Series: Modern Birkhauser Classics
Originally published as volume 107 in the series: Progress in Mathematics
1st ed. 1993. 3rd printing, 2008, XVIII, 302 p. 6 illus., Softcover
ISBN: 978-0-8176-4730-8
Due: December 5, 2007
About this textbook
This book deals with the differential geometry of manifolds, loop spaces, line bundles and groupoids, and the relations of this geometry to mathematical physics.
Various developments in mathematical physics (e.g., in knot theory, gauge theory, and topological quantum field theory) have led mathematicians and physicists to search for new geometric structures on manifolds and to seek a synthesis of ideas from geometry, topology and category theory. In this spirit, this book develops the differential geometry associated to the topology and obstruction theory of certain fiber bundles (more precisely, associated to grebes). The theory is a 3-dimensional analog of the familiar Kostant--Weil theory of line bundles. In particular the curvature now becomes a 3-form.
Applications presented in the book involve anomaly line bundles on loop spaces and anomaly functionals, central extensions of loop groups, Kahler geometry of the space of knots, Cheeger--Chern--Simons secondary characteristics classes, and group cohomology. Finally, the last chapter deals with the Dirac monopole and Dirac’s quantization of the electrical charge.
The book will be of interest to topologists, geometers, Lie theorists and mathematical physicists, as well as to operator algebraists. It is written for graduate students and researchers, and will be an excellent textbook. It has a self-contained introduction to the theory of sheaves and their cohomology, line bundles and geometric prequantization a la Kostant--Souriau.
Table of contents
Introduction.- Complexes of Sheaves and Their Cohomology.- Line Bundles and Geometric Quantization.- Kahler Geometry of the Space of Knots.- Degree 3 Cohomology: The Dixmier-Douady Theory.- Degree 3 Cohomology: Sheave of Groupoids.- Line Bundles over Loop Spaces.- The Dirac Monopole.- Bibliography.- List of Notations.- Index
Series: Modern Birkhauser Classics
Hardcover edition originally published as volume 1 in the series: Progress in Computer Science and Applied Logic (PCS)
3rd ed. 1990. 2nd printing, 2008, X, 132 p., Softcover
ISBN: 978-0-8176-4728-5
About this textbook
Collects some fundamental mathematical techniques that are required for the analysis of algorithms
Very well written: the style and the mathematical exposition make the book pleasant to read
Covers a wide range of topics in an extremely concise manner, including many of the major paradigms used in the analysis of algorithms
Contains a wealth of highly original, instructive problems and solutions taken from actual examinations given at Stanford in various computer science courses
Presents a welcome selection and careful exposition of material that can be (and is) covered in a single course with a group of advanced students well-grounded in undergraduate mathematics and computer science
Details four important topics in algorithm analysis, all from a rudimentary, but highly original, point of view: each of these topics is critical to understanding the modern analysis of algorithms, primarily from the speed of execution perspective
A quantitative study of the efficiency of computer methods requires an in-depth understanding of both mathematics and computer science. This monograph, derived from an advanced computer science course at Stanford University, builds on the fundamentals of combinatorial analysis and complex variable theory to present many of the major paradigms used in the precise analysis of algorithms, emphasizing the more difficult notions. The authors cover recurrence relations, operator methods, and asymptotic analysis in a format that is terse enough for easy reference yet detailed enough for those with little background. Approximately half the book is devoted to original problems and solutions from examinations given at Stanford.
Table of contents
Preface
Binomial Identities.- Summary of Useful Identities.- Deriving the Identities.- Inverse Relations.- Operator Calculus.- Hypergeometric Series.- Identities with the Harmonic Numbers
Recurrence Relations.- Linear Recurrence Relations.- Nonlinear Recurrence Relations
Operator Methods.- The Cookie Monster.- Coalesced Hashing.- Open Addressing: Uniform Hashing.- Open Addressing: Secondary Clustering
Asymptotic Analysis.- Basic Concepts.- Stieltjes Integration and Asymptotics.- Asymptotics from Generating Functions
Bibliography
Appendices.- Schedule of Lectures.- Homework Assignments.- Midterm Exam I and Solutions.- Final Exam I and Solutions.- Midterm Exam II and Solutions.- Final Exam II and Solutions.- Midterm Exam III and Solutions.- Final Exam III and Solutions.- A Qualifying Exam Problem and Solution
Index
2008, Approx. 200 p., Hardcover
ISBN: 978-1-84800-000-1
Due: February 2008
About this book
What are numbers? Why do they exist?
Numbers have fascinated people for centuries. They are familiar to everyone, forming a central pillar of our understanding about the world, yet the number system was not presented to us "gift-wrapped" but, rather, was developed over millennia. Peter Higgins distils these centuries of hard work into a delightful narrative that reviews our progress so far and explains how the different kinds of numbers arose and why they are useful. Full of historical snippets and interesting examples, the book ranges from simple number puzzles and magic tricks to problems that relate to real-world scenarios, such as:
? What are the odds of picking the winning horse at the races; or of being dealt a flush in a hand of poker?
? How certain is it that the political candidate topping the polls will win the final election?
? How do we track the orbit of the moon in order to allow a man to walk upon it?
? Can we develop fail-safe codes for the transfer of sensitive military information, or keep our bank account details secure when shopping over the internet?
This informative book will inspire and entertain readers across a range of abilities: from high school students to anyone with an interest in how numbers impact on our everyday lives. Easy material is blended with more challenging ideas about infinity and complex numbers, and a final chapter "For Connoisseurs" works though some of the particular claims and examples in the book in mathematical language for those who appreciate a complete explanation.
As our understanding of numbers continues to evolve, this book invites us to rediscover the mystery and beauty of numbers and reminds us that the story of numbers is a tale with a long way to run...
Written for:
School-leavers and undergraduates in mathematics;
Maths teachers and lecturers;
General readers with an interest in Mathematics/Popular science;
Table of contents
Preface; The First Numbers; Discovering Numbers; Some Number Tricks; Some Tricky Numbers; Some Useful Numbers; On the Trail of New Numbers; Glimpses of Infinity; Applications of Number: Chance; The Complex History of the Imaginary; From Imaginary to Complex; The Number Line Under the Miscroscope; Applications of Number: Codes and Public Key Cryptography; For Connoisseurs; Further Reading.- Index
Series: Universitext
Original Italian edition published in the Series: UNITEXT
2008, Approx. 470 p., Softcover
ISBN: 978-88-470-0751-2
Due: November 2007
About this textbook
This book is designed as an advanced undergraduate or a first-year graduate
course for students from various disciplines like applied mathematics,
physics, engineering.
The main purpose is on the one hand to train the students to appreciate the
interplay between theory and modelling in problems arising in the applied
sciences; on the other hand to give them a solid theoretical background for
numerical methods, such as finite elements.
Accordingly, this textbook is divided into two parts.
The first one has a rather elementary character with the goal of
developing and studying basic problems from the macro-areas of diffusion,
propagation and transport, waves and vibrations. Ideas and connections with
concrete aspects are emphasized whenever possible, in order to provide
intuition and feeling for the subject.
For this part, a knowledge of advanced calculus and ordinary differential
equations is required. Also, the repeated use of the method of separation of
variables assumes some basic results from the theory of Fourier series,
which are summarized in an appendix.
The main topic of the second part is the
development of Hilbert space methods for the variational formulation and
analysis of linear boundary and initial-boundary value problems\emph{. }%
Given the abstract nature of these chapters, an effort has been made to
provide intuition and motivation for the various concepts and results.
The understanding of these topics requires some basic knowledge of Lebesgue
measure and integration, summarized in another appendix.
At the end of each chapter, a number of exercises at different level of
complexity is included. The most demanding problems are supplied with
answers or hints.
The exposition if flexible enough to allow substantial changes without
compromising the comprehension and to facilitate a selection of topics for a
one or two semester course.
Written for:
Advanced undergraduates, first-year graduates; courses for students from various disciplines like applied mathematics, physics, engineering
Series: Modern Birkhauser Classics
Originally published as volume 90 in the series: Progress in Mathematics
2nd ed. 1996. 3rd printing, 2008, XX, 342 p., Softcover
ISBN: 978-0-8176-4736-0
A Birkhauser book
Due: December 2007
About this textbook
Algebraic K-Theory has become an increasingly active area of research. With its connections to algebra, algebraic geometry, topology, and number theory, it has implications for a wide variety of researchers and graduate students in mathematics. The book is based on lectures given at the author's home institution, the Tata Institute in Bombay, and elsewhere. A detailed appendix on topology was provided in the first edition to make the treatment accessible to readers with a limited background in topology. The second edition also includes an appendix on algebraic geometry that contains the required definitions and results needed to understand the core of the book; this makes the book accessible to a wider audience.
A central part of the book is a detailed exposition of the ideas of Quillen as contained in his classic papers "Higher Algebraic K-Theory, I, II." A more elementary proof of the theorem of Merkujev--Suslin is given in this edition; this makes the treatment of this topic self-contained. An application is also given to modules of finite length and finite projective dimension over the local ring of a normal surface singularity. These results lead the reader to some interesting conclusions regarding the Chow group of varieties.
Table of contents
Preface to the First Edition.- Preface to the Second Edition.- “Classical” K-Theory.- The Plus Construction.- The Classifying Space of a Small Category.- Exact Categories and Quillen's Q-Construction.- The K-Theory of Rings and Schemes.- Proofs of the Theorems of Chapter 4.- Comparison of the Plus and Q-Constructions.- The Merkurjev--Suslin Theorem.- Localization for Singular Varieties.- Appendix A. Results from Topology.- Appendix B. Results from Category Theory.- Appendix C. Exact Couples.- Appendix D. Results from Algebraic Geometry.- Bibliography.