In this article, for the so-called canonical form of a difference scheme under usual positivity conditions on the equation coefficients two-sided estimates for the approximate solution are obtained at the arbitrary non sigh-constant input data of the problem. The obtained results are used both for deriving two-sided estimates of monotone difference schemes, which approximate the initial boundary-value problem for the quasi-linear parabolic convection-diffusion equation, and for studying the correctness of the Gamma equation that is used for describing the option price in financial mathematics.

In the article, the analogue of a RSA-cryptosystem in general quadratic unique factorization domains is obtained. A scheme of digital signature on the basis of the generalized RSA cryptosystem is suggested. The analogue of Wiener’s theorem on low private key is obtained. We prove the equivalence of the problems of generalized RSA-modulus factorization and private key search when the domain of all algebraic integer elements of the quadratic field is Euclidean. A method to secure the generalized RSA-cryptosystem of the iterated encryption cracking is proposed.

The effect of high-fluence ion irradiation of nanostructured coatings based on high-entropy compound (Ti, Hf, Zr, V, Nb) N was studied. The irradiation was performed using helium ions with an energy of 500 keV in the fluence range of 5 · 10¹⁶–3 · 10¹⁷ ion/cm². The thermal annealing at 773 K during 15 min was done with the aim of modeling conditions of this material exploitation in a nuclear reactor. The elemental composition, structure, morphology, as well as mechanical properties of (Ti, Hf, Zr, V, Nb) N coatings were investigated before and after the irradiation. According to the results of this study, there are no significant structural or phase changes in coatings after irradiation, except a strong fragmentation of grains to a size less than 10 nm. Also, there are not observed any changes in the elemental composition. The non-linear behavior of coating hardness on irradiation fluence was detected. It can be argued according to the experimental results that nanostructured coatings (Ti, Hf, Zr, V, Nb)N are radiation-resistant and perspective as fuel claddings in nuclear reactors.

Electrical and magnetotransport in arrays of aligned multi-walled carbon nanotubes are investigated. It is shown that for temperatures below 50 K the 2D weak localization model is the most reliable model for approximation of the experimental data. Electrical resistivity, (1.2–1.5) · 10^–3 Ohm · cm, and the temperature dependence of the phase coherence length, L_th ≈ T^{-p / 2} with p = 0.78, are evaluated from the experimental data in the framework of the 2D weak localization model.

The tetrad-based generalized complex formalism by Majorana–Oppenheimer is applied to examine an electromagnetic field in oscillating de Sitter Universe in nonstatic spherically symmetric coordinates. With the help of Wigner D-functions we separate the angular (Θ, φ) -dependence in the complex vector field E _j (x) + iB _j (x) from the (t, r)-dependence. After that, the system of differential equations in (t, r) variables is solved exactly. Relations between the complex 3-vector Majorana–Oppenheimer formalism and the 10-component Duffin–Kemmer–Petiau approach have been examined. On this basis, electromagnetic waves of magnetic and electric types have been constructed in the both formalisms. In the Duffin–Kemmer–Petiau formalism, the class of gradient-type solutions is constructed in Coulomb and Lorentz gauges.

Quasi-stationary states of the V-configuration three-level system in a driving resonant field is considered in the present article. The main goal of the analysis is to define the conditions when the essential impact on spontaneous emission is possible in this system. It is shown that the spontaneous linewidths substantially change if they are calculated outside the framework of the rotating wave approximation.

The iodine PVA films modified with chrysophenine and IR-806 dyes were tested for resistance to exposure by "white" light. It is established that such anisotropic films are lightfast and efficiently polarize in the near UV and IR regions of the spectrum.

Cytotoxicity and the cellular uptake of different-coated fluorescent semiconductor nanoparticles CdSe/ZnS were studied. The cellular uptake of fluorescent semiconductor nanoparticles in vitro depends on the nanoparticle construction (coating composition and to a lesser extent on the core size), the type of cells, and the stage of cell differentiation. Cytotoxicity of fluorescent semiconductor nanoparticles is determined by the coating composition. Non-specific labeling of cells was demonstrated in vitro using fluorescent semiconductor nanoparticles coated with cysteine, and mercaptoundecanoic and thioglycolic acids. Fibronectin-modified fluorescent semiconductor nanoparticles with carboximethyldextran-shell bind by target tumor cells in vitro. Nanoparticles coated with cysteine are suitable for cell visualization in lymph nodes after intraperitoneal injection.

The genetic transformation of Nicotiana tabacum L. plants with Mn-SOD and Fe-SOD Arabidopsis thaliana L. genes leads to the potentiation of the antioxidant system in cellular compartments, which is manifested not only in a SOD activity increase by its additional amount, but also in an increase of protective components activity involved in hydrogen peroxide detoxification.

674 men of the National Academy of Sciences of Belarus were recruited in effort to be inspected both for evaluation of prostate-specific antigen (PSA) in serum blood and for filling a questionnaire. The group of a prostate cancer risk was selected among examined persons, who had a PSA level >4.0 ng/ml, and the PSA free/PSA total level was <15 %. We found a significant causation between lower physical activity, alcohol consumption, cigarette smoking, and prostate cancer risk.

The analysis of the RUNX3 gene methylation status was conducted in the prospective group of 262 bladder cancer patients. In comparison to normal urothelium, a high frequency (63 %) of RUNX3 promoter hypermethylation was observed in tumor tissues. А statistically significant association of RUNX3 epigenetic abnormalities with a more aggressive tumor phenotype – an advanced tumor stage and grade, as well as a large tumor size – was found. We have shown that RUNX3 hypermethylation is an independent risk factor for progression and cancer-specific survival in the group of patients with non-muscle invasive bladder cancer.

Using the karyometry method, quantitative regularities of pathological changes in nuclei of cells of ductal epithelium of pancreas in adenocarcinoma and chronic pancreatitis were revealed. The expert system for diagnosis of these diseases on the basis of a set of threshold values of quantitative parameters of nuclei in the comparison groups and their weight coefficients has been created. The efficiency of the expert system in the preoperative diagnosis of adenocarcinoma and chronic pancreatitis was 100 % and its substantial contribution to the clarification of the diagnosis in the cases with the cytologic conclusion as "suspicion of cancer! was estimated.

The heart of the geological concept of the terrestrial water origin is based on the idea of internal planetary water resources. Initial ("juvenile") water could only be chemically bounded to a solid protoplanetary substance and released as sub-molecular and molecular particles of water at high temperatures (450–1500 °C) and pressures (0.3–≤120 GPa) mainly in upper mantle, asthenosphere and be higher in the crust-mantle layer. The primary surface waters (protoocean) were characterized by high mineralization but they were the sources of fresh water due to natural distillation by means of evaporation from their surface and condensation in the cooling conditions of the planet. The water delivery to the Earth in the composition of asteroid and comet substances is considered as an additional and less significant water supply.

Human aging causes changes in a number of physiological, anthropometric, and thermal performances, in particular in the level of basal metabolism and tissue blood flow, subcutaneous fat thickness, as well as moisture content and heat conduction of the skin. This work is devoted to assessing the impact of the age factor on the patient’s thermal state using whole body gas cryotherapy. It has been established that there is a very strong dependence of the changes in the skin surface temperature of different compartments of 70-year old patients compared with 20-year old ones due to a decrease in skin heat conduction with age.

This communication is a view of a specialist in study of literature and a writer on the state of-the art and tendencies of development of modern Belarusian literature. Analysis is made of the directions and features of this development, the connection with classical literary heritage, the extension of cognitive and thematic scope and artistic searches, the adequacy of literary activity to modern tasks and demands.

The aim of the research is to find out the origin and to obtain the results of statement of a document at the early stages of its existing in the Grand Duchy of Lithuania. The following conclusions are made. The occurrence and statement of a document were related with the development of global social processes: laicization and pragmatisation of social consciousness and culture, and also with the originating of new cultural values. A document was involved in many complex communicative practices – secular, religious, not only written, but also oral ones. A document statement changed the society structure, the frame of the authority, governance and court, the methods of realisation by the authority its basic functions, the material environment of a user of a document, formed a special kind of social consciousness and culture in which the document was one of the main communication tools.

This article is devoted to compiling a dictionary of the living Belarusian language in the 20s of the twentieth century in comparison with the most important Russian, Polish, Czech, Serbian and Ukrainian dictionaries. Special attention is drawn to the concept of compiling an academic dictionary, in which people’s fundamentals have not been separated from the literary, to the territorial (2nd half of XIX – early XX centuries) and to the chronological boundaries of collecting linguistic materials (all ethnographic Belarus), as well as to the features of the organization of lexicographical work: the publication "Instructions", the support on the work of non-professional lexicographers, the edition of regional dictionaries and so on. The sources of vocabulary materials – folk, literary and technical are discussed in detail.

Announcement has shown that the data of hydrolytic and oxidation enzyme activities may be used for biochemical estimation of the intensity of organic substances mineralization in C- and N-cycles and humification in C-cycle in Luvisol soils. The ratio between mineralization and humification intensities indicates more probable trends of soil fertility status changes under different levels of fertilizer application.