<title>About parallel computing on spatial rotations in spin mesomorphic structures</title>

This paper explains how, following the representation of 3D crystallographic space groups in Clifford's geometric algebra, it is further possible to similarly represent the 162 so called subperiodic groups of crystallography in Clifford's geometric algebra. A new compact geometric algebra group representation symbol is constructed, which allows to read off the complete set of geometric algebra generators. For clarity moreover the chosen generators are stated explicitly. The group symbols are based on the representation of point groups in geometric algebra by versors (Clifford monomials, Lipschitz elements). Keywords: Subperiodic groups, Clifford's geometric algebra, versor representation, frieze groups, rod groups, layer groups .

Three-dimensional image modeling is essential in many scientific disciplines, including computer vision and precision agriculture. So far, various methods of creating three-dimensional (3D) models have been considered. However, the processing of transformation matrices of each input image data is not controlled. Site-specific crop mapping is essential because it helps farmers determine yield, biodiversity, energy, crop coverage, etc. Clifford Geometric Algebraic understanding of signaling and image processing has become increasingly important in recent years. Geometric Algebraic treats multi-dimensional signals in a holistic way to maintain relationship between side sizes and prevent loss of information. This article has used agricultural images acquired by unmanned aerial vehicles (UAVs) to construct three-dimensional models using Clifford geometric algebra. The qualitative and quantitative performance evaluation results show that Clifford geometric algebra can generate a three-dimensional geometric statistical model directly from drones' RGB images. Through peak signal-to-noise ratio (PSNR), structural similarity index measure (SSIM), and visual comparison, the proposed algorithm's performance is compared with latest algorithms. Experimental results show that proposed algorithm is better than other leading 3D modeling algorithms.

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.

This paper introduces the Clifford Support Vector Machines (CSVM) as a generalization of the real- and complex-valued Support Vector Machines using the Clifford geometric algebra. In this framework we handle the design of kernels involving the Clifford or geometric product for linear and nonlinear classification and regression. The major advantage of our approach is that we redefine the optimization variables as multivectors. This allows us to have a multivector as output therefore we can represent multiple classes according to the dimension of the geometric algebra in which we work. We conduct comparisons between CSVM and the most used approaches to solve multi-class classification to show that our approach is more suitable for practical use on certain type of multi-class classification problems.

Computer graphics applications require efficient tools to model geometric objects. The traditional approach based on compute-intensive matrix calculations is error-prone due to a lack of integration between geometric reasoning and matrix-based algorithms. Clifford algebra offers a solution to these issues since it permits specification of geometry at a coordinate-free level. The best way to exploit the symbolic computing power of geometric (Clifford) algebra is supporting its data types and operators directly in hardware. This paper outlines the architecture of S-CliffoSor (Sliced Clifford coprocessor), a parallelizable embedded coprocessor that executes native Clifford algebra operations. S-CliffoSor is a sliced coprocessor that can be replicated for parallel execution of concurrent Clifford operations. A single slice has been designed, implemented and tested on the Celoxica Inc. RC1000 board. The experimental results show the potential to achieve a 3times speedup for Clifford sums and 4times speedup for Clifford products compared to against the analogous operations in the software library generator GAIGEN.

Geometric Clifford algebra <i>G</i>_3,0 is used to describe geometric objects of 3D Euclidean space in an 8D basis with only real numbers. Basis elements are identified by their inner structure. To describe it with only +1 and -1 coefficients, a unit cube representation is used rather, then matrix ones. Two layers of information are distinguished. The first one is contained in a set of real coefficients for basis Clifford numbers. It describes the outer properties of an object. The second binary one is contained in coefficients for elements of their inner structure to implement their geometric properties in algebraic relations. It is shown, that any change of information inside the second (inner) layer preserves information inside the first layer but changes its image. Consequences for information transmission and processing are discussed.

An influence network of events is a view of the universe based on events that may be related to one another via influence. The network of events forms a partially ordered set which, when quantified consistently via a technique called chain projection, results in the emergence of spacetime and the Minkowski metric as well as the Lorentz transformation through changing an observer from one frame to another. Interestingly, using this approach, the motion of a free electron as well as the Dirac equation can be described. Indeed, the same approach can be employed to show how a discrete version of some of the features of Euclidean geometry including directions, dimensions, subspaces, Pythagorean theorem, and geometric shapes can emerge. In this paper, after reviewing the essentials of the influence network formalism, we build on some of our previous works to further develop aspects of Euclidean geometry. Specifically, we present the emergence of geometric shapes, a discrete version of the parallel postulate, the dot product, and the outer (wedge product) in 2+1 dimensions. Finally, we show that the scalar quantification of two concatenated orthogonal intervals exhibits features that are similar to those of the well-known concept of a geometric product in geometric Clifford algebras.

Representations of geometric (Clifford) algebras with real square matrices are reviewed by providing the general theorem as well as examples of lowest dimensions. New definitions for isometry and norm are proposed. Direct and indirect isometries are identified respectively with automorphisms and antiautomorphisms of the geometric algebra, while the norm of every element is defined as the $n^\textit{th}$-root of the absolute value of the determinant of its matrix representation of order $n$. It is deduced in which geometric algebras direct isometries are inner automorphisms (similarity transformations of matrices). Indirect isometries need reversion too. Finally, the most common isometries are reviewed in order to write them in this way.

A Clifford support vector machine (CSVM) learns the decision surface from multi distinct classes of the multiple input points using the Clifford geometric algebra. In many applications, each multiple input point may not be fully assigned to one of these multi-classes. In this paper, we apply a fuzzy membership to each multiple input point and reformulate the CSVM for multiclass classification to make different input points have their own different contributions to the learning of decision surface. We call the proposed method Clifford fuzzy SVM.

In this article, we present a powerful unifying framework for widely linear (WL) adaptive filters building on the concept of geometric algebra (GA), including recently proposed complex-valued (CV), quaternion-valued, and GA WL adaptive filters (WLAFs). We also consider and review WL adaptive filtering methods that feature robustness against impulsive noise, noisy input measurements, partial coefficient updates, subband structures, censoring, and composite structures under the unified framework. Furthermore, we propose innovative WL adaptive filtering algorithms for functional link polynomial (FLP) nonlinear filters, infinite-impulse response (IIR) systems, and kernel-based nonlinear system identification, showcasing the advantages of the unified framework. The article also investigates the relationship among WLAFs, graph filters, and Cayley–Dickson (CD)-valued adaptive filters, offering new insights into how the unified framework can be extended to graph signals and CD numbers. Finally, the article motivates future work on WL adaptive filtering based on GA and its special cases.

This paper introduces the Clifford support vector machines as a generalization of the real- and complex- valued support vector machines using the Clifford geometric algebra. In this framework we handle the design of kernels involving the Clifford or geometric product for linear and nonlinear classification. We present an interesting application to classify mechanical tools using the concept of Clifford moments which is invariant under affine transformations.

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)

Given a quadratic form on a vector space, the geometric algebra of the corresponding pseudo-euclidean space is defined in terms of a simple set of rules which characterizes the geometric product of vectors. We develop geometric algebra in such a way that it augments, but remains fully compatible with, the more traditional tools of matrix algebra. Indeed, matrix multiplication arises naturally from the geometric multiplication of vectors by introducing a spectral basis of mutually annihiliating idempotents in the geometric algebra. With the help of a few more algebraic identities, and given the proper geometric interpretation, the geometric algebra can be applied to the study of affine, projective, conformal and other geometries. The advantage of geometric algebra is that it provides a single algebraic framework with a comprehensive, but flexible, geometric interpretation. For example, the affine plane of rays is obtained from the euclidean plane of points by adding a single anti-commuting vector to the underlying vector space. The key to the study of noneuclidean geometries is the definition of the operations of meet and join, in terms of which incidence relationships are expressed. The horosphere provides a homogeneous model of euclidean space, and is obtained by adding a second anti-commuting vector to the underlying vector space of the affine plane. Linear orthogonal transformations on the higher dimensional vector space correspond to conformal or Mobius transformations on the horosphere. The horosphere was first constructed by F.A. Wachter (1792–1817), but has only recently attracted attention byo ering a host of new computational tools in projective and hyperbolic geometries when formulated in terms of geometric algebra.

Geometric algebra was introduced in the nineteenth century by the English mathematician William Kingdon Clifford (figure 2.1). Clifford appears to have been one of the small number of mathematicians at the time to be significantly influenced by Grassmann's work. Clifford introduced his geometric algebra by uniting the inner and outer products into a single geometric product. This is associative, like Grassmann's product, but has the crucial extra feature of being invertible , like Hamilton's quaternion algebra. Indeed, Clifford's original motivation was to unite Grassmann's and Hamilton's work into a single structure. In the mathematical literature one often sees this subject referred to as Clifford algebra . We have chosen to follow the example of David Hestenes, and many other modern researchers, by returning to Clifford's original choice of name – geometric algebra . One reason for this is that the first published definition of the geometric product was due to Grassmann, who introduced it in the second Ausdehnungslehre . It was Clifford, however, who realised the great potential of this product and who was responsible for advancing the subject. In this chapter we introduce the basics of geometric algebra in two and three dimensions in a way that is intended to appear natural and geometric, if somewhat informal. A more formal, axiomatic approach is delayed until chapter 4, where geometric algebra is defined in arbitrary dimensions.

Optical Mueller matrices in terms of geometric algebra