Asterisque 343 (2012), vii+169 pages
ISBN : 978-2-85629-342-3
Topologie des cordes des champs differentiels
Nous construisons un cadre general pour traiter la topologie des cordes des champs differentiels. En particulier, ce cadre s'applique aussi bien aux lacets libres d'un champ qu'aux lacets fantomes, champs d'inertie. On construit une theorie bivariante (au sens de Fulton et MacPherson) pour les champs topologiques et on en deduit l'existence de morphismes de Gysin compatibles avec les operations standards: produits, produits fibres, recollements. Par ailleurs on demontre une formule d'exces pour les fibres normaux sur des champs differentiels. On definit une notion de champs orientes, qui generalise celle de varietes orientees, qui sont les champs sur lesquels on dispose des operations de la topologie des cordes. En particulier, on demontre que l'homologie du champ des lacets libres d'un champ oriente ainsi que l'homologie de son champ des lacets fantomes sont munies de structures naturelles d'algebres de Frobenius. De plus le morphisme naturel entre ces champs de lacets est un morphisme d'algebres de Frobenius. Par ailleurs, on prouve que l'homologie du champ des lacets libres est muni d'une structure de BV-algebre compatible avec la structure d'algebre de Frobenius au sens ou ces structures sont extraites d'une theorie homologique conforme des champs a bords compacts. On applique egalement nos techniques pour etudier un analogue du produit de Chas-Sullivan, ainsi que des operations puissances compatibles, sur l'homologie des champs de morphismes des spheres dans un champ oriente. Notre cadre permet aussi de construire un produit d'intersection pour les orbifolds quasi-complexes (non-necessairement compacts) qui est, en un sens, le dual de Poincare du produit de Chen et Ruan. On demontre de plus que le produit a la Chas-Sullivan des lacets fantomes d'un orbifold quasi-complexe est isomorphe au produit d'intersection tordu par une classe naturelle. On etudie plusieurs exemples, notamment le cas du champ [*/G] classifiant d'un groupe de Lie compact.
Mots-clefs : Topologie des cordes, champs topologiques, espaces de lacets, champ d'inertie, lacets fantomes, theorie bivariante, morphismes de Gysin, theorie conforme des champs, produit orbifold
We establish the general machinery of string topology for differentiable stacks. This machinery allows us to treat on equal footing free loops in stacks and hidden loops. We construct a bivariant (in the sense of Fulton and MacPherson) theory for topological stacks: it gives us a flexible theory of Gysin maps which are automatically compatible with pullback, pushforward and products. Further we prove an excess formula in this context. We introduce oriented stacks, generalizing oriented manifolds, which are stacks on which we can do string topology. We prove that the homology of the free loop stack of an oriented stack and the homology of hidden loops (sometimes called ghost loops) are a Frobenius algebra which are related by a natural morphism of Frobenius algebras. We also prove that the homology of free loop stack has a natural structure of BV-algebra, which together with the Frobenius structure fits into an homological conformal field theories with closed positive boundaries. We also use our constructions to study an analogue of the loop product for stacks of maps of (-dimensional) spheres to oriented stacks and compatible power maps in their homology. Using our general machinery, we construct an intersection pairing for (non necessarily compact) almost complex orbifolds which is in the same relation to the intersection pairing for manifolds as Chen-Ruan orbifold cup-product is to ordinary cup-product of manifolds. We show that the product of almost complex orbifolds is isomorphic to the orbifold intersection pairing twisted by a canonical class. Finally we gave some examples including the case of the classifying stacks of a compact Lie group.
Keywords: String topology, topological stacks, loop stack, inertia stack, hidden loop, bivariant theory, Gysin maps, conformal field theory, orbifold product
Memoires de la SMF 128 (2012), vi+119 pages
SBN : 978-2-85629-344-7
Repulsion par les resonances
Nous considerons des systemes lents-rapides, dont le mouvement rapide est periodique et le mouvement lent integrable, en presence de resonances faibles ou fortes. En supposant que les phases initiales sont aleatoires et que certaines conditions de non-degenerescence sont satisfaites, nous demontrons que l'evolution effective des invariants adiabatiques est donnee par un processus de Markov. Ce processus de Markov consiste en un mouvement le long des trajectoires d'un champ de vecteurs qui peut presenter des sauts occasionnels. Le generateur du processus limite est calcule a partir de la dynamique du systeme au voisinage des resonances fortes.
Mots-clefs : Moyennisation, Systemes lents-rapides, Processus de Markov, Cones invariants, Resonances
We consider slow-fast systems with periodic fast motion and integrable slow motion in the presence of both weak and strong resonances. Assuming that the initial phases are random and that appropriate non-degeneracy assumptions are satisfied we prove that the effective evolution of the adiabatic invariants is given by a Markov process. This Markov process consists of the motion along the trajectories of a vector field with occasional jumps. The generator of the limiting process is computed from the dynamics of the system near strong resonances.
Keywords: Averaging, Slow-Fast Systems, Markov Processes, Invariant Cones, Resonances
ISBN: 978-1-1180-6330-9
Hardcover
736 pages
August 2012
Combining over 28 years of teaching experience, the authors present a PDE text that is accessible to all students?regardless of their background or mathematical sophistication. The book provides over 150 completely worked problems with solutions and commentaries for both linear partial differential equations and boundary value problems with applications in engineering and biology. Topics covered include a classification of PDEs, heat equation, wave equation, Laplace's equation, separation of variables, Fourier series, classical PDEs, Sturm-Liouville problems, special functions, transform methods, and the method of characteristics for first order PDEs.
ISBN: 978-1-1183-1532-3
Hardcover
536 pages
August 2012
An update of one of the most trusted books on constructing and analyzing actuarial models for the C/4 actuarial exam
This new, abridged edition has been thoroughly revised and updated to include the essential material related to Exam C of the Society of Actuaries' and Casualty Actuarial Society’s accreditation programs. The book maintains an approach to modeling and forecasting that utilizes tools related to risk theory, loss distributions, and survival models. Random variables, basic distributional quantities, the recursive method, and techniques for classifying and creating distributions are also discussed. Both parametric and non-parametric estimation methods are thoroughly covered along with advice for choosing an appropriate model.
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Loss Models, Fourth Edition is an essential resource for students and aspiring actuaries who are preparing to take the SOA and CAS preliminary examinations C/4. It is also a must-have reference for professional actuaries, graduate students in the actuarial field, and anyone who works with loss and risk models in their everyday work.
ISBN: 978-1-1199-4182-8
Hardcover
504 pages
November 2012
Provides an accessible foundation to Bayesian analysis using real world models
This book aims to present an introduction to Bayesian modelling and computation, by considering real case studies drawn from diverse fields spanning ecology, health, genetics and finance. Each chapter comprises a description of the problem, the corresponding model, the computational method, results and inferences as well as the issues that arise in the implementation of these approaches
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Case Studies in Bayesian Statistical Modelling and Analysis is aimed at statisticians, researchers and practitioners who have some expertise in statistical modelling and analysis, and some understanding of the basics of Bayesian statistics, but little experience in its application. Graduate students of statistics and biostatistics will also find this book beneficial.