Functional Analytic Study for Mathematical Information Sciences

数学信息科学的泛函分析研究

• 批准号: 62540135
• 负责人: UMEGAKI Hisaharu
• 金额: $ 1.54万
• 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: 1987
• 资助国家: 日本
• 起止时间: 1987 至 1988
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-62540135/
• 关键词: Functional Analysis; Signal Analysis; Entropy; Crossed Product; Relative and Mutual Entropy; Sampling Space; Optical Communication; Information Theory of Genes; Quantum Fractal Dimension; Conditional Expectation; Martingale; Simulated Anealing

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 … More 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

我们描述了我们的研究在过去两年的结果。数学科学中有许多不同的领域。我们关注占据数学科学主要部分的信息科学，并用数学方法，特别是泛函分析来研究它。我们的研究目的是将许多数学方法应用于信息科学中出现的各种问题，我们认为这些问题一般与数学关系不大。我们的研究结果可分为三个部分。我们用分析、概率论和量子论的方法研究了构成信息论基本部分的熵理论，建立了信息科学的数学结构，并利用这些结果给出了信息传输系统（我们称之为通道）的数学表达式。在此基础上，对信息源构造的信号进行了理论分析， ...更多信息 功能分析的方法。将傅里叶变换理论应用于信号理论，给出了信号函数的一个展开定理。该定理断言了一个显著的性质，即由函数信号空间中的基组成的采样函数与由信号生成的代数中的恒等式的分解完全一致。同时，我们用熵理论分析了谱密度函数，得到了由信号过程生成的von Neumann代数的一个结构定理，为了从数学上阐明量子通信理论，我们研究了量子系统中的熵理论，导出了量子控制通信过程中差错概率的一般公式。此外，我们还利用量子通信理论中的互熵来研究光调制的效率，并将信息论中的一些概念应用到基因的研究中。研究表明，互熵可以用来定义指示两个基因序列之间相似性的度量。一些系统发育树是通过使用这种熵度量来编写的，由此我们看到了信息论在分子进化等基因领域的应用。为了研究这类系统的复杂性和刻画某些动力学系统的性质，本文引入了一般量子系统（包括经典系统和量子系统）的态熵，建立了量子系统的态熵理论本文讨论了量子概率空间中的鞅，讨论了von Neumann代数上的鞅收敛定理。利用非交换L-空间理论，我们得到了鞅收敛的一些结果.模拟退火和神经网络都是求解计算困难的最优化问题的新方法.我们已经重建了它们的数学框架并进行了分析。特别地，我们建立了神经网络的一些稳定性定理。少

项目成果

Masanori Ohya: "information and mathematics I" Suri-Kagaku. 305. 78-83 (1988)

Masanori Ohya：《信息与数学I》Suri-Kagaku。

大矢雅則: 数理科学. No.294. 5-13 (1987)

大谷正德：数学科学第 5-13 期（1987 年）。

大矢雅則: 素粒子論研究. 78. (1988)

大谷正德：基本粒子理论研究78。（1988）

Yoshinori Uesaka: "On stability of neural networks with the energy induced from real-valued functions of two state variables" Tech. Rep. IEICE. 6. 7-14 (1988)

Yoshinori Uesaka：“论神经网络的稳定性，其能量来自两个状态变量的实值函数”Tech。

渡辺昇: Proceedings of International Conference on Opitical Communication Theory. 40-43 (1987)

Noboru Watanabe：国际光通信理论会议论文集 40-43 (1987)。