波集二象性及其作用  

Wave-set Duality and Its Applications

作　　者：李洪兴[1,2] LI Hongxing(Research Center for Applied Mathematics and Interdisciplinary Sciences,Beijing Normal University,Zhuhai 519085,Guangdong;School of Control Science and Engineering,Dalian University of Technology,Dalian 116024,Liaoning)

机构地区：[1]北京师范大学珠海校区应用数学与交叉科学研究中心,广东珠海519085 [2]大连理工大学控制科学与工程学院,辽宁大连116024

出　　处：《四川师范大学学报（自然科学版）》2023年第1期1-11,F0002,共12页Journal of Sichuan Normal University（Natural Science）

基　　金：国家自然科学基金(12071056)。

摘　　要：从Cantor集合的物理意义,引出波集二象性.再从Cantor集合到Fuzzy集合,给出扩展原理.证明Zadeh的扩展原理是一个命题,并不是一个原理,它是我们提出的扩展原理的一个推论.根据扩展原理,又有波Fuzzy集二象性的概念.这样一来,有些数学问题在集合的意义下遇到困难时,在波函数的意义下可能得到顺利的解决.作为波函数的应用,描述导函数的逼近论意义;同时也讨论原函数的逼近论意义.熟知,求某些超越函数的不定积分是困难的,然而使用波函数构造函数序列可以获得这些超越函数在有限论域上的原函数的近似表达.In this paper,wave-set duality is defined based on physical significance of Cantor’s sets.Extension principle is also given considering from Cantor’s sets to fuzzy sets.Based on this extension principle,Zadeh’s extension principle is proved to be a proposition rather than a principle.By using this extension principle,wave-fuzzy-set duality is defined.By means of wave-fuzzy-set duality or wave functions,based on function approximation theory,the approximation theory meanings of derived function and antiderivative functions are discussed.It is well known that it is difficult to find the antiderivative of some transcendental functions.However,the approximate expression of the original function of these transcendental functions on a finite domain can be obtained by using the sequence of wave function constructors.

关 键 词：CANTOR集合 FUZZY集合 Fuzzy推理 波集二象性 波Fuzzy集二象性 导函数 原函数 

分 类 号：O159[理学—数学]