Series: Cambridge Monographs on Mathematical Physics
Hardback (ISBN-13: 9780521889278)
Einsteinfs theory of general relativity is a theory of gravity and, as in the earlier Newtonian theory, much can be learnt about the character of gravitation and its effects by investigating particular idealised examples. This book describes the basic solutions of Einsteinfs equations with a particular emphasis on what they mean, both geometrically and physically. New concepts, such as big bang and big crunch-types of singularities, different kinds of horizons and gravitational waves, are described in the context of the particular space-times in which they naturally arise. These notions are initially introduced using the most simple and symmetric cases. Various important coordinate forms of each solution are presented, thus enabling the global structure of the corresponding space-time and its other properties to be analysed. The book is an invaluable resource both for graduate students and academic researchers working in gravitational physics.
* Describes the basic solutions of Einsteinfs equations emphasising their
geometric and physical meaning * Introduces concepts with simplest cases,
building space-time analyses from basics * Makes explicit the relationships
between different families of solutions to Einsteinfs equations
Preface; 1. Introduction; 2. Basic tools and concepts; 3. Minkowski space-time;
4. de Sitter space-time; 5. Anti-de Sitter space-time; 6. Friedmann-Lemaitre-Robertson-Walker
space-times; 7. Electrovacuum and related background space-times; 8. Schwarzchild
space-time; 9. Space-times related to Schwarzchild; 10. Static axially
symmetric space-times; 11. Rotating black holes; 12. Taub-NUT space-time;
13. Stationary, axially symmetric space-times; 14. Accelerating black holes;
15. Further solutions for uniformly accelerating particles; 16. Pleba*ski-Demia*ski
solutions; 17. Plane and pp-waves; 18. Kundt solutions; 19. Robinson-Trautman
solutions; 20. Impulsive waves; 21. Colliding plane waves; 22. A final
miscellany; Appendix A. 2-spaces of constant curvature; Appendix B. 3-spaces
of constant curvature; References; Index.
Series: Cambridge Monographs on Applied and Computational Mathematics (No. 25)
Hardback (ISBN-13: 9780521864671)
Sure to be influential, this book lays the foundations for the use of algebraic geometry in statistical learning theory. Many widely used statistical models and learning machines applied to information science have a parameter space that is singular: mixture models, neural networks, HMMs, Bayesian networks, and stochastic context-free grammars are major examples. Algebraic geometry and singularity theory provide the necessary tools for studying such non-smooth models. Four main formulas are established: 1. the log likelihood function can be given a common standard form using resolution of singularities, even applied to more complex models; 2. the asymptotic behaviour of the marginal likelihood or ethe evidencef is derived based on zeta function theory; 3. new methods are derived to estimate the generalization errors in Bayes and Gibbs estimations from training errors; 4. the generalization errors of maximum likelihood and a posteriori methods are clarified by empirical process theory on algebraic varieties.
Presents a new statistical theory for singular learning machines Mathematical concepts explained for non-specialists Intended for any student interested in machine learning, pattern recognition, artificial intelligence or bioinformatics
Preface; 1. Introduction; 2. Singularity theory; 3. Algebraic geometry; 4. Zeta functions and singular integral; 5. Empirical processes; 6. Singular learning theory; 7. Singular learning machines; 8. Singular information science; Bibliography; Index.
Series: Encyclopedia of Mathematics and its Applications (No. 109)
Hardback (ISBN-13: 9780521850056)
Like differentiability, convexity is a natural and powerful property of functions that plays a significant role in many areas of mathematics, both pure and applied. It ties together notions from topology, algebra, geometry and analysis, and is an important tool in optimization, mathematical programming and game theory. This book, which is the product of a collaboration of over 15 years, is unique in that it focuses on convex functions themselves, rather than on convex analysis. The authors explore the various classes and their characteristics and applications, treating convex functions in both Euclidean and Banach spaces. The book can either be read sequentially for a graduate course, or dipped into by researchers and practitioners. Each chapter contains a variety of specific examples, and over 600 exercises are included, ranging in difficulty from early graduate to research level.
* Unique focus on the functions themselves, rather than convex analysis
* Contains over 600 exercises showing theory and applications * All material
has been class-tested
Preface; 1. Why convex*; 2. Convex functions on Euclidean spaces; 3. Finer structure of Euclidean spaces; 4. Convex functions on Banach spaces; 5. Duality between smoothness and strict convexity; 6. Further analytic topics; 7. Barriers and Legendre functions; 8. Convex functions and classifications of Banach spaces; 9. Monotone operators and the Fitzpatrick function; 10. Further remarks and notes; References; Index.
Series: Encyclopedia of Mathematics and its Applications (No. 129)
Hardback (ISBN-13: 9780521888318)
This major revision of Berstel and Perrin's classic Theory of Codes has been rewritten with a more modern focus and a much broader coverage of the subject. The concept of unambiguous automata, which is intimately linked with that of codes, now plays a significant role throughout the book, reflecting developments of the last 20 years. This is complemented by a discussion of the connection between codes and automata, and new material from the field of symbolic dynamics. The authors have also explored links with more practical applications, including data compression and cryptography. The treatment remains self-contained: there is background material on discrete mathematics, algebra and theoretical computer science. The wealth of exercises and examples make it ideal for self-study or courses. In sum this is a comprehensive reference on the theory of variable-length codes and their relation to automata.
* Thoroughly updated to reflect developments and results in the field over
the last twenty years * A self-contained treatment containing complete
proofs and over 250 illustrative examples * Exercises and solutions allow
the reader to test their understanding
Preface; 1. Preliminaries; 2. Codes; 3. Prefix codes; 4. Automata; 5. Deciphering delay; 6. Bifix codes; 7. Circular codes; 8. Factorizations of free monoids; 9. Unambiguous monoids of relations; 10. Synchronization; 11. Groups of codes; 12. Factorizations of cyclic groups; 13. Densities; 14. Polynomials of finite codes
Series: Interdisciplinary Statistics
ISBN: 9781420066562
Publication Date: 1/15/2010
Number of Pages: 416
Availability: Not Yet Published
Introduces measurement error models and methods with an emphasis on applications
Covers key topics including misclassification in estimation, measurement error in inferences, predictors, and time series
Includes numerous examples from biostatistics, epidemiology, ecology, and the social sciences
Illustrates methods and examples using available Stata routines and SAS programs
Measurement Error and Misclassification provides an understanding of measurement error, the effects of ignoring it, and how to correct for these effects. The book focuses on the models and methods involved and demonstrates how they can be implemented in practice. Keeping theory to a minimum with an appendix of theoretical background, it presents numerous examples from biostatistics and epidemiology as well as ecology and the social sciences. The author implements these examples using available Stata routines and his own SAS programs. Topics covered include misclassification in estimation, measurement error in inference, predictors, and time series.