### L-CURVE METHOD FOR EVALUATING THE OPTIMAL PARAMETER OF A SMOOTHING CUBIC SPLINE

A universal device for filtering measuring noise is a smoothing cubic spline of defect 1. The magnitude of the filtering error (smoothing) is mostly determined by the value of the smoothing parameter. With the optimal smoothing parameter value, the smoothing error value takes a minimum value. In practice, there is no a priori information on the exact (not noisy) values of the signal, and it is impossible to calculate the value of the optimal smoothing parameter. In this regard, the algorithms used to solve practical problems for choosing the optimal parameter allow for estimating only acceptable smoothing errors, which sometimes significantly exceed the minimum values. The estimation of the optimal smoothing parameter with an unknown variance of the noise measurement of experimental data presents particular difficulty. The current article constructs and examines in detail an algorithm for estimating the optimal parameter based on the L-curve method. This method is used to select the regularization parameter in algorithms for solving incorrectly set tasks. Particular attention is paid to the estimation of the optimal parameter in conditions of correlated noise. Based on the results of these studies, the authors provide practical recommendations for the application of this selection algorithm in the practice of processing experimental data.

Read More### IDEMPOTENT MODEL OF THE CLASSICAL ASSIGNMENT PROBLEM

The main principle of idempotent algebra is observed in this article, the main advantages of idempotent approach are given. An idempotent model of a classical assignment problem is given, this model makes even non-linear assignment problems become linear ones.

Read More### A NEW THEORY OF MAGNETISM

In this article the – criticism of Maxwell’s theory, called the cause of terrestrial magnetism, discovered a new magnetic force, and drew an analogy between the magnetic field and the gravitational field, amended in Newton’s law of universal gravitation.

Read More### THE HYDROACOUSTIC RECEPTION EXTENDED PARAMETRICAL ARRAY

In work use of the flexible extended array on the basis of a wave guide in the form of a flexible wave guide with filling with any liquid is offered, distribution of acoustic fluctuations to a round wave guide is considered. Change of phase speed in a wave guide is analyzed at

influence on it of a low-frequency signal. Schemes of processing signals are considered.

### DEFINITION OF MODE SHAPES OF ACCELEROMETER

The article describes the pendulum accelerometer. The article describes the pendulum accelerometer for which solved the problem of determining the displacement fields of sensing element with the action of electrostatic forces. Eigenfrequencies and mode shapes are identified. It is shown that the first three of them are basis modes shapes.

Read More### BLAST BALANCING

The article covers practical results of industrial blasting seismic waves’effect to crushing of roc and safety of buildings and structures. Results of comparative industrial blasting approved that the application of “blast balance point” method is the most optimal for blasting operations.

Read More### ROLE OF PREVAILING PHASE IN FORMATION OF MECHANICAL PROPERTIES OF ALLOYS OF COMPLEX CHEMICAL COMPOSITION

With the purpose of increasing of forecasting efficienсy of mechanical properties indicators (MPI) of alloys of complex chemical composition the method based on comparison of available experimental data about MPI with calculating state parameters of their most soft, binding and prevailing phase is worked out. Application of the method is perspective for prevision of properties of the metal coverings

received by ionic implantation, deposition, fusion.

### ON THE SOLVING THE SECOND-ORDER VERHULST DIFFERENTIAL EQUATION WITH PERTURBED INITIAL CONDITIONS

The article examines a dynamical system whose behavior is described by a second-order differential equation where the fundamental system of solutions consisting of functions of exponential type is replaced by bounded functions of the Verhulst model. The author investigates the question of the dependence of the behavior of a particular solution of the constructed second-order differential equation on the perturbation of its initial conditions. The general form of the perturbation is assumed to have the form of a polynomial of the second degree, depending on the auxiliary parameter µ. The obtained results are illustrated by a test example with specific numerical data and corresponding graphs. The MathCAD and Wolfram Alpha software packages are used as auxiliary tools for performing calculations.

Read More### DREAMS’ MOTIVTIONS

It was shown that a dream is a mental model of a future state of a part of surrounding world, important for individual, presented in the shape of possible development scenarios. Over a hundred thousand years, creating such scenarios has become one of the basic human needs. A mathematical model for determining an individual’s tendency to dream in different periods of life was elaborated and investigated.

Read More### MODELING OF RECTILINEAR TRAJECTORY-WAVE MOTION OF AN OBJECT

The current study formulates the local variational principle (LVP), the theorem on the necessary and sufficient condition for the existence of a V-function, the wave and trajectory theorem, examines the direct and inverse problems of dynamics in a new formulation, which are the essence of the V-function method. The V-function method indicates that the wave motion of an object is inextricably linked with its trajectory motion. The trajectory motion of an object is described by a system of ordinary differential equations, and the wave motion is modeled by a second-order partial differential equation. On the basis of the V-function method, the optical-mechanical analogy receives a new continuation. It is shown that in the physical sense, the wave amplitude is determined by an action that manifests itself in the movement of an object (particle). It is established that the energy levels obtained using the V-function method for the case of rectilinear uniform motion of a quantum object completely coincide with the classical Schrodinger results for a harmonic oscillator.

Read More### ON THE STUDY OF A SYSTEM CLOSE TO A HARMONIC OSCILLATOR VIA APPROXIMATE POINT MAPPING

The theory of nonlinear oscillations is one of the most important components of modern science. Nonlinear oscillation is a term usually applied to oscillatory phenomena that occur in nonlinear dynamical systems. The most accessible for research are oscillatory systems with low nonlinearity, for which various asymptotic methods have been developed. Moreover, the study of nonlinear systems close to a harmonic oscillator is still of particular interest. The current study assesses the possibility of studying a system close to harmonic oscillators by via approximate point mapping, both equations of which contain nonlinear terms. The article provides explicitly defined functions of the sequence of a point map, in the construction of which asymptotic methods are used, as well as the results of their analysis.

Read More### STM PARAMETERS OF SEMICONDUCTOR COLLOIDAL QUANTUM DOTS

Based on the studies of the tunnel-emission volt-ampere characteristics of colloidal quantum dots CdSe, PbS, InSb and the single-electron transport model, the article determines the most important parameters of quantum dots such as the effective mass of the electron, the permittivity, the energy of the potential tunneling barrier, the linear size of the quantum-limited resonant electron motion (the size of the quantum dot). A comparison of the obtained parameters of quantum dots with similar ones for bulk semiconductors shows their good correspondence, which may indicate both the validity of the obtained parameters and the possibility of using models and properties of bulk semiconductors in relation to quantum dots. Based on the data obtained and with the continuation of the research, it is possible to create a method along with a software and hardware complex for industrial control of the main parameters of quantum dots.

Read More### ON THE DIAGONALIZABILITY CONDITIONS OF A PERTURBED DIFFERENCE OPERATOR IN SOME SPACES

The current paper uses the method of similar operators to obtain the conditions for reducing the matrix of a difference operator of the form 03-09-2021 10-27-01 to a diagonal or block-diagonal form in the standard basis of space 03-09-2021 10-27-11 and estimates its spectral characteristics. The paper presents the main definitions of the method used. In accordance with the method of such operators, the operator A is represented in the form 03-09-2021 10-27-24 where the matrix operator 03-09-2021 10-27-34 has a diagonal structure, and B is the perturbation operator. The conditions for the perturbation operator are considered in cases when this operator belongs to three different spaces. The asymptotic representation of eigenvalues, estimates of eigenvectors, and elements of matrices of spectral projectors are also obtained.

Read More### CALCULATION OF FACTORS IN PONDEROMOTIVE ELEKTROPLASTICHNOSTI MONOCRYSTAL SB.

In this paper the calculations of the intrinsic magnetic field of the current sample of antimony.

Read More### DEVELOPMENT OF CONTENTS AND ACTIVITIES FOR IMPROVING THE EFFICIENCY OF THE PROBLEM OF SOFTWARE AND HARDWARE “DISPLAY STAND”

In the article – the creation of «Display stand», increasing sales efficiency by reducing staff while increasing the quality of the advertising information.

Read More### SOLUTION OF THE EFFICIENCY OF SALE SHOP BY IMPROVING THE HARDWARE “DISPLAY STANDS”

The paper considers the problem of increasing the effectiveness of sales shop and an improved hardware complex “Display stand”.

Read More### OPTIMAL OPERATION MODES OF DUAL MAGNETRONS FOR DEPOSITION TiO2 THIN FILMS

The investigation of optical properties of TiO2 thin films, obtained by a dual magnetron sputtering system (MSS), shows that optical characteristics of films depend on their structure and determine by operation mode and configuration of the magnetic field of the magnetron.

Read More### STRUCTURE AND THERMODYNAMICS PROPERTIES OF EXPLOSIVE TATB, PETN AND TATT

One of the most perspective ways in increasing the safety of explosive works, in particular, in coal-mining industry, is the use of laser initiation technologies, which is able to eliminate the possibility of functioning of the system of initiation in unauthorized cases, and its assured functioning in the sanctioned ones. In this work the substances meeting the practical requirements are being researched. Such substances are TATB, PETN and TATT. The research reveals the geometrical and thermodynamics properties of these molecules.

Read More### DIELECTRIC STUDY OF COMPOSITE MATERIALS FILLED WITH FERROELECTRIC CERAMICS

The paper views the results of the study of temperature and frequency dependence of the dielectric parameters in polymer composites based on polyethylene and barium titanate. With the introduction of ferroelectric into the polymer matrix an increase in the dielectric constant and dielectric loss tangent was found.

Read More### HEAT EXCHANGE BETWEEN THE PLASMA TORCH JET AND THE HEATED SURFACE OF THE ALUMINA MELT FOR NITROGEN, ARGON AND HYDROGEN

The article evaluates the efficiency of heat transfer between the plasma torch jet and the heated surface of the alumina melt in order to select the optimal working gas for the production technology of pure white corundum. This technology is being developed for the purpose of its industrial application with a corundum capacity of 50 kg per hour. According to this technology, corundum crystals are obtained by melting alumina powder in a reactor with a plasma torch jet. Nitrogen, argon, and hydrogen are considered as the working gas of the plasma torch. The calculations use estimates of the thermophysical characteristics of nitrogen, argon, and hydrogen at the operating temperature of the plasma torch, which approximately amounts to 8000K. The jet velocity and Nusselt numbers characterizing heat transfer were calculated taking into account the required performance. According to the results of calculations, the most effective gas for heating and melting alumina is nitrogen because of its incomplete dissociation at the operating temperature of the plasma torch and, as a result, higher thermal conductivity and specific enthalpy.

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