Nonlinear approximationtheoretic methods of numerical table data compression

• Project/Area Number: 22540151
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: General mathematics (including Probability theory/Statistical mathematics)
• Research Institution: Tokyo University of Science
• Principal Investigator: AKASHI Shigeo 東京理科大学, 理工学部情報科学科, 教授 (30202518)
• Project Period (FY): 2010 – 2012
• Project Status: Completed (Fiscal Year 2012)
• Budget Amount: ¥4,160,000 (Direct Cost: ¥3,200,000、Indirect Cost: ¥960,000); Fiscal Year 2012: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000); Fiscal Year 2011: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2010: ¥1,690,000 (Direct Cost: ¥1,300,000、Indirect Cost: ¥390,000)
• Keywords: 情報数理 / Hilbertの第13問題 / ε-エントロピー / 多変数関数 / Collatz予想 / Lee-Yorkの定理 / アトラクター / データ圧縮 / Goedel数 / Hilberの第13問題 / 数値表データ / 数値群データ / 形状認識 / Hilbertの13問題 / Arzela^Ascoliの定理 / 同程度連続性 / ソボレフノルム / フレシュ距離 / 多変数関数の重ね合わせ表現

Research Abstract

The main research results obtained in the research period which began in 2010 and ended in 2012 are the following:
(1). A version of Hilbert’s 13thproblem for infinitely differentiable functions, which remained to be open, has been solved constructively.
(2). Embedding of Collatz mapping into a certain one-dimensional compact dynamical system has enabled us to apply the theory of topological dynamics to its famous conjecture

Research Products

• [Journal Article] On periodic orbits and space-filling orbits in symbolic dynamical systems (2012)
  • Author(s): S. Kodama and S. Akashi
  • Journal Title: Proceedings of the 10thInternational Conference on Fixed Point Theory and Applications Pages: 165-170
  • Peer Reviewed
• [Journal Article] Strong convergence theorems for nonexpansive mappings on star-shaped sets in a Hilbert spaces (2012)
  • Author(s): S. Akashiand W. Takahashi
  • Journal Title: Applied Mathematics and Computations Volume: vol.219 Pages: 2035-2040
  • Peer Reviewed
• [Journal Article] Strong convergence theorems for nonexpansive mappings on star-shaped sets in Hilbert spaces (2012)
  • Author(s): S. Akashi and W. Takahashi (S. Akashi)
  • Journal Title: Applied Mathematics and Computations Volume: 219 Pages: 2035-2040
  • Peer Reviewed
• [Journal Article] On periodic orbits and space-filling orbits in symbolic dynamical systems (2012)
  • Author(s): S. Kodama and S. Akashi(S. Kodama)
  • Journal Title: Proceedings of the10th International Conference on Fixed Point Theory and Applications Volume: 10 Pages: 165-170
  • Peer Reviewed
• [Journal Article] Equidifferentiability of subsets of infinitely differentiable function spaces (2010)
  • Author(s): S. Kodama and S. Akashi
  • Journal Title: Fixed Point theory Volume: 11 Pages: 323-326
  • Peer Reviewed
• [Journal Article] A version of Hilbert's 13thproblem for infinitely differentiable functions (2010)
  • Author(s): S. Akashiand S. Kodama
  • Journal Title: Proceedings of the 9thInternational Conference on Fixed Point Theory and Applications Pages: 1-5
  • Peer Reviewed
• [Journal Article] Equidifferentiability of subsets of finitely differentiable function spaces (2010)
  • Author(s): S. Kodama and S. Akashi
  • Journal Title: Proceedings of the 6thInternational Conference on Nonlinear Analysis and Convex Analysis Pages: 135-138
  • Peer Reviewed
• [Journal Article] Equidifferentiability of subsets of infinitely differentiable function spaces (2010)
  • Author(s): S.Kodama, S.Akashi
  • Journal Title: Fixed Point Theory and Applications Volume: 11 Pages: 323-326
  • Peer Reviewed
• [Journal Article] A version of Hilbert's 13^<th> problem for infinitely differentiable functions (2010)
  • Author(s): S.Akashi, S.Kodama
  • Journal Title: Proceedings of the 6^<th> International Conference on Fixed Point Theory and Applications Pages: 1-5
  • Peer Reviewed
• [Journal Article] Equidifferentiability of subsets of finitely differentiable funcion spaces (2010)
  • Author(s): S.Kodama, S.Akashi
  • Journal Title: Proceedings of the 6^<th> International Conference on Nonlinear Analysis and Convex Analysis Pages: 135-138
  • Peer Reviewed
• [Presentation] The embedding problem of Collatz dynamical system into another dynamical system with compact domain (2013)
  • Author(s): S. Akashi
  • Organizer: International Symposium on Nonlinear Analysis and Optimization 2013
  • Place of Presentation: Pukyong National University, Busan, Korea
  • Year and Date: 2013-01-31
• [Presentation] Fixed point theoretic structures of generalized Collatz mappings (2012)
  • Author(s): S. Akashiand S. Kodama
  • Organizer: The 3rdAsian Conference on Nonlinear Analysis and Optimization
  • Place of Presentation: くにびきメッセ, 島根県松江市
  • Year and Date: 2012-09-06
• [Presentation] The superposition degree problem of Boolean-valued functions of several variables (2012)
  • Author(s): S. Kodama and S. Akashi
  • Organizer: 非線形解析学と凸解析学の研究, 京都大学数理解析研究所
  • Year and Date: 2012-08-31
• [Presentation] Fixed point theoretic approach to the Collatz mapping (2012)
  • Author(s): S. Akashi
  • Organizer: The 10thInternational Conference on Fixed Point Theory and Applications, Babes-Bolyai University
  • Place of Presentation: Cluj-Napoca, Romania
  • Year and Date: 2012-07-09
• [Presentation] FIxed point theoretic approach to the Collatz mapping (Key Note Talk) (2012)
  • Author(s): Shigeo Akashi
  • Organizer: The 10th International Conference on Fixed Point theory and Its Applications
  • Place of Presentation: Babe&#351;-Bolyai University Cluj-Napoca, Romania
  • Invited
• [Presentation] The Superposition Degree Problem of Boolean-valued Functions of Several Variables (2012)
  • Author(s): 児玉賢史, 明石重男(児玉賢史)
  • Organizer: 数理解析研究所共同研究集会「非線形解析学と凸解析学の研究」
  • Place of Presentation: 京都大学数理解析研究所
• [Presentation] Fixed point theoretic structure of generalized Collatz mappings (2012)
  • Author(s): Shigeo Akashi and Satoshi Kodama (Shigeo Akashi)
  • Organizer: The 3rd Asian Conference on Nonlinear Analysis and Optimization
  • Place of Presentation: Kunibiki Messe, Matsue-city, Shimane Prefecture
  • Invited
• [Presentation] The classification problem of one dimensional chaotic dynamical systems (2011)
  • Author(s): S. Akashi
  • Organizer: 2011 Workshop on Nonlinear Analysis and Optimization, National Taiwan Normal University
  • Place of Presentation: Taipei, Taiwan
  • Year and Date: 2011-11-16
• [Presentation] Classification problem of one dimensional chaotic dynamical systems (2011)
  • Author(s): Shigeo Akashi
  • Organizer: 2011 Workshop on Nonlinear Analysis and Optimization
  • Place of Presentation: 国立台湾師範大学(Invited talk)(基調講演)
  • Year and Date: 2011-11-16
• [Presentation] The numbering problem of ε-expansive dynamical systems with totally disconnected compact domains (2011)
  • Author(s): S. Akashi
  • Organizer: The Seventh International Conference on Nonlinear Analysis and Convex Analysis Busan, Korea
  • Year and Date: 2011-08-02
• [Presentation] Nombering problem of ε-expansive dynamcal systems with totally disconnected compact domains (2011)
  • Author(s): Shigeo Akashi
  • Organizer: The Seventh International Conference on Nonlinear Analysis and Convex Analysis
  • Place of Presentation: Pukyong National University(Invited talk)(招待講演)
  • Year and Date: 2011-08-02
• [Presentation] The classification problem of expansive dynamical systems with compact domains (2011)
  • Author(s): S.Akashi
  • Organizer: International Symposium on Nonlinear Analysis and Optimization
  • Place of Presentation: Pukyong National University, Busan, Korea
  • Year and Date: 2011-02-12
• [Presentation] A relation between multidimensional numerical integration and Hiblert's 13thproblem (2010)
  • Author(s): S. Akashi
  • Organizer: The Second Asian Conference on Nonlinear Analysis and Optimization
  • Place of Presentation: Royal Paradise Hotel and Spa, Phuket, Thailand
  • Year and Date: 2010-09-10
• [Presentation] A relation between multidimensional numerical integration and the Hilbert's 13^<th> problem (2010)
  • Author(s): S.Akashi
  • Organizer: The 2^<nd> Asian Conference on Nonlinear Analysis and Optimization
  • Place of Presentation: Royal Paradise Hotel and Spa, Phuket, Thailand
  • Year and Date: 2010-09-10
• [Book] 非線形解析学と凸解析学の研究 (2011)
  • Author(s): 高橋渉, 明石重男
  • Total Pages: 255
  • Publisher: 数理解析研究所講究録
• [Book] 京都大学数理解析研究所講究録1755号「非線形解析学と凸解析学の研究」 (2011)
  • Author(s): 明石重男
  • Total Pages: 225
  • Publisher: 京都大学数理解析研究所
• [Book] Proceedings of the 6thInternational Conference on Nonlinear Analysis and Convex Analysis (2010)
  • Author(s): S. Akashi, Y. Kimura and T. Tanaka (Eds)
  • Total Pages: 434
• [Book] Proceedings of the 6^<th> International Conference on Nonlinear Analysis and Convex Analysis (2010)
  • Author(s): S.Akashi, Y.Kimura, T.Tanaka
  • Total Pages: 434
  • Publisher: Yokohama Publishers
• [Patent(Industrial Property Rights)] 図形に対する内点判別方法 (2011)
  • Inventor(s): 明石重男、付思
  • Industrial Property Rights Holder: 理化工業株式会社
  • Acquisition Date: 2011-06-17
• [Patent(Industrial Property Rights)] 図形に関する内点判別法 (2010)
  • Inventor(s): 明石重男、付思
  • Industrial Property Rights Holder: 理化工業株式会社
  • Industrial Property Rights Type: 特許
  • Filing Date: 2010-05-15
  • Acquisition Date: 2012-06-17