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Harmonic maps are generalisations of the concept of geodesics. They encompass many fundamental examples in differential geometry and have recently become of widespread use in many areas of mathematics and mathematical physics. This is an accessible introduction to some of the fundamental connections between differential geometry, Lie groups, and integrable Hamiltonian systems. The specific goal of the book is to show how the theory of loop groups can be used to study harmonic maps. By concentrating on the main ideas and examples, the author leads up to topics of current research. The book is suitable for students who are beginning to study manifolds and Lie groups, and should be of interest both to mathematicians and to theoretical physicists.

Harmonic maps. --- Loops (Group theory) --- Differential equations. --- 517.91 Differential equations --- Differential equations --- Loop groups --- Group theory --- Maps, Harmonic --- Mappings (Mathematics)

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Topological groups. Lie groups --- Hamiltonian systems --- Loops (Group theory) --- Loop groups --- Group theory --- Hamiltonian dynamical systems --- Systems, Hamiltonian --- Differentiable dynamical systems --- Lacets (théorie des groupes) --- Systèmes hamiltoniens --- Systèmes hamiltoniens.

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Wilson lines (also known as gauge links or eikonal lines) can be introduced in any gauge field theory. Although the concept of the Wilson exponentials finds an enormously wide range of applications in a variety of branches of modern quantum field theory, from condensed matter and lattice simulations to quantum chromodynamics, high-energy effective theories and gravity, there are surprisingly few books or textbooks on the market which contain comprehensive pedagogical introduction and consecutive exposition of the subject. The objective of this book is to get the potential reader acquainted with theoretical and mathematical foundations of the concept of the Wilson loops in the context of modern quantum field theory, to teach him/her to perform independently some elementary calculations with Wilson lines, and to familiarize him/her with the recent development of the subject in different important areas of research. The target audience of the book consists of graduate and postgraduate students working in various areas of quantum field theory, as well as researchers from other fields.

Loops (Group theory) --- Quantum field theory --- Gauge fields (Physics) --- Fields, Gauge (Physics) --- Gage fields (Physics) --- Gauge theories (Physics) --- Field theory (Physics) --- Group theory --- Symmetry (Physics) --- Relativistic quantum field theory --- Quantum theory --- Relativity (Physics) --- Loop groups --- Mathematics.

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Automatic transformation of a sequential program into a parallel form is a subject that presents a great intellectual challenge and promises great practical rewards. There is a tremendous investment in existing sequential programs, and scientists and engineers continue to write their application programs in sequential languages (primarily in Fortran),but the demand for increasing speed is constant. The job of a restructuring compiler is to discover the dependence structure of a given program and transform the program in a way that is consistent with both that dependence structure and the characteristics of the given machine. Much attention in this field of research has been focused on the Fortran do loop. This is where one expects to find major chunks of computation that need to be performed repeatedly for different values of the index variable. Many loop transformations have been designed over the years, and several of them can be found in any parallelizing compiler currently in use in industry or at a university research facility. Loop Transformations for Restructuring Compilers: The Foundations provides a rigorous theory of loop transformations. The transformations are developed in a consistent mathematical framework using objects like directed graphs, matrices and linear equations. The algorithms that implement the transformations can then be precisely described in terms of certain abstract mathematical algorithms. The book provides the general mathematical background needed for loop transformations (including those basic mathematical algorithms), discusses data dependence, and introduces the major transformations. The next volume will build a detailed theory of loop transformations based on the material developed here. Loop Transformations for Restructuring Compilers: The Foundations presents a theory of loop transformations that is rigorous and yet reader-friendly.

Compilers (Computer programs). --- Loops (Group theory). --- Parallel processing (Electronic computers). --- Compilers (Computer programs) --- Loops (Group theory) --- Parallel processing (Electronic computers) --- High performance computing --- Multiprocessors --- Parallel programming (Computer science) --- Supercomputers --- Loop groups --- Group theory --- Compiling programs (Computer programs) --- Computer programs --- Programming software --- Systems software --- Computer science. --- Microprocessors. --- Computer Science. --- Processor Architectures. --- Computer Science, general.

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This volume provides a self-contained introduction to applications of loop representations, and the related topic of knot theory, in particle physics and quantum gravity. These topics are of considerable interest because they provide a unified arena for the study of the gauge invariant quantization of Yang-Mills theories and gravity, and suggest a promising approach to the eventual unification of the four fundamental forces. The book begins with a detailed review of loop representation theory and then describes loop representations in Maxwell theory, Yang-Mills theories as well as lattice techniques. Applications in quantum gravity are then discussed, with the following chapters considering knot theories, braid theories and extended loop representations in quantum gravity. A final chapter assesses the current status of the theory and points out possible directions for future research. First published in 1996, this title has been reissued as an Open Access publication on Cambridge Core.

Gauge fields (Physics) --- Loops (Group theory) --- Knot theory. --- Quantum field theory. --- Quantum gravity --- Mathematics. --- Gravity, Quantum --- General relativity (Physics) --- Gravitation --- Quantum theory --- Relativistic quantum field theory --- Field theory (Physics) --- Relativity (Physics) --- Knots (Topology) --- Low-dimensional topology --- Loop groups --- Group theory --- Fields, Gauge (Physics) --- Gage fields (Physics) --- Gauge theories (Physics) --- Symmetry (Physics)