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Rate of lineage origin explains the diversity anomaly in the World’s mangrove vegetation
The contribution of nonecological factors to global patterns in diversity is evident when species richness differs between regions with similar habitats and geographic area. Mangrove environments in the Eastern Hemisphere harbor six times as many species of trees and shrubs as similar environments in the New World. Genetic divergence of mangrove lineages from terrestrial relatives, in combination with fossil evidence, suggests that mangrove diversity is limited by evolutionary transition into the stressful marine environment, the number of mangrove lineages has increased steadily over the Tertiary with little global extinction, and the diversity anomaly in mangrove vegetation reflects regional differences in the rate of origin of new mangrove lineages
Dense Subgraphs in Random Graphs
For a constant and a graph , let be
the largest integer for which there exists a -vertex subgraph of
with at least edges. We show that if then
is concentrated on a set of two integers. More
precisely, with
,
we show that is one of the two integers closest to
, with high probability.
While this situation parallels that of cliques in random graphs, a new
technique is required to handle the more complicated ways in which these
"quasi-cliques" may overlap
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