Sharp comparison and maximum principles via horizontal normal mapping in the Heisenberg group

In this paper we solve a problem raised by Gutiérrez and Montanari about comparison principles for H−convex functions on subdomains of Heisenberg groups. Our approach is based on the notion of the sub-Riemannian horizontal normal mapping and uses degree theory for set-valued maps. The statement of the comparison principle combined with a Harnack inequality is applied to prove the Aleksandrov-type maximum principle, describing the correct boundary behavior of continuous H−convex functions vanishing at the boundary of horizontally bounded subdomains of Heisenberg groups. This result answers a question by Garofalo and Tournier. The sharpness of our results are illustrated by examples.

A Maximum Radius for Habitable Planets

We compute the maximum radius a planet can have in order to fulfill two constraints that are likely necessary conditions for habitability: 1- surface temperature and pressure compatible with the existence of liquid water, and 2- no ice layer at the bottom of a putative global ocean, that would prevent the operation of the geologic carbon cycle to operate. We demonstrate that, above a given radius, these two constraints cannot be met: in the Super-Earth mass range (1-12 M-earth), the overall maximum that a planet can have varies between 1.8 and 2.3 R-earth. This radius is reduced when considering planets with higher Fe/Si ratios, and taking into account irradiation effects on the structure of the gas envelope.

Stricter regulation boosts exports: the case of Maximum Residue Levels in pesticides

Constructing an original panel on Maximum Residue Levels (MRLs) in pesticides for 50 countries over 2006-2012, this paper studies the effect of heterogeneity in MRL regulation on bilateral trade. We find evidence of regulatory heterogeneity diminishing trade at the extensive margin when the exporter faces more stringent regulation abroad, suggesting compliance costs in entering the destination market. Significantly, however, we also find strong evidence of regulatory heterogeneity increasing trade at the intensive margin for exports coming from countries that set the strictest standards, alluding to the positive informative effect of such regulation.