### ON THE BEST APPROXIMATION BY ABSOLUTELY MONOTONIC FUNCTIONS ON SEMIAXIS

The main result of the paper (Theorem 2) is that in the space C(I) of continuous functions on the interval I = [0, ∞], the cone K⊂C (I) consisting of absolutely monotone functions is Chebyshev, that is, for each continuous function f∈C (I) there is a unique absolutely monotonic function φ∈K of the best uniform approximation on the interval I. In the proof, we use a special criterion for the uniqueness of the best approximation by the wedge (Theorem 1). This criterion can be used in proving the uniqueness of the best approximation for other cones consisting of continuous functions.

Read More### MATHEMATICAL COMPETITIVE MODELS ON THE TROPHIC RESOURCE

The authors have developed mathematical models of operational and interference competition on a linear range based on the systems of equations with distributed parameters. They also have conducted an analysis of stationary states stability. It is shown that the operational competition on the restored trophic resource does not lead to the disappearance of one of the populations due to competition. The model of interference competition contains various options for the effects of competition between the two populations. In both models, for populations with a small number of individuals, the influence of competition is not significant. The assessment of the distribution rates of small populations on the range is given, and the conditions for the existence of an autowave solution on an unbounded straight line are obtained. In order to construct a numerical solution of a boundary value problem for a system of nonlinear differential equations, they have used the grid method with software implementation in the programming environment of the Matlab environment. Numerical results are consistent with analytical results on fine grids.

Read More### COMPARATIVE ANALYSIS OF COMPUTATIONAL METHODS OF SYSTEM OF NONLINEAR EQUATIONS

The consideration of numerous applied problems leads to systems of nonlinear equations, they include boundary problems for ordinary differential equations and partial differential equations (solved by the finite difference method), optimization problems, problems of minimization of functions of many variables, the use of implicit methods for integrating ordinary differential equations, etc. Numerical solution of systems of nonlinear equations in the general case is a more complicated problem than the solution of systems of linear equations, since there are no methods that guarantee the success of solving any problem of this kind. Identifying the optimal method and its further selection allows you to increase the chances of successfully solving systems of nonlinear equations. In connection with the relevance of the above-mentioned, this article presents the algorithms for methods for the numerical solution of systems of nonlinear equations, according to which the root of a typical system for applied problems was searched. According to the obtained results, a comparative analysis was conducted in order to identify the optimal method. The optimal method is the one that found the values of all the roots of the system with the required accuracy in the least number of iterations.

Read More### MAGNETOELECTRIC EFFECT IN THREE LAYER NICKEL – QUARTZ – NICKEL STRUCTURE

The results of theoretical and experimental studies of the magnetoelectric effect in composite three – layer structures of nickel – quartz-nickel are presented. Samples in the form of plates were made by gluing. At the electromechanical resonance region the value of the magnetoelectric voltage coefficient α = 84.7 V / cm Oe, with Q = 93 was observed. Using the quartz as a piezoelectric layer allows to obtain the magnitude of the magnetoelectric effect in composite structures, comparable to the best samples based on lead zirconate-titanate.

Read More### ALGORITHM FOR PRESENTING PROPERTIES OF EXPERT SYSTEM OBJECTS

The article considers the problem of presenting the properties of complex objects in the knowledge bases of expert systems when they are characterized by uncertainty, and there is no other way to estimate ambiguity values of their quantitative characteristics, which complicates the study of the corresponding attribute space. The analysis of the task set testifies to the expediency of the initial orientation of the software for its solution to the multi-valued interpretation from the position of fuzzy and linguistic modeling of the problem area under consideration.

Read More### DEVELOPMENT OF THE METHOD OF ADDITIONAL ARGUMENT FOR A SYSTEM OF NON-LINEAR DIFFERENTIAL EQUATIONS

The paper considers the initial problem for systems of equations and uses the developed method of additional argument to solve the problem. A review of the known results on the method under consideration is presented, and the degree of relevance of the problem under study is substantiated on their basis. The stated initial problem is reduced to a system of integral equations when using certain classes of functions. This developed research technique can be used to prove the existence of a solution to new types of vector-matrix non-linear equations.

Read More### EVALUATING THE CONTRIBUTION OF NON-LINEAR LOADS TO HIGHER-ORDER HARMONICS OF NETWORK VOLTAGE IN SUPPLY VOLTAGE OF HIGHER-ORDER HARMONICS

The following paper proposes a method for calculating current harmonics generated by the non-linear load in the supply voltage of higher-order harmonics of the voltage. A model of non-linear consumer load is also proposed. The conditions for the use of independent circuits for each voltage harmonic are obtained. The effect of higher-order harmonics of non-linear load voltage on higher-order harmonics of the current is considered. This method can be used to calculate the quality of electricity in the network in case of non-linear loads.

Read More### ELASTIC PROPERTIES OF METALS AND ALLOYS BEFORE MARTENSITE TRANSFORMATIONS

A review of the elastic behavior of materials in the temperature region before martensitic transformations (MP) is carried out. The nature of changes in elastic constants before these transformations is extremely informative in studying both the mechanisms and the nature of structural transitions, since elastic modules determine the resistance of the crystal lattice to uniform shifts and influence the stability of structures during martensitic rearrangement. All metals and their alloys by the nature of the change of elastic properties can be divided into three groups. In the 1st group, the temperature dependence of the elastic moduli is normal (the temperature coefficient is less than zero) and phase transformations take place far from the point of loss of stability of metals and intermetallic compounds. The anomalous “softening” of elastic constants of single crystals of alloys before the MP is common for all compounds of the 2nd group. In this case, as a rule, the elastic constant “softens” 11-04-2019 14-37-45, corresponding to the Bane deformation and lattice structural adjustment, it is smaller than all the other constants cij. Alloys of the 3rd group are characterized by an anomalous decrease in the modulus 11-04-2019 14-34-25 and a sharp increase in the elastic anisotropy А = с44/11-04-2019 14-34-25 to dramatic values. The part of the intermetallic compounds is characterized by the type of premartensitic instability, found in B2 compounds based on titanium nickelide and consisting in the simultaneous “softening” of all the shear and longitudinal moduli of elasticity and a decrease in the elastic anisotropy coefficient. As a result, the lattice becomes soft in all major shear systems, which ensures the diversity of structural phase transitions in this group of alloys and the unique properties of shape memory and super elasticity.

Read More### SYNTHESIS, STRUCTURAL AND PHOTOLUMINESCENT PROPERTIES OF COLLOIDAL SEMICONDUCTOR QUANTUM DOTS IN SUSPENSION AND ON A SUBSTRATE

Quantum dots (QDs) of CdSe, PbS, InSb semiconductors are synthesized. The laboratory technology of applying layers on the substrate is tested. It is determined from 3D-ACM-topograms that QDs in a layer on a substrate are aggregated into conglomerates with polygonal flattened shape and consisting of tightly packed smaller particles. The experimental characteristics of the photoluminescence spectra are in good accord with the theoretical ones. The quantum output level at QD-PbS and InSb is 5-10 times lower than the best results for colloidal CdSe/CdS quantum dots. A decrease in the quantum yield is observed when transferring quantum dots from a suspension to a substrate, which may be due to the weakening of the dimensional quantization in QD during their aggregation.

Read More### NUMERICAL STUDY OF SUPER-SOUND JET STRUCTURE IN TWO-DIMENSIONAL NOZZLE DEPENDING ON ADIABAT INDICATOR

Based on the numerical simulation, the structure of a supersonic jet flowing from a flat nozzle is studied in work [8]. A methodical study is conducted in order to assess the influence of turbulence models on the flow structure. The results are compared with experimental data. It is shown that when the NPR parameter is changed, both Mach and regular reflection can be observed. It is revealed that a change in the adiabatic index γ can lead to a displacement of the separation point inside the nozzle.

Read More### SYNTHESIS OF QUANTUM-DIMENSIONAL SEMICONDUCTOR PARTICLES WITH CHEMICAL AND PHOTOCHEMICAL ETCHING METHODS

The conditions of liquid chemical and photochemical etching of submicron powders of the following semiconductors: GaAs, InSb, Si were studied in this work. The analysis of particle size was carried out at various etching stages depending on the composition of the etching solutions, including the additional introduction of a stabilizer – cationic surfactant cetyltrimethylammonium bromide in order to estimate the etching rate and the size of the nanoparticles obtained. The rates of etching processes were determined; photoluminescence was obtained on colloidal nanosized semiconductor particles produced by this method, and its spectral composition was measured.

Read More### CONDITIONS FOR EXISTENCE OF SOLUTIONS OF LINEAR INTEGRODIFFERENTIAL EQUATIONS WITH DELAY WITH ANALYTICAL FUNCTIONS

Previously, with the help of the power series method, the author obtained conditions for the algorithmization of problems of establishing the existence of solutions of certain types of linear integrodifferential equations with analytic functions. This article shows that the applied method can also be used for linear integrodifferential equations with a proportional delay of the argument. A review of the usage of the “algorithm” concept in various branches of mathematics was made.

Read More### OPTICAL PROPERTIES OF NEW PHOTOACTIVE POLYMERS BASED ON PMMA DOPED WITH BORON CHYLATES

The results of the study of linear and nonlinear optical characteristics of new polymeric materials based on polymethyl methacrylate doped with boron chylates compounds are presented in the paper. The obtained samples showed a high yield of one- and two-photon fluorescence and high-coefficients of two-photon absorption. These properties allow using the studied materials for a single- and multiple-photon modification of their optical properties without an optical breakdown in order to obtain photonic elements.

Read More### PHYSICAL ASPECTS OF FRACTAL BEAUTY PHENOMENON

The features of fractal object perception by the human visual system forming a sense of beauty are considered in the paper. The interpretation of a fractal beauty phenomenon and the numerous facts of the positive influence of fractal images on the psychological state of a person described in the literature are presented based on optical and physical concepts using numerical modeling. It is found that regardless of the model of a fractal object and variations in a wide range of its parameters, there is a self-similarity of the spatial spectra of a fractal image in different spectral ranges. We have a reason to believe that the presence of self-similarity, simplifying the processing of optical information in the cerebral cortex, creates a feeling of comfort and excites a sense of beauty.

Read More### STUDY OF TEMPERATURE DEPENDENCE OF SEMICONDUCTOR PHOTORESISTORS

A description is given of a laboratory setup for studying the temperature dependence of the photocurrent of semiconductor photoresistors and the results of experimental measurements of the lifetime of minority carriers of a photocurrent of conductivity depending on the temperature and frequency of light irradiation waves. It is established that the temperature dependence of the lifetime of minority photocurrent carriers in the studied semiconductor photoresistors is linear when illuminated with the green spectrum light. The magnitude of the lifetime monotonously decreases with an increase in temperature up to a statistical distribution of the measured quantity. The dependence of the lifetime of minority carriers on the frequency of light waves is almost linear. The lifetime of the minority carriers of the photocurrent monotonously increases with an increase in the wavelength of the irradiated light.

Read More### STATISTICS OF MODE CONTENT OF VORTEX MATRIX BEAMS DURING DISTRIBUTION IN ATMOSPHERE

This paper analyzes the mode composition of the radiation from vortex matrix laser beams synthesized by a coherent fiber-optic emitter matrix. The energy transfer between the vortex azimuthal radiation modes during propagation in free space and turbulent atmosphere is analyzed on the basis of numerical simulation. It is shown that most of the radiation energy is pumped into the main mode in the far-field zone, which is determined by the topological charge of the synthesized beam. Statistical characteristics of the mode composition of the synthesized vortex matrix beam are similar to those of the Laguerre-Gaussian beam in case of turbulence.

Read More### APPROXIMATION AND INTERPOLATION OF WAVE SIGNALS IN THE BASIS OF BORDERING FUNCTIONS IN THE STUDY OF MECHANICAL AND OTHER PHYSICAL FIELDS

A formula for trigonometric approximation and interpolation of signals based on the bordering functions is proposed, which simplifies the process of modeling and reconstructing changes of wave signals from measured samples. Proposed modifications of the formula for simultaneous parallel filtering of signals based on the use of families of new bordering and filtering functions. The advantages of using the proposed approach and formulas over the known ones are shown. The interrelation of the formula with the known sampling theorem of is shown.

Read More### MAGNON FOCUSING IN CRYSTALS WITH ALPHA-IRON STRUCTURE

The features of the propagation of spin waves (magnons) in crystals with the alpha-iron structure are studied within the framework of a model that takes into account the interaction of an atom with the atoms of the first and second coordination spheres in a layer of a cubic body-centered crystal lattice. It is shown that there is no magnon focusing in long-wave approximation, as for short waves in case of no external constant magnetic field, a focusing effect is observed. The features of focusing are considered with a negative interaction parameter with neighbors.

Read More### MODELING OF DISTRIBUTION DEPENDENCE OF POLLUTANT SUBSTANCES IN THE MOUNTAIN GORGES FROM DAILY CHANGES OF TEMPERATURE

При помощи математической модели в двумерной постановке исследуются течения воздуха и распространение загрязняющих веществ в горных ущельях при учете вызываемых солнечной радиацией суточных изменений температуры поверхности. Рассмотрены ущелье идеализированной формы и ущелье, расположенное в Северной Осетии в районе хвостохранилища вблизи селения Унал в долине реки Ардон. Проведено сравнение динамики течений и распространения загрязняющих веществ. Показано, что учет суточных изменений температуры сказывается на закономерностях распространения загрязняющих веществ в этих ущельях. Также отличия наблюдаются между дневными и ночными режимами проветривания горных ущелий.

Read More### FEATURES OF EXPERIMENTAL RESEARCH ON SUPERSONIC ATMOSPHERIC VACUUM DRIVEN TUNNEL

В настоящее время экспериментальные исследования сверхзвуковых течений проводятся на сложных экспериментальных установках, таких как аэродинамические и ударные трубы. Для исследования сверхзвуковых течений с числами Маха меньше 4 возможно использование вакуумных аэродинамических труб. В статье приводятся результаты исследования процессов, происходящих в вакуумной аэродинамической трубе при пуске. Анализ полученных результатов показал, что реализуемый в трубе стационарный режим сверхзвукового течения имеет продолжительность около 3 секунд, что подтверждается представленными в работе теневыми картинами течения. Это обстоятельство позволяет использовать имеющуюся атмосферно-вакуумную трубу для исследования аэродинамических характеристик летательных аппаратов при числах Маха от 1,5 до 4.

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