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

Impact of wine bottle and glass sizes on wine consumption at home: a within- and between- households randomized controlled trial

Background and aims: Reducing alcohol consumption across populations would decrease the risk of a range of diseases, including many cancers, cardiovascular disease and Type 2 diabetes. The aim of the current study was to estimate the impact of using smaller bottles (37.5- versus 75-cl) and glasses (290 versus 370 ml) on consuming wine at home. Design: Randomized controlled trial of households with cross-over randomization to bottle size and parallel randomization to glass size. Setting: UK households. Participants: A total of 260 households consuming at least two 75-cl bottles of wine each week, recruited from the general population through a research agency. The majority consisted of adults who were white and of higher socio-economic position. Intervention: Households were randomized to the order in which they purchased wine in 37.5- or 75-cl bottles, to consume during two 14-day intervention periods, and further randomized to receive smaller (290 ml) or larger (350 ml) glasses to use during both intervention periods. Measurements: Volume (ml) of study wine consumed at the end of each 14-day intervention period, measured using photographs of purchased bottles, weighed on study scales. Findings: Of the randomized households, 217 of 260 (83%) completed the study as per protocol and were included in the primary analysis. There was weak evidence that smaller bottles reduced consumption: after accounting for pre-specified covariates, households consumed on average 145.7 ml (3.6%) less wine when drinking from smaller bottles than from larger bottles [95% confidence intervals (CI) = –335.5 to 43. ml; −8.3 to 1.1%; P = 0.137; Bayes factor (BF) = 2.00]. The evidence for the effect of smaller glasses was stronger: households consumed on average 253.3 ml (6.5%) less wine when drinking from smaller glasses than from larger glasses (95% CI = –517 to 10 ml; −13.2 to 0.3%; P = 0.065; BF = 2.96). Conclusions: Using smaller glasses to drink wine at home may reduce consumption. Greater uncertainty remains around the possible effect of drinking from smaller bottles

Impact of health warning labels and calorie labels on selection and purchasing of alcoholic and non-alcoholic drinks: A randomized controlled trial

AIMS: To estimate the impact on selection and actual purchasing of (a) health warning labels (text-only and image-and-text) on alcoholic drinks and (b) calorie labels on alcoholic and non-alcoholic drinks. DESIGN: Parallel-groups randomised controlled trial. SETTING: Drinks were selected in a simulated online supermarket, before being purchased in an actual online supermarket. PARTICIPANTS: Adults in England and Wales who regularly consumed and purchased beer or wine online (n = 651). Six hundred and eight participants completed the study and were included in the primary analysis. INTERVENTIONS: Participants were randomized to one of six groups in a between-subjects three [health warning labels (HWLs) (i): image-and-text HWL; (ii) text-only HWL; (iii) no HWL] × 2 (calorie labels: present versus absent) factorial design (n per group 103-113). MEASUREMENTS: The primary outcome measure was the number of alcohol units selected (with intention to purchase); secondary outcomes included alcohol units purchased and calories selected and purchased. There was no time limit for selection. For purchasing, participants were directed to purchase their drinks immediately (although they were allowed up to 2 weeks to do so). FINDINGS: There was no evidence of main effects for either (a) HWLs or (b) calorie labels on the number of alcohol units selected (HWLs: F(2,599)  = 0.406, P = 0.666; calorie labels: F(1,599)  = 0.002, P = 0.961). There was also no evidence of an interaction between HWLs and calorie labels, and no evidence of an overall difference on any secondary outcomes. In pre-specified subgroup analyses comparing the 'calorie label only' group (n = 101) with the 'no label' group (n = 104) there was no evidence that calorie labels reduced the number of calories selected (unadjusted means: 1913 calories versus 2203, P = 0.643). Among the 75% of participants who went on to purchase drinks, those in the 'calorie label only' group (n = 74) purchased fewer calories than those in the 'no label' group (n = 79) (unadjusted means: 1532 versus 2090, P = 0.028). CONCLUSIONS: There was no evidence that health warning labels reduced the number of alcohol units selected or purchased in an online retail context. There was some evidence suggesting that calorie labels on alcoholic and non-alcoholic drinks may reduce calories purchased from both types of drinks

Generalized spacetimes defined by cubic forms and the minimal unitary realizations of their quasiconformal groups

We study the symmetries of generalized spacetimes and corresponding phase spaces defined by Jordan algebras of degree three. The generic Jordan family of formally real Jordan algebras of degree three describe extensions of the Minkowskian spacetimes by an extra "dilatonic" coordinate, whose rotation, Lorentz and conformal groups are SO(d-1), SO(d-1,1) XSO(1,1) and SO(d,2)XSO(2,1), respectively. The generalized spacetimes described by simple Jordan algebras of degree three correspond to extensions of Minkowskian spacetimes in the critical dimensions (d=3,4,6,10) by a dilatonic and extra (2,4,8,16) commuting spinorial coordinates, respectively. The Freudenthal triple systems defined over these Jordan algebras describe conformally covariant phase spaces. Following hep-th/0008063, we give a unified geometric realization of the quasiconformal groups that act on their conformal phase spaces extended by an extra "cocycle" coordinate. For the generic Jordan family the quasiconformal groups are SO(d+2,4), whose minimal unitary realizations are given. The minimal unitary representations of the quasiconformal groups F_4(4), E_6(2), E_7(-5) and E_8(-24) of the simple Jordan family were given in our earlier work hep-th/0409272.Comment: A typo in equation (37) corrected and missing titles of some references added. Version to be published in JHEP. 38 pages, latex fil

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

Dynamics of magnetic assembly of binary colloidal structures

"This is an author-created, un-copyedited version of an article published in Europhysics Letters. IOP Publishing Ltd is not responsible for any errors or omissions in this version of the manuscript or any version derived from it. The Version of Record is available online at 10.1209/0295-5075/111/37002."Magnetic field (MF)-directed assembly of colloidal particles provides a step towards the bottom-up manufacturing of smart materials whose properties can be precisely modulated by non-contact forces. Here, we study the MF-directed assembly in binary colloids made up of strong ferromagnetic and diamagnetic microparticles dispersed in ferrofluids. We present observations of the aggregation of pairs and small groups of particles to build equilibrium assemblies. We also develop a theoretical model capable of solving the aggregation dynamics and predicting the particle trajectories, a key factor to understand the physics governing the MF-directed assembly.The project MINECO FIS2013-41821-R (Spain) is acknowledged for financial support

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

On Symmetries of Extremal Black Holes with One and Two Centers

After a brief introduction to the Attractor Mechanism, we review the appearance of groups of type E7 as generalized electric-magnetic duality symmetries in locally supersymmetric theories of gravity, with particular emphasis on the symplectic structure of fluxes in the background of extremal black hole solutions, with one or two centers. In the latter case, the role of an "horizontal" symmetry SL(2,R) is elucidated by presenting a set of two-centered relations governing the structure of two-centered invariant polynomials.Comment: 1+13 pages, 2 Tables. Based on Lectures given by SF and AM at the School "Black Objects in Supergravity" (BOSS 2011), INFN - LNF, Rome, Italy, May 9-13 201

A dimensionally continued Poisson summation formula

We generalize the standard Poisson summation formula for lattices so that it operates on the level of theta series, allowing us to introduce noninteger dimension parameters (using the dimensionally continued Fourier transform). When combined with one of the proofs of the Jacobi imaginary transformation of theta functions that does not use the Poisson summation formula, our proof of this generalized Poisson summation formula also provides a new proof of the standard Poisson summation formula for dimensions greater than 2 (with appropriate hypotheses on the function being summed). In general, our methods work to establish the (Voronoi) summation formulae associated with functions satisfying (modular) transformations of the Jacobi imaginary type by means of a density argument (as opposed to the usual Mellin transform approach). In particular, we construct a family of generalized theta series from Jacobi theta functions from which these summation formulae can be obtained. This family contains several families of modular forms, but is significantly more general than any of them. Our result also relaxes several of the hypotheses in the standard statements of these summation formulae. The density result we prove for Gaussians in the Schwartz space may be of independent interest.Comment: 12 pages, version accepted by JFAA, with various additions and improvement

The prevalence of vertebral fracture amongst patients presenting with non-vertebral fractures

INTRODUCTION: Despite vertebral fracture being a significant risk factor for further fracture, vertebral fractures are often unrecognised. A study was therefore conducted to determine the proportion of patients presenting with a non-vertebral fracture who also have an unrecognised vertebral fracture. METHODS: Prospective study of patients presenting with a non-vertebral fracture in South Glasgow who underwent DXA evaluation with vertebral morphometry (MXA) from DV5/6 to LV4/5. Vertebral deformities (consistent with fracture) were identified by direct visualisation using the Genant semi-quantitative grading scale. RESULTS: Data were available for 337 patients presenting with low trauma non-vertebral fracture; 261 were female. Of all patients, 10.4% were aged 50–64 years, 53.2% were aged 65–74 years and 36.2% were aged 75 years or over. According to WHO definitions, 35.0% of patients had normal lumbar spine BMD (T-score −1 or above), 37.4% were osteopenic (T-score −1.1 to −2.4) and 27.6% osteoporotic (T-score −2.5 or lower). Humerus (n=103, 31%), radius–ulna (n=90, 27%) and hand/foot (n=53, 16%) were the most common fractures. For 72% of patients (n=241) the presenting fracture was the first low trauma fracture to come to clinical attention. The overall prevalence of vertebral deformity established by MXA was 25% (n=83); 45% (n=37) of patients with vertebral deformity had deformities of more than one vertebra. Of the patients with vertebral deformity and readable scans for grading, 72.5% (58/80) had deformities of grade 2 or 3. Patients presenting with hip fracture, or spine T-score ≤−2.5, or low BMI, or with more than one prior non-vertebral fracture were all significantly more likely to have evidence of a prevalent vertebral deformity (p<0.05). However, 19.8% of patients with an osteopenic T-score had a vertebral deformity (48% of which were multiple), and 16.1% of patients with a normal T-score had a vertebral deformity (26.3% of which were multiple). Following non-vertebral fracture, some guidelines suggest that anti-resorptive therapy should be reserved for patients with DXA-proven osteoporosis. However, patients who have one or more prior vertebral fractures (prevalent at the time of their non-vertebral fracture) would also become candidates for anti-resorptive therapy—which would have not been the case had their vertebral fracture status not been known. Overall in this study, 8.9% of patients are likely to have had a change in management by virtue of their underlying vertebral deformity status. In other words, 11 patients who present with a non-vertebral fracture would need to undergo vertebral morphometry in order to identify one patient who ought to be managed differently. CONCLUSIONS: Our results support the recommendation to perform vertebral morphometry in patients who are referred for DXA after experiencing a non-vertebral fracture. Treatment decisions will then better reflect any given patient’s future absolute fracture risk. The 'Number Needed to Screen' if vertebral morphometry is used in this way would be seven to identify one patient with vertebral deformity, and 14 to identify one patient with two or more vertebral deformities. Although carrying out MXA will increase radiation exposure for the patient, this increased exposure is significantly less than would be obtained if X-rays of the dorso-lumbar spine were obtained