Vol 29, No 146 (2024)

This article is devoted to the study of the properties of recurrent motions of periodic processes defined in a Hausdorff sequentially compact topological space $\Gamma$.

The definition of a recurrent motion of a periodic process is introduced and the main property of the motions is established. This property strictly connects arbitrary motions and recurrent motions in $\Gamma$. Based on this property, it is shown that, in the case of an autonomous process defined in the space $\Gamma$, the classical G. Birkhoff definition of a recurrent motion is equivalent to the definition of a recurrent motion of a periodic process introduced in this article. Besides, it is shown that in $\Gamma \mathrm{,}$ the $\omega$- and $\alpha$-limit sets of each motion of an autonomous process are sequentially compact minimal sets.

The main significance of the results obtained in the article is that they actually establish the interrelation between the motions of periodic processes in the space $\Gamma$.

We consider a problem mixed in boundary conditions for the Laplace equation in a domain that is a part of a cylinder of a rectangular cross-section with homogeneous boundary conditions of the second kind on the side surface of the cylinder. The cylindrical region is limited on one side by surface of a general kind on which the Cauchy conditions are specified, i. e. a function and its normal derivative are given, and the other boundary of the cylindrical region is free. In this case, the problem has the property of instability of the Cauchy problem for the Laplace equation with respect to the error in the Cauchy data, i. e. is ill-posed, and its approximate solution, robust to errors in Cauchy data, requires the use of regularization methods. The problem under consideration is reduced to the Fredholm integral equation of the first kind. Based on the solution of the integral equation obtained in the form of a Fourier series on the eigenfunctions of the second boundary value problem for the Laplace equation in a rectangle, an explicit representation of the exact solution of the problem was constructed. A stable approximate solution to the integral equation was constructed using the Tikhonov regularization method. The extremal of the Tikhonov functional is considered as an approximate solution to the integral equation. Based on the approximate solution of the integral equation, an approximate solution of the boundary value problem as a whole is constructed. A theorem is proved for the convergence of an approximate solution of the problem to the exact one as the error in the Cauchy data tends to zero and the regularization parameter is consistent with the error in the data.

A continuous medium, which is quite common in the industrial sphere, consisting of a set of layers (phases), i.e. a layered unidirectional composite medium (composites), and physical processes in the layers, i.e. transfer processes, wave processes, and changes in the stress-strain state of this this medium, is considered. A mathematical description of the structure of the compositional medium in terms of the layered domain is realized, a Sobolev space of functions with a carrier in the layered domain (together with auxiliary spaces) is constructed to describe the quantitative characteristics of the layers, and the weak solvability of the corresponding boundary value problems is established. At the same time, in the places of mutual adjacency of the layers, the conditions describing the regularities of the transfer process and the wave process, as well as changes in the stress-strain state and the displacement of layer points were determined. The work consists of three parts. The first part contains a description of the structure of the compositional medium, the basic concepts, and a description of the classical spaces of functions with a carrier in the layered domain. The second part is devoted to the construction of auxiliary spaces for the mathematical description of boundary problems of the processes of transfer and wave process, and to obtaining sufficient conditions for their solvability. The third part contains a description of the elastic properties of the composite medium, the problem of stress-strain is formulated for which the space of admissible solutions satisfying the relations describing the laws of displacement of points at the junctions of layers is constructed, the conditions of weak solvability of the specified problem are established. The results of the work are used in the analysis of problems of optimization of physical processes and phenomena in composite materials.

The article considers the question of constructing cubature formulas for the surface of a torus $T$ in ${ℝ}^3$, invariant under the group $G$ generated by reflections of $T$ into itself. For currently known invariant cubature formulas with a degree of accuracy greater than 3, the number of nodes significantly exceeds the minimum possible. The article proposes invariant cubature formulas of degree 5 and 7 for the surface of a torus with a number of nodes close to the minimum. Tables of values of nodes and coefficients of the constructed cubature formulas are given. The dependence of these values on the ratio of the radii of the guide and generatrix of the torus circles is studied. For construction, the method of invariant cubature formulas is used, based on the theorem of S. L. Sobolev.