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MATHEMATICAL FOUNDATIONS OF RENORMALIZATION GROUP THEORY AND ITS APPLICATIONS

Research Project

Project/Area Number 09640304
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field General mathematics (including Probability theory/Statistical mathematics)
Research InstitutionSETSUNAN UNIVERSITY

Principal Investigator

ITO Keiichi R.  SESTUNSN UNIVERSITY,MATHEMATICS DEPARTMENT,PROFESSOR, 工学部, 教授 (50268489)

Co-Investigator(Kenkyū-buntansha) TERAMOTO Yoshiaki  SESTUNSN UNIVERSITY MATHEMATICS DEPARTMENT ASSOCIATE PROFESSOR, 工学部, 助教授 (40237011)
WATARAI Seizo  SESTUNSN UNIVERSITY MATHEMATICS DEPARTMENT ASSOCIATE PROFESSOR, 工学部, 助教授 (20131500)
IKEBE Teruo  SESTUNSN UNIVERSITY MATHEMATICS DEPARTMENT PROFESSOR, 工学部, 教授 (00025280)
中脇 雄治  摂南大学, 工学部, 教授 (60207959)
Project Period (FY) 1997 – 1998
Project Status Completed (Fiscal Year 1998)
Budget Amount *help
¥2,400,000 (Direct Cost: ¥2,400,000)
Fiscal Year 1998: ¥800,000 (Direct Cost: ¥800,000)
Fiscal Year 1997: ¥1,600,000 (Direct Cost: ¥1,600,000)
KeywordsRenormalization Group / Random Walk / Critical Temperature / O (N) Spin Model / Navier-Stokes equaion / Viscosity / Kormogorov law / Dissipation / O(N)Spin Model / くり込み群 / 関数行列式 / 乱歩 / O(N)スピン模型 / polymer exponsicn / ナヴィエ・ストークス方程式 / Kolmogorov則 / モンテカルロ法
Research Abstract

1. ITO and TAMURA (Kanazawa Univ.) studied classical 0(N) symmetric spin model by renormal ization group ( block spin transformation) method. They first showed that the correlation functions can be described by self-avoiding walks which enabled them to obtain almost optimal bounds for the critical temperatures. In the second stage, they argued the integrability of the functional de-terminent det^<N/2>(1+<approximately equal>92iGpsi/ROO<N>) with respect to psi, , where psi is the auxially field introcuced for Fourier Transformation. Using the technique called the polymer (cluster) expansion , they showed that the inverse critical temperature beta_C obeys the bound beta_C <bounded integral>1 N log N in two dimensions, which implies the existence of strong deviation. (beta_C-N for the dimension more than or equal to 3.) This method is expected to establish a complete proof of the conjecture beta_C = *by applying the present method recursively to the model. One typical problem in this approach is that there appear complicated non-local interactions after the transformations. So we need to control the main part of the flow. Some partial results are obtained and will be published soon.
2. Teramoto investigated flow of non-compressible viscous fluid around a cyclinder by using cylindrical coordinate. He established that the equation exhibits global solution in time if the initial condition is sufficiently close to the stational current.
3. Teramoto and Ito investigated properties of turbulence, among them, the Kolmogorov law about the dissipation of energy and deviation from it. They tried to derive the deviation from the Navier-Stokes equation but they could not obtain concrete results this year.

Report

(3 results)
  • 1998 Annual Research Report   Final Research Report Summary
  • 1997 Annual Research Report
  • Research Products

    (24 results)

All Other

All Publications (24 results)

  • [Publications] 伊東恵一: "Representations of O(N) Spin Models by Self-Avoiding Walks" Commun.in Math.Phys.183. 723-737 (1997)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] 伊東恵一: "Deviation of Upper Bounds of Critical Temperatures of Two-Dimensional O(N) Spin Models" Letters in Math.Phys.44. 339-348 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] 伊東恵一: "N Dependence of Upper Bounds of Citical Temperatures of 2D O(N) Spin Models" Common.Math.Phys.(印刷中). (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] 寺本恵昭: "Navier-Stokes Flow Down a Vertical Colomn" Pitman Research Notes in Math.338. 160-173 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] 渡会征三: "Critical Exponents of Ising-Like Heisenberg Ferromagnets" Jounal of Phys.Soc.Japan. 67. 816-824 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] K.R.Ito: "Rigorous Bounds for Critical Temperatures of O (N) Spin Models by Transformations of Block Spin Type" Letters in Math.Phys.37. 349-362 (1996)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] K.R.Ito: "Self-Avoiding Random Walk Representation of O (N) Spin Models" Commun.Math.Phys.183. 723-737 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] K.R.Ito: "Deviations of Upper Bounds of Critical Temperatures of Two-Dimensional O(N) Spin Models" Lett.Math.Phys.44. 339-348 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] K.R.Ito: "N Dependence of Critical Temperatures of Two-Dimensional O (N) Spin Models" Commun.Math.Phys.(in press). (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] K.R.Ito: "New Renormalization Group Flow Two-Dimensional O (N) Spin Model" Phys.Rev.Letters. (submitted). (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] Teruo Ikebe: "Resolvent Convergence of Schroedinger Operators with a Penetrable Sphere Interaction"

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] Yoshiaki Teramoto: "Free Surface Waves and Their Stability" JAERI-memo. 8, No.285. 53-63 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] Yoshiaki Teramoto: "Navier-Stokes Flow Down a Vertical Column" Pitman Res.Notes. 338. 160-173 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] Seizo Watarai: "Critical Exponents of Ising-Like Heisenberg Ferromagnets on the Hexagonal Lattice" J.Phys.Soc.Japan. 67. 816-824 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] 伊 東 恵 一: "Representations of O(N)Spin Models by Self -Avoiding Walks" Commun in Math Phys. 183. 723-737 (1997)

    • Related Report
      1998 Annual Research Report
  • [Publications] 伊 東 恵 一: "Deviation of Upper Bounds of Critical Temperatures of Two-Dimensional O(N) Spin Models" Letters in Math Phys.44. 339-348 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] 伊 東 恵 一: "N Dependence of Upper Bounds of Critical Temperatures of 2D O(N) Spin Models" Commun.in Math.Phys.(印刷中). (1999)

    • Related Report
      1998 Annual Research Report
  • [Publications] 寺 本 恵 昭: "Navier-Stokes Flow Down a Vertical Column" Pitman Research Notes in Math.338. 160-173 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] 渡 会 征 三: "Critical Experients of Ising-Like Heisenberg Fevomagnets" Journal of PhyS.Soc.Japan. 67. 816-824 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] 伊東恵一: "Rigorous Bound for Critical Temperotures of O (N) Spin Models by Transformations of Block Spin Type" Letters in Mathematical physics. 37・3. 349-362 (1996)

    • Related Report
      1997 Annual Research Report
  • [Publications] 伊東恵一: "Representations of O (N) Spin Models by Self Arociding Walks" Common. in Math. Phys.183. 723-737 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] 伊東恵一: "2次元O (N) Spin Modelの臨界点のN依存性" 素粒子論研究. 96・2. 1319-1327 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] 伊東恵一: "Deviation of Upper Bands of Critical Temperature of Two-Dimensional O (N) Spin Models" Letters in Mathematical Physics. (発表予定). (1998)

    • Related Report
      1997 Annual Research Report
  • [Publications] 寺本 恵昭: "自由表面流とその安定性" JAERI-memo. 08・285.

    • Related Report
      1997 Annual Research Report

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Published: 1997-04-01   Modified: 2016-04-21  

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