### ON REPRESENTATION OF A STABLE ELEMENT OF A FREE NILPOTENT GROUP

In order to find a nontrivial stable element of a free nilpotent group, it is necessary to isolate all possible types of basic commutators that are later used to build a candidate for a stable element. The article is devoted to the description of three kinds of switches; each of them is closely related to the other two. For example, if the representation of a stable element includes commutators of one of the types, then the remaining species must be represented in the expansion of this element. We consider a free nilpotent group of rank 3 and of order 12.

Read More### DISSIPATIVE FUNCTION OF COULOMB FRICTION ON POLYGONAL CONTACT PATCH

The paper presents a solution in the quadratures of the problem of determining the principal vector and the principal moment of frictional forces arising from plane motion relative to the rough plane of the area limited by a polygonal contour affected by a uniformly distributed normal pressure. Similar problems arise in the study of traction forces of railway wheel pairs, including the forces of elastic deformations in the contact patch area. The results of the article can be used for comparative evaluation of the tribological characteristics of experimentally obtained lubricants.

Read More### OPTIMIZATION OF LAND-USE MANAGEMENT IN AGROINDUSTRIAL COMPLEX BASED ON MATHEMATICAL METHODS OF FINANCIAL ANALYSIS

The paper presents the model for minimization of irrational land-use based on the data on crop areas and yields for the main agricultural crops grown in the Rostov Region. The regression analysis and nonlinear optimization methods, commonly used in mathematical methods of financial analysis for the optimization of financial portfolios were used in the mathematical analysis and simulation. The model enables to calculate the optimal quantitative distribution of territories, ensuring a maximum average yield with a minimum average risk.

Read More### ELECTRONIC STRUCTURE OF ALKALI METAL ATOMS

The main goal of this paper is to consider the problem of the electronic structure of atoms of the 1st group of the periodic table. Atoms represent the system of nested shells-quasi-spheres, like a Russian Matryoshka. The analysis and justification are given on the formation mechanisms of electron shells of atoms. The conceptual basis of the theory of the electronic structure of atoms is a dipole-shell model. A refined hydrogen-like model of the electronic structure was proposed for the outer shell of alkali metal atoms. The internal electron shells of atoms represent regular geometric figures of different symmetry, including mirror symmetric, trigonal, tetrahedral, hexahedral, etc. Under the solution of the Kepler problem, the equations of motion of electrons on the corresponding shells are given. Formulas for description and calculation of the basic parameters of atoms are obtained. The effective radii of the atoms of the 1st group are determined. Their behavior in the group is considered as well. One of the characteristic features is that the radii obtained are approximately twice smaller in size compared to data from the literature; The explanation to this difference is also given in the paper. The basic parameters of a number of inner shells of atoms – screening constants, eccentricities and large semi-axes of elliptical orbits are also determined.

Read More### MODELING THE IMPACT OF AGRARIAN SECTOR OF ECONOMICS ON THE QUALITY OF ENVIRONMENT

This article presents mathematical models the inclusion of which into the systems of integrated environmental monitoring and production control will enable giving a dynamic forecast of the ecological system state in agro-industrial enterprise area before, during and after the production process.

Stating boundary problems is based on the equations of motion in various coordinate systems. Physical environment where the processes are studied can be characterized by substantially different physical, mechanical and chemical parameters. The construction of models is based on a topological approach, including the theory of block structures and the differential factorization method.

Solutions of model problems are obtained.

Read More### STUDY OF PROBABLE AND QUASI-PROBABLE DISTRIBUTIONS WITH THE HELP OF SPECIALIZED SOFTWARE

The article provides theoretical information on probability and quasi-probability (Wigner functions) distribution functions, as well as on the methods for finding them for various classical and quantum problems, respectively. We developed a computer program that enables the modeling of the evolution of the probability and quasi-probability distribution function for various problems. It is demonstrated that probable and quasi-probable distributions can have both similarities and differences for each particular problem.

Read More### METRIC SPACE OF UNLIMITED CONVEX SETS AND UNLIMITED POLYHEDRON

In the paper there is a definition of metric space H(K) the unlimited closed convex subsets of Banach space X, having the same recessive K. There is a Hausdorff metric that is used as a distance. It is established in this paper that the properties of a metric space H(K) are different from the properties of the metric spaceof convex compacts with Hausdorff metric. It is established that the theorem similar to the theorem of approximation of convex compacts polyhedrons is wrong. That is not each element of metric space H(K) can be approximated by the generalized polyhedrons, which are the analogues of the normalpolyhedrons.

The paper introduces the concept of a generalized polyhedron in the following way. The set of elements H(0)+K are known as generalized polyhedrons.

The criterion of approximation is derived. In order for the element of the space H(K) could be approximated by generalized polyhedrons in the Hausdorff metric it is necessary and sufficient that its basic function was evenly continuous.

Read More### THE CREATION OF THE PHONON SPECTRUM ACCORDING TO THE SPECIFIC HEAT FOR COMPOSITE MATERIALS

The paper proposed a numerical method for determining the phonon density of States of the composite materials according to the experimental data of the heat capacity of composite materials based on polytetrafluoroethylene matrix (PTFE). The maximum boundary values were analyzed and phonon spectra intensity depending on the structure of macromolecules “matrix + filler”. The results obtained by calculations are in good agreement with the experimentally determined phonon spectra for a number of the studied composites filled with carbon fiber.

Read More### CORRECTNESS OF RESTORATION OF MASS PARAMETERS OF ROCKET ENGINE OF SOLID FUEL

The article considers inverse problem of diagnosing the parameters of a rocket engine solid fuel according to known values of the frequencies of small free oscillations. The correctness is shown according to A.N. Tikhonov’s inverse problem of restoring mass parameters of the spring-mass model of a solid propellant rocket. The corresponding theorem is formulated and proved. An algorithm for solving the problem is found. It confirms the uniqueness of the parameters reconstruction from the known values of the four frequencies of the rocket engine oscillations. The decision algorithm is confirmed by a concrete example.

Read More### CURRENT SPECTROSCOPY OF DEFECT STATES IN As-Se GLASSY SYSTEM

Results of a research of an isothermal relaxation of dark current in thin layers of the vitreous As-Se system are presented. Calculation of contact capacity Cc is carried out. Cc is characterized by power law dispersion and reveals exponential dependence on temperature with activation energy Ea ~ 1.3 эВ. Assessment of thickness of the layer corresponding to the contact capacity and electric field strength in contact area gives values for As2Se3 structure dk = 1.24*10-7 m and EC = 8.06*103 V/cm respectively. The appearance of the I-V characteristics of the Al-CSP-Al sandwich structures indicates the influence of the contact phenomena on the polarization processes and the existence of a Schottky barrier whose height is (0.30 … 0.32) eV for As2Se3 TI samples and (0.35 … 0.38) eV for As2Se3 HF samples.

Read More### RESEARCH OF TEMPERATURE CONDITIONS ON THE SURFACE OF FUEL ELEMENTS UNDER DEPARTURE NUCLEATE BOILING

The paper presents the research of the phenomenon of formation and propagation of the film boiling with regard to the real fuel elements. The change in the temperature conditions and the influence of some constructive and operating parameters on the studied processes speed are considered. It is shown that with a sufficient perturbation of the shell temperature a wave of film boiling is formed. A numerical algorithm is used to implement the mathematical model of the process, developed earlier. These issues are of interest in the study of the initial stages of the development of accidents associated with overheating of fuel elements.

Read More### SOME ADDITIONS TO THE PROBLEM OF MULTIDIMENSIONAL INTEGRAL OPERATORS WITH OSCILLATING COEFFICIENTS

The paper presents multidimensional integral operators with homogeneous cores and radial oscillating coefficients of |x|iδ, acting in space L2(Bn). A symbol is defined for operators of this type in the terminology while necessary and sufficient conditions for noethericity and the computational formula for the index are obtained. A criterion for noethericity property and the index formula for a similar class of operators acting in L2(Bn) have been established. In this paper, we give some important additions and refinements to the main results published earlier. We have also considered a special case of an integral operator as a supplement.

Read More### EQUATIONS OF FUNCTIONAL DEPENDENCIES FOR ROTATIONS OF TWO PLANETS

In the article we have provided equations for dependences by modifying the Kepler equation with respect to the orbital motions of the planets and calculated the rotational and orbital motions of the planets with the angles of their rotation. We have established the wave principles of forward and reverse rotation of planets. The established dependencies are manifested at various scale levels of structural interactions, including bio-systems. The equality of angles of the Sun and the Earth rotation through the resonance of the particle-wave mechanism determines the possibility of terrestrial photosynthesis is presented in the paper. The accuracy of the calculations corresponds to the accuracy of experimental data.

Read More### FREQUENCY DEPENDENCE OF INDUCTION CURRENTS LOCALIZATION

The dynamics of changes in the structure of induction currents with increasing frequency of the external quasistationary magnetic field is considered in the paper. A numerical solution of the system of Maxwell’s equations describing the physics of the problem is carried out. The induction currents are determined by the amplitude and the phase shift relative to the external field. Because of the feedbacks caused by Maxwell’s equations, induction current have a complex structure, and its distinctive feature is the presence of radial coordinates and frequencies at which the currents in the skin layer move in reverse phase.

Read More### GENERAL FRAMES AT THE ASSESSMENT OF THE TIME OF FOR SIGNAL RECONSTRUCTION WITHOUT PHASE

Search of fast algorithms for doing signal reconstruction without phase is actual now. Algo-rithms of signal reconstruction are important in handling of various signals, in particular from speech recognition technology , in a tomography. The main property of frames which makes them so useful in applied problems is their redundancy. In general, a carefully chosen frame can provide numerical stability for reconstruction and an ability to capture important signal characteristics. The family of frames signal reconstruction on absolute value of frame coefficients in polynomial time.

Read More### ABOUT THE LAWS OF MOVEMENT OF PLANETS

By modifying the equation of Kepler from the orbital motions of the planets, equations of dependence of rotational and orbital motions of planets are given, their rotation angles are calculated. Wave principles of direct and reverse rotation of planets are established. The established dependencies are demonstrated at different scale levels of structural interactions, in biosystems as well. Equality of the Sun and Earth’s rotation angles through the resonance of the wave-mechanism determines the possibility of terrestrial photosynthesis. The accuracy of calculations corresponds to the accuracy of experimental data.

Read More### CORPUSCULAR-WAVE PROCESSES

Два принципа сложения энергетических характеристик структурных взаимодействий выполняются, если процесс идет или по градиенту потенциала или против него. Трансформируя эти правила на корпускулярно-волной дуализм можно предположить, что корпускулярные взаимодействия имеют место по градиенту потенциала (принцип сложения обратных величин энергий), а волновые процессы идут против градиента потенциала (принцип алгебраического сложения энергий). Такой подход подтверждается эмпирическим уравнением, в котором акт квантового действия сводится к перераспределению энергий в системе частица-волна.

Показано, что угловой вектор вращательно-поступательного движения электронов при квантовых переходах меняется в соответствии с квантовым числом квадрата тангенса этого угла.

### ON THE CORRECTNESS OF INVERSE PROBLEM OF DETERMINING COUPLE {u, σ} IN THE CASE OF A SIMPLE EQUATION FOR THE TRANSFER OF RADIATION

The article outlines the main research questions of the local properties of the collision integral and classical solutions of non-stationary radiative transfer equation, considered in the plain area of R3.

The paper studied and proved the correctness of questions “in the large” number of inverse problems for nonstationary transport equation, considered in a limited plain area of R3, for the simultaneous determination of the pair {u, σ}.

We present the smoothness of the area under consideration, taking into account the area of the prostate, to prove the lemma 1-4, on the basis of the lemma is proved Theorem 1.

Read More### ABOUT THE SPEED OF GRAVITY

The work describes a method to evaluate the speed of gravity interaction using space debris and asteroids. It demonstrates a technical implementation opportunity. Tentative analysis of experimental findings shows Newton’s model can be used in space navigation. The advantage of Newton’s model is confirmed by a seasonal deviation of a number of satellites from their expected location, as detected by a specialized laser location facility in Crimea, which provides high-precision detection of laser impulse direction. This deviation can be explained by Earth’s shift along the orbit while the light travels from the Sun to Earth.

Read More### A METHOD OF FORECASTING THE MOVEMENT OF MATTER

The article considers methods to forecast accelerated inertial movement of items, based on the understanding of the speed of gravity, which is much higher than the speed of light, and “the rain effect”. It offers a possible explanation for thirty phenomena of yet unexplained nature. These phenomena include accelerated expansion of the University; the increase in distance between the Sun and the center of the galaxy, between the Earth and the Moon; uniform explanation of experiments by Michaelson, Sagnac, Fizeau, Bradley; higher rotational speed at the equators of the Sun, Jupiter, Saturn compared to polar areas; spontaneous microparticle disintegration; tunnel effect; abrupt change in inertia moment on cooling solid helium, etc.

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