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Most conformal invariants can be described in terms of extremal properties. Conformal invariants and extremal problems are therefore intimately linked and form together the central theme of this classic book which is primarily intended for students with approximately a year's background in complex variable theory. The book emphasizes the geometric approach as well as classical and semi-classical results which Lars Ahlfors felt every student of complex analysis should know before embarking on independent research. At the time of the book's original appearance, much of this material had never appeared in book form, particularly the discussion of the theory of extremal length. Schiffer's variational method also receives special attention, and a proof of $\vert a_4\vert \leq 4$ is included which was new at the time of publication. The last two chapters give an introduction to Riemann surfaces, with topological and analytical background supplied to support a proof of the uniformization theorem. Included in this new reprint is a Foreword by Peter Duren, F. W. Gehring, and Brad Osgood, as well as an extensive errata.

Critical phenomena arise in a wide variety of physical systems. Classi­ cal examples are the liquid-vapour critical point or the paramagnetic­ ferromagnetic transition. Further examples include multicomponent fluids and alloys, superfluids, superconductors, polymers and fully developed tur­ bulence and may even extend to the quark-gluon plasma and the early uni­ verse as a whole. Early theoretical investigators tried to reduce the problem to a very small number of degrees of freedom, such as the van der Waals equation and mean field approximations, culminating in Landau's general theory of critical phenomena. Nowadays, it is understood that the common ground for all these phenomena lies in the presence of strong fluctuations of infinitely many coupled variables. This was made explicit first through the exact solution of the two-dimensional Ising model by Onsager. Systematic subsequent developments have been leading to the scaling theories of critical phenomena and the renormalization group which allow a precise description of the close neighborhood of the critical point, often in good agreement with experiments. In contrast to the general understanding a century ago, the presence of fluctuations on all length scales at a critical point is emphasized today. This can be briefly summarized by saying that at a critical point a system is scale invariant. In addition, conformal invaTiance permits also a non-uniform, local rescal­ ing, provided only that angles remain unchanged.

The conformal predictions framework is a recent development in machine learning that can associate a reliable measure of confidence with a prediction in any real-world pattern recognition application, including risk-sensitive applications such as medical diagnosis, face recognition, and financial risk prediction. Conformal Predictions for Reliable Machine Learning: Theory, Adaptations and Applications captures the basic theory of the framework, demonstrates how to apply it to real-world problems, and presents several adaptations, including active learning, change detection, and anomaly detection. As practitioners and researchers around the world apply and adapt the framework, this edited volume brings together these bodies of work, providing a springboard for further research as well as a handbook for application in real-world problems.Understand the theoretical foundations of this important framework that can provide a reliable measure of confidence with predictions in machine learningBe able to apply this framework to real-world problems in different machine learning settings, including classification, regression, and clusteringLearn effective ways of adapting the framework to newer problem settings, such as active learning, model selection, or change detection

Filling an important gap in the literature, this comprehensive text develops conformal field theory from first principles. The treatment is self-contained, pedagogical, and exhaustive, and includes a great deal of background material on quantum field theory, statistical mechanics, Lie algebras and affine Lie algebras. The many exercises, with a wide spectrum of difficulty and subjects, complement and in many cases extend the text. The text is thus not only an excellent tool for classroom teaching but also for individual study. Intended primarily for graduate students and researchers in theoretical high-energy physics, mathematical physics, condensed matter theory, statistical physics, the book will also be of interest in other areas of theoretical physics and mathematics. It will prepare the reader for original research in this very active field of theoretical and mathematical physics.

Conformal Field Theory consists of pedagogical lectures delivered at the Feza Gursey Institute, Istanbul, in the summer of 1998 on the following topics: Non-perturbative dynamics of 4-dimensional Field theories” (P. Argyres, Cornell University), 2-Dimensional QCD, Grassmanians and M(atrix) Models” (S. Rajeev, University of Rochester), Affine Kac-Moody algebras and CFT” (M. Walton, Lethbridge), CFT and Vertex Operator algebras” (T. Gannon, York), and Meromorphic CFT” (M. Gaberdiel, Cambridge). The book is suitable for advanced graduate students and researchers in theoretical particle or statistical physics as well as pure mathematicians.

Beginning with a brief survey of some basic mathematical concepts, this graduate-level text proceeds to discussions of a selection of mapping functions, numerical methods and mathematical models, nonplanar fields and nonuniform media, static fields in electricity and magnetism, and transmission lines and waveguides. Other topics include vibrating membranes and acoustics, transverse vibrations and buckling of plates, stresses and strains in an elastic medium, steady state heat conduction in doubly connected regions, transient heat transfer in isotropic and anisotropic media, and fluid flow. Revision of 1991 ed. 247 figures. 38 tables. Appendices.

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Conformal mapping is a field in which pure and applied mathematics are both involved. This book tries to bridge the gulf that many times divides these two disciplines by combining the theoretical and practical approaches to the subject. It will interest the pure mathematician, engineer, physicist, and applied mathematician.The potential theory and complex function theory necessary for a full treatment of conformal mapping are developed in the first four chapters, so the reader needs no other text on complex variables. These chapters cover harmonic functions, analytic functions, the complex integral calculus, and families of analytic functions. Included here are discussions of Green's formula, the Poisson formula, the Cauchy-Riemann equations, Cauchy's theorem, the Laurent series, and the Residue theorem. The final three chapters consider in detail conformal mapping of simply-connected domains, mapping properties of special functions, and conformal mapping of multiply-connected domains. The coverage here includes such topics as the Schwarz lemma, the Riemann mapping theorem, the Schwarz-Christoffel formula, univalent functions, the kernel function, elliptic functions, univalent functions, the kernel function, elliptic functions, the Schwarzian s-functions, canonical domains, and bounded functions. There are many problems and exercises, making the book useful for both self-study and classroom use.The author, former professor of mathematics at Carnegie-Mellon University, has designed the book as a semester's introduction to functions of a complex variable followed by a one-year graduate course in conformal mapping. The material is presented simply and clearly, and the only prerequisite is a good working knowledge of advanced calculus.

Filling an important gap in the literature, this comprehensive text develops conformal field theory from first principles. The treatment is self-contained, pedagogical, and exhaustive, and includes a great deal of background material on quantum field theory, statistical mechanics, Lie algebras and affine Lie algebras. The many exercises, with a wide spectrum of difficulty and subjects, complement and in many cases extend the text. The text is thus not only an excellent tool for classroom teaching but also for individual study. Intended primarily for graduate students and researchers in theoretical high-energy physics, mathematical physics, condensed matter theory, statistical physics, the book will also be of interest in other areas of theoretical physics and mathematics. It will prepare the reader for original research in this very active field of theoretical and mathematical physics.

Filling an important gap in the literature, this comprehensive text develops conformal field theory from first principles. The treatment is self-contained, pedagogical, and exhaustive, and includes a great deal of background material on quantum field theory, statistical mechanics, Lie algebras and affine Lie algebras. The many exercises, with a wide spectrum of difficulty and subjects, complement and in many cases extend the text. The text is thus not only an excellent tool for classroom teaching but also for individual study. Intended primarily for graduate students and researchers in theoretical high-energy physics, mathematical physics, condensed matter theory, statistical physics, the book will also be of interest in other areas of theoretical physics and mathematics. It will prepare the reader for original research in this very active field of theoretical and mathematical physics.

Critical phenomena arise in a wide variety of physical systems. Classi­ cal examples are the liquid-vapour critical point or the paramagnetic­ ferromagnetic transition. Further examples include multicomponent fluids and alloys, superfluids, superconductors, polymers and fully developed tur­ bulence and may even extend to the quark-gluon plasma and the early uni­ verse as a whole. Early theoretical investigators tried to reduce the problem to a very small number of degrees of freedom, such as the van der Waals equation and mean field approximations, culminating in Landau's general theory of critical phenomena. Nowadays, it is understood that the common ground for all these phenomena lies in the presence of strong fluctuations of infinitely many coupled variables. This was made explicit first through the exact solution of the two-dimensional Ising model by Onsager. Systematic subsequent developments have been leading to the scaling theories of critical phenomena and the renormalization group which allow a precise description of the close neighborhood of the critical point, often in good agreement with experiments. In contrast to the general understanding a century ago, the presence of fluctuations on all length scales at a critical point is emphasized today. This can be briefly summarized by saying that at a critical point a system is scale invariant. In addition, conformal invaTiance permits also a non-uniform, local rescal­ ing, provided only that angles remain unchanged.

Conformal field theory is an elegant and powerful theory in the field of high energy physics and statistics. In fact, it can be said to be one of the greatest achievements in the development of this field. Presented in two dimensions, this book is designed for students who already have a basic knowledge of quantum mechanics, field theory and general relativity. The main idea used throughout the book is that conformal symmetry causes both classical and quantum integrability. Instead of concentrating on the numerous applications of the theory, the author puts forward a discussion of the general methods of conformal field theory as a physical theory. Hence the book provides in a self-contained way the necessary knowledge and "conformal" intuition which underline the various applications of conformal field theory. It is aimed to assist students and professionals in the study of the theory from its first principles and in applying the methods in their own research. The first of its kind, this book promises to give a detailed and comprehensive insight into the workings of conformal field theory.

Conformal mapping is a field in which pure and applied mathematics are both involved. This book tries to bridge the gulf that many times divides these two disciplines by combining the theoretical and practical approaches to the subject. It will interest the pure mathematician, engineer, physicist, and applied mathematician.The potential theory and complex function theory necessary for a full treatment of conformal mapping are developed in the first four chapters, so the reader needs no other text on complex variables. These chapters cover harmonic functions, analytic functions, the complex integral calculus, and families of analytic functions. Included here are discussions of Green's formula, the Poisson formula, the Cauchy-Riemann equations, Cauchy's theorem, the Laurent series, and the Residue theorem. The final three chapters consider in detail conformal mapping of simply-connected domains, mapping properties of special functions, and conformal mapping of multiply-connected domains. The coverage here includes such topics as the Schwarz lemma, the Riemann mapping theorem, the Schwarz-Christoffel formula, univalent functions, the kernel function, elliptic functions, univalent functions, the kernel function, elliptic functions, the Schwarzian s-functions, canonical domains, and bounded functions. There are many problems and exercises, making the book useful for both self-study and classroom use.The author, former professor of mathematics at Carnegie-Mellon University, has designed the book as a semester's introduction to functions of a complex variable followed by a one-year graduate course in conformal mapping. The material is presented simply and clearly, and the only prerequisite is a good working knowledge of advanced calculus.

Most conformal invariants can be described in terms of extremal properties. Conformal invariants and extremal problems are therefore intimately linked and form together the central theme of this classic book which is primarily intended for students with approximately a year's background in complex variable theory. The book emphasizes the geometric approach as well as classical and semi-classical results which Lars Ahlfors felt every student of complex analysis should know before embarking on independent research. At the time of the book's original appearance, much of this material had never appeared in book form, particularly the discussion of the theory of extremal length. Schiffer's variational method also receives special attention, and a proof of $\vert a_4\vert \leq 4$ is included which was new at the time of publication. The last two chapters give an introduction to Riemann surfaces, with topological and analytical background supplied to support a proof of the uniformization theorem. Included in this new reprint is a Foreword by Peter Duren, F. W. Gehring, and Brad Osgood, as well as an extensive errata.

The subject matter loosely called "Riemann surface theory" has been the starting point for the development of topology, functional analysis, modern algebra, and any one of a dozen recent branches of mathematics; it is one of the most valuable bodies of knowledge within mathematics for a student to learn.Professor Cohn's lucid and insightful book presents an ideal coverage of the subject in five parts. Part I is a review of complex analysis analytic behavior, the Riemann sphere, geometric constructions, and presents (as a review) a microcosm of the course. The Riemann manifold is introduced in Part II and is examined in terms of intuitive physical and topological technique in Part III. In Part IV the author shows how to define real functions on manifolds analogously with the algebraic and analytic points of view outlined here. The exposition returns in Part V to the use of a single complex variable z. As the text is richly endowed with problem material — 344 exercises — the book is perfect for self-study as well as classroom use.Harvey Cohn is well-known in the mathematics profession for his pedagogically superior texts, and the present book will be of great interest not only to pure and applied mathematicians, but also engineers and physicists. Dr. Cohn is currently Distinguished Professor of Mathematics at the City University of New York Graduate Center.

Conformal Field Theory consists of pedagogical lectures delivered at the Feza Gursey Institute, Istanbul, in the summer of 1998 on the following topics: Non-perturbative dynamics of 4-dimensional Field theories” (P. Argyres, Cornell University), 2-Dimensional QCD, Grassmanians and M(atrix) Models” (S. Rajeev, University of Rochester), Affine Kac-Moody algebras and CFT” (M. Walton, Lethbridge), CFT and Vertex Operator algebras” (T. Gannon, York), and Meromorphic CFT” (M. Gaberdiel, Cambridge). The book is suitable for advanced graduate students and researchers in theoretical particle or statistical physics as well as pure mathematicians.

Conformal field theory is an elegant and powerful theory in the field of high energy physics and statistics. In fact, it can be said to be one of the greatest achievements in the development of this field. Presented in two dimensions, this book is designed for students who already have a basic knowledge of quantum mechanics, field theory and general relativity. The main idea used throughout the book is that conformal symmetry causes both classical and quantum integrability. Instead of concentrating on the numerous applications of the theory, the author puts forward a discussion of the general methods of conformal field theory as a physical theory. Hence the book provides in a self-contained way the necessary knowledge and "conformal" intuition which underline the various applications of conformal field theory. It is aimed to assist students and professionals in the study of the theory from its first principles and in applying the methods in their own research. The first of its kind, this book promises to give a detailed and comprehensive insight into the workings of conformal field theory.

Filling an important gap in the literature, this comprehensive text develops conformal field theory from first principles. The treatment is self-contained, pedagogical, and exhaustive, and includes a great deal of background material on quantum field theory, statistical mechanics, Lie algebras and affine Lie algebras. The many exercises, with a wide spectrum of difficulty and subjects, complement and in many cases extend the text. The text is thus not only an excellent tool for classroom teaching but also for individual study. Intended primarily for graduate students and researchers in theoretical high-energy physics, mathematical physics, condensed matter theory, statistical physics, the book will also be of interest in other areas of theoretical physics and mathematics. It will prepare the reader for original research in this very active field of theoretical and mathematical physics.

The conformal predictions framework is a recent development in machine learning that can associate a reliable measure of confidence with a prediction in any real-world pattern recognition application, including risk-sensitive applications such as medical diagnosis, face recognition, and financial risk prediction. Conformal Predictions for Reliable Machine Learning: Theory, Adaptations and Applications captures the basic theory of the framework, demonstrates how to apply it to real-world problems, and presents several adaptations, including active learning, change detection, and anomaly detection. As practitioners and researchers around the world apply and adapt the framework, this edited volume brings together these bodies of work, providing a springboard for further research as well as a handbook for application in real-world problems.Understand the theoretical foundations of this important framework that can provide a reliable measure of confidence with predictions in machine learningBe able to apply this framework to real-world problems in different machine learning settings, including classification, regression, and clusteringLearn effective ways of adapting the framework to newer problem settings, such as active learning, model selection, or change detection

Building on the foundations laid in his Introduction to Superstrings and M Theory, Professor Kaku discusses such topics as the classification of conformal string theories, knot theory, the Yang-Baxter relation, quantum groups, and the insights into 11-dimensional strings recently obtained from M-theory. New chapters discuss such topics as Seiberg-Witten theory, M theory and duality, and D-branes. Throughout, the author conveys the vitality of the current research and places readers at its forefront. Several chapters reviewing the fundamentals of string theory, making the presentation of the material self-contained while keeping overlap with the earlier book to a minimum.