$(0,1)$-matrices, discrepancy and preservers

Beasley, LeRoy B.. "$(0,1)$-matrices, discrepancy and preservers." Czechoslovak Mathematical Journal 69.4 (2019): 1123-1131. <http://eudml.org/doc/294867>.

@article{Beasley2019,
abstract = {Let $m$ and $n$ be positive integers, and let $R = (r_1, \ldots , r_m)$ and $S = (s_1,\ldots , s_n)$ be nonnegative integral vectors. Let $A(R,S)$ be the set of all $m \times n$$(0,1)$-matrices with row sum vector $R$ and column vector $S$. Let $R$ and $S$ be nonincreasing, and let $F(R)$ be the $m \times n$$(0,1)$-matrix, where for each $i$, the $i$th row of $F(R,S)$ consists of $r_i$ 1’s followed by $(n-r_i)$ 0’s. Let $A\in A(R,S)$. The discrepancy of A, $\{\rm disc\}(A)$, is the number of positions in which $F(R)$ has a 1 and $A$ has a 0. In this paper we investigate linear operators mapping $m\times n$ matrices over the binary Boolean semiring to itself that preserve sets related to the discrepancy. In particular, we show that bijective linear preservers of Ferrers matrices are either the identity mapping or, when $m=n$, the transpose mapping.},
author = {Beasley, LeRoy B.},
journal = {Czechoslovak Mathematical Journal},
keywords = {Ferrers matrix; row-dense matrix; discrepancy; linear preserver; strong linear preserver},
language = {eng},
number = {4},
pages = {1123-1131},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {$(0,1)$-matrices, discrepancy and preservers},
url = {http://eudml.org/doc/294867},
volume = {69},
year = {2019},
}

TY - JOUR
AU - Beasley, LeRoy B.
TI - $(0,1)$-matrices, discrepancy and preservers
JO - Czechoslovak Mathematical Journal
PY - 2019
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 69
IS - 4
SP - 1123
EP - 1131
AB - Let $m$ and $n$ be positive integers, and let $R = (r_1, \ldots , r_m)$ and $S = (s_1,\ldots , s_n)$ be nonnegative integral vectors. Let $A(R,S)$ be the set of all $m \times n$$(0,1)$-matrices with row sum vector $R$ and column vector $S$. Let $R$ and $S$ be nonincreasing, and let $F(R)$ be the $m \times n$$(0,1)$-matrix, where for each $i$, the $i$th row of $F(R,S)$ consists of $r_i$ 1’s followed by $(n-r_i)$ 0’s. Let $A\in A(R,S)$. The discrepancy of A, ${\rm disc}(A)$, is the number of positions in which $F(R)$ has a 1 and $A$ has a 0. In this paper we investigate linear operators mapping $m\times n$ matrices over the binary Boolean semiring to itself that preserve sets related to the discrepancy. In particular, we show that bijective linear preservers of Ferrers matrices are either the identity mapping or, when $m=n$, the transpose mapping.
LA - eng
KW - Ferrers matrix; row-dense matrix; discrepancy; linear preserver; strong linear preserver
UR - http://eudml.org/doc/294867
ER -