The purpose of this paper is to investigate the problem of the classification of finite-dimensional simple central K-algebras with unitary involutions. In this paper, K-isomorphism is proven for weakly ramified finite-dimensional central K-algebras with division and unitary K/k-involutions (where the invariant field k is Henselian).
Earlier, in papers by J.-P. Tignol, V. V. Kursov and V. I. Yanchevskii, generalized Abelian crossed products were defined and the K-isomorphism of generalized Abelian crossed products (D₁, G, (ω, f )) and (D₂, G, (ϖ, g )), was proven for the case D₁ = D₂. In this paper, this criterion is proven when D1 and D2 are different. With the help of this criterion, the main result of this article is obtained.

In this paper, we obtained the primality criteria for ideals of rings of integer algebraic elements of finite extensions of the field Q, which are analogues of Miller and Euler’s primality criteria for rings of integers. Also advanced analogues of these criteria were obtained, assuming the extended Riemann hypothesis. Arithmetic and modular operations for ideals of rings of integer algebraic elements of finite extensions of the field Q were elaborated. Using these criteria, the polynomial probabilistic and deterministic algorithms for the primality testing in rings of integer algebraic elements of finite extensions of the field Q were offered.

The Bose – Chaudhuri – Hocquenghem type of linear cyclic codes (BCH codes) is one of the most popular and widespread error-correcting codes. Their close connection with the theory of Galois fields gave an opportunity to create a theory of the norms of syndromes for BCH codes, namely, syndrome invariants of the G-orbits of errors, and to develop a theory of polynomial invariants of the G-orbits of errors. This theory as a whole served as the basis for the development of effective permutation polynomial-norm methods and error correction algorithms that significantly reduce the influence of the selector problem. To date, these methods represent the only approach to error correction with non-primitive BCH codes, the multiplicity of which goes beyond design boundaries.
This work is dedicated to a special error-correcting code class – generic Bose – Chaudhuri – Hocquenghem codes or simply GBCH-codes. Sufficiently accurate evaluation of the quantity of such codes in each length was produced during our work. We have investigated some properties and connections between different GBCH-codes. Special attention was devoted to codes with constructive distances of 3 and 5, as those codes are usual for practical use. Their almost complete description is given in the range of lengths from 7 to 107. The paper contains a fairly clear theoretical classification of GBCH-codes. Special attention is paid to the corrective capabilities of the codes of this class, namely, to the calculation of the minimal distances of these codes with various parameters. The codes are found whose corrective capabilities significantly exceed those of the well-known GBCH-codes with the same design parameters.

In this paper, we consider a semiclassical approximation of special functional integrals with respect to the conditional Wiener measure. In this apptoximation we use the expansion of the action with respect to the classical trajectory. In so doing, the first three terms of expansion are taken into account. Semiclassical approximation may be interpreted as an expansion in powers of the Planck constant. The novelty of this work is the numerical analysis of the accuracy of the semiclassical approximation of functional integrals. A comparison of the results is used for numerical analysis. Some results are obtained by means of semiclassical approximation, while the other by means of the functional integrals calculation method based on the expansion in eigenfunctions of the Hamiltonian generating a functional integral.

In this paper, we consider the classical problem of the classification of subalgebras of small dimensional Lie algebras. We found all 5-dimentional subalgebras of 6-dimentional nilpotent Lie algebras under the field with the zero characteristic. As is known, up to isomorphism all 6-dimensional nilpotent Lie algebras (their number is 32) were received by V. V. Morosov. However, the standard method based on the Campbell – Hausdorf formula is not effective for the determination of subalgebras of Lie 5- or higher dimensional algebras. In our research, we use a new approach to the solution of the problem of the determination of 5-dimensional subalgeras of indicated 6-dimensional nilpotent Lie algerbas, namely, the application of canonical bases for subspaces of vector spaces.

Herein, taking power series from a real variable that converge on a certain interval to known sums, the authors consider the power series with the same coefficients from an h-complex variable. For such series, the interiors of the regions of convergence are found, and their sums are explicitly expressed in terms of the sums of the original series. Along the way, the problem of isolation conditions for the zeros of the sums of such series is solved.

In this paper, a material system consisting of two spherically symmetric bodies of comparable masses located inside a gas-dust ball with a spherically symmetric distribution of the density of the medium in it is considered. After choosing the corresponding energy-momentum tensor from the Einstein field equations using the Einstein-Infeld approximation procedure, the metric of the corresponding space-time, the gravitational field created by the «two-body – medium» system are found, and then the equations of motion of the bodies and their center of mass are obtained in Newton’s and post-Newtonian approximations of the general theory of relativity. It is proved that in the case of the indicated density of the medium, the following effect should exist already in the Newtonian approximation. The center of mass of two bodies shifts at a variable speed, although it was at rest in the void. This situation is a consequence of the fact that the two-body-medium system is not closed. For the first time, formulas for calculating the displacement value, which is proportional to the density of the medium in the center of the gas-dust ball and the 5th degree of the distance between the bodies, are derived. Therefore, at large distances between bodies, their center of mass has large displacements (it can reach several million kilometers per revolution of bodies around their center of mass). If the masses of the bodies are equal, their center of mass is at rest if it is at rest in the void.

We present the quadratic Hahn algebra QH(3) as an algebra of the hidden symmetry for a certain class of exactly solvable potentials, generalizing the 16D oscillator and its 9D coulomb analogue to the generalized version of the Hurwitz transformation based on SU (1,1)⊕ SU (1,1)  . The solvability of the Schrodinger equation of these problems by the variables separation method are discussed in spherical and parabolic (cylindrical) coordinates. The overlap coefficients between wave functions in these coordinates are shown to coincide with the Clebsch – Gordan coefficients for the SU(1,1) algebra.

In this paper, we considered the method of amplitude electro-optical modulation of radiation using sequences of Fabry-Perot resonators based on the transverse electro-optical effect on the example of lithium niobate LiNbO₃. With this method, it is possible to significantly reduce the voltage of the control electromagnetic field of the electro-optical amplitude modulator operating in the transmission mode of the light beam while maintaining its high efficiency. The reduction of the control voltage is achieved by increasing the number of Fabry-Perot resonators installed in series and the phase shift relative to the extremum of the transmittance function. This method allows to diminish the duration of the received light signals which leads to an increase in the clock frequency while maintaining a high efficiency of the radiation modulation. Diminishing the duration of light signals is achieved by using separate modulation channels of two sequences of electro-optical Fabry-Perot resonators, the first of which works on the transmission and the second one on the reflection. Increasing the clock frequency at the output of the modulator is achieved by summing the signals coming from several modulation channels. It is shown that the value of the control voltage for an amplitude electro-optical modulator based on a sequence of Fabry-Perot resonators made of lithium niobate LiNbO₃, with an operating wavelength of 1.307 microns, can be 4 V in the case when its initial operating point corresponds to the maximum transmittance. The control voltage is 2 V if the initial operating point is shifted in phase relative to the extremum of the transmittance function.

Anion-deficient layered cobaltites Sr_0.75Ln_0.25CoO_3–x (Ln is a lanthanide) have attracted the special attention of the scientists who study the nature of phase transformations in perovskite-like cobaltites, the anomalous behavior of the temperature magnetization of which is still the subject of scientific discussion. The purpose of this work is to investigate the regularity of changes in the elastic, magnetic, and electrical properties of layered cobaltites Sr_1–уY_уCoO_3–x in the composition range 0.2 ≤ y ≤ 0.3 over a wide temperature range. The studied polycrystalline samples were obtained by the known ceramic technology in the air. Electron microscopic studies were performed on a LEO 1455 PV scanning electron microscope. The temperature dependence of the Young’s modulus was studied by the method of resonance vibrations in the frequency range 1000–6000 Hz and in the temperature range 100–450 K. X-ray phase analysis was performed on a DRON-3M diffractometer under Cu-K_α radiation. Magnetic measurements were performed using a physical property measurement system (Cryogenic Ltd.) in the temperature range 5–325 K.
As a result of the studies, it was found that in the temperature range 25–300 K, Sr_1–уY_уCoO_3–x solid solutions (0.2 ≤ y ≤ 0.3) are characterized by the semiconductor-like conductivity. No significant magnetoresistive effect was observed in this temperature range for the studied compositions. It was shown that the Sr_1–уY_уCoO_3–x solid solution (у = 0.25) exhibits two magnetic phase transformations: low-temperature near 220 K and high-temperature at 350 K. The nearby compositions of the concentration range 0.2 ≤ y ≤ 0.3 exhibit magnetic phase transformations at temperatures above room temperature. No low-temperature phase transitions were detected in them. It has been established that magnetic phase transformations are accompanied by structural transitions at corresponding temperatures.

The Mn₃Sb metastable compound is formed at high pressure and temperature and it decomposes upon heating above 420 K into Mn₂Sb and Mn. It has a cubic crystalline structure describing the Pm-3m (№ 221) space group with a lattice parameter of a = 0.400 nm. In the present work, according to the results of neutron diffraction investigations and taking into account magnetometry data, it is shown that Mn₃Sb is an antiferromagnet, and a model of the magnetic structure with a triangular configuration of equal magnetic moments in magnitude is proposed. The magnetic moments of manganese atoms, constituting the basis of a unit magnetic cell, lie in the (111) plane and form an equilateral triangle. According to neutron diffraction data, the magnetic moments of manganese atoms were determined at different temperatures.

In the quasi-classical approximation of quantum mechanics a model for the localization of conduction electrons on the ions of hydrogen-like donors in an external magnetic field was developed. The thermal ionization energy of donors in lightly doped and moderately compensated crystals of gallium arsenide and indium antimonide of n-type was calculated depending on the induction of the external magnetic field. In contrast to the known theoretical works (which use variational methods for solving the Schrödinger equation), a simple analytical expression is proposed for the ionization energy of the donor in the magnetic field, which quantitatively agrees with the known experimental data. It is shown that the magnitude of the magnetic field induced by the orbital motion of the electron around the ion core of the donor is negligible compared to the external field and does not contribute to the ionization energy of donors.