The present paper is devoted to constructing second-order monotone difference schemes for two-dimensional quasi-linear parabolic equation with mixed derivatives. Two-sided estimates of the solution of specific difference schemes for the original problem are obtained, which are fully consistent with similar estimates of the solution of the differential problem, and the a priori estimate in the uniform norm of C is proved. The estimates obtained are used to prove the convergence of difference schemes in the grid norm of L₂.

The present article considers one-parameter families of second-order linear differential systems with a coefficient matrix depending on the real parameter, which is a diagonal matrix at each odd time interval of unit length. The Cauchy matrix is the rotation matrix at each odd time interval, whereas the angle is the sum of a parameter value and some real number. Earlier, it has been has proved that the upper Lyapunov exponent of each such a system, which is considered to be the function of parameter, is positive on the set of the positive Lebesque measure if the diagonal part of the coefficient matrix is independent on a parameter and separated from zero. The proof of this result essentially uses a complex matrix of special type. In recent article, the author has given another way to prove this theorem based on implementing the Parseval equality for trygonometric sums. Besides, the author considers the special case of the above systems. Now the diagonal part of the coefficient matrix is time-independent and is sufficiently big, whereas the rotation angle is defined by a maximum degree of two that divides the number of the corresponding time interval. For such a system, in the case of a continious coefficient dependence on a parameter it is proved that such a value exists, at which the corresponding system is unstable.

In this article it is shown that by performing Levi–Chivita-type transformations in the two-dimensional Helmholtz and Klein–Fock-type equations, it is possible to determine coherent states in a standard way. Moreover, if in the case of the Helmholtz elliptic equation the Levi–Civita transformation is realized by a complex quadratic map, then in the case of the Klein–Foсk-type equation it is realized by an analogue of such a map however defined for functions of a double variable. The coordinate and momentum representations of the coherent state are found. The purpose of constructing coherent states in the described manner is a further development of the hadron model proposed in [1; 2].

The wave equation for a spin 3/2 particle, described by 16-component vector-bispinor, is investigated in spherical coordinates. In the frame of the Pauli–Fierz approach, the complete equation is split into the main equation and two additional constraints, algebraic and differential. The solutions are constructed, on which 4 operators are diagonalized: energy, square and third projection of the total angular momentum, and spatial reflection, these correspond to quantum numbers {ε, j, m, P}. After separating the variables, we have derived the radial system of 8 first-order equations and 4 additional constraints. Solutions of the radial equations are constructed as linear combinations of the Bessel functions. With the use of the known properties of the Bessel functions, the system of differential equations is transformed to the form of purely algebraic equations with respect to three quantities a₁, a₂, a₃. Its solutions may be chosen in various ways by solving the simple linear equation A₁a₁ + A₂a₂ + A₃a₃ = 0 where the coefficients A_i are expressed trough the quantum numbers ε, j. Two most simple and symmetric solutions have been chosen. Thus, at fixed quantum numbers {ε, j, m, P} there exists double-degeneration of the quantum states.

A synthetic scheme for obtaining cyclopentenone synthons for metabolically stable 11-deoxy-3-oxa-3,7-inter-m-phenylene prostaglandin analogues has been developed. The key step of the scheme is the Knoevenagel condensation of cyclopentane-1,3-dione with the readily available 3-(formylphenoxy)acetic acid methyl ester in the presence of Hantzsh ester.

Polymer analogous transformations of a “nitron” fiber were established and a new amino carboxylic fibrous cation exchanger FIBAN X-2 was obtained. It is shown that FIBAN X-2 is an effective sorbent of heavy and non-ferrous metals from multi-ionic aqueous solutions, regardless of the method of fiber amination. The cation exchanger obtained by amination of a “nitron” fiber in the vapor phase absorbs Mn⁺² twice as much as the ion exchanger obtained by amination in the aqueous solution.

In laboratory experiments, salicylates 24-epibrassinolide, 24-epicastasterone and the first synthesized 6-deoxo-24-epicastasterone salicylate act as inducers of plant immunity under biotic stress on the model phytopathosystem of barley-phytopathogenic fungus Helminthosporium teres Sacc. In small-scale field experiments, it was shown that the treatment of plants with brassinosteroid salicylates has a stimulating effect on the formation of protective physiological and biochemical reactions of plants. The most active protective effect exhibited salicylate 24-epibrassinolide.

The sequencing and annotation of the curly birch chloroplast genome were carried out. A high level of similarity of the structural and functional organization of cpDNA among the species of the Betulaceae family was revealed. A set of primers was developed to assess the level of expression of EST markers of the curly birch cpDNA by the real time PCR method.

Еffects of β-1,3-glucan from euglena (Euglena gracilis) were studied in vivo by evaluating pH changes outside and inside the cell in the tissues of the 7-day leaves of barley seedlings with the use of pH-sensitive probes of FITC-dextran and pyranine. It was found that the incubation of barley leaves separated from the roots in the solution of β-1,3-glucan (0.01 %) for 40 min did not cause acidification of cytoplasm as a typical nonspecific plant response to wound stress. The inoculation of intact seedlings with Bipolaris sorokiniana (Sacc.) Shoem. spores resulted in alkalization of apoplast by 1.7 pH units, and pretreatment of seedlings with β-1,3-glucan a day before the fungal infection promoted its acidification (1.04 pH units) compared to the infected variant, indicating an increased performance of ATPase, which pumped protons from cytoplasm into apoplast. The conducted studies contribute to the selection of optimal concentrations of β-1,3-glucan for immunomodulatory mixtures.

The studies on 32 lakes of the National Park “Narochansky” were aimed both at studying quantitative and qualitative characteristics of the autotrophic component of plankton and at finding their correlation with the data of spectral analysis of pigment composition. On the basis of the chlorophyll content in the summer season (July–August), the main part of the sampled lakes can be attributed to the mesotrophic type, 2 lakes – oligotrophic, 10 – eutrophic. The content of chlorophyll-ɑ in the lakes ranged from 0.3 to 55.2 μg/l, seston – from 0.3 to 29.8 mg/l. Pigment indices, reflecting the state of planktonic producers, did not change significantly depending on the trophic state of the lake. However, in more trophic waters, the absorption spectra are written more clearly; accordingly, the indices appear to be more reliable and informative.

The present work investigates the action of food additives: sodium glutamate E-621 (720 mg/kg), sodium benzoate E-211 (30 mg/kg), tartrazine E-102 (45 mg/kg) on the modulation of the heart rate (СR) and the activity of smooth muscles of the stomach and the colon in acute experiments on rats. Narcosis – thiopental sodium, 70 mg/kg is made intraperitoneally. It is established that the unitary administration of each of the additives into the stomach separately or everywhere results in precise effects of increase or decrease of total potentials of smooth muscles (PSM) without CR changes. After introducing glutamate into the stomach intravenously infusion adrenaline (Adr, 10 mkg) causes CR to increase more significantly, if all additives are introduced into the stomach, and then Adr. PSMs after the application of Adr were often more stimulating. Later, a subsequent significant influence on the effects of long use of additives in a diet is established. As it has appeared, the application of adrenaline in finishing acute experiment (after a month of feeding by additives) is accompanied not by amplification(strengthening), and, on the contrary, by the disorder of the heart work: CR gradually falls, arrhythmia arises, fibrillations of a myocardium and then cardiac contractions stopped in 12 of 16 experiments. These processes are associated with decreas ing the background activity on an electrocardiogram of the average area of RST (in the control 149 mkV/s, in a month of experience 80 mkV/s, р < 0.05) and waves Т (82 and 43 mkV/s, respectively, р < 0.01) without significant changes of their duration. The increase or decrease of the RST area is seen and Т is marked at the action of special ADR on the stomach additives. It is supposed that a long use of food additives serves as the precondition for dysfunctions in the heart work in the situations connected to a fast increase in the blood of catecholamines – in the conditions of various stresses, adverse influences of factors of the environment.

An analysis of changes in the CD44 and MMP9 expression during the keratopathy after corneal inflammatory diseases treatment with injected hyaluronic acid is presented.

Following the decision of the International Union of Geological Sciences on the transfer of the lower boundary of the Quaternary System/Period and the Pleistocene Series/Epoch from 1.8 Ma to 2.58 Ma (to the base of the Gelazian Stage/ Age of the Pliocene), the changes have been made in Stratigraphic charts of Neogene and Quaternary Deposits of Belarus (2010). The Neogene – Quaternary boundary in the territory of Belarus is aligned with the International Chronostratigraphic Chart and is made between the Kholmech horizon (analogue of Zanclean and Piacenzian, Pliocene) and the Dvorets horizon (analogue of Gelasian) displaced from the Pliocene to the lower base of the lowermost Pleistocene. Its new position in the sections is substantiated by paleobotanical data. New geological units were introduced into regional and local stratigraphic charts of the Neogene and Quaternary.

The results on the effect of introduction of iron aluminide of various chemical and phase compositions on the structure and mechanical properties of powdered carbon steel and tin bronze are presented. It is shown that the introduction of 0.5 % single-phase iron aluminide Fe₃Al leads to an increase in the strength of powdered carbon steel by 30–40 MPa, of biphase Fe₂Al₅ –FeAl₃ – by 80–90 MPa, 1 % – to an insignificant decrease in strength. When a single-phase iron aluminide in the powder steel structure is introduced, a decrease in cementite, differentiation is observed, aluminum diffusion into the substrate occurs, and when two-phase aluminide is introduced, the structure griding occurs as well. It is established that the introduction of 0.5 % single-phase iron aluminide into powder bronzes makes it possible to increase its strength by 80– 100 MPa, two-phase – leads to a reduction in strength by 40–50 MPa. Introduction of 1 % single-phase iron aluminide and 0.2–1 % biphasic aluminide causes a change in the morphology of the structure of the powder bronze due to alloying the copper with aluminum and iron.

It is shown that in principle, it is possible to obtain bischofite of high purity by converting bischofite lyes purified from heavy metals, iron and sulfate ions into an insoluble form (magnesium oxide, basic magnesium salts) using their hydrohydrolysis method, washing water-soluble impurities and synthesizing bischofite from a purified thermohydrolysis product and salt acid.

The shortcomings of the traditional description of mechanical engineering product are shown, which are based on technological features oriented to single, typical and group operations of the technological process. It was proposed to represent the design of product and its parts with a structured set of relevant technological modules, in the form of a graph of hierarchical structure. The numerical characteristics of the structure of the graph of product, its levels, nodes, branches are considered. The description of the structures of products by a hierarchical graph at the first level makes it possible to identify the functional technological modules of products and, on their basis, to construct a unified classification and to suggest coding products, both for production objects and operation. Representation of parts by a set of modules allows identifying surface modules, working and binding, and on their basis to build a single classification of parts focused on different stages of the product life cycle. The presence of a unified methodological base allows managing the development of constructively complex products, minimizing duplication in the creation of new designs and effectively developing resource-saving technologies for their manufacture.