Applied Mathematics and Control Sciences

Two nonlinear problems in terms of abstract operator equations of the form Bx = f are investigated in this paper. In the first problem the operator B contains a linear differential operator A , the Volterra operator K with kernel of convolution type and the inner product of vectors g ( x )Ф( u ) with nonlinear bounded functionals Φ. The first problem is given by equation Bu ( x ) = Аu ( x ) - Ku ( x ) - g ( x )Ф( u ) = f ( x ) with boundary condition D ( B ) = D ( A ). In the second problem the operator B contains a linear differential operator A and the inner product of vectors g ( x )F( Аu ) with nonlinear bounded on C [ a , b ] functionals F , where F ( Аu ) denotes the nonlinear Fredholm integral. The second problem is presented by equation Bu = Аu - gF ( Au ) = f with boundary condition D ( B ) = D ( A ). A direct method for exact solutions of nonlinear integro-differential Volterra and Fredholm equations is proposed. Specifically, the three theorems about existing exact solutions are proved in this paper. The first theorem is mean that for nonzero constant α0 Volterra integro-differential equation Аu ( x ) - Ku ( x ) = 0 is reducing to Volterra integral equation and has a unique zero solution. During it the operator A - K is closed and continuously invertible. Also, if the functions u ( t ), g ( t ) and f ( t ) are of exponential order α then nonhomogeneous equation Аu ( x ) - Ku ( x ) = f ( x ) for each f ( x ) has a unique solution, shown in this paper. The second theorem is mean that for the first investigated problem with an injective operator A - K , for f ( x ) and g ( x ) from C [ a , b ], the exact solution is given by equation u = ( A - K )- 1 f +( A - K )- 1 g b* for every vector b* = Ф( u ) that solves nonlinear algebraic (transcendental) system of n equations b = Ф(( A - K )- 1 f +( A - K )- 1 g b). And if the last algebraic system has no solution, then investigated problem also has no solution. The third theorem is means that exact solution of the second investigated problem is given by u = A - 1( f+g d*) for every vector d* = F ( Au ) that solves nonlinear algebraic (transcendental) system of n equations d = F ( f + g d). In this case we have same property - if the last algebraic system has no solution, then investigated problem also has no solution. Two particular examples for each considered problem are shown for illustration of exact solutions giving by perform the suggested in this paper methods. In the first example was considered integro-differential Volterra and Fredholm equation and in the second case was considered equation with nonlinear Fredholm integral.

The problem of optimizing the distribution of an integer resource (funds) by tasks (activities, goals) is investigated. The methods investigating this problem relate to the field of combinatorial optimization, namely, to the tasks of assigning goals. The known methods for solving this problem are numerical, selective, approximate, require a large number of iterations, do not involve checking the conditions for the existence of an integer solution, in some cases they can produce a solution not only far from optimal, but also violating the range of acceptable values of variables. The purpose of this work is to develop a new analytical method for solving the problem of the distribution of integer resources by the method of indefinite Lagrange multipliers. To do this, the allocated resources are represented as the sum of the integer and fractional parts of the number. The conditions are formulated and proved when the fractional parts of the variables of the solution of the problem are zero, that is, it is an integer. A theorem (criterion for the existence of an integer solution) is proved, which determines the necessary and sufficient conditions under which the solution of the problem exists and is found according to the algorithm developed in the article. Such conditions include the homogeneity of resources, as well as additional conditions (restrictions on integers and positivity of additional derived formula conditions of the problem). It is shown that the obtained solution of the problem corresponds to the maximum of the objective function. An algorithm for finding an integer solution to the problem of resource allocation by the method of indeterminate Lagrange multipliers is developed and a specific example is analyzed. The method described in this article can be used for the allocation of resources in industrial production, agriculture, organizational management systems, educational process, solving issues of target allocation in military affairs, building information systems, techno sphere security, emergency response, creating systems for the protection of objects and alarm systems. In this case, it is necessary to adapt it to the problems and tasks under consideration. It can also be used for the distribution of life-supporting resources: food, clothing, heat, electricity, gas, water supply.

Social networks began to play an important role in the informatization of society. Experts from all over the world are researching social network data to solve various tasks, such as creating popular content, conducting advertising campaigns, meeting the information needs of society, ensuring state security, etc. The analysis of social networks is understood as the solution of such tasks as determining the tonality of the text, determining the target portrait of the audience, searching for associative rules, calculating community performance indicators and data visualization. The article considers the relevance of solving the problem, analyzes the results of previous work, examines the audience's reaction to content, builds a target portrait of subscribers of various communities, examines the relationship between user interests. The initial data of the study are social networks, or rather informational messages, opinions, subnets and communities, individual users, external nodes.The paper considers the classification of social network analysis systems (such as Brand Analytics, IQBuzz, Agorapulse, Semantic Force, Talkwalker) according to the following criteria: users, analysis methods, objects of analysis, data sources, features.To determine the audience's reaction to the content, the method of determining the tonality of the text was applied by analyzing comments to the content. The cluster analysis method was used to determine the target profile of users in a particular community. To find patterns between the user's interests in the work, the frequency analysis of sets of elements was considered. The search for associative rules was carried out using the Apriori algorithm. As a result, the works are presented in the form of graphs and diagrams. In the course of the work, an integrated approach to solving problems was used, which made it possible to create an automated information and analytical system that can be used as analytical tools in this area.

This paper examines the main generally accepted market concentration indices adapted to the sub-industry structure. These indicators can be metrics for determining the dominant sub-sectors, which can be used to analyze cluster interactions. In fact, indicators can serve as an indicator of the potential for the development of cluster interaction. The current study hypothesizes that if there is one dominant sub-sector in the region, then enterprises that are "representatives" of such a sub-sector, having the most significant weight in the formation of this sub-sector and the industry as a whole, will influence the change in the GRP of the region much more than other enterprises not from the dominant industry. Thus, the paper exploresthe relationship between these metrics and the rate of gross regional product per capita, as one of the key indicators of regional development. At the first stage, a neural network is built and trained to identify a pattern between one of the metrics and the rate of gross regional product. Further, approximating by n-th order polynomials, various specifications of regression equations are considered, between all metrics and the change in the rate of gross regional product. The assumption is made that only the diversification of production does not lead to the socio-economic development of the region, but the creation and development of cluster interaction allows to increase the rate of gross regional product.