| Co-Investigator(Kenkyū-buntansha) |
WATANABE Noboru Science University of Tokyo, Department of Information Sciences, Assistant, 理工学部, 助手 (70191781)
OHYA Masanori Science University of Tokyo, Department of Information Sciences, Professor, 理工学部, 教授 (90112896)
NAGAKURA Yasujiro Science University of Tokyo, Faculty of Science, Professor, 理学部, 教授 (60112900)
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| Research Abstract |
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.
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