Inder K. Rana, Indian Institute of Technology, Powai, Mumbai, India
An Introduction to Measure and Integration: Second Edition
Expected publication date is August 11, 2002
Description
Integration is one of the two cornerstones of analysis. Since the fundamental work of Lebesgue, integration has been interpreted in terms of measure theory. This introductory text starts with the historical development of the notion of the integral and a review of the Riemann integral. From here, the reader is naturally led to the consideration of the Lebesgue integral, where abstract integration is developed via measure theory. The important basic topics are all covered: the Fundamental Theorem of Calculus, Fubini's Theorem, L_p spaces, the Radon-Nikodym Theorem, change of variables formulas, and so on.
The book is written in an informal style to make the subject matter easily accessible. Concepts are developed with the help of motivating examples, probing questions, and many exercises. It would be suitable as a textbook for an introductory course on the topic or for self-study.
For this edition, more exercises and four appendices have been added.
The AMS maintains exclusive distribution rights for this edition in North America and nonexclusive distribution rights worldwide, excluding India, Pakistan, Bangladesh, Nepal, Bhutan, Sikkim, and Sri Lanka.
Contents
Prologue: The length function
Riemann integration
Recipes for extending the Riemann integral
General extension theory
The Lebesgue measure on \mathbb{R} and its properties
Integration
Fundamental theorem of calculus for the Lebesgue integral
Measure and integration on product spaces
Modes of convergence and L_p-spaces
The Radon-Nikodym theorem and its applications
Signed measures and complex measures
Extended real numbers
Axiom of choice
Continuum hypotheses
Urysohn's lemma
Singular value decomposition of a matrix
Functions of bounded variation
Differentiable transformations
Index of symbols
References
Index
Details:
Series: Graduate Studies in Mathematics, Volume: 45
Publication Year: 2002
ISBN: 0-8218-2974-2
Paging: approximately 456 pp.
Binding: Hardcover
Joseph L. Taylor, University of Utah, Salt Lake City, UT
Several Complex Variables with Connections
to Algebraic Geometry and Lie Groups
Expected publication date is June 13, 2002
Description
This text presents an integrated development of core material from several complex variables and complex algebraic geometry, leading to proofs of Serre's celebrated GAGA theorems relating the two subjects, and including applications to the representation theory of complex semisimple Lie groups. It includes a thorough treatment of the local theory using the tools of commutative algebra, an extensive development of sheaf theory and the theory of coherent analytic and algebraic sheaves, proofs of the main vanishing theorems for these categories of sheaves, and a complete proof of the finite dimensionality of the cohomology of coherent sheaves on compact varieties. The vanishing theorems have a wide variety of applications and these are covered in detail.
Of particular interest are the last three chapters, which are devoted to applications of the preceding material to the study of the structure theory and representation theory of complex semisimple Lie groups. Included are introductions to harmonic analysis, the Peter-Weyl theorem, Lie theory and the structure of Lie algebras, semisimple Lie algebras and their representations, algebraic groups and the structure of complex semisimple Lie groups. All of this culminates in Milicic's proof of the Borel-Weil-Bott theorem, which makes extensive use of the material developed earlier in the text.
There are numerous examples and exercises in each chapter. This modern treatment of a classic point of view would be an excellent text for a graduate course on several complex variables, as well as a useful reference for the expert.
Contents
Selected problems in one complex variable
Holomorphic functions of several variables
Local rings and varieties
The Nullstellensatz
Dimension
Homological algebra
Sheaves and sheaf cohomology
Coherent algebraic sheaves
Coherent analytic sheaves
Stein spaces
Frechet sheaves--Cartan's theorems
Projective varieties
Algebraic vs. analytic--Serre's theorems
Lie groups and their representations
Algebraic groups
The Borel-Weil-Bott theorem
Bibliography
Index
Details:
Series: Graduate Studies in Mathematics,Volume: 46
Publication Year: 2002
ISBN: 0-8218-3178-X
Paging: approximately 528 pp.
Binding: Hardcover
S. Kumaresan, University of Mumbai, India
A Course in Differential Geometry and Lie Groups
Description
This book arose out of courses taught by the author. It covers the traditional topics of differential manifolds, tensor fields, Lie groups, integration on manifolds and basic differential and Riemannian geometry. The author emphasizes geometric concepts, giving the reader a working knowledge of the topic. Motivations are given, exercises are included, and illuminating nontrivial examples are discussed.
Important features include the following:
Geometric and conceptual treatment of differential calculus with a wealth of nontrivial examples.
A thorough discussion of the much-used result on the existence, uniqueness, and smooth dependence of solutions of ODEs.
Careful introduction of the concept of tangent spaces to a manifold.
Early and simultaneous treatment of Lie groups and related concepts.
A motivated and highly geometric proof of the Frobenius theorem.
A constant reconciliation with the classical treatment and the modern approach.
Simple proofs of the hairy-ball theorem and Brouwer's fixed point theorem.
Construction of manifolds of constant curvature a la Chern.
This text would be suitable for use as a graduate-level introduction to basic differential and Riemannian geometry.
Contents
Differential calculus
Manifolds and Lie groups
Tensor analysis
Integration
Riemannian geometry
Tangent bundles and vector bundles
Partitions of unity
Bibliography
List of symbols
Index
Details:
Publisher: Hindustan Book Agency
Distributor: American Mathematical Society
Series: Hindustan Book Agency
Publication Year: 2002
ISBN: 81-85931-29-1
Paging: 295 pp.
Binding: Hardcover
Ken Brewer , Consultant Statistician, Australia
Combining Survey Sampling Inferences
The weighing of Basu's elephants
Description:
This text is both a two-semester course for beginners and a balanced guide to the controversial question of survey sampling inferences: whether they should be made primarily in terms of the inclusion probabilities, or whether instead they should be based squarely on estimates of the parameters of a realistic population model.
The author, using his extensive experience in this field, argues cogently that the two approaches are complementary rather than competitive, the former being appropriate for large samples and the latter for small ones. He also shows how they can neatly be combined. In doing so, he unifies the creative results that came out of the randomization approach in the 1940s and early 1950s with the most important of the advances that have been made in both schools since.
To achieve these ends, he uses a story style that brings life to the tools used by the two approaches and shows how they can be fitted together in an expert's hand. The careful back and forth discussions of the paradoxes that arise, depending on which principled approach is used, will be illuminating not only to beginning students but also to experienced practitioners who have used these tools themselves, but perhaps without exploring all their ramifications.
Buy this book if you are teaching or studying a course in survey sampling. A practitioner who wants to understand both sides of the current controversy or a philosopher of science interested in seeing what happens in a discipline undergoing a paradigm shift.
Key Features:
* Design based AND model-based - treated equally seriously and used in combination
* Everything is kept as simple as possible, including the maths
* Written by author with extensive professional sampling experience
Readership:
Senior undergraduates/masters statistics students.
Binding: Paperback
Dimensions: 244 x 172mm
Published: 01/08/2002
ISBN: 0-340-69229-4