Chaotic Dynamics in Generalized Rabinovich System

The article is devoted to the study of the global behavior of the generalized Rabinovich system describing the process of interaction between waves in plasma. For this generalized system, we obtain the positive invariant set (ultimate bound) and globally exponential attractive set using the approach where we transform the initial problem of finding the corresponding set to the conditional extremum problem and solve this problem. Furthermore, the rate of the trajectories going from the exterior of the attractive set to the interior of the attractive set is also obtained. Numerical localization of attractor is presented. Meanwhile, the volumes of the ultimate bound set and the global exponential attractive set are obtained, respectively. The main innovation of this article lays in considering the generalized form of the Rabinovich system with [Formula: see text] and obtaining the results on the ultimate bound set and globally exponential attractive set for this general case.

Optimization of the combined explosion hardening processes

The proposed method for calculating the loading parameters makes it possible to determine the wear parameters after explosion hardening. The calculation method is simple and less time consuming compared with calculation methods that involve the use of nonlinear programming methods. The main methods of increasing the wear resistance of mining equipment parts using explosion methods are generalized. The reserve for increasing the wear resistance consists in the optimization of deformation parameters during the power and thermal intensification of processes and the development of new methods and technologies of hardening. The factors (parameters) of the studied processes: explosive cladding, alloying, hardening, are formulated. Optimization of the processes under consideration is possible by decomposing the process into simpler ones with subsequent optimization of the parameters of these processes and the synthesis of the obtained solutions. For the first time, a solution to the multicriteria problem of two-stage explosion hardening is presented. It is proposed to split the process into simpler ones. Optimization criteria are proposed for each of the simplified processes. The problem is reduced to a conditional extremum problem, which is solved by composing the Lagrange function. By transforming the wear equation, the optimal ratio of strength and ductility for parts operating under abrasive wear conditions is determined.

Ж.-Л. ЛАГРАНЖ КАК ОДИН ИЗ ОСНОВОПОЛОЖНИКОВ ТЕОРИИ ЭКСТРЕМУМОВ ФУНКЦИЙ МНОГИХ ПЕРЕМЕННЫХ

In the article we carried out a detailed analysis of the results obtained by J.-L. Lagrange in his first. The theory of extrema of functions of many variables, as part of mathematical analysis, refers to the mathematical foundations of the study of operations. In turn, many optimization problems are actually problems on the conditional extremum of the function of many variables. The relevance of this topic is determined by the fact that the methods for solving problems on the extremum of the function of many variables obtained in the mid 18th - early 20th centuries are used in solving modern problems. A special place here is occupied by L. Euler and J.-L. Lagrange. The aim of the article is to study the conditions for the maximum and minimum functions of many variables obtained by J.-L. Lagrange, and a comparison of its results with the presentation of this topic in modern textbooks on higher mathematics and mathematical analysis. It was established that in his first printed work he first formulated and proved sufficient conditions for the existence of an extremum of the function of many variables by actually establishing a criterion for the positive (negative) definiteness of quadratic forms, long before it appeared in J. Sylvester in the mid-19th century. A comparative analysis of the results of L. Euler and J.-L. Lagrange. It was found that sufficient conditions for the existence of an extremum of functions of many variables obtained in the first printed work of the young Lagrange are included in all modern textbooks in virtually the same form. The examples shown illustrate his theory. These are tasks of geometric and physical content. Special cases are considered in detail: functions of two and three variables. It is noted that this article became programmatic for the young Lagrange, although it remained unnoticed by his contemporaries. Subsequently, based on the method he obtained, he created the variational calculus, using the principle of least action and the theory of extrema, derived the basic laws of mechanics, the rule of factors for finding the conditional extremum of functions of many variables, which is named after him.

Development of a Modified Self-Organizing Migration Optimization Algorithm (MSOMA)

The modified self-organizing migration optimization algorithm (MSOMA) based on a self-organizing migration algorithm (SOMA) is suggested. An algorithm for solving the problem of finding the global conditional extremum of the objective function on a given set is developed. Examples illustrating the application of the algorithm and created software are given.

SEARCH EXTREME FUNCTIONAL

Examples illustrate the developed procedure for checking the sufficient condition for the extremum of a functional (construction of its mathematical image); in the problem of the calculus of variations on a conditional extremum, the role of the Lagrange multiplier is estimated and the surface of the functional is constructed in terms of the parameters of variations of admissible functions of a normalized linear space.

Statistical method of steepest improvement of response

A new statistical method for response steepest improvement is proposed. This method is based on an initial experiment performed on two-level factorial design and first-order statistical linear model with coded numerical factors and response variables. The factors for the runs of response steepest improvement are estimated from the data of initial experiment and determination of the conditional extremum. Confidence intervals are determined for those factors. The first-order polynomial response function fitted to the data of the initial experiment makes it possible to predict the response of the runs for response steepest improvement. The linear model of the response prediction, as well as the results of the estimation of the parameters of the linear model for the initial experiment and factors for the experiments of the steepest improvement of the response, are used when finding prediction response intervals in these experiments. Kknowledge of the prediction response intervals in the runs of steepest improvement of the response makes it possible to detect the results beyond their limits and to find the limiting values of the factors for which further runs of response steepest improvement become ineffective and a new initial experiment must be carried out.

Optimal Recovery Method of the Solution of a One-Dimensional Wave Equation at Time Τ From Inaccurate Data at t=0 And t = T

The article deals with the problem of optimal recovery of the solution of the wave equation at some time instant by known, but given with some error, functions determining the shape of the string at times t and T. The goal of the paper is to construct an optimal recovery method for the solution of the wave equation from inaccurate data. An important assumption used in the work is the possibility of representing the solution in the form of a Fourier series. The main solution method is the introduction of an auxiliary extremal problem for a conditional extremum, the solution of which determines the optimal recovery method. The result of the work is to find the optimal recovery method among all possible methods. The solution of the restoration problem and the value of the error of optimal recovery are obtained. Cases are indicated when it is possible to reduce the amount of initial information required for solving the problem.