In this paper, we study the approximations of a function |x|α, α > 0 by interpolation rational Lagrange functions on a segment [–1,1]. The zeros of the even Chebyshev – Markov rational functions and a point x = 0 are chosen as the interpolation nodes. An integral representation of an interpolation remainder and an upper bound for the considered uniform approximations are obtained. Based on them, a detailed study is made:

a) the polynomial case. Here, the authors come to the famous asymptotic equality of M. N. Hanzburg;

b) at a fixed number of geometrically different poles, the upper estimate is obtained for the corresponding uniform approximations, which improves the well-known result of K. N. Lungu;

c) when approximating by general Lagrange rational interpolation functions, the estimate of uniform approximations is found and it is shown that at the ends of the segment [–1,1] it can be improved.

The results can be applied in theoretical research and numerical methods. 

This paper considers the mixed problem for a one-dimensional wave equation with second-order derivatives at boundary conditions. Using the method of characteristics, a classical solution to this problem is found in analytical form. Its uniqueness is proved under the relevant compatibility conditions.

High-order finite difference schemes for the time-dependent Schrödinger equation are investigated. Digital signal processing methods allowed proving the conservativeness of high-order finite difference schemes for the unsteady Schrödinger equation. The eighth-order scheme coefficients were found with the help of the proved theoretical results. The conditions for equivalence between the eighth-order finite difference scheme and the scheme in the form of a cascade of allpass first-order filters were found. The numerical analysis of the proposed scheme was made. It was shown that the high-order finite difference schemes gave better results on solving the linear Schrödinger equations comparing to the well-known fourthorder scheme on the six-point stencil, however, the high-order schemes in couple with the second-order splitting algorithm to the nonlinear Schrödinger equation do not lead to a radical improvement in the quality of numerical results. Practical issues implementing the proposed numerical technique are considered. The obtained results can be used to construct efficient solvers for linear and nonlinear Schrödinger-type equations by applying the splitting schemes of adequate accuracy order.

The topic of this paper refers to efficient parallel solvers of block-tridiagonal linear systems of equations. Such systems occur in numerous modeling problems and require usage of high-performance multicore computation systems. One of the widely used methods for solving block-tridiagonal linear systems in parallel is the original block-tridiagonal sweep method. We consider the algorithm based on the partitioning idea. Firstly, the initial matrix is split into parts and transformations are applied to each part independently to obtain equations of a reduced block-tridiagonal system. Secondly, the reduced system is solved sequentially using the classic Thomas algorithm. Finally, all the parts are solved in parallel using the solutions of a reduced system. We propose a modification of this method. It was justified that if known stability conditions for the matrix sweep method are satisfied, then the proposed modification is stable as well.

An algorithm for numerical calculation of the trajectories of electrons emitted by plasma in the electron beam moving in axially symmetric electrostatic and magnetostatic fields is proposed. This algorithm is based on the technology of charged particle beam discretization by current tubes and the decomposition method of the computational domain. Field simulation and numerical solution of equations for particle motion are carried out with the use of quasi-structured grids.

New concepts are introduced in the present work. They are a quite normal index and a quite perfect system of functions. Using these concepts, the uniqueness criterion for solution of two Hermite – Pade problems is proved, the explicit determinant representations of type I and II Hermite – Padé polynomials for an arbitrary system of power series are obtained. The results obtained complement and generalize the well-known result in the theory of Hermite – Padé approximations.

This paper is dedicated to the integrals and integral transformations related to the probability density function of the vector Gaussian distribution and arising in probability applications. Herein, we present three integrals that permit to calculate the moments of the multivariate Gaussian distribution. Moreover, the total probability formula and Bayes formula for the vector Gaussian distribution are given. The obtained results are proven. The deduction of the integrals is performed on the basis of the Gauss elimination method. The total probability formula and Bayes formula are obtained on the basis of the proven integrals. These integrals and integral transformations could be used, for example, in the statistical decision theory, particularly, in the dual control theory, and as table integrals in various areas of research. On the basis of the obtained results, Bayesian estimations of the coefficients of the multiple regression function are calculated.

The generalized Klein – Fock – Gordon equation for a particle with the Darwin–Cox structure allowing for a charge distribution of a particle over a sphere of finite radius is studied with regard to the external Coulomb field. The separation of variables is carried out, the obtained radial equation is significantly more complicated than the equation in the case of ordinary particles, it has essentially singular points r = 0 of rank 3, r = ∞ of rank 2 and 4 regular singular points. In the case of a minimum orbital momentum l = 0, the structure of singularities is simplified: there are essentially singular points r = 0, r = ∞ of rank 2 and 4 regular singular points. Frobenius solutions of this equation are constructed and the structure of the 7-term recurrence relations for the coefficients of the arising power series is investigated. As an analytical quantization condition, the generalized transcendence requirement of solutions is used; it allows one to obtain a fourth-degree algebraic equation for energy levels. The equation has 4 sets of roots depending on the orbital moment l and the main quantum number k = 1,2,3,… . The numerical analysis shows that one of the sets of the roots 0 < εl,k < mc2 can be interpreted as those corresponding to certain bound states of the particle in the Coulomb field.

Some basic properties of acousto-optical (AO) diffraction involving Bessel light and acoustic beams in anisotropic crystals are investigated. Hexagonal symmetry crystals are considered and are optically uniaxial and positive and acoustically transversely isotropic. It is shown that, unlike the case of AO diffraction of plane waves, the transition to Bessel beams allows one to realize a number of new diffraction channels having specific configurations of the wave vectors of interacting waves while maintaining the axial symmetry of the optical scheme as a whole. The diffraction channels for anisotropic scattering are classified and the main parameters of the scattered Bessel light beam and the parameters of the Bessel acoustic beam are calculated for each of them. The possibility of implementing the isotropic-type diffraction was revealed, which makes it possible to increase the efficiency of AO conversion. The parameters of this-type diffraction are determined for two scattering channels, namely, for scattering by a direct Bessel acoustic beam and by a backward propagating acoustic beam.

Due to the appearance of a set of scattering channels and with regard to the fact that Bessel light and acoustic beams have helical wave front dislocations, as well as suppressed diffraction spreading, the study of the features of AO diffraction of such beams in optically positive crystals has both a scientific and practical interest. 

The investigation results of the annealing influence (Тann = 300–800 ºС) on the minority charge currier lifetime tP in the n-base of p-n-structures, manufactured on the base of neutron transmutation doped silicon (NTD) КОФ300, irradiated at room temperature by different fluences (F = 1 · 1014 – 3 · 1016 cm–2) of electrons with the energy of Еe = 4 MeV are presented. It is established that at low electron fluences (F = 1 · 1014 cm–2), the annealing of minority charge currier lifetime tP in the n-base of p-n-structures occurs in two stages: the first – 320–400 ºС and the second – 550–650 ºС. At higher electron fluences (F = 5 · 1015–2 · 1016 cm–2), three annealing stages occur: the first – 400–450 ºС, the second – 520–650 ºС and the third – 710–770 ºС. At this, the structure barrier capacitance C dependences on Тann at high electron fluences show the geometry capacitance up to the annealing temperatures Тann = 400 ºС. In the annealing temperature range of Тann = 420–570 ºС, the increase in С with maximum is seen at Тann = 480 ºС and a subsequent decrease in the geometry capacitance is seen in the annealing temperature range of Тann = 600–670 ºС, and then again the increase in С occurs in the annealing temperature range of Тann = 720–770 ºС reaching the С values corresponding to those of the non-irradiated samples in the annealing temperature range of Тann = 770–800 ºС. The analysis of the DLTS-spectra of the investigated structures has allowed establishing the formation in the annealing process of the deep acceptor level ЕС – 0.68 eV at Тann > 400 ºС, the deep donor level ЕС – 0.32 eV in the annealing temperature range of Тann = 420–570 ºС and the deep acceptor level ЕС – 0.53 eV at Тann > 700 ºС, which satisfactorily explains the dependences of t P and С on Тann obtained in this paper.

The charge build-up in the interface of silicon / buried oxide in n-channel MOS/SOI transistors depending on their geometric parameters and electrical modes during ionizing irradiation is calculated with the use of the software Silvaco. It is shown that the electrical mode is most “harsh”, when during irradiation the voltage of +5 V is applied to drain and source electrodes and 0 V is applied to substrate, gate and channel feeding. The amount of the built-up charge can be substantially reduced by applying a negative bias to the substrate and by decreasing the thickness of the buried oxide layer.