  The characteristic mass M_c in the stellar initial mass function (IMF) is
about constant for most star-forming regions. Numerical simulations
consistently show a proportionality between M_c and the thermal Jeans mass M_J
at the time of cloud fragmentation, but no models have explained how it can be
the same in diverse conditions. Here we show that M_J depends weakly on
density, temperature, metallicity, and radiation field in three environments:
the dense cores where stars form, larger star-forming regions ranging from GMCs
to galactic disks, and the interiors of HII regions and super star clusters. In
dense cores, the quantity T^{3/2}n^{-1/2} that appears in M_J scales with core
density as n^{0.25} or with radiation density as U^{0.1} at the density where
dust and gas come into thermal equilibrium. On larger scales, this quantity
varies with ambient density as n^{-0.05} and ambient radiation field as
U^{-0.033} when the Kennicutt-Schmidt law of star formation determines U(n). In
super star clusters with ionization and compression of pre-stellar globules,
M_J varies as the 0.13 power of the cluster column density. These weak
dependencies on n, U, and column density imply that most environmental
variations affect the thermal Jeans mass by at most a factor of ~2.
Cosmological increases in M_J, which have been suggested by observations, may
be explained if the star formation efficiency is systematically higher at high
redshift for a given density and pressure, if dust grains are smaller at lower
metallicity, and so hotter for a given radiation field, or if small pre-stellar
cores are more severely ionized in extreme starburst conditions.
