A Radical Approach to Lebesgue's Theory of Integration

Author: David M. Bressoud

Publisher: Cambridge University Press

ISBN:

Category: Mathematics

Page: 329

View: 317

Meant for advanced undergraduate and graduate students in mathematics, this introduction to measure theory and Lebesgue integration is motivated by the historical questions that led to its development. The author tells the story of the mathematicians who wrestled with the difficulties inherent in the Riemann integral, leading to the work of Jordan, Borel, and Lebesgue.

A Primer of Lebesgue Integration

Author: H. S. Bear

Publisher: Academic Press

ISBN:

Category: Mathematics

Page: 164

View: 895

This successful text offers a reader-friendly approach to Lebesgue integration. It is designed for advanced undergraduates, beginning graduate students, or advanced readers who may have forgotten one or two details from their real analysis courses. "The Lebesgue integral has been around for almost a century. Most authors prefer to blast through the preliminaries and get quickly to the more interesting results. This very efficient approach puts a great burden on the reader; all the words are there, but none of the music." Bear's goal is to proceed more slowly so the reader can develop some intuition about the subject. Many readers of the successful first edition would agree that he achieves this goal. The principal change in this edition is the simplified definition of the integral. The integral is defined either with upper and lower sums as in the calculus, or with Riemann sums, but using countable partitions of the domain into measurable sets. This one-shot approach works for bounded or unbounded functions and for sets of finite or infinite measure. The author's style is graceful and pleasant to read. The explanations are exceptionally clear. Someone looking for an introduction to Lebesgue integration could scarcely do better than this text. -John Erdman Portland State University This is an excellent book. Several features make it unique. The author gets through the standard canon in only 150 pages and then arranges the material into easily digestible units (a proof hardly ever exceeds three-fourths of a page). The author writes with concision, clarity, and focus. -Robert Burckel Kansas State University This text achieves its worthy goals. The author tends to the business at hand. The short chapter on Lebesgue integration is refreshing and easily understood. One can use a semester covering the book, and the students will be well-grounded in the basics and ready for any of a dozen possible second semesters. -Joseph Diestel Kent State University

Real Analysis for the Undergraduate

With an Invitation to Functional Analysis

Author: Matthew A. Pons

Publisher: Springer Science & Business Media

ISBN:

Category: Mathematics

Page: 409

View: 962

This undergraduate textbook introduces students to the basics of real analysis, provides an introduction to more advanced topics including measure theory and Lebesgue integration, and offers an invitation to functional analysis. While these advanced topics are not typically encountered until graduate study, the text is designed for the beginner. The author’s engaging style makes advanced topics approachable without sacrificing rigor. The text also consistently encourages the reader to pick up a pencil and take an active part in the learning process. Key features include: - examples to reinforce theory; - thorough explanations preceding definitions, theorems and formal proofs; - illustrations to support intuition; - over 450 exercises designed to develop connections between the concrete and abstract. This text takes students on a journey through the basics of real analysis and provides those who wish to delve deeper the opportunity to experience mathematical ideas that are beyond the standard undergraduate curriculum.

Understanding Analysis

Author: Stephen Abbott

Publisher: Springer

ISBN:

Category: Mathematics

Page: 312

View: 162

This lively introductory text exposes the student to the rewards of a rigorous study of functions of a real variable. In each chapter, informal discussions of questions that give analysis its inherent fascination are followed by precise, but not overly formal, developments of the techniques needed to make sense of them. By focusing on the unifying themes of approximation and the resolution of paradoxes that arise in the transition from the finite to the infinite, the text turns what could be a daunting cascade of definitions and theorems into a coherent and engaging progression of ideas. Acutely aware of the need for rigor, the student is much better prepared to understand what constitutes a proper mathematical proof and how to write one. Fifteen years of classroom experience with the first edition of Understanding Analysis have solidified and refined the central narrative of the second edition. Roughly 150 new exercises join a selection of the best exercises from the first edition, and three more project-style sections have been added. Investigations of Euler’s computation of ζ(2), the Weierstrass Approximation Theorem, and the gamma function are now among the book’s cohort of seminal results serving as motivation and payoff for the beginning student to master the methods of analysis.

Varieties of Integration

Author: C. Ray Rosentrater

Publisher: The Mathematical Association of America

ISBN:

Category: Mathematics

Page: 342

View: 185

Varieties of Integration explores the critical contributions by Riemann, Darboux, Lebesgue, Henstock, Kurzweil, and Stieltjes to the theory of integration and provides a glimpse of more recent variations of the integral such as those involving operator-valued measures. By the first year of graduate school, a young mathematician will have encountered at least three separate definitions of the integral. The associated integrals are typically studied in isolation with little attention paid to the relationships between them or to the historical issues that motivated their definitions. Varieties of Integration redresses this situation by introducing the Riemann, Darboux, Lebesgue, and gauge integrals in a single volume using a common set of examples. This approach allows the reader to see how the definitions influence proof techniques and computational strategies. Then the properties of the integrals are compared in three major areas: the class of integrable functions, the convergence properties of the integral, and the best form of the Fundamental Theorems of Calculus.

A Course in Computational Number Theory

Author: David Bressoud

Publisher: Wiley

ISBN:

Category: Mathematics

Page: 384

View: 518

A Course in Computational Number Theory uses the computer as a tool for motivation and explanation. The book is designed for the reader to quickly access a computer and begin doing personal experiments with the patterns of the integers. It presents and explains many of the fastest algorithms for working with integers. Traditional topics are covered, but the text also explores factoring algorithms, primality testing, the RSA public-key cryptosystem, and unusual applications such as check digit schemes and a computation of the energy that holds a salt crystal together. Advanced topics include continued fractions, Pell's equation, and the Gaussian primes.

Choice

Publication of the Association of College and Research Libraries, a Division of the American Library Association

Author:

Publisher:

ISBN:

Category: Academic libraries

Page:

View: 577

Whitaker's Cumulative Book List

A Classified List of Publications...together with an Index to Authors and Titles

Author:

Publisher:

ISBN:

Category: English literature

Page:

View: 968

Issues in General and Specialized Mathematics Research: 2011 Edition

Author:

Publisher: ScholarlyEditions

ISBN:

Category: Mathematics

Page: 862

View: 911

Issues in General and Specialized Mathematics Research: 2011 Edition is a ScholarlyEditions™ eBook that delivers timely, authoritative, and comprehensive information about General and Specialized Mathematics Research. The editors have built Issues in General and Specialized Mathematics Research: 2011 Edition on the vast information databases of ScholarlyNews.™ You can expect the information about General and Specialized Mathematics Research in this eBook to be deeper than what you can access anywhere else, as well as consistently reliable, authoritative, informed, and relevant. The content of Issues in General and Specialized Mathematics Research: 2011 Edition has been produced by the world’s leading scientists, engineers, analysts, research institutions, and companies. All of the content is from peer-reviewed sources, and all of it is written, assembled, and edited by the editors at ScholarlyEditions™ and available exclusively from us. You now have a source you can cite with authority, confidence, and credibility. More information is available at http://www.ScholarlyEditions.com/.