The paper considers the solution of a one-dimensional homogeneous equation of heat conduction in a multilayer. In this paper, the orthogonality of the obtained eigenfunctions is proved.

A linear parabolic equation with nonlocal boundary conditions of the BitsadzeSamarsky type is considered. The existence and uniqueness theorem of the periodic solution is proved.

Let operator G be compact positive operator acting in separable Hilbert space. According with theorem of Hilbert-Schmidt its characteristic numbers μn are positive finite multiple with unique limit point at infinity. In spectral problems of mathematical physics such numbers, as a rule, have power (Weyl’s) asymptotic. Sometimes it is more convenient to use asymptotic of counting function N(r) that is equal to number (taking into account the multiplicity) of characteristic numbers μn in the interval (0; r). For single eigenvalues recalculation of asymptotic formulas is a simple exercise. We prove several theorems on connection between asymptotic of μn and N(r) for an arbitrary compact positive operator G.

On antiperiodic boundary value problem for a semilinear differential inclusion of a fractional order q. The investigation of control systems with nonlinear units forms a complicated and very important part of contemporary mathematical control theory and harmonic analysis, which has numerous applications and attracts the attention of a number of researchers around the world. In turn, the development of the theory of differential inclusions is associated with the fact that they provide a convenient and natural tool for describing control systems of various classes, systems with discontinuous characteristics, which are studied in various branches of the optimal control theory, mathematical physics, radio physics, acoustics etc. One of the best approaches to the study of this kind of problems is provided by the methods of multivalued and nonlinear analysis, which are distinguished as very powerful, effective and useful. However, the solving of these problems within the frameworks of existing theories is often a very difficult problem, since many of them find sufficiently adequate description in terms of differential equations and inclusions with fractional derivatives. The theory of differential equations of fractional order originates from the ideas of Leibniz and Euler, but only by the end of the XX century, interest in this topic increased significantly. In the 70s - 80s, this direction was greatly developed by the works of A.A. Kilbas, S.G. Samko, O.I. Marichev, I. Podlubny, K.S. Miller, B. Ross, R. Hilfer, F. Mainardi, H. M. Srivastava. Notice that the research in this direction will open up prospects and new opportunities for the studying of non-standard systems that specialists encounter while describing the development of physical and chemical processes in porous, rarefied and fractal media. It is known that a periodic boundary value problem is one of the classical boundary value problems of differential equations and inclusions. At the same time, in recent years, along with periodic boundary value problems, antiperiodic boundary value problems are of great interest due to their applications in physics and interpolation problems.
 In this paper, we study an antiperiodic boundary value problem for semilinear differential inclusions with Caputo fractional derivative of order q in Banach spaces. We assume that the nonlinear part is a multivalued map obeying the conditions of the Caratheodory type, boundedness on bounded sets, and the regularity condition expressed in terms of measures of noncompactness. In the first section, we present a necessary information from fractional analysis, Mittag -- Leffler function, theory of measures of noncompactness, and multivalued condensing maps. In the second section, we construct the Green's function for the given problem, then, we introduce into consideration a resolving multivalued integral operator in the space of continuous functions. The solutions to the boundary value problem are fixed points of the resolving multioperator. Therefore, we use a generalization of Sadovskii type theorem to prove their existence. Then, we first prove that the resolving multioperator is upper semicontinuous and condensing with respect to the two-component measure of noncompactness in the space of continuous functions. In a proof of a main theorem of the paper, we show that a resolving multioperator transforms a closed ball into itself. Thus, we obtain that the resolving multioperator obeys all the conditions of the fixed point theorem and we prove the existence of solutions to the antiperiodic boundary value problem.

Mathematical models of diffusion and cathodoluminescence of nonequilibrium minority charge carriers generated by a wide electron beam in homogeneous and multilayer semiconductor materials are considered. The use of wide electron beams makes it possible to reduce these problems to one–dimensional ones and to describe these mathematical models by ordinary differential equations.

A model of motion of a dynamic system with the condition that the trajectory passes through arbitrarily specified points at arbitrarily specified times is constructed. The simulated motion occurs at the expense of the input vector-function, calculated for the first time by the method of indefinite coefficients. The proposed method consists in the formation of the vector function of the trajectory of the system and the input vector function in the form of linear combinations of scalar fractional rational functions with undefined vector coefficients. To change the shape of the trajectory to the specified linear combinations, an exponential function with a variable exponent is introduced as a factor.
 To determine the vector coefficients, the formed linear combinations are substituted directly into the equations describing the dynamic system and into the specified multipoint conditions. As a result, a linear algebraic system is formed.
 The resulting algebraic system has coefficients at the desired parameters only matrices included in the Kalman condition of complete controllability of the system, and similar matrices with higher degrees.
 It is proved that the Kalman condition is sufficient for the solvability of the resulting algebraic system. To form an algebraic system, the properties of finite-dimensional mappings are used:
 decomposition of spaces into subspaces, projectors into subspaces, semi-inverse operators. For the decidability of the system, the Taylor formula is applied to the main determinant.
 For the practical use of the proposed method, it is sufficient to solve the obtained algebraic system and use the obtained linear formulas. The conditions for complete controllability of the linear dynamic system are satisfied. To prove this fact, we use the properties of finite-dimensional mappings. They are used in the decomposition of spaces into subspaces, in the construction of projectors into subspaces, in the construction of semi-inverse matrices. The process of using these properties is demonstrated when solving a linear equation with matrix coefficients of different dimensions with two vector unknowns.
 A condition for the solvability of the linear equation under consideration is obtained, and this condition is equivalent to the Kalman condition. In order to theoretically substantiate the solvability of a linear algebraic system, to determine the sought vector coefficients, the solvability of an equivalent system of linear equations is proved. In this case, algebraic systems arise with the main determinant of the following form: the first few lines are lines of the Wronsky determinant for exponential-fractional-rational functions participating in the construction of the trajectory of motion at the initial moment of time; the next few lines are the lines of the Wronsky determinant for these functions at the second given moment in time, and so on. The number of rows is also related to the Kalman condition - it is the number of matrices in the full rank controllability matrix. Such a determinant for the exponential-fractional-rational functions under consideration is nonzero.
 The complexity of proving the existence of the trajectory and the input vector function in a given form for the system under consideration is compensated by the simplicity of the practical solution of the problem.
 Due to the non-uniqueness of the solution to the problem posed, the trajectory of motion can be unstable. It is revealed which components of the desired coefficients are arbitrary and they should be fixed to obtain motion with additional properties.

The article consists of two parts. The first part is devoted to general questions that are related to uncertainty: causes and sources of uncertainties appearance, classification of uncertainties in economic systems and approach to their assessment. In the second part the concept of maximin, based on the principle of guaranteed result (Wald’s principle) is considered. In this case, maximin is interpreted from viewpoint of two-level hierarchical game. On the basis of the maximin concept, a guaranteed solution in outcomes for K-stage positional single-criterion linear quadratic problem under uncertainty is formalized. An explicit form of the guaranteed solution for this problem is found

In this paper we consider classical Schur transformation and inverse Schur transformation for generalized Nevanlinna functions.

Oligopoly is a basic concept in the theory of competition. This structure is the central object of research in the economics of markets. There are many mathematical models of the market that are formalized in the form of an oligopoly in economic theory. The Cournot oligopoly is an elementary mathematical model of competition. The principle of equilibrium formalizes the non-cooperative nature of the conflict. Each player chooses the equilibrium strategy of behavior that provides the greatest profit, provided that the other competitors adhere to their equilibrium strategies. The Stackelberg model describes a two-level hierarchical model of firm competition. The top-level player (center, leader) chooses his strategy, assuming reasonable (optimal) decision-making by the lower-level players. Lower-level players (agents, followers) recognize the leadership of the center. They consider the center's strategies known. These players choose their strategies, wanting to maximize their payoff functions. This hierarchical structure is from a game point of view a case of a hierarchical game Gamma1. The indefinite uncontrolled factors (uncertainties) are the values for which only the range of possible values is known in this paper. Recently, studies of game models under uncertainty have been actively conducted. In particular, non-coalitional games under uncertainty are investigated. The concepts of risk and regret are formalized in various ways in the theory of problems with uncertainty. At the same time, the decision-maker takes into account both the expected losses and the possibility of favorable actions of factors beyond his control.\nThis article examines the two-level hierarchical structure of decision-making in the problem of firm competition. A linear-quadratic model with two levels of hierarchy is considered. This model uses the concepts of Cournot and Stackelberg under uncertainty. Uncontrolled factors (uncertainties) are identified with the actions of the importing company. The Wald and Savage principles are used to formalize the solution. According to Wald's maximin criterion, game with nature is seen as a conflict with a player who wants to harm the decision-maker as much as possible.\n\nSavage's minimax regret criterion, when choosing the optimal strategy, focuses not on winning, but on regret. As an optimal strategy, the strategy is chosen in which the amount of regret in the worst conditions is minimal. A new approach to decision-making in the game with nature is formalized. It allows you to combine the positive features of both principles and weaken their negative properties. The concept of U-optimal solution of the problem in terms of risks and regrets is considered.\nThe problems of formalization of some types of optimal solutions for a specific linear-quadratic problem with two levels of hierarchy are solved.

Formalizing routing problems of many traveling salesman (mTSP) in complex networks leads to NP-complete pseudobulous conditional optimization problems. The subclasses of polynomially solvable problems are distinguished, for which the elements of the distance matrix satisfy the triangle inequality and other special representations of the original data. The polynomially solvable assignment problem can be used to determine the required number of salesmen and to construct their routes. Uses a subclass of tasks in the form of pseudobulous optimization with disjunctive normal shape (\textit{DNS}) constraints to which the task is reduced mTSP. Problems in this form are polynomially solvable and allow to combine knowledge about network structure, requirements to pass routes by agents (search procedures) and efficient algorithms of logical inference on constraints in the form of \textit{DNS}. This approach is the theoretical justification for the development of multi-agent system management leading to a solution mTSP. Within the framework of intellectual planning, using resources and capabilities, and taking into account the constraints for each agent on the selected clusters of the network, the construction of a common solution for the whole complex network is achieved.

Publications on mathematical game theory with many (not less than 2) players one can conditionally distribute in four directions: noncooperative, hierarchical, cooperative and coalition games. The two last in its turn are divided in the games with side and nonside payments and respectively in the games with transferable and nontransferable payoffs. If the first ones are being actively investigated (St. Petersburg State Faculty of Applied Mathematics and Control Processes, St. Petersburg Economics and Mathematics Institute, Institute of Applied Mathematical Research of Karelian Research Centre RAS), then the games with nontransferable payoffs are not covered. Here we suggest to base on conception of objections and counterobjections. The initial investigations were published in two monographies of E.\;I.\;Vilkas, the Lithuanian mathematician (the pupil of N.\;N.\;Vorobjev, the professor of St. Petersburg University). For the differential games this conception was first applied by E.\;M.\;Waisbord in 1974, then it was continued by the first author of the present article combined with E.\;M.\;Waisbord in the book <<Introduction in the theory of differential games of n-persons and its application>> M.: Sovetskoye Radio, 1980, and in monography of Zhukovskiy <<Equilibrium of objections and counterobjections>>, M.: KRASAND, 2010. However before formulating tasks of coalition games, to which this article is devoted, we return to noncoalition variant of many persons game. Namely we consider noncooperation game in normal form, defined by ordered triple: $$G_N=\langle\mathbb{N}, \{X_i\}_{i \in \mathbb{N}}, \{f_i (x)\}_{i \in \mathbb{N}}\rangle.$$ Here $\mathbb{N}=\{1,2,\ldots , N\ge2\}$ "--- set of ordinal numbers of players, each of them (see later) chooses its strategy $x_i\in X_i\subseteq \mathbb{R}^{n_i}$ (where by the symbol $\R^k$, $k\ge 1$, as usual, is denoted $k$-dimensional real Euclidean space, its elements are ordered sets of $k$-dimensional numbers, as well Euclidean norm $\parallel \cdot \parallel$ is used); as a result situation $x=(x_1,x_2,\ldots,x_N)\in X=\prod \limits_{i\in \mathbb{N}}X_i\subseteq \mathbb{R}^{\sum \limits_{i\in \mathbb{N}}n_i}$ form in the game. Payoff functions $f_i (x)$ are defined on set $X$ of situations $x$ for each players: \begin{gather*} f_1(x)=\sum_{j=1}^{N}x_{j}^{'} D_{1j}x_{j}+2d_{11}^{'} x_1,\\ f_2(x)=\sum_{j=1}^{N}x_{j}^{'} D_{2j}x_{j}+2d_{22}^{'} x_2,\\ \ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\\ f_N(x)=\sum_{j=1}^{N}x_{j}^{'} D_{Nj}x_{j}+2d_{NN}^{'} x_N. \end{gather*} In the opinion of luminaries of game theory to equilibrium as acceptable solution of differential game has to be inherent the property of stability: the deviation from it of individual player cannot increase the payoff of deviated player. The solution suggested in 1949 (at that time by the 21 years old postgraduate student of Prinston University John Forbs Nash (jun.) and later named Nash equilibrium "--- NE) meets entirely this requirement. The NE gained certainly <<the reigning position>> in economics, sociology, military sciences. In 1994 J.\;F.\;Nash was awarded Nobel Prize in economics (in a common effort with John Harsanyi and R.\;Selten) <<for fundamental analysis of equilibria in noncooperative game theory>>. Actually Nash developed the foundation of scientific method that played the great role in the development of world economy. If we open any scientific journal in economics, operation research, system analysis or game theory we certainly find publications concerned NE. However <<And in the sun there are the spots>>, situations sets of Nash equilibrium must be internally and externally unstable. Thus, in the simplest noncoalition game of 2 persons in the normal form $$ \langle\{1,2\},~\{X_i=[-1,1]\}_{i=1,2},~ \{f_i(x_1,x_2)=2x_1x_2-x_i^2\}_{i=1,2}\rangle$$ set os Nash equilibrium situations will be $$X^{e}=\{x^{e}=(x_1^{e},~x_2^{e})=(\alpha, \alpha)~|~\forall \alpha=const\in [-1,1]\},~f_i(x^e)=\alpha^2~(i\in 1,2).$$ For elements of this set (the segment of bisectrix of the first and the third quarter of coordinate angle), firstly, for $x^{(1)}=(0,0)\in X^{(e)}$ and $x^{(2)}=(1,1)\in X^{e}$ we have $f_i(x^{(1)})=0<f_i(x^{(2)})=1~(i=1,2)$ and therefore the set $X^e$ is internally unstable, secondly, $f_i(x^{(1)})=0<f_i(\frac{1}{4},\frac{1}{3})~(i=1,2)$ and therefore the set $X^e$ is externally unstable. The external just as the internal instability of set of Nash equilibrium is negative for its practical use. In the first case there exists situation which dominates NE (for all players), in the second case this situation is Nash equilibrium. Pareto maximality of Nash equilibrium situation would allow to avoid consequences of external and internal instability. However such coincidence is an exotic phenomenon. Thus to avoid trouble connected with external and internal instability then we add the requirement of Pareto maximality to the notion of equilibrium of objections and counterobjections offered below. However we first of all reduce generally accepted solution concepts "--- NE and BE for the game $G_N$. It is proved in the article that in mathematical model both NE and BE are absent but there exist equilibria of objections and conterobjections as well as sanctions and countersanctions and simultaneously Pareto maximality.

In article coefficient criteria of the stability of coalitional structure in differential linear-quadratic positional game of 4 persons are established. Following the approach adopted in the article, it is possible to obtain coefficient criteria of the stability of coalitional structures both in games with a large number of players and for other coalitional structures

This survey focuses on the following problem: it is necessary, observing the behaviour of the object, automatically figure out how to improve (optimize) the quality of his functioning and to identify constraints to the improvement of this quality. In other words, build the objective function (or set of objective functions in multiobjective case) and constraints - i.e. the mathematical model of optimization - by mean machine learning. We present the main developed to date methods and algorithms that enable the automatic construction of mathematical models of planning and management objects by the use of arrays of precedents. The construction of empirical optimization models by reliable case information allows us to obtain an objective control model that reflects real-world processes. This is their main advantage compared to the traditional, subjective approach to the construction of control models. Relevant to the task a set of mathematical methods and information technologies called ``Extraction optimization models from data'', ``BOMD: Building Optimization Models from Data'', ``Building Models from Data'', ``The LION Way: Learning plus Intelligent Optimization'', ``Data-Driven Optimization''. The incompleteness of information and uncertainty are understood in different ways. Significantly different are the problem settings - deterministic, stochastic, parametric, mixed. Therefore, the consideration of a wider range of tasks leads to a variety of (primarily statistical) and other formulations of the problem and interpretations of uncertainty and incompleteness of initial information. The survey contains the following sections: Empirical synthetic of pseudoBoolean models; Empirical linear models with real variables; Empirical neural network optimization models; Iterative models; Models, including statistical statements; Problems, associated with the lack of the training set of points not belonging to the region of feasible solutions.