In 1950 J. Massera proved that a fi rst-order scalar periodic ordinary differential equation has no strongly ira proved that a first-order scalar periodic ordinary differential equation has no strongly irregular periodic solutions, that is, such solutions whose period of solution is incommensurable with the period of equation. For difference equations with discrete time, strong irregularity means that the period of the equation and the period of its solution are relatively prime numbers. It is known that in the case of discrete equations, the above result of J. Massera has no complete analog.

The purpose of this article is to investigate the possibility to realize Massera’s theorem for certain classes of difference equations. To do this, we consider the class of linear difference equations. It is proved that a first-order linear homogeneous non-stationary periodic discrete equation has no strongly irregular non-stationary periodic solutions.

The magnetic properties of perovskite EuMn0.5Co0.5O3, obtained under different synthesis conditions, have been studied. It is shown that the sample obtained at 1500 °C exhibits a metamagnetic transition, whereas in the sample obtained at 1200 °C, the field dependence of magnetization as in a ferromagnet was observed. The both samples have TC = 123 K and approximately equal to magnetization in large magnetic fields. Metamagnetism is associated with a transition from a noncollinear to a collinear magnetic phase, whereas the presence of clusters with the ordering of Co2+ and Mn4+ ions leads to ferromagnetism. The noncollinear phase is formed due to the competition of positive Co2+–Mn4+ and negative Mn4+–Mn4+, Co2+–Co2+ interactions and due to large magnetic anisotropy.

For a Dirac particle, the general mathematical study of the particle tunneling process through an effective potential barrier generated by the Schwarzschild black hole background is done. The study is based on the use of 8 Frobenius’ solutions of the related second-order differential equation with 3 regular and 2 irregular singularities of the rank 2. Solutions of the radial equations are constructed in explicit form, and the convergence of the involved power series is proved in the physical range f the variable (1, ). r∈ +∞ Results for the tunneling effect are significantly different for two situations: one when the particle falls on the barrier from the inside and another when the particle falls from the outside. The mathematical structure of the derived asymptotic relations is exact, however the analytical expressions for the involved convergent powers series are unknown, and a further study of penetration and reflection coefficients should be based on the numerical summation of the power series.

Computer-aided design of the high-affinity inhibitors of aromatase based on 1,2,4-triazole derivatives was performed by molecular modeling tools. The potential biological activity of the designed compounds was evaluated by molecular docking and quantum chemistry calculations. As a result, six hits that form a coordinate bond with an iron atom of an enzyme hem and effectively interact with its substrate-binding site were identified. The intermolecular interactions appearing in the structural complexes of these ligands with aromatase were analyzed and the enthalpies of their formation were calculated. Based on the data obtained, the identified compounds were suggested to present good scaffolds for the development of novel effective drugs against breast cancer.

A number of 1,4- and 1,4,5-triazoles have been synthesized using a modified catalytic system for the Huisgen [3+2] cycloaddition reaction. The bactericidal activity of the synthesized compounds has been studied. The synthesized derivatives of 1H-1,2,3-triazoles have demonstrated higher antibacterial activities in relation to the pathogen Staphylococcus aureus strain which can be comparable with the clinically used antibiotic “Cefotaxime”.

The wettability of composite coatings based on polyvinyl alcohol (PVA) and silicon dioxide formed on silicon by the spin coating method from PVA colloid solutions of SiO2 nanoparticles was studied. These coatings modified with hydrolysed heptadecafluorotetrahydrodecyltrimethoxysilane are found to exhibit superhydrophobic and oleophobic properties. It was found that PVA increases the wear stability of SiO2–PVA coatings.

We estimated the seston elemental composition (C, N, P) and its ratio in pelagic and littoral zones of mesotrophic shallow Lake Obsterno during two contrasting seasons, as well as the influence of macrophyte beds on the seston stoichiometry. In the both pelagial and littoral zones in summer the C: N ratio was small, 4.62 and 7.05 respectively. But it increased slightly during autumn to 5.66 in pelagic samples against 8.33 in littoral ones. The large N : P ratio and the small phosphorus content specially in the macrophyte covered littoral zone equal to 57.7 in July against 22.47 in September suggest a high level of phosphorus limitation in the littoral locations as a possible mediated reason suppressing zooplankton abundance in sumabundance in sum in summer. Our results in the both pelagial and littoral habitat showed a highly P limited situation in which the N : P ratio was larger in littoral with macrophyte than in pelagial zones. The obtained data of littoral seston stoichiometry were recorded for the first time and exceeded the classical Redfield ratio. The elemental imbalance between macrophyte covered littoral and pelagial suggest that nutrients, especially P, are more limiting in macrophyte beds in summer due to the resource competition between phytoplankton and macrophytes for nutrients, a poor food quality, low zooplankton abundance, as well as its poor taxon-specific elemental ratio in summer.

The results of treatment of 290 patients with chronic post-traumatic instability of the shoulder joint treated for the period 2004–2017 are studied. In accordance with the task, the authors identified 3 clinical variants of post-traumatic instability depending on the prevailing direction of the dislocation. Taking into account the revealed variants, the authors studied their morphogenesis and developed a differentiated tactic and ways of their elimination. The proposed approaches based on the identification and elimination of damages of the rotary-bicepital complex, as well as the developed methods of surgical treatment made it possible to improve the subjective feelings of patients on the OOP scale from 56 [48; 58] to 24 [12; 28] (Me [Q25; Q75]) and indicators of objective evaluation of the function on the Rowe scale from 15 [15; 55] to 75 [75; 100] points.

To accurately predict the tumor behavior and individualize the treatment approach, new methods for bladder cancer (BC) prognosis are required. The most promising prognostic markers are the mutational and epigenetic changes of genes involved in maintaining cellular homeostasis. In the present study, we evaluated the influence of p16 promoter hypermethylation on the risk of recurrence, progression and disease outcome in the group of 158 BC patients. p16 epigenetic changes were found in 11.4 % of urothelial carcinomas and did not depend on clinicоmorphological characteristics. However, in the subgroup of patients with non-muscle invasive tumors, p16 abnormal methylation was significantly associated with smoking, and in the subgroup of patients with muscle-invasive BC, it was linked to a high tumor grade (G3). In the multivariate Cox regression analysis, p16 promoter hypermethylation was an independent predictor for bladder cancer progression (HR 6.84; 95 % CI 1.6–29.9; р = 0.011). The use of the data on the p16 methylation status may improve the accuracy of prognosis of the bladder cancer clinical course and the selection of appropriate treatment strategy.

The schemes of modern endogenous, exogenous and technogenic geological processes, intensity, frequency, and the area of their spread, the caused disturbances of natural and man-made complexes, an impact on public health, the level of proposed costs for rehabilitation of territories were used for the analysis of the possible consequences of realization of nature and nature-anthropogeneous hazards of lithosphere class. The zoning of the region with the allocation of areas with low, moderate, increased, high, very high and very high in the zones of active faults, the degrees of adverse consequences of development of extreme types of modern geodynamics was carried out using the mentioned data.

Lower Famenian inter-salt deposits (Belarus) have been studied comprehensively. The microlithofacies are as follows: laminated organic-carbonate-silica rock; massive limestone; radiolarite; algal limestone; massive dolomite; porous dolomite; dolomite of indistinct lumpy texture. Rocks contain pores of dissolved radiolarians, cracks of fluid auto-fracturing
and weakened zones encountered in the secondary transformations. Rocks of Domanic type in the Central oil area of the Pripyat can be oil-bearing and oil-generating ones.

The concentration of 16 polycyclic aromatic hydrocarbons (PAHs) in the gasoline combustion products emitted into the atmosphere by internal combustion engines (ICE) has been measured using the gas chromatography method. The concentrations of PAHs in the exhaust gases sampled behind a catalytic converter has been determined when the ICE operates in five modes: idle mode, high speed mode, load mode, ICE cold start mode (engine warm-up) and transient mode. Using 92 RON, 95 RON and 98 RON gasoline the effect of the octane number of gasoline on the PAHs content in the exhaust gases
has been revealed. The concentration of the most carcinogenic component (benzo(α)pyrene) in the exhaust gases behind a catalytic converter significantly exceeds a reference value of benzo(α)pyrene in the atmospheric air established by the WHO
and the EU for ICE in the load mode.

Quality and reliability of integrated circuits to a great extent depend on the surface condition of silicon wafers. In view of this, great attention is paid on the aspects of their preparation prior to their formation. It is of significant interest to study the possibility of applying rapid thermal treatment for solid phase re-crystallization of a mechanically disrupted layer of the wafer working side. The objective of this work was to establish the behavior regularities of a mechanically disrupted layer subjected to rapid thermal treatment with 2 s light pulses. As samples, there were used the silicon wafers with a diameter
of 100 mm, grade KDB 12 and KEF 4.5, orientation <100> after chemical-mechanical polishing subjected to rapid thermal treatment during 7 s, which ensured their heating up to 1100 °C and without treatment. The application of the methods of Auger-spectroscopy, spectral ellipsometry, X-ray diffraction made it possible to state that such treatment increases the structural flawlessness of the surface layer of silicon wafers due to a decrease in the mechanically disrupted layer, thus ensuring obtaining the atomic-flat surface.

A model for the process of glass microsphere production in a recuperative gas-flame reactor was proposed. Based on the described mathematical model of heating and motion of particles in a high-temperature gas stream, which takes into account conjugate heat exchange between the reactor’s operating environment and the recuperator, the appropriate processes were modeled and optimized by geometric and regime parameters. The particle location time in the reactor at a temperature above 1400 °С, which was determined by data of differential scanning colorimetry, was used as an optimized charac- С, which was determined by data of differential scanning colorimetry, was used as an optimized charac- , which was determined by data of differential scanning colorimetry, was used as an optimized characteristic.
As a result of optimization calculations, the reactor parameters (diameter and height, natural gas flow rate, air flow rate in the recuperator) were found, as well as regime parameters (diameter and flow rate of glass particles), under which microspheres can be formed. The information obtained can be a basis for designing an effective gas-flame reactor for production
of glass microspheres.

Within the framework of the sociological approach, the definition of “civilizational code” is considered, the place and role of its main factors in the system of Belarusian society are determined. Among them is a special place occupied by historical memory as one of the most important elements of civilization. The work reveals the essential features and structural elements of historical memory, its role in preserving and strengthening the socio-spiritual foundations of being, a possible use of historical experience in the activities of people or its return to the sphere of social consciousness. By historical memory are understood the selection, reproduction, and transfer of social experience, which is important and necessary to preserve accumulated knowledge and experience, their transfer from generation to generation, as the perception of history by mass consciousness as a set of ideas about the past existing in society both at a mass and individual level, including cognitive, imaginative, and emotional aspects. It is shown that on the turning points of history, at a transition of society from one qualitative state to another, historical memory can be used as an instrument of ideological struggle and purposeful influence
on mass consciousness and filled with a specific information content at the request of various social groups and elites.

The article presents the results of comparison of the biochemical composition of the muscles of two-year carp of two-way crosses and the breeds and lines of different-origin carp and Amur sazan. The combinations characterized by an increased level of the dry basis content, fat and protein are established in comparison with a mean-population value, a mean level of the Belarusian collection lines, import breeds and Amur sazan bred under the same conditions. The mean mass of two-year carp selected for investigation was 531.3 g within the range from 370.2 to 645.0 g mainly with a low and mean degree ofvariation. The mean level of the dry basis content of all examined crosses was 25.85 %, the maximum value of this parameter of German × three prim was 27.92 %, and the minimum value of Germany ×Lahvin scaleless was 24.08 %. The low values of the variation coefficient (1.0–9.8 %) point to a slight variation of this parameter of different mixed bred crosses.