The crater lake Lake Albano is an increasingly diminishing water resource in terms of volume, the lake level has dropped more than four meters since the 1960s, and water quality resulting from elevated levels of nitrogen and phosphorus.

In this article, a system of nonlinear hyperbolic-elliptic partial differential equations is introduced to model the formation of biofilms. First, a short introduction to some basic concepts about biofilms is given. Then a detailed derivation of the model is presented, which is mainly based on the theory of mixtures, also in comparison with previous models. Adapted numerical schemes will be presented and numerical simulations will be discussed.

Given a finitely generated multiplicative subgroup Γ ⊆ Q * , assuming the Generalized Riemann Hypothesis, we determine an asymptotic formula for average over prime numbers, powers of the order of the reduction group modulo p. The problem was considered in the case of rank 1 by Pomerance and Kurlberg. In the case when Γ contains only positive numbers, we give an explicit expression for the involved density in terms of an Euler product. We conclude with some numerical computations.