Studies on Differential and Difference Equations by Means of Geometric and Algebraic Methods

几何代数方法研究微分方程和差分方程

• 批准号: 09640157
• 负责人: IWASAKI Katsunori
• 金额: $ 2.24万
• 依托单位: KYUSHU UNIVERSITY
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 1997
• 资助国家: 日本
• 起止时间: 1997 至 1998
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-09640157/
• 关键词: Polytopes; Invariants; Harmonic functions; Twisted homology; Hypergeometric functions; Inverse problems; Singular integral equations; Painleve equations; Polytopes; Invanants; Harmonic functions; Twisted homoligy; Hypergeometric functions; ln

1. Polyhedral harmonics and invariant theory : K.Iwasaki settled a longstanding conjecture concerning the finite dimen- sionality of the space of polyhedral harmonic functions. He discovered new basic invariants of finite reflection groups in order to explicitly determine the polyhedral harmonic functions for polytopes with ample symmetry. In collaboration with A.Kenma and K.Matsumoto, he also made an explicit determination of polyhedral harmonic functions for the exceptional regular polytopes. He is planning to write a book on polyhedral harmonics and invariant theory.2. Cohomology for reccurence relations and hypergeometric functions : K.Iwasaki obtained a sharp asymptotic formula for solutions to certain difference equations. Based on this formula, he is planning to develop a cohomology theory for recurrence relations. K.Iwasaki and M.Kita discovered an exterior power structure on the twisted de Rham cohomology groups associated to hypergeometric functions. K.Iwasaki and K.Matsumoto … More obtained a conjecture that the intersection matrix of the twisted cohomology groups associated to generalized Airy functions can be expressed in terms of skew-Schur polynomials. Attempts to prove this are now in progress.3. Invese bifurcation problem and singular integral equations : K.Iwasaki and Y.Kamimura established the solvability of a class of singular integral equations. They applied this result to prove the existence of solutions to the inverse bifurcation problem for nonlinear Sturm-Liouville equations. Y.Kamimura is writing a book on integral equations which gives a detailed account of their results.4. Painlev_ equations and combinatorics : K.Iwasaki and H.Kawamuko discovered a combinatorial formula of Leibniz type associated to the Hamiltonian structure of the fourth Painlev_ equation in several variables. As an application, they obtained a new quadratic relation among Gegenbauer's orthogonal polynomials. H.Watanabe found birational transformations of solutions to the sixth Painlev_ equation. He also determined classical solutions to that equation. Less

1.多面体调和函数和不变理论：岩崎解决了一个长期存在的关于多面体调和函数空间的有限维性的猜想。他发现了新的基本不变量的有限反射群，以明确确定多面体调和函数的多面体具有充分的对称性。在与A.Kenma和K.松本的合作中，他还明确地确定了特殊规则多面体的多面体调和函数。他正计划写一本关于多面体调和函数和不变量理论的书。重现关系和超几何函数的上同调：K.Iwasaki获得了某些差分方程解的精确渐近公式。基于这个公式，他正计划发展递归关系的上同调理论。K.Iwasaki和M.Kita发现了与超几何函数相关的扭曲de Rham上同调群的外部幂结构。K.Iwasaki和K.松本 关于我们 得到了广义Airy函数的扭上同调群的交矩阵可用斜Schur多项式表示的猜想。目前正在努力证明这一点。逆分支问题与奇异积分方程：K.Iwasaki和Y.Kamimura建立了一类奇异积分方程的可解性。他们应用这一结果证明了非线性Sturm-Liouville方程逆分歧问题解的存在性。Y.Kamimura正在写一本关于积分方程的书，其中详细介绍了他们的结果。Painlev_ equations and combinatomics：K.Iwasaki和H.Kawamuko发现了与第四类Painlev_equations的Hamilton结构相关的Leibniz型组合公式。作为应用，他们得到了盖根鲍尔正交多项式之间的一个新的二次关系。H.Watanabe发现了第六个Painlev_方程解的双有理变换。他还确定了该方程的经典解。少

项目成果

Katsunori Iwasaki Yutaka Kamimura: "An Inverse Bifurcation Problem and an Integral Eguation of the Abel Type" Inverse Problems. 13. 1015-1031 (1997)

Katsunori Iwasaki Yutaka Kamimura：“逆分岔问题和阿贝尔型积分方程”逆问题。

K. Iwasaki: "Asymptotic analysis for linear difference equations" Trans. Amer. Math.Soc.349. 4107-4142 (1997)

K. Iwasaki：“线性差分方程的渐近分析” Trans。

K. Iwasaki and Y. Kamimura: "Asymptotic analysis for linear difference equations" Inverse Problems. 13. 1015-1031 (1997)

K. Iwasaki 和 Y. Kamimura：“线性差分方程的渐近分析”反问题。

K. Iwasaki: "Regular simplices, symmetric polynomials and the mean value property" Journal d'Analyse Mathematique. 72. 279-298 (1997)

K. Iwasaki：“正则单纯形、对称多项式和平均值属性”Journal dAnalyse Mathematique。

K. Iwasaki: "Polytopes, invariants and harmonic functions" to appear in Singularities and Arrangements, Sapporo-Tokyo 1998, Proc. Symp. Pure Appl. Math., Kinokuniya and A.M.S.

K. Iwasaki：“多面体、不变量和调和函数”出现在 Singularities and Arrangements，Sapporo-Tokyo 1998，Proc。