  Super-sonic turbulence fragments molecular clouds (MC) into a very complex
density field with density contrasts of several orders of magnitude. A fraction
of the gas is locked into dense and gravitationally bound cores, which collapse
as proto-stars. This process can be studied with numerical simulations of
super-sonic self-gravitating turbulence.
  In this work, we use numerical simulations of magneto-hydrodynamic (MHD),
super-sonic, super-Alfv\'{e}nic and self-gravitating turbulence to compute the
mass distribution of collapsing proto-stellar cores, which are selected as
local density maxima. We find that the mass distribution of collapsing cores is
consistent with the stellar initial mass function (IMF), suggesting that
super-sonic turbulence may be responsible for the generation of the IMF.
  To support this conclusion we also show that the physical properties of the
numerically selected cores are in agreement with the properties of observed
NH$_3$ cores and that their magnetic field strength is consistent with Zeeman
splitting measurements.
  In turbulent MCs, star formation occurs via the gravitational collapse of
super-critical cores, formed by the turbulent flow, sub-critical cores being
irrelevant for the process of star formation.
