Articles

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.

For a fixed period of time, a mathematical model for the treatment of psoriasis is considered, consisting of three differential equations. These equations describe the relationships between the major cell populations that are responsible for the occurrence, progression, and treatment of this disease. The model also contains a bounded control function reflecting the effects of a drug aimed at inhibiting interactions between specific cell populations. The task is to reduce the population of cells directly affecting the disease at the end of a given period of time. The analysis of such an optimal control problem is carried out using the Pontryagin maximum principle. The main attention is paid to clarifying the bang-bang property of the corresponding optimal control, as well as the possibility of its having singular regimens of various orders depending on the relationships between the parameters of the original model.

We study conditions under which components of the distribution of the difference of two independent and identically distributed random variables are determined uniquely up to a shift and reflection. This uniqueness is essential to some characterization problems. An algorithm is presented for estimation of the components when data are given in a symmetrized form.

The paper is devoted to the development of a new method for the approximate construction of the reachability set for a nonlinear control system with discrete time. Pointwise restrictions are imposed on the control parameters. To solve this problem, a technique previously developed and applied for the case of continuous time and differential equations is used. The estimate of the reachability set can be obtained as the level set of a special piecewise affine value function constructed on a grid of simplices in the state space. The paper presents formulas for calculating the coefficients of such a function, which make it possible to analyze the difference between the case with discrete time and the case with continuous time. An example of calculation of piecewise affine value functions and corresponding internal and external estimates of the reachability set is considered.