The structure of equivalent 3-separations in a 3-connected matroid

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The structure of 3-connected matroids of path width three

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Fork-decompositions of matroids

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On matroids of branch-width three

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The structure of 3-connected matroids of path-width three

A 3-connected matroid M is sequential or has path width 3 if its ground set E(M) has a sequential ordering, that is, an ordering (e₁,e₂,...,en) such that ({e₁,e₂,...,ek},{ek+1,ek+2,...,en}) is a 3-separation for all k in {3,4,...,n-3}. In this paper, we consider the possible sequential orderings that such a matroid can have. In particular, we prove that M essentially has two fixed ends, each of which is a maximal segment, a maximal cosegment, or a maximal fan. We also identify the possible structures in M that account for different sequential orderings of E(M). These results rely on an earlier paper of the authors that describes the structure of equivalent non-sequential 3-separations in a 3-connected matroid. Those results are extended here to describe the structure of equivalent sequential 3-separations

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

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

Quartet compatibility and the quartet graph

A collection P of leaf-labelled trees is compatible if there exists a single leaf-labelled tree that displays each of the trees in P. Despite its difficulty, determining the compatibility of P is a fundamental task in evolutionary biology. Attractive characterizations in terms of chordal graphs have been previously given for this problem as well as for the problems of (i) determining if there is a unique tree that displays each of the trees in P, that is 'P is definitive and (ii) determining if there is a tree that displays P and has the property that every other tree that displays P is a refinement of it, that is 'P identifies a leaf-labelled tree. In this paper, we describe new characterizations of each of these problems in terms of edge colourings. Furthermore, for an arbitrary leaf-labelled tree 'T, we also determine the minimum number of 'quartets' required to identify 'T, thus correcting a previously published result

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

Risk factors for failure of outpatient parenteral antibiotic therapy (OPAT) in infective endocarditis

Objectives: To identify risk factors for failure of outpatient antibiotic therapy (OPAT) in infective endocarditis (IE). Patients and methods: We identified IE cases managed at a single centre over 12 years from a prospectively maintained database. ‘OPAT failure’ was defined as unplanned readmission or antibiotic switch due to adverse drug reaction or antibiotic resistance. We analysed patient and disease-related risk factors for OPAT failure by univariate and multivariate logistic regression. We also retrospectively collected follow-up data on adverse disease outcome (defined as IE-related death or relapse) and performed Kaplan–Meier survival analysis up to 36 months following OPAT. Results: We identified 80 episodes of OPAT in IE. Failure occurred in 25/80 episodes (31.3%). On multivariate analysis, cardiac or renal failure [pooled OR 7.39 (95% CI 1.84–29.66), P = 0.005] and teicoplanin therapy [OR 8.69 (95% CI 2.01–37.47), P = 0.004] were independently associated with increased OPAT failure. OPAT failure with teicoplanin occurred despite therapeutic plasma levels. OPAT failure predicted adverse disease outcome up to 36 months (P = 0.016 log-rank test). Conclusions: These data caution against selecting patients with endocarditis for OPAT in the presence of cardiac or renal failure and suggest teicoplanin therapy may be associated with suboptimal OPAT outcomes. Alternative regimens to teicoplanin in the OPAT setting should be further investigated