Formal Verification of Nonlinear Inequalities with Taylor Interval Approximations

We present a formal tool for verification of multivariate nonlinear inequalities. Our verification method is based on interval arithmetic with Taylor approximations. Our tool is implemented in the HOL Light proof assistant and it is capable to verify multivariate nonlinear polynomial and non-polynomial inequalities on rectangular domains. One of the main features of our work is an efficient implementation of the verification procedure which can prove non-trivial high-dimensional inequalities in several seconds. We developed the verification tool as a part of the Flyspeck project (a formal proof of the Kepler conjecture). The Flyspeck project includes about 1000 nonlinear inequalities. We successfully tested our method on more than 100 Flyspeck inequalities and estimated that the formal verification procedure is about 3000 times slower than an informal verification method implemented in C++. We also describe future work and prospective optimizations for our method.Comment: 15 page

Dense Packings of Superdisks and the Role of Symmetry

We construct the densest known two-dimensional packings of superdisks in the plane whose shapes are defined by |x^(2p) + y^(2p)| <= 1, which contains both convex-shaped particles (p > 0.5, with the circular-disk case p = 1) and concave-shaped particles (0 < p < 0.5). The packings of the convex cases with p 1 generated by a recently developed event-driven molecular dynamics (MD) simulation algorithm [Donev, Torquato and Stillinger, J. Comput. Phys. 202 (2005) 737] suggest exact constructions of the densest known packings. We find that the packing density (covering fraction of the particles) increases dramatically as the particle shape moves away from the "circular-disk" point (p = 1). In particular, we find that the maximal packing densities of superdisks for certain p 6 = 1 are achieved by one of the two families of Bravais lattice packings, which provides additional numerical evidence for Minkowski's conjecture concerning the critical determinant of the region occupied by a superdisk. Moreover, our analysis on the generated packings reveals that the broken rotational symmetry of superdisks influences the packing characteristics in a non-trivial way. We also propose an analytical method to construct dense packings of concave superdisks based on our observations of the structural properties of packings of convex superdisks.Comment: 15 pages, 8 figure

BISON Theory Manual The Equations behind Nuclear F

BISON is a finite element-based nuclear fuel perf

Role of the medial prefrontal cortex in the effects of rapid acting antidepressants on decision-making biases in rodents

Major depressive disorder is a significant and costly cause of global disability. Until the discovery of the rapid acting antidepressant (RAAD) effects of ketamine, treatments were limited to drugs that have delayed clinical benefits. The mechanism of action of ketamine is currently unclear but one hypothesis is that it may involve neuropsychological effects mediated through modulation of affective biases (where cognitive processes such as learning and memory and decision-making are modified by emotional state). Previous work has shown that affective biases in a rodent decision-making task are differentially altered by ketamine, compared to conventional, delayed onset antidepressants. This study sought to further investigate these effects by comparing ketamine with other NMDA antagonists using this decision-making task. We also investigated the subtype selective GluN2B antagonist, CP-101,606 and muscarinic antagonist scopolamine which have both been shown to have RAAD effects. Both CP-101,606 and scopolamine induced similar positive biases in decision-making to ketamine, but the same effects were not seen with other NMDA antagonists. Using targeted medial prefrontal cortex (mPFC) infusions, these effects were localised to the mPFC. In contrast, the GABA(A) agonist, muscimol, induced general disruptions to behaviour. These data suggest that ketamine and other RAADs mediate a specific effect on affective bias which involves the mPFC. Non-ketamine NMDA antagonists lacked efficacy and we also found that temporary inactivation of the mPFC did not fully recapitulate the effects of ketamine, suggesting a specific mechanism

Radio Frequency Spectra of 388 Bright 74 MHz Sources

As a service to the community, we have compiled radio frequency spectra from the literature for all sources within the VLA Low Frequency Sky Survey (VLSS) that are brighter than 15 Jy at 74 MHz. Over 160 references were used to maximize the amount of spectral data used in the compilation of the spectra, while also taking care to determine the corrections needed to put the flux densities from all reference on the same absolute flux density scale. With the new VLSS data, we are able to vastly improve upon previous efforts to compile spectra of bright radio sources to frequencies below 100 MHz because (1) the VLSS flux densities are more reliable than those from some previous low frequency surveys and (2) the VLSS covers a much larger area of the sky (declination >-30 deg.) than many other low frequency surveys (e.g., the 8C survey). In this paper, we discuss how the spectra were constructed and how parameters quantifying the shapes of the spectra were derived. Both the spectra and the shape parameters are made available here to assist in the calibration of observations made with current and future low frequency radio facilities.Comment: Accepted to ApJ

Superlattice Magnetophonon Resonances in Strongly Coupled InAs/GaSb Superlattices

We report an experimental study of miniband magnetoconduction in semiconducting InAs/GaSb superlattices. For samples with miniband widths below the longitudinal optical phonon energy we identify a new superlattice magnetophonon resonance (SLMPR) caused by resonant scattering of electrons across the mini-Brillouin zone. This new resonant feature arises directly from the drift velocity characteristics of the superlattice dispersion and total magnetic quantisation of the superlattice Landau level minibands.Comment: 9 pages, 8 figures, submitted to Phys. Rev.

Is there really a debris disc around ζ2 Reticuli\zeta^2\,\mathrm{Reticuli} ?

The presence of a debris disc around the Gyr-old solar-type star $\zeta^2\,\mathrm{Reticuli}$ was suggested by the $\mathit{Spitzer}$ infrared excess detection. Follow-up observations with $\mathit{Herschel}$/PACS revealed a double-lobed feature, that displayed asymmetries both in brightness and position. Therefore, the disc was thought to be edge-on and significantly eccentric. Here we present ALMA/ACA observations in Band 6 and 7 which unambiguously reveal that these lobes show no common proper motion with $\zeta^2\,\mathrm{Reticuli}$. In these observations, no flux has been detected around $\zeta^2\,\mathrm{Reticuli}$ that exceeds the $3\sigma$ levels. We conclude that surface brightness upper limits of a debris disc around $\zeta^2\,\mathrm{Reticuli}$ are $5.7\,\mathrm{\mu Jy/arcsec^2}$ at 1.3 mm, and $26\,\mathrm{\mu Jy/arcsec^2}$ at 870 microns. Our results overall demonstrate the capability of the ALMA/ACA to follow-up $\mathit{Herschel}$ observations of debris discs and clarify the effects of background confusion.Comment: 6 pages, 2 figures, 2 table

Dense sphere packings from optimized correlation functions

Elementary smooth functions (beyond contact) are employed to construct pair correlation functions that mimic jammed disordered sphere packings. Using the g2-invariant optimization method of Torquato and Stillinger [J. Phys. Chem. B 106, 8354, 2002], parameters in these functions are optimized under necessary realizability conditions to maximize the packing fraction phi and average number of contacts per sphere Z. A pair correlation function that incorporates the salient features of a disordered packing and that is smooth beyond contact is shown to permit a phi of 0.6850: this value represents a 45% reduction in the difference between the maximum for congruent hard spheres in three dimensions, pi/sqrt{18} ~ 0.7405, and 0.64, the approximate fraction associated with maximally random jammed (MRJ) packings in three dimensions. We show that, surprisingly, the continued addition of elementary functions consisting of smooth sinusoids decaying as r^{-4} permits packing fractions approaching pi/sqrt{18}. A translational order metric is used to discriminate between degrees of order in the packings presented. We find that to achieve higher packing fractions, the degree of order must increase, which is consistent with the results of a previous study [Torquato et al., Phys. Rev. Lett. 84, 2064, 2000].Comment: 26 pages, 9 figures, 1 table; added references, fixed typos, simplified argument and discussion in Section IV

Deep observations of the Super-CLASS super-cluster at 325 MHz with the GMRT: the low-frequency source catalogue

We present the results of 325 MHz GMRT observations of a super-cluster field, known to contain five Abell clusters at redshift $z \sim 0.2$. We achieve a nominal sensitivity of $34\,\mu$Jy beam$^{-1}$ toward the phase centre. We compile a catalogue of 3257 sources with flux densities in the range $183\,\mu\rm{Jy}\,-\,1.5\,\rm{Jy}$ within the entire $\sim 6.5$ square degree field of view. Subsequently, we use available survey data at other frequencies to derive the spectral index distribution for a sub-sample of these sources, recovering two distinct populations -- a dominant population which exhibit spectral index trends typical of steep-spectrum synchrotron emission, and a smaller population of sources with typically flat or rising spectra. We identify a number of sources with ultra-steep spectra or rising spectra for further analysis, finding two candidate high-redshift radio galaxies and three gigahertz-peaked-spectrum radio sources. Finally, we derive the Euclidean-normalised differential source counts using the catalogue compiled in this work, for sources with flux densities in excess of $223 \, \mu$Jy. Our differential source counts are consistent with both previous observations at this frequency and models of the low-frequency source population. These represent the deepest source counts yet derived at 325 MHz. Our source counts exhibit the well-known flattening at mJy flux densities, consistent with an emerging population of star-forming galaxies; we also find marginal evidence of a downturn at flux densities below $308 \, \mu$Jy, a feature so far only seen at 1.4 GHz.Comment: 25 pages, 18 figures, 10 tables. Accepted for publication in MNRA