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.

We consider a domain Ω with Lipschitz boundary, which is relatively compact in an$n$-dimensional Kähler manifold and satisfies some “logδ-pseudoconvexity” condition. We show that the $$\bar \partial $$ -equation with exact support in ω admits a solution in bidegrees ($p, q$), 1≤$q$≤$n$−1. Moreover, the range of $$\bar \partial $$ acting on smooth ($p, n$−1)-forms with support in $$\bar \Omega $$ is closed. Applications are given to the solvability of the tangential Cauchy-Riemann equations for smooth forms and currents for all intermediate bidegrees on boundaries of weakly pseudoconvex domains in Stein manifolds and to the solvability of the tangential Cauchy-Riemann equations for currents on Levi flat$CR$manifolds of arbitrary codimension.

We introduce a non-linear injective transformation τ from the set of non-vanishing normalized Hausdorff moment sequences to the set of normalized Stieltjes moment sequences by the formula$T$[($a$_{$n$})_{$n$=1}^{∞}]_{$n$}= 1/$a$_{1}...$a$_{$n$}. Special cases of this transformation have appeared in various papers on exponential functionals of Lévy processes, partly motivated by mathematical finance. We give several examples of moment sequences arising from the transformation and provide the corresponding measures, some of which are related to$q$-series.