A new proof of the Vorono\"i summation formula

We present a short alternative proof of the Vorono\"i summation formula which plays an important role in Dirichlet's divisor problem and has recently found an application in physics as a trace formula for a Schr\"odinger operator on a non-compact quantum graph \mathfrak{G} [S. Egger n\'e Endres and F. Steiner, J. Phys. A: Math. Theor. 44 (2011) 185202 (44pp)]. As a byproduct we give a new proof of a non-trivial identity for a particular Lambert series which involves the divisor function d(n) and is identical with the trace of the Euclidean wave group of the Laplacian on the infinite graph \mathfrak{G}.Comment: Enlarged version of the published article J. Phys. A: Math. Theor. 44 (2011) 225302 (11pp

A Parameterized Centrality Metric for Network Analysis

A variety of metrics have been proposed to measure the relative importance of nodes in a network. One of these, alpha-centrality [Bonacich, 2001], measures the number of attenuated paths that exist between nodes. We introduce a normalized version of this metric and use it to study network structure, specifically, to rank nodes and find community structure of the network. Specifically, we extend the modularity-maximization method [Newman and Girvan, 2004] for community detection to use this metric as the measure of node connectivity. Normalized alpha-centrality is a powerful tool for network analysis, since it contains a tunable parameter that sets the length scale of interactions. By studying how rankings and discovered communities change when this parameter is varied allows us to identify locally and globally important nodes and structures. We apply the proposed method to several benchmark networks and show that it leads to better insight into network structure than alternative methods.Comment: 11 pages, submitted to Physical Review

Black holes admitting a Freudenthal dual

The quantised charges x of four dimensional stringy black holes may be assigned to elements of an integral Freudenthal triple system whose automorphism group is the corresponding U-duality and whose U-invariant quartic norm Delta(x) determines the lowest order entropy. Here we introduce a Freudenthal duality x -> \tilde{x}, for which \tilde{\tilde{x}}=-x. Although distinct from U-duality it nevertheless leaves Delta(x) invariant. However, the requirement that \tilde{x} be integer restricts us to the subset of black holes for which Delta(x) is necessarily a perfect square. The issue of higher-order corrections remains open as some, but not all, of the discrete U-duality invariants are Freudenthal invariant. Similarly, the quantised charges A of five dimensional black holes and strings may be assigned to elements of an integral Jordan algebra, whose cubic norm N(A) determines the lowest order entropy. We introduce an analogous Jordan dual A*, with N(A) necessarily a perfect cube, for which A**=A and which leaves N(A) invariant. The two dualities are related by a 4D/5D lift.Comment: 32 pages revtex, 10 tables; minor corrections, references adde

Observations on Integral and Continuous U-duality Orbits in N=8 Supergravity

One would often like to know when two a priori distinct extremal black p-brane solutions are in fact U-duality related. In the classical supergravity limit the answer for a large class of theories has been known for some time. However, in the full quantum theory the U-duality group is broken to a discrete subgroup and the question of U-duality orbits in this case is a nuanced matter. In the present work we address this issue in the context of N=8 supergravity in four, five and six dimensions. The purpose of this note is to present and clarify what is currently known about these discrete orbits while at the same time filling in some of the details not yet appearing in the literature. To this end we exploit the mathematical framework of integral Jordan algebras and Freudenthal triple systems. The charge vector of the dyonic black string in D=6 is SO(5,5;Z) related to a two-charge reduced canonical form uniquely specified by a set of two arithmetic U-duality invariants. Similarly, the black hole (string) charge vectors in D=5 are E_{6(6)}(Z) equivalent to a three-charge canonical form, again uniquely fixed by a set of three arithmetic U-duality invariants. The situation in four dimensions is less clear: while black holes preserving more than 1/8 of the supersymmetries may be fully classified by known arithmetic E_{7(7)}(Z) invariants, 1/8-BPS and non-BPS black holes yield increasingly subtle orbit structures, which remain to be properly understood. However, for the very special subclass of projective black holes a complete classification is known. All projective black holes are E_{7(7)}(Z) related to a four or five charge canonical form determined uniquely by the set of known arithmetic U-duality invariants. Moreover, E_{7(7)}(Z) acts transitively on the charge vectors of black holes with a given leading-order entropy.Comment: 43 pages, 8 tables; minor corrections, references added; version to appear in Class. Quantum Gra

An observational cohort study to produce and evaluate an improved tool to screen older women with back pain for osteoporotic vertebral fractures (Vfrac): study protocol.

The aim of this study is to produce an easy to use checklist for general practitioners to complete whenever a woman aged over 65 years with back pain seeks healthcare. This checklist will produce a binary output to determine if the patient should have a radiograph to diagnose vertebral fracture. PURPOSE: People with osteoporotic vertebral fractures are important to be identified as they are at relatively high risk of further fractures. Despite this, less than a third of people with osteoporotic vertebral fractures come to clinical attention due to various reasons including lack of clear triggers to identify who should have diagnostic spinal radiographs. This study aims to produce and evaluate a novel screening tool (Vfrac) for use in older women presenting with back pain in primary care based on clinical triggers and predictors identified previously. This tool will generate a binary output to determine if a radiograph is required. METHODS: The Vfrac study is a two-site, pragmatic, observational cohort study recruiting 1633 women aged over 65 years with self-reported back pain. Participants will be recruited from primary care in two sites. The Vfrac study will use data from two self-completed questionnaires, a simple physical examination, a lateral thoracic and lateral lumbar radiograph and information contained in medical records. RESULTS: The primary objective is to develop an easy-to-use clinical screening tool for identifying older women who are likely to have vertebral fractures. CONCLUSIONS: This article describes the protocol of the Vfrac study; ISRCTN16550671

Relationships between Parental Education and Overweight with Childhood Overweight and Physical Activity in 9-11 Year Old Children: Results from a 12-Country Study

Background: Globally, the high prevalence of overweight and low levels of physical activity among children has serious implications for morbidity and premature mortality in adulthood. Various parental factors are associated with childhood overweight and physical activity. The objective of this paper was to investigate relationships between parental education or overweight, and (i) child overweight, (ii) child physical activity, and (iii) explore household coexistence of overweight, in a large international sample. Methods: Data were collected from 4752 children (9-11 years) as part of the International Study of Childhood Obesity, Lifestyle and the Environment in 12 countries around the world. Physical activity of participating children was assessed by accelerometry, and body weight directly measured. Questionnaires were used to collect parents' education level, weight, and height. Results: Maternal and paternal overweight were positively associated with child overweight. Higher household coexistence of parent-child overweight was observed among overweight children compared to the total sample. There was a positive relationship between maternal education and child overweight in Colombia 1.90 (1.23-2.94) [odds ratio (confidence interval)] and Kenya 4.80 (2.21-10.43), and a negative relationship between paternal education and child overweight in Brazil 0.55 (0.33-0.92) and the USA 0.54 (0.33-0.88). Maternal education was negatively associated with children meeting physical activity guidelines in Colombia 0.53 (0.33-0.85), Kenya 0.35 (0.19-0.63), and Portugal 0.54 (0.31-0.96). Conclusions: Results are aligned with previous studies showing positive associations between parental and child overweight in all countries, and positive relationships between parental education and child overweight or negative associations between parental education and child physical activity in lower economic status countries. Relationships between maternal and paternal education and child weight status and physical activity appear to be related to the developmental stage of different countries. Given these varied relationships, it is crucial to further explore familial factors when investigating child overweight and physical activity

Radiation exposure in X-ray-based imaging techniques used in osteoporosis

Recent advances in medical X-ray imaging have enabled the development of new techniques capable of assessing not only bone quantity but also structure. This article provides (a) a brief review of the current X-ray methods used for quantitative assessment of the skeleton, (b) data on the levels of radiation exposure associated with these methods and (c) information about radiation safety issues. Radiation doses associated with dual-energy X-ray absorptiometry are very low. However, as with any X-ray imaging technique, each particular examination must always be clinically justified. When an examination is justified, the emphasis must be on dose optimisation of imaging protocols. Dose optimisation is more important for paediatric examinations because children are more vulnerable to radiation than adults. Methods based on multi-detector CT (MDCT) are associated with higher radiation doses. New 3D volumetric hip and spine quantitative computed tomography (QCT) techniques and high-resolution MDCT for evaluation of bone structure deliver doses to patients from 1 to 3 mSv. Low-dose protocols are needed to reduce radiation exposure from these methods and minimise associated health risks

Public perceptions of shale gas in the UK : framing effects and decision heuristics

Using two equivalent descriptions of the shale gas development process, we asked individuals to indicate their levels of support as well as their perceptions of the risks and costs involved. In version 1, shale gas development was framed as ‘fracking’, whereas under version 2 it was framed as ‘using hydraulic pressure to extract natural gas from the ground’. We find that individuals’ support for shale gas development is much lower when using the term ‘fracking’ as opposed to the synonymous descriptive term, and moreover, these differences were substantive. Our analysis suggests that these differences appear to be largely the result of different assessments of the risks associated with ‘fracking’ as opposed to ‘using hydraulic pressure to extract natural gas from the ground’. Our proposed explanation for these differences rests on the idea that shale gas development is a technical and complex process and many individuals will be bounded by the rationality of scientific knowledge when it comes to understanding this process. In turn, individuals may be relying on simple decision heuristics shaped by the way this issue is framed by the media and other interested parties which may constrain meaningful discourse on this topic with the public. Our findings also highlight some of the potential pitfalls when it comes to relying on survey research for assessing the public’s views towards complex environmental issues