The theorems of Koenig and Birkhoff and their connection with the minimization of the duration time of the measurements of automatic telecommunication channels

Szczepan Perz ; Leszek Zaremba

Mathematica Applicanda, Tome 10 (1982) no. 19, pp. 43-50.

Résumé

A problem (P) of minimization of the duration time of the measurements of automatic telecommunication channels is considered. P is a discrete optimization problem solved by graph theory methods. It is defined by (i)-(v), where: (i) for each i, 1≤i≤p, and j, 1≤1≤p, there are given k ij channels to be measured between node ”i” and node ”j”; (ii) measurement of one channel lasts one unit; (iii) there are exactly two devices, say A, B, in each node (the case where there is an arbitrary number of devices A, B in each node may be easily reduced to this case); (iv) the channel between node ”i” and node ”j” may be measured only by use of device A being present in node ”i” and device B in node ”j”; (v) in each time both devices A or B may measure only one channel. To solve P, some knowledge of hypergraphs as well as functional analysis (the Krein-Milman theorem) and linear algebra (the Koenig theorem) is necessary. The Koenig theorem is proved in a simple manner similarly as the dual Koenig theorem (which is a new result). As corollaries the Birkhoff theorem about bistochastic matrices and the dual Birkhoff theorem are deduced.

Détail

DOI : 10.14708/ma.v10i19.1532

Classification : 94A40,93C55,05C65,15A24,15B36,15B51
Mots-clés : Channel models (including quantum), Discrete-time control systems, Hypergraphs, Matrix equations and identities, Matrices of integers, Stochastic matrices