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

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

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

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

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

Yield gap and the shares of climate and crop management in yield and yield variability of staple crops in West Africa. [O-3330b-01]

" Yield gap " (Yg) is a key concept of agricultural science for identifying the room for improvement of yields through better management of the agroecosystem. in rainfed agriculture Yg is the difference between actual yield (Ya) and the theoretical water limited yield (Yw) that would be achieved if solar radiation, temperature and precipitations were the only factor limiting the crop's growth and yield. Changes in Yw over regions and years are due to climate-soil interactions that are not easily modified by crop management, whereas changes in Yg are due to limiting factors that are typically within the scope of crop management such as nutrient availability, weeds, and pests. We provide an example of yield gap estimates in semi-arid a frica, using yield and other agronomic data collected in famers' fields of Senegal in 1990 and 1991 and from 2006 to 2012. i t illustrates how contrarily to what most people would expect climate is not, on average, what most limits yields in that region: yet, actual yields are on average a quarter of water limited yield, and this is due to constraints whose reduction is technically possible albeit subject to the economic and environmental relevance of doing so. Most studies dealing with the impact of climate change on agriculture in West a frica compare Yw under present and future climate as predicted by climate models. t he magnitude of those predicted long term changes in Yw by 2050 is down to –20% in the worst scenario combining a +6°C change with a -20% rainfall change. s uch changes in water limited yields are certainly concerning, but they are remarkably small compared to the potential +390% increase that would result from closing the current yield gap. When considering yield variations observed across plots and years, and not anymore regional averages over a few years, what strikes is the stability of observed yields compared to variations of Yw. We used crop model simulations with historical series of 20 years of weather data to compare yield distributions over years of a crop grown using 3 contrasted levels of fertilisation and no incidence of weeds, pests or diseases. For each fertilisation level, the simulated yield reached a maximum value the 'best year' of the series. t he three fertilisation levels were chosen so that the maximum simulated yield reached 0.25 Yw, 0.5 Yw, and 0.75 Yw respectively. t he resulting simulated yield distributions show that even if management allows increasing the median yield, in many years the climate is the main limiting factor and fertilising has no or a slight impact only. i n other words, the way the current climate limits crop production in this region is by making uncertain the output of investing for high yields. Buying fertilizers or working hard for manure collection, transport and distribution do not translate, a certain number of years, into more production. For farmers struggling for the daily subsistence of their family, that kind of risk may not be justified while alternative use of family resources in cash and labour force provide less risky ways to produce subsistence means. Until recently, in many farming systems of West africa, the growth in food needs due to population growth in rural areas was matched thanks to increases in cultivated or pasted areas rather than increases in crop yields or livestock pressure on land (i.e extension rather than intensification of crop or livestock activities). When rural families reached the limits of this strategy, migrations of many kinds of distance and duration became the adjustment variable to the gap between resources available from farming and population needs. T his suggests that for many, it is less risky to leave home than to intensify cropping or livestock systems. Anyway, as job opportunities for migrants from the rural zones are currently low in West african cities and elsewhere, there are legitimate concerns about the way this strategy may soon reach its limit as well. i n terms of climate change, the worst scenario for farmers of that region would be if crop intensification became even more risky under future climate than at present. t here is thus an urgent need for joint agronomic and climate research to go beyond the prediction of Yw or of yield under unchanged crop management and determine whether or not the future climate will increase the yield risks associated with crop intensification in that region. But this should not divert from designing and implementing policies incentive to such intensification under present climate, as this might be much easier now than later. (Texte intégral

Bloch oscillations of ultracold atoms: a tool for a metrological determination of h/mRbh/m_{Rb}

We use Bloch oscillations in a horizontal moving standing wave to transfer a large number of photon recoils to atoms with a high efficiency (99.5% per cycle). By measuring the photon recoil of $^{87}Rb$, using velocity selective Raman transitions to select a subrecoil velocity class and to measure the final accelerated velocity class, we have determined $h/m_{Rb}$ with a relative precision of 0.4 ppm. To exploit the high momentum transfer efficiency of our method, we are developing a vertical standing wave set-up. This will allow us to measure $h/m_{Rb}$ better than $10^{-8}$ and hence the fine structure constant $\alpha$ with an uncertainty close to the most accurate value coming from the ($g-2$) determination