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This paper is a review of the results on the real-time optimal control problem for linear systems obtained by the Minsk school by mathematical methods of optimal control. We consider optimal control problems for dynamical objects, their deterministic mathematical models and perfect measurements of states, objects with disturbances and imperfect measurements of the observed input and output signals, the problem of optimal decentralized control of groups of interconnected dynamical objects, and application of real-time optimal control problems and the control principle to solving stabilization problems. [ABSTRACT FROM AUTHOR]

The subject of the present paper is the Cauchy problem for the wave equation in a space-time cylinder Ω × ℝ, Ω ⊂ ℝ2, with data on the surface ∂Ω × I, where I is a finite time interval. The algorithm for solving the Cauchy problem with data on S × I, S ⊂ ∂Ω, was obtained previously. Here we adapt this algorithm to the special case S = ∂Ω and show that the solution in this situation is determined with higher stability in comparison with the case S ⊂ ≠ ∂Ω. [ABSTRACT FROM AUTHOR]

The dynamical system under consideration is u tt - u xx + V u = 0 , x > 0 , t > 0 ; u t = 0 = u y t = 0 = 0 , x ≥ 0 ; u x = 0 = f , t ≥ 0 , where V = V (x) is a matrix-valued function (potential); f = f (t) is an ℝN-valued function of time (boundary control); u = uf (x, t) is a trajectory (an ℝN-valued function of x and t). The input/output map of the system is a response operator R : f ↦ u x f (0 , ·) , t ≥ 0. The inverse problem is to determine V from given R. To characterize its data is to provide the necessary and sufficient conditions on R that ensure its solvability. The procedure that solves this problem has long been known and the characterization has been announced (Avdonin and Belishev, 1996). However, the proof was not provided and, moreover, it turned out that the formulation must be corrected. Our paper fills this gap. [ABSTRACT FROM AUTHOR]

In this paper, we examine a Cauchy-type problem for one ordinary differential equation with Riemann–Liouville fractional derivatives. For this problem, based on the Lebesgue space of functions that are summable with an arbitrarily fixed degree, we propose a family of pairs of spaces of elements and right-hand sides that provide the well-posed statement of the problem. In these pairs of spaces, we develop a generalized polynomial projection method for solving the problem considered and justify it from the functional-theoretic point of view. Based on the general results obtained, we prove the convergence of the "polynomial" Galerkin method, the collocation method, and the subdomain method for solving the corresponding Cauchy-type problem. [ABSTRACT FROM AUTHOR]

Recently the author proved that the Hummel–Scheinberg–Zalcman conjecture of 1977 on coefficients of nonvanishing Hp functions is true for all p = 2m, m ϵ ℕ, i.e., for the Hilbertian Hardy spaces H2m. As a consequence, this also implies the proof of the Krzyz conjecture for bounded nonvanishing functions, which originated this direction. In the present paper, we solve the problem for all spaces Hp with p ≥ 2. [ABSTRACT FROM AUTHOR]

The solution of problems on phase transitions in multilayer media when they are heated is of practical interest owing to the increasing use of multilayer materials in engineering and construction under various temperature conditions. In this paper, we assume that the medium contains distributed heat sources caused by physical or chemical processes, which can lead to phase transitions. We describe the method of generalized Bers powers and the matrix method for solving the heat conduction problem in multilayer media and for the search for boundaries of phase transitions. [ABSTRACT FROM AUTHOR]

The paper deals with a plane elasticity problem for a composite half space subjected to frictional contact. The half space is regarded as a microperiodic layered composite consisting of two components. The interfaces of layers are parallel to each other and oblique relative to the boundary of half space. A homogenized model with microlocal parameters is used to solve the considered problem. The solution is obtained in the general form and, specifically, for the case of elliptic loads [4]. On the basis of the obtained results, the maximum tensile stress is analyzed on the boundary of the half space and presented in the graphical form. [ABSTRACT FROM AUTHOR]

A dynamical system u tt − Δ u − ∇ 1 n ρ ⋅ ∇ u = 0 in ℝ + 3 × 0 T , u t = 0 = u t t = 0 = 0 in ℝ + 3 ¯ , u z z = 0 = f for 0 ≤ t ≤ T , is under consideration, where ρ = ρ(x, y, z) is a smooth positive function; f = f(x, y, t) is a boundary control; u = uf (x, y, z, t) is a solution. With the system one associates a response operator R : f ↦ uf|z = 0. The inverse problem is to recover the function ρ via the response operator. A short representation of the local version of the BC-method, which recovers ρ via the data given on a part of the boundary, is provided. If ρ is constant, the forward problem is solved in explicit form. In the paper, the corresponding representations for the solutions and response operator are derived. A way to use them for testing the BC-algorithm, which solves the inverse problem, is outlined. The goal of the paper is to extend the circle of the BC-method users, who are interested in numerical realization of methods for solving inverse problems. [ABSTRACT FROM AUTHOR]

The dynamical system u tt − Δ u − ∇ ln ρ ⋅ ∇ u = 0 in ℝ + 2 × 0 T , u t = 0 = u t t = 0 = 0 in ℝ + 2 , u y y = 0 = f for 0 ≤ t ≤ T , is under consideration, where ℝ + 2 ≔ x y ∈ ℝ 2 y > 0 ; ρ = ρ(x, y) is a smooth positive function; f = f(x, t) is a boundary control; u = uf(x, y, t) is a solution. With the system one associates a response operator R : f ↦ u f y = 0 . The inverse problem is to recover the function ρ via the response operator. A short presentation of the local version of the BC-method, which recovers ρ via the data given on a part of the boundary, is provided. If ρ is constant, the forward problem is solved in explicit form. In the paper, the corresponding representations for the solutions and response operator are derived. A way of making use of them for testing the BC-algorithm, which solves the inverse problem, is outlined. The goal of the paper is to extend the circle of the BC-method users, who are interested in numerical realization of methods for solving inverse problems. [ABSTRACT FROM AUTHOR]

In this paper, we present the formulation and a method for solving the problem of freezing living biological tissue with a flat circular cryoapplicator. The model is a two-dimensional boundary-value problem of Stefan type with nonlinear heat sources of a special type that provide the actually observed spatial localization of the temperature field. Some results of computer simulation are presented. [ABSTRACT FROM AUTHOR]

The paper is devoted to methods of solution of binary additive problems with semiprime numbers, which form sufficiently "rare" subsequences of the natural series. Additional conditions are imposed on these numbers; the main condition is belonging to so-called Vinogradov intervals. We solve two problems that are analogs to the Titchmarsh divisor problem; namely, based on the Vinogradov method of trigonometric sums, we obtain asymptotic formulas for the number of solutions to Diophantine equations with semiprime numbers of a specific form. [ABSTRACT FROM AUTHOR]

This paper is devoted to solving the inverse problem of restoring the hydrometeor distribution function from radar measurements. We use the Tikhonov regularization algorithm based on the minimization of the residual functional by the conjugate gradient method. [ABSTRACT FROM AUTHOR]

This paper is devoted to research in mathematical neurobiology; its main purpose is to establish the connection between approaches to the modeling of neural fields based on continuous and discontinuous equations. The authors also review works on this topic and propose a new method for solving such problems based on Volterra's abstract inclusions, which allows one to generalize some previously obtained results. [ABSTRACT FROM AUTHOR]

Recently the author presented a new approach to solving the coefficient problems for holomorphic functions based on the features of Bers' fiber spaces for punctured Riemann surfaces. The holomorphy of functionals causes strong rigid constrains. This paper extends the previous results to a broad class of plurisubharmonic coefficient functionals and provides new extremal features of the Koebe function. [ABSTRACT FROM AUTHOR]

Recently the author presented a new approach to solving the coefficient problems for holomorphic functions based on the features of Bers' fiber spaces for punctured Riemann surfaces. The holomorphy of functionals causes strong rigid constrains. This paper extends the previous results to a broad class of plurisubharmonic coefficient functionals and provides new extremal features of the Koebe function. [ABSTRACT FROM AUTHOR]

Recently the author has presented a new approach to solving the coefficient problems for holomorphic functions based on the deep features of Teichmüller spaces. It involves the Bers isomorphism theorem for Teichmüller spaces of punctured Riemann surfaces. The aim of the present paper is to provide new applications of this approach and extend the indicated results to more general classes of functions [ABSTRACT FROM AUTHOR]

We use specialized simulators to train drivers of vehicles difficult to drive. The environment created by such simulator for drivers must be as close to reality as possible. To simulate the overloads that occur during the motion of the simulated object, movable platform stands are used. Usually simulator software is developed for a particular imitation stand, which limits the use of such software for other equipment configuration. In addition, often the sensors in the mechanisms of the platform are either not precise enough or absent altogether, and the geometric parameters of the platform change with time due to deformations due to constant loads. These factors negatively affect the management of the platform and, therefore, the accuracy of dynamic simulation and the quality of coordination with visual and other types of simulation. The purpose of this paper is to show that identification of platform parameters and platform positioning can be constructed without using measurements from internal sensors located in the mechanisms of the platform. To solve the problem, the authors developed an algorithm of semi-automatic identification of parameters of a platform model with a system of video analysis. Upon identification of parameters of a platform it is possible to monitor its angular motions without video analysis. Assessment of platform orientation is performed with angular velocity sensors (AVS) and accelerometers. The use of the suggested algorithms enables quick adaptation of stimulator software to any motion platform. [ABSTRACT FROM AUTHOR]

We use specialized simulators to train drivers of vehicles difficult to drive. The environment created by such simulator for drivers must be as close to reality as possible. To simulate the overloads that occur during the motion of the simulated object, movable platform stands are used. Usually simulator software is developed for a particular imitation stand, which limits the use of such software for other equipment configuration. In addition, often the sensors in the mechanisms of the platform are either not precise enough or absent altogether, and the geometric parameters of the platform change with time due to deformations due to constant loads. These factors negatively affect the management of the platform and, therefore, the accuracy of dynamic simulation and the quality of coordination with visual and other types of simulation. The purpose of this paper is to show that identification of platform parameters and platform positioning can be constructed without using measurements from internal sensors located in the mechanisms of the platform. To solve the problem, the authors developed an algorithm of semi-automatic identification of parameters of a platform model with a system of video analysis. Upon identification of parameters of a platform it is possible to monitor its angular motions without video analysis. Assessment of platform orientation is performed with angular velocity sensors (AVS) and accelerometers. The use of the suggested algorithms enables quick adaptation of stimulator software to any motion platform. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of groups from the category M of R-power groups. We examine problems on the commutation of the tensor completion with basic group operations and on the exactness of the tensor completion. Moreover, we introduce the notion of a variety and obtain a description of abelian varieties and some results on nilpotent varieties of A-groups. We prove the hypothesis on irreducible coordinate groups of algebraic sets for the nilpotent R-groups of nilpotency class 2, where R is a Euclidean ring. We state that the analog to the Lyndon result for the free groups (see [10]) holds in this case, whereas the analog to the Myasnikov-Kharlampovich result fails.The paper is dedicated to partial R-power groups which are embeddable to their A-tensor completions. The free R-groups and free R-products are described with usual group-theoretical free constructions. [ABSTRACT FROM AUTHOR]

The paper discusses the de-noising method in a model with an additive Gaussian noise, based on the wavelet-vaguelette decomposition and thresholding of the vaguelette coefficients. For soft and hard thresholding procedures, we derive the values of the thresholds and estimate asymptotically optimal orders of the loss function based on the probabilities of errors in calculating decomposition coefficients. [ABSTRACT FROM AUTHOR]

One of the main problems of noncommutative geometry is the translation of fundamental notions of analysis, topology, and differential geometry onto the language of Banach algebras. In this paper, we present a number of results of this kind focusing the attention on the noncommutative interpretation of the notions of differential and integral. Our presentation is based on the monographs Noncommutative Geometry by A. Connes and Elements of Noncommutative Geometry by J. M. Gracia-Bondia, J. C. Varilly, and H. Figueroa. [ABSTRACT FROM AUTHOR]

The paper deals with energy (weak) solutions u (t; x) of the class of equations with the model representativeup−1ut−Δpu=0,tx∈0T×Ω,Ω∈ℝn,n≥1,p>0,and with the following blow-up condition for the energy:εt≔∫Ωutxp+1dx+∫0t∫Ω∇xuτxp+1dxdτ→∞ast→T,where Ω is a smooth bounded domain. In the case of flat peaking, namely, under the conditionεt≤Fαtω0T−t−α∀t0,α>1p+1,a sharp estimate of the profile of a solution has been obtained in a neighborhood of the blow-up time t = T. [ABSTRACT FROM AUTHOR]

Abstract. This paper deals with an extremal problem concerning hypergraph colorings. Let k be an integer. The problem is to find the value m( n) equal to the minimum number of edges in an n-uniform hypergraph not admitting two-colorings of the vertex set such that every edge of the hypergraph contains at least k vertices of each color. In this paper, we obtain upper bounds of m( n) for small k and n, the exact value of m(8), and a lower bound for m(7). [ABSTRACT FROM AUTHOR]

An example of solving a boundary-value problem for a homogeneous Monge-Ampère equation is given, which produces a Bellman function for an extremal problem on the space BMO. The paper contains a step-by-step instruction for calculation of this function. Cases of rather complicated foliations are considered. This illustrates the technique elaborated in a paper by Ivanishvili, Stolyarov, Vasyunin, and Zatitskiy. Bibliography: 6 titles. [ABSTRACT FROM AUTHOR]

This paper continues another our work, which is the first of two parts where the approach to the processing of quantitative expert evaluations in the process of group decision-making under uncertainty is considered. In the second part, represented by this paper, an approach is proposed for the processing of qualitative expert evaluations in the process of group decision-making. The approach is based on the use of triangular fuzzy numbers. In group decision-making the opinions of experts are expressed by linguistic variables like very bad, not very bad, problematic, good, and so on. The technique of conversion of expert quantitative opinions to triangular fuzzy numbers is considered. A simple method of expressing expert opinions by triangular fuzzy numbers is introduced. A new approach to determining the expert degrees of importance is proposed. The proposed methodology is discussed in full detail and its algorithm is described. An illustrative example is given. [ABSTRACT FROM AUTHOR]

Let X,X, . . . , X be independent, identically distributed random variables. In this paper, we study the behavior of concentration functions of the weighted sums $$ {\displaystyle \sum_{k=1}^n{a}_k{X}_k} $$ with respect to the arithmetic structure of coefficients a. Such concentration results recently became important in connection with the study of singular values of random matrices. In this paper, we formulate and prove some refinements of a result of Vershynin (2014). [ABSTRACT FROM AUTHOR]

Let K be a nonsingular skew-symmetric matrix of even order n = 2m. For such a matrix, the paper proposes a finite algorithm for computing an m-dimensional neutral subspace, which uses arithmetic operations and quadratic radicals only. The necessity of computing neutral subspaces originates in the problem of solving quadratic matrix equations. [ABSTRACT FROM AUTHOR]

In this paper, we show that an overdetermined tropical linear system has a solution if and only if it contains a square subsystem having a stable solution that is a solution of the original system. This leads to a simple algorithm for solving tropical linear systems in time Onmn4, where m is the number of equations and n is the number of variables. For weakly overdetermined systems, this time is polynomial. [ABSTRACT FROM AUTHOR]

Existence and stability problems for relay systems were examined in many research papers, but local methods that play an important role in deep understanding of dynamics have not yet been sufficiently developed. In this paper, we introduce the concept of the monodromy matrix for a cycle and state an analog of the Andronov-Hopf bifurcation theorem. [ABSTRACT FROM AUTHOR]

Control problems in dynamic systems allowing pulse forces due to the presence of unilateral active constraints are considered. This paper gives an analysis of controls used during impact interactions. A new kind of pulse controls arising at the time of impact is introduced. These problems are typical for the so-called vibroimpact systems with numerous applications, whose number is constantly increasing in recent years. The goal of the paper is to describe the phase of a controllable impact and to determine the constraint reactions. [ABSTRACT FROM AUTHOR]

In the present paper, the Bitsadze-Samarski boundary-value problem is considered for a quasi-linear differential equation of first order on the plane and the existence and uniqueness theorem for a generalized solution is proved; the necessary (in the linear case) and sufficient optimality conditions for optimal control problems are found. The optimal control problem is posed, where the behavior of control functions is described by elliptic-type equations with Bitsadze-Samarski nonlocal boundary conditions. The necessary and sufficient optimality conditions are obtained in the form of the Pontryagin maximum principle and the solution existence and uniqueness theorem is proved for the conjugate problem. Nonlocal boundary-value problems and conjugate problems are solved by the algorithm, which reduces nonlocal boundary value problems to a sequence of Dirichlet problems. The numerical method of solution of an optimal control problem by the Mathcad package is presented. [ABSTRACT FROM AUTHOR]

This paper is concerned with the problem of the best (in a precisely defined sense) approximation with given accuracy of periodic functions and functions on the real line from, respectively, a finite tuple of noisy Fourier coefficients or noisy Fourier transform on an arbitrary set of finite measure. [ABSTRACT FROM AUTHOR]

The paper proposes a mathematical formalism for describing artificial sense-of-touch systems. Mathematical models for obtaining, processing, and interpreting tactile information are provided. Problems of encoding and reproducing tactile information are formulated and algorithms for solving these problems are proposed. The problem of interpreting tactile information is considered, and the corresponding simple mathematical model is studied. Within the framework of this simple model, an exact solution of the interpretation problem is obtained for the case of finite deformations, and the insufficiency of the linear method (Hooke’s law) for describing problems of interpreting tactile data is shown. The presentation of mathematical models for the theory of artificial sense-of-touch systems in this paper is the first such detailed presentation in the Russian literature. These models are of interest to mathematicians, mechanicians, physicians, and engineers who construct or use artificial sense-of-touch systems. [ABSTRACT FROM AUTHOR]

The paper continues the series of papers devoted to surveying and developing methods for solving problems for two-parameter polynomial and rational matrices. Different types of factorizations of two-parameter rational matrices (including irreducible and minimal ones), methods for computing them, and their applications to solving spectral problems are considered. Bibliography: 6 titles. [ABSTRACT FROM AUTHOR]

In this paper, we consider linear (in dynamics) conflict control problems (linear antagonistic differential games) with a fixed instant of termination and a continuous terminal cost function. We formulate and prove assertions on sufficient conditions under which such a method guarantees the obtaining of a result close to optimal by the minimizing player and has the stability property. In the concluding part of the paper, we give a brief description of publications devoted to computer modeling by using the proposed control method. [ABSTRACT FROM AUTHOR]

Many problems of conflict-control theory can be reduced to approach-avoidance games with a certain terminal set. One of the main approaches to solution of such problems is the approach suggested by N. N. Krasovskii, which is based on positional constructions. The basis of these constructions consists of the extremal aiming principle at stable bridges. In this connection, the problem of constructing the maximal stable bridge, the set of all positions from which the problem of approaching the terminal set (which is the main task of one of the problem) is solvable, is important. The paper considers the approach-avoidance game on a finite interval of time in which the first player must ensure the attainment by the state vector of the control system of the terminal set on this interval, and the second player must ensure the avoidance of the terminal set. The main subject of the study is the maximal stable bridge, which is the set of positional consumption in this game. The problem of exactly constructing the set of positional consumption is solvable only in simple cases, and it is more realistic to consider and solve the problem of approximate construction of this set. The paper proposes approaches to approximate construction of the set of positional consumption based on sampling the time interval of the game and the technique of backward constructions, which has been developed at the scientific school of N. N. Krasovskii since the 1980s. [ABSTRACT FROM AUTHOR]

The paper deals with studying a connection between the Littlewood-Offord problem and estimating the concentration functions of some symmetric infinitely divisible distributions. Some multivariate generalizations of Arak's results (1980) are given. They establish a relationship of the concentration function of the sum and arithmetic structure of supports of the distributions of independent random vectors for arbitrary distributions of summands. Bibliography: 21 titles. [ABSTRACT FROM AUTHOR]

Let N( m) be the number of points of the integer lattice that belong to a d-dimensional ball of radius m (in the l- and l-norms). The aim of the paper is to study the asymptotic behavior of N( m) as d → ∞, m → ∞. It is shown that if d tends to infinity much faster than m, then the asymptotic is different from the asymptotic volume of a d-dimensional ball of radius m. Bibliography: 6 titles. [ABSTRACT FROM AUTHOR]

The paper considers a connection between weighted norm inequalities for the Hilbert transform with matrix valued weights and an estimating problem. A connection of the vector Muckenhoupt condition on the spectral density of the stationary noise and a possibility to transform a difficult estimating problem to another well-studied problem is established. Bibliography: 12 titles [ABSTRACT FROM AUTHOR]

The paper deals with some problems concerning probabilistic representation and probabilistic approximation for solution of the Cauchy problem for the family of equations $$ \frac{\partial u}{\partial t}=\frac{\sigma^2}{2}\varDelta u $$ with complex parameter σ such that Re σ ≥ 0. This family coincides with the heat equation if Im σ = 0, and with the Schrӧdinger equation if Re σ = 0. Bibliography: 5 titles [ABSTRACT FROM AUTHOR]

The paper has a scientific-methodical nature. The classical soap film shape (minimal surface) problem is considered, the film being stretched between two parallel coaxial rings. An analytic approach based on the relations to the Sturm-Liouville problem is proposed. An interpretation of the energy terms of the classical Goldschmidt condition is discussed. The appearance of a soliton potential when analyzing the second variation is noticed. Bibliography: 3 titles. [ABSTRACT FROM AUTHOR]

Investigation of the shortwave diffraction by elongated bodies of revolution requires a detailed consideration of matching of local asymptotics in the illuminated part of the Fock domain. In the paper, that task is solved by means of a straightforward construction of the reflected wave with the help of the ray method. The main problem on the way, which was judged by V. A. Fock as a rather complicated one, is the calculation of the eikonal and the geometric spreading in curvilinear coordinates used in the boundary layer method in the vicinity of the light-shadow zone. Bibliography: 9 titles. [ABSTRACT FROM AUTHOR]

In the paper, for a Lamé type system the inverse problem of recovering the fast and slow wave velocities from the boundary dynamical data (the response operator) is solved. The velocities are determined in a near-boundary domain, the depth of determination being proportional to the observation time. The BC-method, which is an approach to inverse problems based on their connections with boundary control theory, is used. [ABSTRACT FROM AUTHOR]

In this paper we consider the problem of estimating a function from the noised observations via thresholding its wavelet coefficients and prove the strong consistency of the risk estimator with the adaptive threshold. [ABSTRACT FROM AUTHOR]

Let η be the Atiyah-Patodi-Singer invariant considered on smooth, compact, oriented, threedimensional submanifolds of ℝ, and let A be an additive subgroup of ℝ. The problem of computing the degree of invariants of the form η mod A is examined. Here, the functional definition of invariants of finite degree is used. (A similar approach is used in S. S. Podkorytov's paper 'Quadratic property of the rational semicharacteristic.') The main results are as follows. If 1 ∉ A, then the degree is infinite. If $$ \frac{1}{3}\in A $$, then the degree is equal to 1. Bibliography: 10 titles. [ABSTRACT FROM AUTHOR]

This paper is a review of some previous and new results on integrable cases in the dynamics of a three-dimensional rigid body in a nonconservative field of forces. These problems are stated in terms of dynamical systems with the so-called zero-mean variable dissipation. Finding a complete set of transcendental first integrals for systems with dissipation is a very interesting problem that has been studied in many publications. We introduce a new class of dynamical systems with a periodic coordinate. Since such systems possess some nontrivial groups of symmetries, it can be shown that they have variable dissipation whose mean value over the period of the periodic coordinate vanishes, although in various regions of the phase space there may be energy supply or scattering. The results obtained allow us to examine some dynamical systems associated with the motion of rigid bodies and find some cases in which the equations of motion can be integrated in terms of transcendental functions that can be expressed as finite combinations of elementary functions. [ABSTRACT FROM AUTHOR]

In this paper, we develop a theory of equipped graded graphs (or Bratteli diagrams) and an alternative theory of projective limits of finite-dimensional simplices. An equipment is an additional structure on the graph, namely, a system of 'cotransition' probabilities on the set of its paths. The main problem is to describe all probability measures on the path space of a graph with given cotransition probabilities; it goes back to the problem, posed by E. B. Dynkin in the 1960s, of describing exit and entrance boundaries for Markov chains. The most important example is the problem of describing all central measures, to which one can reduce the problems of describing states on AF-algebras or characters on locally finite groups. We suggest an unification of the whole theory, an interpretation of the notions of Martin, Choquet, and Dynkin boundaries in terms of equipped graded graphs and in terms of the theory of projective limits of simplices. In the last section, we study the new notion of 'standardness' of projective limits of simplices and of equipped Bratteli diagrams, as well as the notion of 'lacunarization.' [ABSTRACT FROM AUTHOR]

Let R, S ∈ ℂ[ x] be two coprime polynomials of the same degree with prescribed multiplicities of their roots. A classical problem of number theory actively studied during the last half-century is, What could be the minimum degree of the difference T = R − S? The theory of dessins d'enfants implies that such a minimum is attained if and only if the rational function f = R/T is a Belyi function for a bicolored plane map all of whose faces except the outer one are of degree 1. Such maps are called weighted trees, since they can be conveniently represented by plane trees whose edges are endowed with positive integral weights. It is well known that the absolute Galois group (the automorphism group of the field $$ \overline{\mathbb{Q}} $$ of algebraic numbers) acts on dessins. An important invariant of this action is the edge rotation group, which is also the monodromy group of a ramified covering corresponding to the Belyi function. In this paper, we classify all weighted trees with primitive edge rotation groups. There are, up to the color exchange, 184 such trees, which are subdivided into (at least) 85 Galois orbits and generate 34 primitive groups (the highest degree is 32). This result may also be considered as a contribution to the classification of covering of genus 0 with primitive monodromy groups in the framework of the Guralnick-Thompson conjecture. [ABSTRACT FROM AUTHOR]

The paper is concerned with the problem of optimal recovery of fractional powers of the Laplace operator in the uniform norm on the multivariate generalized Sobolev class of functions from incomplete data on the Fourier transform of these functions on a ball of radius r centered at the origin. An optimal recovery method is constructed, and a number $$ \widehat{r} $$ > 0 is specified such that for r ≤ $$ \widehat{r} $$ the method makes use of all the information on the Fourier transform, smoothing thereof; and if r > $$ \widehat{r} $$, then the information on the Fourier transform proves superfluous and hence is not used by the optimal method. For fractional powers of the Laplace operator, a sharp inequality is proved. This inequality turns out to be closely related to the recovery problem and is an analogue of Kolmogorov-type inequalities for derivatives. [ABSTRACT FROM AUTHOR]

In this paper, we discuss optimal control problems whose behavior is described by elliptic equations with Bitsadze-Samarski nonlocal boundary conditions. A necessary and sufficient optimality condition is given. The existence and uniqueness of a solution of the conjugate problem are proved. A numerical method of solution of an optimal problem by means of the Mathcad package is presented. [ABSTRACT FROM AUTHOR]