A Bridge to Abstract Mathematics will prepare the mathematical novice to explore the universe of abstract mathematics. Mathematics is a science that concerns theorems that must be proved within the constraints of a logical system of axioms and definitions rather than theories that must be tested, revised, and retested. Readers will learn how to read mathematics beyond popular computational calculus courses. Moreover, readers will learn how to construct their own proofs. The book is intended as the primary text for an introductory course in proving theorems, as well as for self-study or as a reference. Throughout the text, some pieces (usually proofs) are left as exercises. Part V gives hints to help students find good approaches to the exercises. Part I introduces the language of mathematics and the methods of proof. The mathematical content of Parts II through IV were chosen so as not to seriously overlap the standard mathematics major. In Part II, students study sets, functions, equivalence and order relations, and cardinality. Part III concerns algebra. The goal is to prove that the real numbers form the unique, up to isomorphism, ordered field with the least upper bound. In the process, we construct the real numbers starting with the natural numbers. Students will be prepared for an abstract linear algebra or modern algebra course. Part IV studies analysis. Continuity and differentiation are considered in the context of time scales (nonempty, closed subsets of the real numbers). Students will be prepared for advanced calculus and general topology courses. There is a lot of room for instructors to skip and choose topics from among those that are presented.
This volume brings to a close our four volumes on the chronicled life of Augusta during a very turbulent and pivotal period in the History of the United States: the opening-up of the West, the question of whether slavery would prevail nationally with political attempts to legitimize it in the new territories, starting with Kansas; a serious depression brought on by over expansion of our then growth industry, the railroads; the explosive discoveries of gold in most of the Western Territories; one of the worst wars in our history to settle once and for all whether we were to be "one nation indivisible" with slavery or not. Augusta's original three bound journals, which I inherited, with some 2,000 entries, beginning before she was seventeen, records not only her personal and occasionally tragic involvement in all of these events, but the influence these events had on her life at the time. Her journal entries from 1857 to 1860 present a record of the founding (by her father and a few other abolitionists) the town of Eldorado, Kansas that is better and more authentic than any professional early history of the city we've seen. She described in detail these two or three-dozen mostly young pioneers that were willing to go far beyond the Frontier to establish a voting district free of proslavery domination.
Calculus for the Life Sciences is an entire reimagining of the standard calculus sequence with the needs of life science students as the fundamental organizing principle. Those needs, according to the National Academy of Science, include: the mathematical concepts of change, modeling, equilibria and stability, structure of a system, interactions among components, data and measurement, visualization, and algorithms. This book addresses, in a deep and significant way, every concept on that list. The book begins with a primer on modeling in the biological realm and biological modeling is the theme and frame for the entire book. The authors build models of bacterial growth, light penetration through a column of water, and dynamics of a colony of mold in the first few pages. In each case there is actual data that needs fitting. In the case of the mold colony that data is a set of photographs of the colony growing on a ruled sheet of graph paper and the students need to make their own approximations. Fundamental questions about the nature of mathematical modeling—trying to approximate a real-world phenomenon with an equation—are all laid out for the students to wrestle with. The authors have produced a beautifully written introduction to the uses of mathematics in the life sciences. The exposition is crystalline, the problems are overwhelmingly from biology and interesting and rich, and the emphasis on modeling is pervasive. An instructor's 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. Online question content and interactive step-by-step tutorials are available for this title in WebAssign. WebAssign is a leading provider of online instructional tools for both faculty and students.
V. 1. 1813-1835 -- v. 2. 1836-1841 -- v. 3. 1842-1847 -- v. 4. 1848-1855 -- v. 5. 1856-1867 -- v. 6. 1868-1881 -- v. 7. 1807-1844 -- v. 8. 1845-1859. -- v. 9. 1860-1869. -- v. 10. 1870-1881, and an index of proper names for volumes seven to ten.
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