Kramers-Kronig relations beyond the optical approximation

We extend Kramers-Kronig relations beyond the optical approximation, that is to dielectric functions $\varepsilon(q,\omega)$ that depend not only on the frequency but on the wave number as well. This implies extending the notion of causality commonly used in the theory of Kramers-Kronig relations to include the fact that signals cannot propagate faster than light in vacuo. The results we derive do not apply exclusively to electrodynamics but also to other theories of isotropic linear response in which the response function depends on both wave number and frequency.Comment: 1 figur

On the Bell's spaceships paradox and proper length for accelerated bodies

We study the Dewan-Beran-Bell thought experiment of two spaceships connected by a thread that start accelerated motion and discuss the proper length of the thread by means of Born's definition of proper length for arbitrary motion.Comment: 3 figure

Residually free groups do not admit a uniform polynomial isoperimetric function

We show that there is no uniform polynomial isoperimetric function for finitely presented subgroups of direct products of free groups, by producing a sequence of subgroups $G_r\leq F_2^{(1)} \times \dots \times F_2^{(r)}$ of direct products of 2-generated free groups with Dehn functions bounded below by $n^{r}$. The groups $G_r$ are obtained from the examples of non-coabelian subdirect products of free groups constructed by Bridson, Howie, Miller and Short. As a consequence we obtain that residually free groups do not admit a uniform polynomial isoperimetric function.Comment: 9 pages, v2: Improvements to the exposition and minor corrections. Final accepted version, to appear in the Proceedings of the American Mathematical Societ