A Concrete Approach to Abstract Algebra begins with a concrete and thorough examination of familiar objects like integers, rational numbers, real numbers, complex numbers, complex conjugation, and polynomials. The text will be of particular interest to teachers and future teachers as it links abstract algebra to many topics that arise in courses in algebra, geometry, trigonometry, precalculus, and calculus. The final four chapters present the more theoretical material needed for graduate study.

Using methods of Synthetic Differential Geometry, the book utilizes the prolongation spaces (neighborhoods of the diagonal) of manifolds to reformulate several of the algebraic notions of manifold theory into more geometric terms.

Mathematics for Secondary School Teachers discusses topics of central importance in the secondary school mathematics curriculum, including functions, polynomials, trigonometry, exponential and logarithmic functions, number and operation, and measurement. It provides a balance of discovery learning and direct instruction. Activities and exercises address the range of learning objectives appropriate for future teachers. Coauthored with Elizabeth G. Bremigan and John Lorch.

The goal of Time Frequency and Time-Scale Methods is to develop a deeper understanding of the roles of time-frequency or Fourier and Gabor analysis and time-scale or wavelet analysis, when the various tools are properly assembled in a larger context. While researchers at the forefront of developments in time-frequency scale (TFS) analysis are well aware of the benefits of such a unified approach, there remains a gap in the larger community of practitioners concerning precisely the strengths and limitations of Gabor analysis versus wavelets. The book fills this gap by presenting the interface between time-frequency and time-scale methods as a rich area of work.

How the Other Half Thinks: Adventures in Mathematical Reasoning gives the layperson a chance to see what it is to think mathematically. Each of its eight chapters presents an excerpt of advanced mathematics that happens not to use anything beyond sixth-grade arithmetic: no algebra, no trigonometry, no calculus. Each chapter begins by offering the reader a chance to experiment, to get a feel for the problem, and to make a conjecture. Each chapter concludes with a leisurely analysis that settles the problem.

A thorough and mathematically rigorous exposition of single-variable calculus for readers with some previous exposure to calculus techniques but not to methods of proof, Calculus Deconstructed: A Second Course in First Year Calculus is appropriate for a beginning honors calculus course assuming high school calculus or a "bridge course" using basic analysis to motivate and illustrate mathematical rigor. It can serve as a combination textbook and reference book for individual self-study. Standard topics and techniques in single-variable calculus are presented in the context of a coherent logical structure, building on familiar properties of real numbers and teaching methods of proof by example along the way. Numerous examples reinforce both practical and theoretical understanding, and extensive historical notes explore the arguments of the originators of the subject.

No previous experience with mathematical proof is assumed: rhetorical strategies and techniques of proof (reductio ad absurdum, induction, contrapositives, etc.) are introduced by example along the way. Between the text and the exercises, proofs are available for all the basic results of calculus for functions of one real variable.

Need to learn statistics as part of your job, or want some help passing a statistics course? Statistics in a Nutshell is a clear and concise introduction and reference that's perfect for anyone with no previous background in the subject. This book gives you a solid understanding of statistics without being too simple, yet without the numbing complexity of most college texts.

You get a firm grasp of the fundamentals and a hands-on understanding of how to apply them before moving on to the more advanced material that follows. Each chapter presents you with easy-to-follow descriptions illustrated by graphics, formulas, and plenty of solved examples. Before you know it, you'll learn to apply statistical reasoning and statistical techniques, from basic concepts of probability and hypothesis testing to multivariate analysis.

In fictional conversations with Pierre Fermat, the underpinnings and implications of Fermat's Last Theorem are examined using only the mathematical skills and methodology that would be possessed by the accomplished high-school graduate. Although a proof of that theorem is beyond the scope of the book, the objective is to provide sufficient insight so that the reader can appreciate the plausibility of Fermat's Last Theorem.

Mathematical craftwork has become extremely popular, and mathematicians and crafters alike are fascinated by the relationship between their crafts. The focus of this book, written for mathematicians, needleworkers, and teachers of mathematics, is on the relationship between mathematics and the fiber arts (including knitting, crocheting, cross-stitch, and quilting). Each chapter starts with an overview of the mathematics and the needlework at a level understandable to both mathematicians and needleworkers, followed by more technical sections discussing the mathematics, how to introduce the mathematics in the classroom through needlework, and how to make the needlework project, including patterns and instructions.

About this textbook: comprehensive work that covers the vast majority of the material needed for a beginning graduate-level course on complex analysis; wonderfully elegant and economical treatment of complex analysis; provides a real variety of alternative ways of understanding the concept of analyticity.

This book is intended for a graduate course on complex analysis, also known as function theory. The main focus is the theory of complex-valued functions of a single complex variable. This theory is a prerequisite for the study of many current and rapidly developing areas of mathematics including the theory of several and infinitely many complex variables, the theory of groups, hyperbolic geometry and three-manifolds, and number theory. Complex analysis has connections and applications to many other subjects in mathematics and to other sciences. It is an area where the classic and the modern techniques meet and benefit from each other. This material should be part of the education of every practicing mathematician, and it will also be of interest to computer scientists, physicists, and engineers.

The first part of the book is a study of the many equivalent ways of understanding the concept of analyticity. The many ways of formulating the concept of an analytic function are summarized in what is termed the Fundamental Theorem for functions of a complex variable. The organization of these conditions into a single unifying theorem with an emphasis on clarity and elegance is a hallmark of Lipman Bers's mathematical style. Here it provides a conceptual framework for results that are highly technical and often computational. The framework comes from an insight that, once articulated, will drive the subsequent mathematics and lead to new results.

In the second part, the text proceeds to a leisurely exploration of interesting ramifications of the main concepts.

The book covers most, if not all, of the material contained in Bers's courses on first year complex analysis. In addition, topics of current interest such as zeros of holomorphic functions and the connection between hyperbolic geometry and complex analysis are explored.