Mixed empirical poisson random spherical-cap process
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MIXED EMPIRICAL POISSON RANDOM SPHERICAL-CAP PROCESS UDC 519.21
N. G. Semejko
Abstract. A mathematical model of a mixed empirical Poisson random cap process (RCP) on the sphere S 2 is investigated using the theory of mixed empirical marked point processes. The first-order moment measure of the RCP is proposed for spherical sets of special form. Keywords: spherical stochastic geometry, empirical marked point process, Poisson cap process, first-order moment measure. INTRODUCTION ~ A random cap process (RCP) on a two-dimensional Euclidean sphere S 2 is an important object of study in spherical stochastic geometry (SG), mathematical morphology, and it is efficiently used to model and process the results of real experiments in natural sciences such as morphometry of microstructures in biology and medicine (virology, cytology, histology, hematology), selenography, seismology, cosmology, astronomy. Let us present examples of such experiments. ~ Covering a sphere S 2 with randomly arranged hemispheres or hemispherical caps was preferred as a model of neutralization of a virus particle attacked by antibodies. Antibodies are assumed to be attached randomly to the surface of the virus particle (presumably spherical) and can be considered as small hemispherical caps [1, 2]. Of special interest in cytology is the quantitative morphology (morphometry) of the population of pores of the karyolemma of a receptor neuron. The question is the description of the population of pores of the karyolemma (a pseudoconvex surface) of the neuron using quantitative characteristics, which allow making a significant contribution to the development of a forecast and to the analysis of the functional state of a cell and its nucleus in the normal state, in ~ experiment, and in pathology. As a first approximation, karyolemma can be approximated by a sphere S 2 of unit radius, and ~ pore of a RCP on the sphere S 2 [3]. ~ An interesting application of the theory of covering on the sphere S 2 together with computer modeling is encountered in the analysis of lunar craters [4], their models are necessary to obtain the information about the history and structure of the Moon [5]. Neyman and Scott [6] investigated the distribution of stars and galaxies on the heavenly sphere. The need for methods of spherical SG arises also in solving problems related to covering the earth surface with a system of space communication via randomly orbiting artificial earth satellites. The problem of observing a celestial sphere by randomly located observers [7] pertains to the same category. According to the ideas of Davidson and Kendall [8], in many SG problems, a necessary transformation transforms random processes of geometrical objects into the trajectories of marked point processes (MPP) in the corresponding parametric (phase) spaces and are investigated as such. In the present paper, we will use the theory of mixed empirical MPP in a parametric space to first construct the ~ mathematical model of a mixed empirical Poisson random process of hemispherical caps on a sphe
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