We prove two comparison theorems between the time derivative of a real function $u(x, t)$ such that $u(\cdot,t)$ belongs to L$^1 (\Omega)$ for all $t$, and the time derivative of the vector function $\skew2\hat{u}(t) = u(\cdot, t)$.

We show that the order of an automorphism of an arithmetic Riemann surface of genus $g$ is not greater than $2g\,{-}\,2$, provided $g$ is large enough. This bound is an arithmetic analog of the classical Wiman bound. We prove that it is sharp and attained for any genus but in contrast to the general case the automorphisms of maximal order act without fixed points. This allows us to consider the automorphisms which act on arithmetic Riemann surfaces and have a given number of fixed points. For these automorphisms we describe the asymptotic behavior of their orders.

For a class of second order nonlinear differential equations, sufficient conditions are presented to ensure that some, respectively all solutions are asymptotic to lines.

Using the classification by Dotti and Fino [3] we show the existence of an HKT metric on a neighbourhood of the centre of any 8-dimensional nilpotent Lie group $G$ with invariant hypercomplex structure. This metric exists globally if the hypercomplex structure is abelian, and in these cases we construct an HKT structure on a neighbourhood of the zero section of the cotangent bundle $T^{*}G$ extending the HKT metric on $G$.