Random division of spaces

1. 我们研究了三维双曲空间的Poisson-Voronoi镶嵌，并找到了给出顶点的平均数量、边的平均总长度和它们的细胞的平均表面积的显式公式。作为一种特殊情况，这些平均特征包括经典欧几里得情况的相应公式，并且仅取决于双曲空间曲率与泊松过程强度的比值。基于这一结果，我们开发了一种从泊松-沃罗诺伊镶嵌上的数据估计双曲空间曲率的方法。([1],[2]) 2。在三维欧几里德空间中，研究了矩形棒的随机顺序排列问题。假设这些杆平行于笛卡尔坐标系的三个轴中的任何一个。我们找到了一种将问题简化为6维马尔可夫链的方法。利用这种简化进行的大型模拟表明，棒的结构是同位素的，它们的堆积密度等于3/4。([3]) 3。在一维欧几里得空间中，研究了生成自相似概率分布P的内部项的随机顺序填充问题，证明了所得到的填充区间Q的概率分布是自相似但不同于P的，并且当P是均匀分布时，我们确定了不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球问题，并成功地获得了这些球的构型的详细信息。考虑由球心决定的德劳内镶嵌图，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。进一步研究了这些图之间的连续变形。([5])

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])