The article concerns the problem of covering the lateral surface of a right circular cylinder or a cone with equal balls. The surface is required to belong to their union, and the balls’ radius is minimal. The centers of the balls must lie on the covered surface. The problem is relevant for mathematics and for applications since it arises in security and communications. We develop heuristic algorithms for covering construction based on a geodesic Voronoi diagram. The construction of a covering is a non-trivial task since the line of intersection of a cylinder or a cone with a sphere is a closed curve of the fourth order. To compare the numerical results with the known ones, we unroll the surface of revolution onto a plane. Another feature is that, we use both Euclidean distance and a special non-Euclidean metric, which can describe the speed of signal propagation in a heterogeneous medium. We also perform a numerical experiment and discuss its results. Meanwhile, it is shown that with a small number of circles covering a planification of the cylindrical surface, their radius is significantly less than for a similar rectangle.

In this paper, we derive sharp asymptotics for the spectral data (eigenvalues and weight numbers) of the fourth-order linear differential equation with a distribution coefficient and three types of separated boundary conditions. Our methods rely on the recent results concerning regularization and asymptotic analysis for higher-order differential operators with distribution coefficients. The results of this paper have applications to the theory of inverse spectral problems as well as a separate significance.

A necessary and a sufficient condition for solvability of homogeneous difference equations with constant coefficients in the class of rational functions are obtained. The necessary condition is a restriction on the Newton polyhedron of the characteristic polynomial. In the two-dimensional case, this condition is the existence of parallel sides on the polygon. The sufficient condition is the equality to zero of certain sums of the coefficients of the equation. If the sufficient condition is satisfied, the solution is the class of rational functions whose denominators form a subring in the ring of polynomials. This subring can be associated with an edge of the Newton polyhedron of the characteristic polynomial of the equation.

Classifying abelian groups and their elementary theories, a series of characteristics arises that describe certain features of the objects under consideration. Among these characteristics, an important role is played by Szmielew invariants, which define the possibilities of divisibility of elements, orders of elements, dimension of subgroups, and allow describing given abelian groups up to elementary equivalence. Thus, in terms of Szmielew invariants, the syntactic properties of Abelian groups are represented, i.e. properties that depend only on their elementary theories. The work, based on Szmielew invariants, provides a description of the behavior of algebraic and definable closure operators based on two characteristics: degrees of algebraization and the difference between algebraic and definable closures. Thus, possibilities for algebraic and definable closures, adapted to theories of Abelian groups, are studied and described. A theorem on trichotomy for degrees of algebraization is proved: either this degree is minimal, if in the standard models, except for the only two-element group, there are no positively finitely many cyclic and quasi-cyclic parts, or the degree is positive and natural, if in a standard model there are no positively finitely many cyclic and quasi-cyclic parts, except a unique copy of a two-element group and some finite direct sum of finite cyclic parts, and the degree is infinite if the standard model contains unboundedly many nonisomorphic finite cyclic parts or positively finitely many of copies of quasi-finite parts. In addition, a dichotomy of the values of the difference between algebraic closures and definable closures for abelian groups defined by Szmielew invariants for cyclic parts is established. In particular, it is shown that torsion-free abelian groups are quasi-Urbanik.

Infinite groups with finiteness conditions for an infinite system of subgroups are studied. Groups with a condition: the normalizer of any non-trivial finite subgroup is a layer-finite group or the normalizer of any non-trivial finite subgroup has a layer-finite periodic part are studied for beginning in the locally finite class of group, then in the class of periodic groups of Shunkov and finally in the class of Shunkov groups which are contain a strongly embedded subgroup with an almost layer-finite periodic part. The group $G$ is called the Shunkov group if for any prime $p$ and for every finite subgroups $H$ from $G$ any two conjugate elements of order $p$ from the factor-group $N_G(H)/H$ generate a finite subgroup. Results for almost layer-finite groups and groups with almost layer-finite periodic part are transferred to layer-finite groups and groups with layer-finite periodic part. New characterizations of layer-finite groups and groups with layer-finite periodic part are obtained.

This paper is devoted to solving the inverse problem (determining the parameters of a system of ordinary differential equations based on additional information determined at discrete points in time) and analyzing its solution for a mathematical model describing the dynamics of changes in the population and capital of two regions of the world. The inverse problem is reduced to the problem of minimizing the target functional and is solved by the method of differential evolution. A numerical method for solving direct and inverse problems is implemented. The developed method was tested on model and real data for countries such as Russia, China, India and the USA.

The paper considers the problems of macroparameter control in a linear dynamic system. In the authors’ previous paper, such problems in controlling nonlinear systems were investigated. Variations of the considered functionals and necessary optimality conditions were obtained. This paper focuses on the trajectory beam control in the linear case. In this case, the problem of beam control can be reduced to the problem of the individual trajectory control. The developed approach may be of interest in the study of charged particle dynamics in accelerating and focusing structures, as well as in the construction of the velocity field in automatic image processing.