### COMBINATORIAL MEANING OF EULER’S NUMBER

In this paper, we consider a new limit for the number e and give its rigorous proof using the apparatus of mathematical analysis. With the help of this limit, a combinatorial interpretation is given for Euler’s number. It means that Euler’s number is the ratio of the number of permutations (or combinations) of 04-09-2019 10-48-27 by n to the number of permutations (or combinations) of 04-09-2019 10-48-35 by n with an infinitely large number of elements n.

Read More### MAIN PROPERTIES OF A GAS-LIQUID MIXTURE IN PIPES

Recently carried out experimental and theoretical studies with gas-containing liquids have shown that rheological and relaxation properties of gas-liquid systems under pre-transition conditions (i.e., in pressures exceeding the saturation pressure, but close to it) are largely determined by the presence of “micro germs” – the smallest gas bubbles, the cooperative action of which manifests itself when approaching the saturation pressure [1]. When solving many practical problems, it becomes necessary to study the wave propagation with regard to the influence of the interaction between the liquid and the wall of the deformable pipe. In this case, the system consisting of a deformed body, liquid, and gas is obtained; therefore, the study should be carried out with regard to the interaction forces between the deformed body, liquid and gas.

Read More### OPTIMIZATION OF PROCESSING INPUT INFORMATION IN THE INDUSTRIAL FISHERIES MONITORING SYSTEM

The article is devoted to the estimation of the effectiveness of the distributed input data processing model in sectorial fisheries monitoring system (SMS) and the justification of the processing algorithm based on simulation modeling with regard to the performance of the SMS software. Modeling the processing of the input data flow of the SMS enabled the comparison of different processing models in several system operation modes. The results of numerical experiments allowed selecting and implementing the most appropriate scheme based on parallel processing. In general, due to the developed flexible data processing system, SMS has shown its efficiency, resilience to failures, and reliability in providing the users of the system with analytical information.

Read More### MODELING OF IMAGES OF NON-COHERENT OBJECTS IN TURBU-LENT ATMOSPHERE

The application of the pre-ray-tracing method for solving the radiation transfer equation for the numerical simulation of images of incoherent objects in a turbulent atmosphere is dis-cussed. The accuracy of this method is estimated under various conditions of radiation propaga-tion as well as its effectiveness.

Read More### IMPULSE RESPONSE FUNCTION OF THE ACTION POTENTIAL IN NERVE FIBER

Behavior of the impulse signal was analyzed during the propagation process in the nerve fiber. An infinitesimal duration signal which is used as an axon input value has a view of asymmetrical impulse in space and time cross section. The maximum of pulse decreases and width of signal increases when coordinate and time increase.

Read More### ROLE AND CHARGED SURFACES dEBYE TEMPERATURE IN TRIBOLOGY

It is shown that the charged metal surfaces influence on friction coefficient and chemical processes. The adhesive component of friction, the surface parameters of the metal layers are closely related to the Debye temperature of the metal element.

Read More### NONLINEAR OSCILLATOR WITH HYSTERESIS PROPERTIES

The article describes a mathematical model of a nonlinear oscillator with hysteresis properties. We got the qualitative aspects of irregular oscillations in nonlinear dynamical system.

Read More### SOLUTION OF HOMOGENEOUS RIEMANN BOUNDARY-VALUE PROBLEM WITH A CONDITION ON A REAL AXIS AND AN INFINITE INDEX OF LOGARITHMIC ORDER WITH THE NEW METHOD

We consider the homogeneous Riemann boundary-value problem with the boundary condition on the real axis for a function analytic in the complex plane except for points of the real axis. In the boundary condition, the limit value of the desired analytic function at any point on the real axis when approaching from above is represented as the product of the value of a given function called the coefficient, and the limit value of the function at the specified point at the bottom approaching. We assume that the coefficient modulus satisfies the Hölder condition everywhere on the real axis, including the infinitely distant point, and the coefficient argument satisfies the Hölder condition on any finite part of the axis and increases indefinitely as the degree of logarithm coordinates of the axis point with unlimited distance from the origin. The authors derived the formula that determines an analytic function in the upper half-plane the imaginary part of which as the coordinate of the axis point tends to positive infinity is infinitely large of the same order as the argument of the coefficient of the boundary condition. Next, the corresponding function is constructed in the lower half-plane, then analytical functions are introduced the imaginary parts of which turn into the infinity of the same order as the argument of the coefficient of the boundary condition when the points of the negative real axis are removed to infinity. The use of these functions allows us to eliminate the infinite gap of the argument of the coefficient of the boundary condition in the same way as it is done in the case of finite discontinuities of this coefficient. Based on techniques similar to those used by F.D. Gakhov, the problem is reduced to a problem with a boundary condition on the real axis and a finite index. Gakhov’s method is used to solve the last problem. The solution found depends on an arbitrary entire function of order zero, the modulus of which is subject to additional conditions, while in the case of a finite index the solution of the problem depends on an arbitrary polynomial of degree not higher than the index of the problem.

Read More### STUDY OF MN CONCENTRATION EFFECT ON PHASE-FORMATION, DIMENSIONS AND OPTICAL PROPERTIES OF BiFe1-xMnxO3 NANOPARTICLES

Nanoscale multiferroic powders BiFe1-xMnxO3 (x = 0.00, 0.05, 0.075) synthesized by the sol-gel method are the object of the study. Samples are calcined at appropriate temperatures and for a specified period. Phase analysis of the samples is performed with the use of x-ray diffraction (XRD). The morphology of BiFe1-xMnxO3 (x = 0.00, 0.05, 0.075) powder particles is investigated using a scanning electron microscope (SEM). The spectra of UV-visible absorption of the samples are obtained using a spectrophotometric system Cary 5000 UV-Vis-NIR Spectrophotometer. The results show that Mn doping with a certain ratio will help to remove the secondary phase and create samples with single-phase BFO, and this will lead to a decrease in particle size. The study of the absorption spectra in the UV-visible region of the fabricated samples showed that Mn doping expanded and shifted the absorbing edge of the samples towards large wavelength, and reduced the width of the forbidden band. This will increase the photocatalytic activity of the BFO material system, which will make the application more practical.

Read More### SYMMETRIC GROUP AND ITS GENETIC CODE

The article contains the description of the genetic code of the symmetric group. A new approach to the construction of the genetic codes of the symmetric group is proposed. Based on this approach, an unambiguous representation of the elements of the group in the form of a product of cycles is obtained. Using this representation, we studied some properties of the Sn group. The representation of the elements of the group as a monomial allows constructing orthogonal bases in the space of complex-valued functions on the group.

Read More### DEVELOPMENT OF METHODS FOR INCREASING THE LENGTH OF A FIBER-OPTICAL COMMUNICATION CHANNEL OF A QUANTUM-CRYPTOGRAPHIC SYSTEM

The article discusses the method for increasing the length of the fiber-optic communication channel of a quantum-cryptographic system. The article also discusses the technical side of the physical limitation of the length of the fiber-optic communication line: calculation, analysis and visualization in the form of graphs of the key dependences of the visibility and quantum error rate on the length of the fiber communication line were carried out. The installation of the quantum repeater in a quantum cryptosystem is considered as the method to increase the length of the fiber-optic communication channel.

Read More### ALTERNATIVE WAY OF SETTING PERMEABILITY WHILE HISTORY MATCHING IN HYDRODYNAMIC MODEL

The article is devoted to the application of an alternative way of setting permeability while history matching in hydrodynamic model. In this paper, we considered a method of determining the dependence of permeability on porosity based on field data using Dupuit and Joshi formulas. The research showed that this method allowed us to obtain the dependence of permeability on porosity with good accuracy of approximation, which had a positive effect on the first run of the hydrodynamic model, as well as its history matching.

Read More### APPROXIMATIVE PROPERTIES OF PROXIMAL SUBSPACES OF INFINITE DIMENSION

For subspaces L of infinite dimension in a Banach space, the authors obtained the characteristic properties of the existence of elements of the best approximation. As an application, they prove that, in the space 10-06-2019 12-52-57 of continuous functions on a connected Hausdorff compactum T, the Chebyshev subspace 10-06-2019 12-53-43 of infinite dimension, the annihilator 10-06-2019 12-53-56 of which is separable and contains the minimal total subspace, is a hyperplane 10-06-2019 12-54-24 of a strictly positive functional 10-06-2019 12-54-37.

Read More### 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.

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