Consider the abelian category of commutative group schemes of finite type over a field , its full subcategory of finite group schemes, and the associated pro-category (resp. ) of pro-algebraic (resp. profinite) group schemes. When is perfect, we show that the profinite fundamental group is left exact and commutes with base change under algebraic field extensions; as a consequence, the higher profinite homotopy functors vanish for . Along the way, we describe the indecomposable projective objects of over an arbitrary field .
In this paper, we study the nonlinear Schrödinger equation coupled with the Maxwell equation. Using energy methods, we obtain a local existence result for the Cauchy problem.
We consider the weakly asymmetric simple exclusion process on the discrete space , in contact with stochastic reservoirs, both with density at the extremity points, and starting from the invariant state, namely the Bernoulli product measure of parameter . Under time diffusive scaling and for , when the asymmetry parameter is taken of order , we prove that the density fluctuations at stationarity are macroscopically governed by the energy solution of the stochastic Burgers equation with Dirichlet boundary conditions, which is shown to be unique and to exhibit different boundary behavior than the Cole–Hopf solution.
In this article, we study the excursion sets where is a natural real-analytic planar Gaussian field called the Bargmann–Fock field. More precisely, is the centered Gaussian field on with covariance . Alexander has proved that, if , then a.s. has no unbounded component. We show that conversely, if , then a.s. has a unique unbounded component. As a result, the critical level of this percolation model is . We also prove exponential decay of crossing probabilities under the critical level. To show these results, we rely on a recent box-crossing estimate by Beffara and Gayet. We also develop several tools including a KKL-type result for biased Gaussian vectors (based on the analogous result for product Gaussian vectors by Keller, Mossel and Sen) and a sprinkling inspired discretization procedure. These intermediate results hold for more general Gaussian fields, for which we prove a discrete version of our main result.