Fork-decompositions of matroids

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On matroids of branch-width three

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The structure of the 3-separations of 3-connected matroids II

The authors showed in an earlier paper that there is a tree that displays, up to a natural equivalence, all non-trivial 3-separations of a 3-connected matroid. The purpose of this paper is to show that if certain natural conditions are imposed on the tree, then it has a uniqueness property. In particular; suppose that, from every pair of edges that meet at a degree-2 vertex and have their other ends of degree at least three, one edge is contracted. Then the resulting tree is unique

Deposition of general ellipsoidal particles

We present a systematic overview of granular deposits composed of ellipsoidal particles with different particle shapes and size polydispersities. We study the density and anisotropy of such deposits as functions of size polydispersity and two shape parameters that fully describe the shape of a general ellipsoid. Our results show that, while shape influences significantly the macroscopic properties of the deposits, polydispersity plays apparently a secondary role. The density attains a maximum for a particular family of non-symmetrical ellipsoids, larger than the density observed for prolate or oblate ellipsoids. As for anisotropy measures, the contact forces show are increasingly preferred along the vertical direction as the shape of the particles deviates for a sphere. The deposits are constructed by means of an efficient molecular dynamics method, where the contact forces are efficiently and accurately computed. The main results are discussed in the light of applications for porous media models and sedimentation processes.Comment: 7 pages, 8 figure

The structure of the 3-separations of 3-connected matroids

Special Issue Dedicated to Professor W.T. TutteTutte defined a k-separation of a matroid M to be a partition (A,B) of the ground set of M such that ∣A∣,∣B∣ ≥ k and r(A) + r(B) − r(M) < k. If, for all m < n, the matroid M has no m-separations, then M is n-connected. Earlier, Whitney showed that (A,B) is a 1-separation of M if and only if A is a union of 2-connected components of M. When M is 2-connected, Cunningham and Edmonds gave a tree decomposition of M that displays all of its 2-separations. When M is 3-connected, this paper describes a tree decomposition of M that displays, up to a certain natural equivalence, all non-trivial 3-separations of M

Adventures in Invariant Theory

We provide an introduction to enumerating and constructing invariants of group representations via character methods. The problem is contextualised via two case studies arising from our recent work: entanglement measures, for characterising the structure of state spaces for composite quantum systems; and Markov invariants, a robust alternative to parameter-estimation intensive methods of statistical inference in molecular phylogenetics.Comment: 12 pp, includes supplementary discussion of example

The comparative clinical course of pregnant and non-pregnant women hospitalised with influenza A(H1N1)pdm09 infection

Introduction: The Influenza Clinical Information Network (FLU-CIN) was established to gather detailed clinical and epidemiological information about patients with laboratory confirmed A(H1N1)pdm09 infection in UK hospitals. This report focuses on the clinical course and outcomes of infection in pregnancy.Methods: A standardised data extraction form was used to obtain detailed clinical information from hospital case notes and electronic records, for patients with PCR-confirmed A(H1N1)pdm09 infection admitted to 13 sentinel hospitals in five clinical 'hubs' and a further 62 non-sentinel hospitals, between 11th May 2009 and 31st January 2010.Outcomes were compared for pregnant and non-pregnant women aged 15-44 years, using univariate and multivariable techniques.Results: Of the 395 women aged 15-44 years, 82 (21%) were pregnant; 73 (89%) in the second or third trimester. Pregnant women were significantly less likely to exhibit severe respiratory distress at initial assessment (OR?=?0.49 (95% CI: 0.30-0.82)), require supplemental oxygen on admission (OR?=?0.40 (95% CI: 0.20-0.80)), or have underlying co-morbidities (p-trend &lt;0.001). However, they were equally likely to be admitted to high dependency (Level 2) or intensive care (Level 3) and/or to die, after adjustment for potential confounders (adj. OR?=?0.93 (95% CI: 0.46-1.92). Of 11 pregnant women needing Level 2/3 care, 10 required mechanical ventilation and three died.Conclusions: Since the expected prevalence of pregnancy in the source population was 6%, our data suggest that pregnancy greatly increased the likelihood of hospital admission with A(H1N1)pdm09. Pregnant women were less likely than non-pregnant women to have respiratory distress on admission, but severe outcomes were equally likely in both groups

A Note on Encodings of Phylogenetic Networks of Bounded Level

Driven by the need for better models that allow one to shed light into the question how life's diversity has evolved, phylogenetic networks have now joined phylogenetic trees in the center of phylogenetics research. Like phylogenetic trees, such networks canonically induce collections of phylogenetic trees, clusters, and triplets, respectively. Thus it is not surprising that many network approaches aim to reconstruct a phylogenetic network from such collections. Related to the well-studied perfect phylogeny problem, the following question is of fundamental importance in this context: When does one of the above collections encode (i.e. uniquely describe) the network that induces it? In this note, we present a complete answer to this question for the special case of a level-1 (phylogenetic) network by characterizing those level-1 networks for which an encoding in terms of one (or equivalently all) of the above collections exists. Given that this type of network forms the first layer of the rich hierarchy of level-k networks, k a non-negative integer, it is natural to wonder whether our arguments could be extended to members of that hierarchy for higher values for k. By giving examples, we show that this is not the case