12 papers related to Clifford algebra


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

Title: From Vectors to Geometric Algebra

Journal: arXiv:1802.08153 [math.GM]
Authors: Sergio Ramos Ramirez, Jose Alfonso Juarez Gonzalez, Garret Sobczyk
Link: https://arxiv.org/abs/1802.08153
Abstract: Geometric algebra is the natural outgrowth of the concept of a vector and the addition of vectors. After reviewing the properties of the addition of vectors, a multiplication of vectors is introduced in such a way that it encodes the famous Pythagorean theorem. Synthetic proofs of theorems in Euclidean geometry can then be replaced by powerful algebraic proofs. Whereas we largely limit our attention to 2 and 3 dimensions, geometric algebra is applicable in any number of dimensions, and in both Euclidean and non-Euclidean geometries.

Title: Applications of parabolic Dirac operators to the instationary viscous MHD equations on conformally flat manifolds
Journal:
arXiv:1804.09551 [math.AP]
Authors:
Paula Cerejeiras, Uwe Kähler, Rolf Sören Kraußhar
Link:
https://arxiv.org/abs/1804.09551
Abstract:
In this paper we apply classical and recent techniques from quaternionic analysis using parabolic Dirac type operators and related Teodorescu and Cauchy-Bitzadse type operators to set up some analytic representation formulas for the solutions to the time depedendent incompressible viscous magnetohydrodynamic equations on some conformally flat manifolds, such as cylinders and tori associated with different spinor bundles. Also in this context a special variant of hypercomplex Eisenstein series related to the parabolic Dirac operator serve as kernel functions.


Title: Algebraic techniques for Schrödinger equations in split quaternionic mechanics
Journal: Computers & Mathematics with Applications, Volume 75, Issue 7, 1 April 2018, Pages 2217-2222
Authors: Tongsong Jiang, Zhaozhong Zhang, Ziwu Jiang
Link: https://www.sciencedirect.com/science/article/pii/S0898122117307691
Abstract: The split quaternionic Schrödinger equation  plays an important role in split quaternionic mechanics, in which  a split quaternion matrix. This paper, by means of a real representation of split quaternion matrices, studies problems of split quaternionic Schrödinger equation, and gives an algebraic technique for the split quaternionic Schrödinger equation. This paper also derives an algebraic technique for finding eigenvalues and eigenvectors of a split quaternion matrix in split quaternionic mechanics.


Title: On the equivalence of quaternionic contact structures
Journal: Annals of Global Analysis and Geometry, April 2018, Volume 53, Issue 3, pp 331–375
Authors: Ivan Minchev, Jan Slovák
Link: https://link.springer.com/article/10.1007%2Fs10455-017-9580-2
Abstract: Following the Cartan’s original method of equivalence supported by methods of parabolic geometry, we provide a complete solution for the equivalence problem of quaternionic contact structures, that is, the problem of finding a complete system of differential invariants for two quaternionic contact manifolds to be locally diffeomorphic. This includes an explicit construction of the corresponding Cartan geometry and detailed information on all curvature components.

Title: On the global interpolation of motion
Journal: Computer Methods in Applied Mechanics and Engineering, Available online 9 April 2018, In Press, Accepted Manuscript
Authors: Shilei Han, Olivier A. Bauchau
Link: https://www.sciencedirect.com/science/article/pii/S0045782518301725
Abstract: Interpolation of motion is required in various fields of engineering such as computer animation and vision, trajectory planning for robotics, optimal control of dynamical systems, or finite element analysis. While interpolation techniques in the Euclidean space are well established, general approaches to interpolation on manifolds remain elusive. Interpolation schemes in the Euclidean space can be recast as minimization problems for weighted distance metrics. This observation allows the straightforward generalization of interpolation in the Euclidean space to interpolation on manifolds, provided that a metric of the manifold is defined. This paper proposes four metrics of the motion manifold: the matrix, quaternion, vector, and geodesic metrics. For each of these metrics, the corresponding interpolation schemes are derived and their advantages and drawbacks are discussed. It is shown that many existing interpolation schemes for rotation and motion can be derived from the minimization framework proposed here. The problems of averaging of rotation and motion can be treated easily within the same framework. Both local and global interpolation problems are addressed. The proposed interpolation framework can be used with any suitable set of basis functions. Examples are presented with Chebyshev spectral, Fourier spectral, and B-spline basis functions. This paper also introduces one additional approach to the interpolation of motion based on the interpolation of its derivatives. While this approach provides high accuracy, the associated computational cost is high and the approach cannot be used in multi-variable interpolation easily.

Title: On the Learning Machine with Compensatory Aggregation Based Neurons in Quaternionic Domain
Journal: Journal of Computational Design and Engineering, Available online 16 April 2018, In Press, Accepted Manuscript
Authors: Sushil Kumar, Bipin Kumar Tripathi
Link: https://www.sciencedirect.com/science/article/pii/S2288430018300150
Abstract: The nonlinear spatial grouping process of synapses is one of the fascinating methodologies for neuro-computing researchers to achieve the computational power of a neuron. Generally, researchers use neuron models that are based on summation (linear), product (linear) or radial basis (nonlinear) aggregation for the processing of synapses, to construct multi-layered feed-forward neural networks, but all these neuron models and their corresponding neural networks have their advantages or disadvantages. The multi-layered network generally uses for accomplishing the global approximation of input-output mapping but sometimes getting stuck into local minima, while the nonlinear radial basis function (RBF) network is based on exponentially decaying that uses for local approximation to input-output mapping. Their advantages and disadvantages motivated to design two new artificial neuron models based compensatory aggregation functions in quaternionic domain. The net internal potentials of these neuron models are developed with the compositions of basic summation (linear) and radial basis (nonlinear) operations on quaternionic-valued input signals. The neuron models based on these aggregation functions ensure faster convergence, better training, and prediction accuracy. The learning and generalization capabilities of these neurons are verified through various three-dimensional transformations and time series predictions as benchmark problems.

Title: Robust copy–move forgery detection using quaternion exponent moments
Journal: Pattern Analysis and Applications, May 2018, Volume 21, Issue 2, pp 451–467
Authors: Xiang-yang Wang, Yu-nan Liu, Huan Xu, Pei Wang, Hong-ying Yang
Link: https://link.springer.com/article/10.1007%2Fs10044-016-0588-1
Abstract: The detection of forgeries in color images is a very important topic in forensic science. Copy–move (or copy–paste) forgery is the most common form of tampering associated with color images. Conventional copy–move forgeries detection techniques usually suffer from the problems of false positives and susceptibility to many signal processing operations. It is a challenging work to design a robust copy–move forgery detection method. In this paper, we present a novel block-based robust copy–move forgery detection approach using invariant quaternion exponent moments (QEMs). Firstly, original tempered color image is preprocessed with Gaussian low-pass filter, and the filtered color image is divided into overlapping circular blocks. Then, the accurate and robust feature descriptor, QEMs modulus, is extracted from color image block holistically as a vector field. Finally, exact Euclidean locality sensitive hashing is utilized to find rapidly the matching blocks, and the falsely matched block pairs are removed by customizing the random sample consensus with QEMs magnitudes differences. Extensive experimental results show the efficacy of the newly proposed approach in detecting copy–paste forgeries under various challenging conditions, such as noise addition, lossy compression, scaling, and rotation. We obtain the average forgery detection accuracy (F-measure) in excess of 96 and 88% across postprocessing operations, at image level and at pixel level, respectively.

Title: Robust stability analysis of quaternion-valued neural networks via LMI approach
Journal: Advances in Difference Equations, 2018:131
Authors: Xiaofeng Chen, Lianjie Li, Zhongshan Li
Link: https://advancesindifferenceequations.springeropen.com/articles/10.1186/s13662-018-1585-z
Abstract: This paper is concerned with the issue of robust stability for quaternion-valued neural networks (QVNNs) with leakage, discrete and distributed delays by employing a linear matrix inequality (LMI) approach. Based on the homeomorphic mapping theorem, the quaternion matrix theorem and the Lyapunov theorem, some criteria are developed in the form of real-valued LMIs for guaranteeing the existence, uniqueness, and global robust stability of the equilibrium point of the delayed QVNNs. Two numerical examples are provided to demonstrate the effectiveness of the obtained results.

Title: Slice Regular Malmquist–Takenaka System in the Quaternionic Hardy Spaces
Journal: Analysis Mathematica, March 2018, Volume 44, Issue 1, pp 99–114
Authors: M. Pap
Link: https://link.springer.com/article/10.1007%2Fs10476-018-0109-0
Abstract: In this paper the slice regular analogue of the Malmquist–Takenaka system is introduced. It is proved that, under certain restrictions regarding to the parameters of the system, they form a complete orthonormal system in the quaternionic Hardy spaces of the unit ball. The properties of associated projection operator are also studied.

Title: The Standard Model in noncommutative geometry: fundamental fermions as internal forms
Journal: Letters in Mathematical Physics,
May 2018, Volume 108, Issue 5, pp 1323–1340
Authors: Ludwik Dąbrowski, Francesco D’Andrea, Andrzej Sitarz
Link: https://link.springer.com/article/10.1007%2Fs11005-017-1036-x
Abstract: Given the algebra, Hilbert space H, grading and real structure of the finite spectral triple of the Standard Model, we classify all possible Dirac operators such that H is a self-Morita equivalence bimodule for the associated Clifford algebra.

Title: Volumes and distributions for random unimodular complex and quaternion lattices
Journal:
Journal of Number Theory, Available online 22 April 2018, In Press, Corrected Proof
Authors: Peter J. Forrestera, Jiyuan Zhang
Link: https://www.sciencedirect.com/science/article/pii/S0022314X18300970
Abstract: Two themes associated with invariant measures on the matrix groups SLN(F), with F=R,C or H, and their corresponding lattices parametrised by SLN(F)/SLN(O), O being an appropriate Euclidean ring of integers, are considered. The first is the computation of the volume of the subset of SLN(F) with bounded 2-norm or Frobenius norm. Key here is the decomposition of measure in terms of the singular values. The form of the volume, for large values of the bound, is relevant to asymptotic counting problems in SLN(O). The second is the problem of lattice reduction in the case N=2. A unified proof of the validity of the appropriate analogue of the Lagrange–Gauss algorithm for computing the shortest basis is given. A decomposition of measure corresponding to the QR decomposition is used to specify the invariant measure in the coordinates of the shortest basis vectors. With F=C this allows for the exact computation of the PDF of the first minimum (for O=Z[i] and View the MathML source), and the PDF of the second minimum and that of the angle between the minimal basis vectors (for O=Z[i]). It also encodes the specification of fundamental domains of the corresponding quotient spaces. Integration over the latter gives rise to certain number theoretic constants, which are also present in the asymptotic forms of the PDFs of the lengths of the shortest basis vectors. Siegel’s mean value gives an alternative method to compute the arithmetic constants, allowing in particular the computation of the leading form of the PDF of the first minimum for F=H and O the Hurwitz integers, for which direct integration was not possible.

Title: k-Pell, k-Pell–Lucas and modified k-Pell sedenions
Journal:
Asian-European Journal of Mathematics, Online Ready
Authors: Paula Catarino
Link: https://www.worldscientific.com/doi/abs/10.1142/S1793557119500189
Abstract: The aim of this work is to present the k-Pell, the k-Pell–Lucas and the Modified k-Pell sedenions and we give some properties involving these sequences, including the Binet-style formulae and the ordinary generating functions.

Source: Email from N. Velous, nek.valous_AT_nct-heidelberg.de, 30/04/2018 23:17.

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