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Hilbert's 13th Problem and the Data Compression Problem of Multi-dimensional Numerical Tables

Research Project

Project/Area Number 16540120
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field General mathematics (including Probability theory/Statistical mathematics)
Research InstitutionTokyo University of Science

Principal Investigator

AKASHI Shigo  Tokyo University of Science, Faculty of Science and Technology, Professor, 理工学部, 教授 (30202518)

Co-Investigator(Kenkyū-buntansha) YAMAGUCHI Fumihiko  Tokyo University of Science, Faculty of Science and Technology, Assistant, 理工学部, 助手 (60339124)
宮寺 隆之  東京理科大学, 理工学部, 助手 (50339123)
Project Period (FY) 2004 – 2006
Project Status Completed (Fiscal Year 2006)
Budget Amount *help
¥3,500,000 (Direct Cost: ¥3,500,000)
Fiscal Year 2006: ¥1,100,000 (Direct Cost: ¥1,100,000)
Fiscal Year 2005: ¥1,100,000 (Direct Cost: ¥1,100,000)
Fiscal Year 2004: ¥1,300,000 (Direct Cost: ¥1,300,000)
Keywordsepsilon-entropy / numerical tables / Hilbert's 13th problem / approximate dimension / Simpson numerical integral formula / data compression / 強重ね合わせ表現可能性 / 弱重ね合わせ表現可能性 / 弱重ね合わせ表現不可能性 / 強重ね合わせ表現不可能性 / Vituskinの定理 / 形状認識問題 / 重ね合わせ表現可能性 / 同型問題 / Simpsonの公式 / 数値表データ圧縮 / 計算図表 / 重ね合わせ表現 / 記号力学系 / 埋め込み表現
Research Abstract

In the paper entitled "Application of nonlinear approximation method to Simpson's numerical integral formula", a two dimensional version of Simpson's numerical integral formula is given. This formula is applied to prove that, for any smooth function f of two variables, we can obtain sufficiently accurate approximate integral values, even if we use a half of the numerical table corresponding to f.
In the paper entitled "ε-Entropy theoretic aspects of homeomorphism problems of analytic function spaces", epsilon-entropy theoretic solutions to the homeomorphism problem of the entire function spaces equipped with the compact open topology and the homeomorphism problem of the analytic function spaces equipped with the norm topology are given.
It is known that any epsilon-expansive dynamical system with compact and totally disconnected domain can be embedded into a certain symbolic dynamical system. Actually, there exists a dynamical system with a compact and totally disconnected domain which cannot be topologically embedded into any symbolic dynamical system. Exactly speaking, epsilon-expansiveness plays an important role in the embedding problem.
In the paper entitled "Embedding of nonlinear dynamical systems with compact and totally disconnected domains into the product symbolic dynamical systems", the embedding theorem of dynamical systems, which are not epsilon-expansive, is discussed, Moreover, a relation between this result and the topological entropy is given.

Report

(4 results)
  • 2006 Annual Research Report   Final Research Report Summary
  • 2005 Annual Research Report
  • 2004 Annual Research Report
  • Research Products

    (28 results)

All 2006 2005 2004 Other

All Journal Article (20 results) Book (8 results)

  • [Journal Article] Application of nonlinear approximation method to Simpson's numerical integral formula2006

    • Author(s)
      Shigeo Akashi
    • Journal Title

      Proceedings of the Fourth International Conference On Nonlinear Analysis and Convex Analysis 4巻

      Pages: 11-17

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] ε-Entropy theoretic aspects of homeomorphism problems of analytic function spaces2006

    • Author(s)
      Shigeo Akashi
    • Journal Title

      Taiwanese Journal of Mathematics 10巻・2号

      Pages: 431-439

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] 有限回連続微分可能関数空間族の位相同型問題2006

    • Author(s)
      Shigeo Akashi
    • Journal Title

      京都大学数理解析研究所講究録 1484巻

      Pages: 73-78

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] 複雑形状を有する平面図形や空間図形に対する形状認識への複素関数論の応用2006

    • Author(s)
      Shigeo Akashi
    • Journal Title

      京都大学数理解析研究所講究録 1489巻

      Pages: 142-146

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Application of nonlinear approximation method to Simpson's numerical integral formula.2006

    • Author(s)
      Shigeo Akashi
    • Journal Title

      Proceedings of the Fourth International Conference on Nonlinear Analysis and Convex Analysis

      Pages: 11-17

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] ε-Entropy theoretic aspects of homeomorphism problems of analytic function spaces.2006

    • Author(s)
      Shigeo Akashi
    • Journal Title

      Taiwanese Journal of Mathematics vol.10, no. 2

      Pages: 431-439

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Homeomorphism Problems of the family of finitely differentiable function spaces2006

    • Author(s)
      Shigeo Akashi
    • Journal Title

      Proceedings of the Research Institute for Mathematical Science 1484

      Pages: 73-78

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Application of complex analysis to the shape recognition problem.2006

    • Journal Title

      Proceedings of the Research Institute for Mathematical Science 1489

      Pages: 142-146

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] ε-Entropy theoretic aspects of homeomorophism problems of analytic function spaces2006

    • Author(s)
      S.Akashi
    • Journal Title

      Proceedings of the Fourth International Conference on Nonlinear Analysis and COnvex Analysis

      Pages: 11-17

    • Related Report
      2006 Annual Research Report
  • [Journal Article] 有限回連続微分可能関数族の位相同型問題2006

    • Author(s)
      明石重男
    • Journal Title

      京都大学数理解析研究所講究録「非線形解析学と凸解析学の研究」 1484

      Pages: 73-78

    • Related Report
      2006 Annual Research Report
  • [Journal Article] 複雑形状を有する平面図形や空間図形に対する形状認識への複素関数論の応用2006

    • Author(s)
      明石重男
    • Journal Title

      京都大学数理解析研究所講究録「計算理論とアルゴリズムの新展開」 1489

      Pages: 142-146

    • Related Report
      2006 Annual Research Report
  • [Journal Article] ε-Entropy theoretic aspects of homeomorophism problems of analytic function spaces2006

    • Author(s)
      S.Akashi
    • Journal Title

      Taiwanese Journal of Mathematics 10巻・2号

      Pages: 431-439

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Embedding of nonlinear dynamical systems with compact and totally disconnected domains into the product symbolic dynamical systems2005

    • Author(s)
      Shigeo Akashi
    • Journal Title

      Nonlinear Analysis -Theory, Methods and Applications- 63巻

      Pages: 1817-1821

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] A relation between multidimensional data compression and Hilbert's 13th problem2005

    • Author(s)
      Masahiro Yamada, Shigeo Akashi
    • Journal Title

      京都大学数理解析研究所講究録 1415巻

      Pages: 194-197

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Embedding of nonlinear dynamical systems with compact and totally disconnected domains into the product symbolic dynamical systems.2005

    • Author(s)
      Shigeo Akashi
    • Journal Title

      Nonlinear Analysis,---Theory, Methods and Applications… 63

      Pages: 1817-1821

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] A relation between multidimensional data compression and Hilbert's 13th problem.2005

    • Author(s)
      Masahiro Yamada, Shigeo Akashi
    • Journal Title

      Proceedings of the Research Institute for Mathematical Science 1415

      Pages: 194-197

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Application of nonlinear approximation method to Simpson's numerical integral formula

    • Author(s)
      M.Yamada, O.Nakata, M.Watanabe, F.Yanase, S.Akashi
    • Journal Title

      Proceedings of the international conference on nonlinear analysis and convex analysis (To be published)

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Estimation of Complexity for the Ohya-Masuda-Volovich SAT Algorithm

    • Author(s)
      S.Iriyama, S.Akashi
    • Journal Title

      Open Systems and Information Dynamics (To be published)(印刷中)

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Embedding of Nonlinear Dynamical Systems with Compact and Totally Disconnected Domains into the Product Symbolic Dynamical Systems

    • Author(s)
      S.Akashi
    • Journal Title

      Nonlinear Analysis (To be published)(印刷中)

    • Related Report
      2004 Annual Research Report
  • [Journal Article] ε-Entropy Theoretic Aspects of Homeomorphism Problems of Analytic Function Spaces

    • Author(s)
      S.Akashi
    • Journal Title

      Taiwanese Journal of Mathematics (To be published)(印刷中)

    • Related Report
      2004 Annual Research Report
  • [Book] Proceedingsof the Fourth International Conference on Nonlinear Analysis and Convex Analysis2006

    • Author(s)
      Shigeo Akashi
    • Publisher
      Application of nonlinear approximation method to Simpson's numerical integral formula.
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Book] Taiwanese Journal of Mathematics, vol.10, no. 22006

    • Author(s)
      Shigeo Akashi
    • Publisher
      ε-Entropy theoretic aspects of homeomorphism problems of analytic function spaces.
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Book] Proceedings of the Research Institute for Mathematical Science, 14842006

    • Author(s)
      Shigeo Akashi
    • Publisher
      Homeomorophism Problems of the family of finitely differentiable function spaces
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Book] Proceedings of the Research Institute for Mathematical Science, 14892006

    • Publisher
      Application of complex analysis to the shape recognition problem.
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Book] Nonlinear Analysis,---Theory, Methods and Applications…, 632005

    • Author(s)
      Shigeo Akashi
    • Publisher
      Embedding of nonlinear dynamical systems with compact and totally disconnected domains into the product symbolic dynamical systems.
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Book] Proceedings of the Research Institute for Mathematical Science, 14152005

    • Author(s)
      Masahiro Yamada, Shigeo Akashi
    • Publisher
      A relation between multidimensional data compression and Hilbert's 13th problem.
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Book] 応用解析学概論(第5版増刷に伴う改訂版)2004

    • Author(s)
      阿部剛久, 明石重男, 古城知己, 福田博通
    • Total Pages
      216
    • Publisher
      共立出版株式会社
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Book] Kyouritsu Publishing Company, Tokyo, the 5th Edition2004

    • Author(s)
      Takehisa Abe, Shigeo Akashi, Tomoki Kojou, Hiromichi Fukuda
    • Publisher
      The survey of applied analysis
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary

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Published: 2004-04-01   Modified: 2016-04-21  

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