Normal modes of a waveguide as eigenvectors of a self-adjoint operator pencil

A waveguide with a constant, simply connected section S is considered under the condition that the substance filling the waveguide is characterized by permittivity and permeability that vary smoothly over the section S, but are constant along the waveguide axis. Ideal conductivity conditions are assumed on the walls of the waveguide. On the basis of the previously found representation of the electromagnetic field in such a waveguide using 4 scalar functions, namely, two electric and two magnetic potentials, Maxwells equations are rewritten with respect to the potentials and longitudinal components of the field. It appears possible to exclude potentials from this system and arrive at a pair of integro-differential equations for longitudinal components alone that split into two uncoupled wave equations in the optically homogeneous case. In an optically inhomogeneous case, this approach reduces the problem of finding the normal modes of a waveguide to studying the spectrum of a quadratic self-adjoint operator pencil.

On conjugate difference schemes: the midpoint scheme and the trapezoidal scheme

The preservation of quadratic integrals on approximate solutions of autonomous systems of ordinary differential equations x=f(x), found by the trapezoidal scheme, is investigated. For this purpose, a relation has been established between the trapezoidal scheme and the midpoint scheme, which preserves all quadratic integrals of motion by virtue of Coopers theorem. This relation allows considering the trapezoidal scheme as dual to the midpoint scheme and to find a dual analogue for Coopers theorem by analogy with the duality principle in projective geometry. It is proved that on the approximate solution found by the trapezoidal scheme, not the quadratic integral itself is preserved, but a more complicated expression, which turns into an integral in the limit as t0.Thus the concept of conjugate difference schemes is investigated in pure algebraic way. The results are illustrated by examples of linear and elliptic oscillators. In both cases, expressions preserved by the trapezoidal scheme are presented explicitly.

The asymptotic solution of a singularly perturbed Cauchy problem for Fokker-Planck equation

The asymptotic method is a very attractive area of applied mathematics. There are many modern research directions which use a small parameter such as statistical mechanics, chemical reaction theory and so on. The application of the Fokker-Planck equation (FPE) with a small parameter is the most popular because this equation is the parabolic partial differential equations and the solutions of FPE give the probability density function. In this paper we investigate the singularly perturbed Cauchy problem for symmetric linear system of parabolic partial differential equations with a small parameter. We assume that this system is the Tikhonov non-homogeneous system with constant coefficients. The paper aims to consider this Cauchy problem, apply the asymptotic method and construct expansions of the solutions in the form of two-type decomposition. This decomposition has regular and border-layer parts. The main result of this paper is a justification of an asymptotic expansion for the solutions of this Cauchy problem. Our method can be applied in a wide variety of cases for singularly perturbed Cauchy problems of Fokker-Planck equations.

Quantum mereology in finite quantum mechanics

Any Hilbert space with composite dimension can be factored into a tensor product of smaller Hilbert spaces. This allows us to decompose a quantum system into subsystems. We propose a model based on finite quantum mechanics for a constructive study of such decompositions.

Numerical simulation of thermal processes occurring in materials under the action of femtosecond laser pulses

In this work, a numerical study of the solutions of the parabolic and hyperbolic equations of heat conduction with the same physical parameters is carried out and a comparative analysis of the results obtained is carried out. The mathematical formulation of the problem is discussed. The action of the laser is taken into account through the source function, which was chosen as a double femtosecond laser pulse. In the hyperbolic equation, in contrast to the parabolic one, there is an additional parameter that characterizes the relaxation time of the heat flux. In addition, the source of the hyperbolic equation contains an additional term - the derivative of the power density of the source of the parabolic equation. This means that the temperature of the sample is influenced not only by the power density of the source, but also by the rate of its change. The profiles of the sample temperature at different times and its dynamics at different target depths are shown. The calculations were carried out for different time delays between pulses and for different relaxation parameters.

Optimization of mobile device energy consumption in a fog-based mobile computing offloading mechanism

The offloading of computing tasks to the fog computing system is a promising approach to reduce the response time of resource-greedy real-time mobile applications. Besides the decreasing of the response time, the offloading mechanisms may reduce the energy consumption of mobile devices. In the paper, we focused on the analysis of the energy consumption of mobile devices that use fog computing infrastructure to increase the overall system performance and to improve the battery life. We consider a three-layer computing architecture, which consists of the mobile device itself, a fog node, and a remote cloud. The tasks are processed locally or offloaded according to the threshold-based offloading criterion. We have formulated an optimization problem that minimizes the energy consumption under the constraints on the average response time and the probability that the response time is lower than a certain threshold. We also provide the numerical solution to the optimization problem and discuss the numerical results.

Study of the impact of the COVID-19 pandemic on international air transportation

Time Series Forecasting has always been a very important area of research in many domains because many different types of data are stored as time series. Given the growing availability of data and computing power in the recent years, Deep Learning has become a fundamental part of the new generation of Time Series Forecasting models, obtaining excellent results.As different time series problems are studied in many different fields, a large number of new architectures have been developed in recent years. This has also been simplified by the growing availability of open source frameworks, which make the development of new custom network components easier and faster.In this paper three different Deep Learning Architecture for Time Series Forecasting are presented: Recurrent Neural Networks (RNNs), that are the most classical and used architecture for Time Series Forecasting problems; Long Short-Term Memory (LSTM), that are an evolution of RNNs developed in order to overcome the vanishing gradient problem; Gated Recurrent Unit (GRU), that are another evolution of RNNs, similar to LSTM.The article is devoted to modeling and forecasting the cost of international air transportation in a pandemic using deep learning methods. The author builds time series models of the American Airlines (AAL) stock prices for a selected period using LSTM, GRU, RNN recurrent neural networks models and compare the accuracy forecast results.

Calculation of integrals in MathPartner

We present the possibilities provided by the MathPartner service of calculating definite and indefinite integrals. MathPartner contains software implementation of the Risch algorithm and provides users with the ability to compute antiderivatives for elementary functions. Certain integrals, including improper integrals, can be calculated using numerical algorithms. In this case, every user has the ability to indicate the required accuracy with which he needs to know the numerical value of the integral. We highlight special functions allowing us to calculate complete elliptic integrals. These include functions for calculating the arithmetic-geometric mean and the geometric-harmonic mean, which allow us to calculate the complete elliptic integrals of the first kind. The set also includes the modified arithmetic-geometric mean, proposed by Semjon Adlaj, which allows us to calculate the complete elliptic integrals of the second kind as well as the circumference of an ellipse. The Lagutinski algorithm is of particular interest. For given differentiation in the field of bivariate rational functions, one can decide whether there exists a rational integral. The algorithm is based on calculating the Lagutinski determinant. This year we are celebrating 150th anniversary of Mikhail Lagutinski.

To analysis of a two-buffer queuing system with cross-type service and additional penalties

The concept of cloud computing was created to better preserve user privacy and data storage security. However, the resources allocated for processing this data must be optimally allocated. The problem of optimal resource management in the loud computing environment is described in many scientific publications. To solve the problems of optimality of the distribution of resources of systems, you can use the construction and analysis of QS. We conduct an analysis of two-buffer queuing system with cross-type service and additional penalties, based on the literature reviewed in the article. This allows us to assess how suitable the model presented in the article is for application to cloud computing. For a given system different options for selecting applications from queues are possible, queue numbers, therefore, the intensities of transitions between the states of the system will change. For this, the system has a choice policy that allows the system to decide how to behave depending on its state. There are four components of such selection management models, which is a stationary policy for selecting a queue number to service a ticket on a vacated virtual machine each time immediately before service ends. A simulation model was built for numerical analysis. The results obtained indicate that requests are practically not delayed in the queue of the presented QS, and therefore the policy for a given model can be considered optimal. Although Poisson flow is the simplest for simulation, it is quite acceptable for performance evaluation. In the future, it is planned to conduct several more experiments for different values of the intensity of requests and various types of incoming flows.

Calculation of special functions arising in the problem of diffraction by a dielectric ball

To apply the incomplete Galerkin method to the problem of the scattering of electromagnetic waves by lenses, it is necessary to study the differential equations for the field amplitudes. These equations belong to the class of linear ordinary differential equations with Fuchsian singularities and, in the case of the Lneburg lens, are integrated in special functions of mathematical physics, namely, the Whittaker and Heun functions. The Maple computer algebra system has tools for working with Whittaker and Heun functions, but in some cases this system gives very large values for these functions, and their plots contain various kinds of artifacts. Therefore, the results of calculations in the Maple11 and Maple2019 systems of special functions related to the problem of scattering by a Lneburg lens need additional verification. For this purpose, an algorithm for finding solutions to linear ordinary differential equations with Fuchsian singular points by the method of Frobenius series was implemented, designed as a software package Fucsh for Sage. The problem of scattering by a Lneburg lens is used as a test case. The calculation results are compared with similar results obtained in different versions of CAS Maple. Fuchs for Sage allows computing solutions to other linear differential equations that cannot be expressed in terms of known special functions.