集合论在离散数学中的作用探索

在《离散数学》课程中,集合论绝不像表面显现的那么简单,相反地,它可谓一根主线贯穿了整个《离散数学》课程,在该课程的数理逻辑、关系、图论、代数系统等部分均发挥着表达工具或内容支撑的作用.在本文中,我们就集合论在《离散数学》各部分内容中的作用进行了探索,希望所得结论能引起各位《离散数学》授课教师的重视.     

不确定性信息处理的集对论思想与方法

集对论从主体与客体联系层次上较好地实现了不确定性信息的整体处理,集对论的同异反集合和联系数等概念弥补了模糊集合论的缺陷。本文从概念上探讨了集对论的基本原理和思想方法,并与模糊集合论做了对比分析,认为集对论体现了认识过程中的辩证思维,因而具有更强的方法论意义。     

关于Hilbert空间上无限维线性定常系统的典型分解

有限维线性定常系统的典型分解定理是控制理论的重要成果之一 [1] ,也是计算机控制的重要理论基础之一 ,本文将这一结果推广到 Hilbert空间上的无限维线性定常系统 .     

无限Fuzzy关系方程的解集

对(0,1)格上的无限Fuzzy关系方程,给出其可解的充要条件.利用可达解与不可达解的概念,给出方程存在可达解与不可达解的充要条件.进一步,在解集非空时,刻画了方程的解集的结构及性质.     

To the Solution of the Hilbert Problem with Infinite Index

In this paper, we obtain a generalization of the method of regularizing multipliers for the solution of the Hilbert boundary-value problem with finite index in the theory of analytic functions to the case of an infinite power-behaved index. This method is used to obtain a general solution of the homogeneous Hilbert problem for the half-plane, a solution that depends on the existence and the number of entire functions possessing mirror symmetry with respect to the real axis and satisfying some additional constraints related to the singularity characteristic of the index. To solve of the inhomogeneous problem, we essentially use a specially constructed solution of the homogeneous problem whereby we reduce the boundary condition of the Hilbert problem to a Dirichlet problem.     

Properly Discontinuous Isometric Actions on the Unit Sphere of Infinite Dimensional Hilbert Spaces

We study the properly discontinuous and isometric actions on the unit sphere of infinite dimensional Hilbert spaces and we get some new examples of Hilbert manifold with constant positive sectional curvature. We prove some necessary conditions for a group to act isometrically and properly discontinuously, and in the case of finitely generated Abelian groups the necessary and sufficient conditions are given.     

Upper Bounds for the Hausdorff Dimension and Stratification of an Invariant Set of an Evolution System on a Hilbert Manifold

We prove a generalization of the well-known Douady–Oesterlé theorem on the upper bound for the Hausdorff dimension of an invariant set of a finite-dimensional mapping to the case of a smooth mapping generating a dynamical system on an infinite-dimensional Hilbert manifold. A similar estimate is given for the invariant set of a dynamical system generated by a differential equation on a Hilbert manifold. As an example, the well-known sine-Gordon equation is considered. In addition, we propose an algorithm for the Whitney stratification of semianalytic sets on finite-dimensional manifolds.     

对称无限布尔方阵的本原指数集的刻划

研究了对称无限布尔方阵,给出了对称无限布尔方阵为本原阵的一个充分必要条件,证明了具有有限直径d的对称无限布尔方阵的本原指数的上确界为2d,最后证明了直径不超过d的全体对称本原无限布尔方阵的本原指数集是E_d={1,2,…,2d}.     

An Interpretation of the Zermelo-Fraenkel Set Theory and the Kelley-Morse Set Theory in a Positive Theory

An interesting positive theory is the GPK theory. The models of this theory include all hyperuniverses (see [5] for a definition of these ones). Here we add a form of the axiom of infinity and a new scheme to obtain GPK_∞⁺. We show that in these conditions, we can interprete the Kelley-Morse theory (KM) in GPK_∞⁺ (Theorem 3.7). This needs a preliminary property which give an interpretation of the Zermelo-Fraenkel set theory (ZF) in GPK_∞⁺. We also see what happens in the original GPK theory. Before doing this, we first need to study the basic properties of the theory. This is done in the first two sections.     

一类本原无限布尔方阵的本原指数集的刻划

引入了本原无限布尔方阵的概念,给出了无限布尔方阵为本原阵的一个充分必要条件,最后给出了一类本原无限布尔方阵的本原指数集的刻划.     

Ideals of Ultrafilters on the Collection of Finite Subsets of an Infinite Set

Let J be an infinite set and let , i.e., I is the collection of all non empty finite subsets of J. Let denote the collection of all ultrafilters on the set I and let be the compact (Hausdorff) right topological semigroup that is the Stone-Cech Compactification of the semigroup equipped with the discrete topology. This paper continues the study of that was started in [3] and [5]. In [5], Koppelberg established that (where K( S) is the smallest ideal of a semigroup S) and for non empty she established . In this note, we show that for such that is infinite, is a proper subset of and , where .     

Regularity Properties for Solutions of Infinite Dimensional Kolmogorov Equations in Hilbert Spaces

Potential Analysis - In this article we establish regularity properties for solutions of infinite dimensional Kolmogorov equations. We prove that if the nonlinear drift coefficients, the nonlinear...     

粗糙集理论在伪BCK代数上的应用

本文把粗糙集理论应用到伪BCK代数, 作为伪BCK代数的推广, 给出了类代数近似空间上粗伪BCK代数、粗子代数和粗伪滤子的概念, 并讨论了它们的一些相关性质. 进一步研究了伪BCK 代数近似空间上粗子代数和粗伪滤子. 最后给出了几个伪BCK代数、伪滤子和子代数的验证程序.     

The Iteration Formulae of the Maslov-Type Index Theory in Weak Symplectic Hilbert Space

The authors prove a splitting formula for the Maslov-type indices of symplectic paths induced by the splitting of the nullity in weak symplectic Hilbert space. Then a direct proof of the iteration formulae for the Maslov-type indices of symplectic paths is given.