  In this paper, by using weakly \widetilde{\mathcal{D}}^+_J (resp.
\mathcal{D}^+_J )-closed technique firstly introduced by Tan, Wang, Zhou and
Zhu, we will give a characterization of tamed and weakened tamed
four-manifolds, and an almost Kaehler version of Nakai-Moishezon criterion for
almost complex four-manifolds.
