Structural results on convexity relative to cost functions

Mass transportation problems appear in various areas of mathematics, their solutions involving cost convex potentials. Fenchel duality also represents an important concept for a wide variety of optimization problems, both from the theoretical and the computational viewpoints. We drew a parallel to the classical theory of convex functions by investigating the cost convexity and its connections with the usual convexity. We give a generalization of Jensen's inequality for cost convex functions.Comment: 10 page

Monge's transport problem in the Heisenberg group

We prove the existence of solutions to Monge transport problem between two compactly supported Borel probability measures in the Heisenberg group equipped with its Carnot-Caratheodory distance assuming that the initial measure is absolutely continuous with respect to the Haar measure of the group

A glimpse into the differential topology and geometry of optimal transport

This note exposes the differential topology and geometry underlying some of the basic phenomena of optimal transportation. It surveys basic questions concerning Monge maps and Kantorovich measures: existence and regularity of the former, uniqueness of the latter, and estimates for the dimension of its support, as well as the associated linear programming duality. It shows the answers to these questions concern the differential geometry and topology of the chosen transportation cost. It also establishes new connections --- some heuristic and others rigorous --- based on the properties of the cross-difference of this cost, and its Taylor expansion at the diagonal.Comment: 27 page

The Gel'fand problem for the biharmonic operator

We study stable solutions of a fourth order nonlinear elliptic equation, both in entire space and in bounded domains

A Schwarz lemma for K\"ahler affine metrics and the canonical potential of a proper convex cone

This is an account of some aspects of the geometry of K\"ahler affine metrics based on considering them as smooth metric measure spaces and applying the comparison geometry of Bakry-Emery Ricci tensors. Such techniques yield a version for K\"ahler affine metrics of Yau's Schwarz lemma for volume forms. By a theorem of Cheng and Yau there is a canonical K\"ahler affine Einstein metric on a proper convex domain, and the Schwarz lemma gives a direct proof of its uniqueness up to homothety. The potential for this metric is a function canonically associated to the cone, characterized by the property that its level sets are hyperbolic affine spheres foliating the cone. It is shown that for an $n$-dimensional cone a rescaling of the canonical potential is an $n$-normal barrier function in the sense of interior point methods for conic programming. It is explained also how to construct from the canonical potential Monge-Amp\`ere metrics of both Riemannian and Lorentzian signatures, and a mean curvature zero conical Lagrangian submanifold of the flat para-K\"ahler space.Comment: Minor corrections. References adde

A black‐box automated approach to calibrate numerical simulations and optimize cover design: Application to a flow control layer constructed on an experimental waste rock pile

Mining operations often produce large volumes of waste rock to access economically valuable mineralized zones. Waste rock is usually stored in surface piles, the construction and reclamation of which represent a challenge for the industry. A flow control layer (FCL) made of crushed waste rock or sand and constructed on top of each waste rock bench could contribute to control water infiltration, thus improving waste rock pile stability and limiting contamination. An experimental waste rock pile was built and instrumented at the Tio mine (Rio Tinto Fer et Titane, Canada) to evaluate the performance of an FCL in field conditions. Large infiltration tests and rainfall monitoring were carried out, and measured outflow and water contents were used to calibrate numerical simulations. However, data were noisy and sometimes incomplete, and the models were difficult to calibrate. A new automated calibration approach was therefore proposed. An algorithm was developed to automate the numerical simulation calibration, using a black-box method that involves solving an optimization problem on a function without an analytic form. The approach was applied on measurements obtained from large-scale infiltration tests and validated using 2 yr of field monitoring data. Finally, the automated approach was adapted to optimize the design of the FCL, and an optimal design (material properties and layer thickness) was recommended based on local climate conditions. The proposed automated method could contribute to reduce the bias induced by manual calibration and allows for rapid multivariable calibration and optimization for a broad spectrum of mine waste cover system applications

Impacts of air pollutants from rural Chinese households under the rapid residential energy transition

Rural residential energy consumption in China is experiencing a rapid transition towards clean energy, nevertheless, solid fuel combustion remains an important emission source. Here we quantitatively evaluate the contribution of rural residential emissions to PM2.5 (particulate matter with an aerodynamic diameter less than 2.5 μm) and the impacts on health and climate. The clean energy transitions result in remarkable reductions in the contributions to ambient PM2.5, avoiding 130,000 (90,000-160,000) premature deaths associated with PM2.5 exposure. The climate forcing associated with this sector declines from 0.057 ± 0.016 W/m2 in 1992 to 0.031 ± 0.008 W/m2 in 2012. Despite this, the large remaining quantities of solid fuels still contributed 14 ± 10 μg/m3 to population-weighted PM2.5 in 2012, which comprises 21 ± 14% of the overall population-weighted PM2.5 from all sources. Rural residential emissions affect not only rural but urban air quality, and the impacts are highly seasonal and location dependent

Asymptotic behavior of solutions to the σk\sigma_k-Yamabe equation near isolated singularities

$\sigma_k$-Yamabe equations are conformally invariant equations generalizing the classical Yamabe equation. In an earlier work YanYan Li proved that an admissible solution with an isolated singularity at $0\in \mathbb R^n$ to the $\sigma_k$-Yamabe equation is asymptotically radially symmetric. In this work we prove that an admissible solution with an isolated singularity at $0\in \mathbb R^n$ to the $\sigma_k$-Yamabe equation is asymptotic to a radial solution to the same equation on $\mathbb R^n \setminus \{0\}$. These results generalize earlier pioneering work in this direction on the classical Yamabe equation by Caffarelli, Gidas, and Spruck. In extending the work of Caffarelli et al, we formulate and prove a general asymptotic approximation result for solutions to certain ODEs which include the case for scalar curvature and $\sigma_k$ curvature cases. An alternative proof is also provided using analysis of the linearized operators at the radial solutions, along the lines of approach in a work by Korevaar, Mazzeo, Pacard, and Schoen.Comment: 55 page

A renewed rise in global HCFC-141b emissions between 2017???2021

Global emissions of the ozone-depleting gas HCFC-141b (1,1-dichloro-1-fluoroethane, CH3CCl2F) derived from measurements of atmospheric mole fractions increased between 2017 and 2021 despite a fall in reported production and consumption of HCFC-141b for dispersive uses. HCFC-141b is a controlled substance under the Montreal Protocol, and its phase-out is currently underway, after a peak in reported consumption and production in developing (Article 5) countries in 2013. If reported production and consumption are correct, our study suggests that the 2017–2021 rise is due to an increase in emissions from the bank when appliances containing HCFC-141b reach the end of their life, or from production of HCFC-141b not reported for dispersive uses. Regional emissions have been estimated between 2017–2020 for all regions where measurements have sufficient sensitivity to emissions. This includes the regions of northwestern Europe, east Asia, the United States and Australia, where emissions decreased by a total of 2.3 ± 4.6 Gg yr−1, compared to a mean global increase of 3.0 ± 1.2 Gg yr−1 over the same period. Collectively these regions only account for around 30 % of global emissions in 2020. We are not able to pinpoint the source regions or specific activities responsible for the recent global emission rise

A user's guide to optimal transport

This text is an expanded version of the lectures given by the first author in the 2009 CIME summer school of Cetraro. It provides a quick and reasonably account of the classical theory of optimal mass transportation and of its more recent developments, including the metric theory of gradient flows, geometric and functional inequalities related to optimal transportation, the first and second order differential calculus in the Wasserstein space and the synthetic theory of metric measure spaces with Ricci curvature bounded from below