The canonical local definition of the hereditary saturated formation X of all finite groups with P-subnormal cyclic primary subgroups was found in the article. The new criterion of the normal subgroup to be X-hypercentral was obtained using this definition. It was shown that the class X is closed under taking the products of P-subnormal subgroups with coprime indices.

In the article, we consider the equation y′′=(λ-(1/k) y′²)(k/y+ 2y/(1-x²-y²)) in the semicircle 1-x²-y²>0, y=y(x)>0, kλ>0 , to which the generalized Toda lattices with a Hamiltonian containing two exponents reduce.

For sufficiently small in absolute value λ, this equation can be replaced by a simpler equation setting λ = 0. It is proved that the latter has one-dimensional symmetry group and reduces to a differential first-order equation, by means of which one can get an arbitrarily accurate description of the general solution of the simplified second-order equation near the boundary of the semicircle.

In this article we introduce and study a proper subclass of triangle graphs, namely, the domination triangle graphs. A graph G is called a domination triangle graph if for every minimal dominating set D of G and any adjacent vertices u and v not contained in D, there exists a vertex w є D such that the set {u, v, w} induces a triangle in G. We propose a number of characterizations of domination triangle graphs which show in particular that this class of graphs coincides with a well-known class of upper bound graphs and a class of irredundance triangle graphs. We establish the computational complexity and the complexity of approximation for some independence and domination-related graph-theoretical parameters within domination triangle graphs.

In the article it is proved that the analogue of the Khinchine theorem for simultaneous approximations of the points on the plane by algebraic conjugate numbers holds without the monotonicity of the approximation function.

Let (K, D) be a differential ring and T be some set of homomorphism t : K → C (C be ring constants) that for x, y є K, x ≠ y, there exists t є T, tx ≠ ty. Denote the ring K′ of the functions j : t → tx (x є K) and let (δφ)(t) = t(Dx). Some properties of the linear differential equation (δφ)(t) = A(t)φ(t) + ψ(t) over the ring K′ are obtained.

The radar procedure of the distance determination in conformal space-time geometry is considered. It is shown that the space intervals conformal contraction gave rise to an anomalous violet frequency shift during the monochromatic signal propagation over the closed path. Its relative value equals the Hubble constant multiplied by the duration of the signal propagation. The predicted phenomenon is the local manifestation of the cosmologic expansion and, in principle, is accessible to experimental detection.

We demonstrate how the system of two-level emitters coupled to the same dissipative reservoirs can function as a "classical" beam-splitter when operating with single photons. The suggested "classical beam-splitting device" might be useful for devising controlled single-photon emitting devices.

The study of chaotic dynamics in the volume free electron laser (VFEL) under the influence of external incident on the VFEL grid resonator electromagnetic waves is proposed. The possibility of changes in the type and magnitude of the amplitude of the dynamic solutions, as well as the suppression of spurious modes in VFEL is demonstrated.

In the article, the contribution of the monooxygenase component to the carcinogenic activation of a key procarcinogenic derivative of benzo(a)pyrene – 7,8-benzo(a)pyrenediol in lung adenocarcinoma cell line A549 was evaluated. It is shown that the contribution of the monooxygenase process under the experimental conditions in A549 cells is only 13 %, which corresponds to a relatively low level of the constitutive expression of CYP1A1 and CYP1B1. The situation changes significantly when cells are exposed to 20-methylcholanthrene. In this case, the monooxygenase component in the carcinogenic activation of 7,8-benzo(a)pyrenediol has reached 25 %. Moreover, 90 % is accounted for a "complete" carcinogen – diolepoxide-2. As 20-methylcholanthrene is a polycyclic aromatic compound and exists with a benzo(a)pyrene in the tobacco smoke, this suggests that the entry into the body a whole pool of polycyclic aromatic hydrocarbons may significantly affect not only the level but also the direction of the carcinogenic PAH activation of individual compounds.

Film polarizers based on polyvinyl alcohol (PVA), a new dichroic dye – sodium 2-hydroxy-5-((2-methoxy-4-((4-sulfonatophenyl)diazenyl)phenyl)diazenyl) benzoate (M₁₂), and mixtures of dichroic dyes polarizing in a wide spectral range (330–685 nm) are developed.

The work is devoted to the study of the influence of anion cadmium salts (chloride, nitrate and acetate) on the ligand function of the ion (CdBr₄²⁻)-selective electrode (ISE). The graphical form of the ligand functions of film electrodes in a wide concentration range of Cd²⁺-ions (from 1 ∙ 10^–4 to 0.1 M) is described and the influence of anion cadmium salts on the function of the ISE is explained in details. It is established that the increase in the concentration of Cd²⁺-ions has a double impact on the response characteristics of the CdBr₄²⁻-selective electrode. Also, it is established that the most suitable response characteristics are peculiar to the ligand functions of the CdBr₄²⁻-selective electrode based on cadmium acetate. The mathematical model of the ligand functions of the ISE based on trianoniloctadecylammonium is calculated using the system of computer algebra Mathematica 6.0.

Activated ester of a carboxylic derivative of europium diethylenetriaminetetraacetate has been synthesized by the interaction of europium salt of diethylenetriaminepentaacetic acid aminoethylamide with di-N-succinimidyl-p-phthalate. This reagent was applied to introduce rare earth metal ions in animal immunoglobulins (monoclonal antibodies). The proteins labelled with Eu³⁺ provide the required characteristics of fluorescence intensity, background, sensitivity and selectivity in diagnostic systems of lanthanide immunofluorometric assay.

Introduction of molybdenum trioxide to gas sensing materials based on titanium dioxide results in a considerable increase of the output signal in the hydrogen-air environment. It is established that the value of the output signal reaches the maximum at the 1 mol. % MoO₃ content in the composite material. Improved gas sensing characteristics of TiO₂:MoO₃ composite correlate with the structural and phase peculiarities of this system – the inhibition of TiO₂ crystallization in the TiO₂:MoO₃ system, the shift of anatase-rutile phase transition to the higher temperature area in comparison with the temperature of this transition in unloaded TiO₂, and with the presence of different types of tetrahonal distortions in MoO₆ octahedrons, which ensures the MoO₃ lattice high activity in the processes of hydrogen catalytic oxidation, and also with the crystallization of highly dispersed molybdenum oxide with a particle size of 10 nm.

Luminescent properties of Pr_0.1La_0.9InO₃ and Pr_0.0995La_0.9005Cr_0.005In_0.995O₃ solid solutions were investigated. It was shown that Pr_0.1La_0.9InO₃ and Pr_0.0995La_0.9005Cr_0.005In_0.995O₃ have a promising application as phosphors for white LEDs. It was found that Cr³⁺ ion is a good sensitizer of the Pr³⁺ ion photoluminescence in LaInO₃-based phosphors.

The influence of the pileup and mixing of water on the concentration of dissolved nitrate nitrogen, ammonium nitrogen, and phosphorus in the littoral zone of eutrophic Lake Obsterno (Belarus) has been studied.

This concentration was determined for the two types of the littoral zone – with and without water-air vegetation (Scirpus lacustris). After the pileup the concentrations of PO₄–, NO₃–, NH₄+ ions increased. The increasing of PO₄– was higher in the littoral zone without vegetation, but NH₄+ – in the littoral with vegetation.

Thus, the pileup and mixing of water in the littoral zone lead to an increase in biogenic elements, which may influence the phytoplankton development.

We have proposed an original approach for precise identification of bacteria of the genus Bacillus. Random fragment sequencing of bacterial chromosome DNA allowed us to establish a taxonomic membership for four studied microorganisms at the subspecies level (Bacillus amyloliquefaciens subsp. plantarum and B. subtilis subsp. subtilis).

Byelorussian tatars appeared in the territory of modern Belarus not later than the end of the 14th century and they are thought to be descendants of the inhabitants of the Golden Horde. Genetic relationships between Byelorussian Tatars and other Turkic peoples, as well as modern Byelorussians are not well understood. In order to address this question we studied the genetic structure of the population of Byelorussian Tatars using Y-chromosomal biallelic and STR markers. The study revealed the presence of genetic components typical for the populations of Northern and Southern Caucasus, Central Asia, and South Siberia.

The article deals with the application of complex multifractal and wavelet analysis to describe material and surface structures qualitatively and quantitatively, which allows revealing the differences poorly distinguished with other separate methods and estimating system characteristics of structures and characterizing the process of structure self-organization. The formation by tearing friction surface structures for composite materials with additives of Al₂O₃ and SiC is described in comparable thermodynamic conditions. Adding SiC in comparison with Al₂O₃ promotes to a lesser extent the fragmentation and increase in frequency of setting and breaking processes. An additional introduction of graphite in even greater degree strengthens the distinctions between materials with SiC and Al₂O₃ additives, which results in the formation of the least ordered structures of the friction surface. Adding graphite to a reinforced SiC composite material results in increasing the non-equilibrium of thermodynamic conditions for structure formation.

The importance of fundamental research of the scientific sphere functioning for achieving sustainable socio-economic development of the country is revealed. The author emphasizes that the organic interaction of fundamental science with applied research opens the way to practical implementation of scientific and technical developments, including the system of human knowledge. Promising areas of basic and applied research in the field of social sciences and humanities in Belarus are characterized.

The basic histiriographic approaches to the reconstruction of Prussians areal are discussed: conservative (German historical and archeological schools); 2 – ethnos-oriented (Polish historical and archeological schools), 3 – archeology-oriented (Russian archeological school). The source study features of approaches 1 and 2 are found: the orientation mainly to the later Middle Ages writing sources, which makes their definition of Prussians areal (V-XIII centuries) hypothetic, unreliable. The definitions of Russian archeologists (V.I. Kulakov et al.) are distinguished as the most substantiated and reliable, but not enough interpreted historically. A conclusion is made that it is urgent to introduce the material of archeology into the context of historical studies of the early Middle Ages.