Phase transitions in Phylogeny

We apply the theory of markov random fields on trees to derive a phase transition in the number of samples needed in order to reconstruct phylogenies. We consider the Cavender-Farris-Neyman model of evolution on trees, where all the inner nodes have degree at least 3, and the net transition on each edge is bounded by e. Motivated by a conjecture by M. Steel, we show that if 2 (1 - 2 e) (1 - 2e) > 1, then for balanced trees, the topology of the underlying tree, having n leaves, can be reconstructed from O(log n) samples (characters) at the leaves. On the other hand, we show that if 2 (1 - 2 e) (1 - 2 e) < 1, then there exist topologies which require at least poly(n) samples for reconstruction. Our results are the first rigorous results to establish the role of phase transitions for markov random fields on trees as studied in probability, statistical physics and information theory to the study of phylogenies in mathematical biology.Comment: To appear in Transactions of the AM

LAI based trees selection for mid latitude urban developments: A microclimatic study in Cairo, Egypt

To study the leaf area index, LAI, based thermal performance in distinguishing trees for Cairo's urban developments, ENVI-met plants database was used as platform for a foliage modeling parameter, the leaf area density, LAD. Two Egyptian trees: Ficus elastica. and Peltophorum pterocarpum were simulated in 2 urban sites with one having no trees, whilst the second is having Ficus nitida trees. Trees LAD values were calculated using flat leaves' trees LAI definition to produce maximum ground solid shadow at peak time. An empirical value of 1 for LAI is applied to numerically introduce LAD values for ENVI-met. Basically, different meteorological records showed improvements for pedestrian comfort and ambient microclimate of the building using E elastica. About 40-50% interception of direct radiation, reductions in surfaces' fluxes around trees and in radiant temperature T-mrt in comparison to base cases gave preferability to E elastica. The lack of soil water prevented evapotranspiration to take place effectively and the reduced wind speeds concluded negligible air temperature differences from both base cases except slightly appeared with the F elastica. Results show that a flat leaves tree if does not validate LAI of 1, the ground shading would not fulfill about 50% direct radiation interception and this value can be used as a reference for urban trees selection. Further simulations were held to investigate LAI value of maximum direct radiation interception. Performing additional simulations, F elastica of LAI of 3 intercepted almost 84% of direct radiation and revealed implications about urban trees in practice and its actual LAI. (C) 2009 Elsevier Ltd. All rights reserved

On the non-Gaussian fluctuations of the giant cluster for percolation on random recursive trees

We consider a Bernoulli bond percolation on a random recursive tree of size $n\gg 1$, with supercritical parameter $p_n=1-c/\ln n$ for some $c>0$ fixed. It is known that with high probability, there exists then a unique giant cluster of size G_n\sim \e^{-c}, and it follows from a recent result of Schweinsberg \cite{Sch} that $G_n$ has non-gaussian fluctuations. We provide an explanation of this by analyzing the effect of percolation on different phases of the growth of recursive trees. This alternative approach may be useful for studying percolation on other classes of trees, such as for instance regular trees