Renormalization group equations and a phase diagram are derived for a system of two chains with a single-electron hopping between chains in order to correct the results of a recent paper by F. V. Kusmartsev, A. Luther, and A. Nersesyan, Pis’ma v Zh. Exp. Teor. Fiz. 55, 692 (1992) [JETP Lett. 55, 724 (1992)]. In a recent paper [1], the effect of a single-electron hopping (SEH) between chains in ...

I first recount Raychaudhuri’s deep involvement with the singularity problem in general relativity. I then argue that precisely the same situation has arisen today in loop quantum cosmology as obtained when Raychaudhuri discovered his celebrated equation. We thus need a new analogue of the Raychaudhuri equation in quantum gravity. PACS numbers: 04.20.Jb, 04.2.Cv, 98.80 Dr 1 Singularity and AKR ...