1990 Fiscal Year Final Research Report Summary
Nonlinear Differential Equations and Related Topics
| Project/Area Number |
01540125
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| Research Category |
Grant-in-Aid for General Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Research Field |
解析学
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| Research Institution | Kyoto University |
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Principal Investigator |
ASANO Kiyoshi Kyoto University Yoshida College Professor, 教養部, 教授 (90026774)
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| Co-Investigator(Kenkyū-buntansha) |
UE Masaaki Kyoto University Yoshida College Assistant Professor, 教養部, 助教授 (80134443)
KONO Norio Kyoto University Yoshida College Professor, 教養部, 教授 (90028134)
USHIKI Shigehiro Kyoto University Yoshida College Assistant Professor, 教養部, 助教授 (10093197)
UEDA Tetuo Kyoto University Yoshida College Assistant Professor, 教養部, 助教授 (10127053)
MORIMOTO Yoshinori Kyoto University Yoshida College Assistant Professor, 教養部, 助教授 (30115646)
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| Project Period (FY) |
1989 – 1990
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| Keywords | Hypoellipticity / Complex dynamics / Stable manifold / Normalbundle / Arnold's tongue / Swallow's tail / Distribution tail / Boltzmann equations |
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| Research Abstract |
Under our research program we studied various types of differential equations, linear and nonlinear, ordinary and partial, and related topics. We state the summary of our results below :
1. We studied semi-elliptic operators with coefficients degenerating at a point with infinite order and found sufficient conditions for such operators to be hypoelliptic.
2. We found some examples of hypoelliptic operators wich are not micro hypoelliptic
3. We studied the local structure of analytic transformations around its fixed point, and obtained several properties of the stable manifold connected with the fixed point
4. We studied the geometric structure of a neighborhood of a national curve with a node in a two dimensional complex manifold
5. We pursued a long, troublesome numerical experiment to investigate Arnold's tongues and Swallow's tails appearing in 1-dimensional complex dynamical systems, and obtained various interesting figures and pictures.
6. In our study of a Gaussian process with nonconstant variance, we obtained a sharp upper bound of the distribution tail.
7. We found fine structures of the linear and nonlinear collision integrals appearing in the Boltzmann equation. But the paper is not completed yet. In conclusion, we belive we made important steps in the study of several nonlinear phenomena in mathematics. This research should be continued to obtain further results.
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