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

Updates of the nuclear equation of state for core-collapse supernovae and neutron Stars: effects of 3-body forces, QCD, and magnetic fields