The boundary-value problem for analytical functions is investigated. The boundary condition is placed on a closed curve located on the complex plane. The problem belongs to the type of the generalized Riemann boundary-value problems. The boundary condition contains derivatives of the required functions. The problem is reduced to the usual Riemann problem and linear differential equations. The solution is built in closed form. The application of the solved problem to integro-differential equations is indicated.

The introduction presents the main results of previously known papers on Jackson’s singular integral in polynomial and rational cases. Next, we introduce Jackson’s singular integral on the interval [–1, 1] with the kernel obtained by one system of rational Chebyshev–Markov fractions and establish its basic approximative properties: a theorem on uniform convergence of a sequence of Jackson’s singular integrals for an even function is obtained, and conditions are specified that the parameter must satisfy in order for uniform convergence to take place; the approximative properties of sequences of Jackson’s singular integrals on classes of functions satisfying on the interval [–1, 1] the condition of Lipschitz class with constant M. are investigated. The obtained estimates are asymptotically exact as n → ∞; an estimate of deviation of Jackson’s rational singular integral from the function |x|^s, 0 < s < 2 depending on the position of the point on the segment, a uniform estimate of the deviation on the segment [–1, 1] and its asymptotics are found. The optimal value of the parameter is obtained, for which the deviation error of the studied approximation apparatus from the function |x|^s, 0 < s < 2 on the interval [–1, 1] has the highest rate of zero; the order of approximation of the function |x| on the interval [–1, 1] byJackson’s considered singular integral is found. It is shown that with a special choice of the parameter, the velocity of the approximation error tending to zero is higher in comparison with the polynomial case. All results of this article are new. The work is both theoretical and applied. It is possible to apply the results to solve specific problems of computational mathematics and to read special courses at mathematical faculties.

A graph is called K_1,p-restricted (p ≥ 3) if for every vertex of the graph there are at least p - 2 edges between any p neighbours of the vertex. In this article, new sufficient conditions for existence of a perfect matching in K_1,p-restricted graphs are established. In particular, J. Petersen’s classical result that every 2-edge connected 3-regular graph contains
a perfect matching follows from these conditions.

Applied to solving practical problems of extremely accurate laser surgery; the present study examines the conditions of deep, controlled, low-temperature dissection of biological tissues by quasi-continuous laser radiation of the infrared spectrum based on forming a directional wave of environmental enlightment.

The Si layers doped with Te up to the concentrations of (3–5)1020 cm–3 have been formed via ion implantation and pulsed laser melting. It is found, 70–90 % of the embedded impurity atoms are in substitution states in the silicon lattice. These layers have revealed significant absorption (35–66 %) in the wavelength λ range of 1100–2500 nm. In this case, the absorption coefficient increases with the λ growth. The absorption spectra of the implanted layers after pulsed laser melting, equilibrium furnace annealing, and rapid thermal annealing have been compared. It is shown that equilibrium furnace annealing increases the photon absorption by 4 % in the wavelength range of 1100–2500 nm in comparison with virgin Si. After rapid thermal annealing, the photon absorption in the IR-range increases only by 2 %.

In this work, the effect of brassinosteroids on the antitumor activity of classical cytostatic cisplatin in tumor cell lines A549 (human lung carcinoma) and Hep G2 (human hepatocellular carcinoma) was evaluated. Natural brassinosteroids 24-epibrassinolide and 28-homocastasterone, as well as their synthetic analogues (22S,23S)-24-epibrassinolide and (22S,23S)-28-homocastasterone were used. All four compounds with cisplatin inhibited the growth of cancer cells more effectively than cisplatin alone. Combinations with low concentrations of synthetic brassinosteroids were more effecient, and at 1 µM decreased the IC₅₀ of cisplatin by almost 2 times. The results suggest a possible benefit of combinations of classical antitumor drugs with brassinosteroids in overcoming the negative effects of chemotherapy by reducing their effective doses.

Six potential peptidomimetics of the cross-reactive neutralizing anti-HIV-1 antibody N6 that are able to mimic the pharmacophoric features of this immunoglobulin by specific and effective interactions with the CD4-binding site of the viral gp120 protein were identified by virtual screening and molecular modeling. The key role in the interaction of these compounds with gp120 is shown to play multiple van der Waals contacts with conserved residues of the gp120 Phe43 cavity critical for the HIV binding to cellular receptor CD4, as well as hydrogen bonds with Asp-368_gp120 that increase the chemical affinity without activating unwanted allosteric effect. According to the data of molecular dynamics, the complexes of the identified ligands with gp120 are energetically stable and show the lower values of binding free energy compared with the HIV-1 inhibitors NBD-11021 and DMJ-II-121 used in the calculations as a positive control. The identified compounds may be involved in the design of novel antiviral drugs presenting HIV-1 inhibitors that block the early stages of the development of HIV infection.

The adhesion and proliferation of rat adipose-derived mesenchymal stem cells (MSCs) as part of biocomposite with haemostatic drug Spongostan Absorbable Haemostatic Gelatin Powder as well as its regenerative efficiency after its introduction into the periurethral region in rats with experimental urinary incontinence were studied. It was found that MSCs rapidly adhere to the surface of the carrier and actively proliferate. The periurethral injection of biocomposite (Spongostan + MSCs) quickly and effectively recovers the urine continence function in animals that have undergone modeling of incontinence by urethrolysis. Spongostan hydrated in the ratio of 1 :24 inphosphate-saline buffer has a volume-forming effect in the urethra of experimental animals.

The study assessed the genetic diversity of commercial varieties and feral populations of oilseed rape (Brassica napus L.) grown in the Republic of Belarus according to the genotyping data of 7 microsatellite loci – Na12D08, Ol12D04-1, Ol12D04-2, Ra2A05, Na10H03, Na14H11, Ol11B05. The following parameters were calculated: the average number of alleles per locus, the effective number of alleles, the levels of expected and observed heterozygosity, % of polymorphic loci. The results indicate a greater genetic diversity in feral oilseed rape populations. According to the cluster analysis, one feral population fell into a cluster of commercial varieties and demonstrated similarities with the variety Atora, which indicates its recent wildness and the preservation of the genetic characteristics inherent in cultivated varieties. The variety Mercedes was clustered together with samples of feral populations, which may indicate its hybrid origin. An analysis of the structure of the genotype distribution in the Structure software showed the division into three clusters – commercial varieties, feral populations and samples of B. rapa. The established genetic divergence between feral populations and commercial varieties indicates that feral oilseed rape is able to maintain persistent populations inBelarus. In practice, this should be taken into account while assessing the environmental risk when transgenic rape is released into the environment. When cultivating transgenic rapeseed, special attention should be paid to measures to prevent the occurrence of its free-growing populations.

The aim of the study is to improve the results of complex treatment of patients with gingival recession on the
basis of validation of new, highly effective methods using autologous mesenchymal stem cells. A complex treatment was conducted in 93 patients with an anatomical recession of the gum in combination with dentoalveolar anomalies at the age of 20 to 29 years. Patients were divided into 3 groups depending on the treatment used. All patients underwent periodontological treatment. In the 2nd group, patients underwent periodontological treatment including orthodontic equipment, in the 3^rd group, they underwent orthodontic treatment, and a cell transplant based on a mixture of autologous mesenchymal stem cells and sterile bioplastic collagen material “Collost” was used. A comprehensive assessment of the periodontal tissue of the three groups showed that among the patients who received the standard treatment complex, a good condition of the periodontal tissues in 1 to 2 years after treatment was noted only in 16.7 % of the patients of the 1st group. Inclusion of orthodontic treatment made it possible to obtain good results in 25 % of patients in the 2nd group, which is 8 % more than in the 1st group. At the same time, good treatment results were revealed in 94 % of the 3rd group patients, who underwent periodontological and orthodontic treatment using autologous mesenchymal stem cells, which confirms the need for measures aimed to normalize the position of teeth, dentition and bite with the use of small orthodontic long-acting forces, to improve the microcirculation of periodontal tissues, and to restore the contour of marginal dreams. Inclusion of autologous mesenchymal stem cells in the complex treatment of patients with gingival recession is accompanied by a 1.6-fold decrease in the prevalence and gum recession (1.15 mm), and a 2.6-fold increase in the tissue volume, by the stimulation of periodontal microcirculation processes (the capillary periodontal pressure is increased by a factor of 1.1 time and the intensity of periodontic microcirculation is increased by a factor of 3.1 times), as well as by the normalization of the position of teeth, dentition and occlusion, which allowed one to obtain good long-term results in 94 % of patients.

On the basic of consideration of the heat-conduction problem for a semi-bounded space with a temperature profile defined by a parabola with an exponent n, a new concept of construction of constitutive involves the introduction of a local function for a heat flow or for the temperature, with is determined from the heat-conduction equation. The approach proposed made it possible to obtain a number of new integral relation: an improved integral for the temperature momentum, an integral of a quadratic heat flow, and an integral of a quadratic temperature function. Two Schemes of optimizing the exponent n with the use of the error norms E1 and are proposed. As compared to the Langford norm, the indicated error norms made it possible to substantially increase the approximation accuracy of solutions of the problem posed.

The development of mathematical models for design of the thermal technology equipment intended for accelerated hydration of concrete products is an urgent task of industrial heat power engineering. The introduction of mathematical modeling methods can reduce the time and the resources spent for the development of technological regimes that reduce energy consumption in the production of building structures and, ultimately, in the construction of buildings and structures for various purposes. There is not yet a generally accepted mathematical model of thermal processes occurring in thermal technological installations in the case of accelerated hardening of a three-dimensional concrete object. The purpose of this work is to develop an unsteady model of the concrete hydration process applied to a symmetrical three-dimensional reinforced concrete product at software-controlled heating. By using the numerical finite-volume method in the case of a 0.3 × 0.3 ×0.3 mcube, a three-dimensional feld of hydration in a concrete object at a given operation mode of a heater is calculated. The following heating mode was used: “heating–maintaining at a constant temperature–cooling”. The dependences of a temperature difference between reinforced and non-reinforced cubic products on the hardening time at the corresponding space points in the direction from the surface to the center of the product have been obtained. By the example of the numerical simulation results, it is shown that the evolution of the hydration degree at these points during the hardening of non-reinforced concrete differs from the hardening of reinforced concrete. The time dependences of heat treatment of a rate of temperature change and the hydration coefficient at the selected points of a product are presented. The obtained results are analyzed.

The article presents the results of research, showing that the clusters in Western science have not yet gained the unity of understanding; they are attributed to completely different functions and aims. But there are many definitions and similarities. Thus, the organizational and economic base for creation of any cluster organizations is the methodology of cooperation and integration of various business entities, regardless of their feld of activity.