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Adv Appl Math 2019 Jan 14;102:1-17. Epub 2018 Sep 14.

Department of Biology, Stanford University, Stanford, CA 94305 USA.

In mathematical phylogenetics, given a rooted binary leaf-labeled gene tree topology and a rooted binary leaf-labeled species tree topology with the same leaf labels, a coalescent history represents a possible mapping of the list of gene tree coalescences to the associated branches of the species tree on which those coalescences take place. For certain families of ordered pairs (), the number of coalescent histories increases exponentially or even faster than exponentially with the number of leaves . Other pairs have only a single coalescent history. Read More

Adv Appl Math 2018 May 28;96:39-75. Epub 2018 Feb 28.

Department of Mathematics, University of California, 970 Evans Hall #3840, Berkeley, CA 94720-3840, U.S.A.

Given an edge-weighted tree with leaves, sample the leaves uniformly at random without replacement and let , 2 ≤ ≤ , be the length of the subtree spanned by the first leaves. We consider the question, "Can be identified (up to isomorphism) by the joint probability distribution of the random vector (, …, )?" We show that if is known to belong to one of various families of edge-weighted trees, then the answer is, "Yes." These families include the edge-weighted trees with edge-weights in general position, the ultrametric edge-weighted trees, and certain families with equal weights on all edges such as ( + 1)-valent and rooted -ary trees for ≥ 2 and caterpillars. Read More

Adv Appl Math 2010 Feb;44(2):168-184

Department of Mathematics, University College London, Gower Street, London WC1E 6BT, UK and Department of Biological Sciences, University of Essex, Wivenhoe Park, Colchester, CO4 3SQ, UK.

In this paper we discuss the question of how to decide when a general chemical reaction system is incapable of admitting multiple equilibria, regardless of parameter values such as reaction rate constants, and regardless of the type of chemical kinetics, such as mass-action kinetics, Michaelis-Menten kinetics, etc. Our results relate previously described linear algebraic and graph-theoretic conditions for injectivity of chemical reaction systems. After developing a translation between the two formalisms, we show that a graph-theoretic test developed earlier in the context of systems with mass action kinetics, can be applied to reaction systems with arbitrary kinetics. Read More