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, and Leszek Zaremba. "The theorems of Koenig and Birkhoff and their connection with the minimization of the duration time of the measurements of automatic telecommunication channels." Mathematica Applicanda 10.19 (1982): null. <http://eudml.org/doc/292798>.

@article{SzczepanPerz1982,
abstract = {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.},
author = {Szczepan Perz, Leszek Zaremba},
journal = {Mathematica Applicanda},
keywords = {Channel models (including quantum), Discrete-time control systems, Hypergraphs, Matrix equations and identities, Matrices of integers, Stochastic matrices},
language = {eng},
number = {19},
pages = {null},
title = {The theorems of Koenig and Birkhoff and their connection with the minimization of the duration time of the measurements of automatic telecommunication channels},
url = {http://eudml.org/doc/292798},
volume = {10},
year = {1982},
}

TY - JOUR
AU - Szczepan Perz
AU - Leszek Zaremba
TI - The theorems of Koenig and Birkhoff and their connection with the minimization of the duration time of the measurements of automatic telecommunication channels
JO - Mathematica Applicanda
PY - 1982
VL - 10
IS - 19
SP - null
AB - 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.
LA - eng
KW - Channel models (including quantum), Discrete-time control systems, Hypergraphs, Matrix equations and identities, Matrices of integers, Stochastic matrices
UR - http://eudml.org/doc/292798
ER -