This paper is dedicated to studying nonstationary homoclinic solutions with the least energy for a class of fractional reaction-diffusion system \begin{document}$ \begin{eqnarray*} \label{1.1} \left\{\begin{array}{lll} \partial_t u+ (-\Delta)^s u+V(x)u+W(x)v = H_v(t, x, u, v), \\ - v + v+V(x)v+W(x)u H_u(t, |u(t, x)|+|v(t, x)|\rightarrow 0, \ \text{as}\ |t|+|x|\rightarrow \infty, \end{array} \ri...

the submodules with the property of the title ( a submodule $n$ of an $r$-module $m$ is called strongly dense in $m$, denoted by $nleq_{sd}m$, if for any index set $i$, $prod _{i}nleq_{d}prod _{i}m$) are introduced and fully investigated. it is shown that for each submodule $n$ of $m$ there exists the smallest subset $d'subseteq m$ such that $n+d'$ is a strongly dense submodule of $m$...

In many social species, group members form strong bonds. Such bonds are well-known to generate long-term fitness benefits, but they also expected influence short-term behavioural decisions. Here, we use field observations and an experimental manipulation investigate whether variation in social-bond strength (as determined from grooming interactions) influences nearest-neighbour choices while fo...

In this paper, we introduce the dual notion of strongly top modules and study some of the basic properties of this class of modules.