The mixed problem for the one-dimensional Klein – Gordon – Fock type equation with curve derivatives at boundary conditions is considered in the half-strip. 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 functions were proven for these equations when initial functions are smooth enough. It is proven that the fulfillment of the matching conditions on the given functions is necessary and sufficient for the 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 original area of definition to the subdomains. The solution of the subproblem can be constructed in each subdomain with the help of the initial and boundary conditions. Then, the obtained solutions are glued in common points, and the obtained glued conditions are the matching conditions. This approach can be used in constructing as an analytical solution when a solution of the integral equation can be found in an explicit way, so an approximate solution. Moreover, approximate solutions can be constructed in numerical or analytical form. When a numerical solution is built, the matching conditions are essential and they need to be considered while developing numerical methods.

Quadrature linear integro-differential equations on a closed curve located in the complex plane are solved. The equations contain singular integrals which are understood in the sense of the main value and hypersingular integrals which are understood in the sense of the Hadamard finite part. The coefficients of the equations have a special structure.

In the introduction, the object of investigation is indicated – incorrect problems described by first-kind operator equations. The subject of the study is an explicit iterative method for solving first-kind equations. The aim of the paper is to prove the convergence of the proposed method of simple iterations with an alternating step alternately and to obtain error estimates in the original norm of a Hilbert space for the cases of self-conjugated and non self-conjugated problems. The a priori choice of the regularization parameter is studied for a source-like representable solution under the assumption that the operator and the right-hand side of the equation are given approximately. In the main part of the work, the achievement of the stated goal is expressed in four reduced and proved theorems. In Section 1, the first-kind equation is written down and a new explicit method of simple iteration with alternating steps is proposed to solve it. In Section 2, we consider the case of the selfconjugated problem and prove Theorem 1 on the convergence of the method and Theorem 2, in which an error estimate is obtained. To obtain an error estimate, an additional condition is required – the requirement of the source representability of the exact solution. In Section 3, the non-self-conjugated problem is solved, the convergence of the proposed method is proved, which in this case is written differently, and its error estimate is obtained in the case of an a priori choice of the regularization parameter. In sections 2 and 3, the error estimates obtained are optimized, that is, a value is found – the step number of the iteration, in which the error estimate is minimal. Since incorrect problems constantly arise in numerous applications of mathematics, the problem of studying them and constructing methods for their solution is topical. The obtained results can be used in theoretical studies of solution of first-kind operator equations, as well as applied ill-posed problems encountered in dynamics and kinetics, mathematical economics, geophysics, spectroscopy, systems for complete automatic processing and interpretation of experiments, plasma diagnostics, seismic and medicine.

The algorithm implemented on a parallel computer with distributed memory has, as a rule, a tiled structure: a set of operations is divided into subsets, called tiles. One of the modern approaches to obtaining tiled versions of algorithms is a tiling transformation based on information sections of the iteration space, resulting in macro-operations (tiles). The operations of one tile are performed atomically, as one unit of calculation, and the data exchange is done by arrays. The method of construction of tiled computational processes logically organized as a two-dimensional structure for algorithms given by multidimensional loops is stated. Compared to one-dimensional structures, the use of two-dimensional structures is possible in a smaller number of cases, but it can have advantages when implementing algorithms on parallel computers with distributed memory. Among the possible advantages are the reduction of the volume of communication operations, the reduction of acceleration and deceleration of computations, potentially a greater number of computation processes and the organization of data exchange operations only within the rows or columns of processes. The results are a generalization of some aspects of the method of construction of parallel computational processes organized in a one-dimensional structure to the case of a two-dimensional structure. It is shown that under certain restrictions on the structure and length of loops, it is sufficient to perform tiling on three coordinates of a multidimensional iteration space. In the earlier theoretical studies, the parallelism of tiled computations was guaranteed in the presence of information sections in all coordinates of the iteration space, and for a simpler case of a one-dimensional structure, in two coordinates.

An algebraic parameter of a graph – a difference between its maximum degree and its spectral radius is considered in this paper. It is well known that this graph parameter is always nonnegative and represents some measure of deviation of a graph from its regularity. In the last two decades, many papers have been devoted to the study of this parameter. In particular, its lower bound depending on the graph order and diameter was obtained in 2007 by mathematician S. M. Cioabă. In 2017 when studying the upper and the lower bounds of this parameter, M. R. Oboudi made a conjecture that the lower bound of a given parameter for an arbitrary graph is the difference between a maximum degree and a spectral radius of a chain. This is very similar to the analogous statement for the spectral radius of an arbitrary graph whose lower boundary is also the spectral radius of a chain. In this paper, the above conjecture is confirmed for some graph classes.

In this paper, the object of research is Markov’s network with positive and negative customers and unreliable service lines with single-line queuing systems (QS). The discipline of service of customers in the systems – FIFO (“first come first served”) and the service time of customers in each line of the QS network are distributed according to the exponential law with their parameters for each QS. The service lines in each QS are defeated by accidental breakdowns, and the time of correct operation of the service line in each SMO has an exponential distribution, with different parameters for each QS. After the breakdown, the line immediately begins to recover, and the recovery time also has an exponential distribution, the parameters of which are different for each QS. The aim of the study is to find the non-stationary probabilities of network states. To find them, a modified method of successive approximations combined with the method of series is proposed. This method allows one to remove the condition of high load. The properties of successive approximations are proved. On the basis of the obtained data, using a computer, a model example illustrating the finding of the time-dependent probabilities of network states is calculated. The results of this work can be applied to the modeling of various information systems and networks.

Finite difference schemes and iterative methods of solving anisotropic diffusion problems governing multidimensional elliptic PDE with mixed derivatives are considered. By the example of the test problem with discontinuous coefficients, it is shown that the spectral characteristics of the finite difference problem and the efficiency of their preconditioning depend on the mixed derivatives approximation method. On the basis of the comparative numerical analysis, the most adequate approximation formulas for the mixed derivatives providing a maximum convergence rate of the bi-conjugate gradients method with the incomplete LU factorization and the Fourier – Jacobi preconditioners are discovered. It is shown that the monotonicity of the finite difference scheme does not guarantee advantages at their iterative implementation. Moreover, the grid maximum principle is not provided under the conditions of essential anisotropy.

In this paper, the compositional structure of a finite group G is investigated, which has the Sylow 2-subgroup that is permutable with some non p-nilpotent biprimary subgroups, which contain the Sylow р-subgroup of G for all odd simple divisors of the р order of the group G, and such biprimary subgroups are taken one by one for each odd р, and mark the set SB(G). In this work, the existence of the subset SB(G)* in SB(G) is proved, which consists of р-closed subgroups. The main result of this paper is as follows: if the Sylow 2-subgroup of the group G is permutable with all subgroups SB(G)*, then G may have simple non-abelian compositional factors only of L₂ (7) type, if p > 3, and additionally of L₂ (3^f) type, f = 3^a , a ≥ 1, if p = 3.

The paper is devoted to the study of the properties of the Markov – Stieltjes transformation of measures. In the works of J. Anderson, A. A. Pekarsky, N. S. Vyacheslavov, E. P. Mochalina et al., the functions of Markov – Stieltjes type were studied from the point of view of the approximation theory. In the works of A.R. Mirotin and the author, the Markov – Stieltjes transform of functions was studied as an operator in Hardy and Lebesgue spaces. In this paper, the general properties of the Markov – Stieltjes transform of measures are studied, the theorem of analyticity and the uniqueness theorem are proved, the Markov – Stieltjes transformations of positive and complex measures are described, the inversion formula and the continuity theorem are established, the boundary behavior of the given transformation is investigated. In particular, the analogues of the Sokhotsky – Plemelya formulas are established. Applications to the theory of self-conjugate operators are given. In addition, the results obtained can find use in the theory of functions and integral operators, as well as in the theory of information transfer, in particular, in the theory of signal processing.

Young’s modulus of the nc-TiN/a-Si₃N₄ nanocomposite has been calculated depending on a size and a volume fraction of the nanocrystalline phase. The elastic moduli of a-Si₃N₄ matrix and nc-TiN nanocrystals as well as their relation show that they play an important role in the total elastic modulus of the nanocomposite. The kinetics of the defect structure during nanocomposite irradiation was investigated taking into account the recombination processes and sinks on the nanocrystals. 

In this paper, it is shown that under the conditions of total internal reflection of plane homogeneous electromagnetic waves at the interface of a hyperbolic metamaterial and an ordinary isotropic medium, special inhomogeneous electromagnetic waves are excited in certain circumstances near the surface of the metamaterial and their amplitude changes with distance according to the non-exponential law. The existence conditions for such waves are established for the case when the optical axis is located within the interface plane and forms an angle with the plane of incidence. The energy flux and the energy density of special inhomogeneous waves in a hyperbolic metamaterial are determined.

 

Mixed convective heat transfer is very important for a wide class of engineering tasks. However, the experimental study of mixed convection requires significant implementation costs, high-power equipment, as well as large time costs, so it is proposed to expand the scope of experimental studies using numerical simulation. Numerical simulation of the single-row bundle consisting of bimetallic finned tubes at mixed air convection conditions was performed and experimental data were compared. The formulation of the third-dimensional problem for numerical simulation was realized. The conjugated problem for heat exchange modeling from the tube fins to air was solved. In numerical simulation of air momentum it was taken into account that the Reynolds number based on tube diameter and velocity in the space between fins was varied from 100 to 720. Menter’s k–ω shear stress transport model in standard formulation was used to close the Reynolds equations. Flow visualization on the tube surface revealed the transient nature of the air flow. The temperature distribution visualization in the bundle and the exhaust mine made it possible to see the nature of cooling the finned bundle at mixed convection. Results of numerical simulation and experimental investigations are in good agreement and can be used for expansion of the scope of experiments. The experimental data and the numerical simulation results for the single-row bundle consisting of bimetallic finned tubes at mixed air convection are compared in this paper. Flow near tube surfaces was visualized, and the temperature and velocity distributions in a bundle and in the exhaust mine were obtained.