2006 Fiscal Year Final Research Report Summary
Computability of discontinuous functions-Towards its paradigm
| Project/Area Number |
16340028
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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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 | Kyoto Sangyo University |
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Principal Investigator |
YASUGI Mariko Kyoto Sangyo University, Faculty of Science, Professor, 理学部, 教授 (90022277)
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| Co-Investigator(Kenkyū-buntansha) |
TSUJII Yoshiki Kyoto Sangyo University, Faculty of Science, Professor, 理学部, 教授 (90065871)
MORI Takakazu Kyoto Sangyo University, Faculty of Science, Professor, 理学部, 教授 (00065880)
YAMADA Shuji Kyoto Sangyo University, Faculty of Science, Professor, 理学部, 教授 (30192404)
TSUIKI Hideki Kyoto University, Graduate School of Human and Environmental Studies, Associate Professor, 大学院人間・環境学研究科, 助教授 (10211377)
HAYASHI Susumu Kyoto University, Graduate School of Letters, Professor, 大学院文学研究科, 教授 (40156443)
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| Project Period (FY) |
2004 – 2006
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| Keywords | Computable analysis / Effective continuity / Effective sequence of uniformities / limit / Effective sequence of Fine continuous functions / Effective Fine convergence of function sequences / Limit recursion / Proof animation / Fractals with infinite bases |
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| Research Abstract |
In studying computability of some real functions which are discontinuous with respect to the Euclidean metric, the most useful and natural METHOD is uniformization of the domain of a function by isolating the discontinuous points. A general theory of the effective uniform space, especially the space of Fine metric, has been clarified. For the computability problem of a function sequence whose functions have different discontinuity points, a theory of the effective sequence of uniformities and its limit has been developed. Admitting limiting recursive functions in characterizing the image of a computable sequence of elements by a discontinuous function is another theory. In a natural setting, these two theories are equivalent. We can claim that the theory of the effective uniformity (the sequence of effective uniformities) is the fundamental method for the computability of some discontinuous functions. Limiting recursion is equivalent with Sigma^0_1 excluded middle. The relative strength between Sigma^0_1 excluded middle and other semi-constructive principles have been worked out. As for functional analysis, effectivization of various theorems, mainly on the Banach space, has made progress. A representation of real numbers in terms of {0,1,Bottom},characterizing the computability of real numbers has been studied. Some counter-examples have been constructed :a function sequence which is sequentially computable but has no effectively continuous points and a function which is Banach-Mazur computable but is not Markov computable. In applications, dynamics of double rotation maps, description of the inverse problem in multi-sectorial growth theory, the computability problem of fractals with infinitely many contraction maps, reorganization of hardware and software for discovery of non-trivial knots have been studied.
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[Book] ゲーデル不完全性定理2006
Author(s)
林晋, 八杉満利子(訳・解説)
Total Pages
309
Publisher
岩波書店(岩波文庫)
Description
「研究成果報告書概要(和文)」より
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