Volume 517, Nº 1 (2024)

The paper proposes a new approach to control the process of general education. Digital technology tools are used to form spaces of goals, tasks and learning activities, and to record the educational process of each student. Artificial intelligence tools are used when choosing a student’s personal goals and ways to achieve them, to make forecasts and recommendations to participants in the educational process. Big data from the entire education system and big linguistic models are used. The effects of the approach include ensuring the success of each student, objective assessment of the work of teachers and schools, and the adequacy of the succession process to higher education.

Two methods for quantitatively assessing the chirality of a set are considered, the first of which uses the calculation of the area of their symmetric difference of two sets as a measure of the discrepancy between them, and the second uses the Hausdorff distance between them. It is shown that these methods, generally speaking, do not provide a correct quantitative estimate for a fairly wide class of sets, such as bounded Borel sets. Using the example of flat triangles and convex quadrangles, the problem of dividing geometric objects into right-handed and left-handed is considered. For triangles, level lines of two versions of the chirality measure were calculated on the plane of the angular parameters. For a spatial spiral, the values of two versions of the chirality index are found, based respectively on the calculation of the mixed product of vectors and the Hausdorff distance between two sets.

In complex Banach algebra, under the condition of separateness and spectral separateness, the conditions for the reversibility of the Vandermonde matrix are formulated and proved. The necessary and sufficient signs of reversibility of the Vandermonde matrix are given. Analogs of Sylvester's theorem are formulated.

The equality of the coefficients of the interpolation polynomial over a parallelepipedal grid for a multidimensional function to the coefficients of the interpolation polynomial over a uniform grid for a one-dimensional function is proved, for which the fast Fourier transform can be applied according to various schemes.

For the first time, upwind bicompact schemes of third order approximation in space are presented. A formula is obtained for the transition factor of an arbitrary fully discrete bicompact scheme with integration in time by a Runge–Kutta method. Stability and monotonicity of the first-order in time scheme are investigated, dissipative and dispersion properties of the third-order in time scheme are analyzed. Advantages of the new schemes relative to their centered counterparts are demonstrated.

The paper proposes an original method for processing large factor models based on graph condensation using machine learning models and artificial neural networks. The proposed mathematical apparatus can be used in problems of planning and managing complex organizational and technical systems, in optimizing large socio-economic objects on the scale of state sectors, to solve problems of preserving the health of the nation (searching for compatibility when taking medications, optimizing resource provision for healthcare).

The article states the computational algorithm based on the finite volume method with splitting by physical processes for modeling non-stationary problems of laser thermochemistry with catalytic nanoparticles in subsonic gas flows. Two-phase flows in a heated pipe with laser radiation and radical kinetics of non-oxidative methane conversion are simulated. It is shown that the conversion of methane is more than 60 % with the predominant formation of ethylene and hydrogen at the outlet of the pipe.

We define a divisible completion of the solvable Baumslag–Solitar group and prove that under certain restrictions on the elementary theory of this completion is decidable.

The work is devoted to the logistic equation with delay and diffusion with non-classical boundary conditions. The stability of a nontrivial equilibrium state is investigated, and the resulting bifurcations are studied numerically.

The tasks of analyzing and visualizing the dynamics of a viscous incompressible fluid in conditions of complex flow geometry based on traditional grid and projection methods are associated with significant requirements for computer performance to achieve the set goals. To reduce the computational load in solving this class of problems, algorithms for constructing artificial neural networks (ANNs) can be used, using exact solutions of the Navier–Stokes equation system on a given set of spatial regions as training sets. An ANN is implemented to construct flows in areas that are complexes made up of training sets of standard axisymmetric regions (cylinders, balls, etc.). To reduce the amount of calculations in the case of 3-D problems, invariant flow manifolds with smaller dimensions are used. This allows you to identify the detailed structure of solutions. It is established that the typical invariant regions of such flows are rotation figures, in particular, homeomorphic torus, forming the structure of a topological bundle, for example, in a ball, a cylinder and in general complexes composed of such figures. The structures of the flows obtained by approximation by the simplest 3-D vortex unsteady flows are investigated. Classes of exact solutions of the Navier–Stokes system for an incompressible fluid in bounded regions of space based on the superposition of the above topological bundles are distinguished. Comparative computational experiments indicate a significant acceleration of computational work in the case of using the proposed class of ANNs, which allows the use of computing equipment with low performance.