Algorithm for Calculating the Global Minimum of a Smooth Function of Several Variables

Every year the interest of theorists and practitioners in optimisation problems is growing, and extreme problems are found in all branches of science. Local optimisation problems are well studied and there are constructive methods for their solution. However, global optimisation problems do not meet the requirements in practice; therefore, the search for the global minimum remains one of the major challenges for computational and applied mathematics. This study discusses the search for the global minimum of multidimensional and multiextremal problems with high precision. Mechanical quadrature formulas, that is, the formulas for approximate integration were applied to calculate the integrals. Of all the approximate integration formulas, the Sobolev lattice cubature formulas with a regular boundary layer were chosen. In multidimensional examples, the Sobolev formulas are optimal. Computational experiments were carried out in the most popular C++ programming language. Based on the computational experiments, a new algorithm was proposed. In three-dimensional space, the calculations of the global minimum have been described using specific examples. Computational experiments show that the proposed algorithm works for multiextremal problems with the same amount of time as for convex ones.

Multi-Integral Representations for Associated Legendre and Ferrers Functions

For the associated Legendre and Ferrers functions of the first and second kind, we obtain new multi-derivative and multi-integral representation formulas. The multi-integral representation formulas that we derive for these functions generalize some classical multi-integration formulas. As a result of the determination of these formulae, we compute some interesting special values and integral representations for certain particular combinations of the degree and order, including the case where there is symmetry and antisymmetry for the degree and order parameters. As a consequence of our analysis, we obtain some new results for the associated Legendre function of the second kind, including parameter values for which this function is identically zero.

On Some Numerical Integration Formulas on the d-Dimensional Simplex

In this paper, we consider the problem of the approximation of the integral of a function f over a d-dimensional simplex S of $$\mathbb {R}^{d}$$ R d by some quadrature formulas which use only the functional and derivative values of f on the boundary of the simplex S or function data at the vertices of S, at points on its facets and at its center of gravity. The quadrature formulas are computed by integrating over S a polynomial approximant of f which uses functional and derivative values at the vertices of S.

The role of regional integration in the international balance of power The European Union and the Shanghai Organization as a model

The various regional integration formulas, have become prevalent since the second half of the twentieth century under the influence The need dictated by the data of growth and the economic interests of the members of integration, the conviction dictated by economic objectivity based on theories of integration and economic cooperation and its positive economic effect, or data of change and the balance of international powers and the nature of The global system, including. The end of the Cold War helped the power mechanisms witness a temporal and spatial change that made the economic and cognitive power the rest of the power variables. And he identified the new power structures and imposed new rules in the interaction of countries, and produced patterns of relations and different interactions in terms of type, goals and roles, and established a new style in the mechanisms of influence and power and making regional and international poles after pooling capabilities in larger structures in which roles and resources are distributed according to economic logic. Hence, it is possible to focus on two models of integration forms, namely the European Union and the Shanghai Organization. In this framework, two levels of research were focused on the outcomes of integration at the level of members and the bloc in general, and the second focuses on the roles that integration provides in the field of regional and international balance, where it will be imposed New integration force factors are a new pattern of power balance.

TOP EVALUATION FOR THE RA TION FOR THE RATE OF FUNC TE OF FUNCTIONAL OF ERROR AL OF ERROR WEIGHT CUBATURE FORMUL TURE FORMULA IN SP A IN SPACE

An upper bound is obtained for the norm of the error functional of the weight cubature formula in the Sobolev space . The modern formulation of the problem of optimization of approximate integration formulas is to minimize the norm of the error functional of the formula on the selected normalized spaces. In these works, the problem of optimality with respect to some definite space is investigated. Most of the problems are considered in the Sobolev space

Research on Students’ Error Pattern in Solving First Order Differential Equations

First order differential equations can be classified as separable, linear, exact, homogeneous and Bernoulli. Each type has a very systematic method of solution. An analytic method of solution is offered the student for each class of equation whereas integration is essential in the solution process. Hence integration, formulas and steps are important in these kinds of approaches. This study aims to investigate students’ error pattern in solving first order differential equations and focused only to separable, homogeneous and Bernoulli. A test consisting of the three different first-order differential equations was prepared for students. 41 students were asked to solve the equations on the test using appropriate methods. The 41 scripts were examined with a focus on the integration techniques used and the final answers given by students. The results were analyzed using IBM SPSS Statistics 23 and the items such as frequency, mean and standard deviation are used to assess students’ understanding and their ability to solve first order differential equations.

Two approaches for automating analysis of electromagnetic processes in non-linear circuits with valves

This article analyses and compares two approaches related to automating modeling of valve electric circuits with piecewise linear approximation of valve characteristics. The first approach is based on operator equivalent circuits’ analytical formulas and on analytical expressions programming which describe equivalent circuits on each conduction interval of valve elements. The second approach provides implementation of a system for modeling valve converters based on implicit numerical integration formulas.