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

这项研究开始发展指导原则：“潘leve方程是超几何方程的非亚伯类似物”。在这项研究结束时，我们获得了潘leve分析的新方向：“可溶解的可溶解painleve功能”。这类解决方案将在对潘leve分析的未来研究中发挥重要作用。在这三年中，在许多其他领域都发展了各种进展。 painleve分析并不保留在数学物理学中，而是与各个领域的关系，例如Nuber理论，结合理论，概率理论等。现在，这些领域的反馈相反。单型溶剂溶解性是这种相互作用的新关键字。也就是说，在应用中起重要作用的大多数painleve方程解决方案（或单肌保存变形的方程）不是umemura含义的经典解决方案。但是，线性方程的单片与这些溶液相对应……更可解决。单片的溶剂性可能会发展为潘leve功能本身的溶剂性。随着从这项研究得出的研究，我提出了两件重要的事情。我们使用非缔合代数来获得等级4的Darboux-Halphen方程的溶剂溶剂的条件。在等级3的情况下，已知darboux-halphen方程减少到高几幅方程时的类似条件。在等级4的情况下，这一新结果开放了对帕克莱夫分析的应用。在第二个情况下，我们确定了d_7类型的painleve方程的特殊解决方案，即umemura对painleve方程的含义的超然经典解决方案已完全分类。潘勒维VI的代数解决方案的分类仍然存在。宣布了帕尔维维I方程的不可约性的不完善后，它已经过去了100年。我们无法对Umemura含义的所有经典解决方案进行分类，但我认为我们已经找到了潘氏菌分析的新方向。较少的