When does a homogeneous space allow an invariant affine connection? If at least one invariant connection exists, then the space is isotropy-faithful, but the isotropy-faithfulness is insufficient for the space in order to have invariant connections. If a homogeneous space is reductive, then the space allows an invariant connection. The purpose of the work is to describe invariant affine connections on three-dimensional non-reductive homogeneous spaces together with their curvature and torsion tensors, holonomy algebras. We concerned the case, when the Lie group of transformations is unsolvable and a stabilizer is solvable. The basic notions, such as an isotropy-faithful pair, a reductive space, an affine connection, curvature and torsion tensors, holonomy algebra are defined. A local description of three-dimensional non-reductive homogeneous spaces with an unsolvable Lie group of transformations and a solvable stabilizer, allowing affine connections, is given. A local classification of homogeneous spaces is equivalent to that of the effective pairs of the Lie algebras. All invariant affine connections on those spaces are described, curvature and torsion tensors, holonomy algebras are found. Studies are based on the use of properties of the Lie algebras, Lie groups and homogeneous spaces. They are mainly local in character. 

The necessary and sufficient conditions, under which a linear almost periodic differential system has partially irregular almost periodic solutions in the critical resonance case, were obtained. 

Sequential statistical tests for simple hypotheses on parameters of probability distributions of independent observations, as well as of Markov chains are considered in the article. Methods for analysis of performance characteristics (I and II type error probabilities, conditional expected sample sizes) of sequential statistical tests are constructed both on the basis of the approximations of test statistics and on the basis of absorbing Markov chain theory. The proposed methods allow assessing the performance characteristics of sequential statistical tests not only for the hypothetical model of data, but also under deviations from this model, which can be used for robustness analysis of sequential tests. 

An analogue of the Solovay–Strassen primality test in general quadratic Euclidean domains is obtained. We prove that the obtained primality test allows us to prove that N is composite and has a probability of no less than 0.5 in polynomial time with respect to a bit size of the number N. The main part of the Solovay-Strassen primality test analogue is the calculation of the Jacobi symbol. The efficient algorithm for its computing in quadratic unique factorization domains is constructed. The main idea of the proof is to reduce the Jacobi symbol to that in integers. 

In this article we prove a conjecture about a spectral radius of a nonnegative matrix. 

Extremal key-point detectors are presented to describe, analyze and compare images by local descriptors that are determined in neighborhoods of the detected key-points. The proposed detectors select key-points, providing local extremal values of the function that characterizes local properties of the original image at each pixel. The majority of commonly used detecting algorithms are looking for key-points in another way. They mark a pixel as a key-point if the value of the functioncriterion at this pixel exceeds a predetermined threshold value. The remaining known algorithms find key-points that are the local extremal values of the functions defined on gradient transforms of the image. One of the drawbacks of the known detectors (in addition to the use of learning procedures expensive in the computational sense) is a non-uniform distribution of key-points on the image. Often large image areas may be left with no key-points, making their detection or recognition impossible. The proposed extremal detectors allow one in many cases to avoid the appearance of image areas not filled with key-points. 

We have studied the formation of ZnSe and ZnS precipitates in silicon dioxide by ion implantation at 550 °C and subsequent annealing at 900 °C for 30 min in the Ar atmosphere. Two sets of samples have been prepared: the first set of samples has been implanted with Se and Zn ions and the second set of samples – with Se, Zn and S ions. The analysis of XTEM images shows that the “hot” implantation leads to the formation of extended layers containing nanoclusters (with size of 20 nm). Subsequent annealing results in a significant structural transformation of layers with nanoclusters. For annealed samples, large crystallites (up to 90 nm) are observed at the depth of maximum impurity concentration, while small clusters are registered in the subsurface layer of silicon dioxide. Raman spectroscopy proves the formation of ZnSe and ZnSe + ZnS crystal phases for deposited and annealed samples of the first and second sets. 

The feasibility of determination of the 0–0-electronic transition ν0 for diffuse absorption, fluorescence and phosphorescence spectra is proposed on the basic principle: direct and reverse transition probabilities between two elemental molecular states are identical. For the starting state being in thermal equilibrium, the transition cross-sections σ obey the relation ( ( ) / )exp( / (2 )) (| |), σ ν ν ν =ϕ ν−ν h kT 0 ϕ is the mirror function about ν0 with «–» and ν>ν0 for absorption, «+» and ν<ν0 for emission respectively. Examples are given of ν0 determined as the extremum of ϕ ν( ) in comparison to the jet-cool spectroscopy data. As a rule, ν0 differs from the maximum of spectra ( ). ν max The width of the area of the extremum of ϕ ν( ) characterizes the inhomogeneity of the spectra.

The spectral properties of the 5,10,15,20-tetrakis-(3-N-methylpyridyl)-porphyrin were studied in the water solution and the temporal evolution of absorption spectra was revealed. The origin of the spectral changes was found to be due to the solution heterogeneity which is based on the coexistence of atropoisomers. These atropoisomers differ in their pyridyl groups, positioning relative to the mean plane of the tetrapyrrolic macrocycle due to an unrestricted rotation of substituents around the Сm–С1 bond. The observed spectral changes are shown to reflect the redistribution of the atropoisomer concentrations after establishing the equilibrium distribution that is over approximately within 300 hours after the preparation of a solution at a temperature of 290 ± 2 K. The atropoisomerization is suggested to be the inherent property of all 5,10,15,20-tetraarylporphyrins with an asymmetric substitution of aryl groups with respect to the axis passing through the Сm–С1 bond. The prospects for the practical applications of the revealed phenomenon are discussed. 

The present work is devoted to the investigation of the interaction of cytochrome P450cam with 4-phenylimidazole (4-PI) using spectrophotometry and NMR spectroscopy. The data obtained by the STD-NMR method indicate the existence of an intermediate short-lived state of 4-phenylimidazole in the active site of P450cam where 4-phenylimidazole is bound to the inner region of the active site and/or to the substrate access channel without the formation of a coordination bond between the ligand azole group atoms and the heme iron atom. In this article, we first used the STD-NMR method to study the interaction of cytochrome P450 with ligand. The equilibrium dissociation constant of the P450cam-4-PI complex, which was calculated using the dependence of the amplification factor at zero saturation time (10.4 mM), differs from the constant that was calculated at constant saturation time (34.6 mM). This fact indicates the dependence of the Kd determination using STD-NMR at saturation time and concentrations of interacting substances. Comparison of the dissociation energy for the intermediate complex (11.2 kJ) relative to the direct coordination complex (28.5 kJ) suggests that the main contribution to the protein-ligand interaction is related to the hydrophobic interaction of 4-PI with the inner surface of the cavity of the active site of P450cam. The observed intermediate state makes it possible to explain the formation of hydroxylated forms of azole inhibitors during the interaction with cytochromes P450, when an inhibitor is in an intermediate form as a substrate and is not bound by a coordination bond with a heme iron atom. 

We were the first who demonstrated the possibility to produce diploid hybrids (along with triploid ones) while pollinating some wild allotetraploid potato species with the pollen of S. tuberosum diploid clones. It was determined that these hybrids only retained one of the genomes of wild species (genome A). Here, we present the results of revealing diploid interspecific hybrids having genome B of wild species. There were selected four hybrids producing the viable pollen among 235 hybrids between valuable for breeding accessions of S. stoloniferum and fertile diploid clone of S. tuberosum. The analysis of meiosis in pollen mother cells showed that they had 24chromosomes. The DNA marker SolB of genome B of wild species was detected in diploid hybrids, as well as in parental clones of S. stoloniferum and triploid interspecific hybrids. The diploid hybrids inherited the typical for wild species markers RYSC 3 and Yes3-3A of Y-virus resistance genes, as well as the markers of late blight resistance genes R1 and R3b. Two of the diploid hybrids inherited markers of late blight resistance gene Rpi-sto1 presented in parental clones of the wild species lossless. The marker 517/1519 of this gene was not detected in one diploid hybrid and both markers (517/1519 and 1521/518) − in another one. Production of the hybrids with genome B makes the version on a possible mechanism of their formation through the selective elimination of one of the genomes of wild species in ontogenesis of initially triploid interspecific hybrids to be priority. 

The article presents the results of a comparative study of the variability of the biochemical composition of the fruits of six cultivars introduced in Belarus: Oxycoccus macrocarpus (Ait.) Pers. – Stevens, Bain Favorit, Hiliston, Holistar Red, Stankovic and WSU 108 during contrast moistening seasons. It has been established that the least pronounced intersectional differences are characterized by the parameters of accumulation in the fruits of dry substances, ascorbic and hydroxycinnamic acids, as well as the total amount of antoniated pigments and bioflavonoids in general, while of most significance are the sugar-dependent concentration and the content of soluble sugars, pectins, leucoanthocyanins and flavonols. It is shown that in the taxonomic series of large-fruited cranberries the Stankovich cultivar was characterized by the greatest resistance of the biochemical composition of the fruits to weather factors, while the lowest cultivar was Bain Favorit and especially Hiliston. 

With the help of a selected collection of molecular genetic markers for resistance to brown, stem and yellow rust genes: Lr1, Lr9, Lr10, Lr19/Sr25, Lr24/Sr24, Lr26/Sr31/Yr9, Lr34/Yr18/Pm38, Lr35/Sr39, Lr37/Sr38/Yr17, Sr2/Yr30, Sr22, Sr26, Sr1RSAmigo, Sr36, Sr45, Yr10, Yr26, a search was made for resistance donors among 23 varieties of spring soft wheat introduced into the State Register of the Republic of Belarus in 2016 and 46 varieties and lines from the collection nursery “Research and Practical Center of the National Academy of Sciences of the Republic of Belarus for Arable Farming”. As a result, resistance genes were identified in spring wheat varieties: Lr1, Lr10, Lr19/Sr25, Lr24/Sr24, Lr26/Sr31/Yr9/Pm8, Lr34/Yr18/Pm38/Sr57, Lr37/ Sr38/Yr17, Sr1RSAmigo and Yr10. Selected varieties of spring wheat with complex resistance to rusty diseases are identified. 

A systemic administration of lactoferrin per os suppressed the development of Ehrlich’s carcinoma by 39.4 %. In this case, positive actions upon the state of antioxidant and immune systems were observed: decrease in the peroxidation of proteins and increase in the antioxidant activity; the restoration of IL-1β and TGF β1 levels with a statistically reliable increase in the blood serum TNF-α, which, as known, suppresses the growth of tumor cells and regulates a number of metabolic processes, and also the activity of immune response. At the same time, a decrease in the blood serum of the content of testosterone and estradiol is shown. The revealed effect of lactoferrin suppression of the development of tumor is apparently associated with a significant (by a factor of 5.28) increase in the blood serum of the iron concentration, which plays an important role in providing regulation systems of the vital activity of tumor cells. 

The article presents the results of a comparative study of the effect of adjuvant radiotherapy on the survival to biochemical recurrence (BCR) in patients undergoing radical prostatectomy (RP) of prostate cancer (PCa) depending on preand post-operative prognosis factors. Adjuvant radiation therapy following prostatectomy in a total focal dose of 60 Gy in patients with locally advanced prostate cancer (pT3a-b) or the localized process with positive surgical margins (rT2R1) and highly differentiated adenocarcinoma (Gleason score equal to 6 or less) according to a postoperative morphological examination significantly increases a 5-year survival to BCR by 25.7 % (p = 0.047) compared to observation. 

The article discusses the data on the long-time effect of low frequency magnetic fields (IF MF 50 Hz, 0.4 mT, 4h/day, 5 days/week, 26 days) on blood cells and spermatozoa of Wistar rats. It was determined that at different times after exposure (3th, 15th and 30th days) the changes in different quantitative and qualitative indicators of test cells were revealed, some of which are kept in 30 days after exposure. 

The present article studies the structural features of relaxation, mechanical and rheological properties of blends of crystallizing PET and amorphous PET-G containing 0.9 wt. % of a diisocyanate chain extender (CE). The analyzed materials were obtained by melt mixing in a twin-screw reactor-mixer with one-sided rotation of screws. It was shown that at a concentration of PET-G in a blend of ≥70 wt. % by weight, the crystallization of PET added with CE is completely suppressed. According to relaxation spectrometry, PET and PET-G are compatible when they are mixed in melt.