Global existence for semilinear reaction-diffusion systems on evolving domains

We present global existence results for solutions of reaction-diffusion systems on evolving domains. Global existence results for a class of reaction-diffusion systems on fixed domains are extended to the same systems posed on spatially linear isotropically evolving domains. The results hold without any assumptions on the sign of the growth rate. The analysis is valid for many systems that commonly arise in the theory of pattern formation. We present numerical results illustrating our theoretical findings.Comment: 24 pages, 3 figure

Motion around a Monopole + Ring system: I. Stability of Equatorial Circular Orbits vs Regularity of Three-dimensional Motion

We study the motion of test particles around a center of attraction represented by a monopole (with and without spheroidal deformation) surrounded by a ring, given as a superposition of Morgan & Morgan discs. We deal with two kinds of bounded orbits: (i) Equatorial circular orbits and (ii) general three-dimensional orbits. The first case provides a method to perform a linear stability analysis of these structures by studying the behavior of vertical and epicyclic frequencies as functions of the mass ratio, the size of the ring and/or the quadrupolar deformation. In the second case, we study the influence of these parameters in the regularity or chaoticity of motion. We find that there is a close connection between linear stability (or unstability) of equatorial circular orbits and regularity (or chaoticity) of the three-dimensional motion.Comment: 13 pages, 17 figures, to appear in MNRA

Isoperimetric and stable sets for log-concave perturbations of Gaussian measures

Let $\Omega$ be an open half-space or slab in $\mathbb{R}^{n+1}$ endowed with a perturbation of the Gaussian measure of the form $f(p):=\exp(\omega(p)-c|p|^2)$, where $c>0$ and $\omega$ is a smooth concave function depending only on the signed distance from the linear hyperplane parallel to $\partial\Omega$. In this work we follow a variational approach to show that half-spaces perpendicular to $\partial\Omega$ uniquely minimize the weighted perimeter in $\Omega$ among sets enclosing the same weighted volume. The main ingredient of the proof is the characterization of half-spaces parallel or perpendicular to $\partial\Omega$ as the unique stable sets with small singular set and null weighted capacity. Our methods also apply for $\Omega=\mathbb{R}^{n+1}$, which produces in particular the classification of stable sets in Gauss space and a new proof of the Gaussian isoperimetric inequality. Finally, we use optimal transport to study the weighted minimizers when the perturbation term $\omega$ is concave and possibly non-smooth.Comment: final version, to appear in Analysis and Geometry in Metric Space

Systemically Important or “Too Big to Fail” Financial Institutions

Although “too big to fail” (TBTF) has been a perennial policy issue, it was highlighted by the near-collapse of several large financial firms in 2008. Large financial firms that failed or required extraordinary government assistance in the recent crisis included depositories (Citigroup and Washington Mutual), government-sponsored enterprises (Fannie Mae and Freddie Mac), insurance companies (AIG), and investment banks (Bear Stearns and Lehman Brothers).1 In many of these cases, policy makers justified the use of government resources on the grounds that the firms were “systemically important” or “too big to fail.” TBTF is the concept that a firm’s disorderly failure would cause widespread disruptions in financial markets that could not easily be contained. While the government had no explicit policy to rescue TBTF firms, several were rescued on those grounds once the crisis struck. TBTF subsequently became one of the systemic risk issues that policy makers grappled with in the wake of the recent crisis. This report discusses the economic issues raised by TBTF, broad policy options, and policy changes made by the relevant Dodd-Frank provisions. This report also discusses recent legislation addressing the TBTF issue in the 113th Congress. The report ends with an Appendix reviewing the historical experience with TBTF before and during the recent crisis

The Effect of Lattice Vibrations on Substitutional Alloy Thermodynamics

A longstanding limitation of first-principles calculations of substitutional alloy phase diagrams is the difficulty to account for lattice vibrations. A survey of the theoretical and experimental literature seeking to quantify the impact of lattice vibrations on phase stability indicates that this effect can be substantial. Typical vibrational entropy differences between phases are of the order of 0.1 to 0.2 k_B/atom, which is comparable to the typical values of configurational entropy differences in binary alloys (at most 0.693 k_B/atom). This paper describes the basic formalism underlying ab initio phase diagram calculations, along with the generalization required to account for lattice vibrations. We overview the various techniques allowing the theoretical calculation and the experimental determination of phonon dispersion curves and related thermodynamic quantities, such as vibrational entropy or free energy. A clear picture of the origin of vibrational entropy differences between phases in an alloy system is presented that goes beyond the traditional bond counting and volume change arguments. Vibrational entropy change can be attributed to the changes in chemical bond stiffness associated with the changes in bond length that take place during a phase transformation. This so-called ``bond stiffness vs. bond length'' interpretation both summarizes the key phenomenon driving vibrational entropy changes and provides a practical tool to model them.Comment: Submitted to Reviews of Modern Physics 44 pages, 6 figure