Membranes with thin and heavy inclusions: Asymptotics of spectra

We study the asymptotic behaviour of eigenvalues of 2D vibrating systems with mass density perturbed in a vicinity of closed curves. The threshold case in which the resonance frequencies of the membrane and the frequencies of thin inclusion coincide is investigated. The perturbed eigenvalue problem can be realized as a family of self-adjoint operators acting on varying Hilbert spaces. However the so-called limit operator is non-self-adjoint and possesses the Jordan chains of length 2. Apart from the lack of self-adjointness, the operator has non-compact resolvent. As a consequence, its spectrum has a complicated structure, for instance, the spectrum contains a countable set of eigenvalues with infinite multiplicity. The complete asymptotic analysis of eigenvalues has been carried out.

Feasibility of Reducing Operator-to-Passenger Contact for Passenger Screening at the Airport with Respect to the Power Consumption of the System

So far, airport security screening has only been analysed in terms of efficiency, level of service, and protection against any acts of unlawful interference. Screening procedures have not yet addressed the need to limit operator-to-passenger contact. However, the pandemic situation (COVID-19) has shown that it is a factor that can be a key protection for the health of passengers and operators. The purpose of this paper was to analyse the feasibility of reducing contact between operators and passengers in the airport security screening system by process management with respect to the power consumption of the system. Experimental research was conducted on a real system. A computer simulation was applied to estimate system performance and power consumption. The paper identifies the important findings that expand upon previous knowledge. The results showed that there are two key factors: the experience of operators and proper system structure. These factors can significantly reduce the number of operator-to-passenger contacts and, in parallel, provide lower energy consumption of the system. The results obtained in this article showed that proper management improves the process by up to 37%. This approach expands the World Health Organization’s policy of prevention against COVID-19 and helps to ensure sustainable process management.

General methods of convergence and summability

This paper is on general methods of convergence and summability. We first present the general method of convergence described by free filters of $\mathbb{N} $ N and study the space of convergence associated with the filter. We notice that $c(X)$ c ( X ) is always a space of convergence associated with a filter (the Frechet filter); that if X is finite dimensional, then $\ell _{\infty }(X)$ ℓ ∞ ( X ) is a space of convergence associated with any free ultrafilter of $\mathbb{N} $ N ; and that if X is not complete, then $\ell _{\infty }(X)$ ℓ ∞ ( X ) is never the space of convergence associated with any free filter of $\mathbb{N} $ N . Afterwards, we define a new general method of convergence inspired by the Banach limit convergence, that is, described through operators of norm 1 which are an extension of the limit operator. We prove that $\ell _{\infty }(X)$ ℓ ∞ ( X ) is always a space of convergence through a certain class of such operators; that if X is reflexive and 1-injective, then $c(X)$ c ( X ) is a space of convergence through a certain class of such operators; and that if X is not complete, then $c(X)$ c ( X ) is never the space of convergence through any class of such operators. In the meantime, we study the geometric structure of the set $\mathcal{HB}(\lim ):= \{T\in \mathcal{B} (\ell _{\infty }(X),X): T|_{c(X)}= \lim \text{ and }\|T\|=1\}$ HB ( lim ) : = { T ∈ B ( ℓ ∞ ( X ) , X ) : T | c ( X ) = lim and ∥ T ∥ = 1 } and prove that $\mathcal{HB}(\lim )$ HB ( lim ) is a face of $\mathsf{B} _{\mathcal{L}_{X}^{0}}$ B L X 0 if X has the Bade property, where $\mathcal{L}_{X}^{0}:= \{ T\in \mathcal{B} (\ell _{\infty }(X),X): c_{0}(X) \subseteq \ker (T) \} $ L X 0 : = { T ∈ B ( ℓ ∞ ( X ) , X ) : c 0 ( X ) ⊆ ker ( T ) } . Finally, we study the multipliers associated with series for the above methods of convergence.

A Regularization Method for Nonlinear Integro-differential Equations with a Zero Operator of the Differential Part and with Several Rapidly Varying Kernels

A nonlinear integro-differential equation with a zero operator of the differential part and several rapidly changing kernels is considered. The work is a continuation of the research conducted earlier for a single rapidly changing core. The main ideas of this generalization and the subtleties that arise in the development of the corresponding algorithm for the regularization method are fully visible in the case of two rapidly changing kernels, so for the sake of reducing the calculations, this particular case is taken. A similar problem with a single spectral value of the kernel of an integral operator is analyzed in one of the authors ' papers. In this case, the singularities in the solution of the problem are described only by the spectral value of the kernel. However, the influence of the zero operator of the differential part affects the fact that in the first approximation, the asymptotics of the solution of the problem under consideration will not contain the functions of the boundary layer, and the limit operator itself will become degenerate (but not zero). The conditions for the solvability of the corresponding iterative problems, as in the linear case, will not be in the form of differential equations (as was the case in problems with a non-zero operator of the differential part), but integro-differential equations, and the formation of these equations is significantly influenced by nonlinearity. Note that, in contrast to the linear case, there is no inhomogeneity of the corresponding linear problem in the right part of the problem under study. As it was shown earlier, its presence in the problem would lead to the appearance in the asymptotic solution of terms with negative powers of a small parameter, and in the nonlinear case there would be innumerable such powers, and the corresponding formal asymptotic solution would have the form of a Laurent series. This would make the creation of an algorithm for asymptotic solutions problematic, so in this paper, wanting to remain within the framework of asymptotic solutions of the type of Taylor series, inhomogeneity is excluded. In addition, in the nonlinear case, so-called resonances may occur, which significantly complicate the development of the corresponding algorithm for the regularization method. This publication deals with the non-resonant case. It is assumed that the study of an alternative variant (a more complex resonant problem) will be carried out in the future.

Internal boundary layer in a singularly perturbed problem of fractional derivative

This paper is devoted to the study of internal boundary layer. Such motions are often associated with effect of boundary layer, i.e. low flow viscosity affects only in a narrow parietal layer of a streamlined body, and outside this zone the flow is as if there is no viscosity - the so-called ideal flow. Number of exponentials in the boundary layer is determined by the number of non-zero points of the limit operator spectrum. In the paper we consider the case when spectrum of the limit operator vanishes at the point To study the problem the Lomov regularization method is used. The original problem is regularized and the main term of asymptotics of the problem solution is constructed as the low viscosity tends to zero. Numerical results of solutions are obtained for different values of low viscosity.

Singularly Perturbed Cauchy Problem for a Parabolic Equation with a Rational “Simple” Turning Point

The aim of the research is to develop the regularization method. By Lomov’s regularization method, we constructed a uniform asymptotic solution of the singularly perturbed Cauchy problem for a parabolic equation in the case of violation of stability conditions of the limit-operator spectrum. The problem with a “simple” turning point is considered in the case, when the eigenvalue vanishes at t=0 and has the form tm/na(t). The asymptotic convergence of the regularized series is proved.

On the Regularized Asymptotics of a Solution to the Cauchy Problem in the Presence of a Weak Turning Point of the Limit Operator

An asymptotic solution of the linear Cauchy problem in the presence of a “weak” turning point for the limit operator is constructed by the method of S. A. Lomov regularization. The main singularities of this problem are written out explicitly. Estimates are given for ε that characterize the behavior of singularities for ϵ→0. The asymptotic convergence of a regularized series is proven. The results are illustrated by an example. Bibliography: six titles.

Nonlinear Singularly Perturbed Integro-Differential Equations and Regularization Method

The paper considers a nonlinear integro-differential system of singularly perturbed equations. We discuss the question of the spectrum of its operator, which does not coincide with the spectrum of its limit operator and includes an additionally identically zero point. In the case of linear systems, this difference does not play a special role, since the regularization and construction of the space of solutions of the corresponding iterative problems are realized at nonzero points of the spectrum. In the case of nonlinear problems, the identically zero point of the spectrum plays an essential role in the construction of the solution space in the resonance and nonresonance cases (see below); therefore, in most works using the regularization method in nonlinear problems, only the nonresonance case is usually considered. In the paper, for the classical integrodifferential system, regularization (according to Lomov) is carried out and the corresponding algorithm for constructing asymptotic solutions taking into account the zero point of the spectrum is developed.