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($S$)=$g$.

No abstract uploaded!

No abstract uploaded!

We give several characterizations of those sequences of holomorphic self-maps {φ_{$n$}}_{$n$≥1}of the unit disk for which there exists a function$F$in the unit ball $\mathcal{B}=\{f\in H^{\infty}: \|f\|_\infty\leq1\}$ of$H$^{∞}such that the orbit {$F$∘φ_{$n$}:$n$∈ℕ} is locally uniformly dense in $\mathcal{B}$ . Such a function$F$is said to be a $\mathcal{B}$ -universal function. One of our conditions is stated in terms of the hyperbolic derivatives of the functions φ_{$n$}. As a consequence we will see that if φ_{$n$}is the$n$th iterate of a map φ of $\mathbb{D}$ into $\mathbb{D}$ , then {φ_{$n$}}_{$n$≥1}admits a $\mathcal{B}$ -universal function if and only if φ is a parabolic or hyperbolic automorphism of $\mathbb{D}$ . We show that whenever there exists a $\mathcal{B}$ -universal function, then this function can be chosen to be a Blaschke product. Further, if there is a $\mathcal{B}$ -universal function, we show that there exist uniformly closed subspaces consisting entirely of universal functions.

No abstract uploaded!