Comprehensive Study on Discrete Geometry

离散几何综合研究

基本信息

批准号：
08304019
负责人：
MAEHARA Hiroshi
金额：
$ 4.61万
依托单位：
University of the Ryukyus
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (A)
财政年份：
1996
资助国家：
日本
起止时间：
1996 至 1997
项目状态：
已结题

项目摘要

1.On the rigidity of frameworks : We presented a rigid unit-bar-framework in the 3-dimensional space that has no triangle, and a minimum rigid framework in the plane that cannot be constructed from the data of edge-lengths and graph structure. We proved that if a complete bipartite framework K (m, n) (m >= 3, n >= 5) in the plane admits a continuous deformation, then one of the partite-sets lies on a line L and the other partite-set lies on the line perpendicular to L.2.On embeddings of structures : We proved that for any planar graph G = (V,E), there is an emebedding f : V * R^2 such that x, y * V are adijacent if and only if the distance between f (x) and f (y) is an integer. We also proved that every n dimensional inner product space over the rational field can be isometrically embedded into 2n + 1 dimensional Euclidean space.3.On srrangements of spheres : A graph G is said to be representable by balls on the table, if we can place solid balls on a table, one ball for each vertex, so that two balls are tangent only when the corresponding vertices are adjacent. We proved that the family F of graphs representable by balls on a table is different from the family of planar graphs, and that F does not contain any member of the so-called Petersen family, and that the maximum value of the chromatic number of a graph in F is either 5 or 6.4.On random graphs, probability : We determined the probability distribution of the order of the maximum regular tournament in a dominance relation generated by a randam n points on a circle. We extended the classical ruin problem to 3 persons' game, and calculated the probability that a fixed gambler A ruined first, and the probability that A is the sole survivor.

1.在框架的刚度上：我们在没有三角形的3维空间中呈现了一个刚性的单位杆框架，并且在平面中无法从边缘长度和图形结构的数据构造的平面中最小刚性框架。 We proved that if a complete bipartite framework K (m, n) (m >= 3, n >= 5) in the plane admits a continuous deformation, then one of the partite-sets lies on a line L and the other partite-set lies on the line perpendicular to L.2.On embeddings of structures : We proved that for any planar graph G = (V,E), there is an emebedding f : V * R^2 such X，Y * V是及时仅当F（x）和f（y）之间的距离是整数时。我们还证明，在理性字段上的每个n维内部产品空间都可以嵌入2n + 1个维欧几里达空间中。3.在球体的srangements：thaph g可以用桌子上的球表示，如果我们可以将固体球放在桌子上，每个vertex的一个球，那么在两个球上，两个球是差异的。 We proved that the family F of graphs representable by balls on a table is different from the family of planar graphs, and that F does not contain any member of the so-called Petersen family, and that the maximum value of the chromatic number of a graph in F is either 5 or 6.4.On random graphs, probability : We determined the probability distribution of the order of the maximum regular tournament in a dominance relation generated by a randam n points on a circle.我们将经典的废墟问题扩展到了3人的游戏，并计算了固定赌徒首先被毁灭的概率，以及A是唯一的幸存者的可能性。