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The stellar initial mass function (IMF) is commonly assumed to be an invariant probability density distribution function of initial stellar masses being represented by the canonical IMF. As a consequence the galaxy-wide IMF (gwIMF), defined as the sum of the IMFs of all star forming regions, should also be invariant. Recent observational and theoretical results challenge the hypothesis that the gwIMF is invariant. In order to study the possible reasons for this variation we use the IMF determined in resolved star clusters and apply the IGIMF-theory to calculate a grid of gwIMF models for metallicities, -3<[Fe/H]<1, and galaxy-wide star formation rates, $10^{-5}$ 1\,M_\odot$/yr, which is a common condition in the early Universe, we find that the gwIMF is top-heavy (more massive stars), when compared to the canonical IMF. For a SFR $< 1\,\mathrm{M_{\odot}/yr}$ the gwIMF becomes top-light regardless of the metallicity. For metallicities $\mathrm{[Fe/H]} > 0$ the gwIMF can become bottom-heavy regardless of the SFR. The IGIMF models predict that massive elliptical galaxies should have formed with a gwIMF that is top-heavy within the first few hundred Myr of the galaxy's life and that it evolves into a bottom-heavy gwIMF in the metal-enriched galactic center. We study the SFR$-$H$\alpha$ relation, its dependency on metallicity and the SFR, the correction factors to the Kennicutt SFR$_{\rm K}-$H$\alpha$ relation, and provide new fitting functions Late-type dwarf galaxies show significantly higher SFRs with respect to Kennicutt SFRs, while star forming massive galaxies have significantly lower SFRs than hitherto thought. This has implications for the gas-consumption time scales and for the main sequence of galaxies. The Leo P and ultra-faint dwarf galaxies are discussed explicitly. [abridged], Comment: Astronomy and Astrophysics (A&A) in press. 15 pages, 8 figures

Outer-halo globular clusters show large half-light radii and flat stellar mass functions, depleted in low-mass stars. Using N-body simulations of globular clusters on eccentric orbits within a Milky Way-like potential, we show how a cluster's half-mass radius and its mass function develop over time. The slope of the central mass function flattens proportionally to the amount of mass a cluster has lost, and the half-mass radius grows to a size proportional to the average strength of the tidal field. The main driver of these processes is mass segregation of dark remnants. We conclude that the extended, depleted clusters observed in the Milky Way must have had small half-mass radii in the past, and that they expanded due to the weak tidal field they spend most of their lifetime in. Moreover, their mass functions must have been steeper in the past but flattened significantly as a cause of mass segregation and tidal mass loss., Comment: 2 pages, proceedings of IAU Symposium 316 Formation, Evolution, and Survival of Massive Star Clusters