Showing papers on "Geometric algebra published in 2012"
Book•
23 Oct 2012
TL;DR: In this article, a generic framework for image geometry is presented, based on the Clifford Bracket Algebra, which can be used for image recognition in computer vision and robotics applications.
Abstract: Preface Contributors Part I. Algebra and Geometry Point Groups and Space Groups in Geometric Algebra (D. Hestenes) The Inner Products of Geometric Algebra (L. Dorst) Unification of Grassmann's Progressive and Regressive Products using the Principle of Duality (S. Blake) From Unoriented Subspaces to Blade Operators (T.A. Bouma) Automated Theorem Proving in the Homogeneous Model with Clifford Bracket Algebra (H. Li) Rotations in n Dimensions as Spherical Vectors (W.E. Baylis/S. Hadi) Geometric and Algebraic Canonical Forms (N. Gordon) Functions of Clifford Numbers or Square Matrices (J. Snygg) Compound Matrices and Pfaffians: A Representation of Geometric Algebra (U. Prells/M.I. Friswell/S.D. Garvey) Analysis Using Abstract Vector Variables (F. Sommen) A Multivector Data Structure for Differential Forms and Equations (J.A. Chard/V. Shapiro) Jet Bundles and the Formal Theory of Partial Differential Equations (R. Baker/C. Doran) Imaginary Eigenvalues and Complex Eigenvectors Explained by Real Geometry (E.M.S. Hitzer) Symbolic Processing of Clifford Numbers in C++ (J.P. Fletcher) Clifford Numbers and their Inverses Calculated using the Matrix Representation (J.P. Fletcher) A Toy Vector Field Based on Geometric Algebra (A. Rockwood/S. Binderwala) Quadratic Transformations in the Projective Plane (G. Georgiev) Annihilators of Principal Ideals in the Grassmann Algebra (C. Koc/S. Esin) Part II. Applications to Physics Homogeneous Rigid Body Mechanics with Elastic Coupling (D. Hestenes/E.D. Fasse) Analysis of One and Two Particle Quantum Systems using Geometric Algebra (R. Parker/C. Doran) Interaction and Entanglement in the Multiparticle Spacetime Algebra (T.F. Havel/C.J.L. Doran) Laws of Reflection from Two or More Plane Mirrors in Succession (M. Derome) Exact Kinetic Energy Operators for Polyatomic Molecules (J. Pesonen) Geometry of Quantum Computing by Hamiltonian Dynamics of Spin Ensembles (T. Schulte-Herbruggen/K. Huper/U. Helmke/S.J. Glaser) Is the Brain a 'Clifford Algebra Quantum Computer'? (V. Labunets/E. Rundblad/J. Astola) A Hestenes Spacetime Algebra Approach to Light Polarization (Q.M. Sugon/D. McNamara) Quaternions, Clifford Algebra and Symmetry Groups (P.R. Girard) Part III. Computer Vision and Robotics A Generic Framework for Image Geometry (J.J. Koenderink) Color Edge Detection Using Rotors (E. Bayro-Corrochano/S. Flores) Numerical Evaluation of Versors with Clifford Algebra (C.B.U. Perwass/G. Sommer) The Role of Clifford Algebra in Structure-Preserving Transformations for Second-Order Systems (S.D. Garvey/M.I. Friswell/U. Prells) Applications of Algebra of Incidence in Visually Guided Robotics (E. Bayro-Corrochano/P. Lounesto/L.R. Lozano) Monocular Pose Estimation of Kinematic Chains (B. Rosenhahn/O. Granert/G. Sommer) Stabilization of 3D Pose Estimation (W. Neddermeyer/M. Schnell/W. Winkler/A. Lilienthal) Inferring Dynamical Information from 3D Position Data using Geometric Algebra (H. Udugama/G.S. Sajeewa/J. Lasenby) Clifford Algebra Space Singularities of Inline Planar Platforms (M.A. Baswell/R. Ablamowicz/J.N. Anderson) Part IV. Signal Processing and Other Applications Fast Quantum Fourier--Heisenberg--Weyl Transforms (V. Labunets/E. Rundblad/J. Astola) The Structure Multivector (M. Felsberg/G. Sommer) The Application of Clifford Algebra to Calculations of Multicomponent Chemical Composition (J.P. Fletcher) An Algorithm to Solve the Inverse IFS-Problem (E. Hocevar) Fast Quantum n-D Fourier and Radon Transforms (V. Labunets/E. Rundblad/J. Astola)
147 citations
TL;DR: In this paper, the authors argue that what is most important to the clear and concise expression of geometrical ideas is notation, and that a good notation has a subtlety and suggestiveness which at times make it seem almost like a live teacher.
Abstract: I n 1878 William Kingdon Clifford wrote down the rules for his geometric algebra, also known as Clifford algebra. We argue in this paper that in doing so he laid down the groundwork that is profoundly altering the language used by the mathematical community to express geometrical ideas. In the real estate business everyone knows that what is most important is location. We demonstrate here that in the business of mathematics what is most important to the clear and concise expression of geometrical ideas is notation. In the words of Bertrand Russell, ...A good notation has a subtlety and suggestiveness which at times make it seem almost like a live teacher. Heinrich Hertz expressed much the same thought when he said, One cannot escape the feeling that these mathematical formulae have an independent existence and an intelligence of their own, that they are wiser than we are, wiser even than their discoverers, that we get more out of them than we originally put into them. The development of the real and complex number systems represents a hard-won milestone in the robust history of mathematics over many centuries and many different civilizations [5], [29]. Without it mathematics could progress only haltingly, as is evident from the history of mathematics and even the terminology that we use today. Negative numbers were referred to by Rene Descartes (1596–1650) as “fictitious”, and “imaginary” numbers were held up to even greater ridicule, though they were first conceived as early as Heron of
44 citations
Book•
28 Oct 2012
TL;DR: In this paper, the authors describe Modular Number Systems, Complex and Hyperbolic Numbers, Geometric Algebra, Vector Spaces and Matrices, System of Linear Equations, and Linear Transformations on R^n.
Abstract: 1 Modular Number Systems.- 2 Complex and Hyperbolic Numbers.- 3 Geometric Algebra.- 4 Vector Spaces and Matrices.- 5 Outer Product and Determinants.- 6 Systems of Linear Equations.- 7 Linear Transformations on R^n.- 8 Structure of a Linear Operator.- 9 Linear and Bilinear Forms.- 10 Hermitian Inner Product Spaces.- 11 Geometry of Moving Planes.- 12 Representations of the Symmetric Group.- 13 Calculus on m-Surfaces.- 14 Differential Geometry of Curves.- 15 Differential Geometry of k-Surfaces.- 16 Mappings Between Surfaces.- 17 Non-Euclidean and Projective Geometries.- 18 Lie Groups and Lie Algebras.- References.- Symbols.
40 citations
TL;DR: In this article, a unified GIS multidimensionally-unified representation, analysis, and modeling is proposed, it unifies expressions and operation structures, as well as supports the analysis of multidimensional complex scenes.
Abstract: Seamless multidimensional handling and coordinate-free characteristics of geometric algebra (GA) provide means to construct multidimensionally-unified GIS computation models. Using the multivector representation for basic geometric objects within GA, we are able to construct adaptable unified geometric-topological structural models of a multidimensional geographical scene. Multidimensional operators found within the geometry, topology and GIS analysis are developed with basic GA operators. A unified computational framework is proposed, it unifies expressions and operation structures, as well as supports the analysis of multidimensional complex scenes. Finally, we illustrate modelling a three-dimensional residential district, which shows that GA-based multidimensionally-unified computation models can effectively represent and analyze complex and multidimensional geographical scenes. The development of the proposed GIS multidimensionally-unified representation, analysis, and modeling enhances current GIS algorithms and geographical models.
25 citations
TL;DR: A general solution for two-player games is produced and the players' strategy sets remain identical to the ones in the classical mixed-strategy version of the game, and hence the quantum game becomes a proper extension of the classical game, avoiding a criticism of other quantum game frameworks.
Abstract: The framework for playing quantum games in an Einstein-Podolsky-Rosen (EPR) type setting is investigated using the mathematical formalism of geometric algebra (GA) The main advantage of this framework is that the players' strategy sets remain identical to the ones in the classical mixed-strategy version of the game, and hence the quantum game becomes a proper extension of the classical game, avoiding a criticism of other quantum game frameworks We produce a general solution for two-player games, and as examples, we analyze the games of Prisoners' Dilemma and Stag Hunt in the EPR setting The use of GA allows a quantum-mechanical analysis without the use of complex numbers or the Dirac Bra-ket notation, and hence is more accessible to the non-physicist
22 citations
TL;DR: In this article, the authors studied the geometric significance of the imaginary unit in a complex geometric algebra, where the imaginary i is a unit (pseudo) vector with square minus one which anti commutes with all real vectors.
Abstract: The geometric significance of the imaginary unit in a complex geometric algebra has troubled the author for 40 years. In the unitary geometric algebra presented here, the imaginary i is a unit (pseudo) vector with square minus one which anti commutes with all of the real vectors. The resulting natural hermitian inner product and hermitian outer product induce a grading of the algebra into complex k-vectors. Basic orthogonality relationships are studied.
22 citations
Posted Content•
TL;DR: The Fractional Geometric Calculus (FGC) as discussed by the authors is an extension of fractional calculus to any dimension, and it provides a unified language for mathematics, physics and science of complexity of the 21st century.
Abstract: This paper discuss the longstanding problems of fractional calculus such as too many definitions while lacking physical or geometrical meanings, and try to extend fractional calculus to any dimension. First, some different definitions of fractional derivatives, such as the Riemann-Liouville derivative, the Caputo derivative, Kolwankar's local derivative and Jumarie's modified Riemann-Liouville derivative, are discussed and conclude that the very reason for introducing fractional derivative is to study nondifferentiable functions. Then, a concise and essentially local definition of fractional derivative for one dimension function is introduced and its geometrical interpretation is given. Based on this simple definition, the fractional calculus is extended to any dimension and the \emph{Fractional Geometric Calculus} is proposed. Geometric algebra provided an powerful mathematical framework in which the most advanced concepts modern physic, such as quantum mechanics, relativity, electromagnetism, etc., can be expressed in this framework graciously. At the other hand, recent developments in nonlinear science and complex system suggest that scaling, fractal structures, and nondifferentiable functions occur much more naturally and abundantly in formulations of physical theories. In this paper, the extended framework namely the Fractional Geometric Calculus is proposed naturally, which aims to give a unifying language for mathematics, physics and science of complexity of the 21st century.
14 citations
28 Jun 2012
TL;DR: This work studies Bezier-like formulas with weights in geometric algebra for parametrizing a special class of rational surfaces in isotropic 3-space and derives their implicitization formula, which describes bilinear Clifford-Bezier patches as patches on special quartic surfaces called isotropics cyclides.
Abstract: We study Bezier-like formulas with weights in geometric algebra for parametrizing a special class of rational surfaces in isotropic 3-space. These formulas are useful for constructing isotropic-Mobius invariant surfaces that are dual to rational offset surfaces in euclidean 3-space. Our focus is on bilinear Clifford-Bezier patches. We derive their implicitization formula and characterize them as patches on special quartic surfaces called isotropic cyclides. Finally we present one modeling application with rational surfaces admitting rational offsets.
11 citations
TL;DR: The introduced mathematical framework enables to efficiently manipulate and generate protein conformations to any arbitrary degree and several new formulations in the context of rigid body motions are added.
Abstract: Spinor operators in geometric algebra (GA) can efficiently describe conformational changes of proteins by ordered products that act on individual bonds and represent their net rotations. Backward propagation through the protein backbone yields all rotational spinor axes in advance allowing the efficient computation of atomic coordinates from internal coordinates. The introduced mathematical framework enables to efficiently manipulate and generate protein conformations to any arbitrary degree. Moreover, several new formulations in the context of rigid body motions are added. Emphasis is placed on the intimate relationship between spinors and quaternions, which can be recovered from within the GA approach. The spinor methodology is implemented and tested versus the state of the art algorithms for both protein construction and coordinate updating. Spinor calculations have a smaller computational cost and turn out to be slightly faster than current alternatives. © 2012 Wiley Periodicals, Inc.
8 citations
TL;DR: A means for constructing a free-form motion in three dimensions is given and a sparse sequence of prescribed poses is used to guide the motion, and some of these can also be specified to be precision poses through which the motion must pass.
Abstract: A means for constructing a free-form motion in three dimensions is given. A sparse sequence of prescribed poses is used to guide the motion, and some of these can also be specified to be precision poses through which the motion must pass. The resultant motion is given in a sequence of refined poses. Geometric algebra is used to specify these in a way that is analogous to the use of complex numbers in an equivalent approach for free-form curves.
8 citations
1 Jan 2012
TL;DR: The focus of the this work is a simplified integration of algorithms expressed in Geometric Algebra in modern high level computer languages, namely C++, OpenCL and CUDA using a Precompiler that is directly integrated into CMake-based build toolchains.
Abstract: The focus of the this work is a simplified integration of algorithms expressed in Geometric Algebra (GA) in modern high level computer languages, namely C++, OpenCL and CUDA. A high runtime performance in terms of GA is achieved using symbolic simplification and code generation by a Precompiler that is directly integrated into CMake-based build toolchains.
6 Nov 2012
TL;DR: In this article, a translation of cosmological special relativity into the mathematical language of Grassmann and Clifford (Geometric algebra) is given and the physics of Cosmological Special Relation is discussed.
Abstract: Geometric algebra and Clifford algebra are important tools to describe and analyze the physics of the world we live in. Although there is enormous empirical evidence that we are living in four dimensional spacetime, mathematical worlds of higher dimensions can be used to present the physical laws of our world in an aesthetical and didactical more appealing way. In physics and mathematics education we are therefore confronted with the question how these high dimensional spaces should be taught. But as an immediate confrontation of students with high dimensional compactified spacetimes would expect too much from them at the beginning of their university studies, it seems reasonable to approach the mathematics and physics of higher dimensions step by step. The first step naturally is the step from four dimensional spacetime of special relativity to a five dimensional spacetime world. As a toy model for this artificial world cosmological special relativity, invented by Moshe Carmeli, can be used. This five dimensional non-compactified approach describes a spacetime which consists not only of one time dimension and three space dimensions. In addition velocity is regarded as a fifth dimension. This model very probably will not represent physics correctly. But it can be used to discuss and analyze the consequences of an additional dimension in a clear and simple way. Unfortunately Carmeli has formulated cosmological special relativity in standard vector notation. Therefore a translation of cosmological special relativity into the mathematical language of Grassmann and Clifford (Geometric algebra) is given and the physics of cosmological special relativity is discussed.
TL;DR: In this article, a simple and general design procedure is presented for the polarisation diversity of arbitrary conformal arrays; this procedure is based on the mathematical framework of geometric algebra and can be solved optimally using convex optimisation.
Abstract: A simple and general design procedure is presented for the polarisation diversity of arbitrary conformal arrays; this procedure is based on the mathematical framework of geometric algebra and can be solved optimally using convex optimisation. Aside from being simpler and more direct than other derivations in the literature, this derivation is also entirely general in that it expresses the transformations in terms of rotors in geometric algebra which can easily be formulated for any arbitrary conformal array geometry. Convex optimisation has a number of advantages; solvers are widespread and freely available, the process generally requires a small number of iterations and a wide variety of constraints can be readily incorporated. The study outlines a two-step approach for addressing polarisation diversity in arbitrary conformal arrays: first, the authors obtain the array polarisation patterns using geometric algebra and secondly use a convex optimisation approach to find the optimal weights for the polarisation diversity problem. The versatility of this approach is illustrated via simulations of a 7×10 cylindrical conformal array.
TL;DR: In this article, the connection between Mueller matrices and geometrical algebra multivectors is established, and it is shown that starting from 3-dimensional (3D) Cl 3,0 algebra and using isomorphism between Cl 3 0 and even Cl 3 1 + subalgebra, one can generate canonical Mueller matrix and their combinations that describe an optical system.
Posted Content•
TL;DR: It is shown that Clifford's formalism of geometric algebra, provides a significantly more efficient representation than the conventional Bra-ket notation, and that the basis defined by the states of maximum and minimum weight in the Grover search space, allows a simple visualization of theGrover search as the precession of a spin-1/2 particle.
Abstract: The Grover search algorithm is one of the two key algorithms in the field of quantum computing, and hence it is of significant interest to describe it in the most efficient mathematical formalism We show firstly, that Clifford's formalism of geometric algebra, provides a significantly more efficient representation than the conventional Bra-ket notation, and secondly, that the basis defined by the states of maximum and minimum weight in the Grover search space, allows a simple visualization of the Grover search as the precession of a spin-1/2 particle Using this formalism we efficiently solve the exact search problem, as well as easily representing more general search situations
TL;DR: A novel substance identification method via the conformal split based on an optical physical mechanism that can contribute to accurate classification and identification of substances.
Abstract: A terahertz time-domain spectroscopy (THz-TDS) imaging system can obtain high-dimensional signals with substance fingerprint information. By introducing geometric algebra, a novel signal analysis approach to THz-TDS signals is developed based on an optical physical mechanism. Using this approach, signals are represented with vectors in the high-dimensional real vector space. Geometric distribution properties and algebraic relationships of THz-TDS signals are deduced. It is proved that every complex refractive index of substances relates to a unique 2-blade, the vectors corresponding to the samples of the same substance are collinear and belong to the intrinsic 2-blade of the substance. When decomposed through the conformal split with respect to a 2-blade, THz-TDS signals of high dimensionality can be related to vectors in a 2-dimensional subspace. Based on the conformal split properties we deduced, two criteria for substance identification on the basis of THz-TDS signals are proposed. Accordingly, a novel substance identification method via the conformal split is presented. In the method, the 2-blade related to each “known” substance is calculated with two vectors corresponding to THz-TDS signals measured from samples of the substance but with different thicknesses. Using the conformal split with respect to those 2-blades, an identified vector corresponding to a THz-TDS signal is linearly related to the vector in a 2-dimensional subspace. The substance of a sample can be identified using criteria on the projected vectors in the subspaces. This method can contribute to accurate classification and identification. Finally, two experiments are presented that show the feasibility and accuracy of this method.
10 Jun 2012
TL;DR: The main goal of this work is to develop a geometric neural network which can be used as an interface between sensors and robot mechanisms using the conformal geometric algebra framework.
Abstract: The main goal of this work is to develop a geometric neural network which can be used as an interface between sensors and robot mechanisms. For this goal we have developed two new geometric network called Spherical Radial Basis Function Network and Spherical General Regression Network using the conformal geometric algebra framework. The motivation to use circles or spheres as activation functions is due to the fact that the sphere is the computational unity of the conformal geometric algebra, as a result these Networks can be advantageously used as interface between the sensor domain and the robotic mechanism so that all the computing can be done in the same mathematical framework. In fact, there will be no need to abandon the system for the interpolation or reconstruction using this network. This article presents the design principles and a comparison with a standard Radial Basis Function Network and a standard General regression Neural Network. In the area of medical robotics the use of haptics is quite common. This is an interesting domain to apply our network for capturing data with a haptic device and using spheres reconstruct automatically the shape of a human organ. As we shown in other works, the differential robot kinematics can be formulated using lines, planes and spheres using geometric algebra, having the organ tissue modelled also with spheres with the networks, this helps greatly to related the perceptual and the mechanical devices and ensure their control. We show reconstruction results of an organ using both Networks.
TL;DR: In this paper, it was shown that all nonzero vectors and nonzero bivectors in the Clifford algebra are invertible and some conditions for those objects to be element of the Clifford group Γ 0,3 using the corresponding properties in the subalgebra L8 of the matrix algebra.
Abstract: In this paper, we will show that all of nonzero vectors and nonzero bivectors in the Clifford algebra \({\mathcal{C} \ell_{0,3}}\) are invertible and we will find some conditions for those objects to be element of the Clifford group Γ0,3 using the corresponding properties in the subalgebra L8 of the matrix algebra \({M_8 \mathbb{(R)}}\) .
TL;DR: In this article, the authors provide explicit characterizations and formulae for the minimal polynomials of a wide variety of structured 4 × 4 matrices, including symmetric, Hamiltonian and orthogonal matrices.
Abstract: This work provides explicit characterizations and formulae for the minimal polynomials of a wide variety of structured 4 × 4 matrices. These include symmetric, Hamiltonian and orthogonal matrices. Applications such as the complete determination of the Jordan structure of skew-Hamiltonian matrices and the computation of the Cayley transform are given. Some new classes of matrices are uncovered, whose behaviour insofar as minimal polynomials are concerned, is remarkably similar to those of skew-Hamiltonian and Hamiltonian matrices. The main technique is the invocation of the associative algebra isomorphism between the tensor product of the quaternions with themselves and the algebra of real 4 × 4 matrices. Extensions to higher dimensions via Clifford Algebras are discussed.
TL;DR: In this paper, the source localization by utilizing the measurements of a single electromagnetic (EM) vector-sensor is investigated in the framework of the geometric algebra of Euclidean 3-space.
TL;DR: In this paper, the authors follow a common thread to express linear transformations of vectors and bivectors from different fields of physics in a unified way using Clifford products, which have a remarkable similarity when expressed in terms of geometric products.
Abstract: We follow a common thread to express linear transformations of vectors and bivectors from different fields of physics in a unified way. The tensorial representations are coordinate independent and assume a compact form using Clifford products. As specific examples, we present (a) the inertia tensor as a vector-to-vector as well as a bivector-to-bivector linear transformation; (b) the Newtonian tidal acceleration; and (c) the Riemann tensor corresponding to a Schwarzschild black hole as a bivector-to-bivector tensorial transformation. The resulting expressions have a remarkable similarity when expressed in terms of geometric products.
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TL;DR: In this article, it is shown how (3 x 3) permutation matrices can be interpreted as unit vectors and as S3 permutation symmetry is flavour symmetry a unified flavour picture of Geometric algebra will emerge.
Abstract: Quarks are described mathematically by (3 x 3) matrices. To include these quarkonian mathematical structures into Geometric algebra it is helpful to restate Geometric algebra in the mathematical language of (3 x 3) matrices. It will be shown in this paper how (3 x 3) permutation matrices can be interpreted as unit vectors. And as S3 permutation symmetry is flavour symmetry a unified flavour picture of Geometric algebra will emerge.
TL;DR: In this paper, the authors examine one particular type of Clifford algebra that can be used to model mechanical and engineering problems, such as shot-peening, and show how these quantities can be applied to model elastic transformations.
Abstract: Multivectorial algebra is of both academic and technological interest. Its application, however, is not always easy. A distinction must be made between polar and axial vectors and between scalars and pseudo-scalars. Eight element types are often considered even if they are not always identified as multivectors. In some cases, for simplicity’s sake, only vectorial algebra or quaternion algebra is explicitly used for physical and mechanical applications. It would, however, be more convenient to use more complex algebra directly in order to have a wider range of mechanical applications. The aim of this paper is to examine one particular type of Clifford algebra that could solve this problem. The present study focusses on showing how these quantities can be used to model mechanical and engineering problems. First, continuum mechanics in a Cauchy medium is investigated for elastic transformations. Second, a specific type of shot-peening application is studied. Applications are then used to illustrate the scope and efficiency of this type of modeling based on geometric algebra.
27 Oct 2012
TL;DR: This paper presents the implementation of a Multilayer Perceptron (MLP) using a new higher order neuron whose decision region is generated by a conic section (circle, ellipse, parabola, hyperbola) called the hyper-conic neuron.
Abstract: This paper presents the implementation of a Multilayer Perceptron (MLP) using a new higher order neuron whose decision region is generated by a conic section (circle, ellipse, parabola, hyperbola). We call it the hyper-conic neuron. The conic neuron is defined for the conformal space where it can freely work and take advantage of all the rules of Geometric (Clifford) Algebra. The proposed neuron is a non-linear associator that estimates distances from vectors (points) to decision regions. The computational model of the conic neuron is based on the geometric product (an outer product plus an inner product) of geometric algebra in conformal space. The Particle Swarm Optimization (PSO) algorithm is used to find the values of the weights that properly define some MLP for a given classification problem. The performance is presented with a classical benchmark used in neural computing.
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TL;DR: Geometric algebra as mentioned in this paper is an alternative to vector algebra that expands on it in two ways: 1) In addition to scalars and vectors, it defines new objects representing subspaces of any dimension 2) It defines a product that's strongly motivated by geometry and can be taken between any two objects.
Abstract: This is an introduction to geometric algebra, an alternative to traditional vector algebra that expands on it in two ways: 1 In addition to scalars and vectors, it defines new objects representing subspaces of any dimension 2 It defines a product that's strongly motivated by geometry and can be taken between any two objects For example, the product of two vectors taken in a certain way represents their common plane This system was invented by William Clifford and is more commonly known as Clifford algebra It's actually older than the vector algebra that we use today (due to Gibbs) and includes it as a subset Over the years, various parts of Clifford algebra have been reinvented independently by many people who found they needed it, often not realizing that all those parts belonged in one system This suggests that Clifford had the right idea, and that geometric algebra, not the reduced version we use today, deserves to be the standard "vector algebra" My goal in these notes is to describe geometric algebra from that standpoint and illustrate its usefulness The notes are work in progress; I'll keep adding new topics as I learn them myself
1 Dec 2012
TL;DR: In this paper, it was shown that from representations of simple Jordan algebras of Lorentzian type, one can obtain a series of polynomials satisfying (0.1), which are not covered by the theory of prehomogeneous vector spaces.
Abstract: satisfy a functional equation (see [9], [10], [6]). Meanwhile, in [5], Faraut and Koranyi developed a method of constructing poly‐ nomials with the property (0.1), starting from representations of Euclidean (formally real) Jordan algebras. What is remarkable in their result is that, from representations of simple Jordan algebras of Lorentzian type, one can obtain a series of polynomials satisfying (0.1), which are not covered by the theory of prehomogeneous vector spaces (see also Clerc [4]). Thus we got to know that the class of polynomials with the property
1 Jan 2012
TL;DR: The proposed approach exploits the properties of Geometric Algebra rotation operators, called rotors, to code sentences through the rotation of an orthogonal basis of a semantic space.
Abstract: Natural language sentences can be represented as vectors in a high dimensional vector space. Generally, these models are based on bag of words approaches, and therefore they do not fully capture the semantics of sentences which depends both by the semantics of the words, and their order in in the phrase. In this work we propose a sub-symbolic methodology to encode natural language sentences considering both these two aspects. The proposed approach exploits the properties of Geometric Algebra rotation operators, called rotors, to code sentences through the rotation of an orthogonal basis of a semantic space. The methodology is based on three main steps: the construction of a semantic space, the association of ad-hoc rotors to sentence bigrams, and finally the coding of the sentence through the application of the obtained rotors to a standard basis in the
1 Jan 2012
TL;DR: The results suggest that the adaptive template matching method can effectively resolve the structural features of the vector field with different dimensions and can do structure-based classification of vector fields.
Abstract: Taking advantage of the multidimensional unified and simplicity expression of movement characteristics of geometric algebra,an adaptive template matching method for convergence and divergence structure of multi-dimensional vector fields was proposed.The optimal rotor between the original vector field and the standard template is established based on SVD(Singular Value Decomposition).The data adaptive divergence-convergence template generation method is then constructed based on the structure consistency of rotor rotation,and the classification of geometric structure of the vector field based on the rotor rotation angle is proposed.Finally,the adaptive template matching method is constructed based on the geometric convolution.These methods are verified with the wind field of North America.The results suggest that our method can effectively resolve the structural features of the vector field with different dimensions and can do structure-based classification of vector fields.
1 Jan 2012
TL;DR: In this paper, a theory of constrained generalized Killing forms is proposed, which gives a useful geometric translation of supersymmetry conditions in the presence of fluxes, and can be used to formulate a theory for supergravity and string theory.
Abstract: Supersymmetry-preserving backgrounds in supergravity and string theory can be studied using a powerful framework based on a natural realization of Clifford bundles. We explain the geometric origin of this framework and show how it can be used to formulate a theory of ‘constrained generalized Killing forms’, which gives a useful geometric translation of supersymmetry conditions in the presence of fluxes.
5 Sep 2012
TL;DR: The architecture of CliffordCoreDuo, an embedded dual-core coprocessor that offers direct hardware support to four-dimensional (4D) Clifford algebra operations, is introduced and a prototype implementation on an FPGA board is detailed.
Abstract: Geometric or Clifford Algebra (CA) is a powerful mathematical tool that is attracting a growing attention in many research fields such as computer graphics, computer vision, robotics and medical imaging for its natural and intuitive way to represent geometric objects and their transformations. This paper introduces the architecture of CliffordCoreDuo, an embedded dual-core coprocessor that offers direct hardware support to four-dimensional (4D) Clifford algebra operations. A prototype implementation on an FPGA board is detailed. Experimental results show a 1.6x average speedup of CliffordCoreDuo in comparison with the baseline mono-core architecture. A potential cycle speedup of about 40x over Gaigen 2, a geometric algebra software library generator for general-purpose processors, is also demonstrated.