We prove a large sieve statement for the average distribution of Frobenius conjugacy classes in arithmetic monodromy groups over finite fields. As a first application we prove a stronger version of a result of Chavdarov on the “generic” irreducibility of the numerator of the zeta functions in a family of curves with large monodromy.

An approach to the computations of upper bounds on the size of large complete arcs is presented. In particular, we obtain geometrical properties of irreducible envelopes associated to a second largest complete arc provided that the order of the underlying field is large enough.

We review notions of small sets,-irreducibility, etc., and present a simple proof of asymptotic convergence of general state space Markov chains to their stationary distributions.

This is a survey of recent progress on the irreducibility Fermi varieties, rigidity results and embedded eigenvalue problems discrete periodic Schr\"odinger operators.

Given a commensurated subgroup $\Lambda$ of group $\Gamma$, we completely characterize when the inclusion $\Lambda\leq \Gamma$ is $C^*$-irreducible and provide new examples such inclusions. In particular, obtain that $\rm{PSL}(n,\mathbb{Z})\leq\rm{PGL}(n,\mathbb{Q})$ for any $n\in \mathbb{N}$, $C^*$-simple into its abstract commensurator $C^*$-irreducible. The main ingredient use fact action $\...