Some discrete subgroups of the Lorentz group are found using Fedorov’s parametrization by means of complex vector-parameter. It is shown that the discrete subgroups of the Lorentz group, which have no fixed points, are contained in boosts along a spatial direction for time-like and space-like vectors and represent discrete subgroups of group S0(1,1), whereas discrete subgroups of an isotropic vector are subgroups of S0(1,1) х E(1,1). An example of construction of nodes of ‘time-like’ lattice is given.

It is shown that on the real line and in the complex plane there are intervals I of short length and circles K of small radius within which there are no algebraic numbers of small height. If the length of the intervals and the radius of the circles increase, then it is already possible to obtain nontrivial estimates for the number of algebraic numbers in I and in K.

The largest Lyapunov exponents of linear differential systems dx/ dt = µA(t)x, xєℝⁿ, t ≥0, with a real parameter-multiplier as a function of the parameter are considered. It is proved that the largest Lyapunov exponent is a function of a Baire class (*,G_⅋), which vanishes at zero and satisfies one of the two cases: 1) it exceeds the linear function; 2) it is equal to the plus infinity on some real semi-axis. In the first case, the sufficiency of the given necessary conditions is proved.

Опе inverse prob1em of the ana1ytic theory of 1inear differentia1 equations is considered. Name1y, the Fuchs equate with four critica1 points and a given reducib1e monodromy group of dimensiona1ity 2 is constructed.

The method of approximate evaluation of functional integrals with respect to the spin variables of the functionals containing interaction of remote nodes is proposed. This method is based on expansion in large and small parameters which are contained in the functional and the measure.

In this article we consider the complexity of recognizing Hamiltonian graphs with a prescribed local structure: locally connected graphs with a bounded vertex degree, N₂-locally connected K_{1 3}-free graphs and locally connected almost K_{1, 3}-free graphs. We establish the NP-completeness of the problem for each class of graphs under consideration. In particular, the conjecture stated in 2011 on polynomial solvability of the Hamiltonian cycle problem for locally connected graphs of maximum degree at most 6 was disproved. Also, for several known sufficient conditions of Hamiltonicity, it is shown that these conditions cannot be naturally weakened without loss of the polynomial solvability of the Hamiltonian cycle problem.

The algebraic variables separation method, developed prior to Dirack’s relative equation, has been improved and generalized to the system of Maxwell equations in order to construct its accurate solutions.

Statistics, their distributions and criteria for building confidence intervals and checking the hypotheses in parallel factor analysis are presented. Their use is illustrated by the example of computer simulation.

Let K/к be a separable quadratic extension, char K ≠ 2, A a non-commutative finite dimensional central K-division algebra with unitary к-involution. For algebras A of prime index relatively prime to char K we prove the following conjecture: к-linear span of the commutator subgroup of а unitary group of A coincides with A.

Within the framework of the covariant description of the Hopf bundle S³ → S² in quaternions the connection between corresponding projective coordinates and vector-parameters was established, which gives a covariant representation of the potentials of linked electromagnetic fields and elaborates the theory of vector parameterization of the SU (2) group.

Modification of diamond-like carbon (DLC) films by pulsed laser radiation with a wavelength of 532 nm and a pulse duration of 18 ns was investigated. Irradiation of DLC films with laser fluence over 0.4 J/cm² was accompanied by the formation of electrically conducting structures with a surface resistance varying from thousands to a kΩ/sq. The dependence of conductivity of films on the laser fluence was established. A relationship between the energy density of laser radiation and the structural changes in films, their heating and evaporation is discussed.

Magnetic characteristics of MnNi_{1_x}Fe_xGe alloys are studied in the concentration region 0,10 ≤ х ≤0,25. It is shown that all samples under study are ferromagnetics at room temperature and have the Ni2In-type of the crystal structure. At х < 0.20, Fe atoms statistically substitute Mn atoms in Me I and Me II positions of MnNi_{1_x}Fe_xGe. At х ≥ 0.20, Fe atoms replace only Mn in Me I positions and do not replace Ni in Me II positions.

The dielectric permittivity and electrical conductivity of Cu₂ZnSnS4 and Cu₂ZnSnSe₄ single crystals are investigated in the temperature range 100–300 K at the measuring field frequencies of 10³–10⁶ Hz. The values of the majority charge carriers generalized activation energy in these crystals are determined. It is shown, that the absolute values of the studied characteristics increases with the temperature. The dielectric properties dispersion of the studied single crystals is revealed: with the frequency growth the dielectric constant values decreases, and the electrical conductivity increases. It was found that the dielectric constant and conductivity of Cu₂ZnSnSe₄ single crystals more than Cu₂ZnSnS₄.

The crystal structure and electrical properties of ceramics obtained from sonicated powders of lead zirconate-titanate were investigated. It is found that ultrasonic treatment of the initial powders allows to obtain piezoelectric ceramics with the monocrystalline structure and desired physical properties.

Properties of the surface layers of monocrystalline silicon wafers implanted with boron and phosphorus to form a heavily doped "pockets" of CMOS structures was investigated by mass spectrometry of secondary ions, measuring the surface resistivity and microhardness. Near-surface hardening of silicon wafers during implantation was founded. Amorphization of the implanted region of silicon reduces the microhardness of the surface layer. Rapid thermal annealing leads to a softening of the surface layer of the silicon single crystal to a depth of 1 |im and an increase in fracture toughness (K1C and y) at low loads. The experimental results are discussed in terms of the generation of vacances in the process of rapid thermal annealing.

In the 80-300 K temperature range and magnetic fields with induction of up to 2.1 T are studied the characteristics of magnetoresistive properties and Hall effect of Mn_0,55V_0,45S solid solution. It was found that the Mn_0,55V_0,45S composition is a semiconductor with high p-type carrier concentration and low values of their mobility; a magnetoresistive effect is observed; solid solution has a noncollinear antiferromagnetic structure at temperatures ranges T < T_N = 130 K; in the vicinity of the temperature T ~ 180 K in Mn_0,55V_0,45S there is a phase transition of semiconductor-semimetal type due to delocalization of charge carriers and the formation of micro areas with ferromagnetic ordering in an antiferromagnetic matrix. Magnetoresistive effect in this case, most likely is due to the magnetic inhomogeneity and can be interpreted in the framework of the electronic and magnetic phase separation consistent with the theory of current flow in heavily doped semiconductors.

For each Q > 1, the existence of algebraic numbers of third degree is proved in circles of radius _cQ ^-1, where с is a sufficiently large constant.

In the article we obtain a lower bound of the number of integral polynomials of arbitrary degree and bounded height with small values of the derivative at any root of the polynomial.

A combination of decision tree ensemble and genetic algorithm is proposed for the vocal fold pathology diagnosis by acoustic signals. The experimental results show a better performance (higher classification accuracy) of the proposed method in comparison with the others.