It is proved that there exist the completely integrable Pfaffian systems ∂x / ∂ti = Ai (t)x, x∈n , = ( 1, , ) m, t t tm ∈+ i = 1, m, m ≥ 2, with infinitely differentiable bounded coefficients such that the lower characteristic set of these systems is an arbitrary pre-assigned disconnected set and has a positive Lebesgue m-measure. Also, both characteristic and lower characteristic sets for these systems can be arbitrarily pre-assigned. The proofs are constructive.

The necessary and sufficient conditions, under which a reflecting function of a linear nonhomogeneous differential system coincides with that of the corresponding system with a zero matrix, were established. The sufficient condition for the linear nonhomogeneous system, the homogeneous and nonhomogeneous parts of which are periodic with incommensurable periods, has a periodic reflecting function with a period coinciding with that of the nonhomogeneous part. Due to the conditions obtained, the irregular periodic solutions of the quasi-periodic differential system of matrix equations were obtained.

Consider the class 3(Q) of the polynomials P(t)∈[t] of degree 3 and height H(P) ≤ Q, Q >1. Define a subclass S3(Q) in 3(Q) by taking the polynomials P(t) having discriminants not exceeding Q2n−2−2v1 and divisible by the power of the prime number pe , pe > Q2v2 , v1 ≥ 0, v2 ≥ 0, 0 ≤ v1 + v2 < 3 / 2. The upper bound on the number of the elements in S3(Q). is found. It has been proved that for any ε > 0 and Q > Q0 (ε), the inequality 4 5/3( 1 2 ) # 3( ) S Q Q v v < − + +ε is valid.

In this article, we consider the upper bound on the number of points with the integer p-adic conjugate algebraic coordinates in a strip of small measure, near the curve normal by Mahler.

In this article, for the canonical form of vector-difference schemes under the positivity conditions of matrix coefficients the two-sided estimates for an approximate solution at the arbitrary non sign- constant input data of the problem are obtained. The obtained results are used for deriving two-swided estimates and a priori estimates in the norm C of monotone vector-difference schemes that approximate the weakly coupled systems of elliptic and parabolic equations with the Dirichlet foundary conditions.

For the real functions of a real variable, the relations between the notions of ordinary derivatives and NL-derivatives (introduced by the authors), the derivatives of ordinary functions considered as distributions, and the Radon–Nikodym derivatives are studied. The theorems on the NL-derivatives of the sum and product of functions, as well as on the NL-derivatives of composite and inverse functions are given. The reduction of the notion of the NL-derivatives between the Banach spaces to the derivatives of the scalar functions of a real variable is considered as well.

In this article, the order of exponential of growth of analytical function ϕ on the disc is introduced, and the relation between the order of the function ϕ and its coefficients is obtained. An application of this result gives us the description of the behavior of the resolvent  R(B,λ)  of linear bounded operator where λ approaches the spectrum.

In this article, the lower graph order boundary obtained by V. Nikiforov, for which the generalization of the sufficient spectral criterion of Hamiltonicity of a graph is valid, has been reduced twice.

A method for protecting valuable documents against forgery based on the visual ciphering of a high- resolution image (shared secret) and its deciphering through the physical process of imposing noise- like shadow images without using a computer and cryptographic encryption keys is proposed. The cryptographic properties of shadow images are studied. The computer modeling results are presented.

It is shown that the Galileo and Carroll boosts with dimensional parameters are the special cases of vector parameterization of the corresponding transformations of the mathematical Galileo group. In the biquaternions over the algebra of dual numbers, a number of the representations of the Carroll group are realized and two systems of the linear partial differential equations relative to the Carroll bivector fields generated by external sources are constructed. It is shown that the definition of the field strengths in terms of the potentials is possible only if there are the sources of the same type, and only for one system of the equations of Сarroll electrodynamics.

The reversible tunnel migration between nitrogen of the −N = N-group of the initial faint localization of n,π*-excitation on starting nitrogen for n-electron excitation is proposed as the mechanism of extraordinary broad C2v-symmetric cis-azoalkanes fluorescence spectra.

A measuring system for local elemental analysis is constructed on the basis of an ion accelerator AN-2500. It comprises a device of an in-air ion beam extraction through a tapered glass capillary and a PIXE spectrometer. The angular dependence of the proton beam possessing the transmission energy of 150-250 keV through the tapered capillary with the output diameters of 5 and 10μm is determined. The transit of protons through the capillary is found to be in threshold character and to be determined by the charging rate of its internal surface. The characteristics of current and energy of the ion beam extracted into the atmosphere are analyzed. It is found that the value of output current can be varied from 0.5 to 3 nA, while the ion energy is almost equal to the initial beam energy. The PIXE energy spectra of various materials are measured at the atmospheric pressure. It is found that the output beam intensity is enough for an accurate determination of the elemental composition of the samples under analysis.

We propose a model that allows an understanding of the nature of electron traps in π-onjugated polymers that are used in photovoltaic devices. It is assumed that the free-cavity voids in the polar π-conjugated polymer exhibit electron-accepting affinity and are filled with electrons, called  erein as ‘dead’ electrons because they are hold by a static random internal electric field and are not driven to the cathode by external built-in voltage. As a result, the dead electrons into electron-trapping voids are unsuitable for use in the external circuit of organic solar cells. As justified here, the exciton dissociation on the surfaces of voids, the capture of drift electrons by voids, the emerging of the dead electrons and their non-geminate recombination are the main obstacles to create highly efficient polymer solar cells. The model of dead electrons allows explaining the increase in the power conversion efficiency of solar cells caused by the polymer fluorination, side-chain polymer manipulation, and dopant-controlled trap-filling. Some characteristics of hybrid P3HT:CdSe solar cells are also analyzed with the help of this model.

The generalized Duffin–Kemmer equation for a spin 1 particle with the anomalous magnetic moment in the external uniform magnetic field is investigated. The separation of variables in the wave equation is performed on the basis of projective operator techniques. The problem is reduced to a system of differential equations for three independent functions that have been solved in terms of the confluent hypergeometric functions. Three series of the energy levels are found. To assign them the physical sense at all values of the main quantum number n = 0,1, 2, , special restrictions on anomalous magnetic moment values must be imposed – they are formulated in explicit form. Otherwise, only some part of the energy levels corresponds to the bound states. The neutral spin 1 particle is considered as well. In this case, no bound states exist in the systems. The main qualitative manifestation of the anomalous magnetic moment is the space scaling of the arguments of the wave functions in comparison with a particle without such a moment.

A one-step method for preparing associates of magnetite nanocarriers with bioactive 2- rylaminopyrimidine derivatives containing the pharmacophore fragments of protein kinase inhibitors and SH-group was developed. The feature of the method is the presence of dimethyl sulfoxide and 2-arylaminopyrimidines in the medium of formation of magnetite nanoparticles

It is established that thymoquinone (2-isopropyl-5-methyl-1,4-benzoquinone) and 1,4-benzoquinone regulate the intracellular reactive oxygen species (ROS) production and induce the death of tumor cells by different mechanisms. It is shown that the toxic action of 1,4-benzoquinone is associated with the inhibition of electron transfer in the mitochondrial respiratory chain and with the development of cellular oxidative stress. Thymoquinone initiating a lower level of ROS production in comparison with 1,4-benzoquinone is more toxic to tumor cells. It is established that thymoquinone-induced ROS are involved in the redox signaling processes that lead to the opening of mitochondrial transition pores of high permeability and the activation of the programmed death of cells.

To investigate the impact of differential DNA polymorphisms of initial breeding sources on the superior F1 performance, we carried out the screening of their allelic composition and performed the genetic analysis of lines and its hybrids in the test cross design. As a result, it is shown that the overall and locus-specific heterozygosity is important for the formation of the F1 heterotic response, and the evaluation of molecular polymorphisms could be useful for the prediction of promising crosscombinations.

The volumetric activity of the region determined in Quaternary deposits, platform cover and basement rocks was used to compile a diagram representing five types of territories as follows: potentially hazardous in radon contamination, including separate areas potentially hazardous in radon contamination, including local sites potentially hazardous in radon contamination, relatively nonhazardous in radon and nonhazardous in radon.

Gypsum was found in the Silurian deposits of Belarus for the first time. It composes nodules and fills cracks and fenestras in clayey-carbonate rocks revealed by the Davtiuny 3k borehole (Grodno area, Ostrovets region). With the use of the sulphur isotope data, it is shown that gypsum was formed at the diagenetic stage in the Silurian seawater concentrated under interground evaporation. The process was the result of a repeated superimposition of supratidal environment onto clayey- carbonate deposits formed in shallow subtidal and intratidal conditions.

The archeological complex Cordon situated in the Vitebsk Dvina area was investigated for the first time. The complex includes a hill-fort, two settlements, and a burial mound dated back to the middle of IX – first half of X century. A broad spectrum of artifacts of North European and Middle Eastern origin, Arabian and Byzantine coins, metrological materials, and the existence of jewelry production and trade activities put Cordon in line with the important archaeological sites of the Viking Age. The discovery of Cordon proved the leading role of the Vitebsk Dvina area in the establishment of contacts between the North and the South on the way ‘from the Vikings to the Greeks’.