关于实数Cantor级数展开的一个注记

本文主要研究实数的Cantor级数展开式. 通过构造Moran集的方法, 确定了由Cantor级数中不同字符个数的渐近值所定义的一类集合的Hausdorff维数. 本文结果可视为Erdös 和Renyi关于Cantor级数统计性质研究的补充.     

一类齐次Moran集的维数

本文研究了一类特殊的齐次Moran集的维数.将齐次均匀Cantor集通过一系列平移,获得了一类特殊的齐次Moran集并得到了它们维数的精确值,推广了齐次均匀Cantor集维数的计算公式.     

Cantor序列的Hankel行列式

本文给出Cantor序列的Hankel行列式的递推公式,并证明其Hankel行列式是二维的3-自动机序列.利用Cantor序列的Hankel行列式,本文证明Cantor数(即b-进制展式为Cantor序列的超越数)的无理指数等于2.     

关于Cantor三分集定义方式的一个新进展

通过对Cantor三分集本质结构进行深刻洞察,并结合结构化方法对Cantor三分集的定义方式进行研究.定义了?上的无再生三分族、Cantor三分族等概念,证明了Cantor三分集的一个非三进制数模式的新公式,并证明了一个关于三进制数的新公式.为Cantor三分集的"三进制数定义法"提供了一种严密的理论基础,同时也增进了...     

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.     

由Cantor展式定义的Besicovitch-Eggleston子集

本文研究了由Cantor展式所确定的一类Besicovitch-Eggleston子集.应用Billingsley定理,得到了这类集合的维数.并且表明无穷符号空间和有限符号空间上的Besicovitch-Eggleston子集的性质是有区别的.     

Rd中一类齐次Moran集的盒维数

给定Rd 中的Moran集类 ,本文证明了对介于该集类中元素的上盒维数的最大值和最小值之间的任何一个数值s,总存在该集类中的一个元素 ,其上盒维数等于s,对下盒维数、修正的下盒维数也有类似的性质成立 ,从而给文 [1 ]中的猜想 1一个肯定的回答 .此外 ,还讨论了齐次Cantor集和偏次Cantor集盒维数存在性之间的关系 .     

正测度Cantor集与正测度Cantor函数

本文利用类比的方法,将Cantor集上定义的Cantor函数进行了推广.首先给出了正测度Cantor集及正测度Cantor函数的定义;然后通过严格的证明得到了正测度Cantor函数的一些性质,并给出了正测度Cantor函数的一些应用;最后通过实例说明,由于正测度Cantor函数构造的特殊性,可以用来作为一些命题的反例.     

一类非线性Cantor集维数的计算机估值

本文根据连分数性质及压缩变换的特征,给出了一类非线性Cantor集维数的估值算法,得到了其Hausdorff维数的较好上、下界.证明了只要计算机存储量足够,此上、下界可无限逼近维数的准确值.     

β-展开的水平集的维数

本文研究一类由β-展开所形成的水平集的分形结构.通过构造一族非齐次Cantor集,得到了这类水平集的Hausdorff维数,从而给出了这类集合大小的一个刻画.推广了文献[1]和[4]的结果.     

Operation properties and -equalities of complex fuzzy sets

A complex fuzzy set is a fuzzy set whose membership function takes values in the unit circle in the complex plane. This paper investigates various operation properties and proposes a distance measure for complex fuzzy sets. The distance of two complex fuzzy sets measures the difference between the grades of two complex fuzzy sets as well as that between the phases of the two complex fuzzy sets. This distance measure is then used to define δ-equalities of complex fuzzy sets which coincide with those of fuzzy sets already defined in the literature if complex fuzzy sets reduce to real-valued fuzzy sets. Two complex fuzzy sets are said to be δ-equal if the distance between them is less than 1-δ. This paper shows how various operations between complex fuzzy sets affect given δ-equalities of complex fuzzy sets. An example application of signal detection demonstrates the utility of the concept of δ-equalities of complex fuzzy sets in practice.     

Effectively and Noneffectively Nowhere Simple Sets

R. Shore proved that every recursively enumerable (r. e.) set can be split into two (disjoint) nowhere simple sets. Splitting theorems play an important role in recursion theory since they provide information about the lattice ? of all r. e. sets. Nowhere simple sets were further studied by D. Miller and J. Remmel, and we generalize some of their results. We characterize r. e. sets which can be split into two (non) effectively nowhere simple sets, and r. e. sets which can be split into two r. e. non-nowhere simple sets. We show that every r. e. set is either the disjoint union of two effectively nowhere simple sets or two noneffectively nowhere simple sets. We characterize r. e. sets whose every nontrivial splitting is into nowhere simple sets, and r. e. sets whose every nontrivial splitting is into effectively nowhere simple sets. R. Shore proved that for every effectively nowhere simple set A, the lattice L* (A) is effectively isomorphic to ?*, and that there is a nowhere simple set A such that L*(A) is not effectively isomorphic to ?*. We prove that every nonzero r. e. Turing degree contains a noneffectively nowhere simple set A with the lattice L*(A) effectively isomorphic to ?*. Mathematics Subject Classification: 03D25, 03D10.     

A property of creative sets

A new approach to the study of creative sets using the notion of a table is offered. Making use of tables conforming to recursively enumerable sets, novel properties of creative sets are established. Harrington's theorem on the definability of creative sets in the lattice of recursively enumerable sets is proved, and we reprove Lachlan's theorem which states that one of the factors in a direct product of creative sets is again creative. Supported by RFFR grant No. 93-01-16014. Translated fromAlgebra i Logika, Vol. 35, No. 3, pp. 294–307, May–June, 1996.     

Exclusive and essential sets of implicates of Boolean functions

In this paper we study relationships between CNF representations of a given Boolean function f and certain sets of implicates of f. We introduce two definitions of sets of implicates which are both based on the properties of resolution. The first type of sets, called exclusive sets of implicates, is shown to have a functional property useful for decompositions. The second type of sets, called essential sets of implicates, is proved to possess an orthogonality property, which implies that every CNF representation and every essential set must intersect. The latter property then leads to an interesting question, to which we give an affirmative answer for some special subclasses of Horn Boolean functions.     

Friedberg Numberings of Families of n-Computably Enumerable Sets

We establish a number of results on numberings, in particular, on Friedberg numberings, of families of d.c.e. sets. First, it is proved that there exists a Friedberg numbering of the family of all d.c.e. sets. We also show that this result, patterned on Friedberg's famous theorem for the family of all c.e. sets, holds for the family of all n-c.e. sets for any n>2. Second, it is stated that there exists an infinite family of d.c.e. sets without a Friedberg numbering. Third, it is shown that there exists an infinite family of c.e. sets (treated as a family of d.c.e. sets) with a numbering which is unique up to equivalence. Fourth, it is proved that there exists a family of d.c.e. sets with a least numbering (under reducibility) which is Friedberg but is not the only numbering (modulo reducibility).     

Multi-fuzzy sets: An extension of fuzzy sets

In this paper we propose a method to construct more general fuzzy sets using ordinary fuzzy sets as building blocks. We introduce the concept of multi-fuzzy sets in terms of ordered sequences of membership functions. The family of operations T, S, M of multi-fuzzy sets are introduced by coordinate wise t-norms, s-norms and aggregation operations. We define the notion of coordinate wise conjugation of multifuzzy sets, a method for obtaining Atanassov’s intuitionistic fuzzy operations from multi-fuzzy sets. We show that various binary operations in Atanassov’s intuitionistic fuzzy sets are equivalent to some operations in multi-fuzzy sets like M operations, 2-conjugates of the T and S operations. It is concluded that multi-fuzzy set theory is an extension of Zadeh’s fuzzy set theory, Atanassov’s intuitionsitic fuzzy set theory and L-fuzzy set theory.     

A note on non-support points, negligible sets, Gâteaux differentiability and Lipschitz embeddings

This paper first presents a characterization of three classes of negligible closed convex sets (i.e., Gauss null sets, Aronszajn null sets and cube null sets) in terms of non-support points; then gives a generalization of Gâteaux differentiability theorems of Lipschitz mapping from open sets to those closed convex sets admitting non-support points; and as their application, finally shows that a closed convex set in a separable Banach space X can be Lipschitz embedded into a Banach space Y with the Radon–Nikodym property if and only if the closure of its linear span is linearly isomorphic to a closed subspace of Y.     

基于粗糙模糊集的聚类决策及应用

对基于粗糙模糊集的聚类问题进行了研究.首先,定义了粗糙模糊值与粗糙模糊集的相似度,给出了粗糙模糊集相似度的一般表达式;然后利用该公式构建了模糊相似矩阵,进而给出了粗糙模糊集的一种聚类方法;最后以实际算例说明了这种思想.     

Efficient edge domination in regular graphs

An induced matching of a graph G is a matching having no two edges joined by an edge. An efficient edge dominating set of G is an induced matching M such that every other edge of G is adjacent to some edge in M. We relate maximum induced matchings and efficient edge dominating sets, showing that efficient edge dominating sets are maximum induced matchings, and that maximum induced matchings on regular graphs with efficient edge dominating sets are efficient edge dominating sets. A necessary condition for the existence of efficient edge dominating sets in terms of spectra of graphs is established. We also prove that, for arbitrary fixed p≥3, deciding on the existence of efficient edge dominating sets on p-regular graphs is NP-complete.