Let M_{m,n} be the set of all m × n real matrices. A matrix A \in M_{m,n} is said to be row-dense if there are no zeros between two nonzero entries for every row of this matrix. We find the structure of linear functions T: M_{m,n} \rightarrow M_{m,n} that preserve or strongly preserve row-dense matrices, i.e., T(A) is row-dense whenever A is row-dense or T(A) is row-dense if and only if A is row-dense, respectively. Similarly, a matrix A \in M_{m,n} is called a column-dense matrix if every column of A is a column-dense vector. At the end, the structure of linear preservers (strong linear preservers) of column-dense matrices is found., Sara M. Motlaghian, Ali Armandnejad, Frank J. Hall., and Obsahuje seznam literatury
For two vertices $u$ and $v$ of a connected graph $G$, the set $I(u, v)$ consists of all those vertices lying on a $u$–$v$ geodesic in $G$. For a set $S$ of vertices of $G$, the union of all sets $I(u,v)$ for $u, v \in S$ is denoted by $I(S)$. A set $S$ is a convex set if $I(S) = S$. The convexity number $\mathop {\mathrm con}(G)$ of $G$ is the maximum cardinality of a proper convex set of $G$. A convex set $S$ in $G$ with $|S| = \mathop {\mathrm con}(G)$ is called a maximum convex set. A subset $T$ of a maximum convex set $S$ of a connected graph $G$ is called a forcing subset for $S$ if $S$ is the unique maximum convex set containing $T$. The forcing convexity number $f(S, \mathop {\mathrm con})$ of $S$ is the minimum cardinality among the forcing subsets for $S$, and the forcing convexity number $f(G, \mathop {\mathrm con})$ of $G$ is the minimum forcing convexity number among all maximum convex sets of $G$. The forcing convexity numbers of several classes of graphs are presented, including complete bipartite graphs, trees, and cycles. For every graph $G$, $f(G, \mathop {\mathrm con}) \le \mathop {\mathrm con}(G)$. It is shown that every pair $a$, $ b$ of integers with $0 \le a \le b$ and $b \ge 3$ is realizable as the forcing convexity number and convexity number, respectively, of some connected graph. The forcing convexity number of the Cartesian product of $H \times K_2$ for a nontrivial connected graph $H$ is studied.