- PII
- 10.31857/S2686954324010113-1
- DOI
- 10.31857/S2686954324010113
- Publication type
- Article
- Status
- Published
- Authors
- Volume/ Edition
- Volume 515 / Issue number 1
- Pages
- 71-78
- Abstract
- The study of distribution functions (by areas, perimeters) for partitioning a plane (space) by a random field of straight lines (hyperplanes) and for Voronoi mosaics is a classical problem of statistical geometry. Starting from 1972 [1] to the present, moments for such distributions have been investigated. We give a complete solution of these problems for the plane, as well as for Voronoi mosaics. We investigate the following tasks: A random set of straight lines is given on the plane, all shifts are equally probable, and the distribution law has the form F(φ). What is the distribution of the parts of the partition by areas (perimeters)? A random set of points is marked on the plane. Each point A is associated with a “region of attraction”, which is a set of points on the plane to which the point A is the closest of the set marked. The idea is to interpret a random polygon as the evolution of a segment on a moving one and construct kinetic equations. At the same time, it is sufficient to take into account a limited number of parameters: the area covered (perimeter), the length of the segment, the angles at its ends. We will show how to reduce these equations to the Riccati equation using the Laplace transform. (see theorems 1, 1 and 2).
- Keywords
- геометрические вероятности пуассонов процесс прямых мозаики Вороного кинетическое уравнение уравнение Маркова случайные множества стохастическая геометрия распределения случайных величин
- Date of publication
- 15.11.2024
- Year of publication
- 2024
- Number of purchasers
- 0
- Views
- 48
References
- 1. Miles R.E. The random division of space // Advances in Applied Probability. 1972. Vol. 4. P. 243–266.
- 2. Белов А.Я. Cтатистическая геометрия и равновесие блочных массивов // Дисс. … канд. физ.-мат. наук, н. рук. Р.Л. Салганик. М.: МГИ, 1991. С. 190.
- 3. Miles R.E. Poisson flats in Euclidean spaces // Advances in Applied Probability. 1969. Vol. 1. P. 211–237.
- 4. Кендалл M., Моран П. Геометрические вероятности. М.: Наука, 1972.
- 5. Белов А.Я. О случайных разбиениях // Деп. в ВИНИТИ. М., 1991. № 273-B91. С. 26.
- 6. Kanel-Belov A., Golafshan M., Malev S., Yavich R. About random splitting of the plane // Crimean Autumn Mathematical School-Symposium, KROMSH. 2020. P. 294–295.
- 7. Kabluchko Z. Angles of random simplices and face numbers of random polytopes // Advances in Mathematics. 2021. Vol. 380. No. 107612.
- 8. Pierre Calka. An explicit expression for the distribution of the number of sides of the typical Poisson-Voronoi cell // Adv. Appl. Probab. 2003. Vol. 35 (4). P. 863–870.
- 9. Calka P. Precise formulae for the distributions of the principal geometric characteristics of the typical cells of a two-dimensional Poisson-Voronoi tessellation and a Poisson line process // Advances in Applied Probability. 2016. Vol. 35. No. 3. P. 551–562.
- 10. Сентало Д. Интегральная геометрия и геометрические вероятности // М.: Наука, 1983.
- 11. Амбарцумян Р.В., Мекке Й., Штоян Д. Введение в стохастическую геометрию // М.: Наука, 1989.
