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Clifford, Algèbres de. --- Nombres complexes. --- Clifford algebras. --- Numbers, Complex. --- Clifford, Algèbres de.
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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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Algèbre multilinéaire. --- Multilinear algebra --- Clifford algebras --- Clifford, Algèbres de. --- Algèbre multilinéaire. --- Clifford, Algèbres de.
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"The idea behind this book is that geometric algebra is such a fundamental and potentially useful subject that every mathematician should have at least a nodding acquaintance withit. In the words of John Snygg, "the fact taht Clifford algebra (otherwise known as "geometric algebra") is not deeply embedded in our current curriculum is an accident of history." The aim of this book is to provide the reader with evidence with which to judge for himself or herself the validity of this claim." [Back cover]
Algebras, Linear. --- Clifford algebras. --- Geometry, Differential. --- Algèbre linéaire. --- Clifford, Algèbres de. --- Géométrie différentielle. --- Algèbre linéaire. --- Clifford, Algèbres de. --- Géométrie différentielle.
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Group theory --- Ordered algebraic structures --- Clifford algebras. --- Commutative rings. --- Forms, Quadratic. --- Azumaya, Algèbres d'. --- Azumaya algebras. --- Brauer, Groupe de. --- Brauer groups. --- Algèbres associatives --- Algèbres commutatives --- Clifford, Algèbres de --- Formes quadratiques --- Algèbres associatives --- Algèbres commutatives --- Azumaya, Algèbres d'. --- Clifford, Algèbres de
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In this book, Professor Lounesto offers a unique introduction to Clifford algebras and spinors. The beginning chapters could be read by undergraduates; vectors, complex numbers and quaternions are introduced with an eye on Clifford algebras. The next chapters will also interest physicists, and include treatments of the quantum mechanics of the electron, electromagnetism and special relativity with a flavour of Clifford algebras. A new classification of spinors is introduced, one based on bilinear covariants of physical observables. This reveals a new class of spinors, residing between the Weyl, Majorana and Dirac spinors. Scalar products of spinors are classified by involutory anti-automorphisms of Clifford algebras. This leads to the chessboard of automorphism groups of scalar products of spinors. On the algebraic side, Brauer/Wall groups and Witt rings are discussed, and on the analytic, Cauchy's integral formula is generalised to higher dimensions.
Clifford algebras. --- Spinor analysis. --- Algèbre extérieure --- Ausdehnungslehre --- Algèbre extérieure. --- Clifford algebras --- Clifford, Algèbres de --- Algèbre linéaire --- Algebras, Linear --- Algèbre multilinéaire. --- Multilinear algebra --- Algèbre linéaire. --- Clifford, Algèbres de. --- Algèbre multilinéaire --- Multilinear algebra. --- Algèbre linéaire --- Algebras, Linear. --- Clifford, Algèbres de --- Algèbre extérieure --- Ausdehnungslehre. --- Nombres hypercomplexes --- Quaternions
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Mathematical analysis --- Clifford algebras --- Clifford, Algèbres de --- Geometric algebras --- Clifford algebras. --- Clifford, Algèbres de --- Calculus --- 517.986 --- Algebras, Linear --- Analysis (Mathematics) --- Fluxions (Mathematics) --- Infinitesimal calculus --- Limits (Mathematics) --- Functions --- Geometry, Infinitesimal --- 517.986 Topological algebras. Theory of infinite-dimensional representations --- Topological algebras. Theory of infinite-dimensional representations --- Algebra --- Calculus. --- Calcul infinitésimal --- Algèbre linéaire --- Algèbre multilinéaire. --- Multilinear algebra --- Algèbre linéaire. --- Géometrie différentielle --- Algèbre linéaire. --- Algèbre multilinéaire. --- Géometrie différentielle
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517.982 --- Clifford algebras --- Harmonic functions --- Holomorphic functions --- Theory of distributions (Functional analysis) --- Distribution (Functional analysis) --- Distributions, Theory of (Functional analysis) --- Functions, Generalized --- Generalized functions --- Functional analysis --- Functions, Holomorphic --- Functions of several complex variables --- Functions, Harmonic --- Laplace's equations --- Bessel functions --- Differential equations, Partial --- Fourier series --- Harmonic analysis --- Lamé's functions --- Spherical harmonics --- Toroidal harmonics --- Geometric algebras --- Algebras, Linear --- Linear spaces with topology and order or other structures --- Clifford algebras. --- Harmonic functions. --- Holomorphic functions. --- Theory of distributions (Functional analysis). --- 517.982 Linear spaces with topology and order or other structures --- Sphere --- Analyse harmonique (mathématiques) --- Sphère --- Sphère. --- Clifford, Algèbres de --- Clifford, Algèbres de --- Analyse harmonique (mathématiques) --- Sphère.
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This book offers a systematic and comprehensive presentation of the concepts of a spin manifold, spinor fields, Dirac operators, and A-genera, which, over the last two decades, have come to play a significant role in many areas of modern mathematics. Since the deeper applications of these ideas require various general forms of the Atiyah-Singer Index Theorem, the theorems and their proofs, together with all prerequisite material, are examined here in detail. The exposition is richly embroidered with examples and applications to a wide spectrum of problems in differential geometry, topology, and mathematical physics. The authors consistently use Clifford algebras and their representations in this exposition. Clifford multiplication and Dirac operator identities are even used in place of the standard tensor calculus. This unique approach unifies all the standard elliptic operators in geometry and brings fresh insights into curvature calculations. The fundamental relationships of Clifford modules to such topics as the theory of Lie groups, K-theory, KR-theory, and Bott Periodicity also receive careful consideration. A special feature of this book is the development of the theory of Cl-linear elliptic operators and the associated index theorem, which connects certain subtle spin-corbordism invariants to classical questions in geometry and has led to some of the most profound relations known between the curvature and topology of manifolds.
Algebres de Clifford --- Clifford [Algebra's van ] --- Clifford algebras --- Fysica [Mathematische ] --- Fysica [Wiskundige ] --- Mathematische fysica --- Physics -- Mathematics --- Physics [Mathematical ] --- Physique -- Mathématiques --- Physique -- Méthodes mathématiques --- Wiskundige fysica --- Clifford, Algèbres de --- Spin, Nuclear --- Geometric algebras --- Clifford algebras. --- Spin geometry. --- Clifford, Algèbres de --- Spin geometry --- 514.76 --- Algebras, Linear --- 514.76 Geometry of differentiable manifolds and of their submanifolds --- Geometry of differentiable manifolds and of their submanifolds --- Global differential geometry --- Geometry --- Mathematical physics --- Topology --- Nuclear spin --- -Mathematics --- Géométrie --- Physique mathématique --- Spin nucléaire --- Topologie --- Mathematics --- Mathématiques --- Algebraic theory. --- Atiyah–Singer index theorem. --- Automorphism. --- Betti number. --- Binary icosahedral group. --- Binary octahedral group. --- Bundle metric. --- C*-algebra. --- Calabi conjecture. --- Calabi–Yau manifold. --- Cartesian product. --- Classification theorem. --- Clifford algebra. --- Cobordism. --- Cohomology ring. --- Cohomology. --- Cokernel. --- Complete metric space. --- Complex manifold. --- Complex vector bundle. --- Complexification (Lie group). --- Covering space. --- Diffeomorphism. --- Differential topology. --- Dimension (vector space). --- Dimension. --- Dirac operator. --- Disk (mathematics). --- Dolbeault cohomology. --- Einstein field equations. --- Elliptic operator. --- Equivariant K-theory. --- Exterior algebra. --- Fiber bundle. --- Fixed-point theorem. --- Fourier inversion theorem. --- Fundamental group. --- Gauge theory. --- Geometry. --- Hilbert scheme. --- Holonomy. --- Homotopy sphere. --- Homotopy. --- Hyperbolic manifold. --- Induced homomorphism. --- Intersection form (4-manifold). --- Isomorphism class. --- J-invariant. --- K-theory. --- Kähler manifold. --- Laplace operator. --- Lie algebra. --- Lorentz covariance. --- Lorentz group. --- Manifold. --- Mathematical induction. --- Metric connection. --- Minkowski space. --- Module (mathematics). --- N-sphere. --- Operator (physics). --- Orthonormal basis. --- Principal bundle. --- Projective space. --- Pseudo-Riemannian manifold. --- Pseudo-differential operator. --- Quadratic form. --- Quaternion. --- Quaternionic projective space. --- Ricci curvature. --- Riemann curvature tensor. --- Riemannian geometry. --- Riemannian manifold. --- Ring homomorphism. --- Scalar curvature. --- Scalar multiplication. --- Sign (mathematics). --- Space form. --- Sphere theorem. --- Spin representation. --- Spin structure. --- Spinor bundle. --- Spinor field. --- Spinor. --- Subgroup. --- Support (mathematics). --- Symplectic geometry. --- Tangent bundle. --- Tangent space. --- Tensor calculus. --- Tensor product. --- Theorem. --- Topology. --- Unit disk. --- Unit sphere. --- Variable (mathematics). --- Vector bundle. --- Vector field. --- Vector space. --- Volume form. --- Nuclear spin - - Mathematics --- -Clifford algebras.
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