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

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

Analytic Detection Thresholds for Measurements of Linearly Polarized Intensity Using Rotation Measure Synthesis

A fully analytic statistical formalism does not yet exist to describe radio-wavelength measurements of linearly polarized intensity that are produced using rotation measure synthesis. In this work we extend the analytic formalism for standard linear polarization, namely that describing measurements of the quadrature sum of Stokes Q and U intensities, to the rotation measure synthesis environment. We derive the probability density function and expectation value for Faraday-space polarization measurements for both the case where true underlying polarized emission is present within unresolved Faraday components, and for the limiting case where no such emission is present. We then derive relationships to quantify the statistical significance of linear polarization measurements in terms of standard Gaussian statistics. The formalism developed in this work will be useful for setting signal-to-noise ratio detection thresholds for measurements of linear polarization, for the analysis of polarized sources potentially exhibiting multiple Faraday components, and for the development of polarization debiasing schemes.Comment: 14 pages, 6 figures, accepted for publication in MNRA