On the Initial Conditions for Star Formation and the IMF
  Density probability distribution functions (PDFs) for turbulent
self-gravitating clouds should be convolutions of the local log-normal PDF,
which depends on the local average density rho-ave and Mach number M, and the
probability distribution functions for rho-ave and M, which depend on the
overall cloud structure. When self-gravity drives a cloud to increased central
density, the total PDF develops an extended tail. If there is a critical
density or column density for star formation, then the fraction of the local
mass exceeding this threshold becomes higher near the cloud center. These
elements of cloud structure should be in place before significant star
formation begins. Then the efficiency is high so that bound clusters form
rapidly, and the stellar initial mass function (IMF) has an imprint in the gas
before destructive radiation from young stars can erase it. The IMF could arise
from a power-law distribution of mass for cloud structure. These structures
should form stars down to the thermal Jeans mass MJ at each density in excess
of a threshold. The high-density tail of the PDF, combined with additional
fragmentation in each star-forming core, extends the IMF into the Brown Dwarf
regime. The core fragmentation process is distinct from the cloud structuring
process and introduces an independen core fragmentation mass function (CFMF).
The CFMF would show up primarily below the IMF peak.
