The 𝜏 -closed σ-local formations of finite groups are studied, where σ is some partition of the set of all primes, 𝜏  is an arbitrary subgroup functor. The question of the existence of 𝜏  Hσ -critical formations is investigated, i. e., such 𝜏 -closed σ-local formations L H ⊆ for which all proper 𝜏 -closed σ-local subformations are contained in the class of groups H. It is proved that if H is a σ-local formation of classical type, then every 𝜏 -closed σ-local formation F H ⊆ has at least one  H^𝜏 σ - critical subformation.

The generalized Prandtl integro-differential equation is considered, in which the singular integrals contain both the unknown solution and its derivative. Direct numerical methods for solving this problem are constructed, based on representing the solution and the variable coefficients of the equation in the form of Chebyshev interpolation polynomials. The developed approach, which utilizes known spectral relations, makes it possible to obtain analytical expressions for the singular component of the problem and reduces its solution to a system of linear algebraic equations for the coefficients of the interpolation polynomial. Two computational schemes for the solution are constructed using Chebyshev polynomials of the first and second kind, respectively. Using a model problem as an example, it is shown that both schemes ensure good conditioning of the algebraic problem and possess high accuracy, demonstrating an exponential convergence rate of the error.

In this work, we search for infinitesimal equivalence transformations of 16 families of quasi-linear elliptic equations of the second order in two variables. A symmetry reduction has been performed for the families of equations u_tt + u_xx = f(t,x,u), u_tt + u_xx = f(x,u).

The SIR model currently remains the basic model of the epidemic process. This model and its numerous generalizations (SEIR, SEIRD, etc.), as well as some similar models, belong to the class of compartmental models, that describe the dynamics of the epidemic process as the interaction of “compartments”, i. e., various groups into which, according to their role in transmission infections (susceptible, infected, recovered, etc.) break up the entire population of people among whom the epidemic is spreading. Identifying the parameters of these models is often a challenging task. This task is somewhat simplified when conducting retrospective analyses of epidemics, where full incidence data are available at all stages of the process. We propose a simplified method for identifying a stationary SIR model using a minimal set of observed data that allows for relatively reliable determination. Based on the proposed methodology, the inapplicability of the stationary SIR model to describe the first wave of the COVID-19 pandemic in Germany and Italy is shown.

In the present paper, we will focus on the non-relativistic problem for a spin 3/2 particle in magnetic field, applying cylindrical coordinates and two tetrads: Cartesian and cylindrical. Here appear six different presentations for 4-component wave functions: three Cartesian ones L^cart  Ψ^cart  ,Ψ^cart  and provided by using the relevant gauge transformation L^cyl  = S (φ) L ^cart three different presentations in cylindrical tetrad L^cyl   Ψ^cyl   ,Ψ^cyl  First, we specify the non-relativistic equation for a spin 3/2 particle in magnetic field for Cartesian tetrad in bases with non-diagonal and diagonal matrix of the third spin projection. Solutions of two types are found: the first one is associated with the operator of the orbital angular momentum; the second solution relates to the third projection of the total angular momentum. Equations arising here are solved in terms of the confluent hypergeometric functions, and the corresponding energy spectra are found. The gauge transformation is introduced which relates two tetrads: Cartesian and cylindrical; it permits us to transform the system of equations in polar coordinate from Cartesian tetrad to cylindrical one. The rules for gauge transformations of diagonalized operators of the total angular and orbital momentums are found.

Adaptive Optics as a relatively new trend in the modern optical systems allows correcting the perturbations related to light propagation in the turbulent atmosphere obtaining the diffraction-limited images. Because it requires computer processing of large information flow in real time, a key problem lays in finding the mathematical algorithms with reduced computational complexity. The paper proposes the approach, which takes into account structural properties of both atmospheric turbulence and the most active optical system. The possibility was demonstrated of one-dimensional accounting of sparse and convolution-like (Toeplitz) structure of adaptive correction in one algorithm. The analysis is made of Multi-Conjucate Adaptive Optics and the comparison was conducted of the nature complexity of the algorithm induced by this strucute with the complexity of sparse algorithm. It was demonstrated how the structural approach can be applied to the quite laborious tomographic turbulence volume reconstruction phase with obtaining due to this a significant computational speed-up.

We calculate the tensor analyzing power component T₂₀ for coherent photoproduction of pions on the deuteron in the framework of the impulse approximation. The sensitivity of the results to the choice of the deuteron wave function is investigated. We compare the obtained values with the predictions of other models and with experimental data. The noticeable discrepancy between the theory and experiment was found at the photon energy Eγ = 400 MeV.

The study considers the problem of determining geographical relationships between agricultural management zones (MZs) across different rounds of agrochemical surveys of agricultural lands in the Republic of Belarus. A computational algorithm is proposed for identifying relationships between MZs from successive survey rounds, based on the application of the Weiler – Atherton polygon clipping algorithm. The algorithm enables the detection of intersection areas between MZ boundaries and the calculation of the percentage of overlap between MZs of the previous and current survey rounds, thereby allowing the establishment of hierarchical “parent – child” relationships among them. Software developed on the basis of the proposed algorithm was implemented using modern web technologies (PHP, Drupal, MySQL, Bootstrap, Leaflet). The system provides batch processing of large volumes of geospatial data, interactive visualization of MZ boundaries on digital maps, and remote access to analytical results. Experimental validation using geodata from the supposed 15th and 16th agrochemical survey rounds demonstrated the effectiveness of the proposed approach in identifying spatial correspondences between MZs. The developed computational algorithm and software offer new opportunities for the field history maintaining, optimization of fertilizer application rates, and evaluation of their agronomic efficiency, as well as for addressing advanced challenges in the precision agriculture and comprehensive biosphere assessment.