither a wsr-nor an almost semirecursive set. Translated fromMatematicheskie Zametki, Vol. 66, No. 2, pp. 188–193, August, 1999.     

高阶非线性常微分方程组的可积类型

线性微分方程在物理学,力学及控制论中有着广泛应用,近年来在国内外一些重要刊物上发表多篇有关求微分方程精确解的论文。本文在这些论文的基础上,借助于Lcibniz求导公式及变换组法,进而提出了高阶非线性常微分方程组的求解法,于是论证了它的可积性,所得结果是相应文献结果的推广。     

拟正则双序集用矩形双序集的余扩张

称双序集E为双序集F用矩形双序集的余扩张,若存在满双序集态射θ：E→F,使对每个α∈F,αθ－1是E的矩形双序子集．本文讨论了拟正则双序集的这种余扩张的性质,给出了它们的结构．作为应用,证明了拟正则的硬双序集实为正则双序集．     

Some remarks on good sets

It is shown that (1) if a good set has finitely many related components, then they are full, (2) loops correspond one-to-one to extreme points of a convex set. Some other properties of good sets are discussed.