Vol 30, No 3 (2024)

In terms of variations, a sufficient condition for the uniform convergence of sequences of continuous functions is proved. Using this result, we obtain an addition to the classical Helly theorem on the selection of convergent sequences of functions with uniformly bounded variations in the case when the limit function is continuous. Also, by using an example we show that the condition of continuous differentiability of a function, ensuring the differentiability of its variation with the variable upper limit, is in a certain sense sharp.

According to the results of strain measurements, a kinematically hardening body model is used to describe the process of elastic-plastic deformation of 1X18H9T steel. Special attention is paid to the processes occurring in the surface layers. The model takes into account the increasing tightness of shear deformations deep into the material. The increase in the stress of the plastic flow in depth is described by a second-order polynomial. The main parameters of the surface effect were determined experimentally and by calculations using the method of reduction coefficients in the process of successive approximations: depth, coefficient of hardening of the material, stresses of plastic flow on the surface and inside the material. It is shown that in order to study the near-surface effect by the strain gauge method, it is necessary to give preference to bending tests of samples rather than stretching. The presence of the surface effect explains the following facts: the destruction of the sample during the tensile test does not begin from the surface, but from the inside, the origin of fatigue cracks occurs under the surface, the surface effect practically does not affect the deformed state of the elastic body, but very strongly affects the stress state at the surface.

In this paper, we present a new mathematical approach to the analysis of a beam with distributed hysteresis properties. These hysteresis characteristics are described by two methods: phenomenological (Bouc — Wen model) and constructive (Prandtl — Ishlinskii model). The equations for beam are developed using the well-known Hamilton method. We investigate the dynamic response of a hysteresis beam under various external loads, including impulse, periodic and seismic loads. The results of numerical simulations show that the hysteresis beam exhibits differently to external influences as compared to the classical Euler-Bernoulli beam. In particular, under the same external loads, the vibration amplitude and energy characteristics of the hysteresis beam are lower than those of the classical one. These findings can be useful for buildings developers in the design of external load resistant buildings and structures

Various variants of the formulation of optimal control problem of the reorientation of the axis of dynamic symmetry of the spacecraft (spacecraft), which is the axis of rotation, are considered. It is assumed that the solution to this problem should be found in the class of movements with one directional (flat) rotation, provided that before and after the reorientation of the axis of rotation, the angular velocity of the spacecraft is the same. In this case, the angular motion of the spacecraft is controlled according to the "rotary jet engine" scheme, when the control moment is limited by the ellipsoid of rotation. The corresponding mathematical model of spacecraft motion for the control problem under consideration is given. In addition to the formulation of the problem of the steepest reorientation of the axis of rotation of the spacecraft, the optimal control problem is also formulated, for which optimal control for the full control moment is found. In order to analyze the formulation of the problem under consideration, the results of its reduction to the boundary value problem and to the isoperimetric variational problem are also presented.

The article studies the dynamics of pairwise entanglement of three qubits, two of which are trapped in a resonator and interact with a single-mode ideal resonator through single-photon transitions, and the third qubit is outside the resonator. This takes into account the dipole-dipole coupling between the isolated qubit and the qubit in the resonator. We have found a solution to the quantum nonstationary Schrodinger equation for the total wave function of the system for the initial separable and biseparable states of qubits and the thermal initial state of the resonator field. Using these solutions, the criterion of entanglement of qubit pairs - negativity is calculated. The results of numerical simulation of the negativity criterion have shown that the including of a small dipole-dipole coupling between an isolated and one of the trapped qubits can lead to significant entanglement of qubit pairs for all initial states. There is a transition of entanglement from one pair of atoms to other pairs of atoms during the evolution of the system. It is also shown that for some separable and biseparable states, the dipole-dipole interaction can suppress the effect of sudden death of entanglement