Estimates on the asymptotic behaviour of solution to linear integro-differential equations are fundamental in understanding the dynamics occuring in many nonlocal evolution problems. They are usually ...derived by using precise decay estimates on the heat kernel of the considered diffusion process. In this note, we show that for some generic jump diffusion and particular initial data, one can derive a lower bound of the asymptotic behaviour of the solution using a simple PDE argument. This is viewed as an independant preliminary brick to study invasion phenomena in nonlinear reaction diffusion problems.
This article deals with the development of a numerical method for the compressible Euler system valid for all Mach numbers: from extremely low to high regimes. In classical fluid dynamic problems, ...one faces both situations in which the flow is subsonic, and consequently acoustic waves are very fast compared to the velocity of the fluid, and situations in which the fluid moves at high speed and compressibility may generate shock waves. Standard explicit fluid solvers such as Godunov method fail in the description of both flows due to time step restrictions caused by the stiffness of the equations which leads to prohibitive computational costs. In this work, we develop a new method for the full Euler system of gas dynamics based on partitioning the equations into a fast and a low scale. Such a method employs a time step which is independent of the speed of the pressure waves and works uniformly for all Mach numbers. Cell centered discretization on Cartesian meshes is proposed. Numerical results up to the three dimensional case show the accuracy, the robustness and the effectiveness of the proposed approach.
A decade ago OSCAR was introduced as a penalized estimator where the penalty term, the sorted l1 norm, allows to perform clustering selection. More recently, SLOPE was introduced as a penalized ...estimator controlling the False Discovery Rate (FDR) as soon as the hyper-parameter of the sorted l1 norm is properly selected. For both, OSCAR and SLOPE, numerical schemes to compute these estimators are based on the proximal operator of the sorted l1 norm. The main goal of this note is to provide a short and simple formula for this operator. Based on this formula one may observe that the output of the proximal operator has some components equal and thus this formula corroborate that SLOPE as well as OSCAR perform clustering selection. Moreover, our geometric approach to prove the formula for the proximal operator provides insight to show that testing procedures based on SLOPE are more powerful than step-down testing procedures but less powerful than step-up testing procedures.