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.
A numerical analysis of upwind type schemes for the nonlinear nonlocal aggregation equation is provided. In this approach, the aggregation equation is interpreted as a conservative transport equation driven by a nonlocal nonlinear velocity field with low regularity. In particular, we allow the interacting potential to be pointy, in which case the velocity field may have discontinuities. Based on recent results of existence and uniqueness of a Filippov flow for this type of equations, we study an upwind finite volume numerical scheme and we prove that it is convergent at order in Wasserstein distance. The paper is illustrated by numerical simulations that indicate that this convergence order should be optimal.
We consider shape optimization problems for general integral functionals of the calculus of variations, defined on a domain that varies over all subdomains of a given bounded domain of . We show in a rather elementary way the existence of a solution that is in general a quasi open set. Under very mild conditions we show that the optimal domain is actually open and with finite perimeter. Some counterexamples show that in general this does not occur.
We introduce several classes of polytopes contained in and cut out by inequalities involving sums of consecutive coordinates. We show that the normalized volumes of these polytopes enumerate circular extensions of certain partial cyclic orders. Among other things this gives a new point of view on a question popularized by Stanley. We also provide a combinatorial interpretation of the Ehrhart –polynomials of some of these polytopes in terms of descents of total cyclic orders. The Euler numbers, the Eulerian numbers and the Narayana numbers appear as special cases.
We investigate the generalized Poland–Scheraga model, which is used in the bio-physical literature to model the DNA denaturation transition, in the case where the two strands are allowed to be non-complementary (and to have different lengths). The homogeneous model was recently studied from a mathematical point of view in [BGK18, GK17], via a –dimensional renewal approach, with a loop exponent (): it was found to undergo a localization/delocalization phase transition (which corresponds to the denaturation transition) of order , together with (in general) other phase transitions. In this paper, we turn to the disordered model, and we address the question of the influence of disorder on the denaturation phase transition, that is whether adding an arbitrarily small amount of disorder (i.e. inhomogeneities) affects the critical properties of this transition. Our results are consistent with Harris’ predictions for -dimensional disordered systems (here ). First, we prove that when (i.e. ), then disorder is irrelevant: the quenched and annealed critical points are equal, and the disordered denaturation phase transition is also of order . On the other hand, when , disorder is relevant: we prove that the quenched and annealed critical points differ. Moreover, we discuss a number of open problems, in particular the smoothing phenomenon that is expected to enter the game when disorder is relevant.
For closed and oriented hyperbolic surfaces, a formula of Witten establishes an equality between two volume forms on the space of representations of the surface in a semisimple Lie group. One of the forms is a Reidemeister torsion, the other one is the power of the Atiyah–Bott–Goldman–Narasimhan symplectic form. We introduce an holomorphic volume form on the space of representations of the circle, so that, for surfaces with boundary, it appears as peripheral term in the generalization of Witten’s formula. We compute explicit volume and symplectic forms for some simple surfaces and for the Lie group .
We investigate the case of a medium with two inclusions or inhomogeneities with nearly touching corner singularities. We present two different asymptotic models to describe the phenomenon under specific geometrical assumptions. These asymptotic expansions are analysed and compared in a common framework. We conclude by a representation formula to characterise the detachment of the corners and we provide the possible extensions of the geometrical hypotheses.
In this paper we study varieties covered by rational or elliptic curves. First, we show that images of Calabi–Yau or irreducible symplectic varieties under rational maps are almost always rationally connected. Second, we investigate elliptically connected and elliptically chain connected varieties, and varieties swept out by a family of elliptic curves. Among other things, we show that Calabi–Yau or hyperkähler manifolds which are covered by a family of elliptic curves contain uniruled divisors and that elliptically chain connected varieties of dimension at least contain a rational curve, and so do -trivial varieties with finite fundamental group which are covered by elliptic curves.
Let us consider the family of one-dimensional probabilistic cellular automata (PCA) with memory two having the following property: the dynamics is such that the value of a given cell at time is drawn according to a distribution which is a function of the states of its two nearest neighbours at time , and of its own state at time . We give conditions for which the invariant measure has a product form or a Markovian form, and prove an ergodicity result holding in that context. The stationary space-time diagrams of these PCA present different forms of reversibility. We describe and study extensively this phenomenon, which provides families of Gibbs random fields on the square lattice having nice geometric and combinatorial properties. Such PCA naturally arise in the study of different models coming from statistical physics. We review from a PCA approach some results on the -vertex model and on the enumeration of directed animals, and we also show that our methods allow to find new results for an extension of the classical TASEP model. As another original result, we describe some families of PCA for which the invariant measure can be explicitly computed, although it does not have a simple product or Markovian form.
In this work, we are interested in the link between strong solutions of the Boltzmann and the Navier–Stokes equations. To justify this connection, our main idea is to use information on the limit system (for instance the fact that the Navier–Stokes equations are globally wellposed in two space dimensions or when the initial data is small). In particular we prove that the life span of the solutions to the rescaled Boltzmann equation is bounded from below by that of the Navier–Stokes system. We deal with general initial data in the whole space in dimensions 2 and 3, and also with well-prepared data in the case of periodic boundary conditions.
We study irreducible components of the set of polynomial plane differential systems with a center, which can be seen as a modern formulation of the classical center-focus problem. The emphasis is given on the interrelation between the geometry of the center set and the Picard–lefschetz theory of the bifurcation (or Poincaré–Pontryagin–Melnikov) functions. Our main illustrative example is the center-focus problem for the Abel equation on a segment, which is compared to the related polynomial Liénard equation.
Many integrable physical systems exhibit Keplerian shear. We look at this phenomenon from the point of view of ergodic theory, where it can be seen as mixing conditionally to an invariant -algebra. In this context, we give a sufficient criterion for Keplerian shear to appear in a system, investigate its genericity and, in a few cases, its speed. Some additional, non-Hamiltonian, examples are discussed.
We prove that the Poisson–Boolean percolation on undergoes a sharp phase transition in any dimension under the assumption that the radius distribution has a finite moment (in particular we do not assume that the distribution is bounded). To the best of our knowledge, this is the first proof of sharpness for a model in dimension that does not exhibit exponential decay of connectivity probabilities in the subcritical regime. More precisely, we prove that in the whole subcritical regime, the expected size of the cluster of the origin is finite, and furthermore we obtain bounds for the origin to be connected to distance : when the radius distribution has a finite exponential moment, the probability decays exponentially fast in , and when the radius distribution has heavy tails, the probability is equivalent to the probability that the origin is covered by a ball going to distance (this result is new even in two dimensions). In the supercritical regime, it is proved that the probability of the origin being connected to infinity satisfies a mean-field lower bound. The same proof carries on to conclude that the vacant set of Poisson–Boolean percolation on undergoes a sharp phase transition.