This paper describes plurisubharmonic convexity and hulls, and also analytic multifunctions in terms of Jensen measures. In particular, this allows us to get a new proof of Słodkowski's theorem stating that multifunctions are analytic if and only if their graphs are pseudoconcave. We also show that multifunctions with plurisubharmonically convex fibers are analytic if and only if their graphs locally belong to plurisubharmonic hulls of their boundaries. In the last section we prove that minimal analytic multifunctions satisfy the maximum principle and give a criterion for the existence of holomorphic selections in the graphs of analytic multifunctions.

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The modules of principal parts$P$^{$k$}($E$) of a locally free sheaf ε on a smooth scheme$X$is a sheaf of$O$_{$X$}-bimodules which is locally free as left and right$O$_{$X$}-module. We explicitly split the modules of principal parts$P$^{$k$}($O$($n$)) on the projective line in arbitrary characteristic, as left and right$O$_{p1}-modules. We get examples when the splitting-type as left module differs from the splitting-type as right module. We also give examples showing that the splitting-type of the principal parts changes with the characteristic of the base field.

Let$g$be a positive integer. We prove that there are positive integers$n$_{1},$n$_{2},$n$_{3}and$n$_{4}such that the semigroup$S=(n$_{1},$n$_{2},$n$_{3},$n$_{4}) is an irreducible (symmetric or pseudosymmetric) numerical semigroup with g($S$)=$g$.

We study the relation between different spaces of vector-valued polynomials and analytic functions over dual-isomorphic Banach spaces. Under conditions of regularity on$E$and$F$, we show that the spaces of$X$-valued$n$-homogeneous polynomials and analytic functions of bounded type on$E$and$F$are isomorphic whenever$X$is a dual space. Also, we prove that many of the usual subspaces of polynomials and analytic functions on$E$and$F$are isomorphic without conditions on the involved spaces.