Random division of spaces

空间的随机划分

• 批准号: 13640125
• 负责人: ISOKAWA Yukinao
• 金额: $ 1.54万
• 依托单位: Kagoshima University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2001
• 资助国家: 日本
• 起止时间: 2001 至 2002
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-13640125/
• 关键词: Random division; Random packing; The problem of thirteen spheres; ランダムフラクタル

1. We study Poisson-Voronoi tessellations of 3-dimensional hyperbolic spaces, and find explicit formulas that give mean number of vertices, mean total length of edges, and mean surface area of their cells. These mean characteristics comprise, as a particular case, the corresponding formulas for the classical Euclidean case, and depend only on the ratio of curvature of hyperbolic space to intensity of Poisson process. Relying on this result, we develop a method of estimating curvatures of hyperbolic spaces from data on Poisson-Voronoi tessellations. ([1], [2])2. In the 3-dimensional Euclidean spaces, we investigate a problem of random sequential packing of rectangular rods. Assuming that these rods are placed parallel to any of three axes of Cartesian coordinates system. We find a method of reducing the problem to that of 6-dimensional Markov chain. A large simulation using this reduction reveals that configurations of rods are isotopic and their packing density equals 3/4. ([3])3. In the one-dimensional Euclidean spaces, we study a problem of random sequential packing of internals that are generated to a self-similar probability distribution P. Then the resulting probability distribution of packed intervals Q is proved to be self-similar but different from P. Moreover, when P is in particular a uniform distribution, we determine the Hausdorff dimension of the set that are not cover by packed intervals. ([4])4. We study the classical 13 spheres problem, and succeed in obtaining detailed information on the configuration of these spheres. We consider the graph of Delaunay tessellation that are determined by centers of spheres, and prove that only two graphs are possible, that is, the dodecahedron graph and the graph of rhombic dodecahedron. Furthermore we study a continuous deformation of among these graphs. ([5])

1.我们研究了三维双曲空间的Poisson-Voronoi镶嵌，并找到了明确的公式，给出平均顶点数，平均边长和平均表面积的细胞。作为特例，这些平均特征包含了经典欧氏情形的相应公式，并且仅依赖于双曲空间的曲率与泊松过程的强度之比.依靠这一结果，我们开发了一种方法来估计曲率的双曲空间的Poisson-Voronoi镶嵌的数据。([1]，[2]）2.在三维欧氏空间中，研究了矩形杆的随机序列填充问题。假设这些杆平行于笛卡尔坐标系的三个轴中的任何一个放置。我们找到了一种方法，减少问题的6维马尔可夫链。一个大的模拟使用这种减少表明，配置的棒是同位素和他们的包装密度等于3/4。（[3]）3.在一维欧氏空间中，研究了生成自相似概率分布P的内部变量的随机序列填充问题，证明了填充区间Q的概率分布是自相似的，但不同于P的概率分布.此外，当P是均匀分布时，我们确定了未被填充区间覆盖的集合的Hausdorff维数.（[4]）4.我们研究了经典的13个球的问题，并成功地获得详细的信息，这些领域的配置。本文研究了由球心决定的Delaunay镶嵌图，证明了只有两种图是可能的，即十二面体图和菱形十二面体图。此外，我们还研究了这些图中的连续变形。（[5]）

项目成果

Isokawa, Y.: "The problem of thirteen spheres"ISM Symposium on Statistics, Combinatorics and Geometry. 20-20 (2003)

Isokawa, Y.：“十三个球体的问题”ISM 统计、组合学和几何研讨会。

Isokawa, Y.: "Random sequential packing of cuboids with infinite height"Forma. 16. 327-338 (2000)

Isokawa, Y.：“无限高度长方体的随机顺序堆积”Forma。