We give a local criterion in terms of a residue current for strong holomorphicity of a meromorphic function on an arbitrary pure-dimensional analytic variety. This generalizes a result by A. Tsikh for the case of a reduced complete intersection.

In this paper, we compare some deterministic and probabilistic techniques in the study of upper bounds in problems related to certain mean square discrepancie with respect to balls in the$d$-dimensional unit torus, and show that the quality of these techniques depends in an intricate way on the dimension$d$under consideration.

In the present paper we prove, that in the real normed space$X$, having at least three dimensions, the Nordlander’s conjecture about the modulus of convexity of the space$X$is true, i.e. from the validity of Day’s inequality for a fixed real number from the interval (0,2), follows that$X$is an inner product space.

We introduce a basis of the Orlik–Solomon algebra labeled by chambers, the so called chamber basis. We consider structure constants of the Orlik–Solomon algebra with respect to the chamber basis and prove that these structure constants recover D. Cohen’s minimal complex from the Aomoto complex.

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.