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.

We investigate local properties of the Green function of the complement of a compact set$E$υ[0,1] with respect to the extended complex plane. We demonstrate, that if the Green function satisfies the 1/2-Hölder condition locally at the origin, then the density of$E$at 0, in terms of logarithmic capacity, is the same as that of the whole interval [0, 1]..