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The class of generating functions for completely monotone sequences (moments of finite positive measures on $[0,1]$) has an elegant characterization as the class of Pick functions analytic and positive on $(-\infty,1)$. We establish this and another such characterization and develop a variety of consequences. In particular, we characterize generating functions for moments of convex and concave probability distribution functions on $[0,1]$, and we provide a simple analytic proof that for any real $p,r>0$, the Fuss-Catalan or Raney numbers $\frac{r}{pn+r}\binom{pn+r}{n}$, $n=0,1,\ldots$ are the moments of a probability distribution on some interval $[0,\tau]$ if and only if $p\ge1$ and $p\ge r$.

The inflow problem of full compressible NavierStokes equations is considered on the half-line (0,+\infty). First, we give the existence (or nonexistence) of the boundary layer solution to the inflow problem when the right end state (0,+\infty) belongs to the subsonic, transonic, and supersonic regions, respectively. Then the asymptotic stability of not only the single contact wave but also the superposition of the subsonic boundary layer solution, the contact wave, and the rarefaction wave to the inflow problem are investigated under some smallness conditions. Note that the amplitude of the rarefaction wave is not necessarily small. The proofs are given by the elementary energy method.

We will introduce a quantity which measures the singularity of a plurisubharmonic function$φ$relative to another plurisubharmonic function$ψ$, at a point$a$. We denote this quantity by$ν$_{$a$,$ψ$}($φ$). It can be seen as a generalization of the classical Lelong number in a natural way: if$ψ$=($n$−1)log| ⋅ −$a$|, where$n$is the dimension of the set where$φ$is defined, then$ν$_{$a$,$ψ$}($φ$) coincides with the classical Lelong number of$φ$at the point$a$. The main theorem of this article says that the upper level sets of our generalized Lelong number, i.e. the sets of the form {$z$:$ν$_{$z$,$ψ$}($φ$)≥$c$} where$c$>0, are in fact analytic sets, provided that the$weight$$ψ$satisfies some additional conditions.

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