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"Research on prehomogeneous vector spaces and micro-local analysis"

《预齐次向量空间与微局部分析研究》

基本信息

批准号：
11640161
负责人：
MURO Masakazu
金额：
$ 2.24万
依托单位：
Gifu University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1999
资助国家：
日本
起止时间：
1999 至 2000
项目状态：
已结题

项目摘要

(a) (From the abstract of the paper "Hyperfunction solutions of invariant differential equations on the space of real symmetric matrices".) The real special linear group of degree n naturally acts on the vector space of n×n real symmtric matrices. How to determine invariant hyperfunction solutions of invariant linear differential equations with polynomial coefficients on the vector space of n×n real symmtric matrices is discussed in this paper. We observe that every invariant hyperfunction solution is expressed as a linear combination of Laurent expansion coefficients of the complex power of the determinant function with respect to the parameter of the power. Then the problem is reduced to the determination of Laurent expansion coefficients which is needed to express. We give an algorithm to determine them and apply the algorithm in some examples.(b) (From the abstract of the paper "An algorithm th compute the b_P-functions via Grobner bases of invariant differential operators on prehomogeneous vector spaces".) The calculation of b_P-function via Grobner basis for an group invariant differential operator P (x, ∂) on a finite dimensional vector space is considered in this paper. Let (G, V) be a regular prehomogeneous vector space. It is often observed that the space of all G-invariant hyperfunction solutions u (x) to the differential equation P (x, ∂) u (x)=v (x) is determined by its b_P-function, a polynomial associated with the G-invariant differential operator P (x, ∂). We prove in this paper that the b_P-function is computed by an algorithm using Grobner basis of the Weyl algebra on V for a typical class of prehomogeneous vector spaces.

(a)（摘自“实对称矩阵空间上不变微分方程的超函数解”一文的摘要。）n次的实特殊线性群自然作用于n×n实对称矩阵的向量空间。讨论了在n×n实对称矩阵的向量空间上如何确定多项式系数不变线性微分方程的不变超函数解。我们观察到，每个不变的超函数解都被表示为行列式函数的复幂的洛朗展开系数相对于幂参数的线性组合。然后将问题简化为需要表示的洛朗膨胀系数的确定。给出了一种确定它们的算法，并将该算法应用于实例。(b)（摘自论文摘要“利用预齐次向量空间上不变微分算子的Grobner基计算b_p函数的一种算法”。）本文考虑了有限维向量空间上群不变微分算子P （x,∂）的格罗伯纳基b_p函数的计算。设（G， V）是一个正则的预齐次向量空间。我们经常观察到，微分方程P (x,∂)u (x)=v (x)的所有g不变超函数解u (x)的空间是由它的b_p函数决定的，它是一个与g不变微分算子P （x,∂）相关的多项式。本文证明了一类典型的预齐次向量空间的b_p函数是用V上Weyl代数的Grobner基的算法计算出来的。