Despite the seeming simplicity of the steady flow in a pipe of constant radius, the question of the cause and process of the transition remains debatable. Especially since the necessary condition for the stability loss of parabolic profile is not satisfied, and the linear theory of hydrodynamic stability for an axisymmetric Poiseuille flow does not give growing axisymmetric eigen solutions for any Reynolds numbers, since the terms characterizing the interaction of disturbances with the initial velocity profile drop out in the linearized equations of momentum conservation. The report presents the results of the study of stages of convective stability loss for the flow at the initial section of the pipe depending on the variable parameters based on the numerical solution of the three-dimensional system of unsteady Navier-Stokes equations and the equation energy transfer. The variable parameters in this study were: Reynolds number, magnitude and gradient sign of the dynamic viscosity coefficient arising in nonisothermal flows. An analogy of the arising secondary axisymmetric large-scale toroidal vortex structures in the near-wall region to Tollmien-Schlichting waves in the region of the transition of the laminar boundary layer on the plate is shown. The subsequent loss of axisymmetry and stability of these torus-like vortex structures is analyzed, which leads to the formation of fairly regular longitudinal vortex structures downstream, the nonlinear interaction of which leads to chaotization of the flow. The lengths of these sections are determined depending on the Reynolds number, the magnitude and sign of the gradient of the dynamic viscosity coefficient.

Paper provides a research of the mathematical model of a superconducting magnetic suspension in zero gravity. The model consists of a special configuration of the superconducting inductors and uniform magnetic field. The stabilization of suspension in dependence of angle between the magnetic field induction vector and the axis of the inductance coil that is a suspended «free» rigid body is researched. Analysis of the model on stability of equilibrium is carried out and the conditions of spatial magnetic well existence providing the contactless confinement of a rigid body in zero gravity are found.

The results of physical and numerical modeling of a ventilated air cavity behind a streamlined body are presented. The results of laboratory experiments to determine the amount of gas flowing from the ventilated cavity are presented. It is formed behind the cavitator depending on a number of geometric and dynamic parameters. Numerical simulation of non-stationary 3D two-phase flow was performed on the basis of open source software OpenFOAM. The influence of gas blowing parameters on the formation of an air cavity, size, shape and stability has been investigated. Good qualitative agreement with experimental data was obtained. It is shown that the thickness of the ventilated cavity is determined by the diameter of the cavitator regardless of the diameter of the blow hole, and the increase in velocity or gas flow rate has a positive effect on the length and stability of the formed cavity.

The methodology of neural networks is even more often applied in tasks of management and decision-making, including in the sphere of trade and finance. The basis of neural networks is made by nonlinear adaptive systems which proved the efficiency at the solution of problems of forecasting.

The paper develops the theory of existence and necessary optimality conditions for optimal control problems with a vector quality criterion for systems with distributed parameters and generalized impacts. The concept of $(K, e, \epsilon)$-approximate efficiency is investigated. Necessary conditions for $(K, e, \epsilon)$-approximate efficiency of admissible controls in the form of variational inclusions are proved. Methods for solving problems of vector optimization of linear distributed systems with generalized control are proposed. Convergence of algorithms with errors is proved.

We present a general method for analysis of convolutional layers under geometric transformations of the input that are linear with respect to pixel values. We also describe the algorithm for finding all possible types of behaviours of the output of convolutional layers under geometric transformations of the input. We also present a general method for construction of convolutional architectures with desired behaviour under geometric transformations of the input.