This article is devoted to the problem of interpolation of functions defined on sets of matrices with multiplication in the sense of Hadamard and is mainly an overview. It contains some known information about the Hadamard matrix multiplication and its properties. For functions defined on sets of square and rectangular matrices, various interpolation polynomials of the Lagrange type, containing both the operation of matrix multiplication in the Hadamard sense and the usual matrix product, are given. In the case of analytic functions defined on sets of square matrices with the Hadamard multiplication, some analogues of the Lagrange type trigonometric interpolation formulas are considered. Matrix analogues of splines and the Cauchy integral are given on sets of matrices with the Hadamard multiplication. Some of its applications in the theory of interpolation are considered. Theorems on the convergence of some Lagrange interpolation processes for analytic functions defined on a set of matrices with multiplication in the Hadamard sense are proved. The results obtained are based on the application of some well-known provisions of the theory of interpolation of scalar functions. Data presentation is illustrated by a number of examples.

We propose herein a new parsimonious Markov model for a discrete-valued time series with conditional probability distributions of observations lying in the exponential family with the multidimensional parameter. A family of explicit consistent asymptotically normal statistical estimators is constructed for the parameters of the proposed model for increasing length of observed time series, and asymptotically effective estimator is found within this constructed family. The obtained results can be used for robust statistical analysis of discrete-valued time series,and for statistical analysis of discrete-valued spatio-temporal data and random fields.

Graphics Processing Units (GPUs) are considered as the target computer for implementing parallel algorithms. The set of algorithm operations to be implemented on the GPU must be split into computation threads; the threads should be grouped into computation blocks that are performed atomically on stream processors. Threads of a single block are executed on a stream processor in parts-pools called warp; warp threads are executed simultaneously. The efficiency of the parallel algorithm depends on the way the data is stored in the GPU memory. If all warp threads request the same datum when executing the current operator, then it is desirable to place it in a shared or constant GPU memory; in this case, its distribution across the cores of the multiprocessor is actually realized by means of broadcast. If warp threads request data located close to the memory, then in this case there is a spatial locality of data, which makes it advisable to place this data in the GPU’s memory. The implementation of broadcast or spatial locality by placing data in a memory of the appropriate type allows one to significantly reduce traffic when exchanging data between the memory levels of the GPU. This paper formulates and proves the necessary and sufficient conditions under which it is possible to perform a broadcast or there is a spatial locality of data. The conditions are formulated in terms of functions that determine the use of array elements at occurrences in the algorithm operators and functions that define the information dependencies of the algorithm. The results of the work can be used to optimize parallel algorithms when they are implemented on the GPU.

In this article, we study a mixed problem in a quarter-plane for a system of differential equations, which describes vibrations in a string from viscoelastic material, which corresponds to Maxwell material. At the bottom of the boundary, the Cauchy conditions are specified, and one of them has a discontinuity of the first kind at one point. A smooth boundary condition is set at the side boundary. The Klein – Gordon – Fock equation is derived for one of the system’s functions. We find a particular solution in two ways. The first method builds it in an explicit analytical form (with a continuation of one function), and the second one constructs it as a solution of an integral equation using the method of characteristics (without continuation of one function). Conditions are established under which the solution has sufficient smoothness.

The use of the geometry of the Lobachevsky momentum space in the relativistic kinematics of particle collisions is demonstrated by the example of the problem of a special reference system. That system complements the geometric image of the process of elastic scattering of two particles of unequal masses. The speed of a special reference system relative to the center of mass and the angle of scattering of particles in it are determined. The conditions for the existence of such a reference system are analyzed. It is shown that in the case of a process with equal masses, the point corresponding to such a system goes into the ideal region of the extended Lobachevsky space - beyond the cone, and the lines intersecting in it become diverging lines in the sense of Lobachevsky geometry. In this case, the angle between the divergent straight lines (geodesics) of the geometric image is purely imaginary and connected to the minimum length of the segment perpendicular to the diverging straight lines (geodesics).

This paper investigates the degree of influence of the gravitational field of dark matter on the laws of motion of bodies in a medium in a restricted two-body problem, when a test body (planet, asteroid, artificial satellite of a star, in particular, the Sun, etc.) has its own rotation, i. e. own angular momentum impulse. The study was carried out within the framework of the post-Newtonian approximation of the general theory of relativity. In accordance with the latest experimental data, hypotheses about the average densities of dark matter ρD.M. and visible matter ρvis. in planetary systems are accepted. In particular, in the Solar system the following is accepted: ρD.M ≈ 2,8 · 10–19 g · cm–3, ρvis ≈ 3 · 10–20 g · cm–3 and ρΣ = ρvis + ρD.M ≈ 3,1 · 10–19 g · cm–3. In the post-Newtonian approximation of the general theory of relativity, the equation for the trajectory of a rotating test body with respect to ρΣ is derived, and working formulas are obtained that give the laws of secular changes in the direction of the vector of the proper angular momentum impulse of the test body and the modulus of this vector. It is shown that accounting ρD.M changes the magnitude of the periastron shift. For example, in the Solar System when taking into account ρvis, all the planets except Pluto have a directly shifted perihelion in the post-Newtonian approximation of the general theory of relativity. When taking into account ρΣ the planets from Mercury to Saturn included, they have a direct shift of perihelion, and Uranus, Neptune, Pluto have the reverse (against the planets in orbit). There is also a secular change in the eccentricity of the orbit. The formula is derived that can be used to calculate the secular deviation of the translational motion of a rotating body from motion in a plane. Accounting ρΣ enhances deviation. It is emphasized that all the noted effects for planetary systems in the vicinity of neutron stars, radio pulsars and other dense objects can be many orders of magnitude greater than in the solar system.

The cross section of the process e+ e– → π+ π– π0 was measured with the CMD-3 detector at the electron-positron collider VEPP-2000 in the ω meson energy region based on the data collected in 2013–2018 and corresponding to an integrated luminosity of 40 pb–1. The parameters of the ω-meson: Mω = 782.67 ± 0.01 ± 0.1 MeV, Γω = 8.56 ± 0.02 ± 0.07 MeV, σ0(ω → π+ π– π) = 1629 ± 3 ± 36 nb with the precision match or even exceeding the previous data are obtained.

A model of the cell structure of silicon photomultipliers (SiPM) is herein created in the software complex “Silvaco”. The cells are n+–p–p+-structures optically isolated from each other. The optical isolation of the cells is realized by trenches filled with metal after passivation of the walls with a SiO2 layer. Simulations are carried out for two variants of SiPM structures when the trench metal is electrically connected to the n+- (the first design) or p+ (the second design) region of the cell. The cells are irradiated by X-ray quanta with 10 keV energy up to a dose of 105 rads at the reverse bias values of Ub = –30 V (active electrical mode) and Ub = 0 V (passive electric mode). We obtain the distribution of the volume density of the accumulated charge Q in the oxide layer of the separation trench. It is established that the maximal Q value depends upon the irradiation mode. In the passive mode, the Q value is minimal and similar for both variants of the structures. In the active mode, Q increases in comparison to the passive mode by 2.5 times for SiPM with the structure of the second variant and by 5.9 times for the structure of the first variant. The obtained result can be explained by an increase of the hole charge yield under the influence of the appropriately distributed electric fields in the oxide layers of the separating trenches of the investigated SiPM’s cells.