We study the properties of the lattice c ^τ _ω∞ of all τ-closed totally ω-composition formations of finite groups. We prove the modularity of such a lattice of formations for any subgroup functor τ and any nonempty set ω of primes. In particular, we obtain a positive answer to the question of A. N. Skiba and L. A. Shemetkov (2000) about the modularity of the lattice c_∞ ^L of all totally L-composition formations. We establish that the lattice c ^τ _ω∞ is a complete sublattice of the lattice c^ω _∞ of all totally ω-composition formations of finite groups.

The analytical properties of solutions to stationary equations of the second and fourth orders in the hierarchy of the second Painlevé equation are considered. For the second-order equation, it is shown that the Bäcklund transformation in the general case determines the formula of the addition theorem for the Weierstrass elliptic function. For the fourth and sixth-order equations, Bäcklund transformations and special classes of solutions are constructed. It has been established that, for a certain relationship between the parameters, the set of solutions to the first term of the stationary hierarchy is a subset of solutions to the second term and the set of solutions to the second term of the hierarchy is a subset of solutions of the six-order equation of the stationary hierarchy of the second Painlevé equation.

We study the problem of exact reconstruction of the solution from measurements of the observed output for linear autonomous completely regular differential-algebraic systems with commensurate delays. The reconstruction process is implemented using a finite observer, which is the output of a linear autonomous retarded system type with commensurate concentrated and distributed delays and a predetermined finite spectrum. The criterion for the existence of such an observer is obtained.

A finite-difference computational algorithm is proposed for solving a mixed boundary value problem for heat equation given in a two-dimensional domains of complex shape. To solve the problem, generalized curvilinear coordinates are used. The physical domain is mapped to the computational domain (unit square) in the space of generalized coordinates. The original problem is written in curvilinear coordinates and approximated on a uniform grid in the computational domain. The obtained results are mapped on a non-uniform boundary-fitted difference grid in the physical domain. The second-order approximations of mixed Neumann – Dirichlet boundary conditions are constructed. The results of computational experiments are presented. The second order of accuracy of the presented computational algorithm is confirmed.

In essence, primordial black holes generated in the early Universe as a result of a gravitational collapse of the high-density matter are detectors of the processes proceeding in it. As these black holes are generated at high energies (close to the Planck energies) and their radii are small, there is a need to take into consideration the quantum-gravitational corrections for them. In this paper, within the scope of the Generalized Uncertainty Principle, the author continues a study of the quantum-gravitational corrections and their contributions into the inflationary parameters for primordial black holes in the pre-inflationary epoch. Specifically, within this pattern, the author considers a case of Hawking’s radiation (evaporation) for the above-mentioned black holes and derives formulae for the corresponding changes (”shifts”) of the basic inflation parameters. In all cases the expressions for the corresponding correction of е-foldings in an inflation model have been found. In conclusion the main problems for further studies are formulated.

A method based on the Newtonian and relativistic theories of body motion is proposed for calculating the density of dark matter, which, like visible (baryonic) matter, creates a gravitational field. Experimental data obtained by the Pioneer 10 and Pioneer 11 spacecraft and a variety of astronomical observations are used to detect and establish the mass of dark matter in the solar system, which turned out to be approximately equal to the mass of the Sun. Using the equations of motion of test bodies in the Newtonian and post-Newtonian approximations of the general theory of relativity, calculation formulas are obtained for calculating the density of dark matter in three cases: 1) baryonic and dark matter are uniformly distributed in space (their density is constant); 2) they are distributed according to spherically symmetrical laws; 3) baryonic matter is distributed spherically symmetrically, while dark matter is uniformly distributed. In the volume of a sphere with radius of 45 a. u. with the center in the center of gravity of the Sun, on the basis of known experimental data, the average density of the gas-dust and relict matter located in it is calculated, equal to 1,26 · 10^–16 g · cm^–3. In the same volume, the density of dark matter in all three cases varies according to the derived calculation formulas in the range from 3,38 · 10^–16 to 3,34 · 10^–16 g · cm^–3, which gives the superiority of dark matter over baryonic one from 2.68 to 2.72 times. The given numerical estimates may change when the experimental data used change. The paper also contains a brief discussion of other methods for calculating the density of dark matter in space and a comparison with our results.

The optical, morphological and electrophysical properties of silver nanostructures fabricated by electron beam evaporation and annealed at temperatures of 145 and 195 °C were studied. All samples are characterized by the presence of a pronounced surface plasmon absorption resonance band in the visible range and represent close-packed monolayers of nanoparticles, the average sizes of which increase from ~10 nm in the original samples to ~35–40 nm and ~45–60 nm in the annealed ones, depending on the annealing temperature. The influence of various factors on the spectral characteristics of samples, including the size of nanoparticles and electrodynamic interactions between nanoparticles, is discussed. It has been shown that all granular nanostructures studied, both initial and annealed, are highly resistive. It has been established that for the initial and annealed at 145 °C samples, near low values of the applied voltage, a dependence of the current on the irradiation wavelength can be traced, with its value changing up to two orders of magnitude for certain wavelengths.

The study investigated the effect of Co⁶⁰ gamma-quanta on the reverse current-voltage characteristic (IV) of silicon photomultiplier (SiPMs) with 1004 cells, which themselves were optically isolated from each other n⁺ –p–p⁺ -structures. The cells were optically isolated from each other by trenches filled with tungsten after passivation of the walls with SiO₂ and Si3N₄ layers. Two variants of structural design of SiPMs were studied. Two variants were tested for the trench metal connection in the SiPMs: variant BI connected the trench metal to the n⁺-region of the cell through a quenching polysilicon resistor, while variant BII connected it to the p⁺-region. The breakdown voltage of the investigated SiPMs was U_br = 34 ± 1.0 V. The samples were irradiated in both the active electrical mode (avalanche breakdown mode) and the passive mode (reverse bias U_b = 0 V). It was discovered, that at dose of D = 10⁶ rad, the dark current for SiPM (BI) and (BII) increased by 6–7 times when irradiated in passive mode and by 15–16 times for SiPM (BII) when irradiated in active mode. For SiPM (BI) irradiated in the avalanche breakdown mode, the dark current increased by 10⁴ times at D = 10⁵ rad. The research demonstrates that the radiation-induced degradation of the dark current in the SiPMs under study is due to an increase in the generation and, primarily, surface components. This is a result of the accumulation of positive charge in the insulating layer of the separating trenches.