(Hardback)ISBN-13: 978-0-19-921290-3
(Paperback)ISBN-13: 978-0-19-921291-0
Estimated publication date: May 2009
460 pages, 32 line illustrations and 4 halftones, 234x156 mm
Series: Oxford Texts in Applied and Engineering Mathematics number 12
Classical mechanics, one of the oldest branches of science, has undergone a long evolution, developing hand in hand with many areas of mathematics, including calculus, differential geometry, and the theory of Lie groups and Lie algebras. The modern formulations of Lagrangian and Hamiltonian mechanics, in the coordinate-free language of differential geometry, are elegant and general. They provide a unifying framework for many seemingly disparate physical systems, such as n?particle systems, rigid bodies, fluids and other continua, and electromagnetic and quantum systems.
Geometric Mechanics and Symmetry is a friendly and fast-paced introduction to the geometric approach to classical mechanics, suitable for a one- or two- semester course for beginning graduate students or advanced undergraduates. It fills a gap between traditional classical mechanics texts and advanced modern mathematical treatments of the subject. After a summary of the necessary elements of calculus on smooth manifolds and basic Lie group theory, the main body of the text considers how symmetry reduction of Hamilton's principle allows one to derive and analyze the Euler-Poincare equations for dynamics on Lie groups.
Additional topics deal with rigid and pseudo-rigid bodies, the heavy top, shallow water waves, geophysical fluid dynamics and computational anatomy. The text ends with a discussion of the semidirect-product Euler-Poincare reduction theorem for ideal fluid dynamics.
A variety of examples and figures illustrate the material, while the many exercises, both solved and unsolved, make the book a valuable class text.
Readership: Graduate and advanced undergraduate students in mathematics, physics, and engineering.
Preface
Acknowledgements
PART I
1. Lagrangian and Hamiltonian Mechanics
2. Manifolds
3. Geometry on Manifolds
4. Mechanics on Manifolds
5. Lie Groups and Lie Algebras
6. Group Actions, Symmetries and Reduction
7. Euler-Poincare Reduction: Rigid body and heavy top
8. Momentum Maps
9. Lie-Poisson Reduction
10. Pseudo-Rigid Bodies
PART II
11. EPDiff
12. EPDiff Solution Behaviour
13. Integrability of EPDiff in 1D
14. EPDiff in n Dimensions
15. Computational Anatomy: contour matching using EPDiff
16. Computational Anatomy: Euler?Poincare image matching
17. Continuum Equations with Advection
18. Euler?Poincare Theorem for Geophysical Fluid Dynamics
Bibliography
Darryl D. Holm, Professor, Mathematics Department, Imperial College London,
Tanya Schmah, Department of Computer Science, University of Toronto and Department of Mathematics, Macquarie University, Australia, and
Cristina Stoica, Department of Mathematics, Wilfrid Laurier University, Canada
Princeton Mathematical Series
Cloth | March 2009 |
696 pp. | 6 x 9 | 2 halftones. 17 line illus.
This book explores the most recent developments in the theory of planar quasiconformal mappings with a particular focus on the interactions with partial differential equations and nonlinear analysis. It gives a thorough and modern approach to the classical theory and presents important and compelling applications across a spectrum of mathematics: dynamical systems, singular integral operators, inverse problems, the geometry of mappings, and the calculus of variations. It also gives an account of recent advances in harmonic analysis and their applications in the geometric theory of mappings.
The book explains that the existence, regularity, and singular set structures for second-order divergence-type equations--the most important class of PDEs in applications--are determined by the mathematics underpinning the geometry, structure, and dimension of fractal sets; moduli spaces of Riemann surfaces; and conformal dynamical systems. These topics are inextricably linked by the theory of quasiconformal mappings. Further, the interplay between them allows the authors to extend classical results to more general settings for wider applicability, providing new and often optimal answers to questions of existence, regularity, and geometric properties of solutions to nonlinear systems in both elliptic and degenerate elliptic settings.
Kari Astala is the Finnish Academy Professor of Mathematics at the University of Helsinki. Tadeusz Iwaniec is the John Raymond French Distinguished Professor of Mathematics at Syracuse University. Gaven Martin is the Distinguished Professor of Mathematics at Massey University.
Paper | June 2009 | 978-0-691-14039-1
Cloth | June 2009 | 978-0-691-13287-7
1080 pp. | 7 x 10 | 4 line illus.
When first published in 2005, Matrix Mathematics quickly became the essential reference book for users of matrices in all branches of engineering, science, and applied mathematics. In this fully updated and expanded edition, the author brings together the latest results on matrix theory to make this the most complete, current, and easy-to-use book on matrices.
Each chapter describes relevant background theory followed by specialized results. Hundreds of identities, inequalities, and matrix facts are stated clearly and rigorously with cross references, citations to the literature, and illuminating remarks. Beginning with preliminaries on sets, functions, and relations,Matrix Mathematics covers all of the major topics in matrix theory, including matrix transformations; polynomial matrices; matrix decompositions; generalized inverses; Kronecker and Schur algebra; positive-semidefinite matrices; vector and matrix norms; the matrix exponential and stability theory; and linear systems and control theory. Also included are a detailed list of symbols, a summary of notation and conventions, an extensive bibliography and author index with page references, and an exhaustive subject index. This significantly expanded edition of Matrix Mathematics features a wealth of new material on graphs, scalar identities and inequalities, alternative partial orderings, matrix pencils, finite groups, zeros of multivariable transfer functions, roots of polynomials, convex functions, and matrix norms.
Covers hundreds of important and useful results on matrix theory, many never before available in any book
Provides a list of symbols and a summary of conventions for easy use
Includes an extensive collection of scalar identities and inequalities
Features a detailed bibliography and author index with page references
Includes an exhaustive subject index with cross-referencing
Dennis S. Bernstein is professor of aerospace engineering at the University of Michigan.
Cloth | July 2009 | 978-0-691-14062-9
760 pp. | 8 X 10 | 175 line illus.
Probability, Markov Chains, Queues, and Simulation provides a modern and authoritative treatment of the mathematical processes that underlie performance modeling. The detailed explanations of mathematical derivations and numerous illustrative examples make this textbook readily accessible to graduate and advanced undergraduate students taking courses in which stochastic processes play a fundamental role. The textbook is relevant to a wide variety of fields, including computer science, engineering, operations research, statistics, and mathematics.
The textbook looks at the fundamentals of probability theory, from the basic concepts of set-based probability, through probability distributions, to bounds, limit theorems, and the laws of large numbers. Discrete and continuous-time Markov chains are analyzed from a theoretical and computational point of view. Topics include the Chapman-Kolomogorov equations; irreducibility; the potential, fundamental, and reachability matrices; random walk problems; reversibility; renewal processes; and the numerical computation of stationary and transient distributions. The M/M/1 queue and its extensions to more general birth-death processes are analyzed in detail, as are queues with phase-type arrival and service processes. The M/G/1 and G/M/1 queues are solved using embedded Markov chains; the busy period, residual service time, and priority scheduling are treated. Open and closed queueing networks are analyzed. The final part of the book addresses the mathematical basis of simulation.
Each chapter of the textbook concludes with an extensive set of exercises. An instructor's solution manual, in which all exercises are completely worked out, is also available.
Numerous examples illuminate the mathematical theories
Carefully detailed explanations of mathematical derivations guarantee a valuable pedagogical approach
Each chapter concludes with an extensive set of exercises
An instructor's solution manual, in which all exercises are completely worked out, is available
William J. Stewart is professor of computer science at North Carolina State University. He is the author of An Introduction to the Numerical Solution of Markov Chains (Princeton).
"This is an excellent book on the topics of probability, Markov chains, and queuing theory. Extremely well-written, the contents range from elementary topics to quite advanced material and include plenty of well-chosen examples."--Adarsh Sethi, University of Delaware
"Clear, correct, and pleasant to read, this book distinguishes itself from comparable textbooks by its inclusion of the computational aspects of the material."--Richard R. Muntz, University of California, Los Angeles
Series: Annals of Mathematics Studies, vol.170
Paper | August 2009 | 978-0-691-14049-0
Cloth | August 2009 | 978-0-691-14048-3
960 pp. | 6 x 9
Higher category theory is generally regarded as technical and forbidding, but part of it is considerably more tractable: the theory of infinity-categories, higher categories in which all higher morphisms are assumed to be invertible. In Higher Topos Theory, Jacob Lurie presents the foundations of this theory, using the language of weak Kan complexes introduced by Boardman and Vogt, and shows how existing theorems in algebraic topology can be reformulated and generalized in the theory's new language. The result is a powerful theory with applications in many areas of mathematics.
The book's first five chapters give an exposition of the theory of infinity-categories that emphasizes their role as a generalization of ordinary categories. Many of the fundamental ideas from classical category theory are generalized to the infinity-categorical setting, such as limits and colimits, adjoint functors, ind-objects and pro-objects, locally accessible and presentable categories, Grothendieck fibrations, presheaves, and Yoneda's lemma. A sixth chapter presents an infinity-categorical version of the theory of Grothendieck topoi, introducing the notion of an infinity-topos, an infinity-category that resembles the infinity-category of topological spaces in the sense that it satisfies certain axioms that codify some of the basic principles of algebraic topology. A seventh and final chapter presents applications that illustrate connections between the theory of higher topoi and ideas from classical topology.
Jacob Lurie is associate professor of mathematics at Massachusetts Institute of Technology.