This is the first volume of a series of books of problems in mathematical analysis. It is mainly intended for students studying the basic principles of analysis. However, given its organization, level, and selection of problems, it would also be an ideal choice for tutorial or problem-solving seminars, particularly those geared toward the Putnam exam. The volume is also suitable for self-study.

Discusses the Riemann and the Riemann-Stieltjes integrals. This book deals with Lebesgue measure and integration. It is suitable for students studying the basic principles of analysis.

The main notions of set theory (cardinals, ordinals, transfinite induction) are fundamental to all mathematicians, not only to those who specialize in mathematical logic or set-theoretic topology. Basic set theory is generally given a brief overview in courses on analysis, algebra, or topology, even though it is sufficiently important, interesting, and simple to merit its own leisurely treatment.

When information is transmitted, errors are likely to occur. This problem has become increasingly important as tremendous amounts of information are transferred electronically every day. Coding theory examines efficient ways of packaging data so that these errors can be detected, or even corrected.

The calculus of variations is a beautiful subject with a rich history and with origins in the minimization problems of calculus. This book addresses topics such as Morse theory, wave mechanics, minimal surfaces, soap bubbles, and modeling traffic flow.

This book, based on the course given by the author at the College of Mathematics of the Independent University of Moscow, introduces the reader to the language of generating functions, which is nowadays the main language of enumerative combinatorics. It starts with definitions, simple properties, and numerous examples of generating functions. It then discusses topics, such as formal grammars, generating functions in several variables, partitions and decompositions, and the exclusion-inclusion principle. In the final chapter, the author describes applications of generating functions to...

This text is an invitation to the world of variational geometry. A central topic is Plateau's Problem, which is concerned with surfaces that model the behaviour of soap films. When trying to resolve the problem, however, one soon finds that smooth surfaces are insufficient: Varifolds are needed. With varifolds, one can obtain geometrically meaningful solutions without having to know in advance all their possible singularities. This tool makes possible new analyses and new results.

We have been curious about numbers - and prime numbers - since antiquity. One notable new direction of the 20th century in the study of primes has been the influx of ideas from probability. The goal of this book is to provide insights into the prime numbers and to describe how a sequence so tautly determined can incorporate such a striking amount of randomness.

This volume is based on notes from the course developed and taught for more than 30 years at the Department of Mathematics at Leningrad University. The goal of the course was to present quantum mechanics and its mathematical context to students in mathematics.