Connected components of Morse boundaries of graphs of groups

Let a finitely generated group $G$ split as a graph of groups. If edge groups are undistorted and do not contribute to the Morse boundary $\partial_MG$, we show that every connected component of $\partial_MG$ with at least two points originates from the Morse boundary of a vertex group. Under stronger assumptions on the edge groups (such as wideness in the sense of Dru\c{t}u-Sapir), we show that Morse boundaries of vertex groups are topologically embedded in $\partial_MG$

Competing epidemics on complex networks

Human diseases spread over networks of contacts between individuals and a substantial body of recent research has focused on the dynamics of the spreading process. Here we examine a model of two competing diseases spreading over the same network at the same time, where infection with either disease gives an individual subsequent immunity to both. Using a combination of analytic and numerical methods, we derive the phase diagram of the system and estimates of the expected final numbers of individuals infected with each disease. The system shows an unusual dynamical transition between dominance of one disease and dominance of the other as a function of their relative rates of growth. Close to this transition the final outcomes show strong dependence on stochastic fluctuations in the early stages of growth, dependence that decreases with increasing network size, but does so sufficiently slowly as still to be easily visible in systems with millions or billions of individuals. In most regions of the phase diagram we find that one disease eventually dominates while the other reaches only a vanishing fraction of the network, but the system also displays a significant coexistence regime in which both diseases reach epidemic proportions and infect an extensive fraction of the network.Comment: 14 pages, 5 figure

Molecular Analysis of N6-Methyladenine Patterns in \u3cem\u3eTetrahymena thermophila\u3c/em\u3e Nuclear DNA

We have cloned two DNA fragments containing 5\u27-GATC-3\u27 sites at which the adenine is methylated in the macronucleus of the ciliate Tetrahymena thermophila. Using these cloned fragments as molecular probes, we analyzed the maintenance of methylation patterns at two partially and two uniformly methylated sites. Our results suggest that a semiconservative copying model for maintenance of methylation is not sufficient to account for the methylation patterns we found during somatic growth of Tetrahymena. Although we detected hemimethylated molecules in macronuclear DNA, they were present in both replicating and nonreplicating DNA. In addition, we observed that a complex methylation pattern including partially methylated sites was maintained during vegetative growth. This required the activity of a methylase capable of recognizing and modifying sites specified by something other than hemimethylation. We suggest that a eucaryotic maintenance methylase may be capable of discriminating between potential methylation sites to ensure the inheritance of methylation patterns