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

What Happens When Insurers Make Insurance Laws? State Legislative Agendas and the Occupational Makeup of Government

Do the occupational backgrounds of politicians affect the government’s agenda? Businesses have long thought so. The first occupational data on state legislators were collected by the Insurance Information Institute, an interest group representing major insurance companies. In this paper, we test one potential motive for these kinds of efforts: the idea that the occupational makeup of governments affects the agendas they pursue, an argument that has been largely neglected in research on politicians’ occupational backgrounds. We focus here on the insurance industry. Using original data, we find that state legislatures with more former insurers consider fewer bills regulating insurance (negative agenda control), that former insurers play a disproportionate role in drafting the insurance bills that are introduced (positive agenda control), and that the bills former insurers introduce tend to be more favorable to the industry than those that their colleagues introduce (positive agenda control). The occupational makeup of legislatures may indeed affect their agendas, as industry groups have long suspected

A Contractor Based on Convex Interval Taylor

International audienceInterval Taylor has been proposed in the sixties by the interval analysis community for relaxing continuous non-convex constraint systems. However, it generally produces a non-convex relaxation of the solution set. A simple way to build a convex polyhedral relaxation is to select a corner of the studied domain/box as expansion point of the interval Taylor form, instead of the usual midpoint. The idea has been proposed by Neumaier to produce a sharp range of a single function andby Lin and Stadtherr to handle n × n (square) systems of equations. This paper presents an interval Newton-like operator, called X-Newton, that iteratively calls this interval convexification based on an endpoint interval Taylor. This general-purpose contractor uses no preconditioning and can handle any system of equality and inequality constraints. It uses Hansen's variant to compute the interval Taylor form and uses two opposite corners of the domain for every constraint. The X-Newton operator can be rapidly encoded, and produces good speedups in constrained global optimization and constraint satisfaction. First experiments compare X-Newton with affine arithmetic

Faraday rotation spectra of bismuth-substituted ferrite garnet films with in-plane magnetization

Single crystalline films of bismuth-substituted ferrite garnets have been synthesized by the liquid phase epitaxy method where GGG substrates are dipped into the flux. The growth parameters are controlled to obtain films with in-plane magnetization and virtually no domain activity, which makes them excellently suited for magnetooptic imaging. The Faraday rotation spectra were measured across the visible range of wavelengths. To interprete the spectra we present a simple model based on the existence of two optical transitions of diamagnetic character, one tetrahedral and one octahedral. We find excellent agreement between the model and our experimental results for photon energies between 1.77 and 2.53 eV, corresponding to wavelengths between 700 and 490 nm. It is shown that the Faraday rotation changes significantly with the amount of substituted gallium and bismuth. Furthermore, the experimental results suggest that the magnetooptic response changes linearly with the bismuth substitution.Comment: 15 pages, 6 figures, published in Phys. Rev.

Resonant Absorption as Mode Conversion?

Resonant absorption and mode conversion are both extensively studied mechanisms for wave "absorption" in solar magnetohydrodynamics (MHD). But are they really distinct? We re-examine a well-known simple resonant absorption model in a cold MHD plasma that places the resonance inside an evanescent region. The normal mode solutions display the standard singular resonant features. However, these same normal modes may be used to construct a ray bundle which very clearly undergoes mode conversion to an Alfv\'en wave with no singularities. We therefore conclude that resonant absorption and mode conversion are in fact the same thing, at least for this model problem. The prime distinguishing characteristic that determines which of the two descriptions is most natural in a given circumstance is whether the converted wave can provide a net escape of energy from the conversion/absorption region of physical space. If it cannot, it is forced to run away in wavenumber space instead, thereby generating the arbitrarily small scales in situ that we recognize as fundamental to resonant absorption and phase mixing. On the other hand, if the converted wave takes net energy way, singularities do not develop, though phase mixing may still develop with distance as the wave recedes.Comment: 23 pages, 8 figures, 2 tables; accepted by Solar Phys (July 9 2010

Hull Consistency Under Monotonicity

International audienceWe prove that hull consistency for a system of equations or inequalities can be achieved in polynomial time providing that the underlying functions are monotone with respect to each variable. This result holds including when variables have multiple occurrences in the expressions of the functions, which is usually a pitfall for interval-based contractors. For a given constraint, an optimal contractor can thus be enforced quickly under monotonicity and the practical significance of this theoretical result is illustrated on a simple example

Cosmological Implications of Neutrinos

The lectures describe several cosmological effects produced by neutrinos. Upper and lower cosmological limits on neutrino mass are derived. The role that neutrinos may play in formation of large scale structure of the universe is described and neutrino mass limits are presented. Effects of neutrinos on cosmological background radiation and on big bang nucleosynthesis are discussed. Limits on the number of neutrino flavors and mass/mixing are given.Comment: 41 page, 7 figures; lectures presented at ITEP Winter School, February, 2002; to be published in the Proceeding

Mesoscopic Fano Effect in a Quantum Dot Embedded in an Aharonov-Bohm Ring

The Fano effect, which occurs through the quantum-mechanical cooperation between resonance and interference, can be observed in electron transport through a hybrid system of a quantum dot and an Aharonov-Bohm ring. While a clear correlation appears between the height of the Coulomb peak and the real asymmetric parameter $q$ for the corresponding Fano lineshape, we need to introduce a complex $q$ to describe the variation of the lineshape by the magnetic and electrostatic fields. The present analysis demonstrates that the Fano effect with complex asymmetric parameters provides a good probe to detect a quantum-mechanical phase of traversing electrons.Comment: REVTEX, 9 pages including 8 figure

A-dependence of nuclear transparency in quasielastic A(e,e'p) at high Q^2

The A-dependence of the quasielastic A(e,e'p) reaction has been studied at SLAC with H-2, C, Fe, and Au nuclei at momentum transfers Q^2 = 1, 3, 5, and 6.8 (GeV/c)^2. We extract the nuclear transparency T(A,Q^2), a measure of the average probability that the struck proton escapes from the nucleus A without interaction. Several calculations predict a significant increase in T with momentum transfer, a phenomenon known as Color Transparency. No significant rise within errors is seen for any of the nuclei studied.Comment: 5 pages incl. 2 figures, Caltech preprint OAP-73