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; 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”。 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@> d2i@> d2@> d7*<@d7n-1（/）相等的> s <小于或等于> n，小于或相等的构成基于方程式的函数理论。首先，我们使用上述二阶偏微分方程的线性化，使用这种线性（广义cauchy-riemann方程）。有价值的定期函数是广义的欧拉 - 偏见 - 达布克斯方程的解决方案，我们从广义的Euler-Poisson-darboux方程中获得了常规函数的方法，因此我们可以基于总体上基于欧拉 - 帕克斯·帕尔比克斯的函数。轴向对称的潜在理论方程。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。