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2005 Fiscal Year Final Research Report Summary

Dimension, metrics and computability in metric spaces

Research Project

Project/Area Number 16540066
Research Category

Grant-in-Aid for Scientific Research (C)

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

Principal Investigator

HATTORI Yasunao  Shimane University, Interdisciplinary Faculty of Science and Engineering, Professor, 総合理工学部, 教授 (20144553)

Co-Investigator(Kenkyū-buntansha) MAEDA Sadahiro  Shimane University, Interdisciplinary Faculty of Science and Engineering, Professor, 総合理工学部, 教授 (40181581)
KIMURA Makoto  Shimane University, Interdisciplinary Faculty of Science and Engineering, Professor, 総合理工学部, 教授 (30186332)
YOKOI Katsuya  Shimane University, Interdisciplinary Faculty of Science and Engineering, Associate Professor, 総合理工学部, 助教授 (90240184)
KOYAMA Akira  Shizuoka University, Faculty of Science, Professor, 理学部, 教授 (40116158)
TSUIKI Hideki  Kyoto University, Graduate School of Human and Environmental Studies, Associate Professor, 人間・環境学研究科, 助教授 (10211377)
Project Period (FY) 2004 – 2005
KeywordsTopological dimension / Metric spaces / Transfinite dimension / Cohomological diemnsion / Formal balls / Manifolds / Compactness degree / Domain
Research Abstract

We considered the behavior of (transfinite) dimension functions on finite (and countable) unions of closed subsets in separable metrizable spaces. Let K be a class of spaces, α,β ordinal numbers with β<α and X be a space belonging to K with dX=α, where d denotes a dimension function monotonic for closed subspaces. Put m(X,d,β,α)=min{k:X is a union of k-many closed subspaces X1,…, Xk with d(Xk)≦β}, mK(d,β,α)=min(m(X,d,β,α):X is in K} and MK(d,β,α)=sup{m(X,d,β,α):X is in K}. Every ordinal number α can be represented as α=λ(α)+n(α),where λ(α) is a limit ordinal and n(α) is a natural number. For ordinal numbers β<α we put p(β,α)=(n(α)+1)/(n(β)+1) and q(β,α)=the smallest integer ≧p(β,α). Then we have the following : (1)Let P be the class of separable completely metrizable spaces. If a and B are finite ordinals with β<α, then mP(Cmp, β,α)=q(β,α) and MP(Cmp,β,α)=∞. (2)If α and β are infinite ordinals with β<α, then mP(trInd,β,α)=q(β,α) if λ (β) =λ(α), mP(trInd,β,α) does not exist if λ(β)≠λ(α) and MP(trInd,β,α)=∞ if λ(β)=λ(α), MP(trInd,β,α) does not exist if λ(β)≠(α). We also proved that there is no upper bound of trcmp in metrizable spaces. Concern with the theoretical computer science, we proved that the Lawson topology of the space of formal balls in the Hilbert space coincides with the product topology, but the Lawson topology of the space of formal balls in the space of real-valued functions on the closed interval does not coincide with the product topology. We also have several results about topological dyanamics, cohomological dimension of compact metric spaces, ordinal differential equations, and geometry.

  • Research Products

    (41 results)

All 2005 2004 Other

All Journal Article (39 results) Book (2 results)

  • [Journal Article] Holomorphic helix of proper order 3 on a complex hyperbolic plane2005

    • Author(s)
      T.Adachi, S.Maeda
    • Journal Title

      Topology and its Applications 146

      Pages: 201-207

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Characterization of totally umbilic hypersurfaces in a space forms by circles2005

    • Author(s)
      T.Adachi, S.Maeda
    • Journal Title

      Czechoslavak J. Math. 55

      Pages: 203-207

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Real hypersurfaces some of whose geodesics are plane curve in nonflat complex space form2005

    • Author(s)
      T.Adachi, S.Maeda
    • Journal Title

      Tohoku Math. J. 57

      Pages: 223-230

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Asymptotic behavior of nonlinear functional differentiable equations by Schauder's theorem2005

    • Author(s)
      T.A.Burton, T.Furumochi
    • Journal Title

      Nonlinear Studies 12・1

      Pages: 73-84

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Stabilities in FDEs by Schauder's theorem2005

    • Author(s)
      T.Furumochi
    • Journal Title

      Nonlinear Analysis 63

      Pages: 51-59

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Congruence classes of Frenet curves in complex quadrics2005

    • Author(s)
      M.Kimura, M.Ortega
    • Journal Title

      J. Geometry 83

      Pages: 121-136

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] A generalization of Cartan hypersurfaces2005

    • Author(s)
      M.Kimura
    • Journal Title

      Proc. 9th Internationa Workshop on Diff. Geometry, Kyungpook Univ. Korea 9

      Pages: 51-59

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] A class of real hypersurfaces in complex projective space2005

    • Author(s)
      M.Kimura
    • Journal Title

      Proc. 9th Internationa Workshop on Diff. Geometry, Kyungpook Univ. Korea 9

      Pages: 61-67

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Characterization of totally umbilic hypersurfaces in a space forms by circles2005

    • Author(s)
      T.Adachi, S.Maeda
    • Journal Title

      Czechoslovak J.Math. 55

      Pages: 203-207

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Real hypersurfaces some of whose geodesies are plane curve in nonflat complex space form2005

    • Author(s)
      T.Adachi, S.Maeda
    • Journal Title

      Tohoku Math.J. 57

      Pages: 223-230

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Asymptotic behavior of nonlinear functional differentiable equations by Schauder's theorem2005

    • Author(s)
      T.A.Burton, T.Furumochi
    • Journal Title

      Nonlinear Studies 12

      Pages: 73-84

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Congruence classes of Frenet curves in complex quadrics2005

    • Author(s)
      M.Kimura, M.Ortega
    • Journal Title

      J.Geometry 83

      Pages: 121-136

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] A generalization of Cartan hypersurfaces2005

    • Author(s)
      M.Kimura
    • Journal Title

      Proc.9^<th> Internationa Workshop on Diff.Geometry, Kyungpook Univ.Korea 9

      Pages: 51-59

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] A class of real hypersurfaces in complex projective space2005

    • Author(s)
      M.Kimura
    • Journal Title

      Proc.9^<th> Internationa Workshop on Diff.Geometry, Kyungpook Univ.Korea 9

      Pages: 61-67

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Metric spaces2004

    • Author(s)
      Y.Hattori
    • Journal Title

      Encyclopedia of General Topology (Elsevier Publ.)

      Pages: 235-238

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Special metrics2004

    • Author(s)
      Y.Hattori
    • Journal Title

      Encyclopedia of General Topology (Elsevier Publ.)

      Pages: 247-250

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] The behavior of dimension functions on unions of closed subsets2004

    • Author(s)
      M.Charalambous, V.Chatyrko, Y.Hattori
    • Journal Title

      J. Math. Soc. Japan 56

      Pages: 489-501

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] 階数1の対称空間上の幾何学2004

    • Author(s)
      足立俊明, 前田定廣
    • Journal Title

      数学(日本数学会編) 56・1

      Pages: 33-48

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Characterization of totally eta-umbilic real hypersurface in nonflat complex space forms by some inequality2004

    • Author(s)
      T.Itoh, S.Maeda
    • Journal Title

      Proceedings of the Japan Academy 80

      Pages: 61-64

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Stabilities in Volterra difference equations on a Banach space2004

    • Author(s)
      T.Furumochi, S.Murakami, Y.Nagabuchi
    • Journal Title

      Fields Institute Communication 42

      Pages: 159-175

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Volterra difference equations on a Banach space and abstract differential equations with pieacewise continuos delays2004

    • Author(s)
      T.Furumochi, S.Murakami, Y.Nagabuchi
    • Journal Title

      Japan J. Math. 30

      Pages: 387-412

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] A generalization of Wiener's lemma and its application to Volterra difference equations on a Banach space2004

    • Author(s)
      T.Furumochi, S.Murakami, Y.Nagabuchi
    • Journal Title

      J. Difference Equa. Appl. 10

      Pages: 1201-1214

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Recent development of cohomological dimension theory : Existence and applications of Edwards-Walsh resolutions2004

    • Author(s)
      A.Koyama
    • Journal Title

      Sugaku Expositions 17

      Pages: 125-150

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Compact metric spaces as minimal-limit sets in domains of bottomed sequences2004

    • Author(s)
      H.Tsuiki
    • Journal Title

      Math. Structures of Computer Sci. 14

      Pages: 853-878

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Dyadic subbases and efficiency properties of the induced (0,1,⊥)ω-representations2004

    • Author(s)
      H.Tsuiki
    • Journal Title

      Topology Proc. 28

      Pages: 673-687

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] The behavior of dimension functions on unions of closed subsets2004

    • Author(s)
      M.Charalambous, V.Chatyrko, Y.Hattori
    • Journal Title

      J.Math.Soc.Japan 56

      Pages: 489-501

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Geometry of symmetric spaces of rank 12004

    • Author(s)
      T.Adachi, S.Maeda
    • Journal Title

      Sugaku (in Japanese) 56

      Pages: 33-48

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Characterization of totally eta-umbilic real hypersurface in nonflat complex space forms by some inequality2004

    • Author(s)
      T.Adachi, S.Maeda
    • Journal Title

      Proceedings of the Japan Academy 80

      Pages: 61-64

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Volterra difference equations on a Banach space and abstract differential equations with pieacewise continuos delays2004

    • Author(s)
      T.Furumochi, S.Murakami, Y.Nagabuchi
    • Journal Title

      Japan J.Math. 30

      Pages: 387-412

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] A generalization of Wiener's lemma and its application to Volterra difference equations on a Banach space2004

    • Author(s)
      T.Furumochi, S.Murakami, Y.Nagabuchi
    • Journal Title

      J.Difference Equa.Appl. 10

      Pages: 1201-1214

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Compact metric spaces as minimal-limit sets in domains of bottomed sequences2004

    • Author(s)
      H.Tsuiki
    • Journal Title

      Math.Structures of Computer Sci. 14

      Pages: 853-878

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Dyadic subbases and efficiency properties of the induced {0,1,⊥}ω-representations2004

    • Author(s)
      H.Tsuiki
    • Journal Title

      Topology Proc. 28

      Pages: 673-687

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Partitions of spaces by locally compact subspaces

    • Author(s)
      V.Chatyrko, Y.Hattori, H.Ohta
    • Journal Title

      Houston J. Math. (印刷中)

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Totally geodesic immersions of Kohler manifolds and Kahler frenet curves

    • Author(s)
      S.Maeda, H.Tanabe
    • Journal Title

      Math. Z. (印刷中)

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Isoparametric hypersufaces with less than 4 principal curvatures in a sphere

    • Author(s)
      T.Adachi, S.Maeda
    • Journal Title

      Colloqium Math. (印刷中)

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Strong transitivitiy and graph maps

    • Author(s)
      K.Yokoi
    • Journal Title

      Bull. Polish Acad. Sci. (印刷中)

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Partitions of spaces by locally compact subspaces

    • Author(s)
      V.Chatyrko, Y.Hattori, H.Ohta
    • Journal Title

      Houston J.Math. (to appear)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Totally geodesic immersions of Kohler manifolds and Kahler frenet curves

    • Author(s)
      S.Maeda, H.Tanabe
    • Journal Title

      Math.Z. (to appear)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Strong transitivitiy and graph maps

    • Author(s)
      K.Yokoi
    • Journal Title

      Bull.Polish Acad.Sci. (to appear)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Book] Metric spaces, Encyclopedia of General Topology2004

    • Author(s)
      Y.Hattori
    • Total Pages
      235-238
    • Publisher
      Elsevier Publ.
    • Description
      「研究成果報告書概要(欧文)」より
  • [Book] Special metrics, Encyclopedia of General Topology2004

    • Author(s)
      Y.Hattori
    • Total Pages
      247-250
    • Publisher
      Elsevier Publ.
    • Description
      「研究成果報告書概要(欧文)」より

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Published: 2007-12-13  

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