1998 Fiscal Year Final Research Report Summary
MATHEMATICAL FOUNDATIONS OF RENORMALIZATION GROUP THEORY AND ITS APPLICATIONS
| Project/Area Number |
09640304
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
General mathematics (including Probability theory/Statistical mathematics)
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| Research Institution | SETSUNAN UNIVERSITY |
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Principal Investigator |
ITO Keiichi R. SESTUNSN UNIVERSITY,MATHEMATICS DEPARTMENT,PROFESSOR, 工学部, 教授 (50268489)
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| 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)
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| Project Period (FY) |
1997 – 1998
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| Keywords | Renormalization Group / Random Walk / Critical Temperature / O (N) Spin Model / Navier-Stokes equaion / Viscosity / Kormogorov law / Dissipation |
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| 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.
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