Institut Élie Cartan de Lorraine

This book studies observation and control operators for linear systems where the free evolution of the state can be described by an operator semigroup on a Hilbert space. The emphasis is on well-posedness, observability and controllability properties. The abstract results are supported by a large number of examples coming mostly from partial differential equations. These examples are worked out in detail. This book is meant to be an elementary introduction in this theory. The first meaning of "elementary'' is that the text is aimed to be accessible to any reader familiar with linear algebra, calculus, the basics of Hilbert spaces and differential equations. We introduce everything needed on operator semigroups and most of the background used is summarized in the Appendices, often with proofs. The second meaning of "elementary'' is that we only cover results for which we can provide complete proofs. In our bibliographic comments we mention some of the more advanced results, for example those based on microlocal analysis.

This book provides a self contained, thorough introduction to the analytic and probabilistic methods of number theory. The prerequisites being reduced to classical contents of undergraduate courses, it offers to students and young researchers a systematic and consistent account on the subject. It is also a convenient tool for professional mathematicians, who may use it for basic references concerning many fundamental topics. Deliberately placing the methods before the results, the book will be of use beyond the particular material addressed directly. Each chapter is complemented with bibliographic notes, useful for descriptions of alternative viewpoints, and detailed exercises, often leading to research problems. This third edition of a text that has become classical offers a renewed and considerably enhanced content, being expanded by more than 50 percent. Important new developments are included, along with original points of view on many essential branches of arithmetic and an accurate perspective on up-to-date bibliography. The author has made important contributions to number theory and his mastery of the material is reflected in the exposition, which is lucid, elegant, and accurate. -Mathematical Reviews.

Problems linking the shape of a domain or the coefficients of an elliptic operator to the sequence of its eigenvalues are among the most fascinating of mathematical analysis. In this book, we focus on

Abstract We improve the current upper and lower bounds for the normal order of the Erdős–Hooley Δ–function obtaining, for almost all integers n , the inequalities where the exponent γ := (log 2)/log((1−1/log 27)/(1 − 1/log 3)) ≈ 0.33827 is conjectured to be optimal.

Stein's method is a collection of probabilistic techniques that allow one to assess the distance between two probability distributions by means of differential operators. In 2007, the authors discovered that one can combine Stein's method with the powerful Malliavin calculus of variations, in order to deduce quantitative central limit theorems involving functionals of general Gaussian fields. This book provides an ideal introduction both to Stein's method and Malliavin calculus, from the standpoint of normal approximations on a Gaussian space. Many recent developments and applications are studied in detail, for instance: fourth moment theorems on the Wiener chaos, density estimates, Breuer–Major theorems for fractional processes, recursive cumulant computations, optimal rates and universality results for homogeneous sums. Largely self-contained, the book is perfect for self-study. It will appeal to researchers and graduate students in probability and statistics, especially those who wish to understand the connections between Stein's method and Malliavin calculus.

We prove that a holomorphic line bundle on a projective manifold is pseudo-effective if and only if its degree on any member of a covering family of curves is non-negative. This is a consequence of a duality statement between the cone of pseudo-effective divisors and the cone of “movable curves”, which is obtained from a general theory of movable intersections and approximate Zariski decomposition for closed positive <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis 1 comma 1 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(1,1)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -currents. As a corollary, a projective manifold has a pseudo-effective canonical bundle if and only if it is not uniruled. We also prove that a 4-fold with a canonical bundle which is pseudo-effective and of numerical class zero in restriction to curves of a good covering family, has non-negative Kodaira dimension.

The cosmic web is one of the most striking features of the distribution of galaxies and dark matter on the largest scales in the Universe. It is composed of dense regions packed full of galaxies, long filamentary bridges, flattened sheets and vast low-density voids. The study of the cosmic web has focused primarily on the identification of such features, and on understanding the environmental effects on galaxy formation and halo assembly. As such, a variety of different methods have been devised to classify the cosmic web -depending on the data at hand, be it numerical simulations, large sky surveys or other. In this paper, we bring 12 of these methods together and apply them to the same data set in order to understand how they compare. In general, these cosmic-web classifiers have been designed with different cosmological goals in mind, and to study different questions. Therefore, one would not a priori expect agreement between different techniques; however, many of these methods do converge on the identification of specific features. In this paper, we study the agreements and disparities of the different methods. For example, each method finds that knots inhabit higher density regions than filaments, etc. and that voids have the lowest densities. For a given web environment, we find a substantial overlap in the density range assigned by each web classification scheme. We also compare classifications on a halo-by-halo basis; for example, we find that 9 of 12 methods classify around a third of group-mass haloes (i.e. M halo 10 13.5 h -1 M ) as being in filaments. Lastly, so that any future cosmic-web classification scheme can be compared to the 12 methods used here, we have made all the data used in this paper public.

We consider a discrete model that describes a locally regulated spatial population with mortality selection. This model was studied in parallel by Bolker and Pacala and Dieckmann, Law and Murrell. We first generalize this model by adding spatial dependence. Then we give a pathwise description in terms of Poisson point measures. We show that different normalizations may lead to different macroscopic approximations of this model. The first approximation is deterministic and gives a rigorous sense to the number density. The second approximation is a superprocess previously studied by Etheridge. Finally, we study in specific cases the long time behavior of the system and of its deterministic approximation.

In this review article we discuss different techniques to solve numerically the time-dependent Schr\"odinger equation on unbounded domains. We present in detail the most recent approaches and describe briefly alternative ideas pointing out the relations bet\-ween these works. We conclude with several numerical examples from different application areas to compare the presented techniques. We mainly focus on the one-dimensional problem but also touch upon the situation in two space dimensions and the cubic nonlinear case.

This article gives a description, by means of functorial intrinsic fibrations, of the geometric structure (and conjecturally also of the Kobayashi pseudometric, as well as of the arithmetic in the projective case) of compact Kähler manifolds. We first define special manifolds as being the compact Kähler manifolds with no meromorphic map onto an orbifold of general type, the orbifold structure on the base being given by the divisor of multiple fibres. We next show that rationally connected Kähler manifolds or Kähler manifolds with zero Kodaira dimension are special. For any <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>X</mml:mi> </mml:math> , we then construct the unique functorial fibration <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>c</mml:mi> <mml:mi>X</mml:mi> </mml:msub> <mml:mo>:</mml:mo> <mml:mi>X</mml:mi> <mml:mo>→</mml:mo> <mml:mi>C</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> (called its core), such that its general fibre is special, and its orbifold base is either of general type, or a point (the last case occuring if and only if <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>X</mml:mi> </mml:math> is special). We next show that the core has a canonical and functorial decomposition as a tower of fibrations with generic (orbifold) fibres either <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>κ</mml:mi> </mml:math> -rationally generated (a weak version of rational connectedness), or with zero Kodaira dimension. In particular, special manifolds are thus canonically towers of such fibrations. The main technical ingredient in the proofs is an orbifold version of Iitaka’s <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>C</mml:mi> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>,</mml:mo> <mml:mi>m</mml:mi> </mml:mrow> </mml:msub> </mml:math> additivity conjecture, proved here when the orbifold base is of general type. The core of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>X</mml:mi> </mml:math> also gives a very simple conjectural qualitative of description of both the Kobayashi pseudometric and the distribution of its <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>K</mml:mi> </mml:math> -rational points (if <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>X</mml:mi> </mml:math> is projective), description which reduces to Lang’s conjectures when <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>X</mml:mi> </mml:math> is of general type.

International audience

Different types of uniqueness (e.g. pathwise uniqueness, uniqueness in law, joint uniqueness in law) and existence (e.g. strong solution, martingale solution) for stochastic evolution equations driven by a Wiener process are studied and compared. We show

The aim of this paper is to introduce and study quadratic Hom–Lie algebras, which are Hom–Lie algebras equipped with symmetric invariant nondegenerate bilinear forms. We provide several constructions leading to examples and extend the Double Extension Theory to this class of nonassociative algebras. Elements of Representation Theory for Hom–Lie algebras, including adjoint and coadjoint representations, are supplied with application to quadratic Hom–Lie algebras. Centerless involutive quadratic Hom–Lie algebras are characterized. We reduce the case where the twist map is invertible to the study of involutive quadratic Lie algebras. Also, we establish a correspondence between the class of involutive quadratic Hom–Lie algebras and quadratic simple Lie algebras with symmetric involution.

We prove approximate controllability of the bilinear Schrödinger equation in the case in which the uncontrolled Hamiltonian has discrete non-resonant spectrum. The results that are obtained apply both to bounded or unbounded domains and to the case in which the control potential is bounded or unbounded. The method relies on finite-dimensional techniques applied to the Galerkin approximations and permits, in addition, to get some controllability properties for the density matrix. Two examples are presented: the harmonic oscillator and the 3D well of potential, both controlled by suitable potentials. Résumé Nous montrons la contrôlabilité approchée de l'équation de Schrödinger bilinéaire dans le cas où l'hamiltonien non contrôlé a un spectre discret et non-résonnant. Les résultats obtenus sont valables que le domaine soit borné ou non, et que le potentiel de contrôle soit borné ou non. La preuve repose sur des méthodes de dimension finie appliquées aux approximations de Galerkyn du système. Ces méthodes permettent en plus d'obtenir des résultats de contrôlabilité des matrices de densité. Deux exemples sont présentés, l'oscillateur harmonique et le puits de potentiel en dimension trois, munis de potentiels de contrôle adéquats.

INTRODUCTION: In order to study social health inequalities, contextual (or ecologic) data may constitute an appropriate alternative to individual socioeconomic characteristics. Indices can be used to summarize the multiple dimensions of the neighborhood socioeconomic status. This work proposes a statistical procedure to create a neighborhood socioeconomic index. METHODS: The study setting is composed of three French urban areas. Socioeconomic data at the census block scale come from the 1999 census. Successive principal components analyses are used to select variables and create the index. Both metropolitan area-specific and global indices are tested and compared. Socioeconomic categories are drawn with hierarchical clustering as a reference to determine "optimal" thresholds able to create categories along a one-dimensional index. RESULTS: Among the twenty variables finally selected in the index, 15 are common to the three metropolitan areas. The index explains at least 57% of the variance of these variables in each metropolitan area, with a contribution of more than 80% of the 15 common variables. CONCLUSIONS: The proposed procedure is statistically justified and robust. It can be applied to multiple geographical areas or socioeconomic variables and provides meaningful information to public health bodies. We highlight the importance of the classification method. We propose an R package in order to use this procedure.

In this paper we consider second order evolution equations with unbounded feedbacks. Under a regularity assumption we show that observability properties for the undamped problem imply decay estimates for the damped problem. We consider both uniform and non uniform decay properties.

BACKGROUND: Double-blind placebo-controlled food challenge (DBPCFC) is currently considered the gold standard for peanut allergy diagnosis. However, this procedure that requires the hospitalization of patients, mostly children, in specialized centers for oral exposure to allergens may cause severe reactions requiring emergency measures. Thus, a simpler and safer diagnosis procedure is needed. The aim of this study was to evaluate the diagnostic performance of a new set of in vitro blood tests for peanut allergy. METHODS: The levels of IgE directed towards peanut extract and recombinant peanut allergens Ara h 1, Ara h 2, Ara h 3, Ara h 6, Ara h 7, and Ara h 8 were measured in 3 groups of patients enrolled at 2 independent centers: patients with proven peanut allergy (n=166); pollen-sensitized subjects without peanut allergy (n=61), and control subjects without allergic disease (n=10). RESULTS: Seventy-nine percent of the pollen-sensitized patients showed IgE binding to peanut, despite their tolerance to peanut. In contrast, combining the results of specific IgE to peanut extract and to recombinant Ara h 2 and Ara h 6 yielded a peanut allergy diagnosis with a 98% sensitivity and an 85% specificity at a positivity threshold of 0.10 kU/l. Use of a threshold of 0.23 kU/l for recombinant Ara h 2 increased specificity (96%) at the cost of sensitivity (93%). CONCLUSION: A simple blood test can be used to diagnose peanut allergy with a high level of precision. However, DBPCFC will remain useful for the few cases where immunological and clinical observations yield conflicting results.

This paper is concerned with the internal stabilization of the generalized Korteweg--de Vries equation on a bounded domain. The global well-posedness and the exponential stability are investigated when the exponent in the nonlinear term ranges over the interval [1,4). The global exponential stability is obtained whatever the location where the damping is active, confirming positively a conjecture of Perla Menzala, Vasconcellos, and Zuazua [Quart. Appl. Math., 60 (2002), pp. 111-129].

Fouvry and Iwaniec's theorem concerning three-dimensional exponential sums with monomials relies on a spacing lemma whose optimal form is yet unproved. We bypass their spacing lemma via a diophantine problem in four variables and we obtain the expected bound in their theorem. In the problem of abelian groups of a given order, this yields the exponent 1/4 + ε, a result close to a conjecture of H. E. Richert (1952).

Abstract The quadratic variation of a continuous process (when it exists) is defined through a regularization procedure. A large class of finite quadratic variation processes is provided, with a particular emphasis on Gaussian processes. For such processes a calculus is developed with application to the study of some stochastic differential equations. Keywords: Stochastic integrals and differential equationsItô's formulaItôprocessesQuadratic variationGaussian processesAMS Math. Classifications: Primary: 60H05, Secondary: 60G44, 60H10 *Corresponding author. *Corresponding author. Notes *Corresponding author.