Aim. We refine the properties of parallel translations of manifolds with affine connection of dimension greater than two, such that for any three points that are sufficiently close, there exists a two-dimensional autoparallel manifold containing them. Methodology. We use the methods of differentiable universal algebras to describe the properties of certain classes of affine-connected spaces. Results. We prove that in this class of projective flat manifolds with affine connection, the “pseudoline” identity is fulfilled, reflecting the properties of parallel translations. The differential-geometric characteristic of a “pseudoline” identity is derived, that is, if the dimension of the manifold is more than two, then the “pseudoline” identity is equivalent to the fact that the corresponding manifolds of affine connection are projective flat and have a common pseudoconnection (the same concurrency) with the manifold of affine connection with zero torsion. Research implications. Differential geometry has numerous applications in theoretical mechanics, Special and General relativity theory, and other fields of natural sciences. This research can be employed to build a specific mathematical model describing the course of physical processes.

Aim. We have studied the main mechanisms that control the development of gambling. Methodology. Gambling is simulated with a given pot size, a given bet, and a choice of game strategy. The player, in case of winning, receives a double bet. In case of loss, the entire bet is taken from the player. Various game strategies are considered when manipulating the “size” of the pot, the “size” of the bet, and the number of steps (iterations) to achieve success. The finiteness of the game time (the number of iterations) and the discreteness of the ongoing processes are taken into account. We have studied the dependence of the winning frequency on the bet size and the number of steps (iterations) for a given “size” of the pot required for winning. Results. The ways of possible winning are revealed depending on the size of the bet and the number of steps (iterations) for a given “size” of the pot. Research Implications. The paper considers various strategies of the game, focused on the maximum win.

Aim. The dynamics of Bose-condensed atoms in a three-well symmetric chain trap is studied. Methodology. The temporal evolution of the population of atoms in the wells is investigated theoretically. Results. It is shown that under conditions of exact resonance, there are oscillatory regimes of evolution of atoms in the wells of a three-well trap, as well as the rest of the system. Research implications. The dynamics of tunneled Bose-condensed atoms in a three-well trap is determined by the initial number of atoms in the wells and the initial phase.

Aim. We study the grid convergence of the explicit MacCormack method applied to solving the equations of a heterogeneous mathematical model of the dynamics of an electrically charged aerosol. Methodology. The flow of aerosol is described by using a continuous model of the motion of an inhomogeneous medium, which assumes that the motion of each of the mixture components is described by a complete system of equations for the dynamics of a continuous medium. Results. Numerical calculations are carried out on a sequence of refining finite-difference grids. Differences in the calculated solutions decrease as the partition of the computational domain becomes smaller. Research implications. The calculation results demonstrate the convergence of the explicit MacCormack method in modeling the flow of a two-component mixture caused by the movement of the dispersed component. Also, numerical modeling revealed that during the movement of the dispersed phase, the dynamics of the mixture is influenced by both the magnitude of the Coulomb force and the inter-component interaction.

Aim. Geometric properties of ice-phobic surfaces are mathematically simulated to provide the anti-icing effect. Methodology. Numerical calculations of droplet motion in the vicinity of a cylinder simulating the leading edge of a wing relies on the use of previously published mathematical models of physical processes. Results. As applied to the problem of icing of aircrafts, the relief configuration of hydrophobic coatings of a solid is estimated in air flow with supercooled droplets, in which liquid drops do not freeze to the streamlined body as a result of collisions with its surface. Research implications. The results of the study can be used to produce a relief of a hydrophobic coating for a specific range of flying vehicle flight conditions.

Aim. We derive formulas by operator methods for the temperature and concentration profiles around two interacting identical aerosol drops heated by radiation and evaporating in the diffusion regime. Methodology. The temperature and concentration fields are represented as standard expansions in spherical functions, and the undefined coefficients of these expansions are considered as coordinates of vectors of an infinite-dimensional linear normed space. The search for the undefined coefficients is reduced to finding infinite-dimensional vectors from the boundary conditions by means of linear operators. Scalar quantities are represented as linear functionals defined on the infinite-dimensional linear normed space mentioned above. Results. Formulas for temperature and concentration profiles around two identical drops are obtained and corresponding graphs are presented. These graphs are compared with the graphs obtained by the method using the bipolar coordinate system. Research implications. Simple formulas are obtained for temperature and concentration profiles. When solving specific problems, calculation algorithms based on these formulas can be easily implemented in Excel.

Aim. The main purpose of the work is to refine the Navier-Stokes equation for nanoparticles. Methodology. The calculation technique is based on the use of classical Boltzmann kinetic equation. Results. The found equation is a refined Navier-Stokes equation, on the right side of which the components of the higher degrees along the length of the free run of particles are taken into account. Research implications. In all cases where it is necessary to study the hydrodynamic movement of nanoparticles in the flow of a viscous liquid, the resulting equation allows one to calculate all the necessary amendments to any hydrodynamic parameters and, in particular, to the Stokes strength.

Aim. We demonstrate the need to appeal to the main mathematical concepts in the construction of physical models. Methodology. The existing approaches to methods of computing work in physics and corresponding methods of mathematical analysis (methods of higher mathematics) are compared and analyzed. Results. Heuristically reliable results of calculations within the framework of general physical approaches in the learning process require the consideration of the main methods of higher mathematics in the construction of the physical picture of the process. Research implications. The necessity of taking into account the intersubject relations of physics and higher mathematics in the calculation of mechanical work is demonstrated.

Aim. We describe the degeneracy case of basic feasible solutions in the simplex method for use by lecturers both in the classroom and in self-study of students. Methodology. The basic concepts of linear programming are formulated and the problems caused by excessive constraints in the problem conditions are considered. The reasons for the occurrence of such a special case in the simplex method as the degeneracy of basic feasible solutions are presented. The cases of temporal degeneracy and cycling are described. A rule is given to avoid cycling. All of the above is illustrated by concrete examples. Since the transition to the general case raises a problem related to the inability to see mathematical objects, the method of visualization of mathematical objects is used. Results. A detailed description of the degeneracy case of basic feasible solutions in the simplex method is presented. Research implications. The work is of practical significance, since it can be used in the study of one of the four special cases that arise when using the simplex method.