Vol 221 (2023)

In this paper, we formulate the definitions of еру spherical image, the area of the spherical image, and the implementation curvature for a class of polyhedral angles without singularities. Also, we prove a theorem on the equality of the area of the spherical image and the implementation curvature of a polyhedral angle from a distinguished class.

 This paper is devoted to the differential geometry of complexes of two-dimensional planes in the projective space $P^n$ containing a finite number of torsos. We find a necessary condition under which the complex $C^{\rho }$ contains a finite number of torsos, examine the properties of complexes of two-dimensional planes, which are determined by a special configuration of characteristic straight torsos belonging to the complex, and establish the structure and conditions for the existence of such complexes. The self-duality of such complexes is determined.

In this paper, we discuss enumerating some graphs of a special type. New results on the number of spanning forests of graphs playing an important role in information theory are obtained. We consider properties of convergent spanning forests of directed graphs involved in the construction of the quasi-metric of the mean time of the first pass, which is a generalized metric structure closely related to ergodic homogeneous Markov chains. We examine characteristics of spanning root forests and convergent spanning forests of directed and undirected graphs that are used for constructing the matrix of relative forest availability, which is one of the proximity measures of vertices of graphs. The reasonings are illustrated by several simple graph models, including a simple path, a simple cycle, a caterpillar graph, and their oriented analogs.

In this paper, we consider problems of stabilization of stationary motions (equilibrium positions and regular precessions) of a satellite near the center of mass in gravitational and magnetic fields under the assumption that the center of mass moves in a circular orbit. Mathematical models of the problems considered are systems of differential equations with periodic coefficients. We present a rigorous analytical approach to this problem, which allows efficient and correct construction of stabilization algorithms. The method is based on the reducibility of nonstationary systems that describe these problems to stationary systems. Solutions for a number of problems of stabilizing stationary motions of a satellite with the help of magnetic systems are proposed. We present the results of mathematical modeling of the algorithms, which confirm the effectiveness of the developed methodology. This paper is the second part of the work. The first part is: Itogi Nauki i Tekhniki. Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory. — 2023. — 220. — P. 71–85.

The problem of the existence of one class of closed convex polyhedra in $E^3$ — the so-called noncomposite $RR$-polyhedra — is examined. The existence test consists of finding an equation which implies the existence of a polyhedron and allows one to find the angle of rhombuses at a rhombic vertex.

In the second part of the review, we apply theoretical principles developed in the first part to analysing statistical characteristics of clustering the observed distribution of galaxies in the visible part of the Universe. In contrast to the standard approach to solving the dynamic problem of clustering gravitational plasma based on systems of differential equations that describe the plasma as a continuous medium, we use the Ornstein–Zernike integral equation for a system of randomly distributed points whose interaction is described by an appropriate choice of the kernel of the Ornstein–Zernike equation for the two-particle correlation function. Within the framework of this “mesofractal” model, we find a 4-parameter representation of the spectrum of fluctuation power, which allows one to determine statistical parameters of the medium from the observed data. The first part of this work: Itogi Nauki i Tekhniki. Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory. — 2023. — 220. — P. 125–144.