Paperback
Series: Australian Mathematical Society Lecture Series(No. 22)
ISBN:9781107653610
10 b/w illus. 50 exercises
Dimensions: 228 x 152 mm
available from September 2012
This bold and refreshing approach to Lie algebras assumes only modest prerequisites (linear algebra up to the Jordan canonical form and a basic familiarity with groups and rings), yet it reaches a major result in representation theory: the highest-weight classification of irreducible modules of the general linear Lie algebra. The author's exposition is focused on this goal rather than aiming at the widest generality and emphasis is placed on explicit calculations with bases and matrices. The book begins with a motivating chapter explaining the context and relevance of Lie algebras and their representations and concludes with a guide to further reading. Numerous examples and exercises with full solutions are included. Based on the author's own introductory course on Lie algebras, this book has been thoroughly road-tested by advanced undergraduate and beginning graduate students and it is also suited to individual readers wanting an introduction to this important area of mathematics.
1. Motivation: representations of Lie groups
2. Definition of a Lie algebra
3. Basic structure of a Lie algebra
4. Modules over a Lie algebra
5. The theory of SL2-modules
6. General theory of modules
7. Integral GLn-modules
8. Guide to further reading
Appendix: solutions to the exercises
Bibliography
Index.
Hardback
Series: Cambridge Tracts in Mathematics(No. 194)
ISBN:9781107022829
7 b/w illus. 85 exercises
Dimensions: 228 x 152 mm
- available from October 2012
This book is dedicated to the mathematical study of two-dimensional statistical hydrodynamics and turbulence, described by the 2D Navier?Stokes system with a random force. The authors' main goal is to justify the statistical properties of a fluid's velocity field u(t,x) that physicists assume in their work. They rigorously prove that u(t,x) converges, as time grows, to a statistical equilibrium, independent of initial data. They use this to study ergodic properties of u(t,x) ? proving, in particular, that observables f(u(t,.)) satisfy the strong law of large numbers and central limit theorem. They also discuss the inviscid limit when viscosity goes to zero, normalising the force so that the energy of solutions stays constant, while their Reynolds numbers grow to infinity. They show that then the statistical equilibria converge to invariant measures of the 2D Euler equation and study these measures. The methods apply to other nonlinear PDEs perturbed by random forces.
1. Preliminaries
2. Two-dimensional Navier?Stokes equations
3. Uniqueness of stationary measure and mixing
4. Ergodicity and limiting theorems
5. Inviscid limit
6. Miscellanies
7. Appendix
8. Solutions to some exercises.
Hardback
Series: Cambridge Tracts in Mathematics(No. 195)
ISBN:9781107024816
4 b/w illus. 65 exercises
Dimensions: 228 x 152 mm
- available from September 2012
The theory of R-trees is a well-established and important area of geometric
group theory and in this book the authors introduce a construction that
provides a new perspective on group actions on R-trees. They construct
a group RF(G), equipped with an action on an R-tree, whose elements are
certain functions from a compact real interval to the group G. They also
study the structure of RF(G), including a detailed description of centralizers
of elements and an investigation of its subgroups and quotients. Any group
acting freely on an R-tree embeds in RF(G) for some choice of G. Much remains
to be done to understand RF(G), and the extensive list of open problems
included in an appendix could potentially lead to new methods for investigating
group actions on R-trees, particularly free actions. This book will interest
all geometric group theorists and model theorists whose research involves
R-trees
Preface
1. Introduction
2. The group RF(G)
3. The R-tree XG associated with RF(G)
4. Free R-tree actions and universality
5. Exponent sums
6. Functoriality
7. Conjugacy of hyperbolic elements
8. The centralizers of hyperbolic elements
9. Test functions: basic theory and first applications
10. Test functions: existence theorem and further applications
11. A generalization to groupoids
Appendix A. The basics of ĩ-trees
Appendix B. Some open problems
References
Index.
Hardback
ISBN:9780521839402
Paperback
ISBN:9780521548236
1175 exercises
Dimensions: 253 x 177 mm
available from November 2012
Linear algebra and matrix theory are fundamental tools in mathematical and physical science, as well as fertile fields for research. This new edition of this acclaimed text presents results of both classic and recent matrix analyses using canonical forms as a unifying theme and demonstrates their importance in a variety of applications. This thoroughly revised and updated second edition is a text for a second course on linear algebra and has more than 1,100 problems and exercises, new sections on the singular value and CS decompositions and the Weyr canonical form, expanded treatments of inverse problems and of block matrices, and much more.
1. Eigenvalues, eigenvectors, and similarity
2. Unitary similarity and unitary equivalence
3. Canonical forms for similarity, and triangular factorizations
4. Hermitian matrices, symmetric matrices, and congruences
5. Norms for vectors and matrices
6. Location and perturbation of eigenvalues
7. Positive definite and semi-definite matrices
8. Positive and nonnegative matrices
Appendix A. Complex numbers
Appendix B. Convex sets and functions
Appendix C. The fundamental theorem of algebra
Appendix D. Continuous dependence of the zeroes of a polynomial on its coefficients
Appendix E. Continuity, compactness, and Weierstrass's theorem
Appendix F. Canonical pairs.