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| geometric_algebra [2020/07/19 17:34] – [Articles] pbk | geometric_algebra [2020/09/07 19:44] – [Articles] pbk |
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| * [[https://en.wikipedia.org/wiki/Vector_space|Vector space]] | * [[https://en.wikipedia.org/wiki/Vector_space|Vector space]] |
| * [[https://en.wikipedia.org/wiki/Quaternion|Quaternion]] | * [[https://en.wikipedia.org/wiki/Quaternion|Quaternion]] |
| | * [[https://en.wikipedia.org/wiki/Biquaternion|Biquaternion]] |
| * [[https://en.wikipedia.org/wiki/Octonion|Octonion]] | * [[https://en.wikipedia.org/wiki/Octonion|Octonion]] |
| * [[https://en.wikipedia.org/wiki/Spinor|Spinor]] | * [[https://en.wikipedia.org/wiki/Spinor|Spinor]] |
| * [[http://www.siue.edu/~sstaple/index_files/research.html|CliffMath]] - Clifford algebra computations, including zeon, sym-Clifford, and idem-Clifford subalgebras, //George Stacey Staples//. | * [[http://www.siue.edu/~sstaple/index_files/research.html|CliffMath]] - Clifford algebra computations, including zeon, sym-Clifford, and idem-Clifford subalgebras, //George Stacey Staples//. |
| * [[https://github.com/Prograf-UFF/TbGAL|TbGAL]] - Tensor-Based Geometric Algebra C++/Python Library, //Eduardo Vera Sousa, Leandro A. F. Fernandes//. | * [[https://github.com/Prograf-UFF/TbGAL|TbGAL]] - Tensor-Based Geometric Algebra C++/Python Library, //Eduardo Vera Sousa, Leandro A. F. Fernandes//. |
| | * [[https://github.com/vincentnozick/garamon|Garamon Generator]] - Geometric Algebra Recursive and Adaptative Monster is a generator of C++ libraries dedicated to Geometric Algebra, //Vincent Nozick, Stephane Breuils//. |
| ===== Articles ===== | ===== Articles ===== |
| * [[http://geocalc.clas.asu.edu/pdf/OerstedMedalLecture.pdf|Oersted Medal Lecture 2002: Reforming the Mathematical Language of Physics]] (2002) - //David Hestenes// | * [[http://geocalc.clas.asu.edu/pdf/OerstedMedalLecture.pdf|Oersted Medal Lecture 2002: Reforming the Mathematical Language of Physics]] (2002) - //David Hestenes// |
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| * [[https://www.researchgate.net/publication/264423339_An_invitation_to_Clifford_Analysis|Una Invitación al Análisis de Clifford]] (2003) - //Richard Delanghe, Juan Bory-Reyes// | * [[https://www.researchgate.net/publication/264423339_An_invitation_to_Clifford_Analysis|Una Invitación al Análisis de Clifford]] (2003) - //Richard Delanghe, Juan Bory-Reyes// |
| Una panorámica de los tópicos principales y herramientas básicas del Análisis de Clifford se presenta en este artículo, al mismo tiempo, las principales fórmulas integrales relacionadas con la integral tipo Cauchy -- y su versión singular -- son analizadas en un contexto multidimensional, con el uso de las técnicas de álgebras de Clifford. Se incluyen también algunas notas históricas sobre el desarrollo de este campo de investigación. | Una panorámica de los tópicos principales y herramientas básicas del Análisis de Clifford se presenta en este artículo, al mismo tiempo, las principales fórmulas integrales relacionadas con la integral tipo Cauchy --- y su versión singular --- son analizadas en un contexto multidimensional, con el uso de las técnicas de álgebras de Clifford. Se incluyen también algunas notas históricas sobre el desarrollo de este campo de investigación. |
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| * [[http://downloads.hindawi.com/journals/abb/2007/502679.pdf|Surface Approximation using Growing Self-Organizing Nets and Gradient Information]] (2007) - //Jorge Rivera-Rovelo, Eduardo Bayro-Corrochano// | * [[http://downloads.hindawi.com/journals/abb/2007/502679.pdf|Surface Approximation using Growing Self-Organizing Nets and Gradient Information]] (2007) - //Jorge Rivera-Rovelo, Eduardo Bayro-Corrochano// |
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| * [[http://www.cs.ox.ac.uk/people/david.reutter/AtiyahSinger_Essay.pdf|The Heat Equation and the Atiyah-Singer Index Theorem]] (2015) - //David Reutter// | * [[http://www.cs.ox.ac.uk/people/david.reutter/AtiyahSinger_Essay.pdf|The Heat Equation and the Atiyah-Singer Index Theorem]] (2015) - //David Reutter// |
| The Atiyah-Singer index theorem is a milestone of twentieth century mathematics. Roughly speaking, it relates a global analytical datum of a manifold -- the number of solutions of a certain linear PDE -- to an integral of local topological expressions over this manifold. The index theorem provided a link between analysis, geometry and topology, paving the way for many further applications along these lines. | The Atiyah-Singer index theorem is a milestone of twentieth century mathematics. Roughly speaking, it relates a global analytical datum of a manifold --- the number of solutions of a certain linear PDE --- to an integral of local topological expressions over this manifold. The index theorem provided a link between analysis, geometry and topology, paving the way for many further applications along these lines. |
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| * [[http://www.siue.edu/~sstaple/index_files/CODecompAccepted2015.pdf|Clifford algebra decompositions of conformal orthogonal group elements]] (2015) - //G. Stacey Staples, David Wylie// | * [[http://www.siue.edu/~sstaple/index_files/CODecompAccepted2015.pdf|Clifford algebra decompositions of conformal orthogonal group elements]] (2015) - //G. Stacey Staples, David Wylie// |
| * [[https://arxiv.org/pdf/1911.08658|Spinors of real type as polyforms and the generalized Killing equation]] (2019) - //Vicente Cortes, Calin Lazaroiu, C. S. Shahbazi// | * [[https://arxiv.org/pdf/1911.08658|Spinors of real type as polyforms and the generalized Killing equation]] (2019) - //Vicente Cortes, Calin Lazaroiu, C. S. Shahbazi// |
| We develop a new framework for the study of generalized Killing spinors, where generalized Killing spinor equations, possibly with constraints, can be formulated equivalently as systems of partial differential equations for a polyform satisfying algebraic relations in the Kähler-Atiyah bundle constructed by quantizing the exterior algebra bundle of the underlying manifold. At the core of this framework lies the characterization, which we develop in detail, of the image of the spinor squaring map of an irreducible Clifford module Σ of real type as a real algebraic variety in the Kähler-Atiyah algebra, which gives necessary and sufficient conditions for a polyform to be the square of a real spinor. We apply these results to Lorentzian four-manifolds, obtaining a new description of a real spinor on such a manifold through a certain distribution of parabolic 2-planes in its cotangent bundle. We use this result to give global characterizations of real Killing spinors on Lorentzian four-manifolds and of four-dimensional supersymmetric configurations of heterotic supergravity. In particular, we find new families of Einstein and non-Einstein four-dimensional Lorentzian metrics admitting real Killing spinors, some of which are deformations of the metric of AdS_4 space-time. | We develop a new framework for the study of generalized Killing spinors, where generalized Killing spinor equations, possibly with constraints, can be formulated equivalently as systems of partial differential equations for a polyform satisfying algebraic relations in the Kähler-Atiyah bundle constructed by quantizing the exterior algebra bundle of the underlying manifold. At the core of this framework lies the characterization, which we develop in detail, of the image of the spinor squaring map of an irreducible Clifford module Σ of real type as a real algebraic variety in the Kähler-Atiyah algebra, which gives necessary and sufficient conditions for a polyform to be the square of a real spinor. We apply these results to Lorentzian four-manifolds, obtaining a new description of a real spinor on such a manifold through a certain distribution of parabolic 2-planes in its cotangent bundle. We use this result to give global characterizations of real Killing spinors on Lorentzian four-manifolds and of four-dimensional supersymmetric configurations of heterotic supergravity. In particular, we find new families of Einstein and non-Einstein four-dimensional Lorentzian metrics admitting real Killing spinors, some of which are deformations of the metric of AdS_4 space-time. |
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| | * [[https://link.springer.com/content/pdf/10.1007/s00006-019-0987-7.pdf|Garamon: A Geometric Algebra Library Generator]] (2019) - //Stephane Breuils, Vincent Nozick, Laurent Fuchs// |
| | This paper presents both a recursive scheme to perform Geometric Algebra operations over a prefix tree, and Garamon, a C++ library generator implementing these recursive operations. While for low dimension vector spaces, precomputing all the Geometric Algebra products is an efficient strategy, it fails for higher dimensions where the operation should be computed at run time. This paper describes how a prefix tree can be a support for a recursive formulation of Geometric Algebra operations. This recursive approach presents a much better complexity than the usual run time methods. This paper also details how a prefix tree can represent efficiently the dual of a multivector. These results constitute the foundations for Garamon, a C++ library generator synthesizing efficient C++/Python libraries implementing Geometric Algebra in both low and higher dimensions, with any arbitrary metric. Garamon takes advantage of the prefix tree formulation to implement Geometric Algebra operations on high dimensions hardly accessible with state-of-the-art software implementations. |
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| * [[https://arxiv.org/pdf/1911.07145|Geometric Manifolds Part I: The Directional Derivative of Scalar, Vector, Multivector, and Tensor Fields]] (2020) - //Joseph C. Schindler// | * [[https://arxiv.org/pdf/1911.07145|Geometric Manifolds Part I: The Directional Derivative of Scalar, Vector, Multivector, and Tensor Fields]] (2020) - //Joseph C. Schindler// |
