1998 Fiscal Year Final Research Report Summary
Research of Analysis on Clifford Algebra and it's Application
| Project/Area Number |
09640201
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
解析学
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| Research Institution | Fukuoka University of Education |
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Principal Investigator |
NONO Kiyoharu Fukuoka University of Education, Education, Professor, 教育学部, 教授 (10117046)
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| Co-Investigator(Kenkyū-buntansha) |
HARA Takuya Fukuoka University of Education, Education, Associate Professor, 教育学部, 助教授 (50263984)
SAKAMOTO Takanori Fukuoka University of Education, Education, Associate Professor, 教育学部, 助教授 (00162313)
TAMARI Fumikazu Fukuoka University of Education, Education, Professor, 教育学部, 教授 (70036937)
FUKUTAKE Takayoshi Fukuoka University of Education, Education, Professor, 教育学部, 教授 (60036887)
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| Project Period (FY) |
1997 – 1998
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| Keywords | Clifford Function Theory / Clifford Analysis / Clifford Algebra / Euler-Poisson-Darboux Equation / Generalized Cauchy-Riemann Equation |
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| Research Abstract |
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 equal
and 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.
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