In this article we present a local classification of three-dimensional reductive homogeneous spaces allowing a normal connection. We have concerned the case of the unsolvable Lie group of transformations with a solvable stabilizer. We describe all invariant affine connections together with their curvature and torsion tensors, canonical connections and natural torsion-free connections. We have studied the holonomy algebras of homogeneous spaces and have found when the invariant connection is normal.

This article deals with the necessary and sufficient conditions for the operator B, ()1,Bρ= under which the Neumann series converges strongly and on the basis of these conditions, some of the error estimates for the corresponding successive approximations are presented.

The Chebyshev–Markov rational fractions have a number of remarkable properties and are one of the main and, certainly, important elements in the theory of approximation of functions. The Chebyshev–Markov fractions are the integrant apparatus for creation of interpolation rational functions and quadrature formulas. However in the context of the orthogonal Fourier series they are not used as generally they have no property of orthogonality.
The present article considers the system of the Chebyshev–Markov rational fractions with a special choice of the parameters for its definition. In the first part of the present article, the elements of the system are created, some of their representations are specified, and it is proved that there is a weight, at which the studied system is orthogonal on a piece [–1, 1]. In the second part of the work, the Dirichlet integral is created. In the third part of the present article, the coefficients of the Fourier series expansion of the function |x| in the considered system are found in explicit form. In the fourth part, the estimate of the function |x| by means of the partial sums of its Fourier series is investigated. In particular, its accuracy is proved. In the closing part, the asymptotic estimate of the approximation by the partial sums on a piece is obtained as a whole and when the approximation is carried out outside a singular point. Precise constants of these estimates are found.

In this article we consider weighted shift operators generated by irrational rotation. The descreption of the norm of the powers of those operators depending on the properties of the cofficients of the mentioned operators and on the arithmeticals properties of the irrational number yielding an angle of rotation is given.

We derive self-reciprocal twice-relativistic model of one-particle classical dynamics of spatially localized gravitating mass on the basis of Hamilton formalism in complexified extended 8-dimensional phase space taking into account Hibbons’ limit. Mass of particle, being varied in a finite interval, is a unique free parameter of the model. Exact spherically-symmetric solution of the model represents a pulsating massive ball with magnitudes of oscillations in x- and p-space and their frequency defined by the mass, that is connected by a universal relation to a corresponding action. The model has correct Newtonian limit and demonstrates classic analog of Schredinger’s Zitterbewegung. Canonic quantization of the model allows interpretation of self-reciprocal Born operator as quantum operator with eigenvalues of multipes of Planck mass squared. It leads to a model of Dirac oscillator for a fermion with Planck mass.

Colloid-chemical properties of water solutions of compositions of surfactants of different nature are studied. The synergetic effects for the compositions of cationic/anionic surfactants and cationic/non-ionized surfactants at micellization and adsorption are found. It has been found that the maximal synergism of action, showing up in a decrease of surface tension, is offered by the composition of cationic/anionic surfactants in the component 4 : 1. This fact stipulates the efficiency of its dispersive action and its capacity of forming on the surface of particles the dispersible phase of oil slime of the structured layer, possessing maximum elasticity and mechanical strength, as compared to the molecules of cationic and non-ionized surfactants. Water dispersions of oil slime obtained with the use of the composition of cationic and anionic surfactants are the aggregative and kinetically stabilized dispersions of direct type.

The correlation between the structure and phase state of the solutions of polyphenylsulfone (PPSU) and polysulfone (PSF) blends and the transport properties of ultrafiltration membranes, prepared from these solutions, is revealed  : pure water flux of the PPSU/PSF blend membranes passes through the maximum at the most heterogeneous structure of polymer solutions at the PPSU  : PSF blend ratio (30–50) : (70–50). In this study, the maximum pure water flux achieves 80–87  l ∙ · m–2 ∙ · h–1, which significantly exceeds the data reported in the literature up-to-date.

New 6-fluorothymine 2′-deoxy-2′-fluoro-D-arabinofuranosyl nucleosides were prepared by the silyl method starting from persilylated 6-fluorothymine and 3,5-di-O-benzoyl-2-deoxy-2-fluoro-α-D-arabinofuranosyl bromide. A mixture of benzoylated N(1)-β- and α-anomeric 6-fluorothymine 2′-fluorodeoxy arabinonucleosides was obtained by refluxing in CHCl3 with
a 34  % yield. 6-Substituted (OMe, NH2) thymine 2′-deoxy-2′-fluoro-D-arabinonucleosides were prepared by the treatment of individually protected N(1)-α/β-D-arabinosides with methanolic ammonia. It is shown that mild deprotection of the benzoyl groups of intermediate 6-fluorothymine β-nucleoside using LiOH monohydrate in a mixture of acetonitrile-water resulted in the target nucleoside in good yields (82  %). An approach to the synthesis of 2′-fluoro-6,3′-O-α-D-anhydronucleosides was developed as a result of the intramolecular substitution reaction of the fluorine atom at the C(6)-position of the heterocycle by the C(3′)-hydroxyl group of an intermediate deprotected nucleoside during the removal of the protective groups of 6-fluorothymine N(1)-α-arabinonucleoside under the basic reaction conditions. The structures of all synthesized nucleosides were proved by UV-, NMR-, CD- and mass-spectroscopy.

The survival and fecundity of Daphnia longispina were studied in life-table experiments under different Chlorella sp. concentrations (0.08, 0.18 and 0.36 mg/L). The survivorship-weighted fecundity decreased at low concentration of food, but two or more maxima were found at higher concentrations. The “reproductive cost” hypothesis was tested using the data of life-table experiments. It was shown that this hypothesis is confirmed only for low food concentration.

The data on the influence of a high summer temperature on the condition of the population of a rare and protected species – a relic calanoid copepod Limnocalanus macrurus are presented for the first time. It is shown that an increase in surface temperature leads to a decrease in oxygen concentration in the hypolimnion in medium-deep lakes of Belarus and catastrophically reduces the density of the relict, which can lead to its complete extinction.

The analysis of a full nucleotide sequence of bacteriophage Pf-10 as a key constituent of biopesticide “Multiphage” has revealed that its unique genome is composed of a DNA fragment of broad range host phage Phi-S1, where the determinants governing the synthesis of early proteins are localized, and of the DNA fragment of narrow range host phage phiIBB-PF7A containing the genes responsible for the synthesis of late proteins. Low homology of individual genetic determinants and encoded amino acid sequences (namely, the genes determining the synthesis of tail proteins) with those of phages Phi-S1 or phiIBB-PF7A evidences the mutations that emerge in the course of the phage Pf-10 genome formation and are capable to affect its vital important functions.

The DNA ploidy and molecular biological characteristics of adenocarcinoma of sigmoid colon by flow cytometry were studied. In 71  % of the patients, diploid was detected and in 29  % of the patients – aneuploid tumors. In the diploid tumor profile in the distribution of cell cycle phases, significant differences are revealed in the content of cells in G0 / G1 and S phases of the cell cycle, as compared to the aneuploid tumor profile. There is the increased DNA – the synthetic and proliferative activity of tumor cells compared to patients with the aneuploid tumor profile. Patients with the diploid tumor profile have a number of cells in the S phase 5 times more than in patients with the aneuploid tumor profile. It may be assumed that in patients with aneuploid sigmoid cancer, the recurrence rate is higher and the survival rate is lower than in those with the diploid tumor profile.

The change in the radiation situation in the territory of Belarus has been assessed. It was shown that the radiation dose values connected with the Chernobyl accident were decreased and the relative value of the radon factor was increased. It was shown that these radiation factors have different trends in the territory of the Republic. There is a large irregular radon distribution in the territory of Belarus. The difference of the radon dose values in many regions of Belarus was more than in the absolute value of the Chernobyl caesium-137 dose. It is necessary that the radon influence on people must be taken into account
in the studies of biological and medical effects of radiation.

The liquid capture by a moving surface is the most widespread process in chemical engineering along with calendaring, extrusion moulding, pouring, and pressure moulding. The theoretical analysis of the medium capture by a moving surface, which allows revealing the fundamental physical principles and mechanisms of the process over the entire withdrawal speed range realized in practice, was performed for Newtonian, non-Newtonian, and viscoplastic liquids. However, such an analysis of the withdrawal of viscoplastic liquids with a finite yield was not made because of the features of these liquids. Shear flow of viscoplastic liquid is possible only after the stress exceeds its yield. This fact causes serious mathematical difficulties in stating and solving the problem. In the proposed work, such a theory is being developed for viscoplastic liquids.

When the dimension of the problem is high, the Kalman filter becomes difficult to realize in real time due to computational costs. Alternatively, the technique of synthesis of the filter on the basis of the recursive least squares method with solution regularization is considered. The technique allows reducing the computational costs, but it increases the variance of the filtering error. By the example, the extent of this increase is shown and the recommendations for selection of the regularization coefficient are given.

On the basis of direct integration of the generalized equation of heat transfer in a semi-bounded space, the sequences of identical integral equalities defining the features of a differential equation and boundary conditions were obtained for the first time. On the basis of power polynomials with an exponential factor, this made it possible to construct approximate solutions with high convergence. The error in determining parameters over a wide range is hundredths and thousandths of a percent.

Only females are used in technology of caviar aquaculture, and the males shall be rejected as soon as possible. It is believed that sturgeons have no clearly defined external morphological sex character. Because of the dark color of most sturgeons, the external morphological character, the structure and form of which are sex-dependent, have not been noticed earlier. The aim of our work was to study the morphological structure of the derivatives of the corium (dorsal scutes) of different-age sterlet and to identify its sex-dependent structure. The object of this study was the broodstock of 3-year sterlet of the Volga population, with the gonads of the second stage of maturity, as well as the three-month larva and year-old juvenile sterlet. Cultivation was carried out in the re-circulating aquaculture system at an average water temperature of 16 ± 2  °C. Wintering was not used. We were the first in the practice of ichthyology and aquaculture who found that dorsal scutes of sterlet maturing had significant sex-dependent morphological differences. To assess the morphological structure of the dorsal scutes, it was proposed to define two groups of indicators characterizing the scutes shape, as well as the structure of the cusps of scutes. We found that dorsal scutes of males are more extended in width and have a flattened shape. Dorsal scutes of males have long and thin cusps, the number of which is larger than that of females. For the first time it was found that the established dependence on the structure of dorsal scutes of sterlet is saved as in juveniles with an average length of 24.8  ±   1.5 cm and larvae with an average length of 70.3  ± 3.6 mm. Our results form a methodological basis for global aquaculture practice in effort to develop systems of early and very early sex identification of all Acipenseridae, including the early stages of ontogeny.