scholarly journals Exact and approximate limit behaviour of the Yule tree’s cophenetic index

2018 ◽  
Vol 303 ◽  
pp. 26-45 ◽  
Author(s):  
Krzysztof Bartoszek
2017 ◽  
Author(s):  
Krzysztof Bartoszek

AbstractIn this work we study the limit distribution of an appropriately normalized cophenetic index of the pure–birth tree conditioned onncontemporary tips. We show that this normalized phylogenetic balance index is a submartingale that converges almost surely and inL2. We link our work with studies on trees without branch lengths and show that in this case the limit distribution is a contraction–type distribution, similar to the Quicksort limit distribution. In the continuous branch case we suggest approximations to the limit distribution. We propose heuristic methods of simulating from these distributions and it may be observed that these algorithms result in reasonable tails. Therefore, we propose a way based on the quantiles of the derived distributions for hypothesis testing, whether an observed phylogenetic tree is consistent with the pure–birth process. Simulating a sample by the proposed heuristics is rapid, while exact simulation (simulating the tree and then calculating the index) is a time–consuming procedure. We conduct a power study to investigate how well the cophenetic indices detect deviations from the Yule tree and apply the methodology to empirical phylogenies.


1984 ◽  
Vol 21 (03) ◽  
pp. 646-650 ◽  
Author(s):  
Rudolf Mathar

We consider degenerate limit laws for the sequence {Xn, n } n (N of successive maxima of identically distributed random variables. It turns out that the concentration of Xn, n for large n can be determined in terms of a tail ratio of the underlying distribution function F. Applications to the outlier-behaviour of probability distributions are given.


1992 ◽  
Vol 7 (11) ◽  
pp. 2908-2915 ◽  
Author(s):  
Angel L.G. Ponce ◽  
Jose J. Fripiat

Atomic hydrogen obtained from dissociative chemisorption of molecular H2 on Pt particles deposited on the surface of YBa2Cu3O7 reacts with the oxide in producing O vacancies and intercalating H at 82 °C under a H2 pressure of about 400 Torr. An induction period which extends over 1 h is observed as long as the concentration in O vacancies is below 0.1. Above this approximate limit the reaction proceeds quickly until about 1 mol H2 has been consumed. It then slows down progressively, but it is not completed even after 27 h of reaction and ∼1.4 H2 consumed, under these experimental conditions. The enthalpy for the creation of the O vacancy is 143 kcal/g mol O, while the H intercalation enthalpy is −57.5 kcal/g mol H. It appears that the reaction of molecular H2 over YBa2Cu3O7 not coated with Pt proceeds similarly, but the rate is nearly one order of magnitude slower, under identical conditions. The stoichiometry of the reaction agrees with the earlier suggestion that O1 is a labile oxygen which can be replaced by intercalated H. This site and the interstitial vacant sites in the copper chains square plane would be the first ones to be occupied by intercalated H.


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