Stochastic models of nucleotide substitution are playing an increasingly important role in phylogenetic reconstruction through such methods as maximum likelihood. Here, we examine the behaviour of a simple substitution model, and establish some links between the methods of maximum parsimony and maximum likelihood under this model.
Stochastic models of nucleotide substitution are playing an increasingly important role in phylogenetic reconstruction through such methods as maximum likelihood. Here, we examine the behaviour of a simple substitution model, and establish some links between the methods of maximum parsimony and maximum likelihood under this model.
Given a topological space X denote by exp k X the space of non-empty subsets of X of size at most k , topologised as a quotient of X k . This space may be regarded as a union over 1 ≤ l ≤ k of configuration spaces of l distinct unordered points in X . In the special case X = S 1 we show that: (1) exp k S 1 has the homotopy type of an odd dimensional sphere of dimension k or k − 1; (2) The first three results generalise known facts that exp 2 S 1 is a Möbius strip with boundary exp 1 S 1 , and that exp 3 S 1 is the three-sphere with exp 1 S 1 inside it forming a trefoil knot.
The k th finite subset space of a topological space X is the space exp k X of non-empty finite subsets of X of size at most k , topologised as a quotient of X k . The construction is a homotopy functor and may be regarded as a union of configuration spaces of distinct unordered points in X . We calculate the homology of the finite subset spaces of a connected graph Γ, and study the maps (exp k φ) * induced by a map φ : Γ → Γ between two such graphs. By homotopy functoriality the results apply to punctured surfaces also. The braid group B n may be regarded as the mapping class group of an n-punctured disc D n , and as such it acts on H * (exp k D n ). We prove a structure theorem for this action, showing that the image of the pure braid group is nilpotent of class at most (n − 1)/2 .
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