Decomposition of number arrangements in the cube

Abstract

A subset $M\subset {\text{R}}^3$ is called a \emph{basic subset}, if for any funciton $f:M\to \text{R}$ there exist such functions $f_1;f_2;f_3:\text{R}\to \text{R}$ that $f(x_1,x_2,x_3)=f_1(x_1)+f_2(x_2)+f_3(x_3)$ for each point $(x_1,x_2,x_3)\in M$. In this article we prove a criterion for a basic subset for some specific subsets in terms of some graph properties. We also introduce several constructions for mimimal non-basic subsets. The article is written in Russian.

arXiv commentin Russian