1998 Fiscal Year Final Research Report Summary
THEORY OF IDEAL BOUNDARIES OF AN INFINITE NETWORK AND ITS APPLICATION
| Project/Area Number |
09640188
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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 | SHIMANE UNIVERSITY |
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Principal Investigator |
YAMASAKI M. Shimane Univ.Mathematics Professor, 総合理工学部, 教授 (70032935)
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| Co-Investigator(Kenkyū-buntansha) |
SUGIE J. Shimane Univ.Mathematics Professor, 総合理工学部, 教授 (40196720)
FURUMOCHI T. Shimane Univ.Mathematics Professor, 総合理工学部, 教授 (40039128)
AIKAWA H. Shimane Univ.Mathematics Professor, 総合理工学部, 教授 (20137889)
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| Project Period (FY) |
1997 – 1998
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| Keywords | Infinite networks / discrete Kuramochi function / discrete Kuramochi boundary / discrete Laplacian / Hardy's inequality / 固有値問題 / Hardyの不等式 / ヒルベルトネットワーク |
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| Research Abstract |
(1) Solutions of partial difference equations on an infinite network show some interesting behavior at the point at infinity of the network. The aim of this project is to characterize the point at infinity with the aid of potential theoretic tools and to study its effects for the study of the partial difference equations. We succeeded to construct a theory of discrete Kuramochi boundary comparable to the theory of discrete Martin boundary for random walks. Our theory is a discrete analogue to the theory of Kuramoch compactification of Riemann surfaces due to Kuramochi and Ohtsuka. After introducing a discrete Kuramochi function of a network, we define the Kuramochi compactification as the space on which the Kuramochi function can be extended continuously. We obtain results concerning a classification of Kuramochi boundary points and the behavior of Kuramochi potentials and SHS functions
(2) As an application of our study, we give a kind of Hardy's inequality on finite networks. With some numerical experiments, we show that our estimation of the wighted minimum eigenvalue problem for the discrete Laplacian is more effective than the usual theory when the size of the net-work becomes large.
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