This article is devoted to the problem of construction and research of generalized interpolation formulas of Hermite – Birkhoff type for operators given on the function spaces. The construction of operator interpolation formulas is based on interpolation polynomials for scalar functions with respect to the arbitrary Chebyshev system of functions. The given formulas contain the Stieltjes integrals and the Gateaux differentials of an interpolated operator and are invariant for the special-class operator polynomials. An explicit representation of the error of the generalized Hermite – Birkhoff operator interpolation is obtained. On the basis of the generalized interpolation Hermite – Birkhoff formulas the interpolation polynomials for ordinary differential operators of arbitrary order given in the space of continuously differentiable functions are constructed. Some special cases of the Hermite – Birkhoff formulas of this type for various Chebyshev systems of scalar functions are also considered. The obtained results can be used in theoretical research as a basis for constructing approximate methods of solving some nonlinear operator equations that occur in nonlinear dynamics, mathematical physics.

Homogeneous Riemann boundary value problem with meromorphic coefficients for infinitely connected domains is considered. In the closed form the problem is solved in the class of piece-wise analytic functions, possessing meromorphic continuation to the whole complex plane. Special attention is paid to the existence of doubly periodic solutions to the problem with elliptic coefficients. The example of the problem having a unique solution up to an arbitrary constant multiplier is presented, as well as of the problem with a solution depending on a number of arbitrary parameters. The obtained results can be used for solving of an inhomogeneous Riemann boundary value problem with meromorphic coefficients in an infinitely connected domain in the general statement.

The study of the properties of solution mappings in parametrical optimization problems represents an urgent problem. Particularly, considerable efforts are directed to finding the conditions of different types of generalized Lipschitzian continuity of solution mappings, namely their calmness and pseudo-Lipschitzian continuity (also referred to as the Aubin property) [1]. A new interesting approach to investigating the calmness of solution mappings has recently been proposed by Canovas et al. [2] for parametrical linear programming problems and applied to a much wider range of problems by Klatte and Kummer [3]. In this approach, the calmness of solution mappings is related to the calmness of an associated system representing a constraint on the level set of the objective function on the domain of the problem. In our note, we propose to expand the use of the approach [3] for investigating the pseudo-Lipschitzian continuity of solution mappings. Several sufficient conditions for the pseudo-Lipschitzian continuity of solution mappings, as well as the generalization of the Hoffman lemma are presented.

In the article the authors continue to study the polyadic operation η_{s, σ, k} that was earlier defined at the Cartesian power A^k of the nth groupoid < A, η > by the substitution of σ ∈ Sk and the nth operation η. The special case of the polyadic operation ηs, σ, k is the lth operation [ ]_{l, σ, k} that is defined by one of the authors for any integer k ≥ 2, l ≥ 2 and for any substitution of the σ set {1, …, k} at the Cartesian power A^k of the semigroup A. In turn, the special case of the lth operation [ ]_{l, σ, k} consists of two polyadic operations by E. Post, one of which he defined at the Cartesian power of the symmetric group and the second – at the Cartesian power of the general linear group over the field of complex numbers. The properties of the operations η_{s, σ, k} are studied in the article. In particular, a new proof of the associativity of the polyadic operation η_{s, σ, k} was obtained. 

The problem of statistical assignment of realizations of non-stationary time series to the fixed trend models is investigated. The decision rule in a space of trend coefficients determined on the same orthogonal basis is proposed and its efficiency is analytically studied. As an example the case of two alternative trends is studied.

A quadratic (0,1)-matrix of degree n with just n units among its elements will be called a Cameron matrix. The orbits of the natural action of the group S_n× S_n (the square of the symmetric group of degree n) on the set of Cameron matrices of degree n (an independent action on the rows and the columns of matrices) are considered. It is proved that for fixed d < n, the number of such orbits for matrices of rank n – d is constant for n ≥ 3d and grows with the growth of n if n < 3d. For each orbit, its representative in a quasi-Jordan form is indicated. 

The generalized Schrődinger equation for the Cox scalar particle is studied in the presence of the electric field on the background of the Lobachevsky space. Separation of variables is performed. The equation describing the motion along the z axis turns out to be much more complicated than that for the Cox particle in the Minkowski space. It is reduced to the second-order differential equation with two regular singularities and one irregular singularity of rank 2 that is identified as the confluent Heun equation. The nearby singular points of the derived equation correspond to the physical domains z ± ∞. The solutions of the equation are constructed with the help of the power series. The series convergence is examined by the Poincare – Perrone method. These series converge in the whole physical domain z ∈ −∞,+∞ ( ).

The features of the induced changes in the optical density spectra of multilayer Ag-Na₃ AlF₆ nanostructures under a femtosecond laser pulses exitation in the band of surface plasmonic resonance of absorption (SPRA) were studied. The dependence of the amplitude of the induced changes on the thickness of the dielectric Na₃ AlF₆ films separating the monolayeres of silver nanoparticles was registered. A significant increase of the optical response amplitude (up to 80 %) was found for the nanostructures with the quarter-wavelength Na₃ AlF₆ interlayers. For nanostructures with different-thickness dielectric Na₃ AlF₆ interlayers the characteristic relaxation times of induced changes at an excitation energy of 5–10 μJ do not practically vary, are equal to ~2 ps and coincide with the kinetic response time parameters of the used silver nanoparticle monolayers. 

Laser radiation is extensively used for optical diagnostics of various scattering media. In most cases, laser beams having a Gaussian profile are applied for this task. At the same time, other-type light beams have some features that can be used to obtain additional information about investigated objects. In this context, a relevant task is to reveal the penetrability of different-type light beams in a scattering medium with their subsequent application for the nondestructive testing of various objects, including biological tissues. In this article, a comparative analysis is carried out for four different configurations of laser light beams (Gaussian, Laguerre-Gaussian, and zero- and first-order Bessel light beams) in relation to the power stored by them after passing through a scattering medium layer. To form the light beams we used helium-neon laser emitting at a wavelength of 0.633 micrometers, and a modular optical system. This system makes it possible to change the light beam profile by the inclusion / exclusion of the corresponding modules from the light path. As the scattering medium we used plane-parallel layers of semi-transparent silastic with the thickness ranging from 0.17 to 6.61 mm. It is investigated experimentally how the power of the light beam passing through the scattering medium layer depends on the layer thickness. According to the obtained results, the approximating curves are plotted in the form I = exp (–Dx) where D is the attenuation coefficient, I is the total power of the beam, x is the layer thickness. The values of the coefficient D for different-type beams are calculated. The D values for different-type beams scarcely differ (within the standard error) from each other. It means that the beam type in the optical system configuration has almost no effect on the penetration properties of the light beam, and on the total energy of the light passing through the scattering medium layer.

A method is offered for detecting and correcting the errors of information transmitted over communication channels based on the use of well-known in the theory of communication "parity coordinate" of a binary matrix which is the content of the message being transmitted. The parity is commonly used in two coordinates: rows and columns of the matrix. In contrast to the well-established practice of considering only two coordinates, it is proposed to do the error correction on the basis of an extended set of coordinates of matrix elements, among which are rows, columns, as well as main and auxiliary diagonals. Troubleshooting is made by generating a set of possible addresses of erroneous elements and a subsequent analysis of this set to avoid false addresses (non-existent) errors. This takes into account the four coordinates of binary matrix elements, which allows a low cost and fast detection of single, double, and group errors. The effectiveness of the method becomes higher with increase in the ratio "number of columns / rows number" of a binary matrix.

The problem of optimizing the output of multi-product batch and the intensities of its items manufacture in the production line over a number of time intervals is considered. The line has a linearly ordered multiple positions without buffers. Workpieces of the input sequence composed of cyclically repeated identical subsequences (batches) are processed consecutively one by one in each working position in the order of their location in the line. Only a single workpiece is disposed in each position at each time point. The operation of the line consists of takts of simultaneous processing of all workpieces located in respective positions by the sets of tools corresponding to workpieces and positions. The composition of a batch does not vary from interval to interval. The ranges of possible demand quantities for each product and the probability distribution of the demand in these ranges are assumed known for each time interval. The sum of manufacturing cost, costs of storage and/or penalties for unmet demand on products is used as objective function. Manufacturing cost depends on processing intensities to be defined and increases with an increase in the number of batches produced in the current interval. Storage cost of unclaimed product units as well as penalty for product units not supplied to the customer do not decrease with the increase of number of such units. A two-level decomposition method for solving the problem based on the ideas of multi-step optimization is proposed.

Many problems in the theory and practice are reduced to solving the first-kind integral equations with a “weak” kernel, i. e. the kernel goes to the infinity of integrable order when arguments are matching. The success of investigation of such problems often depends on the solution of the explicit equation corresponding to the problem. In some cases, it is possible to get such a solution. In our case, we consider the first-kind equation with a kernel, which represents the square root of a fractional-linear function, on a symmetric interval. Given the equation symmetry, this equation can be reduced to an equivalent two-equation system, each of which is reduced to the Abel equation solution and its generalizations. The solution is written in explicit form. The examples are presented.