David M. Bressoud's Books

A Radical Approach to Real Analysis

Second Edition

By: David Bressoud

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A Radical Approach to Real Analysis

Second Edition

By: David Bressoud

In this second edition of the MAA classic, exploration continues to be an essential component. More than 60 new exercises have been added, and the chapters on Infinite Summations, Differentiability and Continuity, and Convergence of Infinite Series have been reorganized to make it easier to identify the key ideas. A Radical Approach to Real Analysis is an introduction to real analysis, rooted in and informed by the historical issues that shaped its development. It can be used as a textbook, as a resource for the instructor who prefers to teach a traditional course, or as a resource for the student who has been through a traditional course yet still does not understand what real analysis is about and why it was created. The book begins with Fourier's introduction of trigonometric series and the problems they created for the mathematicians of the early 19th century. It follows Cauchy's attempts to establish a firm foundation for calculus and considers his failures as well as his successes. It culminates with Dirichlet's proof of the validity of the Fourier series expansion and explores some of the counterintuitive results Riemann and Weierstrass were led to as a result of Dirichlet's proof.

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Page Count

339

Publisher

American Mathematical Society

Published Date

ISBN 10

1470469049

ISBN 13

9781470469047

Proofs and Confirmations

The Story of the Alternating-Sign Matrix Conjecture

By: David M. Bressoud

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Proofs and Confirmations

The Story of the Alternating-Sign Matrix Conjecture

By: David M. Bressoud

This is an introduction to recent developments in algebraic combinatorics and an illustration of how research in mathematics actually progresses. The author recounts the story of the search for and discovery of a proof of a formula conjectured in the late 1970s: the number of n x n alternating sign matrices, objects that generalize permutation matrices. While apparent that the conjecture must be true, the proof was elusive. Researchers became drawn to this problem, making connections to aspects of invariant theory, to symmetric functions, to hypergeometric and basic hypergeometric series, and, finally, to the six-vertex model of statistical mechanics. All these threads are brought together in Zeilberger's 1996 proof of the original conjecture. The book is accessible to anyone with a knowledge of linear algebra. Students will learn what mathematicians actually do in an interesting and new area of mathematics, and even researchers in combinatorics will find something new here.

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Page Count

292

Publisher

Cambridge University Press

Published Date

ISBN 10

1316582752

ISBN 13

9781316582756

Second Year Calculus

From Celestial Mechanics to Special Relativity

By: David M. Bressoud

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Second Year Calculus

From Celestial Mechanics to Special Relativity

By: David M. Bressoud

Second Year Calculus: From Celestial Mechanics to Special Relativity covers multi-variable and vector calculus, emphasizing the historical physical problems which gave rise to the concepts of calculus. The book guides us from the birth of the mechanized view of the world in Isaac Newton's Mathematical Principles of Natural Philosophy in which mathematics becomes the ultimate tool for modelling physical reality, to the dawn of a radically new and often counter-intuitive age in Albert Einstein's Special Theory of Relativity in which it is the mathematical model which suggests new aspects of that reality. The development of this process is discussed from the modern viewpoint of differential forms. Using this concept, the student learns to compute orbits and rocket trajectories, model flows and force fields, and derive the laws of electricity and magnetism. These exercises and observations of mathematical symmetry enable the student to better understand the interaction of physics and mathematics.

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Page Count

399

Publisher

Springer Science & Business Media

Published Date

ISBN 10

1461209595

ISBN 13

9781461209591

Calculus Reordered

A History of the Big Ideas

By: David M. Bressoud

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Calculus Reordered

A History of the Big Ideas

By: David M. Bressoud

Calculus Reordered takes readers on a remarkable journey through hundreds of years to tell the story of how calculus grew to what we know today. David Bressoud explains why calculus is credited to Isaac Newton and Gottfried Leibniz in the seventeenth century, and how its current structure is based on developments that arose in the nineteenth century. Bressoud argues that a pedagogy informed by the historical development of calculus presents a sounder way for students to learn this fascinating area of mathematics. Delving into calculus's birth in the Hellenistic Eastern Mediterranean--especially Syracuse in Sicily and Alexandria in Egypt--as well as India and the Islamic Middle East, Bressoud considers how calculus developed in response to essential questions emerging from engineering and astronomy. He looks at how Newton and Leibniz built their work on a flurry of activity that occurred throughout Europe, and how Italian philosophers such as Galileo Galilei played a particularly important role. In describing calculus's evolution, Bressoud reveals problems with the standard ordering of its curriculum: limits, differentiation, integration, and series. He contends instead that the historical order--which follows first integration as accumulation, then differentiation as ratios of change, series as sequences of partial sums, and finally limits as they arise from the algebra of inequalities--makes more sense in the classroom environment. Exploring the motivations behind calculus's discovery, Calculus Reordered highlights how this essential tool of mathematics came to be.

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Page Count

242

Publisher

Princeton University Press

Published Date

ISBN 10

0691218781

ISBN 13

9780691218786

Factorization and Primality Testing

By: David M. Bressoud

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Factorization and Primality Testing

By: David M. Bressoud

About binomial theorems I'm teeming with a lot of news, With many cheerful facts about the square on the hypotenuse. " - William S. Gilbert (The Pirates of Penzance, Act I) The question of divisibility is arguably the oldest problem in mathematics. Ancient peoples observed the cycles of nature: the day, the lunar month, and the year, and assumed that each divided evenly into the next. Civilizations as separate as the Egyptians of ten thousand years ago and the Central American Mayans adopted a month of thirty days and a year of twelve months. Even when the inaccuracy of a 360-day year became apparent, they preferred to retain it and add five intercalary days. The number 360 retains its psychological appeal today because it is divisible by many small integers. The technical term for such a number reflects this appeal. It is called a "smooth" number. At the other extreme are those integers with no smaller divisors other than 1, integers which might be called the indivisibles. The mystic qualities of numbers such as 7 and 13 derive in no small part from the fact that they are indivisibles. The ancient Greeks realized that every integer could be written uniquely as a product of indivisibles larger than 1, what we appropriately call prime numbers. To know the decomposition of an integer into a product of primes is to have a complete description of all of its divisors.

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Page Count

252

Publisher

Springer Science & Business Media

Published Date

ISBN 10

1461245443

ISBN 13

9781461245445

A Radical Approach to Lebesgue's Theory of Integration

By: David M. Bressoud

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A Radical Approach to Lebesgue's Theory of Integration

By: David M. Bressoud

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.

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Cambridge University Press

Published Date

ISBN 10

0521884748

ISBN 13

9780521884747

Calculus Reordered

A History of the Big Ideas

By: David M. Bressoud

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Calculus Reordered

A History of the Big Ideas

By: David M. Bressoud

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Page Count

242

Publisher

Princeton University Press

Published Date

ISBN 10

0691218781

ISBN 13

9780691218786

A Radical Approach to Real Analysis

By: David M. Bressoud

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A Radical Approach to Real Analysis

By: David M. Bressoud

Second edition of this introduction to real analysis, rooted in the historical issues that shaped its development.

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Page Count

352

Publisher

MAA

Published Date

ISBN 10

0883857472

ISBN 13

9780883857472

Book's by David M. Bressoud

A Radical Approach to Real Analysis

A Radical Approach to Real Analysis

Proofs and Confirmations

Proofs and Confirmations

Second Year Calculus

Second Year Calculus

Calculus Reordered

Calculus Reordered

Factorization and Primality Testing

Factorization and Primality Testing

A Radical Approach to Lebesgue's Theory of Integration

A Radical Approach to Lebesgue's Theory of Integration

Calculus Reordered

Calculus Reordered

A Radical Approach to Real Analysis

A Radical Approach to Real Analysis

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