2006 Fiscal Year Final Research Report Summary
Hilbert's 13th Problem and the Data Compression Problem of Multi-dimensional Numerical Tables
| Project/Area Number |
16540120
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
General mathematics (including Probability theory/Statistical mathematics)
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| Research Institution | Tokyo University of Science |
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Principal Investigator |
AKASHI Shigo Tokyo University of Science, Faculty of Science and Technology, Professor, 理工学部, 教授 (30202518)
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| Co-Investigator(Kenkyū-buntansha) |
YAMAGUCHI Fumihiko Tokyo University of Science, Faculty of Science and Technology, Assistant, 理工学部, 助手 (60339124)
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| Project Period (FY) |
2004 – 2006
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| Keywords | epsilon-entropy / numerical tables / Hilbert's 13th problem / approximate dimension / Simpson numerical integral formula / data compression |
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| 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.
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