AKASHI Shigeo Faculty of Engineering, Sagami Institute of Technology, 工学部, 講師 (30202518)
WATANABE Noboru Faculty of Science and Technology, Science University of Tokyo, 理工学部, 助手 (70191781)
TSUKADA Makoto Faculty of Science and Technology, Science University of Tokyo, 理工学部, 講師 (10120198)
OHYA Masanori Faculty of Science and Technology, Science University of Tokyo, 理工学部, 教授 (90112896)
UESAKA Yoshinori Faculty of Science and Technology, Science University of Tokyo, 理工学部, 教授 (40019782)
佐藤 元 東京理科大学, 理学部, 助教授 (00162462)
|Budget Amount *help
¥2,400,000 (Direct Cost: ¥2,400,000)
Fiscal Year 1988: ¥1,200,000 (Direct Cost: ¥1,200,000)
Fiscal Year 1987: ¥1,200,000 (Direct Cost: ¥1,200,000)
We describe the results of our studies in the last two years. There are many various areas in mathematical sciences. We pay attention to information science, which occupies the main parts of mathematical science, and investigate it by the methods of mathematics, especially functional analysis.
It has been the purpose of our studies that we have applied many mathematical methods to various problems arising in information sciences which, we think in general, has few relation to mathematics.Our results about these studies can be divided in to three sections. We have investigated entropy theory which construct the basic parts of information theory by the methods of analysis, probability theory and quantum theory, and established the mathematical structure of information sciences.
Using this results, we have given the mathematical formulation of systems of information transmission, which we call channels. We, furthermore, have analized signals constructed by information soruces under the theo
ry of functional analysis. Applying the theory of Fourier transforms to signal theory, we have given an expansion theorem of signal functions. This theorem asserts a remarkable property that the sampling functions, consisting of the basis in the functional signal space, exactly coincide with the resolution of identity in the algebra generated by signals. While, we have analyzed spectral density functions by the theory of -entropy, and have obtained an structure theorem of von Neumann algebras generated by signal processes.
In order to formulate quantum communication theory mathematically, we studiedthe entropy theory in quantum systems and we derived a general formula for the error probability in quantum control communication processes. Moreover, we considered the effeciency of the optical modulations by using the mutual entropy in quantum communication theory.
Some concepts in information theory are applied to the study of genes. It is shown that the mutual entropy is used to define a measure indicating the similarity between two genetic sequences. Some phylogenetic trees are written by using this entropy measure, from which we see the usefulness of the information theory in the field of genes such as molecular evolution.
Three fractal dimensions and the quantum -entropy of states in general quantum systems containing usual classical and quantum systems have been introduced in order to study the complexity of such systems and characterize some dynamical systems.
In order to establish the theory of martingales in quantum probability spaces, we have treated martingale convergence theorems on von Neumann algebras. Using the theory of noncommutative L-spaces,we have got some results of martingale convergence.
Both simulated annealing and neural networks are new methods of numerical solution of computationally hard optimization problems. We have reconstructed their mathematical frameworks and analyze them. In particular, we have established some stability theorems of neural networks. Less