Seminaires et Congres 17 (2009), xiv+182 pages
ISBN: 9782856292396
Le contenu de ce livre est le resultat de la redaction de cours donnes lors de l'ecole CIMPA qui s'est tenue a Damas en mai 2004. Le but de ces cours a ete de presenter differents themes de la recherche actuelle en Analyse ou les problematiques sous-jacentes sont souvent d'origine physique. Ainsi, on y trouve des cours sur les systemes dynamiques etendus (P. Collet), l'analyse semi-classique des operateurs de Schr'odinger en connexion avec la super-conductivite (B. Helffer), les problemes a frontiere libre (R. Monneau) et les EDP fonctionnelles (K. Ezzinbi et M. Adimy).
Mots-clefs : Evolution temporelle, domaines non bornes, instabilites, equations d'amplitude, renormalisation, epsilon entropie, entropie topologique, condition de Hille-Yosida, semi groupe, stabilite, equation carcteristique, formule de variation de la constante, solution bornee, analyse semi-classique, theorie spectrale, operateur de Schrodinger, champ magnetique, frontieres libres, probleme de l'obstacle, blow-up, formule de monotonie, theoreme de Liouville, ensemble singulier
Theoretical and applied aspects of PDEs coming from Geometry or Physics
The content of this book results from the drafting of several courses given at the CIMPA school which was held in Damascus in May 2004. The aim of these courses was to present various topics of current research in Analysis where the underlying problems come from Physics. Thus, one finds courses on extended dynamical systems (P. Collet), the semi-classical analysis of Schr'odinger operators in connection with superconductivity (B. Helffer), free-boundary problems (R. Monneau), and Partial Functional Differential Equations (K. Ezzinbi and M. Adimy).
Keywords: Time evolution, unbounded domains, instabilities, amplitude equation, renormalization, epsilon entropy, topological entropy, Hille-Yosida condition, semigroup, stability, characteristic equations, variation of constants formula, bounded solution, semiclassical analysis, spectral theory, Schrodinger operator, magnetic field, superconductivity, free boundary problems, obstacle problem, blow-up, monotonicity formula, Liouville theorem, singular set
Asterisque 335 (2011), xvi+291 pages
ISBN: 9782856293058
Les proprietes multiplicatives de la filtration par les tranches
Soit un schema noetherien separe de dimension de Krull finie, et la categorie homotopique stable de Morel-Voevodsky. Afin d'obtenir un analogue motivique de la tour de Postnikov, Voevodsky [25] definit la filtration par les tranches dans considerant les smash-produits iterees de le groupe multiplicatif . Nous montrons que la filtration par les tranches est compatible avec le smash-produit dans la categorie de Jardine des -spectres symetriques motiviques [14]. Cette compatibilite a plusieurs consequences interessantes. D'entre eux, sur un corps parfait tous les tranches sont canoniquement modules dans sur le spectre motivique d'Eilenberg-MacLane , et si le corps est de characteristique zero les tranches sont motifs grands au sens de Voevodsky, ce utilise les resultats de Levine [16], Rondigs-Ostvar [22] et Voevodsky [26]. Nous montrons aussi que le smash-produit dans induit des structures multiplicatives sur la suite spectrale motivique de Atiyah-Hirzebruch.
Mots-clefs : Filtration par les tranches, motifs mixtes, suite spectrale motivique de Atiyah-Hirzebruch, theorie homotopique des schemas
Let be a Noetherian separated scheme of finite Krull dimension, and be the motivic stable homotopy category of Morel-Voevodsky. In order to get a motivic analogue of the Postnikov tower, Voevodsky [25] constructs the slice filtration by filtering with respect to the smash powers of the multiplicative group . We show that the slice filtration is compatible with the smash product in Jardine's category of motivic symmetric -spectra [14], and describe several interesting consequences that follow from this compatibility. Among them, we have that over a perfect field all the slices are in a canonical way modules in over the motivic Eilenberg-MacLane spectrum , and if the field has characteristic zero it follows that the slices are big motives in the sense of Voevodsky, this relies on the work of Levine [16], Rondigs-Ostvar [22] and Voevodsky [26]. It also follows that the smash product in induces pairings in the motivic Atiyah-Hirzebruch spectral sequence.
Keywords: Mixed motives, motivic Atiyah-Hirzebruch spectral sequence, motivic homotopy theory, slice filtration
Asterisque 336 (2011), vii+145 pages
ISBN:9782856293065
Le noyau de la chaleur avec condition de Neumann ou de Dirichlet dans les domaines interieurement uniformes
Ce texte traite de l'etude du noyau de la chaleur avec condition de Neumann ou condition de Dirichlet au bord dans les domaines euclidiens non-bornes, mais aussi les domaines non-bornes dans les varietes riemanniennes et, plus generalement, les domaines non-bornes de certain espaces de Dirichlet reguliers locaux. Les travaux de A. Grigor'yan, L. Saloff-Coste et K-T. Sturm, ont montre l'equivalence, dans un large contexte, des proprietes suivantes : itemize l'inegalite de Harnack parabolique, les estimations gaussiennes du noyau de la chaleur, l'inegalite de Poincare et la propriete de doublement du volume. itemize Nous utilisons ce resultat pour obtenir des estimations precises du noyau de la chaleur pour une large classe de domaines definis en termes de leur distance intrinseque et appeles domaines interieurement (ou intrinsequement) uniformes. De facon peut etre surprenante, nous traitons le probleme avec la condition de Neumann au bord et celui avec la condition de Dirichlet au bord par la meme approche, mais avec l'aide supplementaire d'une transformation de Doob dans le cas de la condition de Dirichlet. Les resultats principaux que nous obtenons sont nouveaux meme dans le cas des domaines euclidiens a bord regulier ou ils capturent l'effet de la condition d'uniformite interieure comme, par exemple, dans le cas des domaines qui sont le complement d'un convexe ferme de .
Mots-clefs : Noyau de chaleur, condition aux limites de Dirichlet, domaines interieurs uniformes, espaces de Dirichlet, inegalites de Harnack
This monograph focuses on the heat equation with either the Neumann or the Dirichlet boundary condition in unbounded domains in Euclidean space, Riemannian manifolds, and in the more general context of certain regular local Dirichlet spaces. In works by A. Grigor'yan, L. Saloff-Coste and K-T. Sturm, the equivalence between itemize the parabolic Harnack inequality, the two-sided Gaussian heat kernel estimate, the Poincare inequality and the volume doubling property, itemize is established in a very general context. We use this result to provide precise two-sided heat kernel estimates in a large class of domains described in terms of their inner intrinsic metric and called inner (or intrinsically) uniform domains. Perhaps surprisingly, we treat both the Neumann boundary condition and the Dirichlet boundary condition using essentially the same approach albeit with the additional help of a Doob's h-transform in the case of Dirichlet boundary condition. The main results are new even when applied to Euclidean domains with smooth boundary where they capture the global effect of the condition of inner uniformity as, for instance, in the case of domains that are the complement of a convex set in Euclidean space.
Keywords: Heat kernel, Dirichlet boundary condition, inner uniform domains, Dirichlet spaces, Harnack inequalities
ISBN: 978-0-470-62169-1
Hardcover
384 pages
June 2011
This book, written by three behavioral scientists for other behavioral scientists, addresses common issues in statistical analysis for the behavioral and educational sciences. Modern Statistical & Computing Methods for the Behavioral and Educational Sciences using R emphasizes the direct link between scientific research questions and data analysis. Purposeful attention is paid to the integration of design, statistical methodology, and computation to propose answers to specific research questions. Furthermore, practical suggestions for the analysis and presentation of results, in prose, tables and/or figures, are included. Optional sections for each chapter include methodological extensions for readers desiring additional technical details. Rather than focus on mathematical calculations like so many other introductory texts in the behavioral sciences, the authors focus on conceptual explanations and the use of statistical computing. Statistical computing is an integral part of statistical work, and to support student learning in this area, examples using the R computer program are provided throughout the book. Rather than relegate examples to the end of chapters, the authors interweave computer examples with the narrative of the book. Topical coverage includes an introduction to R, data exploration of one variable, data exploration of multivariate data - comparing two groups and many groups, permutation and randomization tests, the independent samples t-Test, the Bootstrap test, interval estimates and effect sizes, power, and dependent samples.