The problem of construction and research of Hermite interpolation formulas with nodes of arbitrary multiplicity for operators given in functional spaces of one and two variables is considered. The construction of operator interpolation polynomials is based both on interpolation polynomials for scalar functions with respect to an arbitrary Chebyshev system and on identity transformations of functions. The reduced operator formulas contain the Stieltjes integrals and the Gateaux differentials of an interpolated operator and are invariant for a special class of operator polynomials of appropriate degree. For some of the obtained operator polynomials, an explicit representation of the interpolation error is found. Particular cases of Hermite formulas based both on the integral transformations of Hankel, Abel, Fourier and on the Fourier sine (cosine) transform are considered. The application of separate interpolation formulas is illustrated by examples. The presented results can be used in theoretical research as the basis for construction of approximate methods for solving integral, differential and other types of nonlinear operator equations.

For central division algebras D over Henselian fields K with unitary K/k-involutions the Tate cohomology groups of Z/(2)-modules A = N_{Z̅ /K̅}(Nrd_D̅(D̅^*)), where  K̅   ,  D̅    are the residue algebras of K and D, respectively,  Z̅    is the center of  D̅ , and N_{Z̅ / K̅}   is the norm map from  Z̅    to  K̅  , are computed. Moreover, D is assumed to be tamely ramified K-algebra and a field k̅ belongs either to the class of C₁ -fields, or to the class of totally imaginary global fields.

Systems of stochastic differential equations, for which the Riemannian manifold generated by a diffusion matrix has zero curvature, are considered in this article. The method for approximate evaluation of characteristics of the solution of the systems of stochastic differential equations is proposed. This method is based on the representation of the probability density function through the functional integral. To compute functional integrals we use the expansion of action with respect to a classical trajectory, for which the action takes an extreme value. The classical trajectory is found as the solution of the multidimensional Euler – Lagrange equation.

As well known, the classical Rolle and Darboux theorems for a function of one variable establish the existence of a critical point in the behavior of a function at the ends of a closed interval. The question arises of the possibility of extension of the Rolle and Darboux theorems to functions of two variables. More precisely is the existence of a critical point in Ω̅ determined by the behavior of the function f on the boundary of the ∂Ω domain Ω. As shown by A. I. Perov, such extension can be obtained with the help of the concept of rotation. In this article, we establish deeper relations between the Rolle and Darboux theorems and the rotation of a vector field on the boundary ∂Ω. We also present some new formulas for calculating the rotation of a vector field on the boundary, on the basis of which statements about the existence of critical points are formulated.

It is known that the geometry of the Lobachevsky space acts on the fields of particles with spins 0, 1/2, 1 as an ideal mirror distributed in space. The depth of penetration of the field in such a medium increases with increasing field energy. Since the Lobachevsky model is a constituent element in some cosmological models, this property means that in such models it is necessary to take into account the effect of the presence of a “cosmological mirror”; it must lead to a redistribution of the particle density in space. The earlier analysis assumed the static nature of the space-time geometry. In this article, we generalize the research of the spin 1/2 field in the case of the oscillating model of the de Sitter universe. The Dirac equation is solved in the non-static quasi-Cartesian coordinates. At this, we substantially use the diagonalization of a generalized helicity operator. The wave functions of the particle are nontrivially time-dependent; however the effect of a complete reflection of the particles from an effective potential barrier is preserved. For the real Majorana 4-spinor field, the similar results are valid. For the solutions describing the reflection effect to be constructed, we must use linear combinations of solutions with opposite helicities. Such combinations are forbidden for 2-component Weyl particles, for this reason such particles cannot be reflected by the cosmological barrier.

Non-invasive thermographic methods are based on the measurement of IR radiation from human tissue and, by which using Kirchhoff’s law, it restores the temperature of the examined body region. The disadvantage of this method is that the temperature image is noisy due to the thermal influence of the tissue layer between the organ under study inside the medium and the thermal imager. This degrades the quality of the image under consideration. Using the data on the temperature distribution inside the biological medium from a point source, the expression for the temperature from different-shape sources was obtained. The influence of various factors on the brightness increment distribution at the medium surface from a source shaped as a cylinder is studied. These include the source depth in the medium, the radiation wavelength, the absorption index, the heat transfer parameter, the cylinder height and diameter. The brightness increment from the source and the natural brightness of the skin surface are compared. It is shown how the source brightness changes the full brightness of the surface in a wide spectral interval.

Equations describing the generation of medium-wave and long-wave infrared radiation in SRS are obtained in the situation when the frequency of the first or second Stokes components is smaller than the vibration frequency excited by the SRS process. It is shown that such generation occurs due to two-photon emission and four-photon parametric emission. In many cases, these equations make it possible to relatively simple estimate the efficiency of such generation of IR radiation.

The results of investigation of the temperature and frequency dependences of the dielectric characteristics of microwave ceramics of (1–x)((Mg_0,2Zn_0,8)TiO₃–xCaTiO₃ ((1–x)(MZT)–xCT) (0.1 ≤ x< 0.6) compositions synthesized from a mixture of oxides (the first method) and from a mixture of a pre-prepared solid solution of (Mg_0,2 Zn_0,8)TiO₃ and a CaTiO₃  compounds (the second method), as well as ceramics of these compositions doped at the stage of sintering 1–2 % tin and tungsten are presented. It is shown that the synthesized ceramics is a composite consisting of a mixture of phases formed on the basis of solid solutions of (Zn, Mg)₂ TiO₄ , (Zn, Mg)TiO₃  and the CaTiO₃  compounds, the ratio of which in ceramics depends on the composition of the initial mixture and the synthesis conditions. It is established that the dielectric constant (ε) of ceramics synthe sized from a mixture of oxides increases with increasing CaTiO₃ content in the (1–x)((Mg_0,2Zn_0,8)TiO₃–xCaTiO₃ system. This ceramics is characterized by small values of the temperature coefficient of dielectric constant (TK_ε ) and the dielectric loss tangent (tanδ) in the temperature range of 20–200 °C. For ceramics synthesized according to the second method, the high temperature stability of TK_ε and the small value of tanδ are observed in the temperature range of 20–150 °C. Doping ceramics with tin and tungsten oxides leads to an increase in ε and a decrease in dielectric losses.

Thin transparent conductive zinc oxide films are of interest for application in various fields of science and technology, including anti-icing systems for glasses in aircrafts, in coatings that reduce static electric charge on instrument panels, in electrical contacts to liquid crystals, electrochromic and electroluminescent indicators for displays, development of high-efficiency solar cells. Thin zinc oxide films on anodic aluminium oxide substrates are formed on the porous side and on the barrier layer of γ-aluminium oxide under high-frequency pulsed-periodic laser deposition in vacuum. The morphology of the obtained films was studied by atomic-force microscopy and their differences were noted, depending on the side of the substrate. The optical properties of films in the near-IR region, as well as the features of their photoluminescence characteristics were studied experimentally. The substrates Al₂O₃ –ZnO film as a sensitive layer can be used to design sensors and tandem solar cells.

The article presents the results of study of the effect of radiation defects (RD) and thermally stable (up to 873 K) radiation-thermal defects (RTD), created by electron irradiation with an energy of 4 MeV and heat treatment on the static and dynamic characteristics of silicon thyristor p-n-p-n structures. The dependences of turn-on and turn-off current on the life-time of minority charge carriers at a high injection level (in the range of 1.0–10 mks) are obtained in a thick n-base of structures with RD and RTD defects and the same dependences of the minority charge carriers lifetime at a low injection level. DLTS-spectra of the investigated structures and temperature dependences of control current and control voltage in the temperature range of 77–320 K are presented as well.

The two triple-layered SiO₂ /SiN_x /SiO₂  structures with Si-rich and N-rich silicon nitride active layer were fabricated on p-type Si-substrates by chemical vapour deposition. The SiN_x  layer of different composition (x = 0.9 and x = 1.4) was obtained by changing the ratio of the SiH₂ Cl₂ /NH₃ flow rates during deposition of a silicon nitride active layer (8/1 and 1/8, respectively). The spectroscopic ellipsometry and photoluminescence (PL) measurements showed that the refractive index, the absorbance and luminescence properties depend on a chemical composition of silicon nitride layers. The structures with Si-rich and N-rich SiN_x  active layers emit in the red (1.9 eV) and blue (2.6 eV) spectral ranges, respectively. The PL intensities of different structures are comparable. The rapid thermal annealing results in the intensity decrease and in the PL spectra narrowing in the case of SiN_1,4 active layer, whereas the increase in the emission intensity and the PL spectra broadening are observed in the case of the annealed sample with a SiN_0,9 active layer. The PL origin and the effect of annealing treatment have been discussed, taking into account the band tail mechanism of radiative recombination. Multilayered (SiO₂ /SiN_x )_n /Si structures are of practical interest for creation of effective light sources on the basis of current Si technology.

Nanostructural Ti-Al-C-N coatings were produced by reactive magnetron sputtering at substrate temperatures of 220, 340 and 440 °C using mosaic targets with different Al/Ti ratios. Using atomic-force and scanning electron microscopy, it was found that the variation of the elemental composition leads to a change in the morphology of Ti-Al-C-N coatings: at an Al/Ti ratio of 0.39, the films have a mixed columnar-granular structure with no visible defects and a low roughness (3.30–5.86 nm); at an Al/Ti ratio of ~ 0.96, the films show a porous columnar structure with a higher roughness (8.83–11.07 nm) and for an Al/Ti ratio of ~ 1.71, the films have a fine-grained structure and the smallest roughness values (0.48–1.74 nm). Substrate heating from 220 to 440 °C did not significantly affect the elemental composition of Ti-Al-C-N films, but it affected the deposition rate, surface roughness, and the microstructure of the coatings. MTT-test results showed no relationship between the fibroblasts viability, the coating roughness and the coating elemental composition. However, the cells viability and their ability to proliferate on the Ti-Al-C-N coatings surface were preserved.