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Global Analysis of the Spectrum of an Infinite Graph

Research Project

Project/Area Number 10440056
Research Category

Grant-in-Aid for Scientific Research (B).

Allocation TypeSingle-year Grants
Section一般
Research Field Global analysis
Research InstitutionTohoku University

Principal Investigator

URAKAWA Hajime  Tohoku University, Graduate School of Information Sciences, Professor, 大学院・情報科学研究科, 教授 (50022679)

Co-Investigator(Kenkyū-buntansha) KANEKO Makoto  Tohoku University, Graduate School of Information Sciences, Professor, 大学院・情報科学研究科, 教授 (10007172)
UCHIDA Koji  Tohoku University, Graduate School of Information Sciences, Professor, 大学院・情報科学研究科, 教授 (20004294)
ASOH Tohl  Tohoku University, Graduate School of Information Sciences, Associate Professor, 大学院・情報科学研究科, 助教授 (00111352)
ITOH Jin-ichi  Kumamoto University, Faculty of Education, Associate Professor, 教育学部, 助教授 (20193493)
OKADA Masami  Tohoku University, Graduate School of Information Sciences, Professor, 大学院・情報科学研究科, 教授 (00152314)
会田 茂樹  東北大学, 大学院・情報科学研究科, 助教授 (90222455)
Project Period (FY) 1998 – 2000
Project Status Completed (Fiscal Year 2000)
Budget Amount *help
¥13,500,000 (Direct Cost: ¥13,500,000)
Fiscal Year 2000: ¥4,400,000 (Direct Cost: ¥4,400,000)
Fiscal Year 1999: ¥4,400,000 (Direct Cost: ¥4,400,000)
Fiscal Year 1998: ¥4,700,000 (Direct Cost: ¥4,700,000)
Keywordsfirst eigenvalue / Dirichlet eigenvalue problem / spectrum / infinite graph / affine connection / harmonic morphism / Green kernel / graph with boundary / 離散ラプラシアン / チーガー定数 / 熱核 / 離散曲面 / 離散調和射 / 次数 / 射影平坦 / ヤング・ミルズ接続 / ワイル構造 / ディリクレ問題 / バルタの定理 / ファーベル・クラーンの定理 / グラフ
Research Abstract

We have obtained the following results :
(1) We estimated the first eigenvalue and sectional curvature of compact symmetric spaces, and gave counter example of Elworthy-Rosenberg.
(2) We obtained the Faber-Krahn type estimate for the first eigenvalue of the Dirichlet eigenvalue problem for a coonected finite graph with boundary.
(3) We obtained the Barta type estimate of the Dirichlet eigenvalue problem for a graph, and sharp estimate of the infimum of the spectrum of an infinite graph.
(4) We introduce the notion of surgery, and gave its application.
(5) We characterized homogeneous spaces admitting affine projectively flat connections.
(6) We gave the first variational formula of a natural functional on the space ofWeyl structures.
(7) Harmonic morphisms play important roll in differential geometry oh harmonic maps. We consider the discrete harmonic morphisms, and obtained their complete characterization.
(8) We obtained theory of solutions of the inhomogeneous Yang-Mills equation.
(9) We gave a characterization of harmonic morphism between two graphs.
(10) We obtain a general estimation formula of the spectrum of the discrete Laplacian for an infinite graph. Our method is quite new using our new notion of the incresing degree and decreasing one relative to distance from a fixed vertex.

Report

(4 results)
  • 2000 Annual Research Report   Final Research Report Summary
  • 1999 Annual Research Report
  • 1998 Annual Research Report
  • Research Products

    (43 results)

All Other

All Publications (43 results)

  • [Publications] H.Urakawa: "Eigenvalue pinching theorems on compact symmetric spaces"Proceedings of the American Mathematical Society. 126. 3065-3069 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] A.Katsuda: "The first eigenvalue of the discrete Dirichlet problem for a graph"Jouranl of Combinatorial Mathematics Combinatorial Computing. 27. 217-225 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] H.Urakawa: "Eigenvalue comparison theorems of the discrete Laplacians for a graph"Geometriae Dedicata. 74. 95-112 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] A.Katsuda: "The Faber-Krahn type isoperimetric inequalities for a graph"Tohoku Mathematical Journal. 51. 267-281 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] H.Urakawa: "On invariant projectively flat affine connections"Hokkaido Mathematical Journal. 28. 333-356 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] T.Ichiyama: "A conformal gauge invariant functional for Weyl structures and the first variation formula"Tsukuba Journal of Mathematics. 23. 551-564 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] H.Urakawa: "The spetrum of an infinite graph"Canadian Journal of Mathematics. 52(5). 1057-1084 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] H.Urakawa: "A discrete analogue of the harmonic morphism and Green kernel comparison theorems"Glasgow Mathematical Journal. 42. 319-334 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] U.Urakawa: "On the inhomogeneous Yang-Mills equation d_D*R^D=f"Interdisciplinary Information Sciences. 6. 41-52 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] H.Urakawa: "A discrete analogue of the harmonic morphism"Harmonic morphisms, harmonic maps and related topics. 413. 97-108 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] J.Itoh: "The Lipschitz continuity of the distance function to the cut locus"Transactions of the American Mathematical Society. 353. 21-40 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] H.Taya: "Iwasawa invariants and class numbers of quadratic fields for the prime 3"Proceedings of the American Mathematical Society. 128. 1285-1292 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] H.Urakawa: "Eigenvalue pinching theorems on compact symmetric spaces"Proceedings of the American Mathematical Society. 126. 3065-3069 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] A.Katsuda: "The first eigenvalue of the discrete Dirichlet problem for a graph"Journal of Combinatoral Mathematics and Combinatorial Computing. 27. 217-225 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] H.Urakawa: "Eigenvalue comparison theorems of the discrete Laplacians for a graph"Geometriae Dedicata. 74. 95-112 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] A.Katsuda: "The Faber-Krahn type isoperimetric inequalities for a graph"Tohoku Mathematical Journal. 51. 267-281 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] H.Urakawa: "On invariant projectively flat affine connections"Hokkaido Mathematical Journal. 28. 333-356 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] T.Ichiyama: "A conformal gauge invariant functional for Weyl structures and the first variation formula"Tsukuba Journal of Mathematics. 23. 551-564 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] H.Urakawa: "The spetrum of an infinite graph"Canadian Journal of Mathematics. 52(5). 1057-1084 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] H.Urakawa: "A discrete analogue of the harmonic morphism and Green kernel comparison theorems"Glasgow Mathematical Journal. 42. 319-334 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] U.Urakawa: "On the inhomogeneous Yang-Mills equation d_D*R^D=f"Interdisciplinary Information Sciences. 6. 41-52 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] H.Urakawa: "A discrete analogue of the harmonic morphism"Harmonic morphisms, harmonic maps and related topics. 413. 97-108 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] J.Itoh: "The Lipschitz continuity of the distance function to the cut locus"Transactions of the American Mathematical Society. 353. 21-40 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] H.Taya: "Iwasawa invariants and class numbers of quadratic fields for the prime 3"Proceedings of the American Mathematical Society. 128. 1285-1292 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] H.Urakawa: "The spectrum of an infinite graph"Canadian Journal of Mathematics. 52(5). 1057-1084 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] H.Urakawa: "A discrete analogue of the harmonic morphism and Green kernel comparison theorems."Glasgow Mathematical Journal. 42. 319-334 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] H.Urakawa: "On the inhomogeneous Yang-Mills equation ol^*R=f"Interdisciplinary Information Sciences. 6. 41-52 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] H.Urakawa: "A discrete analogue of the harmonic morphism"Harmonic morphisms, harmonic snaps and related topics. 413. 97-108 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] J.Itoh: "The Lipschitz continuity of the distance function to the ait locus"Transactions of American Mathematical Society. 353. 21-40 (2001)

    • Related Report
      2000 Annual Research Report
  • [Publications] H.Taya: "Iwasawa inoariants and class numbers of quadratic fields for the prime 3"Proceedings of American Mathematical Society. 128. 1285-1292 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] H.Urakawa: "On invariant projectively flat affine connections"Hokkaido Mathematical Journal. 28. 333-356 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] H.Urakawa: "Eigenvalue comparison theorems of the discrete Laplacians for a graph"Geometriae Dedicata. 74. 95-112 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] H.Urakawa: "The Faber-Krahn type isoperimetric inequalities for a graph"Tohoku Mathematical Journal. 51. 267-281 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] H.Urakawa: "A discrete analogue of the harmonic morphism"Harmonic morphisms, harmonic maps and related topics. 413. 97-108 (2000)

    • Related Report
      1999 Annual Research Report
  • [Publications] H.Urakawa: "A conformal gauge invariant functional for Weyl structures and the first variational formula"Tsukuba Journal of Mathematics. (印刷中). (2000)

    • Related Report
      1999 Annual Research Report
  • [Publications] H.Urakawa: "A discrete analogue of the harmonic morphism and Green kernel comparison theorems"Glasgow Mathematical Journal. (印刷中). (2000)

    • Related Report
      1999 Annual Research Report
  • [Publications] H.Urakawa: "Eigenvalue pinching theorems on compact symmetric spaces" Proceedings of American Mathematical Society. 126. 3065-3069 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] H.Urakawa: "Open problems in affine differential geometry and related topics" Interdisciplinary Information Sciences. 4. 125-127 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] H.Urakawa: "Eigenvalue comparison theorems of the discrete Laplacians for a graph" Geometriae Dedicata. 74. 95-112 (1999)

    • Related Report
      1998 Annual Research Report
  • [Publications] H.Urakawa: "The first eigenvalue of the discrete Dirichlet problem for a graph" J.Combinatorial Mathematics & Combinatorial Computing. (印刷中). (1999)

    • Related Report
      1998 Annual Research Report
  • [Publications] H.Urakawa: "On invariant projectively flat affine connections" Hokkaido Mathematical J.(印刷中). (1999)

    • Related Report
      1998 Annual Research Report
  • [Publications] H.Urakawa: "On a discrete analogue of harmonic morphisms" Harmonic Morphisms,Harmonic Maps and Related Topics. (印刷中). (1999)

    • Related Report
      1998 Annual Research Report
  • [Publications] 浦川肇: "解析入門-級数/複素関数/ベクトル解析-" 裳華房, 152 (1998)

    • Related Report
      1998 Annual Research Report

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Published: 1998-04-01   Modified: 2016-04-21  

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