  The flavor conversion of a neutrino usually occurs at densities $\lesssim
G_F^{-1} \omega$, whether in the ordinary matter or the neutrino medium, and on
time/distance scales of order $\omega^{-1}$, where $G_F$ is the Fermi weak
coupling constant and $\omega$ is the vacuum oscillation frequency of the
neutrino. In 2005 Sawyer and more recently both he and other groups have shown
that neutrino flavor conversions can occur on scales much shorter than
$\omega^{-1}$ in a very dense, anisotropic neutrino gas such as that in a
core-collapse supernova or a binary neutron star merger. It has also been
suggested that the fast neutrino flavor conversion require a crossing in the
electron lepton number (ELN) angular distribution or spectrum. We study a few
simple analytical models with which we elucidate the roles of the dense matter
and the ELN spectrum crossing in neutrino fast flavor conversions. We show that
a large matter density can induce fast neutrino flavor conversions in certain
astrophysical scenarios such as at the early epoch of a core-collapse
supernova.
