Probability and Measure Theory, Second Edition, is a text for a graduate-level course in probability that includes essential background topics in analysis. It provides extensive coverage of conditional probability and expectation, strong laws of large numbers, martingale theory, the central limit theorem, ergodic theory, and Brownian motion.

Probability and Measure Theory

Science & Math

Robert B. Ash

Catherine A. Doléans-Dade

Mathematics

Academic Press

Complex variables

Academic Press, Inc.

Measure, integration, and functional analysis

Topics in Stochastic Processes

Melvin F. Gardner

Geared toward upper-level undergraduates and graduate students, this text surveys fundamental algebraic structures and maps between these structures. Its techniques are used in many areas of mathematics, with applications to physics, engineering, and computer science as well. Author Robert B. Ash, a Professor of Mathematics at the University of Illinois, focuses on intuitive thinking. He also conveys the intrinsic beauty of abstract algebra while keeping the proofs as brief and clear as possible. The early chapters provide students with background by investigating the basic properties of groups, rings, fields, and modules. Later chapters examine the relations between groups and sets, the fundamental theorem of Galois theory, and the results and methods of abstract algebra in terms of algebraic number theory, algebraic geometry, noncommutative algebra, and homological algebra, including categories and functors. An extensive supplement to the text delves much further into homological algebra than most introductory texts, offering applications-oriented results. Solutions to all problems appear in the text.

Basic Abstract Algebra: For Graduate Students and Advanced Undergraduates (Dover Books on Mathematics)

Dover Publications

This text on complex variables is geared toward graduate students and undergraduates who have taken an introductory course in real analysis. It is a substantially revised and updated edition of the popular text by Robert B. Ash, offering a concise treatment that provides careful and complete explanations as well as numerous problems and solutions. An introduction presents basic definitions, covering topology of the plane, analytic functions, real-differentiability and the Cauchy-Riemann equations, and exponential and harmonic functions. Succeeding chapters examine the elementary theory and the general Cauchy theorem and its applications, including singularities, residue theory, the open mapping theorem for analytic functions, linear fractional transformations, conformal mapping, and analytic mappings of one disk to another. The Riemann mapping theorem receives a thorough treatment, along with factorization of analytic functions. As an application of many of the ideas and results appearing in earlier chapters, the text ends with a proof of the prime number theorem.

Complex Variables: Second Edition (Dover Books on Mathematics)

W. P. Novinger

This introduction to more advanced courses in probability and real analysis emphasizes the probabilistic way of thinking, rather than measure-theoretic concepts. Geared toward advanced undergraduates and graduate students, its sole prerequisite is calculus. Taking statistics as its major field of application, the text opens with a review of basic concepts, advancing to surveys of random variables, the properties of expectation, conditional probability and expectation, and characteristic functions. Subsequent topics include infinite sequences of random variables, Markov chains, and an introduction to statistics. Complete solutions to some of the problems appear at the end of the book.

Basic Probability Theory (Dover Books on Mathematics)

This text for a graduate-level course covers the general theory of factorization of ideals in Dedekind domains as well as the number field case. It illustrates the use of Kummer's theorem, proofs of the Dirichlet unit theorem, and Minkowski bounds on element and ideal norms. 2003 edition.

A Course in Algebraic Number Theory (Dover Books on Mathematics)

Developed by Claude Shannon and Norbert Wiener in the late 1940s, information theory, or statistical communication theory, deals with the theoretical underpinnings of a wide range of communication devices: radio, television, radar, computers, telegraphy, and more. This book is an excellent introduction to the mathematics underlying the theory. Designed for upper-level undergraduates and first-year graduate students, the book treats three major areas: analysis of channel models and proof of coding theorems (chapters 3, 7, and 8); study of specific coding systems (chapters 2, 4, and 5); and study of statistical properties of information sources (chapter 6). Among the topics covered are noiseless coding, the discrete memoryless channel, effort correcting codes, information sources, channels with memory, and continuous channels. The author has tried to keep the prerequisites to a minimum. However, students should have a knowledge of basic probability theory. Some measure and Hilbert space theory is helpful as well for the last two sections of chapter 8, which treat time-continuous channels. An appendix summarizes the Hilbert space background and the results from the theory of stochastic processes necessary for these sections. The appendix is not self-contained but will serve to pinpoint some of the specific equipment needed for the analysis of time-continuous channels. In addition to historic notes at the end of each chapter indicating the origin of some of the results, the author has also included 60 problems with detailed solutions, making the book especially valuable for independent study.

Information Theory (Dover Books on Mathematics)

Computers & Technology

Computer Science

The purpose of this book is to prepare the reader for coping with abstract mathematics. The intended audience is both students taking a first course in abstract algebra who feel the need to strengthen their background, and those from a more applied background who need some experience in dealing with abstract ideas. Learning any area of abstract mathematics requires not only ability to write formally but also to think intuitively about what is going on and to describe that process clearly and cogently in ordinary English. Ash tries to aid intuition by keeping proofs short and as informal as possible, and using concrete examples as illustration. Thus it is an ideal textbook for an audience with limited experience in formalism and abstraction. A number of expository innovations are included, for example, an informal development of set theory which teaches students all the basic results for algebra in one chapter.

A Primer of Abstract Algebra (Classroom Resource Materials)

Reference

The Mathematical Association of America

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Books by Robert B. Ash