A determinantal approach to irrationality

It is a classical fact that the irrationality of a number $\xi\in\mathbb R$ follows from the existence of a sequence $p_n/q_n$ with integral $p_n$ and $q_n$ such that $q_n\xi-p_n\ne0$ for all $n$ and $q_n\xi-p_n\to0$ as $n\to\infty$. In this note we give an extension of this criterion in the case when the sequence possesses an additional structure; in particular, the requirement $q_n\xi-p_n\to0$ is weakened. Some applications are given including a new proof of the irrationality of $\pi$. Finally, we discuss analytical obstructions to extend the new irrationality criterion further and speculate about some mathematical constants whose irrationality is still to be established.Comment: 9 page

A Concept for Molecular Addressing by Means of Far-reaching Electromagnetic Interactions in the Visible

The interaction of molecular optoelectronic components on the molecular scale were studied where the solvent shell indicating the influence of the medium was found to be surprisingly small. The transport of energy as resonant energy transfer covers distances of about 5 nm and was shown not to proceed by a simple to dipole dipole interaction with typical restrictions, but by a more complex mechanism. Furthermore, a novel-type of far-reaching interactions of electronically excited structures until macroscopic dimensions were fond and may be applied for addressing molecular structures by conventional electronics