基于结构元的模糊值函数的一般表示方法

文[1]提出了模糊结构元的概念,并给出了模糊数与模糊值函数的模糊结构元表示,以及一类由模糊结构元线性生成的模糊值函数的微分和积分(黎曼意义下的)的表达形式。本文在文[1]的基础上进一步给出了由模糊结构元生成的模糊值函数的一般表达形式,并得到了一般表达形式下的模糊值函数的连续性和微分、黎曼积分的定义,它们与传统模糊分析中相应定义是等价的。     

模糊数测度空间上模糊值函数的模糊值积分的Fubini定理

建立卡氏积空间 (X×Y,S× T)上的乘积区间值测度和乘积模糊值测度 ,证明模糊值测度空间上模糊值函数的模糊值积分的 Fubini定理。     

模糊数值函数的可导性

把经典分析的方法与模糊分析相结合,利用R.Goetschel和W.Voxman的模糊数值函数的导数的定义,讨论了模糊数值映射的间断点的性质、微分并得到了单调模糊数值函数的可导性与弱导数之间的关系.     

复模糊值函数的收敛性

复模糊值函数是定义在实数集R上取值于F(C)(所有的复模糊数的集合)中的复模糊数的函数.将在新的序关系意义下,定义复模糊值函数的极限,并讨论复模糊值函数的收敛性质及Cauchy收敛判别法等.     

模糊值函数的曲线积分

针对扩张原理在模糊值函数曲线积分中的遍历性问题,根据模糊结构元理论给出了第一型模糊值函数曲线积分的定义.然后通过讨论平面模糊矢量的投影问题,给出了第二型模糊值函数曲线积分定义并对其相关性质进行了讨论.研究结果不仅丰富了模糊分析学理论,而且为具有不确定性因素的工程实践提供了方法依据.     

模糊数值函数积分原函数的可导性问题

基于模糊分析学微分理论的推广和模糊微分方程求解的需要，对模糊数值函数积分原函数的可导性问题进行讨论，完备模糊数值函数微积分的理论体系。     

Two-Parameter Fuzzy-Valued Stochastic Integrals and Equations

This article is concerned with notions of fuzzy-valued stochastic integrals driven by two-parameter martingales and increasing processes. We present their main properties and formulate next two-parameter fuzzy-valued stochastic integral equations. We establish the existence and uniqueness of solutions to such equations as well as their additional properties.     

The Mcshane Integral of Fuzzy-Valued Functions

In this paper, the Mcshane integrals of fuzzy-valued functions are defined, discussed, and characterized. It shows that, in a sense, the Mcshane integrals of interval-valued functions and fuzzy-valued functions are the Riemann-type definitions of the Aumann integral and (K) integral respectively.AMS Subject Classification (2000) 28EI0 (04A72)The authors are supported by the Special Fund of China for Ph.D. Instructors in Universities     

模糊值Choquet积分(Ⅱ)--函数关于模糊值模糊测度的Choquet积分

研究一种取值于模糊数集的Choquet积分，该积分的被积函数是单值函数，所用的测度是模糊值模糊测度。给出其定义、性质和收敛定理。     

Fuzzy-valued integrals based on a constructive methodology

The procedures for constructing a fuzzy number and a fuzzy-valued function from a family of closed intervals and two families of real-valued functions, respectively, are proposed in this paper. The constructive methodology follows from the form of the well-known “Resolution Identity” (decomposition theorem) in fuzzy sets theory. The fuzzy-valued measure is also proposed by introducing the notion of convergence for a sequence of fuzzy numbers. Under this setting, we develop the fuzzy-valued integral of fuzzy-valued function with respect to fuzzy-valued measure. Finally, we provide a Dominated Convergence Theorem for fuzzy-valued integrals.     

模糊分析计算中的结构元方法

提出模糊结构元的概念，研究模糊结构元的性质，给出模糊数和模糊值函数的结构元表现定理。利用模糊数和模糊值函数的结构元表现形式，使得过去必须依赖扩张原理和表现定理来刻画的模糊数运算、模糊值函数的微积分运算等变得更加简单与直观。     

Fuzzy ordinary differential equation (I)

Fuzzy ordinary differential equation has been studied in many ways. In this paper we put forward concepts of ordinary differential equation of a fuzzy-valued function for the first time, discuss the existence and uniqueness of the solution at common points by applying decomposition theorem in fuzzy sets as well, so that a solution is obtained for the equation. Finally the method proves to be feasible and effective by a numerical example.     

局部紧空间上模糊值模糊测度的正则性

研究局部紧Hausdorff空间上模糊值模糊测度。首先,给出模糊值模糊测度的自连续和一致自连续的定义,然后引进模糊值模糊测度正则性的概念,在自连续或一致自连续的条件下,证明了局部紧空间上模糊值模糊测度的一些定理。     

模糊值函数分析学的结构元方法简介(Ⅰ)——结构元与模糊数运算

简述了模糊值函数分析学在具体工程实践应用中存在的困难和障碍,系统地介绍了模糊结构元方法在模糊值函数分析学中的应用,包括模糊结构元的概念、模糊数的模糊结构元表示形式、基于结构元表达形式的模糊数运算与隶属函数确定.模糊结构元方法将复杂的模糊数运算转化为一类单调有界函数的运算,不仅仅为模糊分析计算的简化提供了工具,同时也为模糊值函数分析学应用的研究开创了一条新的途径.     

关于模糊值凸函数的共轭问题的研究

在Goetschel-Voxman所引进的序关系下,首先给出了模糊值凸函数的共轭函数的概念,并证明了模糊值凸函数的共轭函数是模糊值凸函数等相关性质;其次给出了模糊值凸函数的二次共轭函数的概念,并证明了相关性质;最后讨论了模糊值凸函数的共轭与下卷积之间的关系,证明了两个模糊值凸函数的共轭函数与其下卷积的共轭函数之间的等式关系.     

A generalized Borgonovo's importance measure for fuzzy input uncertainty

In this work, the way in which fuzzy uncertainty in a model's output is apportioned to fuzzy uncertainty in model inputs is studied through a sensitivity analysis. Here, an optimization technique is employed to obtain the membership functions of the fuzzy structural response, for which a global sensitivity indicator is introduced to measure the influence of fuzzy input uncertainty on fuzzy output uncertainty. The global sensitivity indicator is the important measure of the fuzzy input uncertainty, which extends Borgonovo's measure. In this study, the mathematical properties of the important measure of the fuzzy input uncertainty are discussed and proved. The results of numerical examples and engineering examples show that the proposed importance measure can effectively describe the effect of fuzzy input uncertainty on fuzzy structural response. When the sensitivity indicator is larger, the basic fuzzy-valued variable becomes more important. The sensitivity indicators of the fuzzy structural response can give an essential importance sequence of all the basic fuzzy-valued variables and identify key contributing fuzzy-valued variables. The sensitivity indicators can provide the availability guidance to reduce the number of basic variables and optimize the fuzzy response model.     

The Karush–Kuhn–Tucker optimality conditions for fuzzy optimization problems

This paper considers optimization problems with fuzzy-valued objective functions. For this class of fuzzy optimization problems we obtain Karush–Kuhn–Tucker type optimality conditions considering the concept of generalized Hukuhara differentiable and pseudo-invex fuzzy-valued functions.     

Hahn-Banach extension theorems over the space of fuzzy elements

The conventional Hahn-Banach extension theorem over a vector space has been widely used to derive many important and interesting results in nonlinear analysis, vector optimization and mathematical economics. Although the space of fuzzy elements is not a real vector space, the Hahn-Banach extension theorems over the space of fuzzy elements and the nonstandard normed space of fuzzy elements are presented in this paper. The work also shows the possible applications of the fuzzy-valued problems to nonlinear analysis, vector optimization and mathematical economics.     

模糊值函数分析学的结构元方法简介(Ⅱ)——模糊值函数及微积分的结构元表述

介绍了利用模糊结构元的模糊值函数的解析表达形式及隶属函数确定,以及基于结构元表示的模糊值函数的微分与黎曼积分的定义、计算与部分性质.同时介绍了模糊值函数拟合的基本思想.     

On the properties of sequences of fuzzy-valued Choquet integrable functions

In this paper, we focus on investigating the properties of sequences of fuzzy-valued Choquet (for short, (C)-) integrable functions. Firstly, the concept of uniform (C)-integrabiliy and other new concepts like uniform absolute continuity and uniform boundedness for sequences of fuzzy-valued (C)-integrable functions are introduced and then the relations among them are discussed. As the applications of these concepts, we also present several convergence theorems for sequences of fuzzy-valued (C)-integrable functions by using uniform (C)-integrability.