Research of Analysis on Clifford Algebra and it's Application

Clifford代数分析及其应用研究

• 批准号: 09640201
• 负责人: NONO Kiyoharu
• 金额: $ 1.47万
• 依托单位: Fukuoka University of Education
• 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-09640201/
• 关键词: Clifford Function Theory; Clifford Analysis; Clifford Algebra; Euler-Poisson-Darboux Equation; Generalized Cauchy-Riemann Equation; Generalized Cauchy-Riemann方程式; Euler-Poisson-Darbour方程式

The purpose of this research is "Composition of theory of functions based on important second order partial differential equation which appears in the mathematical physics". In this research, we focused on the following Generalized Euler-Poisson-Darboux's equations :SIGMA<@D3s(/)i=1@>D3<@D7*<@D12@>D1u(/)*x<@D12@>D1<@D2i@>D2@>D7 - SIGMA<@D3n(/)i=s+1@>D3<@D7*<@D12@>D1u(/)*x<@D12@>D1<@D2I@>D2@>D7 *<@D7n-1(/)x<@D2k@>D2@>D7 <@D7*u(/)*x<@D2k@>D2@>D7=O (O<less than or equal>s<less than or equal>n, l<less than or equaland constituted a function theory based on the equations.At first, we studied on the linearizations of the above second order partial differential equations. Using this linearizations(Generalized Cauchy-Riemann equations), we defined a regularity of functions with vales in Clifford algebra and obtained various properties of regular functions. Also, the compornent functions of a Clifford valued regular function are all solutions of the generalized Euler-Poisson-Darboux's equation. Reversely, we obtained the method to construct regular function from any solutions of the generalized Euler-Poisson-Darboux's equation. Therefore, we could perfectly construct the function theory based on Generalized Euler-Poisson-Darboux's equations.The results obtained in this research include the function theories based on Laplace equation and Generalized axially symmetric potential theory equations. Also, this function theory can be expected to develop in the future.The investigators, Fukutake and Hara obtained the several results on the topology and the operator theory, respectively.

这项研究的目的是“基于重要的二阶部分微分方程的功能理论的组成，该方程出现在数学物理学中”。在这项研究中，我们专注于以下广义的Euler-Poisson-Darboux的方程：Sigma <@d3s（/）i = 1@>@> d3 <@d7*<@d12@> d12@> d1u（/）*x <@d12@d12@> d1 <@d2i@> d2@> d7 -sigma <@d3n（/）i = s+1@> d3 <@d7*<@d12@> d1u（/）*x <@d12@> d12@> d1@> d1 <@d2i @> d2@> d7*<@d7n-1（/）x <@d2k@> d2@> d7 <@d7*u（/）*x <@d2k@> d2@> d2@> d7 = o（o <sily <silly比或等于> s <小于或等于> n，l <小于或等于和等于方程的函数理论。首先，我们研究了上述二阶偏差方程的线性化Cauchy-Riemann方程），我们在Clifford代数中定义了阀门的规律性，并获得了常规功能的各种特性，Clifford Ruality的常规函数​​都是广义的Euler-Poisson-Darboux方程，我们从广义的Euler-Poisson-Darboux方程中获得了定期函数的方法。基于拉普拉斯方程和广义上的轴向对称电位理论方程的理论。同样，这一功能理论可以预期将来发展。研究人员，福生和Hara分别获得了拓扑和操作者理论的几个结果。

项目成果

K.Nono: "Generalized Euler-Poisson-Darboux's Equations in Clifford Analysis" Bull.Fukuoka Univ.Ed.III. 48. 9-31 (1999)

K.Nono：“克利福德分析中的广义欧拉-泊松-达布方程”Bull.Fukuoka Univ.Ed.III。

T.Fukutake: "Nearly Quasi Semi-open Sets II" Bull.Fukuoka Univ.Ed.III. 47. 17-22 (1998)

T.Fukutake：“近准半开放集 II”Bull.Fukuoka Univ.Ed.III。

Takuya Hara: "Monotonicity of the inverse and operators on Krein spaces" Bull.Fukuoka Univ.Ed.III. 47. 1-7 (1998)

Takuy​​a Hara：“Kerin 空间上的逆和算子的单调性”Bull.Fukuoka Univ.Ed.III。

Kiyoharu Nono: "Generaliyed Euler-Poisson-Darbous′s Equations in Clittord Analysis" Bull.Fukuoka Univ.Ed.III. 48. 9-31 (1999)

Kiyoharu Nono：“Clittord 分析中的广义欧拉-泊松-达布斯方程”Bull.Fukuoka Univ.Ed.III. 48. 9-31 (1999)