For various partitions of the set P of all prime numbers, properties of generalized local classes (formations, Fitting classes) of finite groups are investigated. Criteria for σ-locality of an α-local class of groups are proved, where σ and α are some distinct partitions of P. Properties of products of generalized local classes, as well as their algebras, are studied. For a σ-soluble σ-local class, sufficient conditions for the commutativity of the σ-algebra generated by it are obtained.

The proof of the well-posedness of the mixed problem for the string oscillation equation in the half-strip with differential polynoms in the boundary conditions. The conditions of the existence of the unique and smooth enough solution are obtained in the half strip in general. It is shown that it is reduced to the solution of the initial-value problems for the ordinary linear differential equations with variable coefficients. The case when the solution smoothness is reduced during the increasing of the time and the case when it doesn’t happen are studied. For both cases the sufficient conditions for smooth reduction (conservation) are obtained. These conditions are based on the coefficients in boundary conditions. Also, with the help of the characteristics method the necessary and sufficient matching conditions are obtained. These conditions guarantee the existence and uniqueness of the classical solution of the given problem when given functions are smooth enough. The obtained results are given for both homogeneous initial equation and inhomogeneous one.

It is known that the decision problem of t-TOUGHNESS of a graph is coNP-complete in general. Moreover, in many subclasses of graphs, the decision problem of t-TOUGHNESS remains NP-hard, in particular, in the class of r-regular graphs, where r ≥ 3t for any integer number t ≥ 1. The complexity of the decision problem of t-TOUGHNESS for r-regular graphs remains open when 2t ≤ r < 3t, and when r = 2t + 1 the complexity of the decision problem is particularly intriguing. In the latter case it has been conjectured, that it remains NP-hard. In this paper, we establish the validity of this conjecture.

The eigenvalue problem for generalized helicity operator for a spin 3/2 particle in presence of the uniform magnetic field is solved. After separating the variables in the basis of cylindrical coordinates (r, ϕ, z) and the tetrad, the system of 16 first-order differential equations in the variable r is derived. This system is studied with the use of the method of projective operators, constructed with the use of the third projection of the spin for the particle. In accordance with thе method by Fedorov – Gronskiy, all 16 variables may be expressed in terms of only 4 distinguished functions, which are constructed in terms of confluent hypergeometric functions. Further the problem reduces to studying the linear algebraic homogeneous system for 16 algebraic variables. In the end, we derive algebraic equations of the second and the fourth order, their roots determine the possible eigenvalues of the helicity operator.

This paper presents an improved methodology for measuring the transition electromagnetic form factor in the conversion decay ω → π0e+e– using data collected by the CMD-3 detector at the VEPP-2000 e+e– collider. The key improvement involves the application of a kinematic reconstruction technique under two distinct hypotheses: the signal hypothesis (ω → π0e+e–) and the dominant background hypothesis (ω → π+π–π0). This approach allows for a powerful suppression of 3π background, virtually eliminating it, and significantly narrows the invariant mass distribution of two photons from π0 decay in signal events. The refined π0 mass peak enhances the separation of the signal process from the remaining QED background (e+e– → e+e–γγ). To demonstrate the effectiveness of the method, it was applied to a subset of the data with an integrated luminosity of 13 pb⁻¹, accumulated near ω-meson mass. The analysis shows a significant improvement in the precision of the form factor F(q) measurement. The developed methodology paves the way for a more precise determination of the form factor slope parameter Λ ω−2 when applied to the full dataset, which has an integrated luminosity of approximately 50 pb⁻¹.

A detailed analysis of theoretical models used to interpret experiments with parametric X-ray radiation (PXR) emitted by relativistic charged particles in symmetric Bragg geometry is carried out. It is shown that the dynamical theory of PXR is in good agreement with the experimental results obtained at the Sirius synchrotron for 900 MeV electrons. The most important advantage of the PXR dynamical theory over the kinematical one is the correct description of the primary extinction and interference between two types of waves (fast and slow), generated within parametric X-ray radiation in a crystal. At ultrarelativistic energies, an analytical expression is obtained for the total number of quanta emitted by an electron into the solid angle where PXR intensity has a maximum.

The goal of this project is to create a new type of polarizer using printed circuit boards that can convert an incident linearly polarized wave into a reflected circularly polarized wave in the microwave range. This device represents a metamaterial surface consisting of a metal plate array made up of flat copper rectangular Ω-elements on a glass fiber substrate. By optimizing the shape of these elements, we found that they can also be used as absorbers for microwaves in addition to their ability to transform polarization. We showed that this form of Ω-resonators, which make up the metamaterial surface, are universal for use in THz polarizers. Finally, we investigated the polarization-selective properties of a metamaterial based on standard copper-coated fiberglass. This material exhibited polarization-selective properties near the resonant frequency in the microwave range and can be used as an effective polarization converter for microwaves.