Auxiliary SDEs for homogenization of quasilinear PDEs with periodic coefficients

We study the homogenization property of systems of quasi-linear PDEs of parabolic type with periodic coefficients, highly oscillating drift and highly oscillating nonlinear term. To this end, we propose a probabilistic approach based on the theory of forward-backward stochastic differential equations and introduce the new concept of ``auxiliary SDEs.''Comment: Published by the Institute of Mathematical Statistics (http://www.imstat.org) in the Annals of Probability (http://www.imstat.org/aop/) at http://dx.doi.org/10.1214/00911790400000014

Brief of Brian Wolfman, Aderson B. Francois, and Eric Schnapper as Amici Curiae in Support of Petitioner in Peterson v. Linear Controls Incorporated, No. 18-1401 (U.S. Supreme Court June 6, 2019)

In Title VII disparate-treatment, employment-discrimination cases, the term “adverse employment action” originally developed as judicial shorthand for the statute’s text, which broadly prohibits any discriminatory conduct by an employer against an employee based on the employee\u27s race, color, religion, sex, or national origin. See 42 U.S.C. 2000e-2(a)(1). But what started simply as shorthand has taken on a life of its own and now improperly limits the statute’s reach. The Fifth Circuit’s version of the adverse-employment-action rule stands out as especially improper: Only an “ultimate employment decision”—a refusal to hire, a firing, a demotion, or the like—constitutes impermissible discrimination.In this amicus brief, we urge the Supreme Court to grant review and overturn the Fifth Circuit\u27s standard. We argue, first, that the Fifth Circuit\u27s ultimate-employment-decision standard is inconsistent with Title VII’s text and the Supreme Court’s Title VII decisions. Next, we show that the Fifth Circuit’s rule excludes many discriminatory employment practices that are unlawful in its sister circuits. The stories of discrimination victims from these other jurisdictions demonstrate that the Fifth Circuit’s approach is wrong. These individuals suffered discrimination that Title VII prohibits, but the Fifth Circuit’s standard would enable their employers to discriminate without consequence. Finally, we propose a standard consistent with Title VII’s text and Supreme Court precedent: As long as the employer’s intentional, discriminatory conduct imposes meaningful harm on the employee, it is prohibited by, and may be remedied under, Title VII

Primitive geodesic lengths and (almost) arithmetic progressions

In this article, we investigate when the set of primitive geodesic lengths on a Riemannian manifold have arbitrarily long arithmetic progressions. We prove that in the space of negatively curved metrics, a metric having such arithmetic progressions is quite rare. We introduce almost arithmetic progressions, a coarsification of arithmetic progressions, and prove that every negatively curved, closed Riemannian manifold has arbitrarily long almost arithmetic progressions in its primitive length spectrum. Concerning genuine arithmetic progressions, we prove that every non-compact, locally symmetric, arithmetic manifold has arbitrarily long arithmetic progressions in its primitive length spectrum. We end with a conjectural characterization of arithmeticity in terms of arithmetic progressions in the primitive length spectrum. We also suggest an approach to a well known spectral rigidity problem based on the scarcity of manifolds with arithmetic progressions.Comment: v3: 23 pages. To appear in Publ. Ma

Effects of F, B2O3 and P2O5 on the solubility of water in haplogranite melts compared to natural silicate melts

The effects of F, B2O3 and P2O5 on the H2O solubility in a haplogranite liquid (36 wt. % SiO2, 39 wt. % NaAlSi3O8, 25 wt. % KAlSi3O8) have been determined at 0.5, 1, 2, and 3 kb and 800, 850, and 900°C. The H2O solubility increases with increasing F and B content of the melt. The H2O solubility increase in more important at high pressure (2 and 3 kb) than at low pressure (0.5 kb). At 2 kb and 800°C, the H2O solubility increases from 5.94 to 8.22 wt. % H2O with increasing F content in the melt from 0 to 4.55 wt. %, corresponding to a linear H2O solubility increase of 0.53 mol H2O/mol F. With addition of 4.35 wt. % B2O3, the H2O solubility increases up to 6.86 wt. % H2O at 2 kb and 800°C, corresponding to a linear increase of 1.05 mol H2O/mol B2O3. The results allow to define the individual effects of fluorine and boron on H2O solubility in haplogranitic melts with compositions close to that of H2O-saturated thermal minima (at 0.5–3 kb). Although P has a dramatic effect on the phase relations in the haplogranite system, its effect on the H2O solubility was found to be negligible in natural melt compositions. The concominant increase in H2O solubility and F can not be interpreted on the basis of the available spectroscopic data (existence of hydrated aluminofluoride complexes or not). In contrast, hydrated borates or more probably boroxol complexes have been demonstrated in B-bearing hydrous melts

The mean, variance and limiting distribution of two statistics sensitive to phylogenetic tree balance

For two decades, the Colless index has been the most frequently used statistic for assessing the balance of phylogenetic trees. In this article, this statistic is studied under the Yule and uniform model of phylogenetic trees. The main tool of analysis is a coupling argument with another well-known index called the Sackin statistic. Asymptotics for the mean, variance and covariance of these two statistics are obtained, as well as their limiting joint distribution for large phylogenies. Under the Yule model, the limiting distribution arises as a solution of a functional fixed point equation. Under the uniform model, the limiting distribution is the Airy distribution. The cornerstone of this study is the fact that the probabilistic models for phylogenetic trees are strongly related to the random permutation and the Catalan models for binary search trees.Comment: Published at http://dx.doi.org/10.1214/105051606000000547 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Discrete rearranging disordered patterns, part I: Robust statistical tools in two or three dimensions

Discrete rearranging patterns include cellular patterns, for instance liquid foams, biological tissues, grains in polycrystals; assemblies of particles such as beads, granular materials, colloids, molecules, atoms; and interconnected networks. Such a pattern can be described as a list of links between neighbouring sites. Performing statistics on the links between neighbouring sites yields average quantities (hereafter "tools") as the result of direct measurements on images. These descriptive tools are flexible and suitable for various problems where quantitative measurements are required, whether in two or in three dimensions. Here, we present a coherent set of robust tools, in three steps. First, we revisit the definitions of three existing tools based on the texture matrix. Second, thanks to their more general definition, we embed these three tools in a self-consistent formalism, which includes three additional ones. Third, we show that the six tools together provide a direct correspondence between a small scale, where they quantify the discrete pattern's local distortion and rearrangements, and a large scale, where they help describe a material as a continuous medium. This enables to formulate elastic, plastic, fluid behaviours in a common, self-consistent modelling using continuous mechanics. Experiments, simulations and models can be expressed in the same language and directly compared. As an example, a companion paper (Marmottant, Raufaste and Graner, joint paper) provides an application to foam plasticity

Why not a di-NUT? or Gravitational duality and rotating solutions

We study how gravitational duality acts on rotating solutions, using the Kerr-NUT black hole as an example. After properly reconsidering how to take into account both electric (i.e. mass-like) and magnetic (i.e. NUT-like) sources in the equations of general relativity, we propose a set of definitions for the dual Lorentz charges. We then show that the Kerr-NUT solution has non-trivial such charges. Further, we clarify in which respect Kerr's source can be seen as a mass M with a dipole of NUT charges.Comment: 20 pages. v2: minor clarifications in section 4, version to appear in PR