The family of sparse Vilenkin-Chrestenson systems is studied, which are similar to lacunary systems in some properties, but not in others. Problems related to uniqueness and $q$-lacunarity for such systems are considered.

The problem of perturbation of the spectrum of a linear operator in the linear case has been solved long ago and successfully. Important concepts such as holomorphic families of the type $(A)$ and in the sense of Kato have also been introduced. The problems associated with perturbation of the spectrum are also well-known - problems of determining energy levels in quantum mechanics. At the same time, not always built Rayleigh--Schr\"{o}dinger series converge in the usual, and not in the asymptotic sense. In this article, the conditions for ordinary convergence of series in powers of a small parameter were found, representing solutions of one type of nonlinear problems to eigenvalues.

The Lotka-Volterra system of two biological species with constant concentrated delay, describing interspecies interaction, is considered. Any type of interaction between species is allowed, but the values of the reproductive periods are assumed to be equal. Under these conditions, necessary and sufficient conditions for asymptotic stability of this linear system are obtained. These conditions are obtained both in the form of a stability region in a three-dimensional parameter space and in the form of inequalities with respect to the delay value. The case of factorization of the characteristic quasi-polynomial is considered. Stable intervals for the delay values are studied.

The research topic of this paper is at the junction of the theory of Lyapunov exponents and the theory of oscillation. In this paper, we study the spectra (i.e. sets of different values on non-zero solutions) of characteristic frequencies and oscillation exponents of linear homogeneous differential equations with continuous coefficients on the positive semi-axis. For any countable set of non-negative numbers containing zero, an equation with a predetermined order $n>2$ is constructively built, whose spectra of oscillation characteristics coincide with this set. In proving the main result of the work, analytical methods of the qualitative theory of differential equations and the author's method were used, which allows the original differential equation to be varied in a special way so that it has predetermined properties.

In this paper, we use a mixed boundary-value problem to describe the deflection of a plate under longitudinal and transverse loading located on an elastic basis, with the mandatory presence of a Dirichlet boundary condition. We fictitiously continue the boundary-value problem across the boundary with a Dirichlet condition and then approximate the continued problem using the finite element method. The numerical solution to the continued problem is iteratively approximated through numerical solutions to extended problems. The iterative extension method is asymptotically optimal for numerically solving the original problem, meaning it has unimproved asymptotic performance in terms of computation time.

A method for investigation of the weak solvability of the initial-boundary value problem for the Navier-Stokes system in a bounded three-dimensional domain based on high-order parabolic regularization by the artificial viscosity method is presented.

A linear boundary-value problem for a functional-differential system with the Gerasimov-Caputo fractional derivative and a general aftereffect is considered, in which the boundary conditions cover the cases of multi-point and integral conditions. In the case of its unique solvability, relations defining the Green matrix of the boundary value problem are obtained. For the case when the boundary value problem is not everywhere solvable, a representation of solutions is obtained. The main constructions are based on the use of the Cauchy matrix to the system under consideration.

It is shown that the use of refined equations of motion based on the Timoshenko model and constructed taking into account the transverse shear and rotational inertia of the cross-sections for the modal analysis of rods allows us to identify a spectrum of natural frequencies of vibration in which, based on each found low-frequency mode of vibration, called the bending-shear mode, the associated shear-bending mode belonging to the high-frequency spectrum is uniquely determined. A method for its determination is proposed, tested on the construction of a solution to the problem under consideration by the finite element method. Based on the analysis of the results obtained, it is concluded that they can provide an explanation for another mechanism of passage of forced vibrations of the loaded console of an uncut double-ended rod through an elastic section of its attachment to an adjacent unloaded console. Thus, it is proved that the use of refined shear models of dynamic deformation of rods, constructed with mandatory consideration of the inertia of rotation of cross-vj;ysections, makes it possible to identify, along with low-frequency bending-shearing forms, such associated high-frequency forms of shear and bending vibrations, which, passing almost unhindered through deformed fixed sections, can excite in adjacent areas (adjacent) loose and unloaded sections, along with high-frequency forms, also low-frequency forms of vibrations.

In this paper, we consider a model Hamiltonian describing a quantum-mechanical three-particle system on a one-dimensional lattice. The role of the two-particle discrete Schrödinger operator is played by a family of Friedrichs models. We investigate the case in which the kinetic part of the model Hamiltonian has a special structure. Two channel operators are constructed, and their spectra are determined using the spectral properties of the corresponding family of Friedrichs models. It is shown that the essential spectrum of the model Hamiltonian is given by the union of the spectra of the channel operators and consists of at most three intervals. Furthermore, estimates for the lower and upper bounds of these intervals are obtained.