Application of entropy theory to the problem of data compression

熵理论在数据压缩问题中的应用

• 批准号: 10640110
• 负责人: AKASHI Shigeo
• 金额: $ 2.37万
• 依托单位: NIIGATA UNIVERSITY
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 1998
• 资助国家: 日本
• 起止时间: 1998 至 1999
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-10640110/
• 关键词: quantum information; entropy; data compression; compact mapping; reproducing kernel Hilbert space

(1). New locally convex topologies constructed with some families of nuclear operators on a separable Hilbert space which are weaker than the norm topology and stronger than the weak topology are given. These topologies are applied to the classification of nuclear operators. Further, the homeomorphism problem of subspaces with the norms in terms of the ranges of the closed unit ball under nuclear operators is treated. This result is applied to the homeomorphism problem of periodic and continuously differentiable function spaces included by LィイD12ィエD1[0,1].(2). The homeomorphism problems of subspaces with norms in terms of the ranges of the closed unit ball under compact positive operators are examined. These results will be applied to the operator theoretical classification of reproducing kernel Hilbert spaces.(3). It is shown that Ohya's entropy dimension on a WィイD1*ィエD1-dynamical system is time-invariant, and homeomorphism problems of operator algebras equipped with the quasi σ-strong operator topologies are discussed.(4). A formula for estimating ε-entropy of a compact positive operator in terms of the distribution of proper values of such an operator was given by Prosser and Root. In this paper, an inversion formula for estimating the distribution of proper values of a compact positive operator in terms of ε-entropy of such an operator is given.(5). The homeomorphism problem of compact nonlinear mappings on a locally convex topological vector space is studied under the method of dimension theory. First of all, dimension theoretic homeomorphism invariants which are defined on the set of all compact nonlinear mappings are introduced. Next, these invariants are applied to dimension theoretic characterization of the fixed point sets which these compact nonlinear mapping have.

（一）.给出了可分Hilbert空间上由弱于范数拓扑而强于弱拓扑的核算子族构造的新的局部凸拓扑。这些拓扑适用于核算子的分类。进一步，讨论了核算子作用下范数以闭单位球值域表示的子空间的同胚问题。这一结果被应用于L D12 D1[0，1]所包含的周期连续可微函数空间的同胚问题.（二）、研究了在紧正算子作用下，以闭单位球值域为范数的子空间的同胚问题。这些结果将应用于再生核Hilbert空间的算子理论分类。（三）、证明了W-σ-D1* σ-D1-动力系统的Ohya熵维数是时不变的，并讨论了具有拟σ-强算子拓扑的算子代数的同胚问题.（四）、Prosser和Root给出了紧正算子的ε-熵的一个估计公式，该公式是根据该算子的本征值的分布给出的.本文给出了用紧正算子的ε-熵估计其真值分布的反演公式。（五）、利用维数论的方法研究了局部凸拓扑向量空间上紧非线性映射的同胚问题。首先，引入了定义在所有紧非线性映射集合上的维数理论同胚不变量。其次，利用这些不变量对这些紧非线性映射所具有的不动点集进行了维数论刻画。

项目成果

明石重男: "エントロピー解析とその応用"牧野書店. 150 (1998)

Shigeo Akashi：“熵分析及其应用”牧野书店 150 (1998)。

S. Akashi: "Entropy analysis and its applications."Makino Publishers (in Japanese).. 150 (1998)

S. Akashi：“熵分析及其应用。”Makino Publishers（日语）.. 150 (1998)

Shigeo Akashi: "Topological Classification of nuclear operators and its application to the theory of continuously differentiable function spaces"PanAmerican Mathematical Journal. 8巻4号. 89-96 (1998)

Shigeo Akashi：“核算子的拓扑分类及其在连续可微函数空间理论中的应用”，泛美数学杂志，第 8 卷，第 4 期，89-96（1998 年）。

Shigeo Akashi: "An operator trheoretic inversion formula for the distribution of proper values of a compact positive operators"Open Systems and Information Dynamics. 6巻3号. 303-308 (1999)

Shigeo Akashi：“紧凑正算子真值分布的算子理论反演公式”，《开放系统与信息动力学》，第 6 卷，第 3. 303-308 期（1999 年）。

Kensuke Tanaka Y.Kimura and Y.Sawazeki: "A noncooperative equilibrium of n-person game with fractional loss function" to be Published in Proceedings of the International Conference on Nonlines Analysis. (発表予定).

Kensuke Tanaka Y.Kimura 和 Y.Sawazeki：“具有分数损失函数的 n 人博弈的非合作均衡”将发表在国际非线性分析会议记录中（待提交）。