This paper presents a theoretical study of the analytical properties of the Green's function for the internal gravity wave equation for two model density distributions of a stratified, inviscid medium. Integral representations of the solutions are obtained in a linear formulation using the Fourier transform. The selection of a single-valued form for the resulting analytical solutions is discussed. The resulting analytical constructs, using integral convolution, enable the study of wave fields generated by arbitrary nonlocal and nonstationary disturbance sources in real natural stratified media. The obtained asymptotic results enable the investigation of wave disturbances that can be recorded using radar and optical systems. They provide information not only about the sources of generation but also about the characteristics of the marine environment. This is important, among other things, for studying the response of the marine environment to various hydrodynamic disturbances and improving methods for remote sensing of the sea surface. Initial and boundary conditions for specific disturbance sources should be determined from the results of direct numerical modeling of the complete system of hydrodynamic equations or from purely evaluative semi-empirical considerations, allowing for the adequate approximation of real non-local disturbance sources by a certain system of model sources. The resulting analytical solutions enable the calculation of the fundamental amplitude-phase characteristics of the excited far fields of internal gravity waves under certain generation conditions, and, furthermore, the qualitative analysis of the resulting solutions, which is important for the correct formulation of more complex mathematical models of the wave dynamics of real natural stratified media. These model solutions subsequently enable the derivement of representations of wave fields taking into account the actual variability and non-stationarity of such media.

The evaporation of liquid droplets on solid surfaces attracts the attention because it turns out to be decisive in many applied problems: in biology, pesticide spraying, printing, creating films with given properties, OLED technology, nanofabrication, DNA analysis, etc.  The droplet evaporation is complex process, so analytical models can provide insight into process. Free evaporation of sessile liquid non-isothermal drop on solid substrate is analyzed. Exact formulae for temperature and concentration fields are found out as functions of dimensionless parameters. The non-uniform temperature distribution at the drop surface creates the thermocapillar Marangonni forces that change their direction in the vicinity of stagnation points. Direction of the forces and disposition of the stagnation points are derived as function of contact angle and thermodynamic parameters of model. Conditions for the stagnation points to appear are found out. Comparison of calculated and measured temperatures is done. The critical parameters of the ratio of the conductivities of droplet and substrate at which the monotonic temperature distribution changes and the cooling of droplet is possible.

One-dimensional self-similar solutions allows us to establish the fundamental analytical properties for processes. Unlike the well-known Stefan problem concerning the freezing of pure water in contact with ice, the phase transition temperature is not constant and depends on the concentration of the hydrophilic liquid, which is described by a diffusion equation. We use a linear approximation to relate the equilibrium temperature and concentration at the phase boundary. The temperature in the ice and in the liquid is described by the heat conduction equations. Heat and mass transfer occur at the phase boundary when water from the hydrophilic liquid freezes onto the ice surface, increasing its concentration, or conversely, ice melting at the boundary occurs, decreasing the concentration of the hydrophilic liquid. This is determined by the input parameters of the problem. The paper analyzes such solutions for the case of ice - hydrophilic liquids interactions, for example, seawater and an ethanol solution that take place in problems of the interaction of ice shelves with the ocean and in the thermal drilling of boreholes in glaciers. The study also investigates peculiarities of choice of the thermal diffusivity of the liquid in applied problems.

We obtain asymptotic estimates for the interaction forces between topological solitons(kinks) of the Klein–Gordon equation with a polynomial nonlinearity. This equation is the equation of motion for a real scalar field in the Lorentz-invariant (1 + 1)-dimensional φ¹² model, which is important for many physical applications. The model under consideration is not integrable, so it lacks exact two-soliton solutions. Nevertheless, the dynamics of a system consisting of a kink and an antikink located at some distance from each other is important for applications. Such a configuration is not a solution to the equation of motion, but can be constructed from individual soliton solutions. The nonintegrability of the model leads to the presence of an interaction force between the kinks. In this paper, we show that attraction occurs in all cases, and the force decreases exponentially with distance. To obtain expressions for the attractive force, we used the asymptotics of the corresponding kink solutions, which in the model under consideration have an exponential nature, which, in turn, is a consequence of the type of potential of the field-theoretic model that determines the self-interaction of the scalar field.

New classes of Monge – Ampère equations of a fairly general form are described, depending on one to six arbitrary functions of one or two arguments that allow exact linearization in closed form. For linearization, contact Euler and Legendre transformations and special point transformations (including the nonclassical hodograph transformation) of their combinations are used. Special attention is given to the Monge – Ampère equations encountered in meteorology and geophysics. Equivalence transformations of classes of Monge – Ampère equations of a special kind are also considered. For some nonlinear equations, exact solutions were obtained depending on arbitrary functions. Two nonstationary, strongly nonlinear Monge-Ampère type equations with three independent variables, encountered in electron magnetohydrodynamics and geophysical fluid dynamics, were also considered. For these equations, two-dimensional reductions to simpler equations that allow exact linearization were constructed in traveling-wave variables.

We study the nonlinear inverse problem of determining the unknown time-dependent absorption coefficient in a one-dimensional parabolic equation with a weakly degenerate principal part defined in divergence form. The additional observation condition is specified in integral form. Physically, this means, for example, measuring temperature with a finite-size sensor installed at an interior point of the domain. The solution is understood in a generalized sense; in particular, the unknown absorption coefficient is sought in the space L₂(0, T). The equation’s coefficients can depend on both the time and space variables. Degeneracy of the equation is also allowed with respect to both time and space variables. The set of points of degeneracy may be infinite, but must have measure zero. The existence and uniqueness theorems for the solution are proved. The existence of the solution is proven for small T, while the uniqueness theorem is global in nature. Proving the existence of a solution to the inverse problem the latter is reduced to studying the solvability of a certain operator equation, and it is shown that under the conditions imposed in the paper, the operator is contractive

The hydrogen standard currently boasts the highest signal stability and spectral frequency. Transportable hydrogen frequency standards are used for ultra-precise comparisons of remote time and frequency standards in cases where fiber optics or radio communications, such as the GLONASS system, are unavailable. Transportable hydrogen frequency standards may exhibit errors in transmitting the time scale and frequency grid due to external influences, such as vibration, impact, temperature changes, and magnetic fields. Several impact, i.e., short-term impact scenarios, were considered in the study. Damping and shock absorption systems are used for protection, but publications on their effectiveness are currently virtually nonexistent. This article presents the creation of a computer model of a shock-absorbing platform for transporting a hydrogen frequency standard. A comparison of shock-absorbing system models with damping elements made of rubber, silicone, and polyurethane is provided. Natural vibration frequencies were determined for systems with damping elements made of various materials, and their response to impact was examined. Graphs of displacements and accelerations experienced by the transported hydrogen frequency standard were obtained. The system’s response to rocking was examined for various damping coefficients and damping element stiffness coefficients. The dependence of system displacements for different numbers and arrangements of springs was considered. Simulations demonstrated that adding damping blocks at the top and bottom of the shock-absorbing platform for the transported hydrogen frequency standard will improve the system’s resistance to impact.

The theoretical foundations of quantum molecular dynamics and density functional theory (DFT) methods implemented in the Quantum ESPRESSO software package are considered. The main attention is paid to the application of these methods for the atomistic modeling of the properties of substances under extreme conditions. The dissociation energy of hydrogen isotopes (H₂,D₂ ,T₂ ) was calculated, which showed good agreement with the reference data. The hydrogen adsorption process on the aluminum surface was also modeled and the adsorption energy was calculated, the value of which indicates the thermodynamic stability of the resulting system. The verification of the technique was carried out using the example of water adsorption on the surface of lithium hydride, which confirmed its accuracy. The results obtained are of practical importance for hydrogen energy, catalysis, and the development of new materials. The work was performed using computational tools, including VESTA, Avogadro and BURAI, the use of these tools ensured the reliability of the simulation results.

This article, based on a database of initial data on successful patient treatment, proposes two schemes for interpolating optimal ventilation flow parameter values during artificial lung ventilation (ALV) for a given patient. At the mathematical level, selecting optimal ventilation flow parameter values based on the patient's current condition is a task of multivariate nonlinear regression analysis. The first scheme is based on the mathematical apparatus of artificial neural networks. The second scheme is based on the mathematical apparatus of metric analysis, developed at the Department of Applied Mathematics at MEPhI and currently used in mathematical data processing and optimization problems in various applied fields. The implementation of both schemes allows for the use of accumulated data on the successful treatment of patients with similar lung diseases on ventilators for the specific patient in question. Both schemes allow for the adaptation of optimal ventilation flow parameter values to the patient's current condition during treatment. In the future, it is planned to jointly use these two interpolation schemes to obtain a more accurate and reliable final result for solving the above-mentioned optimal interpolation problem.

The issue of high-dose implantation of low-energy helium ions (24 keV) is being considered. In this case, the average projected range is at a depth of 20-30 nm from the sample surface. Implantation of helium ions into silicon is very attractive for creating buried porous layers in silicon during subsequent high-temperature annealing. A natural way to increase the porosity of the buried layer is to increase the implantation dose. However, this is hindered by radiation damage to the surface silicon (blistering and flecking), which makes it impossible to create electronic devices in the latter. As the implantation energy decreases, the implantation profile of embedded helium ions approaches the surface of the substrate, the proportion of vacancies and loosely bound near-surface atoms increases, affecting the course of diffusion processes, and the strength characteristics of the surface layer, including the porous layer, significantly change. It becomes possible to implant significantly higher doses of helium ions without mechanical damage to the surface silicon. The paper presents the results of a study of a buried porous layer after implantation of a dose of 1.75∙1017 He+/cm² at an energy of 24 keV and subsequent high-temperature annealing at a temperature of 1150 ° C for 30 minutes. Huge pores with a diameter of 120-170 nm appear in the area of the initial concentration maximum. The porosity of this layer reaches 50%.