Systems of Holonomic Differential Equations

完整微分方程组

• 批准号: 07454028
• 负责人: TAKANO Kyoichi
• 金额: $ 3.46万
• 依托单位: Kobe University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (B)
• 财政年份: 1995
• 资助国家: 日本
• 起止时间: 1995 至 1996
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-07454028/
• 关键词: confluent hypergeometric function; space of initial values of Painleve; irreducibility of Painleve; classical function; quantum group; q-hypergeometric function; percolation; projective homogeneous space; 射影等質曲面; ホロノミック系; パンルヴェ系; 初期値空間; シンプレクテ

1.Hypergeometric differential equations : (1) We obtained the system of differential equations satisfied by integrals associated with a non-degenerated quadratic hypersurface and n hyperplanes in the (k-1)-dim. complex projective space and we studied certain sym-metries of the system.(2) We made clear geometrically the processes of confluence of general confluent hypergeometric functions on Grassmann manifolds.2.Painleve systems : (1) We found processes of conflunce among the spaces of initial con-ditions of Painleve systems. The processes are compatible with the well known ones.(2) We proved the irreducibility of the second and the fourth Painleve functions exept the known classical functions, by determining ivariant devisors of Hamiltonian vector fields associated with the Painleve systems.3.Quantum groups and q-functions : (1) We realized a family of quantum complex projective spaces as one of quantum homogeneous spaces associated with a family of coideals, and we expressed the zonal spherical functions in terms of Askey-Wilson polynomials. (2) We solved afflrmatively the integrality conjecture of Macdonald for the (q, t)-Kostka coefficients, by constructing raising operators for Macdonald's symmetric polynomials.4.Percolation problem : We obtained the order of the spectral gap in the case where + and - spins are randomly distributed on the boundary. The order is the same as that in the case where there are on spins no the boundary.5.We showed that the affine geometry of surfaces in the 3-dim. projective space works aiso in the case where some invariants are degenerated, and we classified projective homogeoenus surfaces.

1.超几何微分方程：（1）在（k-1）维复射影空间中，得到了非退化二次超曲面和n个超平面的积分所满足的微分方程组，并研究了该方程组的某些对称性。(2)我们从几何上明确了Grassmann流形上一般合流超几何函数的合流过程. 2. Painleve系统：（1）我们发现了Painleve系统的初始条件空间之间的合流过程.这些过程与众所周知的过程是兼容的。(2)通过确定与Painleve系统相关的Hamilton向量场的不变导子，证明了第二类和第四类Painleve函数的不可约性（除了已知的经典函数）。3.量子群和q-函数：（1）我们实现了一个量子复射影空间族作为与一个余理想族相关联的量子齐性空间之一，用Askey-Wilson多项式表示了带状球函数。(2)通过构造Macdonald对称多项式的提升算子，我们近似地解决了Macdonald关于（q，t）-Kostka系数的完整性猜想。4.逾渗问题：我们得到了当+和-自旋随机分布在边界上时谱隙的阶数。5.证明了三维射影空间中曲面的仿射几何在某些不变量退化的情况下也成立，并对射影同胚曲面进行了分类。

项目成果

Kimura, H.: "A normal form of Hamiltonian Systems of several time variables with a regular singularity" J. Differential Equations. 127・2. 337-364 (1996)

Kimura, H.：“具有正则奇点的多个时间变量的哈密顿系统的范式”J. 微分方程 127・2 (1996)。

Sasaki, T.: "Sectional curvature of projective invariant metrics on a strictly convex domain" Tokyo J. Math.19. 419-433 (1996)

Sasaki, T.：“严格凸域上射影不变度量的截面曲率”Tokyo J. Math.19。

Hukuhara, M.et al.: Hadamard "Partial Differential Equations" (translation into Japanese). Kyoritsu, Tokyo, Japan, 480 (1997)

Hukuhara, M.et al.：Hadamard“偏微分方程”（日语翻译）。

Shioda, T.at al.: "On some Hamiltonian structures of Painleye svstems, I" Funkcial. Ekvac.(in press.).

Shioda, T.at al.：“关于 Painleye 系统的一些哈密顿结构，我”Funkcial。

Sasaki,T.: "Sectional curvature of projective invariant metrics on a strictly convex domain" Tokyo J.Math.19. 419-433 (1996)

Sasaki,T.：“严格凸域上射影不变度量的截面曲率”Tokyo J.Math.19。