Cloth | June 2007 | ISBN13: 978-0-691-12526-8
280 pp. | 6 x 9 | 8 color plates. 141 line illus. 2 tables.
By any measure, the Pythagorean theorem is the most famous
statement in all of mathematics, one remembered from high school
geometry class by even the most math-phobic students. Well over
four hundred proofs are known to exist, including ones by a
twelve-year-old Einstein, a young blind girl, Leonardo da Vinci,
and a future president of the United States. Here--perhaps for
the first time in English--is the full story of this famous
theorem.
Although attributed to Pythagoras, the theorem was known to the
Babylonians more than a thousand years before him. He may have
been the first to prove it, but his proof--if indeed he had one--is
lost to us. Euclid immortalized it as Proposition 47 in his
Elements, and it is from there that it has passed down to
generations of students. The theorem is central to almost every
branch of science, pure or applied. It has even been proposed as
a means to communicate with extraterrestrial beings, if and when
we discover them. And, expanded to four-dimensional space-time,
it plays a pivotal role in Einstein's theory of relativity.
In this book, Eli Maor brings to life many of the characters that
played a role in the development of the Pythagorean theorem,
providing a fascinating backdrop to perhaps our oldest enduring
mathematical legacy.
Eli Maor teaches the history of mathematics at Loyola University
in Chicago. He is the author of Venus in Transit, Trigonometric
Delights, e: The Story of a Number, and To Infinity and Beyond: A
Cultural History of the Infinite (all Princeton).
Endorsements:
"There's a lot more to the Pythagorean theorem than a2 + b2
= c2, and you'll find it all in Eli Maor's new book. Destined to
become a classic, this book is written with Maor's usual high
level of skill, scholarship, and attention to detail. He's also
got a sense of humor that will please a range of readers. As we
used to say in the 1950s, 'Miss it and be square!'"--Paul J.
Nahin, author of Chases and Escapes and Dr. Euler1s Fabulous
Formula
"Eli Maor states that the Pythagorean theorem 'is arguably
the most frequently used theorem in all of mathematics.' He then
supports this claim by taking his reader on a journey from the
earliest evidence of knowledge of the theorem to Einstein's
theory of relativity and Wiles's proof of Fermat's last theorem,
from the Babylonians around 1800 BCE to the end of the twentieth
century. I think that the reader who makes the journey with Maor
will be convinced beyond a reasonable doubt. He is the first
author who has sifted through all the mathematics, history of
mathematics, and physics books and collected for us just the
material directly related to the Pythagorean theorem."--Robert
W. Langer, Professor Emeritus, University of Wisconsin, Eau
Claire
Cloth | February 2007 | ISBN13: 978-0-691-12993-8
606 pp. | 6 x 9 | 200 line illus.
This book presents the latest findings on one of the most
intensely investigated subjects in computational mathematics--the
traveling salesman problem. It sounds simple enough: given a set
of cities and the cost of travel between each pair of them, the
problem challenges you to find the cheapest route by which to
visit all the cities and return home to where you began. Though
seemingly modest, this exercise has inspired studies by
mathematicians, chemists, and physicists. Teachers use it in the
classroom. It has practical applications in genetics,
telecommunications, and neuroscience.
The authors of this book are the same pioneers who for nearly two
decades have led the investigation into the traveling salesman
problem. They have derived solutions to almost eighty-six
thousand cities, yet a general solution to the problem has yet to
be discovered. Here they describe the method and computer code
they used to solve a broad range of large-scale problems, and
along the way they demonstrate the interplay of applied
mathematics with increasingly powerful computing platforms. They
also give the fascinating history of the problem--how it
developed, and why it continues to intrigue us.
David L. Applegate is a researcher at AT&T Labs. Robert E.
Bixby is Research Professor of Management and Noah Harding
Professor of Computational and Applied Mathematics at Rice
University. Vasek Chvatal is Canada Research Chair in
Combinatorial Optimization at Concordia University. William J.
Cook is Chandler Family Chair in Industrial and Systems
Engineering at the Georgia Institute of Technology.
Endorsements:
"This book addresses one of the most famous and important
combinatorial-optimization problems--the traveling salesman
problem. It is very well written, with a vivid style that
captures the reader's attention. Many examples are provided that
are very useful to motivate and help the reader to better
understand the results presented in the book."--Matteo
Fischetti, University of Padova
"This is a fantastic book. Ever since the early days of
discrete optimization, the traveling salesman problem has served
as the model for computationally hard problems. The authors are
main players in this area who forged a team in 1988 to push the
frontiers on how good we are in solving hard and large traveling
salesman problems. Now they lay out their views, experience, and
findings in this book."--Bert Gerards, Centrum voor Wiskunde
en Informatica
contents
Paper | January 2007 | ISBN13: 978-0-691-12734-7
Cloth | January 2007 | ISBN13: 978-0-691-12733-0
184 pp. | 8 x 10 | 14 halftones. 6 line illus.
This book describes the theory and applications of discrete
orthogonal polynomials--polynomials that are orthogonal on a
finite set. Unlike other books, Discrete Orthogonal Polynomials
addresses completely general weight functions and presents a new
methodology for handling the discrete weights case.
J. Baik, T. Kriecherbauer, K. T.-R. McLaughlin & P. D. Miller
focus on asymptotic aspects of general, nonclassical discrete
orthogonal polynomials and set out applications of current
interest. Topics covered include the probability theory of
discrete orthogonal polynomial ensembles and the continuum limit
of the Toda lattice. The primary concern throughout is the
asymptotic behavior of discrete orthogonal polynomials for
general, nonclassical measures, in the joint limit where the
degree increases as some fraction of the total number of points
of collocation. The book formulates the orthogonality conditions
defining these polynomials as a kind of Riemann-Hilbert problem
and then generalizes the steepest descent method for such a
problem to carry out the necessary asymptotic analysis.
J. Baik is Associate Professor of Mathematics at the University
of Michigan. T. Kriecherbauer is Professor of Mathematics at Ruhr-Universitat
Bochum in Bochum, Germany. K. T.-R. McLaughlin is Professor of
Mathematics at the University of Arizona. P. D. Miller is
Associate Professor of Mathematics at the University of Michigan.
Cloth | 2007 | ISBN13: 978-0-691-12918-1
264 pp. | 6 x 9
This book represents the first synthesis of the considerable body
of new research into positive definite matrices. These matrices
play the same role in noncommutative analysis as positive real
numbers do in classical analysis. They have theoretical and
computational uses across a broad spectrum of disciplines,
including calculus, electrical engineering, statistics, physics,
numerical analysis, quantum information theory, and geometry.
Through detailed explanations and an authoritative and inspiring
writing style, Rajendra Bhatia carefully develops general
techniques that have wide applications in the study of such
matrices.
Bhatia introduces several key topics in functional analysis,
operator theory, harmonic analysis, and differential geometry--all
built around the central theme of positive definite matrices. He
discusses positive and completely positive linear maps, and
presents major theorems with simple and direct proofs. He
examines matrix means and their applications, and shows how to
use positive definite functions to derive operator inequalities
that he and others proved in recent years. He guides the reader
through the differential geometry of the manifold of positive
definite matrices, and explains recent work on the geometric mean
of several matrices.
Positive Definite Matrices is an informative and useful reference
book for mathematicians and other researchers and practitioners.
The numerous exercises and notes at the end of each chapter also
make it the ideal textbook for graduate-level courses.
Rajendra Bhatia is Professor of Mathematics at the Indian
Statistical Institute in New Delhi. He is the author of five
books, including Matrix Analysis.
Endorsements:
"This is a monograph for mathematicians interested in an
important realm of matrix-analytic ideas. Like the author's
distinguished book, Matrix Analysis, it will be a convenient and
much-quoted reference source. There are many wonderful insights
in a first-rate exposition of important ideas not easily
extracted from other sources. The scholarship is impeccable."--Roger
A. Horn, University of Utah
"I believe that every expert in matrix analysis can find
something new in this book. Bhatia presents some important
material in several topics related to positive definite matrices
including positive linear maps, completely positive maps, matrix
means, positive definite functions, and geometry of positive
definite matrices. There are many beautiful results, useful
techniques, and ingenious ideas here. Bhatia's writing style has
always been concise, clear, and illuminating."--Xingzhi
Zhan, East China Normal University
Contents
Paper | March 2007 | ISBN13: 978-0-691-12810-8
Cloth | March 2007 | ISBN13: 978-0-691-12809-2
200 pp. | 6 x 9 | 15 halftones. 9 line illus.
This book proves an analogue of William Thurston's celebrated
hyperbolic Dehn surgery theorem in the context of complex
hyperbolic discrete groups, and then derives two main geometric
consequences from it. The first is the construction of large
numbers of closed real hyperbolic 3-manifolds which bound complex
hyperbolic orbifolds--the only known examples of closed manifolds
that simultaneously have these two kinds of geometric structures.
The second is a complete understanding of the structure of
complex hyperbolic reflection triangle groups in cases where the
angle is small. In an accessible and straightforward manner,
Richard Evan Schwartz also presents a large amount of useful
information on complex hyperbolic geometry and discrete groups.
Schwartz relies on elementary proofs and avoids quotations of
preexisting technical material as much as possible. For this
reason, this book will benefit graduate students seeking entry
into this emerging area of research, as well as researchers in
allied fields such as Kleinian groups and CR geometry.
Richard Evan Schwartz is Professor of Mathematics at Brown
University
Cloth | July 2007 | ISBN13: 978-0-691-12738-5
416 pp. | 6 x 9 | 6 halftones. ?44 line illus.
To many outsiders, mathematicians appear to think like computers,
grimly grinding away with a strict formal logic and moving
methodically--even algorithmically--from one black-and-white
deduction to another. Yet mathematicians often describe their
most important breakthroughs as creative, intuitive responses to
ambiguity, contradiction, and paradox. A unique examination of
this less-familiar aspect of mathematics, How Mathematicians
Think reveals that mathematics is a profoundly creative activity
and not just a body of formalized rules and results.
Nonlogical qualities, William Byers shows, play an essential role
in mathematics. Ambiguities, contradictions, and paradoxes can
arise when ideas developed in different contexts come into
contact. Uncertainties and conflicts, Byers demonstrates, do not
impede but rather spur the development of mathematics. Creativity
often means bringing apparently incompatible perspectives
together as complementary aspects of a new, more subtle theory.
The secret of mathematics is not to be found only in its logical
structure.
The creative dimensions of mathematical work have great
implications for our notions of mathematical and scientific
truth, Byers argues, and How Mathematicians Think provides a
novel approach to many fundamental questions. Is mathematics
objectively true? Is it discovered or invented? And is there such
a thing as a "final" scientific theory?
Ultimately, How Mathematicians Think shows that the nature of
mathematical thinking can teach us a great deal about the human
condition itself
William Byers is Professor of Mathematics at Concordia University
in Montreal. He has published widely in mathematics journals.
Endorsements:
"This is an important book, one that should cause an epoch-making
change in the way we think about mathematics. While mathematics
is often presented as an immutable, absolute science in which
theorems can be proved for all time in a platonic sense, here we
see the creative, human aspect of mathematics and its paradoxes
and conflicts. This has all the hallmarks of a must-read book."--David
Tall, coauthor of Algebraic Number Theory and Fermat's Last
Theorem
"I strongly recommend this book. The discussions of
mathematical ambiguity, contradiction, and paradox are excellent.
In addition to mathematics, the book draws on other sciences, as
well as philosophy, literature, and history. The historical
discussions are particularly interesting and are woven into the
mathematics."--Joseph Auslander, Professor Emeritus,
University of Maryland
Classics in Applied Mathematics 52
This book focuses on the fundamental ideas of continuum mechanics
by analyzing models of fluid flow and solid deformation and
examining problems in elasticity, water waves, and extremum
principles. Mathematics Applied to Continuum Mechanics gives an
excellent overview of the subject, with an emphasis on clarity,
explanation, and motivation. Extensive exercises and a valuable
section containing hints and answers make this an excellent text
for both classroom use and independent study.
Audience
This is an ideal text for upper-level undergraduate and graduate
students in the fields of applied mathematics, science, and
engineering.
Contents
Foreword to the Classics Edition
Preface
Contents
Conventions
Part A: Geometrical Prerequisites for Three-Dimensional Continuum
Mechanics
Chapter 1: Vectors, Determinants, and Motivation for Tensors
Chapter 2: Cartesian Tensors
Part B: Problems in Continuum Mechanics
Chapter 3: Viscous Fluids
Chapter 4: Foundations in Elasticity
Chapter 5: Some Examples of Static Problems in Elasticity
Chapter 6: Introduction to Dynamic Problems in Elasticity
Part C: Water Waves
Chapter 7: Formulation of the Theory of Surface Waves in an
Inviscid Fluid
Chapter 8: Solution in the Linear Theory
Chapter 9: Group Speed and Group Velocity
Chapter 10: Nonlinear Effects
Part D: Variational Methods and Extremum Principles
Chapter 11: Calculus of Variations
Chapter 12: Characterization of Eigenvalues and Equilibrium
States as Extrema
Bibliography
Hints and Answers
Index.
About the Authors
Lee A. Segel (1932?2005) was the Henry and Bertha Benson
Professor of Mathematics at the Weizmann Institute of Science. He
also served as Head of the Department of Applied Mathematics,
Dean of the Faculty of Mathematical Sciences, and Chairman of the
Scientific Council. Professor Segel taught at institutions
throughout the United States, most recently at the Santa Fe
Institute.
G. H. Handelman is the Amos Eaton Professor Emeritus in the
Department of Mathematical Sciences at Rensselaer Polytechnic
Institute.
Available January 2007 / Approx. xvii + 590 pages / Softcover
ISBN-10: 0-89871-620-9 / ISBN-13: 978-0-898716-20-7