The mixed problem for the one-dimensional Klein – Gordon – Fock type equation with oblique derivatives at boundary conditions in the half-strip is considered. The solution of this problem is reduced to solving the second-type Volterra integral equations. Theorems of existence and uniqueness of the solution in the class of twice continuously differentiable func tions were proven for these equations when initial functions are smooth enough. It is proven that fulfilling the matching conditions on the given functions is necessary and sufficient for existence of the unique smooth solution, when initial functions are smooth enough. The method of characteristics is used for the problem analysis. This method is reduced to splitting the ori ginal definition area into subdomains. The solution of the subproblem can be constructed in each subdomain with the help of the initial and boundary conditions. The obtained solutions are then glued in common points, and the obtained glued сonditions are the matching conditions. Intensification of smoothness requirements for source functions is proven when the di rections of the oblique derivatives at boundary conditions are matched with the directions of the characteristics. This approach can be used in constructing both the analytical solution, when the solution of the integral equation can be found explicitly, and the approximate solution. Moreover, approximate solutions can be constructed in numerical and analytical form. When a numerical solution is constructed, the matching conditions are significant and need to be considered while developing numerical methods.

The conditions of isolation of a zero singular point of plane polynomial fields of third and fourth degree are considered in terms of the coefficients of the components of these fields. The isolation conditions depend on the greatest common divisor of the components of polynomial fields: in some cases only on its degree, and in some cases, additionally,
on the presence of nonzero real zeros. The reasoning, which allows one to write out the isolation conditions, is based on the concept of the resultant and subresultants of components of plane polynomial fields. If the zero singular point is isolated, its index is calculated through the values of subresultants and coefficients of components.

The study of the computational complexity of problems on graphs is an urgent problem. We show that the problem of deciding whether the vertex set of a given split graph of order 3n can be partitioned into induced subgraphs isomorphic to P3 is a polynomially solvable problem. We develop a polynomial-time algorithm based on the method of augmenting graphs. The developed efficient algorithm can be used for solving team formation problems.

The paper considers the issues of constructive analysis of the de la Vallee – Poussin boundary-value problem for the second-order linear matrix differential Lyapunov equation with a parameter and variable coefficients. The initial problem is reduced to an equivalent integral problem, and to study its solvability a modification of the generalized contraction mapping principle is used. A connection between the approach used and the Green’s function method is established. The coefficient sufficient conditions for the unique solvability of this problem are obtained. Using the Lyapunov – Poincaré small parameter method, an algorithm for constructing a solution has been developed. The convergence and the rate of convergence of this algorithm have been investigated, and a constructive estimation of the region of solution localization is given. To illustrate the application of the results obtained, the linear problem of steady heat conduction for a cylindrical wall, as well as
a two-dimensional matrix model problem is considered. With the help of the developed general algorithm, analytical approximate solutions of these problems have been constructed and on the basis of their exact solutions a comparative numerical analysis has been carried out.

The spectral consistency of the finite-difference theta-method for the unsteady Schrödinger equation is investigated. Optimal sampling parameters providing a minimum error for a given spectral range are obtained. It is shown that the op ti mized scheme provides a reduction (by a factor of 5–6) in the error of the approximate solution in comparison with the 4th order accuracy scheme. It is shown that the 4th order scheme provides the best spectral consistency only in the case if the spectral range length tends to zero. The conditions for equivalence between the finite-difference scheme and the scheme in the form of two first-order conjugated IIR filters are found. The obtained scheme is the best scheme in the class of conservative finite difference schemes for solving the Schrödinger equation. Practical issues arising in the process of implementing a numerical solution are considered. The obtained results can be efficiently used for solving linear and non-linear Schrödinger equations.

A complete mathematical model for measuring the parameters of the scattering matrix for a measurement object [Sx] in the form of a four-port network is developed, in which the mathematical model of the eight-port error is described by 16 parameters of the scattering matrix [E]. In addition, in comparison with the 12-parameter model of the eight-port error network, four parameters are included, which allow taking into account leaks, parasitic transmissions of microwave modules under study (microwave microassemblies). Due to the use of matrix analysis methods, the equations in matrix form are obtained that connect the matrices of the measurement object [Sи] and the actual values of the matrix parameters [Sx], with the aim of enabling the solution of these equations instead of the scattering matrix [E] to use a transmition matrix [T] in the form
of cellular matrices [Taa], [Tab], [Tba], [Tbb].

The motion equations for a system of two bodies moving in a medium are derived in the Cartesian coordinate system in the Newtonian theory. The coordinate system is barycentric, that is, the center of mass of the two-body system is immobile. Using the Einstein – Infeld approximation procedure, the gravitational field created by the “two bodies – medium” system was found from the Einstein field equations, and then the equations of motion of the bodies in this field were obtained.

It is shown that in the post-Newtonian approximation of the general theory of relativity, the center of mass of two bodies moving in a gas – dust rarefied medium of constant density, determined by analogy with the Newtonian center of mass, is displaced along the cycloid, although in the Newtonian approximation it is stationary, i.e. the movement along the cycloid occurs with respect to the barycentric Newtonian fixed reference frame. Numerical estimates are given for the magnitude of this displacement. Given a popular value of the medium density ρ = 10–21 g·cm–3 its order can reach 106 km per one rotation of two bodies around their center of mass. In the case of the equality of masses of the bodies, their relativistic center of mass, like their Newtonian center of mass, is immobile.

It has been hypothesized that for any elliptical orbits of two bodies and an inhomogeneous distribution of the gas – dust medium the qualitative picture of motion of the relativistic center of mass of the two bodies will not change.

Five-vectors theory of gravity is proposed, which admits an arbitrary choice of the energy density reference level. This theory is formulated as the constraint theory, where the Lagrange multipliers turn out to be restricted to some class of vector fields unlike the General Relativity (GR), where they are arbitrary. A possible cosmological implication of the proposed model is that the residual vacuum fluctuations dominate during the whole evolution of the universe. That resembles
the universe having a nearly linear dependence of a scale factor on cosmic time.

In this paper, on the basis of the Monte-Carlo simulation results a signal processing algorithm for determination of the energy deposited in real time by incident particles has been developed and implemented in the created electronics prototype of the trigger system for an electromagnetic calorimeter of the COMET experiment. The energy thresholds for trigger cells are determined which make it possible to select signal events – an electron with a momentum of 105 MeV/c, and significantly reduce a rate of background events. The electronics prototype of the trigger system has been verified by testbench measurements and electron beam experiments. The obtained results satisfy the key requirement of the calorimeter – the energy resolution in real time is better than 5 % for the signal electron energy.

Non-invasive (remote) thermographic methods based on IR images are being actively implemented. Using the calculation results of the temperature increment that occurs when a pathological source exists in the person’s skin, a number of ways of solving “inverse problems” are proposed. These include the determination of the depth of the thermal source by measuring the mono or polychrome increment of the normalized brightness of the tissue surface at one point; the source depth and heat transfer parameter by measuring the poly or monochrome one of the normalized brightness (or temperature) at two points; the thermal power of the source by measuring the increment of absolute brightness or temperature at one point; the depth of the thermal source and its size in the near-surface layer by measuring the increment of the normalized brightness at two points. In order to solve these problems, the thermophysical and optical properties of the soft tissues of the biological organism are indicated. Analytical solutions are given for describing the temperature and the glow that arises under its influence from the sources of cylindrical and spherical shape.

The Stockbargard – Bridgman method yielded single crystals Mn0.99Fe0.01As. The effect of an external magnetic field with an intensity of up to 10 T on phase transitions in the single crystal Mn0.99Fe0.01As is studied. It is established that the magnetostructural phase transition in Mn0.99Fe0.01As is accompanied by a change in the entropy ΔSm, which is due to the transformation of the crystal structure. At temperatures above the temperature of the magnetostructural transition Tu = 290 K, the existence of an unstable magnetic structure is obtained. The magnetocaloric characteristics of the material under study are determined by an indirect calculation method based on the Maxwell thermodynamic relations and the Clapeyron – Clausius equation.