  We use a recently-developed analytic model for the ISM structure from scales
of GMCs through star-forming cores to explore how the pre-stellar core mass
function (CMF) and, by extrapolation, stellar initial mass function (IMF)
should depend on both local and galactic properties. If the ISM is
supersonically turbulent, the statistical properties of the density field
follow from the turbulent velocity spectrum, and the excursion set formalism
can be applied to analytically calculate the mass function of collapsing cores
on the smallest scales on which they are self-gravitating (non-fragmenting).
Two parameters determine the model: the disk-scale Mach number M_h (which sets
the shape of the CMF), and the absolute velocity (to assign an absolute scale).
For 'normal' variation in disk properties and core gas temperatures in the MW
and local galaxies, there is almost no variation in the predicted high-mass
behavior of the CMF/IMF. The slope is always close to Salpeter down to <1
M_sun. We predict modest variation in the sub-solar regime, mostly from
variation in M_h, but within the observed scatter in sub-solar IMFs in local
regions. For fixed galaxy properties, there is little variation in shape or
'upper mass limit' with parent GMC mass. However, in extreme starbursts (e.g.
ULIRGs) we predict a bottom-heavy CMF. This agrees with the IMF inferred for
the centers of Virgo ellipticals, believed to form in such a nuclear starburst.
The CMF is bottom heavy despite the gas temperature being an order of magnitude
larger, because M_h is also much larger. Larger M_h values make the 'parent'
cloud mass (turbulent Jeans mass) larger, but promote fragmentation to smaller
scales; this steepens the slope of the low-mass CMF and shifts the turnover
mass. The model may predict a top-heavy CMF for the sub-pc disks around Sgr A*,
but the relevant input parameters are uncertain.
