We consider the spatial distribution of points, whose coordinates are conjugate algebraic numbers of fixed de- gree and bounded height. In the article the main result of a recent joint work by the author and F. Götze, and D. N. Zaporozhets is extended to the case of arbitrary height functions. We prove an asymptotic formula for the number of such algebraic points lying in a given spatial region. We obtain an explicit expression for the density function of algebraic points under an arbitrary height function.

The problem of finding the Lebesgue measure 𝛍 of the set B₁ of the coverings of the solutions of the inequality, ⎸Px⎹ <Q^−w, w>n , Q ∈ N and Q >1, in integer polynomials P (x) of degree, which doesn’t exceed n and the height H (P) ≤ Q , is one of the main problems in the metric theory of the Diophantine approximation. We have obtained a new bound 𝛍B₁ <c(n)Q^−w+n, n<w<n+1, that is the most powerful to date. Even an ineffective version of this bound allowed V. G. Sprindzuk to solve Mahler’s famous problem.

When generalizing the geometrically nonlinear law of Murnaghan elasticity to plasticity, a formally mathematical criterion was introduced for deformational macrofracture (macrocrack appearance) associated with an increase in elastic and plastic anisotropy as a failure cause. The use of the double potentiality of the governing equations in stresses and their velocities made it possible to obtain the reliable information on the structure of the deviatory section of the yield surface, the existence of which is a classical hypothesis in solid mechanics. The normal vector to the surface of the deviatory section is selected from two mutually orthogonal eigenvectors of the constructed operator. There are two families of regular concave surfaces, and a section surface is formed by joining the parts of two representatives of the families at singular points. To select normal vectors, the obtained ratio for them is used for isotropy. In connection with the considered problem of a double simple shift, it is established that multiple eigenvalues appear for the both normal vectors. To unambiguously determine the normal vector at a regular point, it is necessary to exclude the presence of multiple eigenvalues for the both normal vectors at the same time. At a singular point, the appearance of a multiple eigenvalue of one of the normal vectors is still unacceptable. These two conditions are necessary and sufficient to validate the governing equations of the generalized Murnaghan model. Otherwise, a macrocrack occurs. The theoretical construction is supported by the developed software complexes.

A comprehensive study of the crystal structure, magnetic and magnetotransport properties of the La_0.5Sr_0.5Co_1–x Ni_x O_3–d  cobaltite system (x = 0.1–0.16) was carried out. The X-ray measurement results indicate that the unit cell of all solid solutions of the system is cubic and is described by the space group Pm3m. It is found that with an increase in the 540     Doklady of the National Academy of Sciences of Belarus, 2021, vol. 65, no. 5, рр. 539–545 Ni content, the Curie temperature (TC) decreases from 230 to 180 K, as well as magnetization values. The magnetic transition is blurred across the field. The iodometric studies show that the concentration of Co⁴⁺ ions in all samples does not exceed 35 %. The chemical substitution of Co ions by Ni ones does not result in significant modification of the unit cell parameters, which may indicate a spin crossover of Co ions. The temperature dependence of resistivity is metallic in character, which indicates the stability of the main conducting ferromagnetic phase. The nature of exchange interactions of different signs between B-sublattice ions completely determines the behavior of the system. An increase in the content of Ni ions leads both to decrease the component of ferromagnetic exchange interactions between Co³⁺ ions in the intermediate spin state and to increase the fraction of antiferromagnetic and weaker ferromagnetic interactions. In addition, presumably the Co⁴⁺ ion can stabilize the high spin state of the closestCo³⁺ ion and in the next two coordination spheres it can stabilize the Co3+ ion in the low spin state, i. e. the ferromagnetic complexes Co⁴⁺–Co³⁺ (HS) are shielded by the diamagnetic shell of low spin Co3+ ions, which results in decreasing the magnetization values.

 The article discusses the issues of inhibition of spontaneous emission of molecules by using silicon spherical nanoparticles and dimers made from them. It is shown that at different wavelengths of the visible spectral range, the value of the total spontaneous transitions rate in a molecule located at an optimal distance with respect to the structure with silicon nanospheres and at an optimal size of the structure can be up to 5–10 times lower than the transition rate in the case when the nanoparticles are absent.

 A serious drawback of reflective optics is a center without central screening that degrades the image quality. To eliminate it, rotations or displacements of mirrors are introduced, but there appear even-order non-elementary aberrations that must be corrected. The creation of compositions with decentered catoptric elements requires further development of the calculation and methodological base. The exact formulas are obtained for calculation of real rays from the astigmatism and coma correction conditions for the given angles of incidence of the main ray on the mirror surfaces and the “oblique” thickness d, that determines their mutual position. Based on the proposed formulas, a new method for parametric calculation of decentered mirror systems has been created, which allows one to compose algorithms and to design both basic models and complex mirror systems from off-axis mirrors. The development of new algorithms for two- and three-mirror off-center lenses will increase the accumulated potential of computational optics. The scope of the proposed technique can be expanded in terms of the number of components.

The method for synthesis of N-pentofuranosyl oxazolines was developed from the protected 1,2-O-acetonides of D-xylofuranose, -ribofuranose, and -arabinofuranose using boron trifluoride diethyl etherate, acetonitrile, and potassium hydrogen difluoride. A possible mechanism of the catalyzed reaction of acylated acetonides with acetonitrile in the presence of Lewis acid was considered in terms of the activation and cleavage of the 1,3-dioxalane part of the xylose derivative fol- lowed by the conversions of intermediates to α-isooxazoline. The hydrolysis reactions of N-α-glycosyl oxazolines were stud- ied in the acidic and neutral conditions. N-α-xylofuranosyl acetamide derivatives were prepared in high yields as a result of selective hydrolysis of protected α-xylofuranosyl isooxazolines in the neutral conditions.

The analysis of the trajectories of molecular dynamics simulation and spatial structures of homologous models of fusion protein with various linkers was performed to understand the effect of the additional DNA-binding domain of the E. coli SSB protein attached to the truncated and native bovine DNA exotransferase on its stability and activity. It is found that the C-terminus of the enzyme is the preferred end for attachment of the E. coli protein, while the stability of the truncated fusion enzyme is higher than the native one. According to molecular dynamics data, introducing linkers between two proteins for the native (GGGGSGGGSGGGGS, GGGSGGGS, and TCT) and truncated (GGSGGGSGG, GGGGGG, GTGSGT, and 5xGGGGS) forms of the enzyme not only improves its stability, but also increases the mutual mobility of DNA-affinity domains.

The analysis of the trajectories of molecular dynamics On the basis of natural kaolin from the “Dedovka” deposit (Gomel region), aluminosilicate nanotubes 800– 1100 nm in length and 50–60 nm in diameter were obtained for the first time. The synthesis was carried out according to the developed method under relatively mild conditions at a temperature of 60–66 °C, atmospheric pressure, and a low (7.0 mL/g) amount of a reagent (methanol) required for their formation.

Based on the bioinformatic and statistical analysis of the GEO-projects to determine the genome-wide profile of human DNA methylation, a list of 27 CpG dinucleotides with a high predictive potential was formed to create models for prediction of the human age from blood samples. The methylation level was determined for 245 samples of individuals from the Republic of Belarus. The correlation coefficients R were calculated, and the mathematical models for determining the age of an individual were constructed. The average accuracy value of the age prediction from blood samples using 12 CpG-dinucleotides was 3.4 years (for men – 3.3, for women – 3.5). The results obtained will be used as a basis for development of calculators for predicting the age of an individual based on the biological traces for forensic experts.

 COVID-19 has become a planetary emergency which is seriously threatening human health. Comparative studies can shed light on the molecular mechanisms of the formation of humoral antiviral immunity in persons who have undergone the disease and have been vaccinated against the SARS-CoV-2 virus. Serum levels of biomarkers of inflammation and cytokine storm (procalcitonin, sCD14, sLBP, sTREM1, IL-6, IL-8, IL-10, sWnt5a, TNFα and TNFβ) were determined by enzyme immunoassay. We studied the relationship between the content of biomarkers in blood serum with low and high concentrations of antiviral antibodies, indirect indicators of which are the values of the positivity coefficient (CP). A significant increase in the serum levels of sTREM1, IL-6, and procalcitonin was found in persons with a high CP level after infection. A significant increase in IL-6 and PCT was revealed in persons vaccinated with the Sputnik V vaccine, both with a low level of CP and with a high one. In vaccinated individuals with a low CP level, a significant increase in the sTREM1 content was also observed. It has been shown that the content of sCD14 (presepsin) is significantly lower in vaccinated individuals, both with low and high CP, compared with donors. When studying the correlations between CP and serum biomarkers in vaccinated individuals, a weak inverse correlation was found for sLBP (r = –0.465) and IL-6 (r = –0.437), as well as an average inverse correlation for sTREM1 (r = –0.508).

Hypovitaminosis D in both adults and children is detected with high frequency in various geographic zones of our planet. At the same time, vitamin D has a number of positive effects on human health and is an important micronutrient for prevention of some diseases. The purpose of this study is to identify the incidence of vitamin D deficiency and insufficiency in the adult population of the Republic of Belarus. We analyzed the laboratory results on total vitamin D that was produced in 2019 and 2020 in persons over 18 years old living in the Republic of Belarus. The vitamin D status was assessed according to international guidelines. The average level of vitamin D in the surveyed population during the study period did not reach normal values in all age groups in both men and women. The highest frequency of hypovitaminosis D was observed in the autumn-winter period and reached 81.6 % in January 2019 and 77.8 % in January 2020. In the summer months, normal levels of vitamin D were observed in less than 50 % of those surveyed: the maximum values were 40.7 % in July 2019 and 45.8 % in July 2020. Due to the high prevalence of hypovitaminosis D, it is advisable to carry out measures for prevention and treatment of DVD in the residents of the Republic of Belarus at the population level. 

A new direction of system research in the Pripyat Trough to increase the hydrocarbon resource potential of the bowels on the basis of modern geological and geophysical technologies has been justified. Two-type priority objects, which are oil and gas prospecting and research polygons, are distinguished. The first-type object is represented by large oil multifield and prohibitive structures in the highly promising areas and sections. The second-type object includes the structures of the western part of the Central Zone of the Pripyat Trough where a rather high level of direct oil and gas occurrences is established.

The initial phase of a temperature growth in different parts of the Earth has been examined. It is found that climate warming in the continental areas began 6–8 years earlier than in the oceanic areas. Climate warming in the sub-polar cyclonic circulation area in the North Atlantic has been started only after the beginning of the positive quasi-30-year phase of the Atlantic Multidecadal Oscillation (AMO), i. e. since 2000s. On the basis of the Singular Spectrum Analysis of the air temperature time series the continental areas are found, in which summer air temperature changes are synchronous with AMO on a decadal scale. In those regions, climate warming began with a 10–15 year lag in comparison to other continental areas. The highest level of statistical significance of the correlation between quasi-60-year AMO components and summer air temperatures is observed in Central and Eastern Europe (35–60° N, 10–45° E). This oscillation explains the decadal summer temperature changes within the range of ±(0.5–0.6) °C in Belarus. Climate warming intensifies in summer during the ascending AMO segment, while during the descending segment, which has started since around 2010, the decrease in the rate of growth of summer temperatures should be expected.

It is shown that the use of transfer criteria for analysis of the formation processes of structures and phases significantly reduces the amount of experimental research in the design of technological process and equipment using fields and energy flows. It is proposed to use the ratios proportional to the criteria of heat and mass transfer, physicochemical interaction when controlling energy sources. It is shown that the modeling of the combined electromagnetic and thermomechanical effects in the design of technological process and equipment is based on the criteria of heat and mass transfer, electrical and physicochemical interaction, determines the main technological factors, and relates geometric and physical parameters, as well as mechanical quality parameters with the processing performance.

The literary heritage of F. M. Dostoevsky is a rich research material that traces the philosophical and sociological aspect, expressed in the author’s reasoning about the existence of man, his place in the world and society, their interaction through the prism of dichotomy. F.M. Dostoevsky known as a deep and paradoxical thinker. In his works, he asked philosophical questions about man, the relationship between rational and unreasonable principles, his place in the world and society. Dostoevsky F. M. worried about the moral greatness of man and his extreme forms, criticism of social violence and utopian attempts to improve and make people and society happy. The leitmotif of the characters in the novels is the possibility of a dignified, happy life in various social realities.