伪齐次性方程和模态性方程的一致模新解研究

In knowledge-based systems,appropriate aggregation operators that are associated by means of fuzzy functional equations,are used to interpret local behaviors of systems.The effects of information aggregation are determined by the chosen aggregation operators.Therefore,looking for the specific aggregation operators solutions of fuzzy functional equations is an interesting work and can provide more selectable aggregation operators for real-world problem.The modularity and pseudo-homogeneity equations are common fuzzy functional equations.The aim of this project is to look for the new uninorm solutions of modularity and pseudo-homogeneity equations and the main content consists of:(a) In literature,only the most well studied classes of uninorms are considered as solutions of the modularity equation.We will seek for the new uninorm solutions and characterize the structure of uninorm solutions of modularity equation;(b)The non-trivial solutions of pseudo-homogeneity equation are t-norm solutions and only part of continuous t-norm solutions of pseudo-homogeneity equation are known.We will look for the new continuous t-norm solutions and characterize the structure of continuous t-norm solutions of pseudo-homogeneity equation.

在基于知识的系统中,存在诸多局部行为,通常用适当的聚合算子来解释对应的局部行为,但聚合算子的选取不是任意的,须满足一定的模糊函数方程.算子选取的恰当与否直接决定聚合效果的好坏.因此求解模糊函数方程的特定聚合算子解是一项有意义的工作,能为实际问题提供更多的可选聚合算子.常见的模糊函数方程有伪齐次性方程和模态性方程.本项目拟寻找伪齐次性方程和模态性方程的一致模新解,主要内容包括:(a)现有研究仅关注模态性方程的常见类一致模解,拟寻找非常见类一致模新解,刻画一致模通解的数学结构特征;(b)在伪齐次性方程一致模解中,非平凡解必为三角模解.但目前仅得到方程的部分连续三角模解.拟寻找伪齐次性方程的连续三角模新解,刻画连续三角模通解的数学结构特征.

信息融合是智能系统领域的一个关键问题.一致模可作为信息聚合的数学模型,把不同来源的信息综合处理一个数值,该数值能够反映人们所期望的合成信息.一致模的理论研究可为所考虑的问题提供更多的可选聚合算子.一致模理论研究的一个主题是刻画具有某些性质的一致模,该问题通常转化为求解相关的函数方程.拟齐次性方程、分配性方程和交叉迁移性方程是应用中常见的函数方程. . 基于Clifford序和理论,将相关算子分解,使得当算子限制在局部小区域时,具有良好的结构特征,利用算子的这些局部结构,进一步可以描述具有某些性质的算子的结构.采用上述研究思路及方法,本研究(1)描述了基础算子连续一致模的内部结构;(2)刻画了关于连续三角模交叉迁移的连续三角模结构;(3)解决了Stout提出的关于连续三角模分配的一致模的结构问题. . 运用经典代数方法和技巧,分析了聚合算子代数性质之间的相互依赖关系,本研究(4)刻画了几类分配性半一致模的结构;(5)刻画了几类离散聚合算子;(6)描述了伪齐次性聚合算子的结构.