Approximation by trigonometric Fourier series is a well-developed branch of the theory of approximation by polynomials. Methods of approximation by rational trigonometric Fourier series have not been researched so deeply yet. In particular, rational trigonometric operators of the Fejér type have not been used in the rational approximation with free poles. In this paper, we consider the approximation of the function | sin | , (0;2), ∈ s x s by rational trigonometric operators of the Fejér type. An integral representation of the remainder for the above-mentioned approximation is obtained. An estimate of approximations is found in the points of analyticity of the function | sin |s x under the condition that the corresponding system of rational functions is complete. It is shown that the order of uniform approximation in the case of approximation by rational Fejér functions with two geometrically different poles is higher than the order of approximation by trigonometric polynomials. As a result, an asymptotic estimation of the uniform approximation by trigonometric Fejér sums in the polynomial case is obtained. 

The objective of this study is to obtain the stabilizability conditions and a stabilizing composite state feedback control for the exponential stabilization of three-time-scale singularly perturbed linear time-invariant systems with multiple commensurate delays in the slow state variables and with two small parameters of perturbation (TSPLTISD). The stabilizability conditions and the stabilizing feedback do not depend on the small parameters and are valid for all of their sufficiently small values. The approach used in this work is the nondegenerate decoupling transformation that splits the TSPLTISD into three regularly dependent on the small parameters subsystems, which are lower in dimensions than the TSPLTISD. Further, the decoupled subsystems are approximated by three subsystems that do not depend on the small parameters. It is proven that the stabilizability of the approximating subsystems guarantees the robust (with respect to small parameters) stabilizability of the original TSPLTISD. Finally, we obtain a representation of a parameter free composite feedback control for the TSPLTISD, stabilizing it for all sufficiently small values of the parameters. A numerical example is given. 

In this paper, we consider the maximum weight ({K1,K2},k,l)-packing problem in a graph. This problem generalizes a number of well-known problems, for example: maximum induced matching, k-separated matching, connected matching, independent set, dissociating set, k-packing. We show that in the class of cographs, a maximum weight ({K1,K2},k,l)- packing can be computed in O(n + m) time. Let Γ be a class of graphs and Γ* be a class of all simple (with respect to the modular decomposition) induced subgraphs from Γ. It is proven that if the maximum weight ({K1,K2},k,l)-packing problem can be solved in the class of graphs Г* in time O(np ), where p ≥ 2 is a constant, then this problem can be solved in the class of graphs Г in time O(np ). 

Using the representation of bent functions (maximum nonlinear functions) by bent rectangles, that is, special matrices with restrictions on columns and rows, we obtain herein an upper bound on the number of bent functions that improves the previously known bounds in a practical range of dimensions. The core of our method is the following fact based on the recent observation by V. Potapov (arXiv:2107.14583): a 2-row bent rectangle is completely determined by one of its rows and the remaining values in slightly more than half of the columns. 

The initial stage of a gamma-ray burst can be described by the diffusion of radiation in the shell which is ultrarelativistically widening. We have herein considered the case when the initial energy distribution in the shell depends on the depth according to a linear law. At the same time, the slope of the initial energy distribution function determines the effective temperature and the instantaneous spectrum of the gamma-ray burst at the initial time moment, as well as the time-integrated spectrum at high frequencies. Moreover, we investigated the dependence of the instantaneous spectrum of the gamma-ray burst on the time of arrival: if it increases, the flux maximum is observed for a smaller frequency independently of the initial energy distribution in the shell. It the time of arrival is larger than 1 s, the instantaneous spectrum is independent of the initial energy distribution in the shell. The slope of the power-law part of the time-integrated spectrum at lower frequencies is the same for any initial energy distribution in the shell and it is equal to 1.98; at higher frequencies, it becomes steeper with increasing the slope of the function corresponding to the energy distribution in the shell at the initial instant of time, and it has values between –2.38 and –13.73.This allows us to explain with the help of our model a large number of gamma-ray bursts that have a typical Band spectrum.

In this paper, we propose a physics-based method of prediction high-energy solar flares (SFs) with the help of neutrino detectors utilizing coherent elastic neutrino-nucleus scattering (CEνNS). The behavior of neutrino beams passing through coupled sunspots (CSs) being the sources of future SFs is investigated. We consider the evolution of left-handed electron neutrino νeL and muon neutrino νμL beams formed in the convective zone after the passage of the Micheev – Smirnov – Wolfenstein resonance. It is assumed that the neutrinos possess the charge radius, the magnetic and anapole moments while the CS magnetic field is vortex, nonhomogeneous and has twisting. Estimations of the weakening of the neutrino beams after traversing the resonant layers are given. It is shown that for SFs this weakening could be registered by neutrino detectors of the second generation only when neutrinos have the Dirac nature. 

By analyzing the wave equations, the coincidence of the energy reflection and transmission coefficients for the s- and p-polarization waves is herein substantiated when they are incident on the interference coating from opposite directions. The coating can be characterized by an arbitrary spatial profile of the refractive index, the limiting condition is the absence of optical losses in it. Reciprocity relations are obtained for the energy reflection and transmission coefficients of natural light for a structure in the form of a plane-parallel dielectric plate with interference coatings on its opposite sides. It is shown that when a structure with an absorbing plate is illuminated in opposite directions, the energy reflection coefficients can differ, while the energy transmission coefficients always coincide. Reciprocity relations are applied to the calculation of broadband antireflective interference coatings consisting of alternate layers Nb2O5 and SiO2 deposited on a polycarbonate plate. As a result, their correctness is confirmed and it is shown that the optimized antireflection two-sided interference coating provides approximately a five times lower averaged energy reflection coefficient compared to the optimized one-sided coating. 

In this paper, using mathematical programming methods we investigate the problem of optimally planning the inspection and repair of cargo containers at several port facilities over time. Its relevance is conditioned by the necessity of optimization of the processes of planning logistics operations. We represent a formulation of this problem and its reduction to a mixed integer linear programming (MILP). Containers have several types and levels of quality, which determine the cost of their repair. The objective function includes total costs of storage, inspection, repair, transportation of containers and penalties for both container rejection and unsatisfied consumer demand. Using the proposed model, it is possible to achieve reasonable computing time for port operations and minimize underutilization or overutilization of port capacity, preventing financial losses and increasing efficiency. Also, this model can be easily converted to solve optimization problems in other areas of logistics.