Aim. Modeling the properties of hydrophobic and ice-repellent coatings.

Methodology. The methods used are based on the use of known generally accepted laws of mechanics, electrodynamics and thermodynamics. The equations were solved numerically using proprietary computer programs, and the graphs were constructed using known graph plotting programs.

Results. The conditions for superhydrophobicity were formulated. Analytical studies of the features of applying relief to a metal surface using laser ablation were conducted. A method for controlling the spatial period of relief by changing the parameters of the laser system was proposed. Based on published experimental data on laser ablation, the possibility of satisfying the conditions of superhydrophobicity of modified aluminum, titanium and steel surfaces was shown.

Research implications lies in development of methods for modeling hydrophobic and ice-phobic properties, as well as in modeling and optimizing the processes of their creation. The use of such coatings allows for a significant reduction in the intensity of icing of aircraft, the drag of bodies in liquid, and the drag of aircraft in conditions of heavy precipitation.

Aim. The aim of this work is to establish a priori estimates for the integral load of the Kirchhoff equation. This equation models some nonlinear oscillatory processes. Here, the load is the rational degree m / n of a linear function of the norm of the desired solution in the space H¹(Ω).

Methodology. To establish a priori estimates, integral transformations of the terms of the scalar product of the original equation and the time derivative of its solution are performed. The application of some well-known integral inequality leads to the desired estimates.

Results. A priori inequalities limiting the integral load of the Kirchhoff equation to a known function are established. This function depends on the right side of the equation and the initial conditions, depending on the sign and type of exponent. A method is shown for reducing the Kirchhoff equation to a linear equation by replacing the integral load with the right-hand sides of these estimates. An example of such a reduction is given.

Research implications. The described method of establishing a priori estimates and subsequent reduction of a nonlinear equation to a linear one can be applied to a wide class of loaded equations containing the modulus of the integral of the rational degree of the desired function or its derivative.

Aim. To describe an experiment on direct experimental observation of the quantum Zeno effect.

Methodology. The study requires a model of a scanning probe microscope-SPM.

Results. The described theoretical estimates of the study give encouraging results of obtaining stable excited states of atoms by direct measurement of their excited states.

Research implication lies in the prospect of obtaining stable excited states of atoms of various elements.

Aim. The work is performed with the aim of obtaining an approximate expression for the space-time metric of the centrally symmetric gravitational field of a point mass, which is a generalization of the Schwarzschild metric for stationary observers located at any non-zero distance from the point mass.

Methodology. The analysis was carried out, the structure of the Schwarzschild metric and the dependence of the components of the metric tensor on the potential difference of the gravitational field were used, which is one of the main provisions of the general theory of relativity.

Results. An approximate expression for the space-time metric of the centrally symmetric gravitational field of a point mass is obtained, which is a generalization of the Schwarzschild metric for the case of stationary observers located at any non-zero distance from the point mass. The resulting metric asymptotically tends to the Schwarzschild metric as the observer moves away from the point mass and is, at least, the first post-Newtonian approximation to the exact solution.

Research implications. Analysis of the obtained expression for the space-time metric of the centrally symmetric gravitational field of a point mass allows us to conclude that the relative horizons of visibility for observers with finite radial coordinates are located on spheres with radial coordinates smaller than the gravitational radius.

Aim. Search for a variant of the LT system of units that is maximally consistent with the international SI and the subsystems of the CGS system of units.

Methodology. The analysis of the ratios of physical quantities in the international SI, CGS subsystems and Planck LT systems of units is carried out. A method is proposed for determining the values of physical quantities according to the criterion of the maximum degree of consistency between the recommended CODATE values of constants for defining coupling equations.

Results. Conditionally accurate values of the Planck length are obtained ℓ_p = 1.616255272206877 ∙ 10^-35 ∙ 𝑚, the fine structure constant 𝛼 = 7.297352564390205 ∙ 10^-3, and a number of other physical constants were obtained. A variant of the Planck LT system of units is proposed and the conversion coefficients between the electromagnetic quantities of the analyzed systems of units are clarified.

Research implications. It consists in the possibility of using the PLT system of units and conditionally accurate values of a number of physical constants for computational methods and mathematical models of physical processes in various fields of science and technology.