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2000 Fiscal Year Final Research Report Summary

Topological theory of local systems having irregular singularities

Research Project

Project/Area Number 09440065
Research Category

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

Allocation TypeSingle-year Grants
Section一般
Research Field 解析学
Research InstitutionKumamoto University

Principal Investigator

HARAOKA Yoshishige  Faculty of Science, Kumamoto University Associate Professor, 理学部, 助教授 (30208665)

Co-Investigator(Kenkyū-buntansha) OHWAKI Shin-ichi  Faculty of Science, Kumamoto University Professor, 理学部, 教授 (50040506)
KOHNO Mitsuhiko  Faculty of Science, Kumamoto University Professor, 理学部, 教授 (30027370)
KIMURA Hironobu  Graduate School of Science and Technology, Kumamoto University Professor, 大学院・自然科学研究科, 教授 (40161575)
KAMIMURA Yutaka  Tokyo University of Fisheries Faculty of Fisheries Associate Professor, 水産学部, 助教授 (50134854)
TAKADA Yoshikazu  Faculty of Engineering, Kumamoto University Professor, 工学部, 教授 (70114098)
Project Period (FY) 1997 – 2000
Keywordshypergeometric function / confluent hypergeometric function / irrgular singularity / twisted homology / twisted cohomology / rigid local system / arrangement of hyperplanes / accessory parameter
Research Abstract

The aim of this research was to study the topological structure of integral representations of confluent hypergeometric functions, and to clarify the relation between the topological structure and the analytic behavior of the functions.
When the local system corresponding to the integral is defined over a 1-dimensional space, we have constructed good bases of twisted (co) homology group. Since we have constructed them by confluence, the twisted Riemann's period relation for regular singular hypergeometric functions can be extended to ones for confluent hypergeometric functions.
On the relation between twisted cycles and asymptotic behaviors of functions given by the integral over the cycles, we noticed that, if a cycle which give a pure asymptotic behavior suivives in the process of confluence, the cycle after confluence also will give a pure behavior. Using this understanding, we can evaluate several connection coefficients and global behaviors for confluent hypergeometric functions.
For local systems defined over multi-dimensional spaces, we have constructed bases of twisted (co) homology groups by using the exterior power structure. We noticed that the asymptotic behavior of a function given by a vanishing cycles can be discribed by functions corresponding to degenerate arrangements of hyperplanes.
The study of confluent hypergeometric functions corresponding to degenerate arrangements of hypersurfaces will be substantial in future, and they will be closely related to rigid local systems and GKZ hypergeometric functions.

  • Research Products

    (22 results)

All Other

All Publications (22 results)

  • [Publications] Y.Haraoka: "On rigidity of Pfaffian systems coming from Okubo systems"Kyushu J.Math.. 55. 189-205 (2001)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] 原岡喜重: "超幾何関数におけるstratification,外積構造,漸近挙動,接続係数"数理解析研究所講究録. (2001)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Y.Haraoka: "Quadratic relations for confluent hypergeometric function on Z_<2,n+1>"Funkcialaj Ekracioj. 40. 435-490 (1999)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] H.Kimura: "Analogue of flat basis and cohomological intersection number for general hypergeometric functions"J.Math.Sci.Univ.Tokyo. 6. 415-436 (1999)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] M.Kohno: "Integral of Psi-function"J.Difference Eq.Appl.. (2000)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Y.Takada: "A symptotic improvement of the usual confidence set in a multivariate normal distribution with unknown variance"J.Multivariate Anal.. 64. 118-130 (1998)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] M.Kohno: "Gli-al Analysis in Linear Differential Equations"Kluver Academic Publishers. 526 (1999)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Y.Haraoka and T.Yokoyama: "On rigidity of Pfaffian systems coming from Okubo systems"Kyushu Journal of Mathematics. 55 no.1. 189-205 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Y.Haraoka: "Quadratic relations for confluent hypergeometric functions on Z_<2, n+1>, Funkcialaj Ekvacioj"42, no.3. 435-490 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Y.Haraoka: "Monodromy of an Okubo system with non-semisimple exponents"Funkcialaj Ekvacioj. 40, no.3. 435-457 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Y.Haraoka: "Confluence of cycles for hypergeometric functions on Z_<Z2, n+1>"Transactions of the American Mathematical Society. vol.349, no.2. 675-712 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] H.Kimura and M.Taneda: "Analogue of flat basis and cohomological intersection numbers for general hypergeometric functions"J.Math.Sci.Univ.Tokyo.. 6. 415-436 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] H.Kimura: "On the homology group associated with the general Airy integral"Kumamoto J.Math.. 10. 11-29 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] H.Kimura: "On rational de Rham cohomology associated with the generalized Airy function"Annali di Scuola Norm.Sup.di Pisa. 24. 351-366 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] H.Kimura: "On rational de Rham cohomology associated with the generalized confluent hypergeometric function I, P^1 case"Royal Soc.Edinburgh. 127A. 145-155 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] M.Kohno: "Integrals of Psi-function"Journal of Difference Equations and Applications. (to appear). (2000)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Y.Takada: "Asymptotic improvement of the usual confidence set in a multivariate normal distribution with unknown variaqce"J.Multivariate Anal. 64 no.2. 118-130 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Y.Kamimura: "Conductivity identification in the heat equation by the heat flux"J.Math.Anal.Appl.. 235. 192-216 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] K.Iwasaki and Y.Kamimura: "Convolution calculus for a class of singular Volterra integral equation"J.Integral Equations Appl.. 11. 461-499 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] S.Ohwaki and K.Matsuda: "On formulae for correlation coefficients between normally scattered functions II"Kumamoto J.Math.. 13. 83-92 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] S.Ohwaki and K.Matsuda: "On formulae for correlation coefficients between normally scattered functions"Kumamoto J.Math.. 12. 73-79 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] M.Kohno: "Global Analysis in Linear Differential Equations"Kluwer Academic Publishers. 526. (1999)

    • Description
      「研究成果報告書概要(欧文)」より

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Published: 2002-03-26  

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