Project/Area Number  01540125 
Research Category 
GrantinAid for Scientific Research (C).

Research Field 
解析学

Research Institution  Kyoto University 
Principal Investigator 
ASANO Kiyoshi Kyoto University Yoshida College Professor, 教養部, 教授 (90026774)

CoInvestigator(Kenkyūbuntansha) 
西和田 公正 京都大学, 教養部, 助教授 (60093291)
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)
伊達 悦朗 京都大学, 教養部, 助教授 (00107062)
MORIMOTO Yoshinori Kyoto University Yoshida College Assistant Professor, 教養部, 助教授 (30115646)
笠原 皓司 京都大学, 教養部, 教授 (70026748)
KONO Norio Kyoto University Yoshida College Professor, 教養部, 教授 (90028134)
UE Masaaki Kyoto University Yoshida College Assistant Professor, 教養部, 助教授 (80134443)

Project Fiscal Year 
1989 – 1990

Project Status 
Completed(Fiscal Year 1990)

Budget Amount *help 
¥2,200,000 (Direct Cost : ¥2,200,000)
Fiscal Year 1990 : ¥500,000 (Direct Cost : ¥500,000)
Fiscal Year 1989 : ¥1,700,000 (Direct Cost : ¥1,700,000)

Keywords  Hypoellipticity / Complex dynamics / Stable manifold / Normalbundle / Arnold's tongue / Swallow's tail / Distribution tail / Boltzmann equations / 準楕円性 / 複素力学系 / 安定多様体 / ノマル バンドル / アノルドの舌 / 燕の尾 / 分布の裾の評価 / ボルツマン方程式 / 楕円性 / ストレンジ・アトラクタ / 退化シュレディンガ作用素 / 不確定性原理 / SOSモデル / 星状三角形関係 / ボルツマンウィィト / アベル関数方程式 / ホイゲンスの原理 
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 semielliptic 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 1dimensional 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.
