  The stellar initial mass function (IMF) is a key property of stellar
populations. There is growing evidence that the classical star-formation
mechanism by the direct cloud fragmentation process has difficulties
reproducing the observed abundance and binary properties of brown dwarfs and
very-low-mass stars. In particular, recent analytical derivations of the
stellar IMF exhibit a deficit of brown dwarfs compared to observational data.
Here we derive the residual mass function of brown dwarfs as an empirical
measure of the brown dwarf deficiency in recent star-formation models with
respect to observations and show that it is compatible with the substellar part
of the Thies-Kroupa IMF and the mass function obtained by numerical
simulations. We conclude that the existing models may be further improved by
including a substellar correction term that accounts for additional formation
channels like disk or filament fragmentation. The term "peripheral
fragmentation" is introduced here for such additional formation channels. In
addition, we present an updated analytical model of stellar and substellar
binarity. The resulting binary fraction and the dynamically evolved companion
mass-ratio distribution are in good agreement with observational data on
stellar and very-low-mass binaries in the Galactic field, in clusters, and in
dynamically unprocessed groups of stars if all stars form as binaries with
stellar companions. Cautionary notes are given on the proper analysis of mass
functions and the companion mass-ratio distribution and the interpretation of
the results. The existence of accretion disks around young brown dwarfs does
not imply that these form just like stars in direct fragmentation.
