| Project/Area Number |
09640290
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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 |
General mathematics (including Probability theory/Statistical mathematics)
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| Research Institution | Keio University |
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Principal Investigator |
OTA Katsuhiro Keio University, Department of Mathematics, Associate Professor, 理工学部, 助教授 (40213722)
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| Co-Investigator(Kenkyū-buntansha) |
TOKUSHIGE Norihide University of Ryukyus, College of Education, Associate Professor, 教育学部, 助教授 (00217481)
MATSUMOTO Makoto Keio University, Department of Mathematics, Associate Professor, 理工学部, 助教授 (70231602)
MAEDA Yoshiaki Keio University, Department of Mathematics, Professor, 理工学部, 教授 (40101076)
ENOMOTO Hikoe Keio University, Department of Mathematics, Professor, 理工学部, 教授 (00011669)
SHIOKAWA Iekata Keio University, Department of Mathematics, Professor, 理工学部, 教授 (00015835)
神保 雅一 慶應義塾大学, 理工学部, 教授 (50103049)
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| Project Period (FY) |
1997 – 1998
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| Project Status |
Completed (Fiscal Year 1998)
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| Budget Amount *help |
¥3,700,000 (Direct Cost: ¥3,700,000)
Fiscal Year 1998: ¥1,400,000 (Direct Cost: ¥1,400,000)
Fiscal Year 1997: ¥2,300,000 (Direct Cost: ¥2,300,000)
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| Keywords | Triangulation / Quadrangulation / Cycle parity / Mapping class group / Book embedding / 対角変形 / 整数距離グラフ / 超越数 / 幾何学的群論 / 疑似乱数 |
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| Research Abstract |
Triangulations and Quadrangulations of a closed surface with sufficiently fine mesh will approximate the discrete structure of the surface. We first consider the problem to transform one triangulation (or quadrangulation) to another by a sequence of local deformations. As a result for triangulations, if these graphs have same number of vertices and the number is large enough, namely, these graphs are good approximation of the surfacs, then these can be transformed each other by a sequence of diagonal flips. On the other hand, for quadrangulations, there exists an discrete invariant "cycle parity" that is closely related to the structure of the mapping class group of the surface.
In a graph embedded in a closed surface, the degree of a vertex is considered as a discrete curvature on that place. In a 3-connected graph on the sphere, we studied the minimum degree sum of a connected subgraph having prescribed order.
We also consider the problem of embedding graphs in a book-type manifold. We gave good bounds for the pagenumber of complete bipartite graphs, where the pagenumber of a graph is the minimum number of pages in which one can embed the graph without crossing edges and the spine. For general graphs, we gave a lower bound for the number of crossing points of edges and the spine.
Also, we investigate the following topics ; a presentation of mapping class groups in terms of Artin groups, Galois action on fundamental groups, study of difference sets using number theory, searching irreducible polynomials, isometric embedding of simplexes. and the problem to represent an arbitrary graph as an integral distance graph in the plane.
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