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 $K_n$ with the usual operations of addition and multiplication of numbers. Secondly, a transformed (approximated) system in $K_n$ 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 $n$-dimensional cube $K_n$ 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 $K_{n+1}$ 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 $xor\left(x_1, x_2,\dots , x_n\right)$ and $and\left(x_1, x_2,\dots , x_n\right)$. Based on these proven inequalities, infinitely differentiable extensions of the logical functions $xor\left(x_1, x_2,\dots , x_n\right)$ and $and\left(x_1, x_2,\dots , x_n\right)$ were defined for the entire ${ℝ}^n$. 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 ${ℝ}^n$ first into an equivalent system of $m$ smooth rational equations $\left(SmSRE\right)$ 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 $\left(SmPE\right)$. 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 $SmPE$.

Исследована проблема существования и единственности полилинейных продолжений некоторых дискретных функций. Доказано, что для любой булевой функции существует соответствующее полилинейное продолжение и оно единственно. Предложен алгоритм нахождения полилинейного продолжения булевой функции и доказана его корректность. На основе предложенного алгоритма найдены явные формы полилинейных продолжений сначала для булевой функции, а затем для произвольной функции, определенной на множестве вершин n-мерного единичного куба, произвольного куба и параллелепипеда, и в каждом конкретном случае доказана единственность соответствующего полилинейного продолжения. In this paper, we study the existence and uniqueness of polylinear continuations of some discrete functions. It is proved that for any Boolean function, there exists a corresponding polylinear continuation and it is unique. An algorithm for finding a polylinear continuation of a Boolean function is proposed and its correctness is proved. Based on the result of the proposed algorithm, explicit forms of polylinear continuations are found first for a Boolean function and then for an arbitrary function defined only at the vertices of an n-dimensional unit cube, an arbitrary cube, and a parallelepiped, and in each particular case the uniqueness of the corresponding polylinear continuations is proved.

This article explores an elliptic system of n equations where the main part is the Bitsadze operator (the square of the Cauchy–Riemann operator) and the lower term is the product of a given matrix function by the conjugate of the desired vector function. The system was analyzed in the Banach space of vector functions that are bounded and uniformly H¨older continuous in the entire complex plane. It was revealed that the problem of solving the system in the specified space may not be Noetherian. An example of a homogeneous system with an infinite number of linearly independent solutions was given. As is known, for many classes of elliptic systems, the Noetherianity of boundary value problems in a compact domain is equivalent to the presence of a priori estimates in the corresponding spaces. In this regard, it is important to study the issues related to the establishment of a priori estimates for the system under consideration in the above space. In the case of coefficients weakly oscillating at infinity, necessary and sufficient conditions for the validity of the a priori estimate were found. These conditions were written out in the language of the spectrum of limit matrices formed by the partial limits of the coefficient matrix at infinity. Specific examples were provided to illustrate how the limit matrices are constructed and what the above conditions look like.