Stochastic visibility in random fields. Dedicated to the memory of Professor Micha Yadin. Incl. 1 disk
This monograph presents a comprehensive summary of the research on several visibility problems in random fields done by the author in cooperation with M. Yadin over a period of ten years. The level of exposition is such that readers familiar with calculus, elementary geometry/trigonometry and the basic facts from probability theory should be able to understand this book. Of course, a closer familiarity with methods and results from geometric probability and stochastic geometry will facilitate the understanding of the obtained results. The research on visibility in random fields was strongly stimulated (at least at the beginning) by military applications and later by problems in communications and many other fields. This subject is closely connected with coverage problems and first-contact problems in Boolean models and, more generally, in germ-grain models, see e.g. the monographs of \textit{P. Hall} [``Introduction to the theory of coverage processes'' (1988; Zbl 0659.60024)] and \textit{D. Stoyan}, \textit{W. S. Kendall} and \textit{J. Mecke} [``Stochastic geometry and its applications'' (1987; Zbl 0622.60019)] on these subjects. Chapters 1 and 2 present the probability models for visibility problems, introduce some classical problems of geometric probability, and give an elementary treatment of coverage problems. Chapters 3 and 4 are devoted to the determination of visibility probabilities. For example, one is interested in the probability that one observer can be connected by straight lines with a fixed number of target points, where disks with random radii and randomly distributed centres are located in the plane between the observer and the targets. Chapters 5 and 6 deal with distributions of certain random measures of visibility among them the length of partially visible segments. Chapter 7 is devoted to problems and their solutions which illustrate the theory presented in the preceding chapters and demonstrate its applicability. All the material in the book is supported by a large number of figures, and many numerical examples are presented, using the specially designed software.
scientific article; zbMATH DE number 722981