9 papers, 1 preprint related to Clifford Algebra

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

9 Papers:
Title: A dual quaternion algorithm of the Helmert transformation problem
Earth, Planets and Space, 70:26, December 2018
Huaien Zeng, Xing Fang, Guobin Chang, Ronghua Yang
Rigid transformation including rotation and translation can be elegantly represented by a unit dual quaternion. Thus, a non-differential model of the Helmert transformation (3D seven-parameter similarity transformation) is established based on unit dual quaternion. This paper presents a rigid iterative algorithm of the Helmert transformation using dual quaternion. One small rotation angle Helmert transformation (actual case) and one big rotation angle Helmert transformation (simulative case) are studied. The investigation indicates the presented dual quaternion algorithm (QDA) has an excellent or fast convergence property. If an accurate initial value of scale is provided, e.g., by the solutions no. 2 and 3 of Závoti and Kalmár (Acta Geod Geophys 51:245–256, 2016) in the case that the weights are identical, QDA needs one iteration to obtain the correct result of transformation parameters; in other words, it can be regarded as an analytical algorithm. For other situations, QDA requires two iterations to recover the transformation parameters no matter how big the rotation angles are and how biased the initial value of scale is. Additionally, QDA is capable to deal with point-wise weight transformation which is more rational than those algorithms which simply take identical weights into account or do not consider the weight difference among control points. From the perspective of transformation accuracy, QDA is comparable to the classic Procrustes algorithm (Grafarend and Awange in J Geod 77:66–76, 2003) and orthonormal matrix algorithm from Zeng (Earth Planets Space 67:105, 2015. https://doi.org/10.1186/s40623-015-0263-6).

Title: A finite element/quaternion/asymptotic numerical method for the 3D simulation of flexible cables
Journal: Finite Elements in Analysis and Design, Volume 139, February 2018, Pages 14-34
Authors: Emmanuel Cottanceaua, Olivier Thomas, Philipp Véron, Marc Alochet, Renaud Deligny
Link: https://www.sciencedirect.com/science/article/pii/S0168874X17304882
In this paper, a method for the quasi-static simulation of flexible cables assembly in the context of automotive industry is presented. The cables geometry and behavior encourage to employ a geometrically exact beam model. The 3D kinematics is then based on the position of the centerline and on the orientation of the cross-sections, which is here represented by rotational quaternions. Their algebraic nature leads to a polynomial form of equilibrium equations. The continuous equations obtained are then discretized by the finite element method and easily recast under quadratic form by introducing additional slave variables. The asymptotic numerical method, a powerful solver for systems of quadratic equations, is then employed for the continuation of the branches of solution. The originality of this paper stands in the combination of all these methods which leads to a fast and accurate tool for the assembly process of cables. This is proved by running several classical validation tests and an industry-like example.

Title: A system of quaternary coupled Sylvester-type real quaternion matrix equations
Journal: Automatica, Volume 87, January 2018, Pages 25-31
Authors: Zhuo-Heng He, Qing-Wen Wang, Yang Zhang
Link: https://www.sciencedirect.com/science/article/pii/S0005109817304788
In this paper we establish some necessary and sufficient solvability conditions for a system of quaternary coupled Sylvester-type real quaternion matrix equations in terms of ranks and generalized inverses of matrices. An expression of the general solution to this system is given when it is solvable. Also, a numerical example is presented to illustrate the main result of this paper. The findings of this paper widely generalize the known results in the literature. The main results are also valid over the real number field and the complex number field.

Title: Cartan frames and algebras with links to integrable systems differential equations and surfaces
Journal: Journal of Mathematical Physics 59, 021504 (2018)
Authors: Paul Bracken
Link: http://aip.scitation.org/doi/full/10.1063/1.5005816
Moving frames and Clifford algebras will be used to illustrate an interconnected approach to the study of integrable systems, their surfaces, and methods for producing integrable partial differential equations. After a system of one-forms is defined, moving frame equations can be integrated and the resulting equations for the surface can be obtained. Other differential equations which involve quantities relevant to the surface are obtained. For the case of minimal and constant mean curvature surfaces, the coordinate functions can be calculated in closed form. In the case of constant mean curvature, they can be expressed in terms of Jacobi elliptic functions.

Title: Clifford algebra valued boundary integral equations for three-dimensional elasticity
Journal: Applied Mathematical Modelling, Volume 54, February 2018, Pages 246-267
Authors: Li-Wei Liu, Hong-Ki Hong
Link: https://www.sciencedirect.com/science/article/pii/S0307904X17305838
Applications of Clifford analysis to three-dimensional elasticity are addressed in the present paper. The governing equation for the displacement field is formulated in terms of the Dirac operator and Clifford algebra valued functions so that a general solution is obtained analytically in terms of one monogenic function and one multiple-component spatial harmonic function together with its derivative. In order to solve numerically the three-dimensional problems of elasticity for an arbitrary domain with complicated boundary conditions, Clifford algebra valued boundary integral equations (BIEs) for multiple-component spatial harmonic functions at an observation point, either inside the domain, on the boundary, or outside the domain, are constructed. Both smooth and non-smooth boundaries are considered in the construction. Moreover, the singularities of the integrals are evaluated exactly so that in the end singularity-free BIEs for the observation point on the boundary taking values on Clifford numbers can be obtained. A Clifford algebra valued boundary element method (BEM) based on the singularity-free BIEs is then developed for solving three-dimensional problems of elasticity. The accuracy of the Clifford algebra valued BEM is demonstrated numerically.

Title: Coordinate-invariant rigid-body interpolation on a parametric C1 dual quaternion curve
Journal: Mechanism and Machine Theory, Volume 121, March 2018, Pages 731-744
Authors: Felix Allmendinger, Sami Charaf  Eddine, Burkhard Corves
Link: https://www.sciencedirect.com/science/article/pii/S0094114X17312119
We present a method to generate first-order continuous rigid-body motion by interpolation. The input is a sequence of rigid-body poses at given timesteps, which the body is required to pass through (key poses). Different from frequently employed interpolation schemes, the generated rigid-body motion is unique no matter what reference coordinate systems are chosen. Our method is novel in that the user can optionally prescribe key velocity data, too. If key velocities are not prescribed, parametric velocities are computed and incorporated into the interpolating function. The parameters allow to subsequently adjust the rigid-body trajectory. Another purpose of this article is a comprehensive derivation of coordinate-invariant interpolation along with a concise collection of proofs. The derivation enables the reader to straight-forwardly implement this method. Numerical examples are given to highlight the benefits and motivate the implementation.

Title: Dual number algebra method for Green’s function derivatives in 3D magneto-electro-elasticity
Journal: AIP Conference Proceedings 1922, 140005 (2018)
Authors: Grzegorz Dziatkiewicz
Link: http://aip.scitation.org/doi/abs/10.1063/1.5019147
The Green functions are the basic elements of the boundary element method. To obtain the boundary integral formulation the Green function and its derivative should be known for the considered differential operator. Today the interesting group of materials are electronic composites. The special case of the electronic composite is the magnetoelectroelastic continuum. The mentioned continuum is a model of the piezoelectric-piezomagnetic composites. The anisotropy of their physical properties makes the problem of Green’s function determination very difficult. For that reason Green’s functions for the magnetoelectroelastic continuum are not known in the closed form and numerical methods should be applied to determine such Green’s functions. These means that the problem of the accurate and simply determination of Green’s function derivatives is even harder. Therefore in the present work the dual number algebra method is applied to calculate numerically the derivatives of 3D Green’s functions for the magnetoelectroelastic materials. The introduced method is independent on the step size and it can be treated as a special case of the automatic differentiation method. Therefore, the dual number algebra method can be applied as a tool for checking the accuracy of the well-known finite difference schemes.

Title: Generating five-dimensional Myers–Perry black hole solution using quaternions
Journal: Annals of Physics, Volume 389, February 2018, Pages 11-18
Authors: Zahra Mirzaiyan, Behrouz Mirza, Elham Sharifian
Link: https://www.sciencedirect.com/science/article/pii/S0003491617303615
Newman–Janis and Giampieri algorithms are two simple methods to generate stationary rotating black hole solutions in four dimensions. In this paper, we obtain the Myers–Perry black hole from the Schwarzschild solution in five dimensions by using quaternions. Our method generates the Myers–Perry black hole solution with two angular momenta in one fell swoop.

Title: Global μ-stability of quaternion-valued neural networks with mixed time-varying delays
Journal: Neurocomputing, In press, Available online 14 February 2018
Authors: Xingxing You, Qiankun Song, Jing Liang, Yurong Liu, Fuad E. Alsaadid
Link: https://www.sciencedirect.com/science/article/pii/S0925231218301656
In this paper, the problem of global μ-stability for quaternion-valued neural networks with time-varying delays and unbounded distributed delays is investigated. To avoid the non-commutativity of quaternion multiplication, the quaternion-valued neural networks is decomposed into two complex-valued systems. By employing the homomorphic mapping principle, a sufficient condition for the existence and uniqueness of equilibrium point of the considered quaternion-valued neural networks is proposed in the form of linear matrix inequality (LMI) in complex-valued domain. Further, the appropriate Lyapunov-krasovkii functional is constructed in the Hermitian quadratic form, and sufficient condition to ensure the global μ-stability of the equilibrium point is obtained by using inequality technique. Finally, two numerical examples with simulations are provided to verify the effectiveness of the obtained results.

1 Preprint:
Title: New spinor classes on the Graf-Clifford algebra
Journal: arXiv:1802.06413, Submitted on 18 Feb 2018
Authors: R. Lopes, R. da Rocha
Abstract: Pinors and spinors are defined as sections of the subbundles whose fibers are the representation spaces of the Clifford algebra of the forms equipped with the Graf product. In this context, pinors and spinors are here considered and the geometric generalized Fierz identities provide the necessary framework to derive and construct new spinor classes on the space of smooth sections of the exterior bundle, endowed with the Graf product, for prominent specific signatures.

Source: Email from Nek Valous, nek.valous_AT_nct-heidelberg.de, 20 Feb. 2018.


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