Bounding the distinguishing number of infinite graphs and permutation groups

A group of permutations G of a set V is k-distinguishable if there exists a partition of V into k cells such that only the identity permutation in G fixes setwise all of the cells of the partition. The least cardinal number k such that (G, V) is k-distinguishable is its distinguishing number, D(G, V). In particular, a graph X is k-distinguishable if its automorphism group Aut(X) satisfies D(Aut(X), VX) ≤ k. Various results in the literature demonstrate that when an infinite graph fails to have some property, then often some finite subgraph is similarly deficient. In this paper we show first that whenever an infinite connected graph X is not k-distinguishable (for a given cardinal k), then it contains a ball of finite radius whose distinguishing number is at least k. Moreover, this lower bound cannot be sharpened, since for any integer k ≥ 3 there exists an infinite, locally finite, connected graph X that is not k-distinguishable but in which every ball of finite radius is k-distinguishable. In the second half of this paper we show that a large distinguishing number for an imprimitive permutation group G is traceable to a high distinguishing number either of a block of imprimitivity or of the action induced by G on the corresponding system of imprimitiv ity. An immediate application is to automorphism groups of infinite imprimitive graphs. These results are companion to the study of the distinguishing number of infinite primitive groups and graphs in a previous paper by the authors together with T. W. Tucker

Distinguishability of infinite groups and graphs

The distinguishing number of a group G acting faithfully on a set V is the least number of colors needed to color the elements of V so that no non-identity element of the group preserves the coloring. The distinguishing number of a graph is the distinguishing number of its automorphism group acting on its vertex set. A connected graph Gamma is said to have connectivity 1 if there exists a vertex alpha \in V\Gamma such that Gamma \setminus \{\alpha\} is not connected. For alpha \in V, an orbit of the point stabilizer G_\alpha is called a suborbit of G. We prove that every nonnull, primitive graph with infinite diameter and countably many vertices has distinguishing number 2. Consequently, any nonnull, infinite, primitive, locally finite graph is 2-distinguishable; so, too, is any infinite primitive permutation group with finite suborbits. We also show that all denumerable vertex-transitive graphs of connectivity 1 and all Cartesian products of connected denumerable graphs of infinite diameter have distinguishing number 2. All of our results follow directly from a versatile lemma which we call The Distinct Spheres Lemma

The burden of antenatal heart disease in South Africa: a systematic review

<p>Abstract</p> <p>Background</p> <p>Maternal mortality in South Africa is rising, and heart conditions currently account for 41 per cent of indirect causes of deaths. Little is known about the burden of heart disease in pregnant South Africans.</p> <p>Methods</p> <p>We systematically reviewed the contemporary epidemiology and peripartum outcomes of heart disease in South African women attending antenatal care. Searches were performed in PubMed, ISI Web of Science, the EBSCO Africa-Wide database, the South African Union Catalogue, and the Current and Completed Research database (South Africa). References of included articles were also hand-searched. Studies reporting epidemiologic data on antenatal heart disease in South Africa were included. Data on morbidity and mortality were also collected.</p> <p>Results</p> <p>Seven studies were included in the systematic review. The prevalence of heart disease ranged from 123 to 943 per 100,000 deliveries, with a median prevalence of 616 per 100,000. Rheumatic valvular lesions were the commonest abnormalities, although cardiomyopathies were disproportionately high in comparison with other developing countries. Peripartum case-fatality rates were as high as 9.5 per cent in areas with limited access to care. The most frequent complications were pulmonary oedema, thromboembolism, and major bleeding with warfarin use. Perinatal mortality ranged from 8.9 to 23.8 per cent, whilst mitral lesions were associated with low birth weight. Meta-analysis could not be performed due to clinical and statistical heterogeneity of the included studies.</p> <p>Conclusion</p> <p>Approximately 0.6 per cent of pregnant South Africans have pre-existing cardiac abnormalities, with rheumatic lesions being the commonest. Maternal and perinatal morbidity and mortality continue to be very high. We conclude this review by summarising limitations of the current literature and recommending standard reporting criteria for future reports.</p

Interpreting structural geometry in fold-thrust belts : Why style matters

The Fold-Thrust Research Group is funded by InterOil, Oil Search and Santos. We thank David Ferrill and Chris Morley for robust reviews of an early draft of this paper. We also thank Bill Dunne for his patience and his usual editorial rigor – although of course authors alone are responsible for the views expressed here.Peer reviewedPostprin

Fold-thrust structures : where have all the buckles gone?

Special publication title: Folding and Fracturing of Rocks: 50 Years of Research since the Seminal Text Book of J. G. Ramsay We dedicate the paper to the memory of Martin Casey (1948-2008), who did much through good-humored argument to ensure that buckling ideas were not lost to what he called “the Ramping Club” (the thrust belt community). The Fold – Thrust Research Group has been funded by InterOil, OilSearch and Santos. We thank Paul Griffiths and anonymous referee for comments together with Hermann Lebit for scientific editing. The views expressed here of course remain those of the authors.Peer reviewedPostprin

Natural fracture patterns at Swift Reservoir anticline, NW Montana : the influence of structural position and lithology from multiple observation scales

Acknowledgements We gratefully acknowledge constructive reviews by Amerigo Corradetti and an anonymous reviewer and thank Stefano Tavani for editorial handling. Adam J. Cawood is grateful to David Ferrill, Kevin Smart, and Paul Gillespie for helpful conversations about fracture patterns, although the data and interpretations shown here are of course the sole responsibility of the authors. This study was carried out as part of a University of Aberdeen doctoral programme supported by the Natural Environment Research Council (NERC) Centre for Doctoral Training in Oil and Gas. Additional funding for fieldwork was provided by the University of Aberdeen Fold–Thrust Research Group. Petroleum Experts (formerly Midland Valley Exploration) is acknowledged for allowing the academic use of Move 2016.1 software. Financial support This research has been supported by the Natural Environment Research Council (grant no. NE/M00578X/1).Peer reviewedPublisher PD