### VISUALIZATION OF MATRIX OF ELASTIC PROPERTIES OF CRYSTALS USING SPECIALIZED GRAPHIC PACKETS

A review of the means of visualizing elastic properties of cubic crystals by the matrix of elastic constants is performed. As an example, the measurement data for cij cubic TiNi single crystals is used. It is known for its martensitic transformations, shape memory effects and super-elasticity, as well as TiFe crystals isomorphic to nickelide titanum. The matrix of elastic constants of crystals was visualized using the program of computer algebra Mathcad, ELATE calculation and graphics programs: Elastic tensor analysis and SC-EMA: Self-Consistent Elasticity of Multi-phase Aggregates. The characteristic surfaces of Young’s modulus, the shift, and the Poisson’s ratio are obtained.

Read More### FEATURES OF NANOSTRUCTURAL STATES OF MECHANO-SYNTHESIZED POWDER STEELS ALLOYED BY CR AND SI

The paper studies the influence of alloying elements of substitution (Cr, Si) on the formation of nanostructures in powder nanocrystalline steels based on Fe-1mas. of % C obtained by means of mechano-alloyage of the initial powders of iron and graphite. The study was carried out by X-ray diffraction and magnetic structuroscopy. Carbon is distributed between the volumes of ferrite nanograin and grain boundary segregations in the nanostructure of steels. The carbon concentration in the ferrite varies between 0.2 and 0.37 at. % depending on the alloyage. Cr and Si increase the concentration of carbon in the ferrite as compared to non-alloy steel. The carbon concentration in grain boundary segregations varies within the range (1.1 – 1.7)•10-5 mol/m2. Cr lowers the concentration of carbon in segregations, while Si causes little change. The concentration of carbon in segregations is determined mainly by the grain size and the associated extent of the boundaries obtained by mechano-alloyage.

Read More### MIXED PROBLEM FOR ONE QUASILINEAR PARABOLIC EQUATION WITH HYSTERESIS

The paper considers an initial-boundary value problem for a quasilinear parabolic equation with a memory operator in a bounded domain with a sufficiently smooth boundary. A theorem on the existence of solutions of the initial-boundary value problem with a memory operator is proved. We used the method of discretization with respect to time to prove this theorem. The uniqueness of the solutions of this problem is also proved if the memory operator is a hysteresis nonlinearity of the generalized backlash type.

Read More### ON SOLVABILITY OF CAUCHY PROBLEM FOR SYSTEMS OF NON-LINEAR INTEGRO-DIFFERENTIAL EQUATIONS IN PARTIAL DERIVATIVES WITH PARAMETERS

It is possible to carry out the method of transforming solutions to study the problem of solvability of the Cauchy problem for non-linear integro-differential partial differential equations. The essence of this approach is the transformation of the initial Cauchy problem into an equivalent Volterra integral equation of the second kind, to which one can apply the topological method – the principle of condensed mappings. Sufficient conditions are defined for given functions for which the original problem is solvable from the conditions of contraction of the operator u.

In this paper we study the solvability of the Cauchy problem for systems of non-linear integro-differential partial differential equations of the first order with a parameter and an integral representation of the solutions obtained. Further, for a new class of systems of non-linear integro-differential partial differential equations of the third order, sufficient conditions for the existence of solutions of the Cauchy problem are found, and, in addition, an integral representation of such solutions is constructed. In view of the non-linearity of the initial problems, sufficient conditions do not guarantee the uniqueness of the solutions obtained.

Read More### CONSTRUCTION OF LABELED MULTITUDES USING POTENTIAL FUNCTIONS

In this paper, based on earlier papers, we constructed a labeled area using potential functions. The definitions and designations of labeled areas are given, specific cases of labeled areas are considered. The paper also introduces the concept of oriented labeled areas. Examples of oriented labeled areas are given. We consider a linear singularly perturbed ordinary differential equation of the first order as an example of the application of labeled area. A labeled area is constructed to study the asymptotic behavior of the solution of the initial problem. It is proved that there is a part of the labeled area that is the area of the attraction of the solution of the degenerate equation.

Read More### EXTREME VALUES OF ELASTIC MODULI AND POISSON’S RATIO OF TiFe AND TiNi WITH SHAPE MEMORY

Among the macroscopic characteristics of solids, elastic properties play a crucial role in the analysis of the loss of stability of the crystal lattice of materials to phase transitions. For shear-type transitions, such as martensitic transformations in metals and alloys, the study of the anisotropy of the parameters of crystal structures: elastic moduli and constants, Poisson’s ratio, and others is of particular importance. In this paper, the surfaces of Young’s and shear moduli of crystals, as well as the Poisson’s ratio and their central sections are constructed. Extreme values of the moduli and the Poisson’s ratio of the crystals are calculated. Transformation of surfaces and their central sections is given in the context of loss of stability of alloys to martensitic transformations.

Read More### CONSTRUCTION OF DOMAINS OF ATTRACTION AT DEGENERATION OF SINGULARLY PERTURBED EQUATIONS

The paper presents the analysis of singularly perturbed systems of ordinary differential equations. The overview of known results on the considered issue is given and the degree of relevance of the problem under investigation is substantiated based on them. We consider a system of singularly perturbed ordinary differential equations with analytic functions in the complex domain. A degenerate system corresponding to the system under consideration loses its uniqueness under degeneracy. In order to analyze the solution of the initial problem with respect to a small parameter, we introduce the notion of the domain of attraction of a solution of a degenerate system. The problem is reduced to finding the domains of attraction. Domains are constructed with the use of the level line of harmonic functions in the complex domain, and it is proved that they are domains of attraction of the solutions of the degenerate system under consideration.

Read More### SYNTHESIS AND LUMINESCENCE OF QUANTUM DOTS OF ZINC SULFIDE DOPED WITH MANGANESE

The work presents the results of the optical characteristics study of quantum dots of zinc sulfide and cadmium in various organic shells. The possibility of controlling the main characteristics of luminescence by various synthetic approaches and post-synthetic processing is demonstrated. The ratios of the components of the reaction mixture ensuring the highest luminescence intensity are determined. It was found that the conclusion of a quantum dot in a glutathione shell leads to the emergence of two luminescence bands with maxima near 420 and 590 nm. For the shell of mercaptosuccinic acid, one luminescence band comprises 590 nm. In turn, a cysteine shell there is one luminescence maximum of 500 nm.

Read More### INVESTIGATION OF LASER-INDUCED FLUORESCENCE SPECTRA OF DIFFERENT GRADES OF PETROLEUM PRODUCTS DISSOLVED IN SEA WATER WITH DOUBLE-FREQUENCY EXCITATION BY PULSES OF FEMTOSECOND LENGTH

The time dynamics of the laser-induced fluorescence spectra of various grades of petroleum products dissolved in seawater was studied for two-frequency excitation at wavelengths of 266 and 400 nm with pulses of duration about 100 fs. The minimum detection limits for concentrations of MFO fuel oil samples and TCM gas engine fuel were determined. The conducted studies showed that the method used has good sensitivity and can be used to analyze traces of hydrocarbons in seawater, both anthropogenic and natural origin.

Read More### DEVELOPMENT OF ATMOSPHERE LIDAR SOUNDING METHODS WITH FEMTOSECOND IMPULSES

Data were obtained for lidar sounding of the atmosphere in the continent-ocean transition zone by three modifications of a femtosecond lidar, based on a titanium-sapphire laser with chirped power amplification: elastic scattering lidar, Raman scattering lidar, and white light lidar. In the lidar mode of white light, the emission lines of the first positive system of nitrogen molecules were registered. A comparison of the obtained data with the results of lidar sounding using laser pulses of nanosecond duration is presented.

Read More### SYNTHESIS OF FE:MGAL2O4 NANOPOWDERS INTO LASER PLUM

The features of production of Fe:MgAl2O4 nanopowders by evaporation of targets made from a simple oxide mixture (Fe2O3, MgO, Al2O3) by repetitively pulsed CO2 laser radiation with I=1.6 MW/cm2 peak power density and Paver=600 W average radiation power as well as by ytterbium fiber laser radiation (I=0.4 MW/cm2 and Paver=300 W) were studied. It was demonstrated that the nanopowder produced with the use of the CO2 laser has the specific surface of 56 m2/g and contains two crystalline phases, i.e. MgAl2O4 (98.2 wt%) and MgO (1.8 wt%) with Fe ions dissolved in them. At the average radiation power of 600 W the output of the nanopowder was 16 g/h. For the nanopowder produced using the ytterbium fiber laser twofold increase of the specific surface (105 m2/g) was observed. This nanopowder contains four phases, i.e. MgAl2O4 (67.5 wt%), γ-Al2O3 (24.8 wt%), Fe3O4 (3.2 wt%) and MgO (4.5 wt%). In this case the output of the nanopowder was 2.7 g/h due to formation of a “forest-like” array of 4–5 mm high spikes covered with a semitransparent melt layer. Significant differences in the phase compositions of the nanopowders obtained using these lasers are associated with a higher rate of the laser plume cooling for the ytterbium fiber laser.

Read More### FEATURES OF EMISSION IN NANOGRAIN STRUCTURE OF SEMICONDUCTORS

The experimental study and theoretical analysis of the possible mechanisms of field emission in the nanograin structure of the most widely used semiconductors (Si, GaAs, InAs, InSb) are carried out in this work. A model scheme of electronic processes is proposed. The parameters of the electronic spectrum of the structures studied are calculated. Qualitative and quantitative agreement of the experimental results with a theoretical estimate is obtained, which confirms the legitimacy of the formulated model representations. The carried out research allows asserting that emitters on the basis of narrow-band semiconductors. А3В5 are much more effective than those based on metals, carbon, silicon.

Read More### PHOTOLUMINESCENCE AND PLASMA REFLECTION SPECTRA OF COLLOIDAL QUANTUM DOTS CdSe, PbS, GaAs

A simple technique for depositing colloidal quantum dots (QD) in relatively thick (up to 1 μm) layers on a glass substrate was tested. According to 3D-AFM topograms, it is concluded that QD is aggregated into conglomerates, which consist of densely-packed smaller particles that have the form of faceted plates. The experimental characteristics of the photoluminescence spectra are in good agreement with the theoretical ones. When the QD is transferred from the suspension to the substrate, a decrease in the quantum yield is observed. A resonance reflection was detected on the QD-PbS in the spectral region ~ 8 μm and QD-CdSe/CdS ~ 2 μm.

Read More### ON PERIODIC SOLUTIONS OF BOUNDARY VALUE PROBLEM FOR QUASILINEAR INTEGRAL VOLTERRA EQUATIONS

The following problem is studied: under what conditions the periodic function is a solution of the Volterra integral equation with periodic coefficients. In this paper, we find sufficient conditions for the existence of periodic solutions of the boundary value problem for quasilinear integral Volterra equations that tend to the solution of a periodic boundary value problem for the generating equation. The principle of condensed mappings and the conditions for the analyticity of given functions are applied. The solution of the Volterra quasilinear integral equations is constructed in the space of continuous functions.

Read More### ON INVESTIGATION OF FORCED SYNCHRONIZATION BY THE METHOD OF APPROXIMATED POINT MAPPINGS

The article considers the possibility of investigating the synchronization of a quasiharmonic oscillator with nonlinearity of the type of a cubic parabola by the method of approximate point mappings. The synchronization of a quasiharmonic oscillator comes to solving the problem of the existence of fixed points on a point map, while the method of successive approximations is applied for their construction. The proposed method of investigation is an asymptotic method; therefore, the applicability of the results of an approximate investigation at specific values of the small parameter is also important. In this paper, we propose to consider the problem of applicability of the results of an approximate investigation, estimating the degree of closeness of the approximate point mapping to the exact mapping.

Read More### POISSON’S RATIO OF DENTIN AS ANISOTROPIC MEDIUM WITH HETEROGONAL SYMMETRY

The Poisson’s ratio (transverse deformation) plays an important role in the deformation behavior of materials. Along with the Young’s module, it constitutes a pair of independent and most informative material constants of solids. For hard tissues of the tooth (enamel and dentin), the Poisson’s ratio should correspond to the Poisson’s ratio of restorative materials in order to avoid overvoltages at the boundary of the sections restorative material-enamel and restoration material-dentin. In addition, the value of the Poisson’s ratio affects the deformation strength of enamel and dentin, namely, crack resistance, when they occur in a stressed-deformed state. In this paper, the orientational dependence of the Poisson’s ratio of dentin teeth on the basis of matrices of elastic constants and the compliance coefficients of hexagonal crystals, such as crystals of dentine hydroxyapatite, was obtained for the first time. The results of calculating the Poisson’s ratios of dentin as a crystalline system with a hexagonal structure are presented in the form of tables and diagrams in the polar and Cartesian coordinate systems. The minimum and maximum coefficients for the corresponding directions of the longitudinal and transverse deformations in the crystallographic coordinate system are also calculated. It is shown that the maximum value of the Poisson’s ratio of dentin (0,53) is greater than the upper limit for the Poisson’s ratio of isotropic materials, including known restoration materials, which in some cases may reduce the quality of restorations. It is noted that a similar analysis can be performed for tooth enamel.

Read More### ON EFFECT OF LONGITUDINAL DUST MOTION ON STABILITY OF HOMOGENEOUS STATE OF MAGNETIC DUST PLASMA

The propagation of low-frequency electromagnetic waves of small amplitude in a low-pressure dust plasma is considered in the paper. Dust is considered to be cold and has both a transverse and longitudinal velocity relative to the direction of the external magnetic field. The fourth-degree dispersion equation for the phase velocities of waves is analytically studied. Limits on the unperturbed plasma parameters with stable homogeneous state of plasma are obtained. It is shown that in the absence of sufficiently large values of the velocity component of the dust component along the magnetic field, the homogeneous state of plasma is unstable concerning small perturbations.

Read More### CALCULATION OF VISCOSITY DIFFERENCES AND DIFFUSION OF DISPERSED BINARY GAS MIXTURES OF CARBON DIOXIDE WITH ETHANOL AND PROPANE

The method for calculating viscosity and mutual diffusion coefficients of rarefied binary gas mixtures for various temperatures and compositions of the mixture is proposed on the basis of the molecular-kinetic theory of gases. The results of the calculation of viscosity of carbon dioxide with ethane and carbon dioxide with propane in the temperature range 250-1200 K for different mixtures compositions and also for the equimolar mixtures of these gases in the same temperature range are presented. The obtained values of the properties are compared with the calculated and experimental results of other authors.

Read More### ON THE QUASIRELATIVISTIC EQUATIONS FOR THE SOLAR CORONA PLASMA AND A VARIANT OF RATIONALE OF ITS HIGH TEMPERATURE

The quasi-relativistic equations are derived from nonrelativistic equations bringing amendments in them. They comprise of the continuity equations, and the conservation laws for impulses and energies both for protons and for electrons. It is so because the hydrodynamic and electric forces acting on them are similar, so, their impulses are also similar. The rife proton velocity in solar corona is 500 km/s and the consistent electron velocity for such impulse is 0.953 c, hence, the electrons must be taken into account and they are relativistic. Expressions for the collisional terms are also adapted to relativistic velocities. Heat emitting to warm protons due to friction between electron and proton flows is sufficient to heat plasma. Concentration ne = 1010 cm-3 and amount of scattering energy compensate large free path time.

Read More### TO PROJECTIVE PROPERTIES OF THE PHYSICAL SPACE-TIME. PART II. MEASURES AND CURVATURE IN CLASSICAL THE GEOMETRY OF LOBACHEVSKY ‒ BOLYAI

Under the assumption that 4-dimensional physical space (space-time) is projective, and its geometry the classical non-Euclidean geometry of Lobachevsky ‒ Bolyai (hyperbolic geometry) the following tasks: 1) the rationale for using projective geometry for the existence in the geometry of Lobachevsky ‒ Bolyai’s two main non-Euclidean measures of distance − additive classical non-Euclidean and non-Euclidean non-additive measure which is a generalization of the physical interval between the events; 2) derivation of the formulas describing the transformation of coordinates between two autopolarity coordinate systems – the case of the mutual arrangement of the two autopolarity coordinate systems are 4-dimensional projective hyperbolic space, when the time axis and one spatial coordinate axes of both systems lie in the same plane, and the other two axis systems are pairwise parallel; 3) justification of curvature of the flat non-Euclidean geometry as curvature measures; 4) derivation of the formulas describing the change with time of distance, speed and acceleration between inertial systems in 4-dimensional case.

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