In this paper, Lax’s equivalence theorem, which states that stability is a necessary and sufficient condition for its convergence in the presence of an approximation of a difference scheme, is generalized to abstract nonlinear difference problems with operators acting in finite dimensional Banach spaces. In contrast to linear finite-difference methods, such a criterion in the nonlinear case can be established only for unconditionally stable computational methods, when the corresponding a priori estimates take place for sufficiently small |h| ≤ h₀. In this case, the value of h₀ depends both on the consistency of discrete and continuous norms in Banach spaces, and on the magnitude of the perturbation of the input data of the problem. The proven convergence criterion is used to study the stability of difference schemes approximating quasilinear parabolic equations with nonlinearities of unbounded growth with respect to the initial data.

In the metric theory of Diophantine approximations, one of the main problems leading to exact characteristics in the classifications of Mahler and Koksma is to estimate the Lebesgue measure of the points x ∈ B ⊂ I from the interval I such as the inequality | P (x) | < Q^-w, w > n, Q >1 for the polynomials P(x) ∈ Z[x], deg P ≤ n, H(P) ≤Q is satisfied. The methods of obtaining estimates are different at different intervals of w change. In this article, at w > n +1 we get the estimate µ B< c₁(n) Q ^– (^w-1/_n). The best estimate to date was c₂(n) Q ^–(^{w- n}/_n).

The problem of radar detection of small-sized targets using the traditional methods of selection of signals embedded in background noise is considered. It is shown that for a false alarm rate of 10^–5, which provides for 1–2 false alarms within the entire coverage of the modern 3D radar, the probability of detection of a small-sized target is getting unacceptably low. Repeatedly decreasing the threshold can provide an acceptable level of the detection probability at ultra-low signal-tonoise ratio (SNR) values. At the same time, decreasing the threshold will result in an unacceptable increase of the false alarm rate. A new target detection procedure using the “track before detect” method (TBD) is proposed. In the TBD procedure, the target is considered detected when two conditions are met: the signal exceeds once a definite threshold; the target is detected within a strictly defined observation area (acquisition or tracking gate). For low SNR values in the range of 3–8 dB and equal false alarm rate, the detection probability increases by 20–50 % compared to the traditional detection method. The simulation results showed a strong dependence of efficacy of the TBD algorithm on the threshold value and the decision rule. The possibility is noted of adaptive control over the threshold due to the use the detection results in the preceding scanning cycles, as well as the introduction of matrix radar surveillance not only by the target coordinates and parameters, but also by the detection threshold, decision rules, etc. Examination of these issues is the subject of further research.

Nanostructured Ti_xCr_1–xN coatings of various compositions 0.58 ≤ x ≤ 0.8 on the substrates made of AISI 304 stainless steel and monocrystalline silicon were formed by the method of separable vacuum-arc deposition. The elemental composition was studied by Rutherford backscattering spectrometry of helium ions. The structural-phase state and the morphology were examined by X-ray diffraction, optical and scanning electron microscopy, tribomechanical tests of the initial coatings were also carried out. The radiation tolerance of the nanostructured Ti_xCr_1–xN coatings within 0.58 ≤ x ≤ 0.8 under He⁺ ion irradiation with an energy of 500 keV in the fluence range of 5·10¹⁶–3·10¹⁷ ions/cm² was studied for the first time. It was found that the Ti_xCr_1–xN coatings within 0.58 ≤ x ≤ 0.8 withstand irradiation without significant changes in the structure up to a fluence of 2·1017 ions/cm², when a partial coating flaking (exfoliation) up to a depth of the projected range of helium ions takes place. A decrease in the average size of crystallites of coatings and the crystal lattice period reduction after radiation exposure were detected. The decrease in the microhardness of the Ti_xCr_1–xN coatings of all compositions after irradiation was found.

A comparison of the capabilities of two methods for extraction-chromatographic determination of narcotic drugs and psychotropic substances in various objects (syrups, ointments, tablets, herbal mixtures, etc.), the screening methanol method commonly used in the practice of customs and forensic examination and the proposed extraction method based on the use of water phase (in particular, water solutions of salts) for preliminary effective separation of analytes from matrix components was made. In extraction systems of hexane–water, hexane–aqueous solutions of inorganic salts (sodium chloride, dipotassium phosphate and potassium carbonate), chloroform–water, the method of gas chromatography – mass spectrometry was used to determine the distribution constants (P) of organic bases – narcotic drugs and psychotropic substances (N-methylephedrine, methamphetamine, amphetamine, methadone, dihydrocodeine, hydrocodone, oxycodone, ketamine, cocaine zolpidem, fentanyl, harmine, harmaline). Based on the obtained extraction characteristics of analytes and matrix components, the optimal extractants and extraction conditions, which ensure the maximum efficiency of separation of analytes and matrix components, as well as the concentration of the analyte were substantiated. The use of extraction using water and aqueous solutions of salts provides both a sharp decrease in the detection limit (from 10^–3 to 10^–7 mol/dm³) in comparison with the methanol method and stable working of chromatographic equipment due to the removal of thermally unstable matrix components was shown.

Composite membranes for pervaporation were prepared by forming a selective layer based on cross-linked polyvinyl alcohol (PVA) on the porous membrane-substrate surface in the dynamic mode (via PVA solution ultrafiltration). It was found that the pressure growth results in increasing the thickness of the composite membrane selective layer. Composite membrane contact angle, flux, water content in permeate in ethanol/water (mass ratio 90/10) pervaporation were revealed to have maximum values at 3–4 atm depending on the PVA concentration in the feed solution. It was shown that the revealed dependence of the contact angle, selectivity, and permeability on the pressure of the selective layer formation is due to the compaction of the polymer matrix-substrate under the action of the transmembrane pressure and its relaxation after pressure release. When using elevated pressures (more than 3–4 atm), the relaxation of the polymer matrix causes the microdefect to form as a result of deformation of the selective layer.

Norway spruce (Picea abies (L.) Karst.) is one of the main forest-forming species in Belarus. It plays important economic, ecological and social roles. The spruce forest of the region is characterized by a complex history of the formation of its population genetic structure. The aim of this study was the genogeographic analysis of P. abies populations and the description of regional features of its gene pool in Belarus. Molecular genetic analysis of microsatellite (SSR) loci of chloroplast DNA and mt15-D02 locus of mitochondrial DNA of Norway spruce was carried out for samples from 57 naturally originated forest stands. We identified 19 allelic variants of the Pt63718, Pt26081, Pt71936 cpDNA loci and two allelic variants of mt15-D02 mtDNA. The geographical distribution of the alleles has been described and the regional features of the genogeographic differentiation of the spruce forests have been considered. The southern border of the continuous distribution area of P. abies and its island localities lie in the south of Belarus (Brest and Gomel regions). The frequency of occurrence of individual allelic variants of cpDNA SSR loci has the most pronounced deviations from the average values for the whole country in those regions. Analysis of cpDNA showed the presence of certain regional features of the genogeographic structure of the spruce forest in the “south-north” and “west-east” directions. According to mitochondrial DNA analysis higher concentration of P. abies trees of southern (Carpathian) origin is observed in the southwest of Belarus. On the contrary northern (Boreal) origin dominates in the rest of the country. The results of performed genogeographic analysis of Norway spruce populations can serve as a basis for improving the forest seed zoning of the tree species.

Engineering a three-dimensional scaffold opens up great prospects for creation of manufacturing biological artificial organs. The article presents a method of perfusion decellularization of a rat liver, with the main problems and options for their solution being analyzed. Perfusion of a donor liver with 0.1 % a sodium dodecyl sulfate (SDS) solution allows obtaining a high-quality cell-free matrix characterized by preserved hepatic architectonics, patent vascular bed, residual DNA of less than 1 %, no signs of collagen fibers destruction and tissue edema. The obtained scaffold can be used for recellularization by allogeneic cell cultures when creating volumetric tissue-engineered designs.

The sensitivity of expression at the level of transcription of genes encoding proteins involved in calcium signal transduction to gravistimulation was revealed using real-time RT-PCR. For three genes SCA2, РВР2, САМ2, the increase in the transcript formation was shown at early response stages, starting from 15–60 minute gravistimulus. The treatment of plants before the start of gravistimulation with an ethephon (source of exogenous ethylene) led to a change in the modulation of expression of the studied genes in response to gravistimulus. The role of calcium metabolism in realization of final steps of gravitropism reaction is considered.

A comparative analysis of three single nucleotide variations of the proteasomal genes PSMA3 (rs2348071) and PSMA6 (rs2277460 and rs1048990) was carried out in the groups of children from 1 to 16 years old with the autoimmune rheumatic disease - juvenile idiopathic arthritis (JIA; n = 199), with articular syndrome of non-autoimmune etiology (n = 229) and in the clinical control group with neither autoimmune nor chronic inflammatory diseases (n = 379). PCR, PCR–RFLP and real-time PCR were used for genotyping. It was found that the CG genotype and G allele of rs10489990 polymorphism (OR = = 1.93; 95 % CI 1.29-2.90; p = 0.002 and OR = 1.51; 95 % CI 1.11-2.04; p = 0.008 respectively), as well as the AA genotype of rs2348071 polymorphism (OR = 1.89; 95 % CI 1.02–3.49; p = 0.044) are associated with the JIA susceptibility, but not with articular syndrome. The established JIA risk genotypes may indicate the involvement of PSMA3 and PSMA6 genes in the development of an autoimmune reaction. In combination with other risk DNA markers, they can be proposed to assess a genetic predisposition to JIA. It was also revealed that the frequencies of risk genotypes and alleles for JIA in the group of patients with articular syndrome as a whole occupy an intermediate position between JIA and control group frequencies. This may indicate an increased JIA risk in some patients with articular syndrome.

The anti-tumor properties of common lichens of Belarus: Hypogymnia physodes, Ramalina pollinaria, Cladonia arbuscula, Evernia prunastri were studied using the models of spontaneous and induced carcinogenesis in mice Af lungs. Without exposing to a carcinogen in 20 weeks after taking aqueous lichen extracts (1 g/l) of 4 species for 28 days, only C. arbuscula showed the pro-carcinogenic properties almost doubling the average number of pulmonary adenomas per mouse. R. pollinaria and E. prunastri did not provoke spontaneous carcinogenesis and had a corrective effect significantly reducing the number of adenomas per mouse by 21 and 45 %, respectively. To carry out induced carcinogenesis, after 2 weeks of taking aqueous extracts, mice were injected intraperitoneally with urethane (1 g/kg). The analysis of the total number of adenomas per mouse showed a pro-tumor tendency for C. arbuscula, and the analysis of the number of adenomas of 1 mm and more showed that for H. physodes. The extracts do not affect the proportion of urethane adenomas of various sizes in their total number, but increase the incidence of high-yield adenomas of various sizes; at the same time, H. physodes has a much greater pro-carcinogenic activity according to the incidence of high-yield adenomas of 1 mm or more. The analysis of the total number of urethane adenomas per mouse, as well as adenomas of 1 mm showed an anti-tumor tendency for R. pollinaria and a pro-tumor tendency for E. prunastri. Taking E. prunastri increased the incidence of high-yield adenomas of various sizes, along with a notable increase in the incidence of low-yield adenomas. In the case of R. pollinaria, the increase in the incidence of low-yield adenomas was supplemented by a sharp decrease in the incidence of high-yield adenomas. As a result of the experiment, E. prunastri has exhibited both the pro-carcinogenic activity and the anti-tumor properties, but R. pollinaria has revealed a pronounced anti-tumor potential.

The geological structure and the physical-mechanical properties of rocks composing and overlapping the Third potash formation in the areas of the mine field mine No. 4 JSC Belaruskali, which are scheduled to practice, are studied. The geological sections of the Third potash layer and its roof were built, the thickness of sylvinite and halite layers was measured, the roof falling of the rocks of the layer with a height of up to 20 m was assessed, and the type of immediate roof was determined by the composition and structure of its constituent rocks. It was established that the strength under uniaxial compression of both sylvinite and halite layers for the areas under consideration differs slightly. Therefore, to solve engineering problems of ensuring the stability of mine workings and to calculate fastening parameters, strength values averaged over the groups of layers can be used that correlate well with the percentage of weak and clay interlayers. The same applies to Poisson’s ratio, the values of which for various layers of formation are slightly different and can be taken by their averaging. It is shown that the main factor that negatively affects the strength of rocks occurring in the roof of the Third potash seam is the presence of weak and clay interlayers, which increases with depth. The introduction of the obtained results allows us to improve the calculation accuracy of securing mining parameters, to reduce the costs of their maintenance, and to improve the safety of mining operations in developing the Third potash formation in complex geological and mining conditions.

The first results of study of stable isotopes of oxygen (δ¹⁸O) and hydrogen (δD) in the snow samples taken on the islands of Marguerite Bay (Antarctic Peninsula), in the Vecherny Oasis (Enderby Land), and Larsemann Hills (Princess Elizabeth Land) by the participants of the 12^th Belarusian Antarctic Expedition (January–March 2020) are presented. The concentration of water isotopes: deuterium (D) and oxygen-18 (¹⁸O) in the samples was determined using a laser isotope composition analyzer Picarro L2130. A total of 32 snow samples were analyzed. The statistical parameters of the isotopic composition of snow were estimated, and the main differences in the content of δ¹⁸O and δD between the study areas were shown. A decrease in the content of heavy oxygen and hydrogen isotopes in the newly fallen snow to the old snow of the surface horizons is shown. The maximum values of δ¹⁸O and δD are typical for the Maritime Antarctica, decreasing towards the coastal zone and further – towards its continental part. The possible factors affecting the isotope content are described. It is shown that the monitoring of the isotope composition can be an integral part of the monitoring of climatic changes within the area of operation of the Belarusian Antarctic Expedition. The study of the isotopic composition of surface snow is important for the reconstruction of the paleoclimate of the marginal zone of the Antarctic ice sheet based on the ice cores study.

The main methods of classification and coding in mechanical engineering are studied from the systemic point of view, their main characteristics are considered. When analyzing the systemology of classification of machines of various functional purposes, the absence of a unified methodological approach in constructing the classification and in coding the products is shown. In the existing classifications, products are considered constructively only as objects of operation, and the other stages of their life cycle are not affected. As a result of the system analysis, a unified methodological approach is proposed for constructing classifications, coding and unification of products for various functional purposes, reflecting the features of all stages of their life cycle. For the production stages of the life cycle, a system of classifications using various coding methods is considered: a description of structural and technological elements formed by the tool edges, a sequence of numbers, taking into account signs; a description of the surfaces formed by tool movements, in numbers, indicating the directions of feed movements; and to describe complex-profile surfaces and a combination of long-range elements, their combination is used. The expediency of using each of the classifications in real production conditions is determined by the statistics of use of structural and technological elements and surfaces, as well as of tools that form them.