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A central object of this book is the discrete
Laplace operator on finite and infinite graphs. The eigenvalues of the
discrete Laplace operator have long been used in graph theory as a
convenient tool for understanding the structure of complex
graphs. They can also be used in order to estimate the rate of
convergence to equilibrium of a random walk (Markov chain) on finite
graphs. For infinite graphs, a study of the heat kernel allows to
solve the type problem—a problem of deciding whether the random walk
is recurrent or transient.
This book starts with elementary properties of the eigenvalues on
finite graphs, continues with their estimates and applications, and
concludes with heat kernel estimates on infinite graphs and their
application to the type problem.
The book is suitable for beginners in the subject and accessible to
undergraduate and graduate students with a background in linear
algebra I and analysis I. It is based on a lecture course taught by
the author and includes a wide variety of exercises. The book will
help the reader to reach a level of understanding sufficient to start
pursuing research in this exciting area.
Undergraduate and graduate students and researchers interested in random walks on graphs and groups.
Anybody who has ever read a mathematical text of the author would agree that his way of presenting complex material is nothing short of marvelous. This new book showcases again the author's unique ability of presenting challenging topics in a clear and accessible manner, and of guiding the reader with ease to a deep understanding of the subject.
-- Matthias Keller, University of Potsdam
A TeXas Style Introduction to Proof is an IBL textbook designed for a one-semester course on proofs (the “bridge course”) that also introduces TeX as a tool students can use to communicate their work. As befitting “textless” text, the book is, as one reviewer characterized it, “minimal.” Written in an easy-going style, the exposition is just enough to support the activities, and it is clear, concise, and effective. The book is well organized and contains ample carefully selected exercises that are varied, interesting, and probing, without being discouragingly difficult.
A lovely little book for beginning mathematics majors and other students encountering proofs for the first time. Students should find the text appealing, and it contains many good exercises that a professor can build a course around. … Overall, a most satisfying book for a beginning class in mathematical proofs.
-- Curt Bennett, Professor of Mathematics at Loyola Marymount University and 2010 Haimo Award Winner
A TeXas Style Introduction to Proof by Ron Taylor and Patrick X. Rault is truly delightful-full of humanizing charm that softens the hard edge of mathematical rigor. It is gentle, lively, clear, and warm. … From this book, students and their instructors will find many proofs of the joy of mathematics.
-- Michael Starbird, University Distinguished Teaching Professor ofMathematics at The University of Texas at Austin and 2007 Haimo Award Winner
Taylor and Rault skillfully guide students through basic proof-writing techniques so that the student createsand discovers the content. The book is well-written, the integration of LaTeX is unique, and the authors have a fantastic sense of humor.
-- Amanda Croll, Assistant Professor of Mathematics, Concordia University, Irvine
Common Sense Mathematics is a text for a one
semester college-level course in quantitative literacy. The text
emphasizes common sense and common knowledge in approaching real
problems through popular news items and finding useful mathematical
tools and frames with which to address those questions.
We asked ourselves what we hoped our students would remember about
this course in ten years' time. From that ten year perspective,
thoughts about syllabus—"what topics should we
cover?"—seemed much too narrow. What matters more is our wish to
change the way our students' minds work the way they approach a
problem, or, more generally, the way they approach the world. Most
people skip the numbers in newspapers, magazines, on the web, and,
more importantly, even in financial information. We hope that in ten
years our students will follow the news, confident in their ability to
make sense of the numbers they find there and in their daily
lives.
Most quantitative reasoning texts are arranged by mathematical topics
to be mastered. Since the mathematics is only a part of what we hope
students learn, we've chosen another strategy. We look at real life
stories that can be best understood with careful reading and a little
mathematics.
An instructor's manual is freely available electronically: click here.
A solutions manual for this title is available
electronically to those instructors who have adopted the textbook for classroom use. Please send email to textbooks@ams.org for more
information.
The authors present a remarkable collection of problem situations gleaned from public media and supplemented by topics of personal finance. The problem situations are not only central to students' everyday lives but also provide a venue for continuing learning beyond a course and beyond school. Based on years of teaching quantitative reasoning, the book offers a clear model for such a course for the inexperienced teacher as well as opportunities for localizing and extending the material for the more experienced teacher.
-- Bernard L. Madison, University of Arkansas, Fayetteville
Where did math come from? Who thought up all those algebra symbols, and why? What's the story behind \(\pi\)? … negative numbers? … the metric system? … quadratic equations? … sine and cosine? The 25 independent sketches in Math through the Ages answer these questions and many others in an informal, easygoing style that's accessible to teachers, students, and anyone who is curious about the history of mathematical ideas. Each sketch contains Questions and Projects to help you learn more about its topic and to see how its main ideas fit into the bigger picture of history. The 25 short stories are preceded by a 56-page bird's-eye overview of the entire panorama of mathematical history, a whirlwind tour of the most important people, events, and trends that shaped the mathematics we know today. Reading suggestions after each sketch provide starting points for readers who want to pursue a topic further.
'This is a beautiful and important book, a pleasure to read, in which the history recounted fully illuminates the mathematical ideas, and the ideas themselves are superbly explained: a wonderful accomplishment.'
-- Barry Mazur, Harvard University
Math through the Ages is a treasure, one of the best history of math books at its level ever written. Somehow, it manages to stay true to a surprisingly sophisticated story, while respecting the needs of its audience.
-- Glen van Brummelen, Quest University President (2012.14), Canadian Society for History and Philosophy of Mathematics
This book offers the beginning undergraduate
student some of the vista of modern mathematics by developing and
presenting the tools needed to gain an understanding of the arithmetic
of elliptic curves over finite fields and their applications to modern
cryptography. This gradual introduction also makes a significant
effort to teach students how to produce or discover a proof by
presenting mathematics as an exploration, and at the same time, it
provides the necessary mathematical underpinnings to investigate the
practical and implementation side of elliptic curve cryptography
(ECC).
Elements of abstract algebra, number theory, and affine and
projective geometry are introduced and developed, and their interplay
is exploited. Algebra and geometry combine to characterize congruent
numbers via rational points on the unit circle, and group law for the
set of points on an elliptic curve arises from geometric intuition
provided by Bézout's theorem as well as the construction of projective
space. The structure of the unit group of the integers modulo a prime
explains RSA encryption, Pollard's method of factorization,
Diffie–Hellman key exchange, and ElGamal encryption, while the group
of points of an elliptic curve over a finite field motivates Lenstra's
elliptic curve factorization method and ECC.
The only real prerequisite for this book is a course on
one-variable calculus; other necessary mathematical topics are
introduced on-the-fly. Numerous exercises further guide the
exploration.
Undergraduate and graduate students interested in elliptic curves with applications to cryptography.
The main objective of this book, which is mainly aimed at undergraduate students, is to explain the arithmetic of elliptic curves defined over finite fields and to show how those curves can be used in cryptography. In order to do that, the author purposely avoids complex mathematical demonstrations and, instead, presents the concepts in a more descriptive way, suggesting some topics for further exploration by the reader.
-- Victor Gayoso Martíinez, Mathematical Reviews
Hassler Whitney was a giant of twentieth-century mathematics. This
biography paints a picture of him and includes dozens of revealing
anecdotes. Mathematically, he had a rare detector that went off
whenever he spotted a piece of mathematical gold, and he would then
draw countless pictures, gradually forging a path from hunch to proof.
This geometric path is seldom reflected in the rigor of his formal
papers, but thanks to a close friendship and many conversations over
decades, author Kendig was able to see how he actually worked. This
book shows this through accessible accounts of his major mathematical
contributions, with figures copiously supplied.
Whitney is probably best known for introducing the grandfather of
today's innumerable embedding theorems—his strong embedding theorem
stating that any smooth manifold can be smoothly embedded in a
Euclidean space of twice the manifold's dimension. This in turn led
to several standard techniques used every day in algebraic topology.
Whitney also established the fundamentals of graph theory, the
four-color problem, matroids, extending smooth functions, and
singularities of smooth functions. He almost never used complicated
technical machinery, so most of his work is accessible to a general
reader with a modest mathematical background.
His math-music connection was intense: He played piano, violin, and
viola and won “best composition of the year” while earning
a Bachelor's degree in music at Yale. He was an accomplished mountain
climber, and as a tinkerer, at age fourteen he built the large-format
camera used to take this book's cover photograph. Whitney's family
generously provided dozens of photographs appearing here for the very
first time. This biography is a revealing portrait of a fascinating
personality and a titan of twentieth-century mathematics.
Undergraduate and graduate students and researchers interested in history, biography, and the history of topology.
The main part of this book describes the first semester of the
existence of a successful and now highly popular program for
elementary school students at the Berkeley Math Circle. The topics
discussed in the book introduce the participants to the basics of many
important areas of modern mathematics, including logic, symmetry,
probability theory, knot theory, cryptography, fractals, and number
theory. Each chapter in the first part of this book consists of two
parts. It starts with generously illustrated sets of problems and
hands-on activities. This part is addressed to young readers who can
try to solve problems on their own or to discuss them with adults.
The second part of each chapter is addressed to teachers and
parents. It includes comments on the topics of the lesson, relates
those topics to discussions in other chapters, and describes
the actual reaction of math circle participants to the proposed
activities.
The supplementary problems that were discussed at workshops of Math
Circle at Kansas State University are given in the second part of the
book.
The book is richly illustrated, which makes it attractive to its young
audience.
In the interest of fostering a greater awareness and appreciation of
mathematics and its connections to other disciplines and everyday life, MSRI
and the AMS are publishing books in the Mathematical Circles Library series as
a service to young people, their parents and teachers, and the mathematics
profession.
Titles in this series are co-published with the Mathematical Sciences Research Institute (MSRI).
Math educators, teachers, mathematicians, instructors of math circles, and parents interested in general mathematical education.
The book is richly illustrated, which makes it attractive to its young audience.
-- Zentralblatt MATH
Why are there so few math circles, particularly for younger children? One of the reasons is the belief that very young kids are simply not ready for complex math. Another reason is that finding deep and engaging math activities, adapted for this younger audience, is itself a challenge. Natasha Rozhkovskaya's new book, Math Circles for Elementary School Students, helps deal with both these difficulties.
-- Moebuis Noodles
Mathematicians is a remarkable
collection of ninety-two photographic portraits, featuring a selection
of the most impressive mathematicians of our time. Acclaimed
photographer Mariana Cook captures the exuberance and passion of
these brilliant thinkers. The superb images are accompanied by
autobiographical texts written by each mathematician. Together, the
photographs and words illuminate a diverse group of men and women
dedicated to the absorbing pursuit of mathematics.
The compelling black-and-white portraits introduce readers to
mathematicians who are both young and old and from notably diverse
backgrounds. They include Fields Medal winners, those at the beginning
of major careers, and those who are long-established celebrities in
the discipline. Their candid personal essays reveal unique and
wide-ranging thoughts, opinions, and humor. The mathematicians discuss
how they became interested in mathematics, why they love the subject,
how they remain motivated in the face of mathematical challenges, and
how their greatest contributions have paved new directions for future
generations. Mathematicians in the book include Jean-Pierre Serre,
Henri Cartan, Karen Uhlenbeck, David Blackwell, Eli Stein, John
Conway, Timothy Gowers, Frances Kirwan, Peter Lax, William Massey,
John Milnor, Cathleen Morawetz, John Nash, Pierre Deligne, and James
Simons.
This book conveys the beauty and joy of mathematics to readers
outside the field as well as those in it. These pictures and their
texts are an inspiration, and a perfect gift for those who love
mathematics as well as for those who think they can't do it!
Undergraduate and graduate students and researchers interested in biographies of mathematicians.
"Mathematicians, An Outer View of the Inner World," is an irresistible book. It consists of ninety-two positively stunning portraits by Mariana Cook (evidently the last pupil (or protégé) of Ansel Adams) accompanied by brief introspective essays by her subjects on the facing pages...The choice of subjects is outstanding; almost everyone in the mathematical world has met or corresponded with at least some of these representative scholars, and many among the latter are household words. So it is that the personal reflections accompanying the marvelous photographic portraits more often than not come across as chats with friends...This is much more than a coffee-table book.
-- Michael Berg, MAA Review
Pay a visit to the Infinite Farm!
In Life on the Infinite Farm, mathematician and
award-winning children's book author Richard Schwartz teaches about
infinity and curved space through stories of whimsical farm
animals. Join Gracie, the shoe-loving cow with infinitely many feet,
Hammerwood, the gum-loving crocodile with an endless mouth, and their
friends as they navigate the challenges that come with being
infinitely large.
Children as young as 5 will enjoy the lighthearted illustrations
and the fanciful approach to infinity. Older students (and even adult
professional mathematicians) will also appreciate the more advanced
ideas and geometric references. The two approaches are woven together
to appeal to a wide audience, from budding mathematicians to hardcore
geometers.
A Guide for
Teachers and Parents is freely available with both simple and
advanced lesson plans.
Additionally, page-by-page
Notes on the
Infinite Farm are also available. These notes are intended for a
sophisticated mathematical audience but, perhaps, will be of some
interest to people who are not mathematicians.
For children ages 5 and up interested in new ways to think of space and geometry.
Selected as a 2018 CHOICE Outstanding Academic Title!
2018 PROSE Awards Honorable Mention
An Illustrated Theory of Numbers gives a
comprehensive introduction to number theory, with complete proofs,
worked examples, and exercises. Its exposition reflects the most
recent scholarship in mathematics and its history.
Almost 500 sharp illustrations accompany elegant proofs, from prime
decomposition through quadratic reciprocity. Geometric and dynamical
arguments provide new insights, and allow for a rigorous approach with
less algebraic manipulation. The final chapters contain an extended
treatment of binary quadratic forms, using Conway's topograph to solve
quadratic Diophantine equations (e.g., Pell's equation) and to study
reduction and the finiteness of class numbers.
Data visualizations introduce the reader to open questions and
cutting-edge results in analytic number theory such as the Riemann
hypothesis, boundedness of prime gaps, and the class number 1
problem. Accompanying each chapter, historical notes curate primary
sources and secondary scholarship to trace the development of number
theory within and outside the Western tradition.
Requiring only high school algebra and geometry, this text is
recommended for a first course in elementary number theory. It is also
suitable for mathematicians seeking a fresh perspective on an ancient
subject.
Undergraduate and graduate students interested in number theory.
This book is an introduction to number theory like no other. It covers the standard topics of a first course in number theory from integer division with remainder to representation of integers by quadratic forms. Nearly 500 illustrations elucidate proofs, provide data visualization, and give fresh new insights...The page layout is exquisite...Each chapter begins with a figure on the left side and text on the right side of a two-page spread. Chapters end with historical notes and exercises, each exactly filling two facing pages. The historical notes reference original sources, often outside of Western tradition.
-- Samuel S. Wagstaff, Jr., Mathematical Reviews
It is rare that a mathematics book can be described with this word, but Weissman's 'An Illustrated Theory of Numbers' is gorgeous! Weissmann (Univ. of California, Santa Cruz) not only wrote a great textbook on number theory but also did so in a visually stunning way. The work is full of hundreds of beautiful visuals that complement the otherwise difficult subject matter. Any reader with a high school geometry and algebra background will be prepared to read, understand, and enjoy this text...most readers will love this work because they will be able to see numbers for the first time.
-- A. Misseldine, CHOICE
This is a meticulously written and stunningly laid-out book influenced not only by the classical masters of number theory like Fermat, Euler, and Gauss, but also by the work of Edward Tufte on data visualization. Assuming little beyond basic high school mathematics, the author covers a tremendous amount of territory, including topics like Ford circles, Conway's topographs, and Zolotarev's lemma which are rarely seen in introductory courses. All of this is done with a visual and literary flair which very few math books even strive for, let alone accomplish.
-- Matthew Baker, Georgia Institute of Technology
'An Illustrated Theory of Numbers' is a textbook like none other I know; and not just a textbook, but a work of practical art. This book would be a delight to use in the undergraduate classroom, to give to a high school student in search of enlightenment, or to have on your coffee table, to give guests from the world outside mathematics a visceral and visual sense of the beauty of our subject.
-- Jordan Ellenberg, University of Wisconsin-Madison, author of “How Not to Be Wrong: the Power of Mathematical Thinking”
Weissman's book represents a totally fresh approach to a venerable subject. Its choice of topics, superb exposition and beautiful layout will appeal to professional mathematicians as well as to students at all levels.
-- Kenneth A. Ribet, University of California, Berkeley