Bücher der Reihe Cambridge Tracts in Mathematics

Originally published in 1913 as number fourteen in the Cambridge Tracts in Mathematics and Mathematical Physics series, this book provides a concise account regarding the properties of the twisted cubic. A bibliography and appendix section are also included.

Originally published in 1932 as number twenty-seven in the Cambridge Tracts in Mathematics and Mathematical Physics series, this book provides a concise account of the theory of modular invariants as embodied in the work of Dickson, Glenn and Hazlett. Appendices are included.

First published in 1913, as the second edition of a 1905 original, this book is the first volume in the Cambridge Tracts in Mathematics and Mathematical Physics Series. The text provides a concise account regarding volume and surface integrals used in physics.

Originally published in 1915 as number eighteen in the Cambridge Tracts in Mathematics and Mathematical Physics series, and here reissued in its 1952 reprinted form, this book contains a condensed account of Dirichlet's Series, which relates to number theory.

Originally published in 1928, this book deals with the physical and mathematical aspects of the symmetrical optical system. Steward's explanation is taken in part from lectures delivered to students of mathematics and physics at the University of Cambridge in the early twentieth century.

Originally published in 1908 in the Cambridge Tracts in Mathematics and Mathematical Physics series, this book provides a concise account regarding the invariant theory connected with a single quadratic differential form. This book will be of value to anyone with an interest in quadratic differential forms and the history of mathematics.

Originally published in 1914 as number sixteen in the Cambridge Tracts in Mathematics and Mathematical Physics series, this book provides a concise account regarding the theory of linear associative algebras. Textual notes are incorporated throughout. This book will be of value to anyone with an interest in algebra and the history of mathematics.

Originally published in 1914 as number fifteen in the Cambridge Tracts in Mathematics and Mathematical Physics series, this book provides a concise proof of Cauchy's Theorem, along with some applications of the theorem to the evaluation of definite integrals.

Originally published in 1936 as part of the Cambridge Tracts in Mathematics and Mathematical Physics series, this book provides a concise account regarding the rational quartic curve in space of three and four dimensions. Textual notes are also included.

In the past several decades the classical Perron-Frobenius theory for nonnegative matrices has been extended to obtain remarkably precise and beautiful results for classes of nonlinear maps. This nonlinear Perron-Frobenius theory has found significant uses in computer science, mathematical biology, game theory and the study of dynamical systems. This is the first comprehensive and unified introduction to nonlinear Perron-Frobenius theory suitable for graduate students and researchers entering the field for the first time. It acquaints the reader with recent developments and provides a guide to challenging open problems. To enhance accessibility, the focus is on finite dimensional nonlinear Perron-Frobenius theory, but pointers are provided to infinite dimensional results. Prerequisites are little more than basic real analysis and topology.

The authors survey the state of current research on the invariant subspace problem for linear operators, a major unsolved problem in mathematics that has inspired much research in the past fifty years. The subject is presented at a level suitable for postgraduate students and established researchers in mathematics.

This self-contained monograph presents rigidity theory for a large class of dynamical systems, differentiable higher rank hyperbolic and partially hyperbolic actions. This first volume describes the subject in detail and develops the principal methods presently used in various aspects of the rigidity theory. Part I serves as an exposition and preparation, including a large collection of examples that are difficult to find in the existing literature. Part II focuses on cocycle rigidity, which serves as a model for rigidity phenomena as well as a useful tool for studying them. The book is an ideal reference for applied mathematicians and scientists working in dynamical systems and a useful introduction for graduate students interested in entering the field. Its wealth of examples also makes it excellent supplementary reading for any introductory course in dynamical systems.

Convexity is important in theoretical aspects of mathematics and also for economists and physicists. In this monograph the author provides a comprehensive insight into convex sets and functions including the infinite-dimensional case and emphasizing the analytic point of view. Chapter one introduces the reader to the basic definitions and ideas that play central roles throughout the book. The rest of the book is divided into four parts: convexity and topology on infinite-dimensional spaces; Loewner's theorem; extreme points of convex sets and related issues, including the Krein-Milman theorem and Choquet theory; and a discussion of convexity and inequalities. The connections between disparate topics are clearly explained, giving the reader a thorough understanding of how convexity is useful as an analytic tool. A final chapter overviews the subject's history and explores further some of the themes mentioned earlier. This is an excellent resource for anyone interested in this central topic.

Cho-Ho Chu discusses recent advances of Jordan theory in differential geometry and analysis, including applications of Jordan methods in infinite dimensional manifolds, complex function theory and Banach spaces. Containing numerous examples and historical notes, this book is a useful reference as well as an introduction for beginning researchers.

Ideas of projective geometry keep reappearing in seemingly unrelated fields of mathematics. The authors' main goal in this 2005 book is to emphasize connections between classical projective differential geometry and contemporary mathematics and mathematical physics. They also give results and proofs of classic theorems. Exercises play a prominent role: historical and cultural comments set the basic notions in a broader context. The book opens by discussing the Schwarzian derivative and its connection to the Virasoro algebra. One-dimensional projective differential geometry features strongly. Related topics include differential operators, the cohomology of the group of diffeomorphisms of the circle, and the classical four-vertex theorem. The classical theory of projective hypersurfaces is surveyed and related to some very recent results and conjectures. A final chapter considers various versions of multi-dimensional Schwarzian derivative. In sum, here is a rapid route for graduate students and researchers to the frontiers of current research in this evergreen subject.

The author presents an accessible and self-contained introduction to harmonic map theory and its analytical aspects, covering recent developments in the regularity theory of weakly harmonic maps. The book begins by introducing these concepts, stressing the interplay between geometry, the role of symmetries and weak solutions. The reader is then presented with a guided tour into the theory of completely integrable systems for harmonic maps, followed by two chapters devoted to recent results on the regularity of weak solutions. A self-contained presentation of 'exotic' functional spaces from the theory of harmonic analysis is given and these tools are then used for proving regularity results. The importance of conservation laws is stressed and the concept of a 'Coulomb moving frame' is explained in detail. The book ends with further applications and illustrations of Coulomb moving frames to the theory of surfaces.

The first book containing a rigorous construction and uniqueness proof for the largest and most famous sporadic simple group, the Monster. The author provides a systematic exposition of the theory of the Monster group, which so far remains largely unpublished, and explores the theory of groups generated by Majorana involutions.

This book studies normal approximations by means of two powerful probabilistic techniques: the Malliavin calculus and Stein's method. Largely self-contained it is perfect for self-study and will appeal both to researchers and to graduate students in probability and statistics.

Assuming only basic knowledge of probability theory and functional analysis, this book is an ideal introduction to the field. The author's careful exposition, which is neither too abstract nor too theoretical, makes it accessible to graduate students, as well as to researchers who are interested in the author's techniques.

This book lays the foundations for an exciting new area of research in descriptive set theory. It develops a robust connection between two active topics: forcing and analytic equivalence relations. This in turn allows the authors to develop a generalization of classical Ramsey theory. Given an analytic equivalence relation on a Polish space, can one find a large subset of the space on which it has a simple form? The book provides many positive and negative general answers to this question. The proofs feature proper forcing and Gandy-Harrington forcing, as well as partition arguments. The results include strong canonization theorems for many classes of equivalence relations and sigma-ideals, as well as ergodicity results in cases where canonization theorems are impossible to achieve. Ideal for graduate students and researchers in set theory, the book provides a useful springboard for further research.

Hardy's Z-function, related to the Riemann zeta-function I (s), was originally utilised by G. H. Hardy to show that I (s) has infinitely many zeros of the form 1/2+it. It is now amongst the most important functions of analytic number theory, and the Riemann hypothesis, that all complex zeros lie on the line 1/2+it, is perhaps one of the best known and most important open problems in mathematics. Today Hardy's function has many applications; among others it is used for extensive calculations regarding the zeros of I (s). This comprehensive account covers many aspects of Z(t), including the distribution of its zeros, Gram points, moments and Mellin transforms. It features an extensive bibliography and end-of-chapter notes containing comments, remarks and references. The book also provides many open problems to stimulate readers interested in further research.

This book introduces the reader to powerful methods of critical point theory and details successful contemporary approaches to many problems, some of which had proved resistant to attack by older methods. Topics covered include Morse theory, critical groups, the minimax principle, various notions of linking, jumping nonlinearities and the Fucik spectrum in an abstract setting, sandwich pairs and the cohomological index. Applications to semilinear elliptic boundary value problems, p-Laplacian problems and anisotropic systems are given. Written for graduate students and research scientists, the book includes numerous examples and presents more recent developments in the subject to bring the reader up to date with the latest research.

A treatment of cutting-edge research on the distribution modulo one of sequences and related topics, much of it from the last decade. There are numerous exercises to aid student understanding of the topic, and researchers will appreciate the notes at the end of each chapter, extensive references and open problems.

This 1999 book investigates the high degree of symmetry that lies hidden in integrable systems. Differential equations arising from classical mechanics, such as the KdV equation and the KP equations, are used here by the authors to introduce the notion of an infinite dimensional transformation group acting on spaces of integrable systems.

The theme of this book is the distribution of integer powers modulo a prime number. It provides numerous new links between number theory and computer science as well as other areas of mathematics. Applications include (but are not limited to) complexity theory, random number generation, cryptography, and coding theory.

The modern version of the bilinear, or Hirota's direct, method is described here using relatively simple mathematics. As the only account in book form of the modern form of the theory, it will be essential reading for all those working in soliton theory.

Minuscule representations occur in a variety of contexts in mathematics and physics. They are typically much easier to understand than representations in general, which means they give rise to relatively easy constructions of algebraic objects such as Lie algebras and Weyl groups. This book describes a combinatorial approach to minuscule representations of Lie algebras using the theory of heaps, which for most practical purposes can be thought of as certain labelled partially ordered sets. This leads to uniform constructions of (most) simple Lie algebras over the complex numbers and their associated Weyl groups, and provides a common framework for various applications. The topics studied include Chevalley bases, permutation groups, weight polytopes and finite geometries. Ideal as a reference, this book is also suitable for students with a background in linear and abstract algebra and topology. Each chapter concludes with historical notes, references to the literature and suggestions for further reading.

This book gives a comprehensive treatment of the singularities that appear in the minimal model program and in the moduli problem for varieties. The study of these singularities and the development of Mori's program have been deeply intertwined. Early work on minimal models relied on detailed study of terminal and canonical singularities but many later results on log terminal singularities were obtained as consequences of the minimal model program. Recent work on the abundance conjecture and on moduli of varieties of general type relies on subtle properties of log canonical singularities and conversely, the sharpest theorems about these singularities use newly developed special cases of the abundance problem. This book untangles these interwoven threads, presenting a self-contained and complete theory of these singularities, including many previously unpublished results.

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 formulas for the Dirichlet integral, reproducing kernels, boundary behaviour and capacity, zero sets and uniqueness sets, multipliers, interpolation, Carleson measures, composition operators, local Dirichlet spaces, shift-invariant subspaces, and cyclicity. Special features include a self-contained treatment of capacity, including the strong-type inequality. The book will be valuable to researchers in function theory, and with over 100 exercises it is also suitable for self-study by graduate students.

Group cohomology reveals a deep relation between algebra and topology, and its recent applications have provided important insights into the Hodge conjecture and algebraic geometry more broadly. This book presents a coherent suite of computational tools for the study of group cohomology and algebraic cycles.