Translation Lengths in Out(F n)

We prove that all elements of infinite order in Out(F _n) have positive translation lengths; moreover, they are bounded away from zero. As a consequence we get a new proof that solvable subgroups of Out(F _n) are finitely generated and virtually Abelian.     

Extending the McClelland formula for total $$\pi $$-electron energy

The McClelland formula, based on the upper bound \(\sqrt{2mn}\), is capable of reproducing over 99.5% of the total \(\pi \)-electron energy (\(E_\pi \)) of a conjugated hydrocarbon, whose molecules possess n carbon atoms and m carbon–carbon bonds. Its weak point is that it predicts equal \(E_\pi \)-values for all isomers. We now show how this failure can be overcome, offering a general strategy for extending McClelland’s formula. By means of one of these extensions, \(E_\pi \) is related with the energy of the highest occupied molecular orbital, and the error of the new formula is diminished by more than \(50\%\) relative to the standard McClelland approximation.