Series: Contributions to Statistics
2008, XX, 304 p. 33 illus., Hardcover
ISBN: 978-3-7908-2061-4
An increasing number of statistical problems and methods involve
infinite-dimensional aspects. This is due to the progress of
technologies which allow us to store more and more information while
modern instruments are able to collect data much more effectively
due to their increasingly sophisticated design. This evolution directly
concerns statisticians, who have to propose new methodologies while
taking into account such high-dimensional data (e.g. continuous processes,
functional data, etc.). The numerous applications (micro-arrays, paleo-
ecological data, radar waveforms, spectrometric curves, speech recognition,
continuous time series, 3-D images, etc.) in various fields (biology,
econometrics, environmetrics, the food industry, medical sciences, paper
industry, etc.) make researching this statistical topic very worthwhile.
This book gathers important contributions on the functional and operatorial statistics fields
Series: Understanding Complex Systems
2008, XV, 844 p. 124 illus., Hardcover
ISBN: 978-3-540-79356-4
Due: June 18, 2008
Complex Nonlinearity: Chaos, Phase Transitions, Topology Change and Path Integrals is a book about prediction & control of general nonlinear and chaotic dynamics of high-dimensional complex systems of various physical and non-physical nature and their underpinning geometro-topological change.
The book starts with a textbook-like expose on nonlinear dynamics, attractors and chaos, both temporal and spatio-temporal, including modern techniques of chaos?control. Chapter 2 turns to the edge of chaos, in the form of phase transitions (equilibrium and non-equilibrium, oscillatory, fractal and noise-induced), as well as the related field of synergetics. While the natural stage for linear dynamics comprises of flat, Euclidean geometry (with the corresponding calculation tools from linear algebra and analysis), the natural stage for nonlinear dynamics is curved, Riemannian geometry (with the corresponding tools from nonlinear, tensor algebra and analysis). The extreme nonlinearity ? chaos ? corresponds to the topology change of this curved geometrical stage, usually called configuration manifold. Chapter 3 elaborates on geometry and topology change in relation with complex nonlinearity and chaos. Chapter 4 develops general nonlinear dynamics, continuous and discrete, deterministic and stochastic, in the unique form of path integrals and their action-amplitude formalism. This most natural framework for representing both phase transitions and topology change starts with Feynman’s sum over histories, to be quickly generalized into the sum over geometries and topologies. The last Chapter puts all the previously developed techniques together and presents the unified form of complex nonlinearity. Here we have chaos, phase transitions, geometrical dynamics and topology change, all working together in the form of path integrals.
The objective of this book is to provide a serious reader with a serious scientific tool that will enable them to actually perform a competitive research in modern complex nonlinearity. It includes a comprehensive bibliography on the subject and a detailed index. Target readership includes all researchers and students of complex nonlinear systems (in physics, mathematics, engineering, chemistry, biology, psychology, sociology, economics, medicine, etc.), working both in industry/clinics and academia.
Original French edition published by L' Ecole Polytechnique, 2006
2009, Approx. 205 p. 10 illus., Softcover
ISBN: 978-0-387-78865-4
Due: June 2009
- Combines material from many branches of mathematics, including algebra, geometry, and analysis, so students see connections between these areas
- Applies material to chemistry and physics, so students appreciate the applications of abstract mathematics
- Assumes only linear algebra and calculus, making an advanced subject accessible to undergraduates
- Includes 142 exercises, many with hints or complete solutions, so text may be used in the classroom or for self study
The theory of group representations is an elegant subject at the intersection of algebra, geometry, and analysis, with applications to crystallography, chemistry, and the physics of atomic and subatomic particles. This book is an introduction to both the theory and applications of group representations. The author lends coherence to this vast subject by focusing on the connections between group representations and the study of symmetry.
The book begins with a brisk review of the basic definitions and fundamental results of group theory. Definitions are illustrated with examples important to the study of symmetry, such as the symmetric group and the Lie groups SU(2) and SO(3). The representation theory of finite groups is introduced in Chapter 2, and the principle results generalized to compact groups in Chapter 3. In later chapters Lie groups are introduced, with the Lie groups SU(2) and SO(3) and their representations studied in detail. Chapter 7 on spherical harmonics connects SO(3) to applications in electrodynamics and quantum mechanics, while Chapter 8 applies SU(2) to the study of quarks.
With only linear algebra and calculus as prerequisites, this book will be accessible to both advanced undergraduates and beginning graduate students. An extensive collection of exercises, many with answers or complete solutions, makes this an ideal text for the classroom or independent study.
Introduction.- Groups.- Representations of Finite Groups.- Representations of Compact Groups.- Lie algebras and Lie groups.- Lie groups SU(2) and SO(3).- Representation of SU(2) and SO(3).- Problems and Solutions
Series: Publications of the Scuola Normale Superiore
Subseries: CRM Series , Vol. 5
2008, Approx. 140 p., Softcover
ISBN: 978-88-7642-329-1
Due: June 2008
The main aim is to present at the level of beginners several modern tools of micro-local analysis which are useful for the mathematical study of nonlinear partial differential equations. The core of these notes is devoted to a presentation of the para-differential techniques, which combine a linearization procedure for nonlinear equations, and a symbolic calculus which mimics or extends the classical calculus of Fourier multipliers. These methods apply to many problems in nonlinear PDE’s such as elliptic equations, propagation of singularities, boundary value problems, shocks or boundary layers. However, in these introductory notes, we have chosen to illustrate the theory on two selected and relatively simple examples, which allow becoming familiar with the techniques. They concern the well posed-ness of the Cauchy problem for systems of nonlinear PDE's, firstly hyperbolic systems and secondly coupled systems of Schrodinger equations which arise in various models of wave propagation.
I. Introduction to Systems - 1. Notations and Examples - 2. Constant Coefficient Systems. Fourier Synthesis - 3. The Method of Symmetrizers.- II. The Para-Differential Calculus - 1. Pseudo-Differential Operators 2. Para-Differential Operators - 3. Symbolic Calculus.-III. Applications - 1. Nonlinear Hyperbolic systems - 2. Systems of Schrodinger Equations
Series: Lecture Notes in Mathematics
Subseries: Ecole d'Ete Probabilit.Saint-Flour , Vol. 1950
2008, VIII, 190 p. 20 illus., Softcover
ISBN: 978-3-540-68895-2
Queueing networks constitute a large family of stochastic models, involving jobs that enter a network, compete for service, and eventually leave the network upon completion of service. Since the early 1990s, substantial attention has been devoted to the question of when such networks are stable.
This volume presents a summary of such work. Emphasis is placed on the use of fluid models in showing stability, and on examples of queueing networks that are unstable even when the arrival rate is less than the service rate.
The material of this volume is based on a series of nine lectures given at the Saint-Flour Probability Summer School 2006.
Lectures were also given by Alice Guionnet and Steffen Lauritzen.