a galaxy $m$ to the mass of its dark matter halo $M$ we need to specify the SHM ratio. A direct comparison of the halo mass function $n(M)$ and the galaxy mass function $\phi(m)$ helps to constrain the stellar-to-halo mass function. If we assume that every host (sub) halo contains exactly one central (satellite) galaxy and that each system has exactly the same SHM ratio $m/M$, the galaxy stellar mass function can be derived trivially from the halo mass function and has the same features. The galaxy mass function derived for $m/M =
0.05$ is compared to the observed SDSS galaxy mass function in Figure \[f:fig1\]. The observed galaxy mass function is steeper for high masses and shallower for low masses than the one derived from the halo mass function. Thus, for a constant SHM ratio there will inevitably be too many galaxies at the low and high mass end.

This implies that the actual SHM ratio $m/M$ is not constant, but increases with increasing mass, reaches a maximum around $m^*$ and then decreases again. Hence we adopt the following parametrization, similar to the one used in @yang03: $$\frac{m(M)}{M} = 2 \left( \frac{m}{M}\right)_0 \left[\left(\frac{M}{M_1}\right)^{-\beta} + \left(\frac{M}{M_1}\right)^{\gamma}\right]^{-1}
\label{eqnmmap}$$ It has four 