In this work, we consider two coupled initial-boundary value problems, which, based on the Saint-Venant theory, model the longitudinal impact phenomena in a semi-infinite rod. The mathematical formulation of the problem is two mixed problems for the one-dimensional wave equation with conjugation conditions. The Cauchy conditions are specified on the spatial half-line. The initial condition for the partial derivative with respect to the time variable has a discontinuity of the first kind at one point. The boundary condition, which includes the unknown function and its first- and second-order partial derivatives, is specified on the time half-line. The solution is constructed by the method of characteristics in an explicit analytical form. The uniqueness of the solution is proved, and the conditions under which a piecewise-smooth solution exists are established. The classical solution to a mixed problem with matching conditions is considered.

We consider the problem of studying the fracture of a composite macrolayer synthesized by additive technology on a solid substrate, to which the boundary of the lower microlayer is rigidly bonded. The materials of the microlayers are homogeneous and isotropic. The thicknesses of microlayers and the total thickness of the macrolayer are the control parameters of synthesis. The physical and mechanical characteristics of the microlayers and their combination in the macrolayer are also controlling parameters. The layer is fractured by a crack, which may appear in the top layer (or any other layer) and move toward the substrate or free surface, perpendicular to the boundaries of the microlayers. The crack can stop at the boundaries between the layers (then it is treated as a separate state) or pass through the boundaries without stopping. Based on the theory of Markov chains, the average lifetime of a stratified layer synthesized on a substrate using additive technology and the variance of the number of cycles (lifetime) characterizing the quality of the synthesized structure are calculated. On the basis of the materials available for the synthesis of the layered coating, the problems of optimal design of the composite body with maximum average resource, minimum resource dispersion, maximum specific strength for given geometric and physical-mathematical constraints on the composite parameters are solved. To find optimal solutions, dynamic programming algorithms are used, implemented on circuits (graphs) of sequential selection of a combination of layer materials according to their synergistic properties.

A new linear integro-differential equation of order n ≥ 3, given on a closed curve located in the complex plane, is investigated. Integrals in the equation are understood in the sense of the finite part according to Hadamard. A characteristic feature of the equation is the presence of quadratic functions of a special kind in its coefficients. The equation is reduced first to the boundary value problem of linear conjugation for analytical functions. In the case of its solvability, two linear differential equations should be further solved in the domains of the complex plane with some additional conditions for the solution. All conditions for the solvability of the original equation are explicitly specified. When they are executed, the desired solution is constructed in a closed form. An example is given.

We begin with some known results of the 50-component theory for a spin-2 field described in cylindrical coordinates. This theory is based on the use of a 2nd-rank symmetric tensor and a 3rd-rank tensor symmetric in two indices. In the massive case, this theory describes a spin-2 particle with an anomalous magnetic moment. According to the Fedorov – Gronskiy method, which is based on projective operators, all 50 functions involved in the description of the spin-2 field for the case of the free particle can be expressed in terms of only 7 different functions constructed from Bessel functions. This leads to a homogeneous system of linear algebraic equations for 50 numerical parameters. We have found 6 independent solutions to these equations. Additionally, we have obtained explicit expressions for 4 guage solutions defined in accordance with the Pauli – Fierz approach. These solutions are exact and correspond to non-physical states that do not affect observable quantities, such as the energy-momentum tensor. Finally, we have constructed two classes of solutions that represent physically observable states.

The Bragg diffraction of displaced Gaussian light beams on a slow shear ultrasonic wave in paratellurite crystals, in which the beam energy is efficiently transferred from the zero diffraction order to the first, is investigated. It is established that circularly polarized light beams should be used to optimize the energy exchange of diffracted light beams in the considered geometry of acousto-optic interaction. It is shown that the diffracted beams have the form of displaced Gaussian beams, the spatial structure of which is determined by the displacement parameter of the incident beam and the ultrasound power.

We propose a structural and electrical schemes of a capacitor based on a 3 μm thick a-Si:H (amorphous hydrogenated silicon) layer separated from the metal plates by 0.3 μm thick dielectric layers of SiO₂ (silicon dioxide). We consider room temperatures (T ≈ 300 K) when in the absence of illumination for a-Si:H the hopping mechanism of electron migration via point defects of the structure prevails. For such a capacitor, the dependencies of the capacitance on the frequency of the measuring signal ω/2π in the range from 0.1 to 300 Hz are calculated for the a-Si:H layer with stationary hopping electrical conductivity σ_dc ≈ 1 ∙ 10⁻¹⁰ (Ohm ∙ cm)⁻¹. It is assumed that there is no end-to-end electron transfer between the a-Si:H layer, dielectric layers and capacitor plates in the small-signal mode of capacitance measurement. It is shown that the real part of the capacitance of the capacitor decreases with increasing angular frequency ω, and the imaginary part is negative and depends non-monotonically on ω. The decrease in the real part of the device capacitance to the geometric capacitance of the series-connected oxide layers and the a-Si:H layer with increasing ω is due to a decrease in the electrical resistance of the capacitor. As a result, with increasing ω, the imaginary part of the capacitance is shunted by the hopping electrical conductivity of the capacitor. The phase shift for a sinusoidal electrical signal supplied to the capacitor is determined depending on the frequency ω/2π in the range of 0.1–300 Hz for the values of electrical conductivities of the hydrogenated amorphous silicon layer σ_dc ≈ 1 ∙ 10⁻¹¹, 1 ∙ 10⁻¹⁰, and 1 ∙ 10⁻⁹ (Ohm ∙ cm)−1 at the temperature 300 K. With an increase in the electrical conductivity σ_dc of the a-Si:H layer, the minimum absolute value of the phase shift angle (≈65°) shifts to the high- frequency region (from 1 to 100 Hz). The proposed low-frequency capacitor can find application in electrical circuits for detecting low-frequency electrical signals for the purposes of biomedicine.

The development of thin-film thermal control coatings for small spacecraft is relevant. Coatings based on titanium nitride are capable of functioning in unfavorable conditions of near and deep space, due to their high resistance to the irradiation by high-energy particles. Using the reactive magnetron sputtering method, the nanostructured TiAlSiN and TiAlSiCN coatings were formed on the substrates of silicon oxide (SiO₂), glass-ceramic CT-1 and single-crystalline silicon (Si(100)). A study of the electrophysical and optical properties of the formed coatings was carried out. The deposited coatings demonstrate a good reflectivity in the infrared range of spectrum (700–2000 nm), what is important for reducing the overheating of the spacecraft (SC) under the influence of the direct sunlight. In the visible range of spectrum (400–700 nm), a low level of total R_total reflection is observed. This is promising for satellites designed to observe the Earth’s surface. The values of solar absorption coefficients α_s, emissivity coefficients ε₀, ratios α_s/ε₀, as well as the equilibrium temperature Tр for the samples under study were obtained. The values of resistivity ρ and surface resistance R□, electron concentration N and electron mobility μ were determined. It has been discovered that TiAlSiN, TiAlSiCN films are electrically conductive: ρ_TiAlSiN = (92÷4260) ∙ 10^–7 Ω ∙ m, ρ_TiAlSiCN = (51÷2360) ∙ 10^–7 Ω ∙ m. It has been found that adding carbon to the coating composition reduces the resistance. The obtained nanostructured coatings of TiAlSiN nitride and TiAlSiCN carbonitride can be used as temperature control coatings for small spacecrafts.