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| geometric_algebra [2020/07/10 17:18] – [Articles] pbk | geometric_algebra [2020/07/19 17:23] – [Articles] pbk |
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| * [[https://en.wikipedia.org/wiki/Multivector|Multivector]] | * [[https://en.wikipedia.org/wiki/Multivector|Multivector]] |
| * [[https://en.wikipedia.org/wiki/Paravector|Paravector]] | * [[https://en.wikipedia.org/wiki/Paravector|Paravector]] |
| | * [[https://en.wikipedia.org/wiki/Outermorphism|Outermorphism]] |
| * [[https://en.wikipedia.org/wiki/Projective_geometry|Projective Geometry]] | * [[https://en.wikipedia.org/wiki/Projective_geometry|Projective Geometry]] |
| * [[https://en.wikipedia.org/wiki/Pl%C3%BCcker_coordinates|Plücker coordinates]] | * [[https://en.wikipedia.org/wiki/Pl%C3%BCcker_coordinates|Plücker coordinates]] |
| * [[http://cs.unitbv.ro/~acami/what_kc_can_do.pdf|What the Kähler Calculus can do that other calculi cannot]] (2016) - //Jose G. Vargas// | * [[http://cs.unitbv.ro/~acami/what_kc_can_do.pdf|What the Kähler Calculus can do that other calculi cannot]] (2016) - //Jose G. Vargas// |
| Underlied by Clifford algebra of differential forms --- like tangent Clifford algebra underlies the geometric calculus --- it brings about a fresh new view of quantum mechanics. This view arises, almost without effort, from the equation which is in Kähler Calculus what the Dirac equation is in traditional quantum mechanics. | Underlied by Clifford algebra of differential forms --- like tangent Clifford algebra underlies the geometric calculus --- it brings about a fresh new view of quantum mechanics. This view arises, almost without effort, from the equation which is in Kähler Calculus what the Dirac equation is in traditional quantum mechanics. |
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| | * [[https://www.mit.edu/~fengt/282C.pdf|The Atiyah–Singer index theorem]] (2015) - //Dan Berwick-Evans, via Tony Feng// |
| | Lecture notes about the Atiyah-Singer index theorem. |
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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://vixra.org/pdf/1911.0127v1.pdf|Robust Quaternion Estimation with Geometric Algebra]] (2019) - //Mauricio C. Lopez// | * [[https://vixra.org/pdf/1911.0127v1.pdf|Robust Quaternion Estimation with Geometric Algebra]] (2019) - //Mauricio C. Lopez// |
| Robust methods for finding the best rotation aligning two sets of corresponding vectors are formulated in the linear algebra framework, using tools like the SVD for polar decomposition or QR for finding eigenvectors. Those are well established numerical algorithms which on the other hand are iterative and computationally expensive. Recently, closed form solutions has been proposed in the quaternion’s framework, those methods are fast but they have singularities i.e., they completely fail on certain input data. In this paper we propose a robust attitude estimator based on a formulation of the problem in Geometric Algebra. We find the optimal eigen-quaternion in closed form with high accuracy and with competitive performance respect to the fastest methods reported in literature. | Robust methods for finding the best rotation aligning two sets of corresponding vectors are formulated in the linear algebra framework, using tools like the SVD for polar decomposition or QR for finding eigenvectors. Those are well established numerical algorithms which on the other hand are iterative and computationally expensive. Recently, closed form solutions has been proposed in the quaternion’s framework, those methods are fast but they have singularities i.e., they completely fail on certain input data. In this paper we propose a robust attitude estimator based on a formulation of the problem in Geometric Algebra. We find the optimal eigen-quaternion in closed form with high accuracy and with competitive performance respect to the fastest methods reported in literature. |
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| | * [[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. |
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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// |
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| * [[https://link.springer.com/content/pdf/10.1007/s00006-020-1046-0.pdf|A 1d Up Approach to Conformal Geometric Algebra]] (2020) - //Anthony N. Lasenby// | * [[https://link.springer.com/content/pdf/10.1007/s00006-020-1046-0.pdf|A 1d Up Approach to Conformal Geometric Algebra]] (2020) - //Anthony N. Lasenby// |
| We discuss an alternative approach to the conformal geometric algebra (CGA) in which just a single extra dimension is necessary, as compared to the two normally used. This is made possible by working in a constant curvature background space, rather than the usual Euclidean space. A possible benefit, which is explored here, is that it is possible to define cost functions for geometric object matching in computer vision that are fully covariant, in particular invariant under both rotations and translations, unlike the cost functions which have been used in CGA so far. An algorithm is given for application of this method to the problem of matching sets of lines, which replaces the standard matrix singular value decomposition, by computations wholly in Geometric Algebra terms, and which may itself be of interest in more general settings. Secondly, we consider a further perhaps surprising application of the 1d up approach, which is to the context of a recent paper by Joy Christian published by the Royal Society, which has made strong claims about Bell's Theorem in quantum mechanics, and its relation to the sphere S^7 and the exceptional group E_8, and proposed a new associative version of the division algebra normally thought to require the octonians. We show that what is being discussed by Christian is mathematically the same as our 1d up approach to 3d geometry, but that after the removal of some incorrect mathematical assertions, the results he proves in the first part of the paper, and bases the application to Bell's Theorem on, amount to no more than the statement that the combination of two rotors from the Clifford Algebra Cl(4, 0) is also a rotor. | We discuss an alternative approach to the conformal geometric algebra (CGA) in which just a single extra dimension is necessary, as compared to the two normally used. This is made possible by working in a constant curvature background space, rather than the usual Euclidean space. A possible benefit, which is explored here, is that it is possible to define cost functions for geometric object matching in computer vision that are fully covariant, in particular invariant under both rotations and translations, unlike the cost functions which have been used in CGA so far. An algorithm is given for application of this method to the problem of matching sets of lines, which replaces the standard matrix singular value decomposition, by computations wholly in Geometric Algebra terms, and which may itself be of interest in more general settings. Secondly, we consider a further perhaps surprising application of the 1d up approach, which is to the context of a recent paper by Joy Christian published by the Royal Society, which has made strong claims about Bell's Theorem in quantum mechanics, and its relation to the sphere S^7 and the exceptional group E_8, and proposed a new associative version of the division algebra normally thought to require the octonians. |
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| * [[https://arxiv.org/pdf/2004.06655|Dimensional scaffolding of electromagnetism using geometric algebra]] (2020) - //Xabier Prado Orbán, Jorge Mira// | * [[https://arxiv.org/pdf/2004.06655|Dimensional scaffolding of electromagnetism using geometric algebra]] (2020) - //Xabier Prado Orbán, Jorge Mira// |
