  We present two adaptations of an argument of Sonin, which is known to be a
powerful tool for obtaining both qualitative and quantitative information about
special functions. Our particular applications are as follows:
  (i) We give a rigorous formulation and proof of the following assertion about
focusing NLS in any dimension: The spatial envelope of a spherically symmetric
soliton in a repulsive potential is a non-increasing function of the radius.
  (ii) Driven by the question of determining the most stably singular matrix,
we determine the location of the maximal eigenvalue density of an $n\times n$
GUE matrix. Strikingly, in even dimensions, this maximum is \emph{not} at zero.
