Let $S= \{e_1,\,e_2‎, ‎\ldots,\,e_m\}$ be an ordered subset of edges of a connected graph $G$‎. ‎The edge $S$-representation of an edge set $M\subseteq E(G)$ with respect to $S$ is the‎ ‎vector $r_e(M|S) = (d_1,\,d_2,\ldots,\,d_m)$‎, ‎where $d_i=1$ if $e_i\in M$ and $d_i=0$‎ ‎otherwise‎, ‎for each $i\in\{1,\ldots‎ , ‎k\}$‎. ‎We say $S$ is a global forcing set for maximal matchings of $G$‎ ‎if $...

let $r$ be an arbitrary ring and $t$ be a submodule of an $r$-module $m$. a submodule $n$ of $m$ is called $t$-small in $m$ provided for each submodule $x$ of $m$, $tsubseteq x+n$ implies that $tsubseteq x$. we study this mentioned notion which is a generalization of the small submodules and we obtain some related results.

the crossing number of a graph  is the minimum number of edge crossings over all possible drawings of  in a plane. the crossing number is an important measure of the non-planarity of a graph, with applications in discrete and computational geometry and vlsi circuit design. in this paper we introduce vertex centered crossing number and study the same for maximal planar graph.

background: maximal volume oxygen is an accepted standard for measuring cardiorespiratory readiness. this factor indicates the power of physiological adoption for increasing activity time and efficiency of cardiovascular system. the objective of this study was to find out an optimal correlation between maximal volume oxygen and maximal voluntary ventilation indicators and their effect on cardio...