Series: Cambridge Monographs on Applied and Computational Mathematics (No. 26)
Hardback (ISBN-13: 9780521857017)
50 b/w illus. 160 exercises
Page extent: 280 pages
This book explains recent results in the theory of moving frames that concern the symbolic manipulation of invariants of Lie group actions. In particular, theorems concerning the calculation of generators of algebras of differential invariants, and the relations they satisfy, are discussed in detail. The author demonstrates how new ideas lead to significant progress in two main applications: the solution of invariant ordinary differential equations and the structure of Euler-Lagrange equations and conservation laws of variational problems. The expository language used here is primarily that of undergraduate calculus rather than differential geometry, making the topic more accessible to a student audience. More sophisticated ideas from differential topology and Lie theory are explained from scratch using illustrative examples and exercises. This book is ideal for graduate students and researchers working in differential equations, symbolic computation, applications of Lie groups and, to a lesser extent, differential geometry.
* Contains original proofs and details two significant applications * Worked
examples, explanations and illustrations provide access to the topic for
graduate students * Provides a helpful introduction to differential topology
and Lie theory
Preface; Introduction to invariant and equivariant problems; 1. Actions galore; 2. Calculus on Lie groups; 3. From Lie group to Lie algebra; 4. Moving frames; 5. On syzygies and curvature matrices; 6. Invariant ordinary differential equations; 7. Variational problems with symmetry; References; Index.
Series: London Mathematical Society Lecture Note Series (No. 373)
Paperback (ISBN-13: 9780521125727)
Page extent: 160 pages
Size: 228 x 152 mm
Written to complement standard texts on commutative algebra, this short
book gives complete and relatively easy proofs of important results, including
the standard results involving localisation of formal smoothness (M. Andre)
and localisation of complete intersections (L. Avramov), some important
results of D. Popescu and Andre on regular homomorphisms, and some results
from A. Grothendieck's EGA on smooth homomorphisms. The authors make extensive
use of the Andre*Quillen homology of commutative algebras, but only up
to dimension 2, which is easy to construct, and they deliberately avoid
using simplicial methods. The book also serves as an accessible introduction
to some advanced topics and techniques. The only prerequisites are a basic
course in commutative algebra and the first definitions in homological
algebra.
* Perfectly complements standard books in the field * Provides more elementary
proofs, avoiding simplicial techniques * Useful resource for graduate students
as well as researchers who are unfamiliar with simplicial methods
Introduction; 1. Definition and first properties of (co-)homology modules; 2. Formally smooth homomorphisms; 3. Structure of complete noetherian local rings; 4. Complete intersections; 5. Regular homomorphisms: Popescu's theorem; 6. Localization of formal smoothness; Appendix: some exact sequences; Bibliography; Index.
Series: London Mathematical Society Lecture Note Series (No. 374)
Paperback (ISBN-13: 9780521128223)
Page extent: 135 pages
Size: 228 x 152 mm
Its self-contained presentation and edo-it-yourself' approach make this the perfect guide for graduate students and researchers wishing to access recent literature in the field of nonlinear wave equations and general relativity. It introduces all of the key tools and concepts from Lorentzian geometry (metrics, null frames, deformation tensors, etc.) and provides complete elementary proofs. The author also discusses applications to topics in nonlinear equations, including null conditions and stability of Minkowski space. No previous knowledge of geometry or relativity is required.
* No prerequisites - easily accessible to analysts in the field of PDEs
* Elementary proofs serve as exercises for the reader * Provides all the
necessary mathematical tools of Lorentzian geometry
Preface; 1. Introduction; 2. Metrics and frames; 3. Computing with frames; 4. Energy inequalities and frames; 5. The good components; 6. Pointwise estimates and commutations; 7. Frames and curvature; 8. Nonlinear equations, a priori estimates and induction; 9. Applications to some quasilinear hyperbolic problems; References; Index.
Series: Cambridge Studies in Advanced Mathematics (No. 123)
Hardback (ISBN-13: 9780521519182)
7 b/w illus. 85 exercises
Page extent: 300 pages
Random walks are stochastic processes formed by successive summation of independent, identically distributed random variables and are one of the most studied topics in probability theory. This contemporary introduction evolved from courses taught at Cornell University and the University of Chicago by the first author, who is one of the most highly regarded researchers in the field of stochastic processes. This text meets the need for a modern reference to the detailed properties of an important class of random walks on the integer lattice. It is suitable for probabilists, mathematicians working in related fields, and for researchers in other disciplines who use random walks in modeling.
* Suitable for researchers from a variety of fields who use random walks
in modeling * Extensive bibliography assists further reading * Contains
over 80 exercises
Preface; 1. Introduction; 2. Local central limit theorem; 3. Approximation by Brownian motion; 4. Green's function; 5. One-dimensional walks; 6. Potential theory; 7. Dyadic coupling; 8. Additional topics on simple random walk; 9. Loop measures; 10. Intersection probabilities for random walks; 11. Loop-erased random walk; Appendix; Bibliography; Index of symbols; Index.
Series: Cambridge Studies in Advanced Mathematics (No. 125)
Hardback (ISBN-13: 9780521768795)
1 b/w illus. 135 exercises
Page extent: 280 pages
Over the last 50 years the theory of p-adic differential equations has grown into an active area of research in its own right, and has important applications to number theory and to computer science. This book, the first comprehensive and unified introduction to the subject, improves and simplifies existing results as well as including original material. Based on a course given by the author at MIT, this modern treatment is accessible to graduate students and researchers. Exercises are included at the end of each chapter to help the reader review the material, and the author also provides detailed references to the literature to aid further study.
* Assumes only an undergraduate-level course in abstract algebra * Up-to-date
treatment including previously unpublished results * Class-tested by the
author
Preface; Introductory remarks; Part I. Tools of p-adic Analysis: 1. Norms on algebraic structures; 2. Newton polygons; 3. Ramification theory; 4. Matrix analysis; Part II. Differential Algebra: 5. Formalism of differential algebra; 6. Metric properties of differential modules; 7. Regular singularities; Part III. p-adic Differential Equations on Discs and Annuli: 8. Rings of functions on discs and annuli; 9. Radius and generic radius of convergence; 10. Frobenius pullback and pushforward; 11. Variation of generic and subsidiary radii; 12. Decomposition by subsidiary radii; 13. p-adic exponents; Part IV. Difference Algebra and Frobenius Modules: 14. Formalism of difference algebra; 15. Frobenius modules; 16. Frobenius modules over the Robba ring; Part V. Frobenius Structures: 17. Frobenius structures on differential modules; 18. Effective convergence bounds; 19. Galois representations and differential modules; 20. The p-adic local monodromy theorem: Statement; 21. The p-adic local monodromy theorem: Proof; Part VI. Areas of Application: 22. Picard-Fuchs modules; 23. Rigid cohomology; 24. p-adic Hodge theory; References; Index of notation; Index.