Logarithmically Slow Expansion of Hot Bubbles in Gases

We report logarithmically slow expansion of hot bubbles in gases in the process of cooling. A model problem first solved, when the temperature has compact support. Then temperature profile decaying exponentially at large distances is considered. The periphery of the bubble is shown to remain essentially static ("glassy") in the process of cooling until it is taken over by a logarithmically slowly expanding "core". An analytical solution to the problem is obtained by matched asymptotic expansion. This problem gives an example of how logarithmic corrections enter dynamic scaling.Comment: 4 pages, 1 figur

MAESTRO: An Adaptive Low Mach Number Hydrodynamics Algorithm for Stellar Flows

Many astrophysical phenomena are highly subsonic, requiring specialized numerical methods suitable for long-time integration. In a series of earlier papers we described the development of MAESTRO, a low Mach number stellar hydrodynamics code that can be used to simulate long-time, low-speed flows that would be prohibitively expensive to model using traditional compressible codes. MAESTRO is based on an equation set derived using low Mach number asymptotics; this equation set does not explicitly track acoustic waves and thus allows a significant increase in the time step. MAESTRO is suitable for two- and three-dimensional local atmospheric flows as well as three-dimensional full-star flows. Here, we continue the development of MAESTRO by incorporating adaptive mesh refinement (AMR). The primary difference between MAESTRO and other structured grid AMR approaches for incompressible and low Mach number flows is the presence of the time-dependent base state, whose evolution is coupled to the evolution of the full solution. We also describe how to incorporate the expansion of the base state for full-star flows, which involves a novel mapping technique between the one-dimensional base state and the Cartesian grid, as well as a number of overall improvements to the algorithm. We examine the efficiency and accuracy of our adaptive code, and demonstrate that it is suitable for further study of our initial scientific application, the convective phase of Type Ia supernovae.Comment: Accepted to Astrophysical Journal Suppliment (http://iop.org). 56 pages, 15 figures

Passing to the Limit in a Wasserstein Gradient Flow: From Diffusion to Reaction

We study a singular-limit problem arising in the modelling of chemical reactions. At finite {\epsilon} > 0, the system is described by a Fokker-Planck convection-diffusion equation with a double-well convection potential. This potential is scaled by 1/{\epsilon}, and in the limit {\epsilon} -> 0, the solution concentrates onto the two wells, resulting into a limiting system that is a pair of ordinary differential equations for the density at the two wells. This convergence has been proved in Peletier, Savar\'e, and Veneroni, SIAM Journal on Mathematical Analysis, 42(4):1805-1825, 2010, using the linear structure of the equation. In this paper we re-prove the result by using solely the Wasserstein gradient-flow structure of the system. In particular we make no use of the linearity, nor of the fact that it is a second-order system. The first key step in this approach is a reformulation of the equation as the minimization of an action functional that captures the property of being a curve of maximal slope in an integrated form. The second important step is a rescaling of space. Using only the Wasserstein gradient-flow structure, we prove that the sequence of rescaled solutions is pre-compact in an appropriate topology. We then prove a Gamma-convergence result for the functional in this topology, and we identify the limiting functional and the differential equation that it represents. A consequence of these results is that solutions of the {\epsilon}-problem converge to a solution of the limiting problem.Comment: Added two sections, corrected minor typos, updated reference

Novae Ejecta as Colliding Shells

Following on our initial absorption-line analysis of fifteen novae spectra we present additional evidence for the existence of two distinct components of novae ejecta having different origins. As argued in Paper I one component is the rapidly expanding gas ejected from the outer layers of the white dwarf by the outburst. The second component is pre-existing outer, more slowly expanding circumbinary gas that represents ejecta from the secondary star or accretion disk. We present measurements of the emission-line widths that show them to be significantly narrower than the broad P Cygni profiles that immediately precede them. The emission profiles of novae in the nebular phase are distinctly rectangular, i.e., strongly suggestive of emission from a relatively thin, roughly spherical shell. We thus interpret novae spectral evolution in terms of the collision between the two components of ejecta, which converts the early absorption spectrum to an emission-line spectrum within weeks of the outburst. The narrow emission widths require the outer circumbinary gas to be much more massive than the white dwarf ejecta, thereby slowing the latter's expansion upon collision. The presence of a large reservoir of circumbinary gas at the time of outburst is suggestive that novae outbursts may sometime be triggered by collapse of gas onto the white dwarf, as occurs for dwarf novae, rather than steady mass transfer through the inner Lagrangian point.Comment: 12 pages, 3 figures; Revised manuscript; Accepted for publication in Astrophysics & Space Scienc

Amenability of groups and GG-sets

This text surveys classical and recent results in the field of amenability of groups, from a combinatorial standpoint. It has served as the support of courses at the University of G\"ottingen and the \'Ecole Normale Sup\'erieure. The goals of the text are (1) to be as self-contained as possible, so as to serve as a good introduction for newcomers to the field; (2) to stress the use of combinatorial tools, in collaboration with functional analysis, probability etc., with discrete groups in focus; (3) to consider from the beginning the more general notion of amenable actions; (4) to describe recent classes of examples, and in particular groups acting on Cantor sets and topological full groups

Genetic loci of Staphylococcus aureus associated with anti-neutrophil cytoplasmic autoantibody (ANCA)-associated vasculitides

The proteinase 3 (PR3)-positive anti-neutrophil cytoplasmic autoantibody (ANCA)-associated vasculitis (AAV) granulomatosis with polyangiitis (GPA) has been associated with chronic nasal S. aureus carriage, which is a risk factor for disease relapse. The present study was aimed at comparing the genetic make-up of S. aureus isolates from PR3-ANCA-positive GPA patients with that of isolates from patients suffering from myeloperoxidase (MPO)-ANCA-positive AAV, and isolates from healthy controls. Based on a DNA microarray-based approach, we show that not only PR3-ANCA-positive GPA patients, but also MPO-ANCA-positive AAV patients mainly carried S. aureus types that are prevalent in the general population. Nonetheless, our data suggests that MPO-ANCA-associated S. aureus isolates may be distinct from healthy control- and PR3-ANCA-associated isolates. Furthermore, several genetic loci of S. aureus are associated with either PR3-ANCA- or MPO-ANCA-positive AAV, indicating a possible role for pore-forming toxins, such as leukocidins, in PR3-ANCA-positive GPA. Contrary to previous studies, no association between AAV and superantigens was detected. Our findings also show that a lowered humoral immune response to S. aureus is common for PR3-ANCA- and MPO-ANCA-positive AAV. Altogether, our observations imply that the presence or absence of particular virulence genes of S. aureus isolates from AAV patients contributes to disease progression and/or relapse