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

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

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

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

A Systematic Analysis of Temporal Trends in the Handgrip Strength of 2,216,320 Children and Adolescents Between 1967 and 2017

Objective: To estimate national and international temporal trends in handgrip strength for children and adolescents, and to examine relationships between trends in handgrip strength and trends in health-related and sociodemographic indicators. Methods: Data were obtained through a systematic search of studies reporting temporal trends in the handgrip strength for apparently healthy 9–17 year-olds, and by examining large national fitness datasets. Temporal trends at the country-sex-age level were estimated by sample-weighted regression models relating the year of testing to mean handgrip strength. International and national trends were estimated by a post-stratified population-weighting procedure. Pearson’s correlations quantified relationships between trends in handgrip strength and trends in health-related/sociodemographic indicators. Results: 2,216,320 children and adolescents from 13 high-, 5 upper-middle-, and 1 low-income countries/special administrative regions between 1967 and 2017 collectively showed a moderate improvement of 19.4% (95%CI: 18.4 to 20.4) or 3.8% per decade (95%CI: 3.6 to 4.0). The international rate of improvement progressively increased over time, with more recent values (post-2000) close to two times larger than those from the 1960s/1970s. Improvements were larger for children (9–12 years) compared to adolescents (13–17 years), and similar for boys and girls. Trends differed between countries, with relationships between trends in handgrip strength and trends in health-related/sociodemographic indicators negligible-to-weak and not statistically significant. Conclusions: There has been a substantial improvement in absolute handgrip strength for children and adolescents since 1967. There is a need for improved international surveillance of handgrip strength, especially in low- and middle-income countries, to more confidently determine true international trends. PROSPERO registration number: CRD42013003657

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