The initial mass function of star clusters that form in turbulent
  molecular clouds
  We simulate the formation and evolution of young star clusters using the
combination of SPH simulations and direct N-body simulations. We start by
performing SPH simulations of the giant molecular cloud with a turbulent
velocity field, a mass of $4\times10^4$ to $5\times10^6M_{\odot}$, and a
density between $1.7\times10^3$ and $170cm^{-3}$. We continue the SPH
simulations for a free-fall time scale, and analyze the resulting structure of
the collapsed cloud. We subsequently replace a density-selected subset of SPH
particles with stars by adopting a local star-formation efficiency proportional
to $\rho^{1/2}$. As a consequence, the local star formation efficiency exceeds
30 %, whereas globally only a few % of the gas is converted to stars. The
stellar distribution by the time gas is converted to stars is very clumpy, with
typically a dozen bound conglomerates that consist of 100 to $10^4$ stars. We
continue to evolve the stars dynamically using the collisional N-body method,
which accurately treats all pairwise interactions, stellar collisions and
stellar evolution. We analyze the results of the N-body simulations when the
stars have an age of 2 Myr and 10 Myr. During the dynamical simulations,
massive clusters grow via hierarchical merging of smaller clusters. The shape
of the cluster mass function that originates from an individual molecular cloud
is consistent with a Schechter function with a power-law slope of -1.73 at 2
Myr and -1.67 at 10 Myr, which fits to observed cluster mass function of the
Carina region. The superposition of mass functions have a power-law slope of <
-2, which fits the observed mass function of star clusters in the Milky Way,
M31 and M83. We further find that the mass of the most massive cluster formed
in a single molecular cloud with a mass of $M_g$ scales with $6.1M_g^{0.51}$
which also agrees with recent observation of the GMC and young clusters in M51.
