Vittarioideae worthy of subgeneric rank, while "Rheopteris" had not yet been discovered. "Haplopteris" he explicitly synonymized with "Vittaria" in 1914. Carl Christensen used the name "Vittarioideae" in Verdoorn's "Manual of Pteridology" in 1938, but did not include a description, leaving it nomenclaturally invalid. Ren-Chang Ching raised Vittariaceae to the rank of a family in 1940. The first well-sampled molecular phylogenetic study of the vittarioids was based on the chloroplast gene rbcL. In this study, it was found that the type species of "Monogramma" is embedded in "Haplopteris"; the segregation of "Vaginularia" from "Monogramma" was also supported, as members of "Vaginularia" formed
Subshift of finite type metric on a shift space by considering two points to be "close" if they have many initial symbols in common; this is the "p"-adic metric. In fact, both the one- and two-sided shift spaces are compact metric spaces. A subshift of finite type may be endowed with any one of several different measures, thus leading to a measure-preserving dynamical system. A common object of study is the Markov measure, which is an extension of a Markov chain to the topology of the shift. A Markov chain is a pair ("P",π) consisting of the transition matrix, an formula_14 matrix formula_15 for