The problem of finding birth-death fixation probabilities for configurations of normal and mutants on an N-vertex graph is formulated in terms of a Markov process on the 2 N -dimensional state space of possible configurations. Upper and lower bounds on the fixation probability after any given number of iterations of the birth-death process are derived in terms of the transition matrix of this process. Consideration is then specialized to a family of graphs called circular flows, and we present a summation formula for the complete bipartite graph, giving the fixation probability for an arbitrary configuration of mutants in terms of a weighted sum of the singlevertex fixation probabilities. This also yields a closedform solution for the fixation probability of bipartite graphs. Three entropy measures are introduced, providing information about graph structure. Finally, a number of examples are presented, illustrating cases of graphs that enhance or suppress fixation probability for fitness r > 1 as well as graphs that enhance fixation probability for only a limited range of fitness. Results are compared with recent results reported in the literature, where a positive correlation is observed between vertex degree variance and fixation probability for undirected graphs. We show a similar correlation for directed graphs, with correlation not directly to fixation probability but to the difference between fixation probability for a given graph and a complete graph.
This article provides simple details of what to do with digital images once they have been captured. Inspection of the images as 'thumbnails' using Exif viewer is described, as well as zooming in to check detail contained on the images. Storage of every orthodontic patient using the popular programme, Dentofacial Showcase is described in some detail. For more formal verbal presentations or written material intended for display Microsoft Powerpoint is the programme of choice. Transfer of the images between the three programmes is described in detail, as well as recommended layouts for written and verbal presentations.
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