This article considers a classical solution of the boundary problem for the four-order strictly hyperbolic equation with four different characteristics. Note that the well-posed statement of mixed problems for hyperbolic equations not only depends on the number of characteristics, but also on their location. The operator appearing in the equation involves a composition of first-order differential operators. The equation is defined in the half-strip of two independent variables. There are Cauchy’s conditions at the domain bottom and periodic conditions at other boundaries. Using the method of characteristics, the analytic solution of the considered problem is obtained. The uniqueness of the solution is proved. We have also noted that the solution in the whole given domain is a composition of the solutions obtained in some subdomains. Thus, for the obtained classical solution to possess required smoothness, the values of these piecewise solutions, as well as their derivatives up to the fourth order must coincide at the boundary of these subdomains. A classical solution is understood as a function that is defined everywhere at all closure points of a given domain and has all classical derivatives entering the equation and the conditions of the problem.

This article is devoted to the problem of construction and research of the generalized Hermite – Birkhoff interpolation formulas for arbitrary-order partial differential operators given in the space of continuously differentiable functions of many variables. The construction of operator interpolation polynomials is based both on interpolation polynomials for scalar functions with respect to an arbitrary Chebyshev system, and on the generalized Hermite – Birkhoff interpolation formulas obtained earlier by the authors for general operators in functional spaces. The presented operator formulas contain the Stieltjes integrals and the Gateaux differentials of an interpolated operator. An explicit representation of the error of operator interpolation was obtained. Some special cases of the generalized Hermite – Birkhoff formulas for partial differential operators are considered. The obtained results can be used in theoretical research as the basis for constructing approximate methods for solution of some nonlinear operator-differential equations found in mathematical physics.

The problem of the positive integer solution of the equation Xn = A for different-order matrices is important to solve a large range of problems related to the modeling of economic and social processes. The need to solve similar problems also arises in areas such as management theory, dynamic programming technique for solving some differential equations.
In this connection, it is interesting to question the existence of positive and positive integer solutions of the nonlinear equations of the form Xn = A for different-order matrices in the case of the positive integer n. The purpose of this work is to explore the possibility of using analytical methods to obtain positive integer solutions of nonlinear matrix equations of the form Xn = A where A, X are the third-order matrices, n is the positive integer. Elements of the original matrix A are integer and positive numbers. The present study found that when the root of the nth degree of the third-order matrix will have zero diagonal elements and nonzero and positive off-diagonal elements, the root of the nth degree of the third-order matrix will have two zero diagonal elements and nonzero positive off-diagonal elements. It was shown that to solve the problem of finding positive integer solutions of the matrix equation for third-order matrices in the case of the positive integer n, the analytical techniques can be used. The article presents the formulas that allow one to find the roots of positive integer matrices for n = 3,…,5. However, the methodology described in the article can be adopted to find the natural roots of the third-order matrices for large n. 

In this article the problem of a sequential test for the model of independent non-identically distributed observations is considered. Based on recursive calculation a new numerical approach to approximate test characteristics for a sequential probability ratio test (SPRT) and a truncated SPRT (TSPRT) is constructed. The problem of robustness evaluation is also studied when the contamination is presented by the distortion of the distributions of all increments of the log-likelihood ratio statistics. The two-side truncated functions are proposed to be used for constructing the robustified SPRT. An algorithm to choose the thresholds of these truncated functions is indicated. The results are applied for a sequential test on parameters
of time series with trend. Some kinds of the contaminated models of time series with trend are used to study the robustness of the truncated SPRT. Numerical examples confirming the theoretical results mentioned above are given.

In the present acticle we consider finite-dimensional stochastic differential equations with fractional Brownian motions having different Hurst indices larger than 1/3 and a drift. These heterogeneous components of the equations are combined into a single process. The solutions of the equations are understood in the integral sense, and the integrals in turn
are Gubinelli’s rough path integrals [1] realizing the well-known approach of the rough paths theory [2]. The existence
and uniqueness conditions of the solutions of these stochastic differential equations are specified. Such conditions are sufficient to obtain the results related the continuous dependence on the initial data. In this article, we have first proved a continuous dependence on the initial conditions and the right-hand sides of the solutions of the stochastic differential equations under consideration for almost all their trajectories. The result obtained does not depend on the probabilistic properties of fractional Brownian motions, and therefore it can be easily generalized to the case of arbitrary Holder-continuous processes with an exponent greater than 1/3. In this case, the constant arising in the estimates appears to be exponentially dependent on the norms of fractional Brownian motions. Taking into account the last fact and the proved result, an expected logarithmic continuous dependence on the initial conditions and the right-hand sides of the solutions of the stochastic differential equations con - si dered is subsequently derived. This is the major result of this article.

For an eye-safe optical parametric oscillator (OPO) built on the basis of a three-mirror ring cavity, each section of which (the space between adjacent plane mirrors) contains a x-cut KTiOPO4 (KТР) crystal having a size of 15(х) × 7(y) × 7(z) mm3, the thermal effects due to idler wave absorption in KTP crystals were investigated. These thermal effects were evaluated by means of experimental measurement of the change in the OPO performance (divergence of the output beam and pulse energy) when transferring the OPO from the mode of generation of occasional single pulses to the mo de of generation of periodically repetitive pulses. It was found when the eye-safe OPO pumped by multimode YAG: Nd laser radiation generates 8-ns pulses with a repetition rate of 10 Hz and an energy of 30–35 mJ, thermal distortions of KTP crystals placed in metal holders at their natural air-cooling, are moderate. The total effect of positive thermolenses induced in nonlinear crystals leads to an increase in the divergence of the beam of the eye-safe OPO by 10 % and to a decrease in the efficiency of the OPO by 0.76 %, by virtue of fact that the induced thermal lenses are not ideal and thereby introduce additional aberration losses into the OPO cavity. The theoretical simulation of the OPO operation in the plane-wave approximation with the use of a system of three coupled first-order abridged differential equations showed that among three KTP crystals the KTP crystal placed first in the path of pump radiation in the OPO is the largest thermal load and the action of the most intense beams.

With the use of deep level transient spectroscopy (DLTS) the effect of injection of minority charge carriers (electrons) on an annealing rate of self di-interstitial – oxygen (I2O) complex in silicon has been studied. The complex has been formed by irradiation of epitaxial boron-doped n+–p diode structures with alpha-particles at room temperature. It has been shown that the disappearance of this complex at room temperature begins at a direct current density of ~1.5 A/cm2. This characteristic current density has been found for 10 W·cm p-type silicon when the total radiation defect density was less than 15 % of the initial boron concentration, a divalent hole trap with energy levels of Ev + 0.43 eV and Ev + 0.54 eV has been found to appear as a result of recombination-enhanced annealing of the I2O. When the I2O complex is annealed thermally, the concurrent appearance of an electron trap with an energy level of Ec – 0.35 eV has been observed. It has been shown that the divalent hole trap represents a metastable configuration (BH-configuration) of the bistable defect, whereas the electron trap is stab le in the p-Si configuration (ME-configuration). From the comparison of DLTS signals related to different defect configurations it is found that the ME-configuration of this bistable defect can be characterized as a center with negative correlation energy. It has been shown that the injection-stimulated processes make it very difficult to obtain reliable data on the formation kinetics of the bistable defect in the BH-configuration when studying the thermal annealing of the I2O complex.

The quaternary semiconductors Cu2CdSnS4, Cu2CdSnSe4 and Cu2CdSn(SxSe1–x)4 solid solutions were synthesized by the one-temperature method from the elementary components. The X-ray diffraction method showed that the obtained polycrystalline samples are single-phased. The unit cell parameters of the synthesized compounds and Cu2CdSn(SxSe1–x)4 solid solutions were determined from diffraction spectra by the full-profile analysis using the Rietveld method with the Fullprof software package. It has been established that with an increase in sulfur concentration, the unit cell parameters decrease smoothly linearly in accordance with the Vegard rule, which indicates the formation of a continuous series of solid solutions in the Cu2CdSn(SxSe1–x)4 system within the range 0 ≤ x ≤ 1. The parameter of crystal lattice tetragonal distortions h of the investigated compounds is calculated. The h values are close to 1 for all the compositions studied, which indicates a small crystal lattice distortion of the obtained samples.

The method of transformation of information from one spectral range to another based on Fabry – Perot microresonators is offered. The method uses incident radiation of an object as affecting a microresonator material (a microresonator material must absorb this radiation), and visible radiation of the optical part of the spectrum as sensing, or reading radiation (a microresonator material should not absorb this radiation). The absorbed energy of incident radiation leads to a change in a microcavity temperature, which results in a change in the optical base of the resonator. The high sensitivity of the Fabry – Perot microcavities is a consequence of the fact that the principle of their operation is based on the physical phenomenon of multipath interference. A common shortcoming of the Fabry – Perot standards is their sensitivity to operating conditions, for example, to a change in the ambient temperature, which also leads to a change in the optical base of the resonator, as well as the influence of IR radiation. This leads to a shift in the spectral characteristics of transmittance or reflection of the Fabry – Perot standards, which worsens their performance characteristics. The method allows one to minimize the environmental temperature fluctuation influence on characteristics of the Fabry – Perot microresonator, which is an element that transforms the information from one spectral range to another. Minimization is performed when the starting temperature point of the microresonator corresponds to a maximum change in the probing radiation intensity due to the temperature.

An increase in the informative content of the calculated values of the reliability measure (RM) of objects, whose reliability is ensured by the redundancy of structural elements, is considered in the article. The increase of the informative content is ensured using the interval estimates of the RM. In the normal reliability calculation, the calculated value of the object’s RM is unambiguous, and for an interval reliability estimate, the value range is obtained, which can be quite appreciated as the increase in the informative content. The choice of on-board equipment for small spacecrafts as an object of research in this work is determined as follows: at present, the vast majority of spacecrafts can be classified as small spacecrafts; since the reliability
of small spacecrafts is high, it is necessary to use redundancy; the Belarusian spacecraft for remote sen-sing of the Earth belongs to the category of small spacecrafts. As a result of research, the formulas for calculation of interval estimation results are established for the linear and nonlinear dependence of the object’s RM on the RM of its elements. Structural reliability schemes (SSR) are used as an object (system) reliability model, which includes blocks of elements without redundancy (simple) and blocks with different-type redundancy (complex). The object’s RM is a reliability measure determined by its SSR. Therefore, for an interval estimation of the object’s RM to be obtained, the interval estimates of the RM of its blocks must be made. RM interval estimates of simple and complex SSR blocks are obtained in the article. Complex blocks were considered as a set of parallel circuits provi- ding: continuous redundancy for all loaded circuits; non-continuous redundancy of loaded and unloaded circuits; standby redun-dancy; redundancy by voting. The formulas for interval estimation of the object’s RM represented by the SSR and the example of using the methodology on the component part of a real on-board information system are given in the article. The boundary values of the interval estimates of the example can be taken as optimistic and pessimistic estimates.