权益分类	功能权益	普通用户	{{item.name}}会员
{{category.name}}	{{benefitItem.name}}

Non-Abelian gauge fields end Painleve functions

非阿贝尔规范场结束 Painleve 函数

基本信息

批准号：
12640174
负责人：
OHYAMA Yousuke
金额：
$ 2.37万
依托单位：
Osaka University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2000
资助国家：
日本
起止时间：
2000 至 2002
项目状态：
已结题

项目摘要

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

本研究开始发展的指导原则“Painleve方程是超几何方程的非阿贝尔模拟”。在研究的最后，我们得到了Painleve分析的一个新方向：“单值可解Painleve函数”。这类解将在Painleve分析的未来研究中发挥重要作用。在这三年中，许多其他领域也取得了各种进展。Painleve分析并不局限于数学物理，而是与许多领域都有联系，如数论、组合论、概率论等。单值可解性是这类相互作用中的一个新的关键词。也就是说，在实际应用中起重要作用的Painleve方程（或单值保持变形方程）的解大多不是Umemura意义下的经典解。但是对应于这些解的线性方程组的单值性是 ...更多信息可解的monodromy的可解性可以发展为Painleve函数本身的可解性，作为研究的延伸，我提出了两个重要的问题。本文利用非结合代数得到了秩为4的Darboux-Hegenen方程的可解性条件。在秩为3的情形下，已知Darboux-Heglien方程约化为超几何方程的类似条件. Painleve分析的应用程序已经打开了这个新的结果在秩4的情况下。第二部分确定了D_7型Painleve方程的特解，从而对Painleve方程的Umemura意义下的超越经典解进行了完整的分类。Painleve VI方程的代数解的分类至今仍在，自从Painleve I方程不可约性的一个不完善的证明被公布后，已经过去了100年。我们不能把梅村意义下的所有经典解都分类，但我认为我们已经找到了Painleve分析的一个新方向。少