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Functional Analysis in Mathematical Sciences

数学科学中的泛函分析

基本信息

批准号：
01540155
负责人：
UMEGAKI Hisaharu
金额：
$ 1.02万
依托单位：
Science University of Tokyo
依托单位国家：
日本
项目类别：
Grant-in-Aid for General Scientific Research (C)
财政年份：
1989
资助国家：
日本
起止时间：
1989 至 1990
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-01540155/
关键词：
Fourier Transforms Radon Transforms CT-Scanner Signal Analysis Sampling Functions Optical Modulations Genetic Codes Entropy Fourier変換 Fourir変換 CTスキャナ

项目摘要

During the last ten years, the field of mathematics has been rapidly developed to various branches. In particular, the effect of the applications of mathematics to the field of information sciences are remarkable. These are actually an important extension of mathematics, and are called to be mathematical sciences or mathematical information sciences. In this research, making the most use of the functional analysis method, we have aimed at further new branches. In the following we describe them by the items (1)-(5).(1) Development of Fourier analysis by functional analysis. Several fundamental theorems of operator algebras are adapted to constructions of Gelfand representation and Fourier transforms. Under these constructions, all fundamental theorems in harmonic analysis are expressed.(2) Applying the results in (1) to Radon transforms, we developed mathematical analysis of CT-Scanner, which is one of the most important principle in new medical diagnosis.(3) A fundamental method in signal analysis is sampling expansion theorem, where the main tool is sampling function. We have developed the functional analysis around the mathematical treatments. The formulations were done by von Neumann algebras and spectral theory. By these invesligations, the mathematical theory of signal has been clarified.While, we have adapted Shannon theory of entropy to optical communication theory, analysis of genes, quantum fractal theory and several topics, and discussed the usefulness of entropy. We describe the following results :(4) By using the functional analytic methods for the mutual entropy and the construction of quantum channel, we discussed the efficiency of some modulations and the error probability in optical communication processes.(5) We made the phylogenetic tree by using the techniques of information theory, and discussed the evolution of organisms.

在过去的十年中，数学领域迅速发展成各个分支。特别是，数学在信息科学领域的应用效果是显著的。这些实际上是数学的重要延伸，被称为数学科学或数学信息科学。在本研究中，充分利用泛函分析方法，进一步开拓新的分支。在下文中，我们用(1)-(5)项来描述它们。(1)通过泛函分析发展傅里叶分析。算子代数的几个基本定理适用于Gelfand表示和傅里叶变换的构造。在这些结构下，给出了谐波分析中的所有基本定理。(2)将(1)的结果应用于Radon变换，建立了ct扫描仪的数学分析，这是新医学诊断的重要原理之一。(3)信号分析的基本方法是采样展开定理，其主要工具是采样函数。我们围绕数学处理发展了泛函分析。这些公式是由冯·诺伊曼代数和谱理论完成的。通过这些研究，明确了信号的数学理论。同时，我们将香农熵理论应用于光通信理论、基因分析、量子分形理论等多个领域，并讨论了熵的有用性。(4)利用互熵的泛函分析方法和量子信道的构造，讨论了光通信过程中某些调制的效率和误差概率。(5)运用信息论的方法构建了系统发生树，讨论了生物的进化过程。