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

Hippocampal Infusion of Zeta Inhibitory Peptide Impairs Recent, but Not Remote, Recognition Memory in Rats.

Spatial memory in rodents can be erased following the infusion of zeta inhibitory peptide (ZIP) into the dorsal hippocampus via indwelling guide cannulas. It is believed that ZIP impairs spatial memory by reversing established late-phase long-term potentiation (LTP). However, it is unclear whether other forms of hippocampus-dependent memory, such as recognition memory, are also supported by hippocampal LTP. In the current study, we tested recognition memory in rats following hippocampal ZIP infusion. In order to combat the limited targeting of infusions via cannula, we implemented a stereotaxic approach for infusing ZIP throughout the dorsal, intermediate, and ventral hippocampus. Rats infused with ZIP 3-7 days after training on the novel object recognition task exhibited impaired object recognition memory compared to control rats (those infused with aCSF). In contrast, rats infused with ZIP 1 month after training performed similar to control rats. The ability to form new memories after ZIP infusions remained intact. We suggest that enhanced recognition memory for recent events is supported by hippocampal LTP, which can be reversed by hippocampal ZIP infusion

Helical Tubes in Crowded Environments

When placed in a crowded environment, a semi-flexible tube is forced to fold so as to make a more compact shape. One compact shape that often arises in nature is the tight helix, especially when the tube thickness is of comparable size to the tube length. In this paper we use an excluded volume effect to model the effects of crowding. This gives us a measure of compactness for configurations of the tube, which we use to look at structures of the semi-flexible tube that minimize the excluded volume. We focus most of our attention on the helix and which helical geometries are most compact. We found that helices of specific pitch to radius ratio 2.512 to be optimally compact. This is the same geometry that minimizes the global curvature of the curve defining the tube. We further investigate the effects of adding a bending energy or multiple tubes to begin to explore the more complete space of possible geometries a tube could form.Comment: 10 page

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

The VLA Low-frequency Sky Survey

The Very Large Array (VLA) Low-frequency Sky Survey (VLSS) has imaged 95% of the 3*pi sr of sky north of declination = -30 degrees at a frequency of 74 MHz (4 meter wavelength). The resolution is 80" (FWHM) throughout, and the typical RMS noise level is ~0.1 Jy/beam. The typical point-source detection limit is 0.7 Jy/beam and so far nearly 70,000 sources have been catalogued. This survey used the 74 MHz system added to the VLA in 1998. It required new imaging algorithms to remove the large ionospheric distortions at this very low frequency throughout the entire ~11.9 degree field of view. This paper describes the observation and data reduction methods used for the VLSS and presents the survey images and source catalog. All of the calibrated images and the source catalog are available online (http://lwa.nrl.navy.mil/VLSS) for use by the astronomical community.Comment: 53 pages, including 3 tables and 15 figures. Has been accepted for publication in the Astronomical Journa

Certification of Bounds of Non-linear Functions: the Templates Method

The aim of this work is to certify lower bounds for real-valued multivariate functions, defined by semialgebraic or transcendental expressions. The certificate must be, eventually, formally provable in a proof system such as Coq. The application range for such a tool is widespread; for instance Hales' proof of Kepler's conjecture yields thousands of inequalities. We introduce an approximation algorithm, which combines ideas of the max-plus basis method (in optimal control) and of the linear templates method developed by Manna et al. (in static analysis). This algorithm consists in bounding some of the constituents of the function by suprema of quadratic forms with a well chosen curvature. This leads to semialgebraic optimization problems, solved by sum-of-squares relaxations. Templates limit the blow up of these relaxations at the price of coarsening the approximation. We illustrate the efficiency of our framework with various examples from the literature and discuss the interfacing with Coq.Comment: 16 pages, 3 figures, 2 table

‘Distancers’ and ‘non-distancers’? The potential social psychological impact of moralizing COVID-19 mitigating practices on sustained behaviour change

COVID‐19 mitigating practices such as ‘hand‐washing’, ‘social distancing’, or ‘social isolating’ are constructed as ‘moral imperatives’, required to avert harm to oneself and others. Adherence to COVID‐19 mitigating practices is presently high among the general public, and stringent lockdown measures supported by legal and policy intervention have facilitated this. In the coming months, however, as rules are being relaxed and individuals become less strict, and thus, the ambiguity in policy increases, the maintenance of recommended social distancing norms will rely on more informal social interactional processes. We argue that the moralization of these practices, twinned with relaxations of policy, may likely cause interactional tension between those individuals who do vs. those who do not uphold social distancing in the coming months: that is, derogation of those who adhere strictly to COVID‐19 mitigating practices and group polarization between ‘distancers’ and ‘non‐distancers’. In this paper, we explore how and why these processes might come to pass, their impact on an overall societal response to COVID‐19, and the need to factor such processes into decisions regarding how to lift restriction

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