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 the zeta function is a rational function of "z": Subshift of finite type In mathematics, subshifts of finite type are used to model dynamical systems, and in particular are the objects of study in symbolic dynamics and ergodic theory. They also describe the set of all possible sequences executed by a finite state machine. The most widely studied shift spaces are the subshifts of finite type. Let formula_1 be a finite set of formula_2 symbols (alphabet). Let "X" denote the set formula_3 of all bi-infinite sequences of elements of "V" together with the shift operator "T". We endow "V" with