Vol 220 (2023)
Статьи
Vadim Fedorovich Kirichenko
3-16
Differential geometry of (n - m)m-dimensional complexesin n-dimensional projective space
Abstract
We consider an (n — m)m-dimensional complex in the projective space Pn. In the principal bundle associated with this complex, we construct a fundamental-group connection and calculate the curvature and torsion of this connection. We examine this complex by the Cartan-Laptev method. We prove that the fundamental object of the 1th order of this complex is a pseudoquasitensor, the curvature is a pseudotensor, and the torsion is a geometric object only in combination with the connection subobject and the fundamental object. We perform the compositional framing of the (n — m)m-dimensional complex. Also, we prove that this framing induces connections of three types in the principal bundle associated with the complex.
17-27
Infinite products of binomials with increasing degree
Abstract
In this paper, we considered infinite products of binomials with increasing degree of the variable. We present formulas for calculating power coefficients in infinite products representing smooth functions and propose a condition for the convergence of infinite products with increasing degree.
28-32
On a generalization of the quaternion algebra
Abstract
In this paper, we consider a generalization of quaternion algebras and hypercomplex Clifford algebras over the field of complex numbers in which the mth power of a vector of the underlying space is the sum of the mth powers of the coordinates of this vector.
33-37
38-43
Secular condition for the McKean system
Abstract
In this paper, we study the McKean kinetic system for two groups of particles with periodic initial data in the weight space. The system is reduced to an integro-differential operator containing nonintegrable terms. We find a secularity condition that allows one to eliminate the nondissipative part and hence reduce the problem to a nonlinear equation in a Hilbert space; this is the main step towards proving the stabilization of the solution.
44-48
Features of the problem on synchronization of two Van der Pol-Duffing oscillators in the case of a direct connection and the presence of symmetry
Abstract
Two coupled van der Pol-Duffing oscillators are considered in the case of direct symmetric coupling. We show that synchronization of oscillations (i.e., the presence of stable limit cycles) is typical for self-oscillating systems. Asymptotic formulas for the corresponding solutions are obtained. It is found that the behavior of solutions is not affected by the presence or absence of resonances of eigenfrequencies in the linearized problem.
49-60
On infinite products of euler type
Abstract
In this paper, we consider recurrent and general formulas expressing the number of representations of a natural number n by sums of other natural numbers in terms of sums of divisors of a given natural number.
61-70
Stabilization of stationary motions of a satellite near the center of mass in a geomagnetic field. I
Abstract
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. The work is published with a continuation.
71-85
Criteria for the straightness of a curve
Abstract
The following three criteria for the straightness of the a are proved:
- A curve in an affine n-dimensional space is rectilinear if and only if each of its chords has a common point with the arc contracted by it, which is different from their common ends.
- A curve in a Euclidean 3-dimensional space is rectilinear if and only if any two of its oriented arcs are similar.
- A rectifiable curve in a Euclidean n-dimensional space is rectilinear if and only if any two of its oriented arcs are similar.
86-98
On the structure of an affine connection object and the torsion tensor in the bundle of linear frames
Abstract
In this paper, we study affine connections in the bundle of linear frame over a smooth manifold based on the structural equations of this bundle. The structure of the components of an affine connection in the bundle of frames over a two-dimensional manifold is obtained by using the layer coordinates whose coefficients are functions of the base coordinates of a point of the manifold. We construct expressions for the comp onents of the torsion tensor for two- and three-dimensional manifolds by using the first-order layer coordinates and functions of the base coordinates. Also, we find expressions for the ob ject of flat connection in terms of the coordinates of absolutely parallel vectors and their Pfaffian derivatives and expressions for the ob ject of symmetric flat connection in terms of the coordinates of absolutely parallel covectors.
99-112
Statistical structures on manifolds and their immersions
Abstract
An important example of structures of information geometry is a statistical structure. This is a Riemannian metric g on a smooth manifold M with a completely symmetric tensor field K of type (2, 1). On a manifold endowed with the statistical structure (g, K), a one-parameter family of α-connections ∇α = D + α • K is defined invariantly, where D is the Levi-Civita connection of the metric g and α is a parameter. In this paper, we characterize conjugate symmetric statistical structures and their particular case—structures of constant α-curvature. As an example, a description of a structure with α-connection of constant curvature on a two-dimensional statistical Pareto model is given. We prove that the two-dimensional logistic model has a 2-connection of constant negative curvature and the two-dimensional Weibull—Gnedenko model has a 1-connection of constant positive curvature. Both these models possess conjugate symmetric statistical structures. For the case of a manifold with a torsion-free linear connection immersed in a Riemannian manifold with statistical structure (g,K), a criterion is obtained that a statistical structure with an appropriate а-connection is induced on the preimage.
113-124
Spontaneous clustering in markov chains. I. Fractal dust
Abstract
The review is devoted to the description of the statistical properties of a set of isolated points randomly distributed in space, which are nodes of one (or a family of independent) realizations of a Markov chain. The purpose of the analysis of this model is to study the conditions for the emergence of clusters in the set of these nodes and to describe their characteristics. This (first) part of the review introduces the basic concepts of statistics of point distributions: generating functionals, many-particle densities, factorial moments, Markov chains, correlation functions. The part ends with a description of one-dimensional self-similar (in the statistical sense) sets generated by a Poisson-fractional random process and a demonstration of the clustering phenomenon.
125-144