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p-adic Numbers
p-adic numbers are of great theoretical importance in number theory, since they allow the use of the language of analysis to study problems relating toprime numbers and diophantine equations. Further, they offer a realm where one can do things that are very similar to classical analysis, but with results that are quite unusual. The book should be of use to students interested in number theory, but at the same time offers an interesting example of the many connections between different parts of mathematics. The book strives to be understandable to an undergraduate audience. Very little background has been assumed, and the presentation is leisurely. There are many problems, which should help readers who are working on their own (a large appendix with hints on the problem is included). Most of all, the book should offer undergraduates exposure to some interesting mathematics which is off the beaten track. Those who will later specialize in number theory, algebraic geometry, and related subjects will benefit more directly, but all mathematics students can enjoy the book.
Math through the Ages: A Gentle History for Teachers and Others Expanded Second Edition
Where did math come from? Who thought up all those algebra symbols, and why? What is the story behind π π? … negative numbers? … the metric system? … quadratic equations? … sine and cosine? … logs? The 30 independent historical sketches in Math through the Ages answer these questions and many others in an informal, easygoing style that is accessible to teachers, students, and anyone who is curious about the history of mathematical ideas. Each sketch includes Questions and Projects to help you learn more about its topic and to see how the main ideas fit into the bigger picture of history. The 30 short stories are preceded by a 58-page bird's-eye overview of the entire panorama of ...
p-adic Numbers
There are numbers of all kinds: rational, real, complex, p-adic. The p-adic numbers are less well known than the others, but they play a fundamental role in number theory and in other parts of mathematics. This elementary introduction offers a broad understanding of p-adic numbers. From the reviews: "It is perhaps the most suitable text for beginners, and I shall definitely recommend it to anyone who asks me what a p-adic number is." --THE MATHEMATICAL GAZETTE
A Guide to Groups, Rings, and Fields
Insightful overview of many kinds of algebraic structures that are ubiquitous in mathematics. For researchers at graduate level and beyond.
A Short Book on Long Sums
This concise textbook introduces calculus students to power series through an informal and captivating narrative that avoids formal proofs but emphasizes understanding the fundamental ideas. Power series—and infinite series in general—are a fundamental tool of pure and applied mathematics. The problems focus on ideas, applications, and creative thinking instead of being repetitive and procedural. Calculus is about functions, so the book turns on two fundamental ideas: using polynomials to approximate a function and representing a function in terms of simpler functions. The derivative is reinterpreted in terms of linear approximations, which then leads to Taylor polynomials and the questi...
Arithmetic of p-adic Modular Forms
The central topic of this research monograph is the relation between p-adic modular forms and p-adic Galois representations, and in particular the theory of deformations of Galois representations recently introduced by Mazur. The classical theory of modular forms is assumed known to the reader, but the p-adic theory is reviewed in detail, with ample intuitive and heuristic discussion, so that the book will serve as a convenient point of entry to research in that area. The results on the U operator and on Galois representations are new, and will be of interest even to the experts. A list of further problems in the field is included to guide the beginner in his research. The book will thus be of interest to number theorists who wish to learn about p-adic modular forms, leading them rapidly to interesting research, and also to the specialists in the subject.
Catalan's Conjecture
Eugène Charles Catalan made his famous conjecture – that 8 and 9 are the only two consecutive perfect powers of natural numbers – in 1844 in a letter to the editor of Crelle’s mathematical journal. One hundred and fifty-eight years later, Preda Mihailescu proved it. Catalan’s Conjecture presents this spectacular result in a way that is accessible to the advanced undergraduate. The author dissects both Mihailescu’s proof and the earlier work it made use of, taking great care to select streamlined and transparent versions of the arguments and to keep the text self-contained. Only in the proof of Thaine’s theorem is a little class field theory used; it is hoped that this application will motivate the interested reader to study the theory further. Beautifully clear and concise, this book will appeal not only to specialists in number theory but to anyone interested in seeing the application of the ideas of algebraic number theory to a famous mathematical problem.
Statisticians of the Centuries
Written by leading statisticians and probabilists, this volume consists of 104 biographical articles on eminent contributors to statistical and probabilistic ideas born prior to the 20th Century. Among the statisticians covered are Fermat, Pascal, Huygens, Neumann, Bernoulli, Bayes, Laplace, Legendre, Gauss, Poisson, Pareto, Markov, Bachelier, Borel, and many more.
Problems in Abstract Algebra
This is a book of problems in abstract algebra for strong undergraduates or beginning graduate students. It can be used as a supplement to a course or for self-study. The book provides more variety and more challenging problems than are found in most algebra textbooks. It is intended for students wanting to enrich their learning of mathematics by tackling problems that take some thought and effort to solve. The book contains problems on groups (including the Sylow Theorems, solvable groups, presentation of groups by generators and relations, and structure and duality for finite abelian groups); rings (including basic ideal theory and factorization in integral domains and Gauss's Theorem); linear algebra (emphasizing linear transformations, including canonical forms); and fields (including Galois theory). Hints to many problems are also included.
