In recent years, various methods and directions for solving a system of Boolean algebraic equations have been invented, and now they are being very actively investigated. One of these directions is the method of transforming a system of Boolean algebraic equations, given over a ring of Boolean polynomials, into systems of equations over a field of real numbers, and various optimization methods can be applied to these systems. In this paper, we propose a new transformation method for Solving Systems of Boolean Algebraic Equations (SBAE). The essence of the proposed method is that firstly, SBAE written with logical operations are transformed (approximated) in a system of harmonic-polynomial equations in the unit n-dimensional cube Kn  with the usual operations of addition and multiplication of numbers. Secondly, a transformed (approximated) system in Kn  is solved by using the optimization method. We substantiated the correctness and the right to exist of the proposed method with reliable evidence. Based on this work, plans for further research to improve the proposed method are outlined.

In connection with applications, the solution of a system of logical equations plays an important role in computational mathematics and in many other areas. As a result, many new directions and algorithms for solving systems of logical equations are being developed. One of these directions is transformation into the real continuous domain. The real continuous domain is a richer domain to work with because it features many algorithms, which are well designed. In this study, firstly, we transformed any system of logical equations in the unit -dimensional cube into a system of polylinear–polynomial equations in a mathematically constructive way. Secondly, we proved that if we slightly modify the system of logical equations, namely, add no more than one special equation to the system, then the resulting system of logical equations and the corresponding system of polylinear–polynomial equations in is equivalent. The paper proposes an algorithm and proves its correctness. Based on these results, further research plans are developed to adapt the proposed method.

In this paper, two not-difficult inequalities are invented and proved in detail, which adequately describe the behavior of discrete logical functions xor(x1, x2,…, xn) and and(x1, x2,…, xn). Based on these proven inequalities, infinitely differentiable extensions of the logical functions xor(x1, x2,…, xn) and and(x1, x2,…, xn) were defined for the entire ℝn. These suitable extensions were applied to systems of logical equations. Specifically, the system of m logical equations in a constructive way without adding any equations (not field equations and no others) is transformed in ℝn first into an equivalent system of m smooth rational equations (SmSRE) so that the solution of SmSRE can be reduced to the problem minimization of the objective function, and any numerical optimization methods can be applied since the objective function will be infinitely differentiable. Again, we transformed SmSRE into an equivalent system of m polynomial equations (SmPE). This means that any symbolic methods for solving polynomial systems can be used to solve and analyze an equivalent SmPE. The equivalence of these systems has been proved in detail. Based on these proofs and results, in the next paper, we plan to study the practical applicability of numerical optimization methods for SmSRE and symbolic methods for SmPE.

The problem of solving a system of linear ordinary differential equations with constant coefficients is one of the most important problems in both the theory of ordinary differential equations and linear algebra. Therefore, on the one hand, new methods and algorithms are being developed for such systems, and on the other hand, existing methods and algorithms for solving such systems are being improved. One of the most well-known methods for solving a system of linear ordinary differential equations with constant coefficients is the method of reducing a system of linear equations to a single higher-order equation, which makes it possible to find solutions to the original system in the form of linear combinations of derivatives of only one unknown function.In this paper, we consider a refinement of the method for reducing a system of linear ordinary differential equations with constant coefficients to a single higher-order equation, which makes it possible to find a general solution to the original system; namely, we study the expressibility of all functions of the system of linear homogeneous differential equations with constant coefficients x^' (t)= A⋅x(t) in the form of linear combinations of derivatives of only one unknown function x_k (t), which is part of this system. For any matrix A, all of whose eigenvalues are not multiples, a new simple criterion for expressibility in terms of matrix ranks is formulated, and its correctness is proved. The result obtained can also be applied in the study of solutions of the system x^' (t)= A⋅x(t) for periodicity and in the study of linear systems for complete observability.

Исследуется задача выразимости всех функций x1(t), x2(t), . . . , xn(t), входящих в заданную однородную систему линейных дифференциальных уравнений с постоянными коэффициентами x′(t) = A·x(t), в виде линейных комбинаций производных только одной неизвестной функции xk(t), входящей в эту систему. Найден простой критерий выразимости всех функций системы x′(t) = A·x(t) в виде линейных комбинаций производных xk(t) и доказана его корректность. На основе доказанного критерия разработан соответствующий алгоритм и обоснована его корректность. In this paper, we study the problem of expressibility of all functions x1(t), x2(t), . . . , xn(t) included in a given homogeneous system of linear differential equations with constant coefficients x′(t) = A·x(t), in the form of linear combinations of derivatives of only one unknown function xk(t) included in this system. A simple criterion is found for the expressibility of all functions of the system x′(t) = A·x(t), in the form of linear combinations of derivatives xk(t), and its correctness is proved. Based on the proven criterion, an appropriate algorithm was developed and its correctness was substantiated.