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sp-Groups form an interesting and informative class of Abelian mixed groups. In this paper, we systematically study self-small sp-groups of finite rank and their endomorphism rings.

Using examples of modules and a number of familiar Abelian groups, we demonstrate the Kaplansky method of proving isomorphism theorems for endomorphism rings.

The paper contains both known and new results on automorphism-invariant modules, automorphism-coinvariant modules and modules which are invariant or coinvariant with respect to idempotent endomorphisms of its hull and its cover, respectively. The main results are given with proofs.

In this paper, we describe the technique of extremely monotonic functions in the theory of computable linear orders. The basic definitions of extremely monotonic functions and their generalizations are given, and a number of their basic properties and applications are presented.

The categoricity spectrum of a computable structure S is the set of all Turing degrees capable of computing isomorphisms among arbitrary computable presentations of S. The degree of categoricity of S is the least degree in the categoricity spectrum of S. This paper is a survey of results on categoricity spectra and degrees of categoricity for computable structures. We focus on the results about degrees of categoricity for linear orders and Boolean algebras. We construct a new series of examples of degrees of categoricity for linear orders.

This paper is a survey of results on countable families with natural degree spectra. These results were obtained by a modification of the methodology proposed by Wechner, who first found a family of sets with the spectrum consisting precisely of nonzero Turing degrees. Based on this method, many researchers obtained examples of families with other natural spectra. In addition, in this paper we extend these results and present new examples of natural spectra. In particular, we construct a family of finite sets with the spectrum consisting of exactly non-K-trivial degrees and also we find new sufficient conditions on $$ {\Delta}_2^0 $$ -degree a, which guarantees that the class {x : x ≰ a} is the degree spectrum of some family. Finally, we give a survey of our recent results on the degree spectra of α-families, where α is an arbitrary computable ordinal.

In this work, we review results of the last years related to the development of the structural theory of n-c.e. Turing degrees for n > 1. We also discuss possible approaches to solution of the open problems.

We investigate qualitative properties of solutions of special classes of convolution type nonlinear integral equations on the whole line. We study the asymptotic properties, continuity, and monotonicity of arbitrary nontrivial bounded solutions. Depending on the properties of the kernel of the equation, we find out whether there exist nontrivial bounded solutions with a finite limit at ±∞. Based on the obtained results, we establish uniqueness theorems for large classes of bounded functions. The results obtained are illustrated by examples from applications.

We prove that any linear partial integral operator acting from the space of continuous functions to the Lebesgue class $$ {L}_p\left({D}_{\beta}^{\left({n}_2\right)}\right),1

We formulate and prove sufficient epsilon-optimality conditions for stochastic optimal control dynamic systems with random quantization period and incomplete information about the state.

We study the problem of searching periodic solutions to the Euler–Bernoulli equation governing vibrations of a beam with the boundary conditions corresponding to the case of rigidly sealed beam ends. The nonlinear term satisfies the nonresonance condition at infinity. We establish the existence and uniqueness of a solution. To prove the results, we use topological methods (the Leray–Schauder principle) as well as variational methods (the mountain pass theorem).

We study quasiperiodic nonconservative perturbations of two-dimensional Hamiltonian systems with nonmonotone rotation. We study the behavior of solutions in neighborhoods of resonance energy levels close to degenerate ones. We find conditions for the existence of resonance quasiperiodic solutions (m-dimensional invariant tori in the extended phase space) and study bifurcations in resonance zones. The results are illustrated by an example of asymmetric Duffing equation for which we also solve the problem of constructing solutions of unperturbed equations.

We consider the Veselov system corresponding to a Veselov–Chalykh deformation of the root system such that some component of some solution to the Veselov system is a Lauricella function. We express the parameters of the Veselov system in terms of the parameters of the Lauricella function.

We obtain conditions for the existence and uniqueness of a solution to the Fredholm equation with multi-dimensional integral operators with partial integrals in anisotropic Lebesgue spaces.

We consider the classical Euler equations governing the 3D unsteady isothermal motion of an ideal compressible medium in the absence of external mass forces and barotropy, where the pressure, density, and velocity components are unknown. We propose a procedure for reducing the original problem to a family of simpler problems by introducing new associated unknowns.

We consider the Dirichlet problem for the inhomogeneous Lame system in the plane with constant leading coefficients in a (finite or infinite) domain, bounded by a Lyapunov contour. For domains of finite diameter we use the weighted Holder class of functions with power behavior at infinity. We propose an equivalent reduction of the problem to a system of Fredholm integral equations in the domain and on the boundary contour.

We consider a time-optimal control problem for a hybrid system of variable dimension. Continuous motion of the system alternates with discrete variations (switchings) at which the dimension of the state space changes. The change of the space of states of the system can occur if the number of control objects changes, which is typical, for example, for the control problems for groups of a variable number of aircrafts. We obtain sufficient conditions for the time-optimal achievement of all purposes by a group of control objects. The application of optimality conditions is demonstrated by academic examples.

We consider the system of Maxwell equations and Lorentz torque equation which describes the motion of a charge in an electromagnetic field. Under certain symmetry conditions on the charge distribution and initial fields, the center of mass of the charge remains fixed, but the charge rotates. The system admits stationary soliton type solutions. We study the Lyapunov stability and the orbital stability of solitons exploiting the energy conservation law for the system.

We prove upper estimates for differences between the solution to the Cauchy problem for the heat equation with initial function u0(x) and its Stieltjes means

We solve the problem of electromagnetic positioning on the basis of two magnetic dipoles with the same center. We construct a navigation and orientation algorithm for a remote object.

We study an implicit inhomogeneous linear differential-difference equation in the module of formal generalized functions over a commutative ring. We prove the well-posedness of the equation and find the fundamental solution. We obtain a representation of a unique solution to the equation in the form of the convolution of the fundamental solution and a given formal generalized function. We also consider the inhomogeneous second order differential equation over an arbitrary commutative ring.

We consider a system of nonlinear differential equations describing the antiferromagnetism victor in a ferrite film. The system contains two parameters and admits a family of exact solutions with plane phase trajectories. Perturbation of such solutions leads to the Hill equation with periodic coefficient. We study the stability of the equilibrium state of the Hill equation.

We construct and justify an efficient model arising when homogenizing the Poisson equation in an e-periodically perforated domain along an (n − 1)-dimensional manifold by sets of an arbitrary shape and critical size on the boundary of which we impose a nonlinear dynamic condition containing an absorption coefficient of the form e−k, where k takes the critical value (n − 1)/(n − 2), n ≥ 3. We show that the transmission conditions on the manifold contain a nonlocal monotone operator.

We present sufficient conditions for the uniform convergence of interpolation processes constructed on the basis of solutions to Cauchy problems in terms of the one-sided modulus of continuity and modulus of variation of the approximated function on a compact connected subset of the domain.

We establish the unique solvability of a radiative-conductive heat transfer problem in a system of grey bodies with semitransparent inclusions and prove the comparison theorem. We show that higher summability of the data improves the properties (exponential summability and boundedness) of the solution.

We propose an algorithm for linearizing systems of partial differential equations at constant solutions. The algorithm is based on an isomorphism constructed between the ring of linearized functions and the ring of special matrices, which makes it possible to simplify calculations in the process of linearization. The algorithm is illustrated by applying it to the quasigasdynamic system.

We study homogenization of fourth order elliptic operators Ae in divergence form with e-periodic coefficients in ℝd and obtain an e2-order approximation of the resolvents (Ae + 1)−1 in the energy operator (L2→H2)-norm as e → 0.

We study variations of the first nontrivial eigenvalue of the two-dimensional p-Laplace operator, p > 2, generated by measure preserving quasiconformal mappings. The study is based on the geometric theory of composition operators in Sobolev spaces and sharp embedding theorems. Using a sharp version of the reverse Holder inequality, we obtain a lower estimate for the first nontrivial eigenvalue in the case of Ahlfors type domains.

Let Qn denote the space of all n × n skew-symmetric matrices over the complex field ℂ. It is proved that for n = 4, there are no linear maps T : Q4 → Q4 satisfying the condition dχ' (T (A)) = dχ(A) for all matrices A ∈ Q4, where χ, χ' ∈ {1, ∈, [2, 2]} are two distinct irreducible characters of S4. In the case χ = χ' = 1, a complete characterization of the linear maps T : Q4 → Q4 preserving the permanent is obtained. This case is the only one corresponding to equal characters and remaining uninvestigated so far.

The paper characterizes linear operators converting column majorization into weak, directional, and strong majorizations. An example of a linear converter from weak, directional, and strong majorizations to column majorization preserving none of these majorizations is provided.

In this paper, the length of the group algebra of a dihedral group in the modular case is computed under the assumption that the order of the group is a power of two. Various methods for studying the length of a group algebra in the modular case are considered. It is proved that the length of the group algebra of a dihedral group of order 2k+1 over an arbitrary field of characteristic 2 is equal to 2k.

The paper presents a matrix representation of filter banks corresponding to spline wavelets with shifted supports. The matrix form of the decomposition and reconstruction filters simplifies the representation of nonuniform nonstationary wavelet transforms built on nonuniform grids on a finite interval. Such a representation of filters is used, for example, in constructing error-correcting codes.

An algorithm for computing Jordan semilattices of vectors corresponding to a multiple point of the spectrum of a singular multiparameter polynomial matrix, which is based on quasiresultant matrices, is suggested. An illustration of the algorithm implementation is provided.

The paper considers parallel preconditioned iterative methods in Krylov subspaces for solving systems of linear algebraic equations with large sparse symmetric positive-definite matrices resulting from grid approximations of multidimensional problems. For preconditioning, generalized block algorithms of symmetric successive over-relaxation or incomplete factorization type with matching row sums are used. Preconditioners are based on variable-triangular matrix factors with multiple alternations in triangular structure. For three-dimensional grid algebraic systems, methods are based on nested factorizations, as well as on two-level iterative processes. Successive approximations in Krylov subspaces are computed by applying a family of conjugate direction algorithms with various orthogonality and variational properties, including preconditioned conjugate gradient, conjugate residual, and minimal error methods.

A finite computational process using arithmetic operations only is called a rational algorithm. Presently, no rational algorithm for checking the congruence of arbitrary complex matrices A and B is known. The situation can be different if both A and B belong to a special matrix class. For instance, there exist rational algorithms for the cases where both matrices are Hermitian, unitary, or accretive. This paper proposes a rational algorithm for checking whether two involutive matrices A and B are congruent.

New general upper and lower bounds for the Perron root of a nonnegative matrix, which involve nonempty proper subsets of the index set and the matrix sparsity pattern, are suggested, and some special cases are considered. Also the nonsingularity criteria related to the upper bounds presented, which generalize some known results on subclasses of nonsingular ℋ-matrices, are derived.

Conjugate gradient and conjugate residual methods for multiple solution of systems of linear algebraic equations (SLAEs) with the same matrices but different successively determined right-hand sides are considered. In order to speed up the iterative solution of the second and subsequent SLAEs, deflation algorithms are applied. These algorithms use the direction vectors obtained in the course of solving the first system as the basis ones. Results of numerical experiments for model examples, illustrating the efficiency of the approaches under consideration, are provided.

Let 𝕊 denote the algebra of sedenions and let ΓO(𝕊) denote its orthogonality graph. One can observe that every pair of zero divisors in 𝕊 generates a double hexagon in ΓO(𝕊). The set of vertices of a double hexagon can be extended to a basis of 𝕊 that has a convenient multiplication table. The set of vertices of an arbitrary connected component of ΓO(𝕊) is described, and its diameter is found. Then, the bijection between the connected components of ΓO(𝕊) and the lines in the imaginary part of the octonions is established. Finally, the commutativity graph of the sedenions is considered, and it is shown that all the elements whose imaginary part is a zero divisor belong to the same connected component, and its diameter lies in between 3 and 4.

We analyze the following general version of the deterministic Hats game. Several sages wearing colored hats occupy the vertices of a graph. Each sage can have a hat of one of k colors. Each sage tries to guess the color of his own hat merely on the basis of observing the hats of his neighbors without exchanging any information. A predetermined guessing strategy is winning if it guarantees at least one correct individual guess for every assignment of colors. We present an example of a planar graph for which the sages win for k = 14. We also give an easy proof of the theorem about the Hats game on “windmill” graphs.

We discuss some results and problems related to the Newton–Puiseux algorithm and its generalization for nonzero characteristic obtained by the author earlier. A new method is suggested for obtaining efficient estimates of the roots of a polynomial in the field of fractional power series in the case of arbitrary characteristic.

We show that if a distribution does not depend on the vertical coordinates, then the Schouten curvature tensor coincides with the Riemannian curvature. The Schouten curvature tensor and the nonholonomicity tensor are used to write the Jacobi equation for the distribution. This leads to a study of second-order optimality conditions for horizontal geodesics in sub-Riemannian geometry. We study conjugate points for horizontal geodesics on the Heisenberg group as an example.

A criterion for the maximum possible pointwise convergence rate in Birkhoff’s ergodic theorem for ergodic semiflows in a Lebesgue space is obtained. It is proved that higher rates of convergence in this theorem are impossible.

We study the Gnedin–Kingman graph, which corresponds to Pieri’s rule for the monomial basis {Mλ} in the algebra QSym of quasisymmetric functions. The paper contains a detailed announcement of results concerning the classification of indecomposable semifinite harmonic functions on the Gnedin–Kingman graph. For these functions, we also establish a multiplicativity property, which is an analog of the Vershik–Kerov ring theorem.

We suggest an algorithm for constructing all the isolated primary components of a given polynomial ideal. At the output, they are determined by systems of generators up to embedded components, and also as kernels of some homomorphisms. The complexity of this algorithm is subexponential in the size of the input data.

The following general variant of deterministic “Hats” game is analyzed. Several sages wearing colored hats occupy the vertices of a graph, the kth sage can have hats of one of h(k) colors. Each sage tries to guess the color of his own hat merely on the basis of observing the hats of his neighbors without exchanging any information. A predetermined guessing strategy is winning if it guarantees at least one correct individual guess for every assignment of colors. For complete graphs and cycles, the problem of describing the function h(k) for which the sages win is solved in the present paper. A “theory of constructors,” i.e., a collection of theorems demonstrating how one can construct new graphs for which the sages win is developed. A new game “Rook check ” equivalent to the Hats game on a 4-cycle is introduced and completely analyzed.

An embedded system of algebras of vertex subsets of weighted digraph is constructed. The properties of minimal spanning forests restricted to atoms of the corresponding algebras are investigated.

Labeled and unlabeled Hamiltonian cycles in complete n-partite graphs Kd,d,...,d having exactly d vertices in each part are enumerated. Recurrence relations for the exact values $$ {b}_n^{(d)} $$ of such cycles for arbitrary n and d are obtained.

The goal of the paper is to study vertices of degree 6 of minimal and contraction critical 6-connected graph, i.e., a 6-connected graph that looses 6-connectivity both upon removal and upon contraction of any edge. It is proved that if x and z are adjacent vertices of degree 6, then x and z have at least 4 common neighbors. In addition, a detailed description of the neighborhood of the set {x, z} is given. An infinite series of examples of minimal and contraction critical 6-connected graphs for which the fraction of vertices of degree 6 equals $$ \frac{11}{17} $$ is constructed.

We examine the unique solvability of an initial-value problem for a certain higher-order quasilinear partial integro-differential equation with a degenerate kernel. Expressing the higher-order partial integro-differential operator as the superposition of first-order partial differential operators, we represent the integro-differential equation considered as an ordinary integro-differential equations that describes the change of the unknown function along characteristics. Using the method of successive approximations, we prove the unique solvability of the initial-value problem and obtain an estimate for the convergence rate of the Picard iterative process.

Using a modified method of boundary-layer functions, we construct a complete uniform asymptotic solution of the singularly perturbed Cauchy problem for a second-order, linear, inhomogeneous, ordinary differential equation with a multiple turning point on the real axis.