In this paper, we prove a mean ergodic theorem for an almost surely bounded strongly continuous semigroup of random linear operators on a random reflexive random normed module, which generalizes and improves several known important results.

For a mixed characteristic complete discrete valuation field $K$ which contains a $p^{n}$-th root of unity, we determine the graded quotients of the filtration on the Milnor $K$-groups $K_{q}^{M}(K)$ modulo $p^{n}$ in terms of differential forms of the residue field of $K$.

We prove that the growth rates of three-dimensional generalized simplex reflection groups, i.e. three-dimensional non-compact hyperbolic Coxeter groups with four generators are always Perron numbers.