Bøger i Cambridge Mathematical Textbooks serien

"Emphasizing the creative nature of mathematics, this conversational textbook guides students through the process of discovering a proof as they transition to advanced mathematics. Using several strategies, students will develop the thinking skills needed to tackle mathematics when there is no clear algorithm or recipe to follow"--

This upper undergraduate abstract algebra text covers classical themes on groups, rings and fields in depth, augmented with a strong emphasis on irreducible polynomials, a fresh approach to modules and linear algebra, a fresh take on Groebner theory, and a group theoretic treatment of Rejewski's deciphering of the Enigma machine.

Abstract Algebra with Applications provides a friendly and concise introduction to algebra, with an emphasis on its uses in the modern world. The first part of this book covers groups, after some preliminaries on sets, functions, relations, and induction, and features applications such as public-key cryptography, Sudoku, the finite Fourier transform, and symmetry in chemistry and physics. The second part of this book covers rings and fields, and features applications such as random number generators, error correcting codes, the Google page rank algorithm, communication networks, and elliptic curve cryptography. The book's masterful use of colorful figures and images helps illustrate the applications and concepts in the text. Real-world examples and exercises will help students contextualize the information. Intended for a year-long undergraduate course in algebra for mathematics, engineering, and computer science majors, the only prerequisites are calculus and a bit of courage when asked to do a short proof.

This lively undergraduate combinatorics text covers all essential topics, with few prerequisites. Mini-projects, warm-up problems, and 1200+ exercises encourage active student participation. Students will get a glimpse into current research trends and open problems as well as some of the history and global origins of the subject.

This conversational introduction to abstract algebra takes a modern, rings-first approach. In addition to its unconventional order of classical material, another key feature is the treatment of topics often neglected in undergraduate textbooks, such as modules. More than 400 exercises are included, 150 of which are carefully worked out.

This classroom-tested textbook is an introduction to probability theory, with the right balance between mathematical precision, probabilistic intuition, and concrete applications. Introduction to Probability covers the material precisely, while avoiding excessive technical details. After introducing the basic vocabulary of randomness, including events, probabilities, and random variables, the text offers the reader a first glimpse of the major theorems of the subject: the law of large numbers and the central limit theorem. The important probability distributions are introduced organically as they arise from applications. The discrete and continuous sides of probability are treated together to emphasize their similarities. Intended for students with a calculus background, the text teaches not only the nuts and bolts of probability theory and how to solve specific problems, but also why the methods of solution work.

A recent surge in computer-based experimental approaches to pure mathematics is revolutionizing the way research mathematicians work. As the first of its kind, this textbook provides students with an introduction to the ends and means of experimental mathematics using the popular computer algebra system Maple.

This user-friendly textbook offers an introduction to complex analysis. Unlike other textbooks, it follows Weierstrass' approach, and includes several elegant proofs that were recently discovered. Classroom-tested and self-contained, it is for beginning graduate or advanced undergraduate students with a modest undergraduate real analysis background.

This book offers a concise and modern introduction to differential topology, the study of smooth manifolds and their properties, at the advanced undergraduate/beginning graduate level. The treatment throughout is hands-on, including many concrete examples and exercises woven into the text with hints provided to guide the student.

Linear Algebra offers a unified treatment of both matrix-oriented and theoretical approaches to the course. Written for students in pure and applied mathematics, as well as physics, engineering, and computer science, it is designed to facilitate the transition from calculus to advanced mathematics.

Assuming minimal prerequisites, this rigorous yet accessible text is intended for a year-long analysis or advanced calculus course for advanced undergraduate or beginning graduate students. It clearly and concisely explains differentiation and integration of functions of one, and several variables and covers the Green, Gauss, and Stokes theorems.

The transition from predominantly computational courses to upper-level math requires the development of skills, including reading and writing mathematical proofs, and creating illuminating examples and insights. Exploring Mathematics supports students by covering core topics and having them actively develop theorems through exercises and projects.

This undergraduate text is a rigorous introduction to dynamical systems and an accessible guide for those transitioning from calculus to advanced mathematics. It has many student-friendly features, such as graded exercises ranging from straightforward to more difficult with hints, and includes applications of real analysis and metric space theory.

Mathematicians have shown that virtually all mathematical concepts and results can be formalized within set theory. This textbook covers the fundamentals of abstract sets and develops these theories within the framework of axiomatic set theory. The proofs presented are rigorous, clear, and suitable for undergraduate and graduate students.

This classroom-tested undergraduate textbook is intended for a general education course in game theory at the freshman or sophomore level. While it starts off with the basics and introduces the reader to mathematical proofs, this text also presents several advanced topics, including accessible proofs of the Sprague-Grundy theorem and Arrow's impossibility theorem.