A linear control system with a periodic matrix of coefficients and program control is considered. The matrix under control is constant, rectangular (the number of columns does not exceed the number of rows) and its rank is not maximum, i. e. less than the number of columns. It is assumed that the control is periodic, and the module of its frequencies, i. e. the smallest additive group of real numbers, including all Fourier exponents of this control, is contained in the frequency module of the coefficient matrix. The following task is posed: to construct such a control from an admissible set that switches the system to asynchronous mode, i. e. the system must have periodic solutions such that the intersection of the frequency moduli of the solution and the coefficient matrix is trivial. The problem posed is called the problem of synthesis of asynchronous mode. The solution to the formulated problem significantly depends on the structure of the average value of the coefficient matrix. In particular, its solution is known for systems with zero average. In addition, solvability conditions were obtained in the case when the matrix under control has zero rows, the averaging of the coefficient matrix is reduced to the form with upper left diagonal block and with zero remaining blocks. In this paper we consider a more general case with a nontrivial left lower block. Assuming an incomplete column rank of the matrix function composed from the rows of oscillation path of the coefficient matrix, we construct the control explicitly. This control switches the system to asynchronous mode.

In the article, the theory of the Fourier series on the orthogonal multidimensional-matrix (mdm) polynomials is developed. The known results from the theory of the orthogonal polynomials of the vector variable and the Fourier series are given and the new results are presented. In particular, the known results of the Fourier series theory are extended to the case of the mdm functions, what allows us to solve more general approximation problems. The general case of the approximation of the mdm function of the vector argument by the Fourier series on the orthogonal mdm polynomials is realized programmatically as the program function and its efficiency is confirmed. The analytical expressions for the coefficients of the second degree orthogonal polynomials and Fourier series for possible analytical studies are obtained.

In the paper a (1 + 1)-dimension equation of motion for the artificial axon is considered. The artificial axon is a dynamical structure like a neuron. They are widely used in biophysics, for example, in studying the physiological processes. A topological non-trivial solution of one-kink type for this equation is constructed in an analytical form. The modified direct Hirota method for solving the nonlinear partial derivatives equations is applied. The special cases are considered for different voltages on the contacts of axon.

The quantum-mechanical problem of a harmonic oscillator on a hyperbola as a one-dimensional space of constant negative curvature is considered in this article. A generalization to the singular oscillator model in the context of one-dimensional Cayley – Klein geometries is given by the factorization method. The energy spectrum and wave functions of stationary states are found having the curvature of space as a parameter. For the energy levels of the singular oscillator, the effect of non-zero curvature is clearly manifested through a positive or negative term, depending on the sign of the curvature, which is quadratic in the level number. The results obtained are consistent with those previously published. The dynamical symmetry of the problem is shown explicitly as a quadratic Hahn algebra QH(3) or its isomorphic Higgs algebra.

Within the framework of the generalization of Freund – Nambu scalar-tensor theory of gravity, a massless scalar field is considered, the source of which is the trace of its own energy-momentum tensor. For the cosmological problem, numerical solutions of field equations were obtained, with the help of which the dependencies of the Hubble parameter and the photometric distance to the observed sources on red-shift were constructed. To the consistency of the models with observational data, contours of confidence intervals for model parameters were constructed.

The conversion decay w® p⁰e⁺e^- with the CMD-3 detector on the VEPP-2000 electron-positron collider VEPP-2000 at the Budker Institute of Nuclear Physics of the Siberian Branch of the Russian Academy of Sciences was investigated. The data from the first scan were used in the work, which corresponded to an integral brightness of around 10 pb^–1 in the energy range of 660 to 840 MeV in the center of mass system. The 1113 ± 37 signal events were detected. The product of the relative decay probability and ω-meson width G(w® e⁺e^- ) × Br(w® p⁰e⁺e^- ) = (4, 20 ± 0,12 ± 0, 25) ×10^-7 MeV, as well branching ratio Br(w® p⁰e⁺e^- ) = (6, 78 ± 0,19 ± 0, 40) ×10^-4 are measured with better precision than the world average.

This paper presents methods for the formation and properties of light fields that controllably vary in time and space, obtained as a result of the interference of three or four coherent light beams using refractive optical elements. The formation of three-beam and four-beam interference fields is carried out using trihedral and tetrahedral glass pyramids respectively. The possibility of the interference field displacement in the transverse plane is ensured by devices for controlled phase changes of at least two of the interfering beams. The directions of propagation of these beams do not lie in the same plane with the optical axis of the original beam incident on the pyramid. In threeand four-beam dynamic interference fields the peak intensity values are higher than in the original laser beams and two-beam interference fields, so it is advisable to use them for processing flat objects with laser radiation, moving the interference maxima along the surface of the object. With a pairwise azimuthal displacement of the propagation directions of four interfering beams around the longitudinal axis, a dynamic interference field is formed, the periodically structured maxima of which cyclically smoothly change their shape from cells to band and back. At different speeds of pairs of directions the interference structure of the maxima rotates around the longitudinal axis. Therefore, this field can be used for therapeutic effects on biological tissues and for mixing microparticles in suspensions and emulsions. Since the local maxima of the intensity of all these interference fields have dimensions of the order of several micrometers while exceeding in value the maximum intensity of the initial light beam, these fields in the cross section are gradient and therefore can be used not only for laser exposure, but also for moving ensembles of microparticles including for sorting and changing concentration.