  We summarize several new developments in the nuclear equation of state for
supernova simulations and neutron stars. We discuss an updated and improved
Notre-Dame-Livermore Equation of State (NDL EoS) for use in supernovae
simulations. This Eos contains many updates. Among them are the effects of 3-
body nuclear forces at high densities and the possible transition to a QCD
chiral and/or super-conducting color phase at densities. We also consider the
neutron star equation of state and neutrino transport in the presence of strong
magnetic fields. We study a new quantum hadrodynamic (QHD) equation of state
for neutron stars (with and without hyperons) in the presence of strong
magnetic fields. The parameters are constrained by deduced masses and radii.
The calculated adiabatic index for these magnetized neutron stars exhibit rapid
changes with density. This may provide a mechanism for star-quakes and flares
in magnetars. We also investigate the strong magnetic field effects on the
moments of inertia and spin down of neutron stars. The change of the moment of
inertia associated with emitted magnetic flares is shown to match well with
observed glitches in some magnetars. We also discuss a perturbative calculation
of neutrino scattering and absorption in hot and dense hyperonic neutron-star
matter in the presence of a strong magnetic field. The absorption
cross-sections show a remarkable angular dependence in that the neutrino
absorption strength is reduced in a direction parallel to the magnetic field
and enhanced in the opposite direction. The pulsar kick velocities associated
with this asymmetry comparable to observed pulsar velocities and may affect the
early spin down rate of proto-neutron star magnetars with a toroidal field
configuration.
