65 research outputs found
Markov models for accumulating mutations
We introduce and analyze a waiting time model for the accumulation of genetic changes. The continuous-time conjunctive Bayesian network is defined by a partially ordered set of mutations and by the rate of fixation of each mutation. The partial order encodes constraints on the order in which mutations can fixate in the population, shedding light on the mutational pathways underlying the evolutionary process. We study a censored version of the model and derive equations for an em algorithm to perform maximum likelihood estimation of the model parameters. We also show how to select the maximum likelihood partially ordered set. The model is applied to genetic data from cancer cells and from drug resistant human immunodeficiency viruses, indicating implications for diagnosis and treatmen
Separability problem for multipartite states of rank at most four
One of the most important problems in quantum information is the separability
problem, which asks whether a given quantum state is separable. We investigate
multipartite states of rank at most four which are PPT (i.e., all their partial
transposes are positive semidefinite). We show that any PPT state of rank two
or three is separable and has length at most four. For separable states of rank
four, we show that they have length at most six. It is six only for some
qubit-qutrit or multiqubit states. It turns out that any PPT entangled state of
rank four is necessarily supported on a 3x3 or a 2x2x2 subsystem. We obtain a
very simple criterion for the separability problem of the PPT states of rank at
most four: such a state is entangled if and only if its range contains no
product vectors. This criterion can be easily applied since a four-dimensional
subspace in the 3x3 or 2x2x2 system contains a product vector if and only if
its Pluecker coordinates satisfy a homogeneous polynomial equation (the Chow
form of the corresponding Segre variety). We have computed an explicit
determinantal expression for the Chow form in the former case, while such
expression was already known in the latter case.Comment: 19 page
Matrix Schubert varieties and Gaussian conditional independence models
Seth Sullivant was partially supported by the David and Lucille Packard Foundation and the US National Science Foundation (DMS 0954865)
Integrality gaps of integer knapsack problems
We obtain optimal lower and upper bounds for the (additive) integrality
gaps of integer knapsack problems. In a randomised setting, we show that the integrality
gap of a “typical” knapsack problem is drastically smaller than the integrality
gap that occurs in a worst case scenario
Rev. Isidore Nahayewsky, History of Ukraine. Philadelphia: “America” Publishing House of the “Providence” Association of Ukrainian Catholics in America, 1962. 295 pp. $5.00. - Isidor Nahaievs'kyi, Starodavnia Ukraïna v svitli istorychnykh pam'iatnykiv. Yorkton, Saskatchewan: Redeemer's Voice Press, 1961. 216 pp.
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