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Discovering Discrete Dynamical Systems is a mathematics textbook designed for use in a student-led, inquiry-based course for advanced mathematics majors. Fourteen modules each with an opening exploration, a short exposition and related exercises, and a concluding project guide students to self-discovery on topics such as fixed points and their classifications, chaos and fractals, Julia and Mandelbrot sets in the complex plane, and symbolic dynamics. Topics have been carefully chosen as a means for developing student persistence and skill in exploration, conjecture, and generalization while at the same time providing a coherent introduction to the fundamentals of discrete dynamical systems. This book is written for undergraduate students with the prerequisites for a first analysis course, and it can easily be used by any faculty member in a mathematics department, regardless of area of expertise. Each module starts with an exploration in which the students are asked an open-ended question. This allows the students to make discoveries which lead them to formulate the questions that will be addressed in the exposition and exercises of the module. The exposition is brief and has been written with the intent that a student who has taken, or is ready to take, a course in analysis can read the material independently. The exposition concludes with exercises which have been designed to both illustrate and explore in more depth the ideas covered in the exposition. Each module concludes with a project in which students bring the ideas from the module to bear on a more challenging or in-depth problem. A section entitled "To the Instructor" includes suggestions on how to structure a course in order to realize the inquiry-based intent of the book. The book has also been used successfully as the basis for an independent study course and as a supplementary text for an analysis course with traditional content.
The American Journal of Mathematics is the premier journal in its field, featuring research articles and notes of the highest quality. Topics covered include algebraic geometry, topology, and differential equations, among many others. Whether you are an established mathematician or an up-and-coming scholar, the American Journal of Mathematics is an essential resource for staying current with the latest developments in the field. This work has been selected by scholars as being culturally important, and is part of the knowledge base of civilization as we know it. This work is in the "public domain in the United States of America, and possibly other nations. Within the United States, you may freely copy and distribute this work, as no entity (individual or corporate) has a copyright on the body of the work. Scholars believe, and we concur, that this work is important enough to be preserved, reproduced, and made generally available to the public. We appreciate your support of the preservation process, and thank you for being an important part of keeping this knowledge alive and relevant.
Mathematics for Social Justice offers a collection of resources for mathematics faculty interested in incorporating questions of social justice into their classrooms. The book begins with a series of essays from instructors experienced in integrating social justice themes into their pedagogy; these essays contain political and pedagogical motivations as well as nuts-and-bolts teaching advice. The heart of the book is a collection of fourteen classroom-tested modules featuring ready-to-use activities and investigations for the college mathematics classroom. The mathematical tools and techniques used are relevant to a wide variety of courses including college algebra, math for the liberal arts, calculus, differential equations, discrete mathematics, geometry, financial mathematics, and combinatorics. The social justice themes include human trafficking, income inequality, environmental justice, gerrymandering, voting methods, and access to education. The volume editors are leaders of the national movement to include social justice material into mathematics teaching. Gizem Karaali is Associate Professor of Mathematics at Pomona College. She is one of the founding editors of The Journal of Humanistic Mathematics, and an associate editor for The Mathematical Intelligencer and Numeracy ; she also serves on the editorial board of the MAA's Carus Mathematical Monographs. Lily Khadjavi is Associate Professor of Mathematics at Loyola Marymount University and is a past co-chair of the Infinite Possibilities Conference. She has served on the boards of Building Diversity in Science, the Barbara Jordan-Bayard Rustin Coalition, and the Harvard Gender and Sexuality Caucus.
Cover -- Title page -- Contents -- Preface -- Acknowledgments -- Photograph and Figure Credits -- Chapter 1. An overview of American mathematics: 1776-1876 -- Chapter 2. A new departmental prototype: J.J. Sylvester and the Johns Hopkins University -- Chapter 3. Mathematics at Sylvester's Hopkins -- Chapter 4. German mathematics and the early mathematical career of Felix Klein -- Chapter 5. America's wanderlust generation -- Chapter 6. Changes on the horizon -- Chapter 7. The World's Columbian exposition of 1893 and the Chicago mathematical congress -- Chapter 8. Surveying mathematical landscapes: The Evanston colloquium lectures -- Chapter 9. Meeting the challenge: The University of Chicago and the American mathematical research community -- Chapter 10. Epilogue: Beyond the threshold: The American mathematical research community, 1900-1933 -- Bibliography -- Subject Index -- Back Cover
At the turn of the twentieth century, mathematical scholarship in the United States underwent a stunning transformation. In 1890 no American professor was producing mathematical research worthy of international attention. Graduate students were then advised to pursue their studies abroad. By the start of World War I the standing of American mathematics had radically changed. George David Birkhoff, Leonard Dickson, and others were turning out cutting edge investigations that attracted notice in the intellectual centers of Europe. Harvard, Chicago, and Princeton maintained graduate programs comparable to those overseas. This book explores the people, timing, and factors behind this rapid advance. Through the mid-nineteenth century most American colleges followed a classical curriculum that, in mathematics, rarely reached beyond calculus. With no doctoral programs of any sort in the United States until 1860, mathematical scholarship lagged far behind that in Europe. After the Civil War, visionary presidents at Harvard and Johns Hopkins broadened and deepened the opportunities for study. The breakthrough for mathematics began in 1890 with the hiring, in consecutive years, of William F. Osgood and Maxime Bôcher at Harvard and E. H. Moore at Chicago. Each of these young men had studied in Germany where they acquired vital mathematical knowledge and taste. Over the next few years Osgood, Bôcher, and Moore established their own research programs and introduced new graduate courses. Working with other like-minded individuals through the nascent American Mathematical Society, the infrastructure of meetings and journals were created. In the early twentieth century Princeton dramatically upgraded its faculty to give the United States the stability of a third mathematics center. The publication by Birkhoff, in 1913, of the solution to a famous conjecture served notice that American mathematics had earned consideration with the European powers of Germany, France, Italy, England, and Russia.
