| Project/Area Number |
12640174
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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 |
Basic analysis
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| Research Institution | Osaka University |
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Principal Investigator |
OHYAMA Yousuke Osaka University, Graduate School of Information Science and Technology, Associate Professor, 大学院・情報科学研究科, 助教授 (10221839)
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| Co-Investigator(Kenkyū-buntansha) |
MIYANISHI Masayoshi Osaka University, Graduate School of Science, Professor, 大学院・理学研究科, 教授 (80025311)
HIBI Takayuki Osaka University, Graduate School of Information Science and Technology, Professor, 大学院・情報科学研究科, 教授 (80181113)
KOTANI Shin-ichi Osaka University, Graduate School of Science, Professor, 大学院・理学研究科, 教授 (10025463)
CHAWANYA Tsuyoshi Osaka University, Graduate School of Information Science and Technology, Associate Professor, 大学院・情報科学研究科, 助教授 (80294148)
YAMAMOTO Yoshihiko Osaka University, Graduate School of Science, Professor, 大学院・理学研究科, 教授 (90028184)
鹿野 忠良 大阪大学, 大学院・理学研究科, 講師 (80028183)
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| Project Period (FY) |
2000 – 2002
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| Project Status |
Completed (Fiscal Year 2002)
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| Budget Amount *help |
¥3,700,000 (Direct Cost: ¥3,700,000)
Fiscal Year 2002: ¥1,100,000 (Direct Cost: ¥1,100,000)
Fiscal Year 2001: ¥1,100,000 (Direct Cost: ¥1,100,000)
Fiscal Year 2000: ¥1,500,000 (Direct Cost: ¥1,500,000)
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| Keywords | the Painleve equation / Twisotr thoery / non-associative algebras / パンルベ函数 / τ函数 / パンルベ方程式 / 可積分系 / 超幾何微分方程式 / モノドロミ保存変形 |
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| Research Abstract |
This research started to develop the guiding principle "The Painleve equations are non-Abelian analogue of the hypergeometric equations". At the end of this research we obtain a new direction in Painleve analysis : "monodromy-solvable Painleve functions". This class of solutions will play an important role in future study of the Painleve analysis.
In this three years, various progress was developped in many other fields. The Painleve analysis does not remain in mathematical physics, but has a relation on various fields, such as nuber theory, a combination theory, probability theory and more. Now feedbach from those fields is performed conversely.
The monodromy solvablity is a new keyword in such interaction. Namely, most of solutions of the Painleve equations (or the equations of monodromy preservation deformations) which play the important role in applications are not classical solutions in Umemura's meaning. But the monodromy of the linear equations corresponding to these solutions is
… More
solvable. The solvablity of the monodromy may be developped into the solvablity of the Painleve functions themselves.
As research derived from this research, I raise two important things. We obtain the conditions of the solvablity of the Darboux-Halphen equations of rank 4 using non-associative algebras. In the case of the rank 3, the similar conditions when the Darboux-Halphen equations reduce to the hypergeometric equations were known. An application to Painleve analysis have opened by this new result in the case of the rank 4. In the second, we determined specials solutions of the Painleve equation of D_7 type, Thus the transcendent classical solutions of Umemura's meaning of the Painleve equations were completely classified. Classification of algebra solutions of the Painleve VI still remains.
After an imperfect proof on the irreducibility of the Painleve I equation is announced, it passed 100 years. We could not classify all of classical solutions of Umemura's meaning, but I think that we have found out a new direction of the Painleve analysis. Less
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