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After a classical presentation of quadratic mappings and Clifford algebras over arbitrary rings (commutative, associative, with unit), other topics involve more original methods: interior multiplications allow an effective treatment of deformations of Clifford algebras; the relations between automorphisms of quadratic forms and Clifford algebras are based on the concept of the Lipschitz monoid, from which several groups are derived; and the Cartan-Chevalley theory of hyperbolic spaces becomes much more general, precise and effective.

Clifford algebras. --- Forms, Quadratic. --- Quadratic forms --- Diophantine analysis --- Forms, Binary --- Number theory --- Geometric algebras --- Algebras, Linear --- Algebra. --- Mathematics --- Mathematical analysis

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Geometric algebra is a powerful mathematical language with applications across a range of subjects in physics and engineering. This book is a complete guide to the current state of the subject with early chapters providing a self-contained introduction to geometric algebra. Topics covered include new techniques for handling rotations in arbitrary dimensions, and the links between rotations, bivectors and the structure of the Lie groups. Following chapters extend the concept of a complex analytic function theory to arbitrary dimensions, with applications in quantum theory and electromagnetism. Later chapters cover advanced topics such as non-Euclidean geometry, quantum entanglement, and gauge theories. Applications such as black holes and cosmic strings are also explored. It can be used as a graduate text for courses on the physical applications of geometric algebra and is also suitable for researchers working in the fields of relativity and quantum theory.

Clifford algebras. --- Mathematical physics. --- Physical mathematics --- Physics --- Geometric algebras --- Algebras, Linear --- Mathematics --- Clifford, Algèbres de --- Physique mathématique

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The author defines “Geometric Algebra Computing” as the geometrically intuitive development of algorithms using geometric algebra with a focus on their efficient implementation, and the goal of this book is to lay the foundations for the widespread use of geometric algebra as a powerful, intuitive mathematical language for engineering applications in academia and industry. The related technology is driven by the invention of conformal geometric algebra as a 5D extension of the 4D projective geometric algebra and by the recent progress in parallel processing, and with the specific conformal geometric algebra there is a growing community in recent years applying geometric algebra to applications in computer vision, computer graphics, and robotics. This book is organized into three parts: in Part I the author focuses on the mathematical foundations; in Part II he explains the interactive handling of geometric algebra; and in Part III he deals with computing technology for high-performance implementations based on geometric algebra as a domain-specific language in standard programming languages such as C++ and OpenCL. The book is written in a tutorial style and readers should gain experience with the associated freely available software packages and applications. The book is suitable for students, engineers, and researchers in computer science, computational engineering, and mathematics.

Clifford algebras -- Data processing. --- Clifford algebras. --- Clifford algebras --- Electrical & Computer Engineering --- Engineering & Applied Sciences --- Technology - General --- Electrical Engineering --- Applied Physics --- Data processing --- Geometry --- Data processing. --- Geometric algebras --- Computer science. --- Computer graphics. --- Geometry. --- Applied mathematics. --- Engineering mathematics. --- Computer Science. --- Computer Imaging, Vision, Pattern Recognition and Graphics. --- Appl.Mathematics/Computational Methods of Engineering. --- Algebras, Linear --- Computer vision. --- Mathematical and Computational Engineering. --- Engineering --- Engineering analysis --- Mathematical analysis --- Mathematics --- Euclid's Elements --- Machine vision --- Vision, Computer --- Artificial intelligence --- Image processing --- Pattern recognition systems --- Optical data processing. --- Optical computing --- Visual data processing --- Bionics --- Electronic data processing --- Integrated optics --- Photonics --- Computers --- Optical equipment

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Differential geometry is the study of curvature and calculus of curves and surfaces.  Because of an historical accident, the Geometric Algebra devised by William Kingdom Clifford (1845–1879) has been overlooked in favor of the more complicated and less powerful formalism of differential forms and tangent vectors to deal with differential geometry.  Fortuitously a student who has completed an undergraduate course in linear algebra is better prepared to deal with the intricacies of Clifford algebra than with the formalism currently used.  Clifford algebra enables one to demonstrate a close relation between curvature and certain rotations.  This is an advantage both conceptually and computationally—particularly in higher dimensions. Key features and topics include: * a unique undergraduate-level approach to differential geometry; * brief biographies of historically relevant mathematicians and physicists; * some aspects of special and general relativity accessible to undergraduates with no knowledge of Newtonian physics; * chapter-by-chapter exercises. The textbook will also serve as a useful classroom resource primarily for undergraduates as well as beginning-level graduate students; researchers in the algebra and physics communities may also find the book useful as a self-study guide.

Clifford algebras. --- Geometry, Differential. --- Geometry. --- Clifford algebras --- Geometry, Differential --- Mathematics --- Physical Sciences & Mathematics --- Algebra --- Geometry --- Differential geometry --- Geometric algebras --- Mathematics. --- Algebra. --- Applied mathematics. --- Engineering mathematics. --- Differential geometry. --- Mathematical physics. --- Physics. --- Differential Geometry. --- Mathematical Methods in Physics. --- Mathematical Physics. --- Mathematics, general. --- Applications of Mathematics. --- Algebras, Linear --- Global differential geometry. --- Mathematical analysis --- Math --- Science --- Physical mathematics --- Physics --- Natural philosophy --- Philosophy, Natural --- Physical sciences --- Dynamics --- Engineering --- Engineering analysis

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The purpose of this volume is to bring forward recent trends of research in hypercomplex analysis. The list of contributors includes first rate mathematicians and young researchers working on several different aspects in quaternionic and Clifford analysis. Besides original research papers, there are papers providing the state-of-the-art of a specific topic, sometimes containing interdisciplinary fields. The intended audience includes researchers, PhD students, postgraduate students who are interested in the field and in possible connection between hypercomplex analysis and other disciplines, i.

Functions of complex variables. --- Mathematical analysis. --- Clifford algebras. --- Geometric algebras --- Algebras, Linear --- 517.1 Mathematical analysis --- Mathematical analysis --- Complex variables --- Elliptic functions --- Functions of real variables --- Clifford algebras --- Analytic functions --- Differential equations, partial. --- Mathematical physics. --- Harmonic analysis. --- Numerical analysis. --- Partial Differential Equations. --- Mathematical Methods in Physics. --- Abstract Harmonic Analysis. --- Numerical Analysis. --- Analysis (Mathematics) --- Functions, Potential --- Potential functions --- Banach algebras --- Calculus --- Mathematics --- Bessel functions --- Fourier series --- Harmonic functions --- Time-series analysis --- Physical mathematics --- Physics --- Partial differential equations --- Partial differential equations. --- Physics. --- Natural philosophy --- Philosophy, Natural --- Physical sciences --- Dynamics