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2006 Fiscal Year Final Research Report Summary

Structures of sets related to differentiability

Research Project

Project/Area Number 16540087
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Geometry
Research InstitutionKinki University

Principal Investigator

IZUMI Shuzo  Kinki University, Science and engineering, professor, 理工学部, 教授 (80025410)

Co-Investigator(Kenkyū-buntansha) KOIKE Satoshi  Hyogo University of Teacher Education, Science, Technology, and Human Life, professor, 学校教育研究科, 教授 (60161832)
FUKUI Toshizumi  Saitama University, Science, professor, 理学部, 教授 (90218892)
SAKUMA Kazuhiro  Kinki University, Science and engineering, assistant-professor, 理工学部, 助教授 (80270362)
Project Period (FY) 2004 – 2006
Keywordsparatangent bundle / smooth function / Whitney's extension problem / interpolation
Research Abstract

Our main problem is an answer to Whitney's problem 1934, the problem of finding a condition for a function defined on a closed subset of a Euclidean space to be extendible to smooth function on the ambient space. We have given a partial answer to this problem in the paper Ann Inst. Fourier, Grenoble 2004. Namely we have shown that a certain kind of self similar sets have full higher order tangent space. Then, by a conjecture posed by Bierstone-Milman-Pawlucki, we get an positive answer to Whitney's problem for these sets.
Later, this problem has been completely solved by Fefferman (Annals of Mathematics 2007). Hence we have shifted the subject a little. We studied the higher order tangent space of a set itself. We have found that the interpolation theory plays an important role in the study of smooth functions. This yields a survey paper on interpolation theory in the book.

  • Research Products

    (7 results)

All 2007 2006 2005 2004

All Journal Article (6 results) Book (1 results)

  • [Journal Article] 1. On the realization of a map of certain class as a desingularization map, 2. Introduction to algebraic theory of multivariate interpolation, 3. Fundamental properties of germs of analytic mappings of analytic sets and related topics, 4. Existence problem for fold maps2007

    • Author(s)
      S.Izumi, S.Koike, K.Sakuma, T.Fukui
    • Journal Title

      Real and Complex Singularities (Proceedings of the Australian-Japanese workshop)

      Pages: 33-45, 85-108, 109-123, 342-387

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Mapping degree and Euler characteristic2006

    • Author(s)
      Fukui, T., Khovanskii, A.
    • Journal Title

      Kodai Math. J. 29

      Pages: 144-162

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Mapping degree and Euler characteristic2006

    • Author(s)
      T.Fukui, A.Khovanskii
    • Journal Title

      Kodai Mathematical Journal vol. 29-1

      Pages: 144-162

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Isolated roundings and flattenings of submanifolds in Eucledean spaces2005

    • Author(s)
      T.Fukui, J.J.Nuno-Ballesteros
    • Journal Title

      Tohoku Mathematical Journal vol. 57

      Pages: 469-503

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Restrictions of smooth functions to a closed subset2004

    • Author(s)
      Shuzo Izumi
    • Journal Title

      Ann. Inst. Fourier, Grenoble 54-6

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Restrictions of smooth maps to a closed subset2004

    • Author(s)
      S.Izumi
    • Journal Title

      Annales de l'Institut Fourier Grenoble, Tome 54-6

      Pages: 1811-1826

    • Description
      「研究成果報告書概要(欧文)」より
  • [Book] Real and Complex Singularities ( Proc. Australian-Japanese Workshop)2007

    • Author(s)
      L.Paunescu, (S.Izumi, T.Fukui, S.Koike, K.Sakuma 他)
    • Total Pages
      459
    • Publisher
      World Scientific
    • Description
      「研究成果報告書概要(和文)」より

URL: 

Published: 2008-05-27  

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