Consider the abelian category $𝒞$ of commutative group schemes of finite type over a field $k$, its full subcategory $ℱ$ of finite group schemes, and the associated pro-category $\mathrm{Pro}\left(𝒞\right)$ (resp. $\mathrm{Pro}\left(ℱ\right)$) of pro-algebraic (resp. profinite) group schemes. When $k$ is perfect, we show that the profinite fundamental group ${\varpi }_1:\mathrm{Pro}\left(𝒞\right)\to \mathrm{Pro}\left(ℱ\right)$ is left exact and commutes with base change under algebraic field extensions; as a consequence, the higher profinite homotopy functors ${\varpi }_i$ vanish for $i\ge 2$. Along the way, we describe the indecomposable projective objects of $\mathrm{Pro}\left(𝒞\right)$ over an arbitrary field $k$.

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We show that for any group $G$ that is hyperbolic relative to subgroups that admit a proper affine isometric action on a uniformly convex Banach space, then $G$ acts properly on a uniformly convex Banach space as well.

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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.

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We consider the weakly asymmetric simple exclusion process on the discrete space $\{1,\cdots ,n-1\}(n\in \mathbb{N})$, in contact with stochastic reservoirs, both with density $\rho \in (0,1)$ at the extremity points, and starting from the invariant state, namely the Bernoulli product measure of parameter $\rho$. Under time diffusive scaling $t{n}^2$ and for $\rho =\frac{1}{2}$, when the asymmetry parameter is taken of order $1/\sqrt{n}$, 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.

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In this article, we study the excursion sets ${𝒟}_p=f^{-1}\left(\right[-p,+\infty \left[\right)$ where $f$ is a natural real-analytic planar Gaussian field called the Bargmann–Fock field. More precisely, $f$ is the centered Gaussian field on ${ℝ}^2$ with covariance $(x,y)\mapsto exp(-\frac{1}{2}|x-y{|}^2)$. Alexander has proved that, if $p\le 0$, then a.s. ${𝒟}_p$ has no unbounded component. We show that conversely, if $p>0$, then a.s. ${𝒟}_p$ has a unique unbounded component. As a result, the critical level of this percolation model is $0$. 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.

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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 $1/2$ in Wasserstein distance. The paper is illustrated by numerical simulations that indicate that this convergence order should be optimal.

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We consider shape optimization problems for general integral functionals of the calculus of variations, defined on a domain $\Omega$ that varies over all subdomains of a given bounded domain $D$ of ${ℝ}^d$. 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.

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We introduce several classes of polytopes contained in ${[0,1]}^n$ 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 $h^{*}$–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.

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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 $2$–dimensional renewal approach, with a loop exponent $2+\alpha$ ($\alpha >0$): it was found to undergo a localization/delocalization phase transition (which corresponds to the denaturation transition) of order $\nu =min{(1,\alpha )}^{-1}$, 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 $d$-dimensional disordered systems (here $d=2$). First, we prove that when $\alpha <1$ (i.e. $\nu >d/2$), then disorder is irrelevant: the quenched and annealed critical points are equal, and the disordered denaturation phase transition is also of order $\nu ={\alpha }^{-1}$. On the other hand, when $\alpha >1$, 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.

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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 ${\mathrm{SL}}_N\left(ℂ\right)$.

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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.

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In this paper, we give a complete topological, as well as geometrical classification of closed $3$-dimensional Lorentz manifolds admitting a noncompact isometry group.

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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 $2$ contain a rational curve, and so do $K$-trivial varieties with finite fundamental group which are covered by elliptic curves.

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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 $t+1$ is drawn according to a distribution which is a function of the states of its two nearest neighbours at time $t$, and of its own state at time $t-1$. 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 $8$-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.

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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.

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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.

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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 $\sigma$-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.

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We prove that the Poisson–Boolean percolation on ${ℝ}^d$ undergoes a sharp phase transition in any dimension under the assumption that the radius distribution has a $5d-3$ 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 $d\ge 3$ 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 $n$: when the radius distribution has a finite exponential moment, the probability decays exponentially fast in $n$, 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 $n$ (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 ${ℝ}^d$ undergoes a sharp phase transition.

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We examine how the measure and the number of vertices of the convex hull of a random sample of $n$ points from an arbitrary probability measure in ${ℝ}^d$ relate to the wet part of that measure. This extends classical results for the uniform distribution from a convex set proved by Bárány and Larman in 1988. The lower bound of Bárány and Larman continues to hold in the general setting, but the upper bound must be relaxed by a factor of $logn$. We show by an example that this is tight.

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This article is devoted to the characterization of the minimal null control time for abstract linear control problem. More precisely we aim at giving a precise answer to the following question: what is the minimal time needed to drive the solution of the system starting from any initial condition in a given subspace to zero? Our setting will encompass a wide variety of systems of coupled one dimensional linear parabolic equations with a scalar control.

Following classical ideas we reduce this controllability issue to the resolution of a moment problem on the control and provide a new block resolution technique for this moment problem. The obtained estimates are sharp and hold uniformly for a certain class of operators. This uniformity allows various applications for parameter dependent control problems and permits us to deal naturally with the case of algebraically multiple eigenvalues in the underlying generator.

Our approach sheds light on a new phenomenon: the condensation of eigenvalues (which can cause a non zero minimal null control time in general) can be somehow compensated by the condensation of eigenvectors. We provide various examples (some are abstract systems, others are actual PDE systems) to highlight those new situations we are able to manage by the block resolution of the moment problem we propose.

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Growth-fragmentation processes describe the evolution of systems of cells which grow continuously and fragment suddenly; they are used in models of cell division and protein polymerisation. Typically, we may expect that in the long run, the concentrations of cells with given masses increase at some exponential rate, and that, after compensating for this, they arrive at an asymptotic profile. Up to now, this question has mainly been studied for the average behavior of the system, often by means of a natural partial integro-differential equation and the associated spectral theory. However, the behavior of the system as a whole, rather than only its average, is more delicate. In this work, we show that a criterion found by one of the authors for exponential ergodicity on average is actually sufficient to deduce stronger results about the convergence of the entire collection of cells to a certain asymptotic profile, and we find some improved explicit conditions for this to occur.

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Consider an ergodic stationary random field $A$ on the ambient space ${ℝ}^d$. In order to establish concentration properties for nonlinear functions $Z\left(A\right)$, it is standard to appeal to functional inequalities like Poincaré or logarithmic Sobolev inequalities in the probability space. These inequalities are however only known to hold for a restricted class of laws (product measures, Gaussian measures with integrable covariance, or more general Gibbs measures with nicely behaved Hamiltonians). In this contribution, we introduce variants of these inequalities, which we refer to as multiscale functional inequalities and which still imply fine concentration properties, and we develop a constructive approach to such inequalities. We consider random fields that can be viewed as transformations of a product structure, for which the question is reduced to devising approximate chain rules for nonlinear random changes of variables. This approach allows us to cover most examples of random fields arising in the modelling of heterogeneous materials in the applied sciences, including Gaussian fields with arbitrary covariance function, Poisson random inclusions with (unbounded) random radii, random parking and Matérn-type processes, as well as Poisson random tessellations. The obtained multiscale functional inequalities, which we primarily develop here in view of their application to concentration and to quantitative stochastic homogenization, are of independent interest.

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A well-known theorem of Wolpert shows that the Weil–Petersson symplectic form on Teichmüller space, computed on two infinitesimal twists along simple closed geodesics on a fixed hyperbolic surface, equals the sum of the cosines of the intersection angles. We define an infinitesimal deformation starting from a more general object, namely a balanced geodesic graph, by which any tangent vector to Teichmüller space can be represented. We then prove a generalization of Wolpert’s formula for these deformations. In the case of simple closed curves, we recover the theorem of Wolpert.

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L’approximation d’une racine isolée multiple est un problème difficile. En effet la racine peut même être répulsive pour une méthode de point fixe comme la méthode de Newton. La littérature sur le sujet est vaste mais les réponses proposées pour résoudre ce problème ne sont pas satisfaisantes. Des méthodes numériques qui permettent de faire une analyse locale de convergence sont souvent élaborées sous des hypothèses particulières. Ce point de vue privilégiant l’analyse numérique néglige la géométrie et la structure de l’algèbre locale. C’est ainsi qu’ont émergé des méthodes qualifiés de symboliques-numériques. Mais l’analyse numérique précise de ces méthodes pourtant riches d’enseignement n’a pas été faite. Nous proposons dans cet article une méthode de type symbolique-numérique dont le traitement numérique est certifié. L’idée générale est de construire une suite finie de systèmes admettant la même racine, appelée suite de déflation, telle que la multiplicité de la racine chute strictement entre deux systèmes successifs. La racine devient ainsi régulière lors du dernier système. Il suffit alors d’en extraire un système carré régulier pour obtenir ce que nous appelons système déflaté. Nous avions déjà décrit la construction de cette suite de déflation quand la racine est connue. L’originalité de cette étude consiste d’une part à définir une suite de déflation à partir d’un point proche de la racine et d’autre part à donner une analyse numérique de cette méthode. Le cadre fonctionnel de cette analyse est celui des systèmes analytiques constitués de fonctions de carré intégrable. En utilisant le noyau de Bergman, noyau reproduisant de cet espace fonctionnel, nous donnons une $\alpha$-théorie à la Smale de cette suite de déflation. De plus nous présentons des résultats nouveaux relatifs à la détermination du rang numérique d’une matrice et à celle de la proximité à zéro de l’application évaluation. Comme conséquence importante nous donnons un algorithme de calcul d’une suite de déflation qui est libre de $\epsilon$, quantité-seuil qui mesure l’approximation numérique, dans le sens que les entrées de cet algorithme ne comportent pas la variable $\epsilon$.

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We initiate the theory of ${\ell }^p$-improving inequalities for arithmetic averages over hypersurfaces and their maximal functions. In particular, we prove ${\ell }^p$-improving estimates for the discrete spherical averages and some of their generalizations. As an application of our ${\ell }^p$-improving inequalities for the dyadic discrete spherical maximal function, we give a new estimate for the full discrete spherical maximal function in four dimensions. Our proofs are analogous to Littman’s result on Euclidean spherical averages. One key aspect of our proof is a Littlewood–Paley decomposition in both the arithmetic and analytic aspects. In the arithmetic aspect this is a major arc-minor arc decomposition of the circle method.

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In this article, we examine how the structure of soluble groups of infinite torsion-free rank with no section isomorphic to the wreath product of two infinite cyclic groups can be analysed. As a corollary, we obtain that if a finitely generated soluble group has a defined Krull dimension and has no sections isomorphic to the wreath product of two infinite cyclic groups then it is a group of finite torsion-free rank. There are further corollaries including applications to return probabilities for random walks. The paper concludes with constructions of examples that can be compared with recent constructions of Brieussel and Zheng.

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We consider the totally asymmetric simple exclusion process with soft-shock initial particle density, which is a step function increasing in the direction of flow and the step size chosen small to admit KPZ scaling. The initial configuration is deterministic and the dynamics create a shock.

We prove that the fluctuations of a particle at the macroscopic position of the shock converge to the maximum of two independent GOE Tracy–Widom random variables, which establishes a conjecture of Ferrari and Nejjar. Furthermore, we show the joint fluctuations of particles near the shock are determined by the maximum of two lines described in terms of these two random variables. The microscopic position of the shock is then seen to be their difference.

Our proofs rely on determinantal formulae and a novel factorization of the associated kernels.

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In this paper, we are interested in path-dependent stochastic differential equations (SDEs) which are controlled by Brownian motion and its delays. Within this non-Markovian context, we give a Hörmander-type criterion for the regularity of solutions. Indeed, our criterion is expressed as a spanning condition with brackets. A novelty in the case of delays is that noise can “flow from the past” and give additional smoothness thanks to semi-brackets.

The proof follows the general lines of Malliavin’s probabilistic proof, in the Markovian case. Nevertheless, in order to handle the non-Markovian aspects of this problem and to treat anticipative integrals in a path-wise fashion, we heavily invoke rough path integration.

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We study a Schrödinger equation which describes the dynamics of an electron in a crystal in the presence of impurities. We consider the regime of small wave-lengths comparable to the characteristic scale of the crystal. It is well-known that under suitable assumptions on the initial data and for highly oscillating potential, the wave function can be approximated by the solution of a simpler equation, the effective mass equation. Using Floquet–Bloch decomposition, as it is classical in this subject, we establish effective mass equations in a rather general setting. In particular, Bloch bands are allowed to have degenerate critical points, as may occur in dimension strictly larger than one. Our analysis leads to a new type of effective mass equations which are operator-valued and of Heisenberg form and relies on Wigner measure theory and, more precisely, to its applications to the analysis of dispersion effects.

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We construct a semi-orthogonal decomposition on the category of perfect complexes on the blow-up of a derived Artin stack in a quasi-smooth centre. This gives a generalization of Thomason’s blow-up formula in algebraic K-theory to derived stacks. We also provide a new criterion for descent in Voevodsky’s cdh topology, which we use to give a direct proof of Cisinski’s theorem that Weibel’s homotopy invariant K-theory satisfies cdh descent.

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We study the monopole Floer homology of a $\mathsf{Solv}$ rational homology sphere $Y$ from the point of view of spectral theory. Applying ideas of Fourier analysis on solvable groups, we show that for suitable $\mathsf{Solv}$ metrics on $Y$, small regular perturbations of the Seiberg–Witten equations do not admit irreducible solutions; in particular, this provides a geometric proof that $Y$ is an $L$-space.

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Consider a cubic surface satisfying the mild condition that it may be described in Sylvester’s pentahedral form. There is a well-known Enriques or Coble surface $S$ with K3 cover birationally isomorphic to the Hessian surface of this cubic surface. We describe the nef cone and $(-2)$-curves of $S$. In the case of pentahedral parameters $(1,1,1,1,t\ne 0)$ we compute the automorphism group of $S$. For $t\ne 1$ it is the semidirect product of the free product ${(\mathbb{Z}/2)}^{*4}$ and the symmetric group ${𝔖}_4$. In the special case $t=\frac{1}{16}$ we study the action of $Aut\left(S\right)$ on an invariant smooth rational curve $C$ on the Coble surface $S$. We describe the action and its image, both geometrically and arithmetically. In particular, we prove that $Aut\left(S\right)\to Aut\left(C\right)$ is injective in characteristic $0$ and we identify its image with the subgroup of ${PGL}_2$ coming from the isometries of a regular tetrahedron and the reflections across its facets.

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The purpose of this paper is to investigate the spectral effects of an interface between vacuum and a negative-index material (NIM), that is, a dispersive material whose electric permittivity and magnetic permeability become negative in some frequency range. We consider here an elementary situation, namely, 1) the simplest existing model of NIM: the non dissipative Drude model, for which negativity occurs at low frequencies; 2) a two-dimensional scalar model derived from the complete Maxwell’s equations; 3) the case of a simple bounded cavity: a polygonal domain partially filled with a portion of Drude material. Because of the frequency dispersion (the permittivity and permeability depend on the frequency), the spectral analysis of such a cavity is unusual since it yields a nonlinear eigenvalue problem. Thanks to the use of an additional unknown, we linearize the problem and we present a complete description of the spectrum. We show in particular that the interface between the NIM and vacuum is responsible for various resonance phenomena related to various components of an essential spectrum.

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In this paper we prove that the Cauchy problem for first-order quasi-linear systems of partial differential equations is ill-posed in Gevrey spaces, under the assumption of an initial ellipticity. The assumption bears on the principal symbol of the first-order operator. Ill-posedness means instability in the sense of Hadamard, specifically an instantaneous defect of Hölder continuity of the flow from $G^{\sigma }$ to $L^2$, where $\sigma \in (0,1)$ depends on the initial spectrum. Building on the analysis carried out by G. Métivier [Remarks on the well-posedness of the nonlinear Cauchy problem, Contemp. Math. 2005], we show that ill-posedness follows from a long-time Cauchy–Kovalevskaya construction of a family of exact, highly oscillating, analytical solutions which are initially close to the null solution, and which grow exponentially fast in time. A specific difficulty resides in the observation time of instability. While in Sobolev spaces, this time is logarithmic in the frequency, in Gevrey spaces it is a power of the frequency. In particular, in Gevrey spaces the instability is recorded much later than in Sobolev spaces.

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We consider the semilinear damped wave equation

In this article, we obtain the first results concerning the stabilization of this semilinear equation in cases where $\gamma$ does not satisfy the geometric control condition. When some of the geodesic rays are trapped, the stabilization of the linear semigroup is semi-uniform in the sense that $\parallel e^{At}{A}^{-1}\parallel \le h\left(t\right)$ for some function $h$ with $h\left(t\right)\to 0$ when $t\to +\infty$. We provide general tools to deal with the semilinear stabilization problem in the case where $h\left(t\right)$ has a sufficiently fast decay.

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We show that the continuous $L^p$-cohomology of locally compact second countable groups is a quasi-isometric invariant. As an application, we prove partial results supporting a positive answer to a question asked by M. Gromov, suggesting a classical behaviour of continuous $L^p$-cohomology of simple real Lie groups. In addition to quasi-isometric invariance, the ingredients are a spectral sequence argument and Pansu’s vanishing results for real hyperbolic spaces. In the best adapted cases of simple Lie groups, we obtain nearly half of the relevant vanishings.

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A quadrature rule of a measure $\mu$ on the real line represents a conic combination of finitely many evaluations at points, called nodes, that agrees with integration against $\mu$ for all polynomials up to some fixed degree. In this paper, we present a bivariate polynomial whose roots parametrize the nodes of minimal quadrature rules for measures on the real line. We give two symmetric determinantal formulas for this polynomial, which translate the problem of finding the nodes to solving a generalized eigenvalue problem.

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We discuss the Berezin transform, a Markov operator associated to positive operator valued measures (POVMs), in a number of contexts including the Berezin–Toeplitz quantization, Donaldson’s dynamical system on the space of Hermitian inner products on a complex vector space, representations of finite groups, and quantum noise. In particular, we calculate the spectral gap for quantization in terms of the fundamental tone of the phase space. Our results confirm a prediction of Donaldson for the spectrum of the $Q$-operator on Kähler manifolds with constant scalar curvature, and yield exponential convergence of Donaldson’s iterations to the fixed point. Furthermore, viewing POVMs as data clouds, we study their spectral features via geometry of measure metric spaces and the diffusion distance.

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We prove that the law of a random walk $X_n$ is determined by the one-dimensional distributions of $max(X_n,0)$ for $n=1,2,...$, as conjectured recently by Loïc Chaumont and Ron Doney. Equivalently, the law of $X_n$ is determined by its upward space-time Wiener–Hopf factor. Our methods are complex-analytic.

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For a complex projective manifold that is rationally connected, resp. rationally simply connected, every finite subset is connected by a rational curve, resp. the spaces parameterizing these connecting rational curves are themselves rationally connected. We prove that a projective scheme over a global function field has a rational point if it deforms to a rationally simply connected variety in characteristic $0$ with vanishing elementary obstruction. This gives new, uniform proofs over these fields of the Period-Index Theorem, the quasi-split case of Serre’s “Conjecture II”, and Lang’s $C_2$ property.

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