611 pages 70 line diagrams 158 exercises 57 worked examples
Hardback | ISBN: 0-521-80615-1 | - available from October 2004
Collocation based on piecewise polynomial approximation
represents a powerful class of methods for the numerical solution
of initial-value problems for functional differential and
integral equations arising in a wide spectrum of applications,
including biological and physical phenomena. The present book
introduces the reader to the general principles underlying these
methods and then describes in detail their convergence properties
when applied to ordinary differential equations, functional
equations with (Volterra type) memory terms, delay equations, and
differential-algebraic and integral-algebraic equations. Each
chapter starts with a self-contained introduction to the relevant
theory of the class of equations under consideration. Numerous
exercises and examples are supplied, along with extensive
historical and bibliographical notes utilising the vast annotated
reference list of over 1300 items. In sum, Hermann Brunner has
written a treatise that can serve as an introduction for
students, a guide for users, and a comprehensive resource for
experts.
Contents
1. The collocation method for ODEs: an introduction; 2. Volterra
integral equations with smooth kernels; 3. Volterra integro-differential
equations with smooth kernels; 4. Initial-value problems with non-vanishing
delays; 5. Initial-value problems with proportional (vanishing)
delays; 6. Volterra integral equations with weakly singular
kernels; 7. VIDEs with weakly singular kernels; 8. Outlook:
integral-algebraic equations and beyond; 9. Epilogue.
200 pages 33 exercises
Paperback | ISBN: 0-521-54649-4 | - available from November 2004
Reissued in the Cambridge Mathematical Library this classic book
outlines the theory of thermodynamic formalism which was
developed to describe the properties of certain physical systems
consisting of a large number of subunits. It is aimed at
mathematicians interested in ergodic theory, topological
dynamics, constructive quantum field theory, the study of certain
differentiable dynamical systems, notably Anosov diffeomorphisms
and flows. It is also of interest to theoretical physicists
concerned with the conceptual basis of equilibrium statistical
mechanics. The level of the presentation is generally advanced,
the objective being to provide an efficient research tool and a
text for use in graduate teaching. Background material on
mathematics has been collected in appendices to help the reader.
Extra material is given in the form of updates of problems that
were open at the original time of writing and as a new preface
specially written for this new edition by the author.
Index
264 pages 11 line diagrams 502 exercises
Hardback | ISBN: 0-521-84072-4 | available from November 2004
Paperback | ISBN: 0-521-60047-2 | available from November 2004
This self-contained text, suitable for advanced undergraduates,
provides an extensive introduction to mathematical analysis, from
the fundamentals to more advanced material. It begins with the
properties of the real numbers and continues with a rigorous
treatment of sequences, series, metric spaces, and calculus in
one variable. Further subjects include Lebesgue measure and
integration on the line, Fourier analysis, and differential
equations. In addition to this core material, the book includes a
number of interesting applications of the subject matter to areas
both within and outside the field of mathematics. The aim
throughout is to strike a balance between being too austere or
too sketchy, and being so detailed as to obscure the essential
ideas. A large number of examples and 500 exercises allow the
reader to test understanding, practise mathematical exposition
and provide a window into further topics.
Table of Contents
480 pages 34 line diagrams 5 tables
Hardback | ISBN: 0-521-83733-2 | - available from October 2004
Quantum gravity is perhaps the most important open problem in
fundamental physics. It is the problem of merging quantum
mechanics and general relativity, the two great conceptual
revolutions in the physics of the twentieth century. The loop and
spinfoam approach, presented in this book, is one of the leading
research programs in the field. The first part of the book
discusses the reformulation of the basis of classical and quantum
Hamiltonian physics required by general relativity. The second
part covers the basic technical research directions. Appendices
include a detailed history of the subject of quantum gravity,
hard-to-find mathematical material, and a discussion of some
philosophical issues raised by the subject. This fascinating text
is ideal for graduate students entering the field, as well as
researchers already working in quantum gravity. It will also
appeal to philosophers and other scholars interested in the
nature of space and time.
Contents
Preface; Acknowledgements; Terminology and notation; Part I.
Relativistic Foundations: 1. General relativity; 2. General
relativity; 3. Relativistic mechanics; 4. Hamiltonian general
relativity; 5. Quantum mechanics; Part II. Loop Quantum Gravity;
6. Quantum space; 7. Quantum spacetime: the Hamiltonian operator;
8. Matter; 9. Applications; 10. Quantum spacetime: spinfoams; 11.
Discussion; Part III. Appendices: A. Mathematical appendices; B.
History; C. On method and truth; References.
Reviews
eIn spite of its sociological success string theory is still
far from a solution of the problem of quantum gravity which
should be considered as wide open. The book of Carlo Rovelli
provides the basis, both at the technical and the conceptual
level, for research in this fundamental problem of physics. The
basic issues are clearly and deeply analyzed without any dogmatic
stand and with great freedom of thoughts resulting in an
unvaluable opportunity to learn and think for both mathematicians
and physicists.f Alain Connes is currently a Professor at the
College de France, I.H.E.S. and Vanderbilt University
320 pages 73 line diagrams
Hardback | ISBN: 0-521-84248-4 | -available from January 2005
What is the best way to divide a 'cake' and allocate the pieces among some finite collection of players? In this book, the cake is a measure space, and each player uses a countably additive, non-atomic probability measure to evaluate the size of the pieces of cake, with different players generally using different measures. The author investigates efficiency properties (is there another partition that would make everyone at least as happy, and would make at least one player happier, than the present partition?) and fairness properties (do all players think that their piece is at least as large as every other player?s piece?). He focuses exclusively on abstract existence results rather than algorithms, and on the geometric objects that arise naturally in this context. By examining the shape of these objects and the relationship between them, he demonstrates results concerning the existence of efficient and fair partitions.
Contents
0. Preface; 1. Notation and preliminaries; 2. Geometric object #1a: the individual pieces set (IPS) for two players; 3. What the IPS tells us about fairness and efficiency in the two-player context; 4. The general case of n players; 5. What the IPS and the FIPS tell us about fairness and efficiency in the n-player context; 6. Characterizing Pareto optimality: introduction and preliminary ideas; 7. Characterizing Pareto optimality I: the IPS and optimization of convex combinations of measures; 8. Characterizing Pareto optimality II: partition ratios; 9. Geometric object #2: The Radon-Nikodym set (RNS); 10. Characterizing Pareto optimality III: the RNS, Weller?s construction, and w-association; 11. The shape of the IPS; 12. The relationship between the IPS and the RNS; 13. Other issues involving Weller?s construction, partition ratios, and Pareto optimality; 14. Strong Pareto optimality; 15. Characterizing Pareto optimality using hyperreal numbers; 16. The multi-cake individual pieces set (MIPS): symmetry restored.