Michael A. Bean

Probability:
The Science of Uncertainty: with Applications to Investments, Insurance, and Engineering

Pure and Applied Undergraduate Texts, Volume: 6
2001; 448 pp; hardcover
ISBN-13: 978-0-8218-4792-3
This book covers the basic probability of distributions with an emphasis on applications from the areas of investments, insurance, and engineering. Written by a Fellow of the Casualty Actuarial Society and the Society of Actuaries with many years of experience as a university professor and industry practitioner, the book is suitable as a text for senior undergraduate and beginning graduate students in mathematics, statistics, actuarial science, finance, or engineering as well as a reference for practitioners in these fields. The book is particularly well suited for students preparing for professional exams, and for several years it has been recommended as a textbook on the syllabus of examinations for the Casualty Actuarial Society and the Society of Actuaries.
In addition to covering the standard topics and probability distributions, this book includes separate sections on more specialized topics such as mixtures and compound distributions, distributions of transformations, and the application of specialized distributions such as the Pareto, beta, and Weibull. The book also has a number of unique features such as a detailed description of the celebrated Markowitz investment portfolio selection model. A separate section contains information on how graphs of the specific distributions studied in the book can be created using MathematicaTM.
The book includes a large number of problems of varying difficulty. A student manual with solutions to selected problems is available. For more information regarding the student manual, please contact AMS Member and Customer Services at cust-serv@ams.org.
An instructor's manual with complete solutions to all the problems as well as supplementary material is available to teachers using the book as the text for the class. To receive it, send e-mail to textbooks@ams.org.

Textbook: Beginning Topology, by Sue E. Course Description: An introduction to topology, including Euclidean spaces, point-set topology, surfaces. For further information, see Article VI of the Faculty Handbook or the M-Book. Students who have questions concerning the quality of instruction in this class may. ZAlerts allow you to be notified by email about the availability of new books according to your search query. A search query can be a title of the book, a name of the author, ISBN or anything else.

Readership

Undergraduate and graduate students in mathematics, finance, or engineering interested in probability and its applications.

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Table of Contents

Introduction
A survey of some basic concepts through examples
Classical probability
Random variables and probability distributions
Special discrete distributions
Special continuous distributions
Transformations of random variables
Sums and products of random variables
Mixtures and compound distributions
The Markowitz investment portfolio selection model
Appendixes
Answers to selected exercises
Index

Ronald Solomon, Ohio State University, Columbus, OH

Abstract Algebra

Pure and Applied Undergraduate Texts, Volume: 9
2003; 227 pp; hardcover
ISBN-13: 978-0-8218-4795-4
This undergraduate text takes a novel approach to the standard introductory material on groups, rings, and fields. At the heart of the text is a semi-historical journey through the early decades of the subject as it emerged in the revolutionary work of Euler, Lagrange, Gauss, and Galois. Avoiding excessive abstraction whenever possible, the text focuses on the central problem of studying the solutions of polynomial equations. Highlights include a proof of the Fundamental Theorem of Algebra, essentially due to Euler, and a proof of the constructability of the regular 17-gon, in the manner of Gauss. Another novel feature is the introduction of groups through a meditation on the meaning of congruence in the work of Euclid. Everywhere in the text, the goal is to make clear the links connecting abstract algebra to Euclidean geometry, high school algebra, and trigonometry, in the hope that students pursuing a career as secondary mathematics educators will carry away a deeper and richer understanding of the high school mathematics curriculum. Another goal is to encourage students, insofar as possible in a textbook format, to build the course for themselves, with exercises integrally embedded in the text of each chapter.

Readership

Undergraduate students interested in abstract algebra.

Table of Contents

Background
Geometry
Polynomials
Numbers
The grand synthesis
Index

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Sue E. Goodman, University of North Carolina, Chapel Hill, NC

Beginning Topology

Pure and Applied Undergraduate Texts, Volume: 10
2005; 236 pp; hardcover
ISBN-13: 978-0-8218-4796-1
Beginning Topology is designed to give undergraduate students a broad notion of the scope of topology in areas of point-set, geometric, combinatorial, differential, and algebraic topology, including an introduction to knot theory. A primary goal is to expose students to some recent research and to get them actively involved in learning. Exercises and open-ended projects are placed throughout the text, making it adaptable to seminar-style classes.
The book starts with a chapter introducing the basic concepts of point-set topology, with examples chosen to captivate students' imaginations while illustrating the need for rigor. Most of the material in this and the next two chapters is essential for the remainder of the book. One can then choose from chapters on map coloring, vector fields on surfaces, the fundamental group, and knot theory.
A solid foundation in calculus is necessary, with some differential equations and basic group theory helpful in a couple of chapters. Topics are chosen to appeal to a wide variety of students: primarily upper-level math majors, but also a few freshmen and sophomores as well as graduate students from physics, economics, and computer science. All students will benefit from seeing the interaction of topology with other fields of mathematics and science; some will be motivated to continue with a more in-depth, rigorous study of topology.

Readership

Undergraduate students interested in topology.

Table of Contents

Introduction to point set topology
Surfaces
The Euler characteristic
Maps and graphs
Vector fields on surfaces
The fundamental group
Introduction to knots
Bibliography and reading list
Index

Victor Goodman and Joseph Stampfli, Indiana University, Bloomington, IN

The Mathematics of Finance: Modeling and Hedging

Pure and Applied Undergraduate Texts, Volume: 7
2001; 250 pp; hardcover
ISBN-13: 978-0-8218-4793-0
This book is ideally suited for an introductory undergraduate course on financial engineering. It explains the basic concepts of financial derivatives, including put and call options, as well as more complex derivatives such as barrier options and options on futures contracts. Both discrete and continuous models of market behavior are developed in this book. In particular, the analysis of option prices developed by Black and Scholes is explained in a self-contained way, using both the probabilistic Brownian Motion method and the analytical differential equations method.
The book begins with binomial stock price models, moves on to multistage models, then to the Cox-Ross-Rubinstein option pricing process, and then to the Black-Scholes formula. Other topics presented include Zero Coupon Bonds, forward rates, the yield curve, and several bond price models. The book continues with foreign exchange models and the Keynes Interest Rate Parity Formula, and concludes with the study of country risk, a topic not inappropriate for the times.
In addition to theoretical results, numerical models are presented in much detail. Each of the eleven chapters includes a variety of exercises.
An instructor's manual with complete solutions to all the problems as well as supplementary material is available to teachers using the book as the text for the class. To receive it, send e-mail to textbooks@ams.org.

Readership

Undergraduate students interested in financial engineering.

Table of Contents

I. Martin Isaacs, University of Wisconsin, Madison, WI

Geometry for College Students

Pure and Applied Undergraduate Texts, Volume: 8
2001; 222 pp; hardcover
ISBN-13: 978-0-8218-4794-7
One of the challenges many mathematics students face occurs after they complete their study of basic calculus and linear algebra, and they start taking courses where they are expected to write proofs. Historically, students have been learning to think mathematically and to write proofs by studying Euclidean geometry. In the author's opinion, geometry is still the best way to make the transition from elementary to advanced mathematics.
The book begins with a thorough review of high school geometry, then goes on to discuss special points associated with triangles, circles and certain associated lines, Ceva's theorem, vector techniques of proof, and compass-and-straightedge constructions. There is also some emphasis on proving numerical formulas like the laws of sines, cosines, and tangents, Stewart's theorem, Ptolemy's theorem, and the area formula of Heron.
An important difference of this book from the majority of modern college geometry texts is that it avoids axiomatics. The students using this book have had very little experience with formal mathematics. Instead, the focus of the course and the book is on interesting theorems and on the techniques that can be used to prove them. This makes the book suitable to second- or third-year mathematics majors and also to secondary mathematics education majors, allowing the students to learn how to write proofs of mathematical results and, at the end, showing them what mathematics is really all about.

Readership

Undergraduate students interested in Euclidean geometry.

Table of Contents