The paper studies perturbed initial-boundary value problems of conjugation generated by a sesquilinear form. The principle of superposition allows us to represent the solution of the original problem as a sum of solutions of auxiliary problems containing inhomogeneity either in the equation or in one of the boundary conditions. The original initial-boundary value problems are reduced to Cauchy problems for first-order integro-differential and differential operator equations in a Hilbert space.

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For large systems expected to come, such as, for example,

long beam with a payload at the end, control modes are proposed

active vibration damping. Self extinguishing turn

internal mass (damper) along the guide, vertical beam. Control

consists in choosing the coefficients of a linear function, which is combined for

rejection of the absorber and payload.

In this paper, we consider a model spectral problem that preserves all the features of the real problem of normal oscillations of the hydrodynamic

a system consisting of two incompressible homogeneous liquids placed in an arbitrary vessel. In this case, a liquid of higher density is viscous,

and the smaller one is ideal. The study of the spectrum of the problem was carried out on the basis of the study of the transcendental characteristic equation with respect to

complex damping decrement of normal oscillations.

Let $\mathcal H$~--- be an infinite-dimensional complex Hilbert space, let $(\mathcal B(\mathcal H), \|\cdot\|_\infty)$~--- $C^\star$ - algebra of all bounded linear operators acting in $\mathcal H$, and let \ $\mathcal C_E$ \ l be the symmetric

The problems with changing phase space are a subclass of the so-called hybrid (composite)

systems. They are characterized by the fact that at different time intervals they are described by

different differential systems and certain links for the connection of the trajectories. The systems

can have the similar dimensionality and also the transfer both from the dimension with the higher

dimensionality to the lower dimensionality and vice versa. The original source of such problems

were the multistage processes of space flights.

The paper considers a problem of competition between three manufacturing firms in the

market of homogeneous infinitely divisible products. It is assumed that the nature of the

interaction of manufacturing firms in the market has a hierarchical structure. Namely, one of

the companies, the leader company, is the leading manufacturer and is the first to decide on the

volume of product deliveries to the market. While the other two firms decide on the number of

The low-pressure RFI model is considered as a nonlinear eigenvalue problem with a parameter for a system that includes the electron balance equations and the Maxwell equations with mixed boundary conditions. The free parameter of the problem is the value of the electron density at the center of the plasma bunch $n_{e0}$.

The problem of stability of stationary solutions of traditional initial-boundary value problems for systems of equations describing the growth and distribution of a substance is considered. A positive effect of migration (diffusion) processes on stability in small areas is noted.

** Keywords: ** diffusion model, initial boundary value problem, stationary solution (state), stability, sufficient stability condition.

The article presents the method of generalized powers (OS) for constructing a sequence of basic solutions for a system of linear differential equations of the first order, known as the Moisil-Teodorescu systems. To accomplish this task, the quaternion form of the Moisil-Teodorescu equation is translated into a matrix form. With the help of a certain operation called joining, the system is reduced to a form that allows the use of the OS method.

In this paper, we consider the class $G$ of orientation-preserving Morse-Smale diffeomorphisms defined on a closed 3-manifold whose nonwandering set consists of exactly four points of pairwise distinct Morse indices. It is known that the two-dimensional saddle separatrices of any such diffeomorphism always intersect, and their intersection necessarily contains non-compact heteroclinic curves, but can also contain compact ones.

It is known that the contemporary approach in the theory of control systems and mathematical physics leads to models that are conveniently described by using differential equations and inclusions. Recently, the attention of many researchers (see [1]–[3] and the references therein) was attracted to generalizations of differential and functional differential equations and inclusions, namely to the class of functional equations and inclusions with causal operators. The term causal operator or Volterra operator in the sense of A. N.

The paper considers the dynamics and stability of homogeneous equilibrium states of a mathematical model of a nonlinear generator of optical radiation with a stretching operator for the spatial coordinates of a light wave and a time delay in the feedback loop. The mathematical model is an initial-boundary value problem for a parabolic type equation in a circle with a spatial argument stretching operator and a time delay in a nonlinear feedback functional.

The stability of rotation of a symmetrical rigid body on a vertical flexible rod (shaft) is investigated. Both ends of the shaft are secured from offsets. The lower end of the shaft is supported by a bearing, and the bearing of the upper end is attached to an elastic ring plate, pinched along the outer diameter. This is one of the designs of a real-life centrifuge. The shaft material is considered inherently viscoelastic. The equations of motion of the flexible shaft are constructed within the framework of the linear Euler–Bernoulli model.

In the paper, one constructs the examples of polar Morse–Smale systems (diffeomorphisms and flows) with a sink fixed point, source fixed point and two saddles fixed points on n-dimensional sphere \(S^n\),\(n ≥ 3\). To prove this result, we construct different decompositions of the n-dimensional sphere \(S^n\). Moreover, the Morse index of a saddle fixed point can be any value between \(1\) and \(n − 1\), and the Morse indexes of the saddles fixed points are always different.

The paper presents a method for constructing point and interval estimates of the regression coefficient (RC) of nonlinear regression in a passive, in a certain sense, experiment. Passivity is understood through the role of the experimenter regarding the content of the initial data. The role is passive and does not affect the method of collecting information. The initial data for the experimenter is an unchangeable given.

Applied network tasks of multiagent routing or (applied network tasks of multiagent routing или \(mTSP\)) arise in many application areas and lead to various models of pseudo-Boolean optimization. Such problems, as a rule, are \(NP\)--hard, for them exact algorithms are applicable only in the case of a small dimension of the original network (graph). Multiagency can be contained in the initial formulation or arise as a result of simplifying and reducing the dimension of the problem (decomposition, clustering).

A sufficient condition for the consistency of nonparametric estimator for regression function based on the partial Fourier–Lagrange sums is proved.

**Keywords:** nonparametric regression, consistency, estimator, orthogonal series.

The article explores the practical necessity of using elliptic functions. The history

of the origin of the concept of an elliptic function is considered in detail. Clear conclusions on

the formation of the apparatus of the theory of elliptic functions in the works of Abel, Jacobi,

Weierstrass and Somov are proposed. Based on the proof of Abel’s theorem, a representation of

elliptic functions in terms of theta functions is shown.

The introduction and use of elliptic and hyperelliptic functions bring the problems of control

In a $n$-dimensional unitary space $ \mathrm{U}_{}^{n} $ $(n > 4)$ there are three series of regular polytopes: the regular simplex $ \mathrm{\alpha}_{n}^{} $ , the generalized cross polytopes $ \mathrm{\beta}_{n}^{m} $ and the generalized $n$-cube $ \mathrm{\gamma}_{n}^{m} $. The generalized $n$-cube has $ m^{n} $ vertices:

$ (\mathrm{\Theta}_{}^{\mathrm{k}_{1}^{}}, \mathrm{\Theta}_{}^{\mathrm{k}_{2}^{}}, ..., \mathrm{\Theta}_{}^{\mathrm{k}_{n}^{}} ) $,

In this paper we consider periodic homeomorphism $ \varphi $, which acts on genus $ p $ surface. Homeomorphism is called * periodic *, if exists $ n \in \mathbb {N} $ such that $ \varphi ^ {n} \equiv \mathrm {id}$. We study connections of such homeomorphisms with 3-dimensional topology. More accurately, we have established the condition that given 3-dimensional Seifert manifold is realised as mapping torus of some periodic homeomorphism $ \varphi $. Moreover, this periodic homeomorphism is almost fully determined by topology of its mapping torus.

In recent years, there has been a revival of interest in the history of mathematics in Russia, especially domestic. However, the inclusion of young researchers in this scientific work is very difficult. One of the reasons for this is the violation of the continuity of scientific generations of historians of mathematics, caused, in turn, by the "brain drain"and the departure of researchers to other fields of activity since the 1990s in the absence of sufficient motivation.

We consider the energy operator of four electron systems in the impurity Hubbard model and investigated the structure of essential spectra and discrete spectrum of the system

in the quintet state of the system. It is shown that there are such situations:

a). the essential spectrum of the four-electron quintet state operator is consists of the union of four segments, and the discrete spectrum of the four-electron quintet state operator is consists of single eigenvalue;

In this paper, we consider with a class of system of differential equations whose argument transforms are involutions. In this an initial value problem for a differential equation with involution is reduced to an initial value problem for a higher order ordinary differential equation. Then either two initial conditions are necessary for a solution; the equation is then reduced to a boundary value problem for a higher order ODE.

**Keywords: ** involution, linear differential equation, fixed point, boundary value problem

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.

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.

**Keywords**: nonlocal problem, parabolic equation, monotone operator.

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