The effective ensure of sustainable development of regions, including the Tula region, characterized by an unfavorable demographic situation, the presence of environmental problems and stagnation of industrial production, without the use of economic and mathematical apparatus and tools for data analysis and decision-making is difficult to implement. At the same time, most decisions are made on the basis of expert assessments, and this is typical for most Russian regions. The development and implementation of modern decision-making methods based on multi-criteria optimization will increase the validity of such decisions at various levels of management, and which can be applied not only in the Tula region, but also in other regions to solve their problems. The purpose of the study is to develop and test a method for optimizing the results of the functioning of socio-ecological-economic systems on the example of the regions of the Central Federal District and the district as a whole on the basis of the author's methodology within the framework of a multi-level optimization approach. The results of the study show that due to changes in values of factors included in the optimization model of the socio-ecological-economic systems functioning, which was developed on the basis of the author's methodology. It is possible to improve the target indicators of the development of the Tula region. The presented methodology can be used for other regions, which expands the scope of its application.

At the beginning of the twentieth century, the fundamental problems in group theory from the point of view of the existence of algorithms took shape. These include the problems of words, as well as the conjugacy of arbitrary words (formulated by M. Den). In addition, the problem of isomorphism for groups is posed (formulated by G. Titze). Russian algebraists P.S. Novikov and S.I. Adyan were able to obtain a proof of the unsolvability of these problems for all groups belonging to finitely defined groups. After the publication of these results, algorithmic problems, together with their various generalizations, are investigated already in fixed classes of groups. As one of the generalizations of the problem for conjugacy of arbitrary words, the question of conjugacy of arbitrary subgroups is posed, namely, the question of finding an algorithm that, by arbitrary subgroups given by a finite set of generating and defining words of some group, could determine whether they are conjugate in this group or not conjugate. As another generalization of the conjugacy of words, we can consider generalized conjugacy for words, which allows us to determine an algorithm capable of determining from arbitrary finite sets of words from a certain group whether they will be conjugated in it or not. If both of these generalizations are combined into one, then we get a generalization of the conjugacy problem of subgroups. Artin groups appeared a long time ago, they include wellknown braid groups that have been studied from an algebraic point of view since the twenties of the last century. The solution of the problems under consideration in Artin's groups caused great difficulties, which led to the allocation of various subclasses. The article proves the solvability of generalization of the conjugacy problem of subgroups in Artin groups on two generating ones.

This article introduces a new class of games, which we will call G-games. One of the representatives of this class is the famous Banach–Mazur game. Some properties of convexity and smoothness of Banach spaces in connection with G-games are analyzed. An overview of mainly dynamic games developed by a number of authors in recent years is presented, the application of which can be expanded beyond the scope of the situations under consideration and used to analyze other processes.