2006 Fiscal Year Final Research Report Summary
Relations of geometric structure of manifolds and graphs, spectre, asymptotic analysis and their applications
| Project/Area Number |
16540068
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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 |
Geometry
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| Research Institution | OKAYAMA UNIVERSITY |
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Principal Investigator |
KATSUDA Atsushi Okayama University, Graduate School of Natural Science, Associate Professor, 大学院自然科学研究科, 助教授 (60183779)
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| Co-Investigator(Kenkyū-buntansha) |
KIYOHARA Kazuyoshi Okayama University, Graduate School of Natural Science, Professor, 大学院自然科学研究科, 教授 (80153245)
TAMURA Hideo Okayama University, Graduate School of Natural Science, Professor, 大学院自然科学研究科, 教授 (30022734)
SHIMAKAWA Kazuhisa Okayama University, Graduate School of Natural Science, Professor, 大学院自然科学研究科, 教授 (70109081)
YOSHIOKA Iwao Okayama University, Graduate School of Natural Science, Associate Professor, 大学院自然科学研究科, 助教授 (70033199)
IKEDA Akira Okayama University, Graduate School of Natural Science, Professor, 教育学部, 教授 (30093363)
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| Project Period (FY) |
2004 – 2006
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| Keywords | inverse problem / stability / boundary distance |
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| Research Abstract |
We studied inverse problems from geometric view points. A boundary distance representation of a Riemannian manifold with boundary is the set of functions which represent the distance from the given point in a manifold to points in the boundary. We study the question whether this representation determines the Riemannian manifolds in a stable way if this manifold satisfies some a priori geometric bounds. The answer is affermative, moreover, given a discrete set of approximate boundary distance functions, we construct a finite metric space that approximates the manifold in the Gromov-Hausdorff topology.
In applications, the boundary distance representation appears in many inverse problems, where measurements are made on the boundary of the object under investigations. For example, for the heat equation with unknown heat conductivity, the boundary measurements determine the boundary distance representation of the Riemannian metric which corresponds to this conductivity. The Gel'fand inverse problem, which asks whether a Riemannian manifold with boundaries can be determined by the eigenvalues and boundary values of the eigenfunctions of the Laplacian, first the boundary distance representation is determined and second they determines the Riemannian metric of the interior. Analogus problems exists for Maxwell and Dirac equations.
Besides collaborating the above works, each investigator studied own works. Kiyohara studied behavior of geodesics on the Liouville manifolds, Tamura studied scattering under two solenoidal magnetic fields, Shimakawa studied topology of configuration spaces, Sakai studied isodiadic inequlities, Yoshioka studies g-fuctions, Tanaka studied semigroups of operators. Takeuchi studied p-harmonic functions on graphs.
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