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geometric_algebra [2020/04/23 07:33] – [Articles] pbkgeometric_algebra [2020/04/23 08:09] – [Articles] pbk
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 Recently suggested scheme of quantum computing uses g-qubit states as circular polarizations from the solution of Maxwell equations in terms of geometric algebra, along with clear definition of a complex plane as bivector in three dimensions. Here all the details of receiving the solution, and its polarization transformations are analyzed. The results can particularly be applied to the problems of quantum computing and quantum cryptography. The suggested formalism replaces conventional quantum mechanics states as objects constructed in complex vector Hilbert space framework by geometrically feasible framework of multivectors. Recently suggested scheme of quantum computing uses g-qubit states as circular polarizations from the solution of Maxwell equations in terms of geometric algebra, along with clear definition of a complex plane as bivector in three dimensions. Here all the details of receiving the solution, and its polarization transformations are analyzed. The results can particularly be applied to the problems of quantum computing and quantum cryptography. The suggested formalism replaces conventional quantum mechanics states as objects constructed in complex vector Hilbert space framework by geometrically feasible framework of multivectors.
  
-  * [[https://arxiv.org/ftp/arxiv/papers/1807/1807.08603.pdf|State/observable interactions using basic geometric algebra solutions of the Maxwell equation]] (2018) - //Alexander Soiguine//+  * [[https://arxiv.org/pdf/1807.08603|State/observable interactions using basic geometric algebra solutions of the Maxwell equation]] (2018) - //Alexander Soiguine//
 Maxwell equation in geometric algebra formalism with equally weighted basic solutions is subjected to continuously acting Clifford translation. The received states, operators acting on observables, are analyzed with different values of the Clifford translation time factor and through the observable measurement results. Maxwell equation in geometric algebra formalism with equally weighted basic solutions is subjected to continuously acting Clifford translation. The received states, operators acting on observables, are analyzed with different values of the Clifford translation time factor and through the observable measurement results.
  
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   * [[https://pastel.archives-ouvertes.fr/tel-02085820/document|Algorithmic structure for geometric algebra operators and application to quadric surfaces]] (2018) - //Stephane Breuils//   * [[https://pastel.archives-ouvertes.fr/tel-02085820/document|Algorithmic structure for geometric algebra operators and application to quadric surfaces]] (2018) - //Stephane Breuils//
 Geometric Algebra is considered as a very intuitive tool to deal with geometric problems and it appears to be increasingly efficient and useful to deal with computer graphics problems. The Conformal Geometric Algebra includes circles, spheres, planes and lines as algebraic objects, and intersections between these objects are also algebraic objects. More complex objects such as conics, quadric surfaces can also be expressed and be manipulated using an extension of the conformal Geometric Algebra. However due to the high dimension of their representations in Geometric Algebra, implementations of Geometric Algebra that are currently available do not allow efficient realizations of these objects. In this thesis, we first present a Geometric Algebra implementation dedicated for both low and high dimensions. Geometric Algebra is considered as a very intuitive tool to deal with geometric problems and it appears to be increasingly efficient and useful to deal with computer graphics problems. The Conformal Geometric Algebra includes circles, spheres, planes and lines as algebraic objects, and intersections between these objects are also algebraic objects. More complex objects such as conics, quadric surfaces can also be expressed and be manipulated using an extension of the conformal Geometric Algebra. However due to the high dimension of their representations in Geometric Algebra, implementations of Geometric Algebra that are currently available do not allow efficient realizations of these objects. In this thesis, we first present a Geometric Algebra implementation dedicated for both low and high dimensions.
 +
 +  * [[https://arxiv.org/pdf/1809.09706|Notes on Plucker's relations in Geometric Algebra]] (2018) - //Garret Sobczyk//
 +Grassmannians are of fundamental importance in projective geometry, algebraic geometry, and representation theory. A vast literature has grown up utilizing using many different languages of higher mathematics, such as multilinear and tensor algebra, matroid theory, and Lie groups and Lie algebras. Here we explore the basic idea of the Plucker relations in Clifford's geometric algebra. We discover that the Plucker Relations can be fully characterized in terms of the geometric product. 
  
   * [[https://arxiv.org/pdf/1908.08110.pdf|On the Clifford Algebraic Description of the Geometry of a 3D Euclidean Space]] (2019) - //Jaime Vaz, Stephen Mann//   * [[https://arxiv.org/pdf/1908.08110.pdf|On the Clifford Algebraic Description of the Geometry of a 3D Euclidean Space]] (2019) - //Jaime Vaz, Stephen Mann//
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 In this paper, we address the stability of a broad class of discrete-time hypercomplex-valued Hopfield-type neural networks. To ensure the neural networks belonging to this class always settle down at a stationary state, we introduce novel hypercomplex number systems referred to as real-part associative hypercomplex number systems. Real-part associative hypercomplex number systems generalize the well-known Cayley-Dickson algebras and real Clifford algebras and include the systems of real numbers, complex numbers, dual numbers, hyperbolic numbers, quaternions, tessarines, and octonions as particular instances. Apart from the novel hypercomplex number systems, we introduce a family of hypercomplex-valued activation functions called B-projection functions. Broadly speaking, a B-projection function projects the activation potential onto the set of all possible states of a hypercomplex-valued neuron. Using the theory presented in this paper, we confirm the stability analysis of several discrete-time hypercomplex-valued Hopfield-type neural networks from the literature. Moreover, we introduce and provide the stability analysis of a general class of Hopfield-type neural networks on Cayley-Dickson algebras. In this paper, we address the stability of a broad class of discrete-time hypercomplex-valued Hopfield-type neural networks. To ensure the neural networks belonging to this class always settle down at a stationary state, we introduce novel hypercomplex number systems referred to as real-part associative hypercomplex number systems. Real-part associative hypercomplex number systems generalize the well-known Cayley-Dickson algebras and real Clifford algebras and include the systems of real numbers, complex numbers, dual numbers, hyperbolic numbers, quaternions, tessarines, and octonions as particular instances. Apart from the novel hypercomplex number systems, we introduce a family of hypercomplex-valued activation functions called B-projection functions. Broadly speaking, a B-projection function projects the activation potential onto the set of all possible states of a hypercomplex-valued neuron. Using the theory presented in this paper, we confirm the stability analysis of several discrete-time hypercomplex-valued Hopfield-type neural networks from the literature. Moreover, we introduce and provide the stability analysis of a general class of Hopfield-type neural networks on Cayley-Dickson algebras.
  
-   * [[https://arxiv.org/pdf/1901.05873.pdf|Projective geometric algebra: A new framework for doing euclidean geometry]] (2019) - //Charles G. Gunn//+  * [[https://arxiv.org/pdf/1903.02444|Efficient representation and manipulation of quadratic surfaces using Geometric Algebras]] (2019) - //Stéphane Breuils, Vincent Nozick, Laurent Fuchs, Akihiro Sugimoto// 
 +Quadratic surfaces gain more and more attention among the Geometric Algebra community and some frameworks were proposed in order to represent, transform, and intersect these quadratic surfaces. As far as the authors know, none of these frameworks support all the operations required to completely handle these surfaces. Some frameworks do not allow the construction of quadratic surfaces from control points when others do not allow to transform these quadratic surfaces. However , if we consider all the frameworks together, then all the required operations over quadratic are covered. This paper presents a unification of these frameworks that enables to represent any quadratic surfaces either using control points or from the coefficients of its implicit form. The proposed approach also allows to transform any quadratic surfaces and to compute their intersection and to easily extract some geometric properties.  
 + 
 +  * [[https://arxiv.org/pdf/1901.05873.pdf|Projective geometric algebra: A new framework for doing euclidean geometry]] (2019) - //Charles G. Gunn//
 A tutorial introduction to projective geometric algebra (PGA), a modern, coordinate-free framework for doing euclidean geometry. PGA features: uniform representation of points, lines, and planes; robust, parallel-safe join and meet operations; compact, polymorphic syntax for euclidean formulas and constructions; a single intuitive sandwich form for isometries; native support for automatic differentiation; and tight integration of kinematics and rigid body mechanics. Inclusion of vector, quaternion, dual quaternion, and exterior algebras as sub-algebras simplifies the learning curve and transition path for experienced practitioners. On the practical side, it can be efficiently implemented, while its rich syntax enhances programming productivity. A tutorial introduction to projective geometric algebra (PGA), a modern, coordinate-free framework for doing euclidean geometry. PGA features: uniform representation of points, lines, and planes; robust, parallel-safe join and meet operations; compact, polymorphic syntax for euclidean formulas and constructions; a single intuitive sandwich form for isometries; native support for automatic differentiation; and tight integration of kinematics and rigid body mechanics. Inclusion of vector, quaternion, dual quaternion, and exterior algebras as sub-algebras simplifies the learning curve and transition path for experienced practitioners. On the practical side, it can be efficiently implemented, while its rich syntax enhances programming productivity.
 +
 +  * [[https://arxiv.org/pdf/1909.02408|A Low-Memory Time-Efficient Implementation of Outermorphisms for Higher-Dimensional Geometric Algebras]] (2019) - //Ahmad Hosny Eid//
 +Many important mathematical formulations in GA can be expressed as outermorphisms such as versor products, linear projection operators, and mapping between related coordinate frames. (...) This work attempts to shed some light on the problem of optimizing software implementations of outermorphisms for practical prototyping applications using geometric algebra. The approach we propose here for implementing outermorphisms requires orders of magnitude less memory compared to other common approaches, while being comparable in time performance, especially for high-dimensional geometric algebras.
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 +  * [[http://downloads.hindawi.com/journals/cin/2019/9374802.pdf|Evaluating a Semi-Autonomous Brain-Computer Interface Based on Conformal Geometric Algebra and Artificial Vision]] (2019) - //Mauricio Ramirez-Moreno, David Gutiérrez//
 +We evaluate a semi-autonomous brain-computer interface (BCI) for manipulation tasks. In such system, the user controls a robotic arm through motor imagery commands. (...) We take a semi-autonomous approach based on a conformal geometric algebra model that solves the inverse kinematics of the robot on the fly, then the user only has to decide on the start of the movement and the final position of the effector (goal-selection approach). Under these conditions, we implemented pick-and-place tasks with a disk as an item and two target areas placed on the table at arbitrary positions.
 +
 +  * [[https://arxiv.org/pdf/1912.11198|Geometric Obstructions on Gravity]] (2019) - //Yuri Martins, Rodney Biezuner//
 +These are notes for a short course and some talks gave at Departament of Mathematics and at Departament of Physics of Federal University of Minas Gerais, based on the author's paper. (...) We present obstructions to realize gravity, modeled by the tetradic Einstein-Hilbert-Palatini (EHP) action functional, in a general geometric setting.
  
   * [[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//
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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//
 This is the first entry in a planned series aiming to establish a modified, and simpler, formalism for studying the geometry of smooth manifolds with a metric, while remaining close to standard textbook treatments in terms of notation and concepts. The key step is extending the tangent space at each point from a vector space to a geometric algebra, which is a linear space incorporating vectors with dot and wedge multiplication, and extending the affine connection to a directional derivative acting naturally on fields of multivectors (elements of the geometric algebra). (...) The theory that results from this extension is simpler and more powerful than either differential forms or tensor methods, in a number of ways. This is the first entry in a planned series aiming to establish a modified, and simpler, formalism for studying the geometry of smooth manifolds with a metric, while remaining close to standard textbook treatments in terms of notation and concepts. The key step is extending the tangent space at each point from a vector space to a geometric algebra, which is a linear space incorporating vectors with dot and wedge multiplication, and extending the affine connection to a directional derivative acting naturally on fields of multivectors (elements of the geometric algebra). (...) The theory that results from this extension is simpler and more powerful than either differential forms or tensor methods, in a number of ways.
 +
 +  * [[https://arxiv.org/pdf/2002.11313|Computational Aspects of Geometric Algebra Products of Two Homogeneous Multivectors]] (2020) - //Stephane Breuils, Vincent Nozick, Akihiro Sugimoto//
 +Studies on time and memory costs of products in geometric algebra have been limited to cases where multivectors with multiple grades have only non-zero elements. This allows to design efficient algorithms for a generic purpose; however, it does not reflect the practical usage of geometric algebra. Indeed, in applications related to geometry, multivectors are likely to be full homogeneous, having their non-zero elements over a single grade. In this paper, we provide a complete computational study on geometric algebra products of two full homogeneous multivectors, that is, the outer, inner, and geometric products of two full homogeneous multivectors. We show tight bounds on the number of the arithmetic operations required for these products.
 +
 +  * [[https://arxiv.org/pdf/2001.00656|Two-State Quantum Systems Revisited: A Geometric Algebra Approach]] (2020) - //Pedro Amao, Hernán Castillo//
 +We revisit the topic of two-state quantum systems using Geometric Algebra (GA) in three dimensions G3. In this description, both the quantum states and Hermitian operators are written as elements of G3. By writing the quantum states as elements of the minimal left ideals of this algebra, we compute the energy eigenvalues and eigenvectors for the Hamiltonian of an arbitrary two-state system. The geometric interpretation of the Hermitian operators enables us to introduce an algebraic method to diagonalize these operators in GA. We then use this approach to revisit the problem of a spin-1/2 particle interacting with an external arbitrary constant magnetic field, obtaining the same results as in the conventional theory. However, GA reveals the underlying geometry of these systems, which reduces to the Larmor precession in an arbitrary plane of G3.
 +
 +  * [[https://arxiv.org/abs/2003.07159|Periodic Table of Geometric Numbers]] (2020) - //Garret Sobczyk//
 +Perhaps the most significant, if not the most important, achievements in chemistry and physics are the Periodic Table of the Elements in Chemistry and the Standard Model of Elementary Particles in Physics. A comparable achievement in mathematics is the Periodic Table of Geometric Numbers discussed here. In 1878 William Kingdon Clifford discovered the defining rules for what he called geometric algebras. We show how these algebras, and their coordinate isomorphic geometric matrix algebras, fall into a natural periodic table, sidelining the superfluous definitions based upon tensor algebras and quadratic forms.
  
   * [[https://arxiv.org/pdf/2002.05993|Projective Geometric Algebra as a Subalgebra of Conformal Geometric Algebra]] (2020) - //Jaroslav Hrdina, Ales Navrat, Petr Vasik, Dietmar Hildenbrand//   * [[https://arxiv.org/pdf/2002.05993|Projective Geometric Algebra as a Subalgebra of Conformal Geometric Algebra]] (2020) - //Jaroslav Hrdina, Ales Navrat, Petr Vasik, Dietmar Hildenbrand//
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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//
-Using geometric algebra and calculus to express the laws of electromagnetism we are able to present magnitudes and relations in a gradual way, escalating the number of dimensions. In the one-dimensional case, charge and current densities, the electric field E and the scalar and vector potentials get a geometric interpretation in spacetime diagrams. The geometric vector derivative applied to these magnitudes yields simple expressions leading to concepts like displacement current, continuity and gauge or retarded time, with a clear geometric meaning. As the geometric vector derivative is invertible, we introduce simple Green's functions and, with this, it is possible to obtain retarded Liénard-Wiechert potentials propagating naturally at the speed of light. In two dimensions, these magnitudes become more complex, and a magnetic field B appears as a pseudoscalar which was absent in the one-dimensional world. The laws of induction reflect the relations between E and B, and it is possible to arrive to the concepts of capacitor, electric circuit and Poynting vector, explaining the flow of energy. The solutions to the wave equations in this two-dimensional scenario uncover now the propagation of physical effects at the speed of light. This anticipates the same results in the real three-dimensional world, but endowed in this case with a nature which is totally absent in one or three dimensions. Electromagnetic waves propagating entirely at the speed of light can thus be viewed as a consequence of living in a world with an odd number of spatial dimensions. Finally, in the real three-dimensional world the same set of simple multivector differential expressions encode the fundamental laws and concepts of electromagnetism. +Using geometric algebra and calculus to express the laws of electromagnetism we are able to present magnitudes and relations in a gradual way, escalating the number of dimensions. In the one-dimensional case, charge and current densities, the electric field E and the scalar and vector potentials get a geometric interpretation in spacetime diagrams. The geometric vector derivative applied to these magnitudes yields simple expressions leading to concepts like displacement current, continuity and gauge or retarded time, with a clear geometric meaning. As the geometric vector derivative is invertible, we introduce simple Green's functions and, with this, it is possible to obtain retarded Liénard-Wiechert potentials propagating naturally at the speed of light. In two dimensions, these magnitudes become more complex, and a magnetic field B appears as a pseudoscalar which was absent in the one-dimensional world. The laws of induction reflect the relations between E and B, and it is possible to arrive to the concepts of capacitor, electric circuit and Poynting vector, explaining the flow of energy. The solutions to the wave equations in this two-dimensional scenario uncover now the propagation of physical effects at the speed of light. (...) Electromagnetic waves propagating entirely at the speed of light can thus be viewed as a consequence of living in a world with an odd number of spatial dimensions. Finally, in the real three-dimensional world the same set of simple multivector differential expressions encode the fundamental laws and concepts of electromagnetism.
  
 ===== Books ===== ===== Books =====
geometric_algebra.txt · Last modified: 2023/12/30 00:23 by pbk

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