Combinatorial decompositions and homogeneous geometrical processes
The author deals with linear processes and random mosaics, which are assumed invariant under the group of translations of \(R^ 2\) (Poissinity is not assumed). Let \(\pi_ k(t,\alpha)\) be the probability, that an interval of length t taken on a `typical line of direction \(\alpha\) ' has k hits with other lines of the processes. An equation for \(\pi_ k(t,\alpha)\) is obtained. In the isotropic case, the solution is the family of Poisson distributions. Secondary (for the case of random mosaics), the distribution of a length of a `typical edge having direction \(\alpha\) ' in terms of the process \(\{\) \({\mathcal P}_ i,\psi_ i\}\) is found, here \({\mathcal P}_ i\) is the point process of intersections of edges of the mosaic with a fixed line of direction \(\alpha\) and the mark \(\psi_ i\) is the intersection angle at \({\mathcal P}_ i\). The method is based on the results of combinatorial integral geometry, e.g. the main combinatorial decompositions for the space G of lines in the plane.
scientific article; zbMATH DE number 4020085