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Study on a theory of Fourirer transformation by using the thory of prehomogeneous vector paces

利用预齐次矢量步理论研究傅里叶变换理论

基本信息

批准号：
15540044
负责人：
KOGISO Takeyoshi
金额：
$ 1.79万
依托单位：
Josai University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2003
资助国家：
日本
起止时间：
2003 至 2006
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-15540044/
关键词：
Fourier Transformation Prehomogeneous Vector Spaces Relative invariants b-functions Igusa local zeta functions Clifford Algebra Jordan algebra Higher degree elementary function

项目摘要

The first half of the research period was able to compose all rational indices corresponding to a relative invariability type of two simple prehomogeneous vector spaces with Tatsuo Kimura and Kazunari Sugiyama of Tsukuba University. Moreover, respectively the Dynkin-Kostant type prehomogeneous vector space of the E8 type was studied by a joint research with Satoshi Wakatsuki of Kanazawa University in parallel with it, and a basic, relative invariability type of the space, b-function, and Igusa local zeta function were able to be composed specifying it.Moreover, the explanation of a recent research of Bhargava of the Princeton university in the United States was able to be received from Takashi Taniguchi of The University of Tokyo in detail research for informations about this research, and acquire very significant information.Moreover, an original purpose was able to be achieved in some degree in the latter half of the research period by guidance and the cooperation of Fumihiro Sato of Rikkyo University. It succeeded in obtaining a lot of polynomials by which the function equation was filled though it was not a relative invariant of prehomogeneous vector space and new classification problem of prehomogeneous vector spaces was able to be found. These researches are progressing now. We list up the results in this research period as follows :(1)Explicitly calculation of relative invariants and thir rarional character of 2-sipmle prehomogeneous vector spaces.(2)Explicitly calculation relative invariants b-function, and Igusa local zeta function of E8, Dynkin-Kostant type prehomogeneous vector spaces.(3)The re-construction of Clerc's resut by using representations of Clifford algebra.(4)A lot of examples of polynomials which satisfy the functional equation from the view point of representatio of Clifford ring Cl(p,0) and Cl(q,0) n (n=p+q).(5)Our system have the infinity class of real form of some prehomogeneous vector spaces.(6)Combinatric observation of our research.

研究期间的前半段与筑波大学的 Tatsuo Kimura 和 Kazunari Sugiyama 一起构建了与两个简单预齐次向量空间的相对不变性类型相对应的所有有理指数。此外，与金泽大学的Satoshi Wakatsuki并行研究，分别对E8型的Dynkin-Kostant型预齐次向量空间进行了研究，并且能够具体地组成该空间的基本相对不变型、b函数和Igusa局部zeta函数。此外，普林斯顿大学的Bhargava的最近研究的解释美国大学能够从东京大学的谷口隆那里得到关于这项研究的详细研究，并获得了非常重要的信息。而且，在研究期间的后半段，在立教大学的佐藤文宏的指导和合作下，在一定程度上实现了最初的目的。它成功地获得了许多填充函数方程的多项式，尽管它不是预齐次向量空间的相对不变量，并且能够发现预齐次向量空间的新分类问题。这些研究目前正在取得进展。我们将本期研究成果罗列如下：(1)显式计算2-sipmle预齐次向量空间的相对不变量及其有理数特征。(2)显式计算E8、Dynkin-Kostant型预齐次向量空间的相对不变量b函数和Igusa局部zeta函数。(3)Clerc结果的重构 (4)从Clifford环Cl(p,0)和Cl(q,0) n (n=p+q)表示的角度来看满足函数方程的多项式的例子。(5)我们的系统具有某些预齐次向量空间的实数形式的无穷类。(6)我们研究的组合观察。