A formation F is called a Baer – Shemetkov formation in a class X of groups if in any finite X-group the intersection of all F-maximal subgroups coincides with the F-hypercenter. It is proved that for a non-empty hereditary saturated formation F there exists the greatest by inclusion hereditary saturated formation BSF such that F is a Baer – Shemetkov formation in BSF. The connection of this result with the solution of Agrawal’s (1976) and Shemetkov’s (1995) questions is discussed. For the class U of all supersolvable groups the class BSU is described and the algorithm for its recognition is presented.

In the article we develop a simple and uniform method that allows one to calculate the number and localization of real solutions of three-term (trinomial) algebraic equations of arbitrary degree with real coefficients. The method is based on the fact that using certain substitutions, a three-term equation is reduced to an auxiliary equation with one parameter, represented as an explicit function of the coefficients of the initial equation, and the properties of the solutions of the initial equation depend only on the values of this parameter.

This article is devoted to the problem of interpolation of operators defined on Banach algebras of functions. Interpolation operator polynomials in the Lagrange and Newton forms of arbitrary fixed degree, containing the operation of multiplying elements of functional algebra, are obtained as solutions of the corresponding problems of single operator interpolation. The construction of interpolation formulas of the Newton’s structure is based on apparatus of operator separated differences. Classes of operator polynomials are indicated that are typical for functional Banach algebras under consideration, with respect to which the presented interpolation formulas are invariant. Explicit representations of the operator interpolation error are obtained. Special cases of linear interpolation formulas are considered, when the multiplication operation is given by various rules for convolution of elements of functional algebra. Corresponding first-order interpolation formulas are constructed, which contain the Fourier or Laplace transforms and are exact for linear operator polynomials of a special form.

The paper explains the shortcomings of the existing definition of the strong NP-completeness. An alternative definition is proposed, that preserves all existing results on proving the strong NP-completeness of decision problems. Besides, the definition of NP-completeness and the alternative definition of the strong NP-completeness have the same structure. 

Purpose of the research is to explain the peculiarities of the dependence of the Bose – Einstein correlation radius (size of the production region) on the multiplicity of produced charged particles at the energy in the center of mass system 7 и 13 ТeV in the processes of pp-collisions at the Large Hadron Collider in the ATLAS project experiment based on the hypothesis according to which protons are coherent states of partons, considered as excitations on the horosphere of relativistic momentum space. It is shown that in the framework of the model, a general form of the dependence of the correlation radius on the charged particle multiplicity is established, allowing the description of experimental data.

In order to improve the energy and coordinate resolution of the neutral trigger of the СMD-3 detector, it is proposed to use a clustering algorithm based on overlapping “windows” of trigger cells. The window of 2 × 2 crystals is considered to be a single trigger cell, and its signal amplitude value is sent for processing by the trigger system. Based on MonteCarlo simulation, the optimal number of trigger cells to assign to the cluster for energy reconstruction of particles hitting the calorimeter in real time was determined.

Transformation of ordinary laser beams of Gaussian type into the ones of Bessel type gives a number of advantages. In particular, it may improve a multi-mode beam energy distribution in the far zone and keep it stable over a long range of propagation distances. A usual method of the Bessel beam formation is based on the use of one or a pair of axicons, at that the latter is used for creating annular beams with small conic angles optimized for long range propagation. The disadvantage of axicons is their fixed geometry not allowing the beam parameter adjustment. An additional Bessel beam improvement allowing to reach a finer energy distribution comes from injection into it a controllable amount of spherical aberration. This requires development of a specialized optical system, which adds adjustable spherical aberration to the laser beam. Efficient design of such an optical system has been created and experimentally tested on the transformation of a multi-mode laser beam into Bessel beam.

In this paper, we propose and investigate a scheme for optical nanolithography of the interference type based on the use of excitation of counter-propagating surface plasmon-polaritons at a flat interface of a metal-dielectric nanostructure. A detailed calculation of the optical nanolithography scheme designed to form sinusoidal diffraction gratings is performed. It is shown that the use of an input prism with a large refractive index allows increasing the gain of the light field formed in the photoresist by more than an order of magnitude. It is found, that by changing the thickness of the layers of the metal-dielectric structure it is possible to change the wave number at which the plasmon resonance condition is realized, and thereby to control the period of the formed gratings and the depth of field penetration into the photoresist. The proposed scheme may be used to create two-dimensional, circular gratings, as well as gratings of arbitrary shape with an appropriate choice of the shape of input prism.

The effect of annealing at 350–500 °C on the structural-phase state, nanohardness, elastic modulus and critical peel load Lc of vacuum-arc TiN coatings deposited on a substrate made of pre-annealed 90CrSi steel was investigated. It has been established that the coating contains TiN and Ti phases. The nanohardness of the coating is 29 GPa, and the elastic modulus is 485 GPa. It was concluded that the increased values of the nanohardness and coating elastic modulus are associated with the presence of a large number of crystal lattice defects in it. It is shown that as the annealing temperature of the coating increases from 350 to 500 °C, the values of the nanohardness and elastic modulus of TiN coatings decrease, while an increase in the crystal lattice parameter, as well as a decrease in the dislocation density and TiN dispersion are recorded. An increase in the crystal lattice parameter of TiN during annealing of the coating is associated with the formation of vacancy complexes in it. It was found that as a result of annealing TiN coatings, the peeling load increases from 12.8 to 21.1 N. It was concluded that the increase in the peeling load of the coating during annealing is associated with the formation of an oxide film, which prevents the nucleation (generation) of dislocations during the introduction and movement of the indenter in the coating and, thus, slows down the formation of microcracks at the boundary of the coating and the substrate.