A Circular Hamming Distance, Circular Gumbel Distribution, RNA Relics and Primitive Genome

If we compare a RNA or DNA chain or ring sequence to a reference ring sequence or to random ring sequences, we need the circular version of some distances and distributions like the Hamming and Gumbel one´s. We define here these circular versions and then we apply to the comparison of RNA relics like micro-RNAs and tRNAs to reference RNA rings coming from a primitive genome whose structure is described as solution of a variational problem.