Project/Area Number  14540108 
Research Category 
GrantinAid for Scientific Research (C)

Allocation Type  Singleyear Grants 
Section  一般 
Research Field 
General mathematics (including Probability theory/Statistical mathematics)

Research Institution  Tokyo University of Science(2003) Niigata University(2002) 
Principal Investigator 
AKASHI Shigeo Tokyo University of Science, Faculty of Science and Engineering, Professor, 理工学部, 教授 (30202518)

CoInvestigator(Kenkyūbuntansha) 
OHYA Masanori Tokyo University of Science, Faculty of Science and Engineering, Professor, 理工学部, 教授 (90112896)
TAKAHASHI Wataru Tokyo Institute of Technology, Faculty of Science, Professor, 理学部, 教授 (40016142)
ISOGAI Eiichi Niigata University, Faculty of Science, Professor, 理学部, 教授 (40108014)
MIYADERA Takayuki Tokyo University of Science, Faculty of Science and Engineering, Assistant, 理工学部, 助手 (50339123)
鈴木 智成 新潟大学, 大学院・自然科学研究科, 助手 (00303173)
室伏 俊明 東京工業大学, 大学院・情報理工学研究科, 助教授 (40200315)

Project Period (FY) 
2002 – 2003

Project Status 
Completed(Fiscal Year 2003)

Budget Amount *help 
¥3,500,000 (Direct Cost : ¥3,500,000)
Fiscal Year 2003 : ¥1,700,000 (Direct Cost : ¥1,700,000)
Fiscal Year 2002 : ¥1,800,000 (Direct Cost : ¥1,800,000)

Keywords  epsilon entropy / data compression / Hilbert's 13th problem / uperposit ion representation / low band pass filter / Poisoon kernels / Approximate identity / Gibbs penomena / 重ねあわせ表現 / コンパクト楕円体の被覆問題 / Baireのカテゴリー定理 / イプシロネントロピー / データ圧縮 / Stachelberg均衡点 / コンパクト集合値映像 
Research Abstract 
The contents of the researchers published during two years from 2002 till 2003 can be classified into two parts, which are satated as follows : (1).The solution to an open problem related to Hubert's 13th problem. The 13th problem formulated by Hubert, which asks if any continuous functions of several variables can be represented as superpositions constructed from several continuous functions of fewer variables, was solved by Kolmogorov and Arnold after about fifty years.Actually, theentire function theoretic problem asking if any entire functions of several variables can be represented as superpositions constructed from several entire functions of fewer variables has remained to be solved.The representative of this research project has succeeded in giving the solution to this open problem which is based on entropy theoretic methods. (2).A relation between the superposition representation proble'm and the theory of nomographs. Primarily, it has been shown that the concept of superposition irrepresentability can be classified. into two concepts which is called strong superposition irrepresentability and weak superposition irepresentability, respectively.Moreover, there does not exist any nomographs enabling to estimate the values of the functions of several variables if they are strongly irrepresentable, and, for a certain positive integer k, there does not exist any nomographs enabling to estimate the values of the functions of several variables by at least ktime operations if they are weakly irrepresentable.
