%0 Journal Article
%T On a Property of a Three-Dimensional Matrix
%A David Blokh
%J Journal of Discrete Mathematics
%D 2013
%I Hindawi Publishing Corporation
%R 10.1155/2013/797249
%X Let be the symmetrical group acting on the set and . Consider the set The main result of this paper is the following theorem. If the number of set entries is more than , then there exist entries such that , , and . The application of this theorem to the three-dimensional assignment problem is considered. 1. Introduction Let be the set of -matrices over the field of real numbers. Three-dimensional matrix not only is an interesting mathematical object [1¨C3], but also has applications in many fields, such as theoretical physics [4] and operational research [5, 6]. Let be the symmetrical group acting on the set , , and The main result of this paper is the following theorem. Theorem 1. If , and the number of set entries is more than , then there exist entries , and such that the entry . We give another formulation of Theorem 1. Consider the set Theorem 2. If the number of set entries is more than , then there exist entries such that , , and . 2. Proof of Theorem 1 We prove the theorem by contradiction. The set of matrix entries with one index fixed and the two others having values from 1 to will be called a layer. We denote a layer by , where indicates the location of a fixed index and indicates its value. For example, . Furthermore, entries from will be called basic, entries from will be called nonbasic; will be termed a trajectory; the layer containing a basic trajectory entry will be termed a basic layer and the layer containing a nonbasic trajectory entrywill be termed a nonbasic layer. If is a nonbasic layer, then one layer in the pair , is basic. Suppose to the contrary that layers , , and are nonbasic. Let , , and be the trajectory entries of , , layers. Replace , , and with , , and . The nonbasic trajectory entries , , and are replaced with entries , , and , among which there is a basic one. Similar assertions may be proved for the layer and the following layer pairs: ; ; ; ; ; and . Three nonbasic layers may not be consecutive. Suppose that layers , , and are nonbasic. One of the layers or is basic. In these layers, the basic entries may be , , , , , , and for the layer , and , , , , , , for the layer . However, this contradicts the assumption that the layers , , and are nonbasic. The fact that the two first and the two last layers may not be nonbasic is proved in a similar way. All assertions given below represent conditions that prevent replacing nonbasic trajectory entries with entries that include a basic one. Consider the sequence of layers , and . Given below are possible arrangements of layers in this sequence. A basic layer is denoted by 1;
%U http://www.hindawi.com/journals/jdm/2013/797249/