PRECOMPACTNESS IN FUZZY UNIFORM SPACES

Abstract

The notion of a precompact fuzzy set in a fuzzy uniform space is defined and it is shown that this is a good extension of the standard notion. A theory of precompact fuzzy sets is developed using the previously defined notion of a Cauchy prefilter in a fuzzy uniform space and this theory generalises standard theory.     

Fuzzy metric spaces

In this paper, fuzzy metric spaces are redefined, different from the previous ones in the way that fuzzy scalars instead of fuzzy numbers or real numbers are used to define fuzzy metric. It is proved that every ordinary metric space can induce a fuzzy metric space that is complete whenever the original one does. We also prove that the fuzzy topology induced by fuzzy metric spaces defined in this paper is consistent with the given one. The results provide some foundations for the research on fuzzy optimization and pattern recognition.     

L-拓扑空间的弱局部强F紧性及单点强F紧化

以R.Lowen的强F紧性为基础,定义了L-拓扑空间的弱局部强F紧性及单点强F紧化,推广了有关弱局部紧拓扑空间和拓扑空间的单点紧化的若干结果,证明了L-拓扑空间的弱局部强F紧性是拓扑空间的弱局部紧性的L-推广。     

Fuzzy discriminant analysis in fuzzy groups

This paper deals with discriminant problems to classify samples with fuzzy multi-attribute into fuzzy groups. In this problem, the objective is to determine the linear discriminant function that provides the maximum separation of fuzzy groups in a real space. For this purpose, we define the fuzzy variance ratio and employ maximization of the fuzzy variance ratio as a criterion. Furthermore, a partial correlation coefficient in the fuzzy groups is defined to estimate the influence of each attribute itself on the discrimination between fuzzy groups.     

Fuzzy bounded linear operators

In this paper, a notion of boundedness of a linear operator from a fuzzy normed linear space to another fuzzy normed linear space is introduced and two types (strong and weak) of fuzzy bounded linear operators are defined. Relation between fuzzy continuity and fuzzy boundedness is studied. Definitions of fuzzy bounded linear functionals are given and the notions of fuzzy dual spaces are developed. The Hahn-Banach theorem, the Open mapping theorem, the Closed graph theorem and the Uniform boundedness principle theorem are established.     

Pareto Simulated Annealing for Fuzzy Multi-Objective Combinatorial Optimization

The paper presents a metaheuristic method for solving fuzzy multi-objective combinatorial optimization problems. It extends the Pareto simulated annealing (PSA) method proposed originally for the crisp multi-objective combinatorial (MOCO) problems and is called fuzzy Pareto simulated annealing (FPSA). The method does not transform the original fuzzy MOCO problem to an auxiliary deterministic problem but works in the original fuzzy objective space. Its goal is to find a set of approximately efficient solutions being a good approximation of the whole set of efficient solutions defined in the fuzzy objective space. The extension of PSA to FPSA requires the definition of the dominance in the fuzzy objective space, modification of rules for calculating probability of accepting a new solution and application of a defuzzification operator for updating the average position of a solution in the objective space. The use of the FPSA method is illustrated by its application to an agricultural multi-objective project scheduling problem.     

Elementary fuzzy calculus

In [2] Dubois and Prade introduced the notion of fuzzy numbers and defined the basic operations of addition, subtraction, multiplication, and division. A slightly modified definition of fuzzy numbers was presented in [4], and in that paper a metric was defined for this family of fuzzy sets. Another less restrictive definition of fuzzy numbers can be found in [1].In the present paper we consider fuzzy numbers from a somewhat different perspective. Basically, we shall view fuzzy numbers in a topological vector space setting. Using the customary vector space operations together with the metric given in [4] we will define differentiation and integration of fuzzy-valued functions in ways that parallel closely the corresponding definitions for real differentiation and integration.     

Fuzzy dynamical systems

A fuzzy dynamical system on an underlying complete, locally compact metric state space X is defined axiomatically in terms of a fuzzy attainability set mapping on X. This definition includes as special cases crisp single and multivalued dynamical systems on X. It is shown that the support of such a fuzzy dynamical system on X is a crisp multivalued dynamical system on X, and that such a fuzzy dynamical system can be considered as a crisp dynamical system on a state space of nonempty compact fuzzy subsets of X. In addition fuzzy trajectories are defined, their existence established and various properties investigated.     

Imprecise set and fuzzy valued probability

Set valued probability and fuzzy valued probability theory is used for analyzing and modeling highly uncertain probability systems. In this paper the set valued probability and fuzzy valued probability are defined over the measurable space. They are derived from a set and fuzzy valued measure using restricted arithmetics. The range of set valued probability is the set of subsets of the unit interval and the range of fuzzy valued probability is the set of fuzzy sets of the unit interval. The expectation with respect to set valued and fuzzy valued probability is defined and some properties are discussed. Also, the fuzzy model is applied to binomial model for the price of a risky security.     

模糊多属性决策的投影折衷方法

基于矢量投影的思想,导出了分量为L-R型梯形模糊数的模糊矢量投影的计算公式。通过将加权后的方案矢量投影到理想解上,再将负理想解投影到方案矢量上,进而在两个投影的基础上构建方案与理想解的相对贴近度,用以确定多属性决策方案的优劣次序。同时,本文以实例对这一决策方法进行了说明。     

模糊多属性决策的加权投影折衷方法

基于矢量投影的思想,建立了分量为L-R型梯形模糊数的模糊矢量间投影的计算公式,把加权后的方案矢量投影到理想解上,负理想解投影到方案矢量上,以这两个投影构造方案与理想解的相对贴近度,来确定方案的优劣次序.并通过实例对这一决策方法进行说明.     

模糊一致环

本文定义了模糊一致环概念,研究了它与模糊拓扑环的关系及它与模糊一致空间的关系;给出了借助于环的模糊子集族对模糊一致环的刻画,还引入了模糊一致子环,模糊一致剩余类环与模糊一致环的直积;并讨论了它们的分离性.     

L-Smooth紧

在模糊格上讨论Smooth拓扑空间的Smooth紧性,它定位于每一个模糊集,并兼顾整个Smooth拓扑空间。     

有关拓扑学与覆盖粗糙集的一些结论

引入了拓扑覆盖的概念,并结合最小描述元对有限论域上的拓扑覆盖加于研究,得出了拓扑覆盖的最简覆盖和基与最小描述元之间的关系.介绍了在基于有限论域U上的覆盖,构造U上的一个拓扑的方法.并且在最小描述元的基础上将划分下的粗糙隶属函数推广至一般覆盖下的粗糙隶属函数,而后介绍了其相关运用.     

模糊一致体

定义了模糊一致体概念,研究了它与模糊拓扑体的关系及它与模糊一致空间的关系;给出了借助于体的模糊子集族对模糊一致体的刻画,还引入了模糊一致子体,模糊一致商体与模糊一致体的直积;并讨论了它们的分离性.     

Lusin's theorem on fuzzy measure spaces

In this paper, we show that weakly null-additive fuzzy measures on metric spaces possess regularity. Lusin's theorem, which is well-known in classical measure theory, is generalized to fuzzy measure space by using the regularity and weakly null-additivity. A version of Egoroff's theorem for the fuzzy measure defined on metric spaces is given. An application of Lusin's theorem to approximation in the mean of measurable function on fuzzy measure spaces is presented.     

Fuzzy random renewal process and renewal reward process

So far, there have been several concepts about fuzzy random variables and their expected values in literature. One of the concepts defined by Liu and Liu (2003a) is that the fuzzy random variable is a measurable function from a probability space to a collection of fuzzy variables and its expected value is described as a scalar number. Based on the concepts, this paper addresses two processes—fuzzy random renewal process and fuzzy random renewal reward process. In the fuzzy random renewal process, the interarrival times are characterized as fuzzy random variables and a fuzzy random elementary renewal theorem on the limit value of the expected renewal rate of the process is presented. In the fuzzy random renewal reward process, both the interarrival times and rewards are depicted as fuzzy random variables and a fuzzy random renewal reward theorem on the limit value of the long-run expected reward per unit time is provided. The results obtained in this paper coincide with those in stochastic case or in fuzzy case when the fuzzy random variables degenerate to random variables or to fuzzy variables.     

Fuzzy probability measures

In [4] Höhle has defined fuzzy measures on G-fuzzy sets [2] where G stands for a regular Boolean algebra. Consequently, since the unit interval is not complemented, fuzzy sets in the sense of Zadeh [8] do not fit in this framework in a straightforward manner. It is the purpose of this paper to continue the work started in [5] which deals with [0,1]-fuzzy sets and to give a natural definition of a fuzzy probability measure on a fuzzy measurable space [5]. We give necessary and sufficient conditions for such a measure to be a classical integral as in [9] in the case the space is generated. A counterexample in the general case is also presented. Finally it is shown that a fuzzy probability measure is always an integral (if the space is generated) if we replace the operations ∧ and ∨ by the t-norm T_o and its dual S₀ (see [6]).     

拟合模糊观测数据的线性回归模型

本文讨论了实验观测数据为一般模糊数的线性最优拟合问题，通过定义模糊数空间中的距离，建立了模糊数空间到模糊数空间的回归模型，证明了最小二乘问题的解与其正则方程组的解的一致性，进而由正则方程组导出了问题的显式解。本模型的计算简便，具有实用价值。