Series: Encyclopedia of Mathematics and its Applications (No. 112)
Hardback (ISBN-13: 9780521865166 | ISBN-10: 0521865166)
The classical fields are the real, rational, complex and p-adic numbers. Each of these fields comprises several intimately interwoven algebraical and topological structures. This comprehensive volume analyzes the interaction and interdependencies of these different aspects. The real and rational numbers are examined additionally with respect to their orderings, and these fields are compared to their non-standard counterparts. Typical substructures and quotients, relevant automorphism groups and many counterexamples are described. Also discussed are completion procedures of chains and of ordered and topological groups, with applications to classical fields. The p-adic numbers are placed in the context of general topological fields: absolute values, valuations and the corresponding topologies are studied, and the classification of all locally compact fields and skew fields is presented. Exercises are provided with hints and solutions at the end of the book. An appendix reviews ordinals and cardinals, duality theory of locally compact Abelian groups and various constructions of fields.
* First book to comprehensively discuss the abstract structural properties of the classical number systems of mathematics
* Discusses in detail the interrelations between real, rational, complex and p-adic numbers
* Contains over 200 exercises, hints and solutions
Contents
Preface; 1. Real numbers; 2. Non-standard numbers; 3. Rational numbers; 4. Completion; 5. The p-adic numbers; 6. Appendix; Hints and solutions; Bibliography; Notation; Index.
Hardback (ISBN-13: 9780521872072 | ISBN-10: 0521872073)
Combinatorics is an area of mathematics involving an impressive
breadth of ideas, and it encompasses topics ranging from codes
and circuit design to algorithmic complexity and algebraic graph
theory. In a highly distinguished career Bela Bollobas has made,
and continues to make, many significant contributions to
combinatorics, and this volume reflects the wide range of topics
on which his work has had a major influence. It arises from a
conference organized to mark his 60th birthday and the thirty-one
articles contained here are of the highest calibre. That so many
excellent mathematicians have contributed is testament to the
very high regard in which Bela Bollobas is held. Students and
researchers across combinatorics and related fields will find
that this volume provides a wealth of insight to the state of the
art.
* Volume celebrating the 60th birthday of Bela Bollobas,
highlighting the significant contributions he has made during his
distinguished career
* Contributors of the highest calibre present the current state
of the art in combinatorics and related fields
* Encompasses wide range of topics, ranging from codes and
circuit design to algorithmic complexity and algebraic graph
theory
Contents
1. Measures of pseudorandomness for finite sequences: minimal
values N. Alon, Y. Kohayakawa, C. Mauduit, and V. R. Rodl; 2.
MaxCut in H-Free graphs Noga Alon, Michael Krivelevich and Benny
Sudakov; 3. A tale of three couplings: Poisson-Dirichlet and GEM
approximations for random permutations Richard Arratia, A. D.
Barbour and Simon Tavare; 4. Positional games Jozsef Beck; 5.
Degree distribution of competition-induced preferential
attachment graphs N. Berger, C. Borgs, J. T. Chayes, R. M.
D'Souza and R. D. Kleinberg; 6. On two conjectures on packing of
graphs Bela Bollobas, Alexandr Kostochka and Kittikorn Nakprasit;
7. Approximate counting and quantum computation M. Bordewich, M.
Freedman, L. Lovasz and D. Welsh; 8. Absence of zeros for the
chromatic polynomial on bounded degree graphs Christian Borgs; 9.
Duality in infinite graphs Henning Bruhn and Reinhard Diestel; 10.
Homomorphism-homogeneous relational structures Peter J. Cameron
and Jaroslav Ne?etril; 11. A spectral Turan theorem Fan Chung; 12.
Automorphism groups of metacirculant graphs of order a product of
two distinct primes Edward Dobson; 13. On the number of
Hamiltonian cycles in a tournament Ehud Friedgut and Jeff Kahn;
14. The game of JumbleG Alan Frieze, Michael Krivelevich, Oleg
Pikhurko and Tibor Szabo; 15. 2-Bases of quadruples Zoltan Furedi
and Gyula O. H. Katona; 16. On triple systems with independent
neighbourhoods Zoltan Furedi, Oleg Pikhurko and Miklos
Simonovits; 17. Quasirandomness, counting and regularity for 3-uniform
hypergraphs W. T. Gowers; 18. Triangle-free hypergraphs Ervin
Gyori; 19. Odd independent transversals are odd Penny Haxell and
Tibor Szabo; 20. The first eigenvalue of random graphs Svante
Janson; 21. On the number of monochromatic solutions of x + y = z2
Ayman Khalfalah and Endre Szemeredi; 22. Rapid Steiner
symmetrization of most of a convex body and the slicing problem B.
Klartag and V. Milman; 23. A note on bipartite graphs wthout 2k-cycles
Assaf Naor and Jacques Verstraete; 24. Book Ramsey numbers and
quasi-eandomness V. Nikiforov, C. C. Rousseau and R. H. Schelp;
25. Homomorphism and dimension Patrice Ossona de Mendez and
Pierre Rosenstiehl; 26. The distance of a permutation from a
subgroup of Sn Richard G.E. Pinch; 27. On dimensions of a random
solid diagram Boris Pittel; 28. The small giant component in
scale-free random graphs Oliver Riordan; 29. A Dirac-type theorem
for 3-uniform hypergraphs Vojtech Rodl, Andrzej Rucinski and
Endre Szemeredi; 30. On dependency graphs and the lattice gas
Alexander D. Scott and Alan D. Sokal; 31. Solving sparse random
instances of max cut and max 2-CSP in linear expected time
Alexander D. Scott and Gregory B. Sorkin.
Series: London Mathematical Society Lecture Note Series (No.
341)
Paperback (ISBN-13: 9780521699648 | ISBN-10: 0521699649)
Random matrix theory is an area of mathematics first developed by
physicists interested in the energy levels of atomic nuclei, but
it can also be used to describe some exotic phenomena in the
number theory of elliptic curves. The purpose of this book is to
illustrate this interplay of number theory and random matrices.
It begins with an introduction to elliptic curves and the
fundamentals of modelling by a family of random matrices, and
moves on to highlight the latest research. There are expositions
of current research on ranks of elliptic curves, statistical
properties of families of elliptic curves and their associated L-functions
and the emerging uses of random matrix theory in this field. Most
of the material here had its origin in a Clay Mathematics
Institute workshop on this topic at the Newton Institute in
Cambridge and together these contributions provide a unique in-depth
treatment of the subject.
* The first book on random matrix theory and elliptic curves
* Gives an overview of the entire subject, making it suitable as
an introduction
* Many papers by leading experts in the field, presenting the
very latest research findings
Contents
Introduction J. B. Conrey, D. W. Farmer, F. Mezzadri and N. C.
Snaith; Part I. Families: 1. Elliptic curves, rank in families
and random matrices E. Kowalski; 2. Modeling families of L-functions
D. W. Farmer; 3. Analytic number theory and ranks of elliptic
curves M. P. Young; 4. The derivative of SO(2N +1) characteristic
polynomials and rank 3 elliptic curves N. C. Snaith; 5. Function
fields and random matrices D. Ulmer; 6. Some applications of
symmetric functions theory in random matrix theory A. Gamburd;
Part II. Ranks of Quadratic Twists: 7. The distribution of ranks
in families of quadratic twists of elliptic curves A. Silverberg;
8. Twists of elliptic curves of rank at least four K. Rubin and A.
Silverberg; 9. The powers of logarithm for quadratic twists C.
Delaunay and M. Watkins; 10. Note on the frequency of vanishing
of L-functions of elliptic curves in a family of quadratic twists
C. Delaunay; 11. Discretisation for odd quadratic twists J. B.
Conrey, M. O. Rubinstein, N. C. Snaith and M. Watkins; 12.
Secondary terms in the number of vanishings of quadratic twists
of elliptic curve L-functions J. B. Conrey, A. Pokharel, M. O.
Rubinstein and M. Watkins; 13. Fudge factors in the Birch and
Swinnerton-Dyer Conjecture K. Rubin; Part III. Number Fields and
Higher Twists: 14. Rank distribution in a family of cubic twists
M. Watkins; 15. Vanishing of L-functions of elliptic curves over
number fields C. David, J. Fearnley and H. Kisilevsky; Part IV.
Shimura Correspondence, and Twists: 16. Computing central values
of L-functions F. Rodriguez-Villegas; 17. Computation of central
value of quadratic twists of modular L-functions Z. Mao, F.
Rodriguez-Villegas and G. Tornaria; 18. Examples of Shimura
correspondence for level p2 and real quadratic twists A. Pacetti
and G. Tornaria; 19. Central values of quadratic twists for a
modular form of weight H. Rosson and G. Tornaria; Part V. Global
Structure: Sha and Descent: 20. Heuristics on class groups and on
Tate-Shafarevich groups C. Delaunay; 21. A note on the 2-part of
X for the congruent number curves D. R. Heath-Brown; 22. 2-Descent
tThrough the ages P. Swinnerton-Dyer.
Series: London Mathematical Society Student Texts (No. 69)
Paperback (ISBN-13: 9780521688970 | ISBN-10: 0521688973)
Hardback (ISBN-13: 9780521868914 | ISBN-10: 0521868912)
Kahler geometry is a beautiful and intriguing area of
mathematics, of substantial research interest to both
mathematicians and physicists. This self-contained graduate text
provides a concise and accessible introduction to the topic. The
book begins with a review of basic differential geometry, before
moving on to a description of complex manifolds and holomorphic
vector bundles. Kahler manifolds are discussed from the point of
view of Riemannian geometry, and Hodge and Dolbeault theories are
outlined, together with a simple proof of the famous Kahler
identities. The final part of the text studies several aspects of
compact Kahler manifolds: the Calabi conjecture, Weitzenbock
techniques, Calabi-Yau manifolds, and divisors. All sections of
the book end with a series of exercises and students and
researchers working in the fields of algebraic and differential
geometry and theoretical physics will find that the book provides
them with a sound understanding of this theory.
* The first graduate-level text on Kahler geometry, providing a
concise introduction for both mathematicians and physicists with
a basic knowledge of calculus in several variables and linear
algebra
* Over 130 exercises and worked examples
* Self-contained and presents varying viewpoints including
Riemannian, complex and algebraic
Contents
Introduction; Part I. Basics on Differential Geometry: 1. Smooth
manifolds; 2. Tensor fields on smooth manifolds; 3. The exterior
derivative; 4. Principal and vector bundles; 5. Connections; 6.
Riemannian manifolds; Part II. Complex and Hermitian Geometry: 7.
Complex structures and holomorphic maps; 8. Holomorphic forms and
vector fields; 9. Complex and holomorphic vector bundles; 10.
Hermitian bundles; 11. Hermitian and Kahler metrics; 12. The
curvature tensor of Kahler manifolds; 13. Examples of Kahler
metrics; 14. Natural operators on Riemannian and Kahler
manifolds; 15. Hodge and Dolbeault theory; Part III. Topics on
Compact Kahler Manifolds: 16. Chern classes; 17. The Ricci form
of Kahler manifolds; 18. The Calabi-Yau theorem; 19. Kahler-Einstein
metrics; 20. Weitzenbock techniques; 21. The Hirzebruch-Riemann-Roch
formula; 22. Further vanishing results; 23. Ricci-flat Kahler
metrics; 24. Explicit examples of Calabi-Yau manifolds;
Bibliography; Index.
Series: London Mathematical Society Student Texts (No. 70)
Hardback (ISBN-13: 9780521876599 | ISBN-10: 0521876591)
Paperback (ISBN-13: 9780521700153 | ISBN-10: 0521700159)
Dependence is a common phenomenon, wherever one looks: ecological
systems, astronomy, human history, stock markets - but what is
the logic of dependence? This book is the first to carry out a
systematic logical study of this important concept, giving on the
way a precise mathematical treatment of Hintikka's independence
friendly logic. Dependence logic adds the concept of dependence
to first order logic. Here the syntax and semantics of dependence
logic are studied, dependence logic is given an alternative game
theoretic semantics, and results about its complexity are proven.
This is a graduate textbook suitable for a special course in
logic in mathematics, philosophy and computer science
departments, and contains over 200 exercises, many of which have
a full solution at the end of the book. It is also accessible to
readers, with a basic knowledge of logic, interested in new
phenomena in logic.
* The first book on the logic of dependence as well as the first
to give a precise mathematical presentation of the independence
friendly logic of Jaakko Hintikka
* Based on the novel idea of taking dependence as a basic notion,
making it possible to give simple and exact proofs of the basic
results of this area
* The text contains many examples and over 200 exercises, many
with solutions provided at the end of the book
Contents
Preface; 1. Introduction; 2. Preliminaries; 3. Dependence logic;
4. Examples; 5. Game theoretic semantics; 6. Model Theory; 7.
Complexity; 8. Team logic; 9. Solutions to selected exercises by
Ville Nurmi; References; Index.
Hardback (ISBN-13: 9780521876254 | ISBN-10: 0521876257)
Description Logics are embodied in several knowledge-based
systems and are used to develop various real-life applications.
The Description Logic Handbook provides a thorough account of the
subject, covering all aspects of research in this field, namely
theory, implementation, and applications. Its appeal will be
broad, ranging from more theoretically-oriented readers, to those
with more practically-oriented interests who need a sound and
modern understanding of knowledge representation systems based on
Description Logics. As well as general revision throughout the
book, this new edition presents a new chapter on ontology
languages for the semantic web, an area of great importance for
the future development of the web. In sum, the book will serve as
a unique reference for the subject, and can also be used for self-study
or in conjunction with Knowledge Representation and Artificial
Intelligence courses.
* Only comprehensive introduction to Description Logics - this
new edition includes a chapter on ontology languages for the
semantic web
* Full coverage of all aspects of the subject: theory,
implementation and applications
* A modern perspective on knowledge (frame) based systems
Contents
1. An introduction to description logics D. Nardi and R. J.
Brachman; Part I. Theory: 2. Basic description logics F. Baader
and W. Nutt; 3. Complexity of reasoning F. M. Donini; 4.
Relationships with other formalisms U. Sattler, D. Calvanese and
R. Molitor; 5. Expressive description logics D. Calvanese and G.
De Giacomo; 6. Extensions to description logics F. Baader, R.
Kusters and F. Wolter; Part II. Implementation: 7. From
description logic provers to knowledge representation systems D.
L. McGuinness and P. F. Patel-Schneider; 8. Description logics
systems R. Moller and V. Haarslev; 9. Implementation and
optimisation techniques I. Horrocks; Part III. Applications: 10.
Conceptual modeling with description logics A. Borgida and R. J.
Brachman; 11. Software engineering C. Welty; 12. Configuration D.
L. McGuinness; 13. Medical informatics A. Rector; 14. Ontology
languages for the semantic web I. Horrocks, P. F. Patel-Schneider,
D. L. McGuinness and C. Welty; 15. Natural language processing E.
Franconi; 16. Description logics for data bases A. Borgida, M.
Lenzerini and R. Rosati; Appendix. Description logic terminology
F. Baader; Bibliography.
Series: Cambridge Tracts in Mathematics (No. 170)
Hardback (ISBN-13: 9780521829205 | ISBN-10: 0521829208)
The behaviour of vanishing cycles is the cornerstone for
understanding the geometry and topology of families of
hypersurfaces, usually regarded as singular fibrations. This self-contained
tract proposes a systematic geometro-topological approach to
vanishing cycles, especially those appearing in non-proper
fibrations, such as the fibration defined by a polynomial
function. Topics which have been the object of active research
over the past 15 years, such as holomorphic germs, polynomial
functions, and Lefschetz pencils on quasi-projective spaces, are
here shown in a new light: conceived as aspects of a single
theory with vanishing cycles at its core. Throughout the book the
author presents the current state of the art. Transparent proofs
are provided so that non-specialists can use this book as an
introduction, but all researchers and graduate students working
in differential and algebraic topology, algebraic geometry, and
singularity theory will find this book of great use.
* Self-contained and accessible tract proposing a systematic
geometro-topological approach to vanishing cycles appearing in
non-proper fibrations
* The latest research on new topics such as topology of
singularities of meromorphic functions and non-generic Lefschetz
pencils is discussed in detail
* Presents the current state of the art in this field
Contents
Preface; Part I. Singularities at Infinity of Polynomial
Functions: 1. Regularity conditions at infinity; 2. Detecting
atypical values via singularities at infinity; 3. Local and
global fibrations; 4. Families of complex polynomials; 5.
Topology of a family and contact structures; Part II. The Impact
of Global Polar Varieties: 6. Polar invariants and topology of
affine varieties; 7. Relative polar curves and families of affine
hypersurfaces; 8. Monodromy of polynomials; Part III. Vanishing
Cycles of Non-Generic Pencils: 9. Topology of meromorphic
functions; 10. Slicing by pencils of hypersurfaces; 11. Higher
Zariski-Lefschetz theorems; Notes; References; Bibliography;
Appendix 1. Stratified singularities; Appendix 2. Hints to
exercises; Index.