In the present paper, geometry of the Boolean space B<sup>n</sup> in terms of Hausdorff distances between subsets and subset sums is investigated. The main results are the algebraic and analytical expressi...In the present paper, geometry of the Boolean space B<sup>n</sup> in terms of Hausdorff distances between subsets and subset sums is investigated. The main results are the algebraic and analytical expressions for representing of classical figures in B<sup>n</sup> and the functions of distances between them. In particular, equations in sets are considered and their interpretations in combinatory terms are given.展开更多
In many problems of combinatory analysis, operations of addition of sets are used (sum, direct sum, direct product etc.). In the present paper, as well as in the preceding one [1], some properties of addition operatio...In many problems of combinatory analysis, operations of addition of sets are used (sum, direct sum, direct product etc.). In the present paper, as well as in the preceding one [1], some properties of addition operation of sets (namely, Minkowski addition) in Boolean space B<sup>n</sup> are presented. Also, sums and multisums of various “classical figures” as: sphere, layer, interval etc. are considered. The obtained results make possible to describe multisums by such characteristics of summands as: the sphere radius, weight of layer, dimension of interval etc. using the methods presented in [2], as well as possible solutions of the equation X+Y=A, where , are considered. In spite of simplicity of the statement of the problem, complexity of its solutions is obvious at once, when the connection of solutions with constructions of equidistant codes or existence the Hadamard matrices is apparent. The present paper submits certain results (statements) which are to be the ground for next investigations dealing with Minkowski summation operations of sets in Boolean space.展开更多
文摘In the present paper, geometry of the Boolean space B<sup>n</sup> in terms of Hausdorff distances between subsets and subset sums is investigated. The main results are the algebraic and analytical expressions for representing of classical figures in B<sup>n</sup> and the functions of distances between them. In particular, equations in sets are considered and their interpretations in combinatory terms are given.
文摘In many problems of combinatory analysis, operations of addition of sets are used (sum, direct sum, direct product etc.). In the present paper, as well as in the preceding one [1], some properties of addition operation of sets (namely, Minkowski addition) in Boolean space B<sup>n</sup> are presented. Also, sums and multisums of various “classical figures” as: sphere, layer, interval etc. are considered. The obtained results make possible to describe multisums by such characteristics of summands as: the sphere radius, weight of layer, dimension of interval etc. using the methods presented in [2], as well as possible solutions of the equation X+Y=A, where , are considered. In spite of simplicity of the statement of the problem, complexity of its solutions is obvious at once, when the connection of solutions with constructions of equidistant codes or existence the Hadamard matrices is apparent. The present paper submits certain results (statements) which are to be the ground for next investigations dealing with Minkowski summation operations of sets in Boolean space.
