The Dirichlet space is one of the three fundamental Hilbert spaces of holomorphic functions on the unit disk. It boasts a rich and beautiful theory, yet at the same time remains a source of challenging open problems and a subject of active mathematical research. This book is the first systematic account of the Dirichlet space, assembling results previously only found in scattered research articles, and improving upon many of the proofs. Topics treated include: the Douglas and Carleson form...

This book offers a first course in analysis for scientists and engineers. It can be used at the advanced undergraduate level or as part of the curriculum in a graduate program. The book is built around metric spaces. In the first three chapters, the authors lay the foundational material and cover the all-important “four-C’s”: convergence, completeness, compactness, and continuity. In subsequent chapters, the basic tools of analysis are used to give brief introductions to differential and i...

The theory of elliptic curves involves a pleasing blend of algebra, geometry, analysis, and number theory. This volume stresses this interplay as it develops the basic theory, thereby providing an opportunity for advanced undergraduates to appreciate the unity of modern mathematics. At the same time, every effort has been made to use only methods and results commonly included in the undergraduate curriculum. This accessibility, the informal writing style, and a wealth of exercises make

This valuable book focuses on a collection of powerful methods of analysis that yield deep number-theoretical estimates. Particular attention is given to counting functions of prime numbers and multiplicative arithmetic functions. Both real variable (”elementary”) and complex variable (”analytic”) methods are employed. The reader is assumed to have knowledge of elementary number theory (abstract algebra will also do) and real and complex analysis. Specialized analytic techniques, including...

Beginning graduate students in mathematics and other quantitative subjects are expected to have a daunting breadth of mathematical knowledge. But few have such a background. This book will help students to see the broad outline of mathematics and to fill in the gaps in their knowledge. The author explains the basic points and a few key results of all the most important undergraduate topics in mathematics, emphasizing the intuitions behind the subject. The topics include linear algebra, vec...

For over three decades, this best-selling classic has been used by thousands of students in the United States and abroad as a must-have textbook for a transitional course from calculus to analysis. It has proven to be very useful for mathematics majors who have no previous experience with rigorous proofs. Its friendly style unlocks the mystery of writing proofs, while carefully examining the theoretical basis for calculus. Proofs are given in full, and the large number of well-chosen examp...

For the second edition of this very successful text, Professor Binmore has written two chapters on analysis in vector spaces. The discussion extends to the notion of the derivative of a vector function as a matrix and the use of second derivatives in classifying stationary points. Some necessary concepts from linear algebra are included where appropriate. The first edition contained numerous worked examples and an ample collection of exercises for all of which solutions were provided at th...

This book serves as an introduction to calculus on normed vector spaces at a higher undergraduate or beginning graduate level. The prerequisites include basic calculus and linear algebra, as well as a certain mathematical maturity. All the important topology and functional analysis topics are introduced where necessary.In its attempt to show how calculus on normed vector spaces extends the basic calculus of functions of several variables, this book is one of the few textbooks to br...

Sure to be influential, this book lays the foundations for the use of algebraic geometry in statistical learning theory. Many widely used statistical models and learning machines applied to information science have a parameter space that is singular: mixture models, neural networks, HMMs, Bayesian networks, and stochastic context-free grammars are major examples. Algebraic geometry and singularity theory provide the necessary tools for studying such non-smooth models. Four main formulas ar...

This book is intended for a graduate course in complex analysis, where the main focus is the theory of complex-valued functions of a single complex variable. This theory is a prerequisite for the study of many areas of mathematics, including the theory of several finitely and infinitely many complex variables, hyperbolic geometry, two- and three-manifolds, and number theory. Complex analysis has connections and applications to many other subjects in mathematics and to other sciences. Thus ...

Quaternions are a number system that has become increasingly useful for representing the rotations of objects in three-dimensional space and has important applications in theoretical and applied mathematics, physics, computer science, and engineering. This is the first book to provide a systematic, accessible, and self-contained exposition of quaternion linear algebra. It features previously unpublished research results with complete proofs and many open problems at various levels, as well...

Probability theory on compact Lie groups deals with the interaction between “chance” and “symmetry,” a beautiful area of mathematics of great interest in its own sake but which is now also finding increasing applications in statistics and engineering (particularly with respect to signal processing). The author gives a comprehensive introduction to some of the principle areas of study, with an emphasis on applicability. The most important topics presented are: the study of measures via the ...