Articles

The matrix equation XAX = AXA is called the equation of the Young–Baxter type. We examine this equation for matrices of order 2 under the assumption that A is a nonsingular matrix, and we are only interested in the nonsingular solutions. Using a unified rule, one can associate with every such solution a matrix commuting with A. In other words, this matrix is an element of the centralizer MA of A. No obvious reasons exist for two distinct solutions X1 and X2 to generate the same element of MA. Nevertheless, all the solutions (and there are infinitely many of them) generate one and the same matrix in the centralizer. We give an explanation of this surprising fact.

The paper considers an approach to solving the problem of removing noise in a large data set from the sparsity class mp under condition of weak dependence. The approach is based on the method of controlling the false discovery rate (FDR). An upper asymptotic bound for the mean-square risk is obtained.

A method for approximate construction of the boundary of attainability domain is developed for a class of nonsmooth controlled dynamical systems on the plane that arise in economics. The method is based on an explicit procedure for smoothing the system and using the apparatus of the Pontryagin maximum principle. As an example, we consider the problem of constructing the boundary of the attainability domain for a controlled version of the well-known Kaldor’s model of business cycle.

This paper discusses generative intelligence systems for image synthesis, provides a detailed description of one of them (DALL·E 2), and presents known examples of such systems in use. The rationale for preparing such a review is due to the current situation in the field of generative intelligence, with many inflated expectations and even fears, and a practical lack of description and analysis of their business use cases. The article will be useful to anyone who would like to understand the real possibilities and limitations of such systems.

A diffusion logistic model of information dissemination in a social network in the form of a one-dimensional unsteady parabolic equation is considered. The problem of parametric identification is posed as an extreme problem for finding a parameter in the form of a spatially distributed function of network bandwidth. Gradient optimization methods are applied. The results obtained demonstrated uniform convergence to the exact solution in the method with an adjustable descent direction.

The problem of a non-publicized agreement between buyers of unlicensed spectrum in a spectrum auction for the coordinated formation of their price bids is considered in terms of game theory and operations research. Such auction participants are potential free riders, they hope on free access to the frequencies being sold, which causes their non-standard behavior. We propose two options for organizing an agreement for Vickrey auction in the case of complete information about the amounts of participants’ income from the use of the frequency purchased with shared expenses. It is shown that the lack of information leads to an equalizing distribution of payment between contracting buyers, and this significantly reduces their competitive advantage in the spectrum auction. Being a mechanism stimulating the true preferences uncovering, the Clark–Groves mechanism is analyzed, and its modified version is developed. Unfortunately, according to the results of the study, its application to the considered problem seems inappropriate. An alternative possibility of choosing a joint decision based on the Germeier–Vatel model is discussed.

The inverse problem of reconstructing the coefficient in the nonlinear equation of the model of the development of a homogeneous biological age-structured population is considered. The model takes into account the dependence of the parameters of the vital activity of individuals on the size of the population. The coefficients of the model are non-local and have an integral structure. Conditions are established to ensure the uniqueness of the solution of the inverse problem.

The implicatively implicit extensions of all 27 single functions of three-valued logic are characterized. It is established that among them there are both extensions that coincide with the known implicatively closed classes, and extensions that are not closed with respect to the superposition operation. In addition, it is shown that for any k > 3, any implicatively implicit extension in Pk contains the class Hk of all homogeneous functions from Pk.

The article considers an attack-defense model, in which the attack has the ability to strike several blows on defense points in order to inflict the most damage. A method for constructing a solution in mixed strategies of the zero-sum game has been developed.

In this paper, we construct a complete classification of linear codes, which are obtained from codimension 1 subcodes of Reed-Muller codes using Hadamard’s product.

The additive schemes (splitting schemes) are the basis for the construction of efficient computational algorithms for the approximate solution of initial boundary value problems for non-stationary partial differential equations. Usually, splitting schemes are developed for the additive representation of the basic operator of the problem. Also of interest are problems where the operator splits at the time derivative of the solution. We propose splitting schemes for first order evolution equations. These schemes are based on the transformation of the original equation into an equivalent system of equations.