20 papers related to Clifford algebra

Note: Communicated by Nek Valous, National Center for Tumor Diseases (NCT), Heidelberg.

Title: The Boolean SATisfiability Problem in Clifford algebra
arXiv:1704.02942 [math-ph]
Marco Budinich

We present a formulation of the Boolean Satisfiability Problem in spinor language that allows to give a necessary and sufficient condition for unsatisfiability. With this result we outline an algorithm to test for unsatisfiability with possibly interesting theoretical properties.

Title: Bochner-Martinelli formula in superspace
Journal: arXiv:1805.00338 [math.CV]
Authors: Juan Bory Reyes, Alí Guzmán Adán, Frank Sommen
Link: https://arxiv.org/abs/1805.00338
In a series of recent papers, a harmonic and hypercomplex function theory in superspace has been established and amply developed. In this paper, we address the problem of establishing Cauchy integral formulae in the framework of Hermitian Clifford analysis in superspace. This allows us to obtain a successful extension of the classical Bochner-Martinelli formula to superspace by means of the corresponding projections on the space of spinor-valued superfunctions.

Title: Frame Multipliers for discrete frames on Quaternionic Hilbert Spaces
Journal: arXiv:1805.01594 [math.FA]
Authors: M. Khokulan, K. Thirulogasanthar
Link: https://arxiv.org/abs/1805.01594
In this note, following the complex theory, we examine discrete controlled frames, discrete weighted frames and frame multipliers in a non-commutative setting, namely in a left quaternionic Hilbert space. In particular, we show that the controlled frames are equivalent to usual frames under certain conditions. We also study connection between frame multipliers and weighted frames in the same Hilbert space.

Title: On complex representations of Clifford algebra
Journal: arXiv:1805.03126 [math-ph]
Authors: Marco Budinich
Link: https://arxiv.org/abs/1805.03126
We show that complex representations of Clifford algebra can always be reduced either to a real or to a quaternionic algebra depending on signature of complex space thus showing that complex spinors are unavoidably either real Majorana spinors or quaternionic spinors. We use this result to support (1,3) signature for Minkowski space.

Title: Octonions, exceptional Jordan algebra and the role of the group F_4 in particle physics
Journal: arXiv:1805.06739 [hep-th]
Authors: Ivan Todorov, Svetla Drenska
Link: https://arxiv.org/abs/1805.06739
Normed division rings are reviewed in the more general framework of composition algebras that include the split (indefinite metric) case. The Jordan – von Neumann – Wigner classification of finite dimensional Jordan algebras is outlined with special attention to the 27 dimensional exceptional Jordan algebra J. The automorphism group F_4 of J and its maximal Borel – de Siebenthal subgroups are studied in detail and applied to the classification of fundamental fermions and gauge bosons. Their intersection in F_4 is demonstrated to coincide with the gauge group of the Standard Model of particle physics.

Title: A multi-channel approach through fusion of audio for detecting video inter-frame forgery
Journal: Computers & Security, Volume 77, August 2018, Pages 412-426
Authors: Tianqiang Huang, Xueli Zhang, Wei Huang, Lingpeng Lin, Weifeng Su
Link: https://www.sciencedirect.com/science/article/pii/S0167404818304243
The forgery operation of digital video in the temporal domain is often accompanied by the synchronization of the audio channel operation. In this paper, we proposed a fusion of audio forensics detection methods for video inter-frame forgery. First, the audio channel of the video is extracted, and discrete wavelet packet decomposition and analysis of singularity points of audio signals are used to locate the forged singularity points. Next, features of each frame of the video are extracted with the perceptual hash and used to calculate the similarity between consecutive frames, to locate the forgery position in the video frame sequence. We fused the results of the audio channel and the video frame sequence channel. The QDCT feature is used to further fine detect the suspected forgery location. Our method can position replication source locations for copy-move forgery. Experiments show that our method has higher accuracy and better performance in comparison with similar methods, especially on the delete forgery operation.

Title: A new real structure-preserving quaternion QR algorithm
Journal: Journal of Computational and Applied Mathematics, Volume 343, 1 December 2018, Pages 26-48
Authors: Zhigang Jia, Musheng Wei, Mei-Xian Zhao, Yong Chen
Link: https://www.sciencedirect.com/science/article/pii/S0377042718301870
New real structure-preserving decompositions are introduced to develop fast and robust algorithms for the (right) eigenproblem of general quaternion matrices. Under the orthogonally JRS-symplectic transformations, the Francis JRS-QR step and the JRS-QR algorithm are firstly proposed for JRS-symmetric matrices and then applied to calculate the Schur form of quaternion matrices. A novel quaternion Givens matrix is defined and utilized to compute the QR factorization of quaternion Hessenberg matrices. An implicit double shift quaternion QR algorithm is presented with a technique for automatically choosing shifts and within real operations. Numerical experiments are provided to demonstrate the efficiency and accuracy of newly proposed algorithms.

Title: Bézier motions with end-constraints on speed
Journal: Computer Aided Geometric Design, Volume 63, July 2018, Pages 135-148
Authors: Glen Mullineux, Robert J. Cripps, Ben Cross
Link: https://www.sciencedirect.com/science/article/pii/S0167839618300499
A free-form motion can be considered as a smoothly varying rigid-body transformation. Motions can be created by establishing functions in an appropriate space of matrices. While a smooth motion is created, the geometry of the motion itself is not always immediately clear. In a geometric algebra environment, motions can be created using extensions of the ideas of Bézier and B-spline curves and the geometric significance of the construction is clearer. A motion passing through given precision poses can be obtained by direct analogy with the curve approach. This paper considers the more difficult problem of dealing additionally with velocity constraints at the ends of the motion: here the analogy is less obvious. A geometric construction for the end pairs of control poses is established and is demonstrated by creating motions satisfying given pose and velocity constraints.

Title: Cauchy’s Formula on Surfaces Embedded in the Quaternions
Journal: Complex Analysis and Operator Theory, June 2018, Volume 12, Issue 5, pp 1337–1349
Authors: Hee Chul Pak
Link: https://link.springer.com/article/10.1007/s11785-018-0778-5
Cauchy’s theorem for analytic functions on complex numbers is extended to analytic functions on the quaternions. For this purpose, we carefully define the notions of differentiation and integration on two or three dimensional manifolds embedded in the quaternions

Title: Diffeological Clifford algebras and pseudo-bundles of Clifford modules
Journal: Linear and Multilinear Algebra, Published online: 15 May 2018
Authors: Ekaterina Pervova
Link: https://www.tandfonline.com/doi/full/10.1080/03081087.2018.1472202
We consider the diffeological version of the Clifford algebra of a diffeological finite dimensional vector space; we start by commenting on the notion of a diffeological algebra (which is the expected analogue of the usual one) and that of a diffeological module (also an expected counterpart of the usual notion). After considering the natural diffeology of the Clifford algebra, and considering which of its standard properties re-appear in the diffeological context (most of them), we turn to our main interest, which is constructing the pseudo-bundles of Clifford algebras associated to a given (finite dimensional) diffeological vector pseudo-bundle, and those of the usual Clifford modules (the exterior algebras). The substantial difference that emerges with respect to the standard context, and paves the way to various questions that do not have standard analogues, stems from the fact that the notion of a diffeological pseudo-bundle is very different from the usual bundle, and this under two main respects: it may have fibres of different dimensions, and even if it does not, its total and base spaces frequently are not smooth, or even topological, manifolds.

Title: Estimates of automorphic cusp forms over quaternion algebras
Journal: International Journal of Number Theory, Volume 14, Issue 04, May 2018
Authors: Anilatmaja Aryasomayajula, Baskar Balasubramanyam
Link: https://www.worldscientific.com/doi/abs/10.1142/S1793042118500719
In this paper, using methods from geometric analysis and theory of heat kernels, we derive qualitative estimates of automorphic cusp forms defined over quaternion algebras. Using which, we prove an average version of the holomorphic QUE conjecture. We then derive quantitative estimates of classical Hilbert modular cusp forms. This is a generalization of the results from [A. Aryasomayajula, Heat kernel approach for sup-norm bounds for cusp forms of integral and half-integral weight, Arch. Math. 106(2) (2016) 165–173; J. S. Friedman, J. Jorgenson and J. Kramer, Uniform sup-norm bounds on average for cusp forms of higher weights, in Arbeitstagung Bonn 2013, Progress in Mathematics, Vol. 319 (Birkhäuser/Springer, Cham, 2016), pp. 127–154] to higher dimensions.

Title: Galerkin Lie-group variational integrators based on unit quaternion interpolation
Journal: Computer Methods in Applied Mechanics and Engineering, Volume 338, 15 August 2018, Pages 333-361
Authors: Thomas Leitz, Sigrid Leyendecker
Link: https://www.sciencedirect.com/science/article/pii/S0045782518302019
Lie-group variational integrators of arbitrary order are developed using the Galerkin method, based on unit quaternion interpolation. To our knowledge, quaternions have not been used before for this purpose, though they allow a very simple and efficient way to perform the interpolation. The resulting integrators are symplectic and structure preserving, in the sense that certain symmetries in the Lagrangian of the mechanical system are carried over to the discrete setting, which leads to the preservation of the corresponding momentum maps. The integrators furthermore exhibit a very good long time energy behavior, i.e. energy is neither dissipated nor gained artificially. At the same time, the Lie-group structure is preserved by carefully defining the variations, the interpolation method and by solving the non-linear system of equations directly on the manifold, rather than constraining it in a surrounding space using Lagrange multipliers. As a consequence, we are able to show that Lie-group variational integrators based on the special orthogonal group, are equivalent to the variational integrators for constrained systems using the discrete null-space method employed e.g. in DMOCC (discrete mechanics and optimal control of constrained systems). We show new numerical results on the convergence rates, which are substantially higher than the known theoretical bounds, and on the relation between accuracy and computational cost.

Title: Involutions in split semi‐quaternions
Journal: Mathematical Methods in the Applied Sciences, First published: 09 May 2018
Authors: Murat Bekar, Yusuf Yayli
Link: https://onlinelibrary.wiley.com/doi/abs/10.1002/mma.4910
A map is an involution (resp, anti‐involution) if it is a self‐inverse homomorphism (resp, antihomomorphism) of a field algebra. The main purpose of this paper is to show how split semi‐quaternions can be used to express half‐turn planar rotations in 3‐dimensional Euclidean space R3 and how they can be used to express hyperbolic‐isoclinic rotations in 4‐dimensional semi‐Euclidean space R3,1. We present an involution and an anti‐involution map using split semi‐quaternions and give their geometric interpretations as half‐turn planar rotations in R3. Also, we give the geometric interpretation of nonpure unit split semi‐quaternions, which are in the form p = coshθ + sinhθi + 0j + 0k = coshθ + sinhθi, as hyperbolic‐isoclinic rotations in R3,1.

Title: On generalized inverses of dual matrices
Journal: Mechanism and Machine Theory, Volume 123, May 2018, Pages 89-106
Authors: Domenico de Falco, Ettore Pennestrì, Firdaus E.Udwadia
Link: https://www.sciencedirect.com/science/article/pii/S0094114X17312831
The present paper offers theoretical and numerical insights regarding the types of dual generalized inverses that can be computed from a recently proposed formula. Moreover, the usefulness of the formula is demonstrated solving different kinematic problems. In particular, thanks to the dual matrix generalized inverse formula availability, the computation of infinite and infinitesimal screw parameters motion from redundant point and line features is obtained within a unified theoretical treatment. Numerical examples and comparison with the results from previous investigations are provided.

Title: Quaternion generalized Chebyshev-Fourier and pseudo-Jacobi-Fourier moments for color object recognition
Journal: Optics & Laser Technology, Volume 106, October 2018, Pages 234-250
Authors: Chandan Singh, Jaspreet Singh
Link: https://www.sciencedirect.com/science/article/pii/S0030399217316365
In this paper, the classical generalized Chebyshev-Fourier moments (G-CHFMs) and generalized pseudo–Jacobi-Fourier moments (G-PJFMs) have been extended to represent color images using quaternion algebra. The proposed quaternion G-CHFMs (QG-CHFMs) and quaternion G-PJFMs (QG-PJFMs) are characterized by a parameter a, called free parameter, which distinguishes them from the conventional Chebyshev-Fourier moments (CHFMs) and pseudo-Jacobi-Fourier moments (PJFMs). All these moments are rotation-invariant and orthogonal. The effect of the parameter a on image reconstruction and object recognition is studied in detail and its optimal values have been obtained for these two image processing tasks. It is shown that the choice of a influences significantly the image reconstruction capability and the object recognition performance of the proposed QG-CHFMs and QG-PJFMs moments. Extensive experiments are conducted to demonstrate the behavior of these moments on image reconstruction and object recognition under normal condition and under rotation, scaling, and noise using COIL-100, SIMPLIcity and Corel datasets of color objects.

Title: Radiation Conditions and Integral Representations for Clifford Algebra-Valued Null-Solutions of the Helmholtz Operator
Journal: Journal of Mathematical Sciences, June 2018, Volume 231, Issue 3, pp 367–472
Authors: E. Marmolejo-Olea, I. Mitrea, D. Mitrea, M. Mitrea
Link: https://link.springer.com/article/10.1007/s10958-018-3826-9
The goal of this paper is to develop a unified approach to radiation conditions for the entire class of null-solutions of the Helmholtz operator which are Clifford algebra-valued. The latter is an algebraic context which permits the simultaneous consideration of scalar valued and vector-valued functions, as well as differential forms of any mixed degree. In such a setting, we provide a multitude of novel radiation conditions which naturally contain the classical Sommerfeld and Silver–Müller radiation conditions in the case of null-solutions for the scalar Helmholtz operator and the Maxwell system respectively, and which also encompass as a particular case the radiation condition introduced by McIntosh and Mitrea for perturbed Dirac operators.

Title: Solutions to matrix equations X – AXB = CY + R and X – AXB = CY + R
Journal: Journal of Computational and Applied Mathematics, Available online 9 May 2018, In Press, Accepted Manuscript
Authors: Caiqin Song, Guoliang Chen
Link: https://www.sciencedirect.com/science/article/pii/S0377042718302735
The present work proposed an alternative approach to find the closed-form solutions of the nonhomogeneous Yakubovich matrix equation X – AXB = CY + R. Based on the derived closed-form solution to the nonhomogeneous Yakubovich matrix equation, the solutions to the nonhomogeneous Yakubovich quaternion j-conjugate matrix equation X – AXB = CY + R are obtained by the use of the real representation of a quaternion matrix. The existing complex representation method requires the coefficient matrix A to be a block diagonal matrix over complex field. In contrast in this publication we allow a quaternion matrix of any dimension. As an application, eigenstructure assignment problem for descriptor linear systems is considered.

Title: Some Old Orthogonal Polynomials Revisited and Associated Wavelets: Two-Parameters Clifford-Jacobi Polynomials and Associated Spheroidal Wavelets
Journal: Acta Applicandae Mathematicae, June 2018, Volume 155, Issue 1, pp 177–195
Authors: Sabrine Arfaoui, Anouar Ben Mabrouk
Link: https://link.springer.com/article/10.1007/s10440-017-0150-1
In the present paper, new classes of wavelet functions are developed in the framework of Clifford analysis. Firstly, some classes of orthogonal polynomials are provided based on two-parameters weight functions generalizing the well known Jacobi and Gegenbauer classes when relaxing the parameters. The discovered polynomial sets are next applied to introduce new wavelet functions. Reconstruction formula as well as Fourier-Plancherel rule have been proved.

Title: The Standard Model Algebra – Leptons, Quarks, and Gauge from the Complex Clifford Algebra Cl6
Journal: arXiv:1702.04336 [hep-th]
Authors: Ovidiu Cristinel Stoica
Link: https://arxiv.org/abs/1702.04336
A simple geometric algebra is shown to contain automatically the leptons and quarks of a family of the Standard Model, and the electroweak and color gauge symmetries, without predicting extra particles and symmetries. The algebra is already naturally present in the Standard Model, in two instances of the Clifford algebra Cℓ6, one being algebraically generated by the Dirac algebra and the weak symmetry generators, and the other by a complex three-dimensional representation of the color symmetry, which generates a Witt decomposition which leads to the decomposition of the algebra into ideals representing leptons and quarks. The two instances being isomorphic, the minimal approach is to identify them, resulting in the model proposed here. The Dirac and Lorentz algebras appear naturally as subalgebras acting on the ideals representing leptons and quarks. The resulting representations on the ideals are invariant to the electromagnetic and color symmetries, which are generated by the bivectors of the algebra. The electroweak symmetry is also present, and it is already broken by the geometry of the algebra. The model predicts a bare Weinberg angle θW given by sin2θW=0.25. The model shares common ideas with previously known models, particularly with Chisholm and Farwell, 1996, Trayling and Baylis, 2004, and Furey, 2016.

Title: The horizontal heat kernel on the quaternionic anti de-Sitter spaces and related twistor spaces
Journal: arXiv:1805.06796 [math.DG]
Authors: Fabrice Baudoin, Nizar Demni, Jing Wang
Link: https://arxiv.org/abs/1805.06796
The geometry of the quaternionic anti-de Sitter fibration is studied in details. As a consequence, we obtain formulas for the horizontal Laplacian and subelliptic heat kernel of the fibration. The heat kernel formula is explicit enough to derive small time asymptotics. Related twistor spaces and corresponding heat kernels are also discussed and the connection to the quaternionic magnetic Laplacian is done.

Source: Email from N. Velous, nek.valous_AT_nct-heidelberg.de, 23/05/2018 23:32.


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