We characterize model polynomials that are cyclic in Dirichlet-type spaces in the unit ball of $\mathbb C^n$, and we give a sufficient capacity condition in order to identify noncyclic vectors.

Under the local Langlands correspondence, the conductor of an irreducible representation of $\text{G}{{\text{l}}_{n}}\left( F \right)$ is greater than the Swan conductor of the corresponding Galois representation. In this paper, we establish the geometric analogue of this statement by showing that the conductor of a categorical representation of the loop group is greater than the irregularity of the corresponding meromorphic connection.

In this paper, we generalize a result recently obtained by the author. We characterize the cyclic vectors in $L_{a}^{p}\,\left( \mathbb{C},\,\phi \right)$ . Let $f\,\in \,L_{a}^{p}\,\left( \mathbb{C},\,\phi \right)$ and $fC$ be contained in the space. We show that $f$ is non-vanishing if and only if $f$ is cyclic.

We prove that a random function in the Hardy space $H^2$ is a non-cyclic vector for the backward shift operator almost surely. The question of existence of a local pseudocontinuation for a random analytic function is also studied.