Series

Dolciani Mathematical Expositions (No. 31)

Print Publication Year: 2007
Print ISBN: 9780883853375

Online Publication Date: April 2012
Online ISBN: 9781614442097

Book DOI: http://dx.doi.org/10.7135/UPO9781614442097

The derivative and the integral are the fundamental notions of calculus. Though there is essentially only one derivative, there are a variety of integrals, developed over the years for a variety of purposes, and this book describes them. No other single source treats all of the integrals of Cauchy, Riemann, Riemann-Stieltjes, Lebesgue, Lebesgue-Steiltjes, Henstock-Kurzweil, Weiner, and Feynman. The basic properties of each are proved, their similarities and differences are pointed out, and the reason for their existence and their uses are given. Historical information is plentiful. Advanced undergraduate mathematics majors, graduate students, and faculty members are the audience for the book. Even experienced faculty members are unlikely to be aware of all of the integrals in the Garden of Integrals and the book provides an opportunity to see them and appreciate the richness of the idea of integral. Professor Burke's clear and well-motivated exposition makes this book a joy to read.

Dolciani Mathematical Expositions (No. 32)

Print Publication Year: 2008
Print ISBN: 9780883853382

Online Publication Date: January 2012
Online ISBN: 9780883859148

Book DOI: http://dx.doi.org/10.5948/UPO9780883859148

A Guide to Complex Variables gives the reader a quick and accessible introduction to the key topics. While the coverage is not comprehensive, it certainly gives the reader a solid grounding in this fundamental area. There are many figures and examples to illustrate the principal ideas, and the exposition is lively and inviting. An undergraduate wanting to have a first look at this subject or a graduate student preparing for the qualifying exams, will find this book to be a useful resource. In addition to important ideas from the Cauchy theory, the book also includes the Riemann mapping theorem, harmonic functions, the argument principle, general conformal mapping and dozens of other central topics. Readers will find this book to be a useful companion to more exhaustive texts in the field. It is a valuable resource for mathematicians and non-mathematicians alike. Steven Krantz is well-known for his skill in expository writing and this volume confirms it. He is the author of more than 50 books, and more than 150 scholarly papers. The MAA has awarded him both the Beckenbach Book Prize and the Chauvenet Prize.

Dolciani Mathematical Expositions (No. 33)

Print Publication Year: 2008
Print ISBN: 9780883853399

Online Publication Date: January 2012
Online ISBN: 9781614442073

Book DOI: http://dx.doi.org/10.5948/UPO9781614442073

Sink or Float: Thought Problems in Math and Physics is a collection of problems drawn from mathematics and the real world. Its multiple-choice format forces the reader to become actively involved in deciding upon the answer. The book's aim is to show just how much can be learned by using everyday common sense. The problems are all concrete and understandable by nearly anyone, meaning that not only will students become caught up in some of the questions, but professional mathematicians, too, will easily get hooked. The more than 250 questions cover a wide swath of classical math and physics. Each problem's solution, with explanation, appears in the answer section at the end of the book. A notable feature is the generous sprinkling of boxes appearing throughout the text. These contain historical asides or little-known facts. The problems themselves can easily turn into serious debate-starters, and the book will find a natural home in the classroom.

Dolciani Mathematical Expositions (No. 36)

Print Publication Year: 2009
Print ISBN: 9780883853429

Online Publication Date: January 2012
Online ISBN: 9781614442028

Book DOI: http://dx.doi.org/10.5948/UPO9781614442028

Inequalities permeate mathematics, from the Elements of Euclid to operations research and financial mathematics. Yet too often, especially in secondary and collegiate mathematics, the emphasis is on things equal to one another rather than unequal. While equalities and identities are without doubt important, they don’t possess the richness and variety that one finds with inequalities. The objective of this book is to illustrate how the use of visualization can be a powerful tool for better understanding some basic mathematical inequalities. Drawing pictures is a well-known method for problem solving, and the authors will convince you that the same is true when working with inequalities. They show how to produce figures in a systematic way for the illustration of inequalities and open new avenues to creative ways of thinking and teaching. In addition, a geometric argument cannot only show two things unequal, but also help the observer see just how unequal they are. The concentration on geometric inequalities is partially motivated by the hope that secondary and collegiate teachers might use these pictures with their students. Teachers may wish to use one of the drawings when an inequality arises in the course. Alternatively, When Less Is More might serve as a guide for devoting some time to inequalities and problem solving techniques, or even as part of a course on inequalities.

Dolciani Mathematical Expositions (No. 37)

Print Publication Year: 2009
Print ISBN: 9780883853436

Online Publication Date: January 2012
Online ISBN: 9780883859155

Book DOI: http://dx.doi.org/10.5948/UPO9780883859155

A Guide to Advanced Real Analysis is an outline of the core material in the standard graduate-level real analysis course. It is intended as a resource for students in such a course as well as others who wish to learn or review the subject. On the abstract level, it covers the theory of measure and integration and the basics of point set topology, functional analysis, and the most important types of function spaces. On the more concrete level, it also deals with the applications of these general theories to analysis on Euclidean space: the Lebesgue integral, Hausdorff measure, convolutions, Fourier series and transforms, and distributions. The relevant definitions and major theorems are stated in detail. Proofs, however, are generally presented only as sketches, in such a way that the key ideas are explained but the technical details are omitted. In this way a large amount of material is presented in a concise and readable form.

Dolciani Mathematical Expositions (No. 38)

Print Publication Year: 2009
Print ISBN: 9780883853443

Online Publication Date: January 2012
Online ISBN: 9780883859162

Book DOI: http://dx.doi.org/10.5948/UPO9780883859162

A Guide to Real Variables provides aid and conceptual support for the student studying for the qualifying exam in real variables. Beginning with the foundations of the subject, the text moves rapidly but thoroughly through basic topics like completeness, convergence, sequences, series, compactness, topology and the like. All the basic examples like the Cantor set, the Weierstrass nowhere differentiable function, the Weierstrass approximation theory, the Baire category theorem, and the Ascoli-Arzela theorem are treated. The book contains over 100 examples, and most of the basic proofs. It illustrates both the theory and the practice of this sophisticated subject. Graduate students studying for the qualifying exams will find this book to be a concise, focused and informative resource. Professional mathematicians who need a quick review of the subject, or need a place to look up a key fact, will find this book to be a useful resource too. Steven Krantz is well-known for his skill in expository writing and this volume confirms it. He is the author of more than 50 books, and more than 150 scholarly papers. The MAA has awarded him both the Beckenbach Book Prize and the Chauvenet Prize.

Dolciani Mathematical Expositions (No. 40)

Print Publication Year: 2009
Print ISBN: 9780883853467

Online Publication Date: January 2012
Online ISBN: 9780883859179

Book DOI: http://dx.doi.org/10.5948/UPO9780883859179

A Guide to Topology is an introduction to basic topology. It covers point-set topology as well as Moore-Smith convergence and function spaces. It treats continuity, compactness, the separation axioms, connectedness, completeness, the relative topology, the quotient topology, the product topology, and all the other fundamental ideas of the subject. The book is filled with examples and illustrations. Graduate students studying for the qualifying exams will find this book to be a concise, focused and informative resource. Professional mathematicians who need a quick review of the subject, or need a place to look up a key fact, will find this book to be a useful research too. Steven Krantz is well-known for his skill in expository writing and this volume confirms it. He is the author of more than 50 books, and more than 150 scholarly papers. The MAA has awarded him both the Beckenbach Book Prize and the Chauvenet Prize.

Dolciani Mathematical Expositions (No. 41)

Print Publication Year: 2009
Print ISBN: 9780883853474

Online Publication Date: January 2012
Online ISBN: 9780883859186

Book DOI: http://dx.doi.org/10.5948/UPO9780883859186

CHOICE Award winner! A Guide to Elementary Number Theory is a 140-page exposition of the topics considered in a first course in number theory. It is intended for those who may have seen the material before but have half-forgotten it, and also for those who may have misspent their youth by not having a course in number theory and who want to see what it is about without having to wade through a traditional text, some of which approach 500 pages in length. It will be especially useful to graduate student preparing for the qualifying exams. Underwood Dudley received the Ph.D. degree (number theory) from the University of Michigan in 1965. He taught at the Ohio State University and at DePauw University, from which he retired in 2004. He is the author of three books on mathematical oddities, The Trisectors, Mathematical Cranks, and Numerology all published by the Mathematical Association of America. He has also served as editor of the College Mathematics Journal, the Pi Mu Epsilon Journal, and two of the Mathematical Association of America's book series.

Dolciani Mathematical Expositions (No. 42)

Print Publication Year: 2010
Print ISBN: 9780883853481

Online Publication Date: October 2011
Online ISBN: 9781614442011

Book DOI: http://dx.doi.org/10.5948/UPO9781614442011

Theorems and their proofs lie at the heart of mathematics. In speaking of the purely aesthetic qualities of theorems and proofs, G. H. Hardy wrote that in beautiful proofs 'there is a very high degree of unexpectedness, combined with inevitability and economy.' Charming Proofs present a collection of remarkable proofs in elementary mathematics that are exceptionally elegant, full of ingenuity, and succinct. By means of a surprising argument or a powerful visual representation, the proofs in this collection will invite readers to enjoy the beauty of mathematics, to share their discoveries with others, and to become involved in the process of creating new proofs. Charming Proofs is organized as follows. Following a short introduction about proofs and the process of creating proofs, the authors present, in twelve chapters, a wide and varied selection of proofs they consider charming, Topics include the integers, selected real numbers, points in the plane, triangles, squares, and other polygons, curves, inequalities, plane tilings, origami, colorful proofs, three-dimensional geometry, etc. At the end of each chapter are some challenges that will draw the reader into the process of creating charming proofs. There are over 130 such challenges. Charming Proofs concludes with solutions to all of the challenges, references, and a complete index. As in the authors’ previous books with the MAA (Math Made Visual and When Less Is More), secondary school and college and university teachers may wish to use some of the charming proofs in their classrooms to introduce their students to mathematical elegance. Some may wish to use the book as a supplement in an introductory course on proofs, mathematical reasoning, or problem solving.

Dolciani Mathematical Expositions (No. 43)

Print Publication Year: 2010
Print ISBN: 9780883853498

Online Publication Date: October 2011
Online ISBN: 9781614442004

Book DOI: http://dx.doi.org/10.5948/UPO9781614442004

This book is an eclectic compendium of the essays solicited for the 2010 Mathematics Awareness Month web page on the theme of Mathematics and Sports. In keeping with the goal of promoting mathematics awareness to a broad audience, all of the articles are accessible to college level mathematics students and many are accessible to the general public. The book is divided into sections by the kind of sports. The section on football includes an article that evaluates a method for reducing the advantage of the winner of a coin flip in an NFL overtime game; the section on track and field examines the ultimate limit on how fast a human can run 100 meters; the section on baseball includes an article on the likelihood of streaks; the section on golf has an article that describes the double-pendulum model of a golf swing, and an article on modeling Tiger Wood's career. The articles provide source material for classroom use and student projects. Many students will find mathematical ideas motivated by examples taken from sports more interesting than the examples selected from traditional sources.

Dolciani Mathematical Expositions (No. 44)

Print Publication Year: 2011
Print ISBN: 9780883853511

Online Publication Date: January 2012
Online ISBN: 9780883859674

Book DOI: http://dx.doi.org/10.5948/UPO9780883859674

Linear algebra occupies a central place in modern mathematics. This book provides a rigorous and thorough development of linear algebra at an advanced level, and is directed at graduate students and professional mathematicians. It approaches linear algebra from an algebraic point of view, but its selection of topics is governed not only for their importance in linear algebra itself, but also for their applications throughout mathematics. Students in algebra, analysis, and topology will find much of interest and use to them, and the careful treatment and breadth of subject matter will make this book a valuable reference for mathematicians throughout their professional lives. Topics treated in this book include: vector spaces and linear transformations; dimension counting and applications; representation of linear transformations by matrices; duality; determinants and their uses; rational and especially Jordan canonical form; bilinear forms; inner product spaces; normal linear transformations and the spectral theorem; and an introduction to matrix groups as Lie groups. The book treats vector spaces in full generality, though it concentrates on the finite dimensional case. Also, it treats vector spaces over arbitrary fields, specializing to algebraically closed fields or to the fields of real and complex numbers as necessary.

Dolciani Mathematical Expositions (No. 45)

Print Publication Year: 2011
Print ISBN: 9780883853528

Online Publication Date: October 2011
Online ISBN: 9780883859865

Book DOI: http://dx.doi.org/10.5948/UPO9780883859865

Icons of mathematics are certain geometric diagrams that play a crucial role in visualizing mathematical proofs, and in the book the authors present 20 of them and explore the mathematics that lies within and that can be created. The authors devote a chapter to each icon, illustrating its presence in real life, its primary mathematical characteristics and how it plays a central role in visual proofs of a wide range of mathematical facts. Among these are classical results from plane geometry, properties of the integers, means and inequalities, trigonometric identities, theorems from calculus, and puzzles from recreational mathematics.

Dolciani Mathematical Expositions (No. 46)

Print Publication Year: 2011
Print ISBN: 9780883853535

Online Publication Date: October 2011
Online ISBN: 9781614442035

Book DOI: http://dx.doi.org/10.5948/UPO9781614442035

This book can be used in a one semester undergraduate course or senior capstone course, or as a useful companion in studying algebraic geometry at the graduate level. This Guide is a friendly introduction to plane algebraic curves. It emphasizes geometry and intuition, and the presentation is kept concrete. You'll find an abundance of pictures and examples to help develop your intuition about the subject, which is so basic to understanding and asking fruitful questions. Highlights of the elementary theory are covered, which for some could be an end in itself, and for others an invitation to investigate further. Proofs, when given, are mostly sketched, some in more detail, but typically with less. References to texts that provide further discussion are often included. Computer algebra software has made getting around in algebraic geometry much easier. Algebraic curves and geometry are now being applied to areas such as cryptography, complexity and coding theory, robotics, biological networks, and coupled dynamical systems. Algebraic curves were used in Andrew Wiles' proof of Fermat's Last Theorem, and to understand string theory, you need to know some algebraic geometry. There are other areas on the horizon for which the concepts and tools of algebraic curves and geometry hold tantalizing promise. This introduction to algebraic curves will be appropriate for a wide segment of scientists and engineers wanting an entrance to this burgeoning subject.

Dolciani Mathematical Expositions (No. 48)

Print Publication Year: 2012
Print ISBN: 9780883853559

Online Publication Date: April 2013
Online ISBN: 9781614442110

Book DOI: http://dx.doi.org/10.5948/UPO9781614442110

This Guide offers a concise overview of the theory of groups, rings, and fields at the graduate level, emphasizing those aspects that are useful in other parts of mathematics. It focuses on the main ideas and how they hang together. It will be useful to both students and professionals. In addition to the standard material on groups, rings, modules, fields, and Galois theory, the book includes discussions of other important topics that are often omitted in the standard graduate course, including linear groups, group representations, the structure of Artinian rings, projective, injective and flat modules, Dedekind domains, and central simple algebras. All of the important theorems are discussed, without proofs but often with a discussion of the intuitive ideas behind those proofs. Those looking for a way to review and refresh their basic algebra will benefit from reading this Guide, and it will also serve as a ready reference for mathematicians who make use of algebra in their work.

Dolciani Mathematical Expositions (No. 49)

Print Publication Year: 2013
Print ISBN: 9780883853573

Online Publication Date: May 2013
Online ISBN: 9781614442134

Book DOI: http://dx.doi.org/10.5948/9781614442134

The purpose of A Guide to Functional Analysis is to introduce the reader with minimal background to the basic scripture of functional analysis. Readers should know some real analysis and some linear algebra. Measure theory rears its ugly head in some of the examples and also in the treatment of spectral theory. The latter is unavoidable and the former allows us to present a rich variety of examples. The nervous reader may safely skip any of the measure theory and still derive a lot from the rest of the book. Apart from this caveat, the book is almost completely self-contained; in a few instances we mention easily accessible references. A feature that sets this book apart from most other functional analysis texts is that it has a lot of examples and a lot of applications. This helps to make the material more concrete, and relates it to ideas that the reader has already seen. It also makes the book more accessible to a broader audience.