On Hypergeometric Functions and its Applications
| Project/Area Number |
10640163
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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 |
Basic analysis
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| Research Institution | Shiga University of Medical Science, Faculty of Medicine |
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Principal Investigator |
TERADA Toshiaki University of Medical Science, Faculty of Medicine, Department of Mathematics, Professor, 医学部, 教授 (80025402)
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| Project Period (FY) |
1998 – 2001
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| Project Status |
Completed (Fiscal Year 2001)
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| Budget Amount *help |
¥1,500,000 (Direct Cost: ¥1,500,000)
Fiscal Year 2001: ¥400,000 (Direct Cost: ¥400,000)
Fiscal Year 2000: ¥300,000 (Direct Cost: ¥300,000)
Fiscal Year 1999: ¥400,000 (Direct Cost: ¥400,000)
Fiscal Year 1998: ¥400,000 (Direct Cost: ¥400,000)
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| Keywords | Hypergeometric Function / Riemann Problem / Braid Group / Hypergeometric Representation / Complete Quadrilateral / 組紐群 / 群の線型表現 / 保型関数 / 色付組紐群 / Burau表現 |
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| Research Abstract |
(1) We presented some conditions under which the Wonskian of a finite sequence of functions does not vanish identically, and, by using them, we solved Riemann's problems for Larricella's F_D and Appell's F_4 without the non-vanishing of the Wonskian. They were solved by the author and respectively by Kato with stronger conditions about the orders of zeros at singular loci, which essentially assure the non-vanishing of the Wronskian.
(2) At first, we intended also to prove the faithfulness of the hypergeometric representation of the braid group, but it have been proved not to be true. But we made a sketch of the proof of the faithfulness of that of the pure braid group.
The procedure is as the following: the hypergeomtric representation of the pure braid group is the monodromy representation of the system of partial differential equations witch F_D satisfies and, if every parameters is rational, the solutions are periods of the algebraic curve υ^p = II^^<n+1>__<i=0>(u-a_i)^<pi>. So the problem of the faithfulness is reduced to the problem : For a sequence of curves on a complex plain, if, every lift on every algebraic curve as above is 0-homologuous, are they homotopically trivial under some conditions?
At present, it is hard to declare it is solved, but it will be necessary only to polish up the details. And, using this, the conjugacy problem of the braid group is probably reduced to calculations of matrices, and there will be many contributions to the research of the hypergeometric functions.
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Report
(5 results)
Research Products
(7 results)
