  Observational studies are showing that the galaxy-wide stellar initial mass
function are top-heavy in galaxies with high star-formation rates (SFRs).
Calculating the integrated galactic stellar initial mass function (IGIMF) as a
function of the SFR of a galaxy, it follows that galaxies which have or which
formed with SFRs > 10 Msol yr^-1 would have a top-heavy IGIMF in excellent
consistency with the observations. Consequently and in agreement with
observations, elliptical galaxies would have higher M/L ratios as a result of
the overabundance of stellar remnants compared to a stellar population that
formed with an invariant canonical stellar initial mass function (IMF). For the
Milky Way, the IGIMF yields very good agreement with the disk- and the
bulge-IMF determinations. Our conclusions are that purely stochastic
descriptions of star formation on the scales of a pc and above are falsified.
Instead, star formation follows the laws, stated here as axioms, which define
the IGIMF theory. We also find evidence that the power-law index beta of the
embedded cluster mass function decreases with increasing SFR. We propose
further tests of the IGIMF theory through counting massive stars in dwarf
galaxies.
