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

Potential and timescales for oxygen depletion in coastal upwelling systems: A box-model analysis

A simple box model is used to examine oxygen depletion in an idealized ocean-margin upwelling system. Near-bottom oxygen depletion is controlled by a competition between flushing with oxygenated offshore source waters and respiration of particulate organic matter produced near the surface and retained near the bottom. Upwelling-supplied nutrients are consumed in the surface box, and some surface particles sink to the bottom where they respire, consuming oxygen. Steady states characterize the potential for hypoxic near-bottom oxygen depletion; this potential is greatest for faster sinking rates, and largely independent of production timescales except in that faster production allows faster sinking. Timescales for oxygen depletion depend on upwelling and productivity differently, however, as oxygen depletion can only be reached in meaningfully short times when productivity is rapid. Hypoxia thus requires fast production, to capture upwelled nutrients, and fast sinking, to deliver the respiration potential to model bottom waters. Combining timescales allows generalizations about tendencies toward hypoxia. If timescales of sinking are comparable to or smaller than the sum of those for respiration and flushing, the steady state will generally be hypoxic, and results indicate optimal timescales and conditions exist to generate hypoxia. For example, the timescale for approach to hypoxia lengthens with stronger upwelling, since surface particle and nutrient are shunted off-shelf, in turn reducing subsurface respiration and oxygen depletion. This suggests that if upwelling winds intensify with climate change the increased forcing could offer mitigation of coastal hypoxia, even as the oxygen levels in upwelled source waters decline

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

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

Phase field approach to optimal packing problems and related Cheeger clusters

In a fixed domain of $\Bbb{R}^N$ we study the asymptotic behaviour of optimal clusters associated to $\alpha$-Cheeger constants and natural energies like the sum or maximum: we prove that, as the parameter $\alpha$ converges to the "critical" value $\Big (\frac{N-1}{N}\Big ) _+$, optimal Cheeger clusters converge to solutions of different packing problems for balls, depending on the energy under consideration. As well, we propose an efficient phase field approach based on a multiphase Gamma convergence result of Modica-Mortola type, in order to compute $\alpha$-Cheeger constants, optimal clusters and, as a consequence of the asymptotic result, optimal packings. Numerical experiments are carried over in two and three space dimensions

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

Synchrotron radiation and absence of linear polarization in the colliding wind binary WR 146

Context. Several massive early-type binaries exhibit non-thermal emission which has been attributed to synchrotron radiation from particles accelerated by diffusive shock acceleration (DSA) in the wind-collision region (WCR). If the magnetic field in the strong shocks is ordered, its component parallel to the shock front should be enhanced, and the resultant synchrotron radiation would be polarized. However, such polarization has never been measured. Aims. We aim to determine the percentage of linearly polarized emission from the well-known non-thermal radio emitter WR 146, a WC6+O8 system. Methods. We performed spatially-unresolved radio continuum observations of WR 146 at 5 cm and 20 cm with the Karl G. Jansky Very Large Array. We constructed a numerical model to investigate a scenario where particles are accelerated by turbulent magnetic reconnection (MR), and we performed a quantitative analysis of possible depolarization effects. Results. No linearly polarized radio emission was detected. The data constrain the fractional linear polarization to less than 0.6% between 1 to 8 GHz. This is compatible with a high level of turbulence and a dominant random component in the magnetic field. In this case the relativistic particles could be produced by turbulent magnetic reconnection. In order for this scenario to satisfy the required non-thermal energy budget, the strength of the magnetic field in the WCR must be as high as ∼ 150 mG. However, if the magnetic field is ordered and DSA is ongoing, then a combination of internal and external Faraday rotation could equally account for the depolarization of the emission. Conclusions. The absence of polarization could be caused by a highly turbulent magnetic field, other depolarization mechanisms such as Faraday rotation in the stellar wind, or a combination of these processes. It is not clear whether it is possible to develop the high level of turbulence and strong magnetic fields required for efficient MR in a long-period binary such as WR 146. This scenario might also have trouble explaining the low-frequency cutoff in the spectrum. We therefore favor a scenario where particles are accelerated through DSA and the depolarization is produced by mechanisms other than a large ratio between random to regular magnetic fields.Facultad de Ciencias Astronómicas y GeofísicasInstituto Argentino de Radioastronomí

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