Seven new Clifford algebra papers


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

F. Torunbalcı Aydın, Bicomplex Fibonacci quaternions, Chaos, Solitons & Fractals, Volume 106, January 2018, Pages 147-153, URL:
https://www.sciencedirect.com/science/article/pii/S0960077917304812, Note, free PDF download also available from this URL.

Abstract: In this paper, bicomplex Fibonacci quaternions are defined. Also, some algebraic properties of bicomplex Fibonacci quaternions which are connected with bicomplex numbers and Fibonacci numbers are investigated. Furthermore, Binet’s formula, Cassini’s identity, Catalan’s identity for these quaternions and real representation of these quaternions are given.

Keywords: Bicomplex number, Fibonacci number, Fibonacci quaternion, Bicomplex quaternion, Bicomplex Fibonacci quaternion

Pei Dang, José Mourão, João P. Nunes, TaoQ ian, Clifford coherent state transforms on spheres, Journal of Geometry and Physics, Volume 124, January 2018, Pages 225-232, URL: https://www.sciencedirect.com/science/article/pii/S0393044017302735

Abstract: We introduce a one-parameter family of transforms, , , from the Hilbert space of Clifford algebra valued square integrable functions on the –dimensional sphere, , to the Hilbert spaces, , of solutions of the Euclidean Dirac equation on which are square integrable with respect to appropriate measures, . We prove that these transforms are unitary isomorphisms of the Hilbert spaces and are extensions of the Segal–Bargman coherent state transform, , to higher dimensional spheres in the context of Clifford analysis. In Clifford analysis it is natural to replace the analytic continuation from to as in (Hall, 1994; Stenzel, 1999; Hall and Mitchell, 2002) by the Cauchy–Kowalewski extension from to . One then obtains a unitary isomorphism from an –Hilbert space to a Hilbert space of solutions of the Dirac equation, that is to a Hilbert space of monogenic functions. (Note: mathematical symbols missing in this text.)

Keywords: Clifford analysis, Coherent state transforms, Cauchy–Kowalewski extension

Guangbin Ren, Zeping Zhu, Discrete Complex Analysis in Split Quaternions, Complex Analysis and Operator Theory, February 2018, Volume 12, Issue 2, pp 415–438, URL: https://link.springer.com/article/10.1007/s11785-017-0664-6

Abstract: A natural question in discrete complex analysis is whether the Taylor series of a discrete holomorphic function is convergent to itself in the whole grid ℤ2h. In this paper we answer this question in the affirmative in the setting of a new kind of discrete holomorphic function on the square grid ℤ2h with values in split quaternions based on the methods of Sheffer sequences. On the other hand, we also establish the integral theory for this new kind of discrete holomorphic functions, including the discrete Green theorem and the Cauchy integral formula. In contrast to the discrete Clifford analysis, we obtain a new version of the discrete Cauchy integral formula without the extra error term.

Keywords: Discrete holomorphic function, Sheffer sequence, Taylor series expansion, Cauchy–Pompeiu formula

Yuri Grigor’ev, Quaternionic Functions and Their Applications in a Viscous Fluid Flow, Complex Analysis and Operator Theory, February 2018, Volume 12, Issue 2, pp 491–508, URL: https://link.springer.com/article/10.1007/s11785-017-0715-z

Abstract: We used a quaternion function method for the Moisil–Theodoresco system (MTS). Solutions of the MTS are (left-) regular quaternion functions f(r)=f0(r)+f(r)=f0(x,y,z)+ifx(x,y,z)+jfy(x,y,z)+rfz(x,y,z) of a reduced quaternion variable r=ix+jy+kz. Here we present the quaternion three-dimensional representation of a general solution of the Stokes system for the slow flow of a viscous fluid in star-shaped domain. It is shown that in particular cases of plane and axially symmetric flows this representation goes into the representations by means of analytical and generalized analytical functions of complex variables. As applications the main problems the Stokes flow in a ball are solved.

Keywords:
Regular quaternion functions, Moisil–Theodoresco system, Viscous fluid flow Stokes system

Agnieszka Szczęsna, Verification of the blobby quaternion model of human joint limits, Biomedical Signal Processing and Control, Volume 39, January 2018, Pages 130-138. URL: https://www.sciencedirect.com/science/article/pii/S1746809417301611
, Note, free PDF download also available from this URL.

Abstract: The quaternion blobby model for constraining the joint range of motions based on real captured data has been proposed. The boundary of the feasible region is modeled using a geometric approach. The proposed method aims at generating an implicit representation of quaternion volume field boundaries which represent the space of all possible and permitted orientations in the joint. The implicit surface is generated as an isosurface of quaternion volume. This approximation volume is determined based on data captured by the optical motion capture system and transformed to unit quaternions. The isosurface is generated from the blobby model which is popular as a solid object modeling tool in computer graphics. The obtained quaternion orientation space represents valid orientations and allows to reproject any orientation to the nearest valid ones. The model was verified based on motion captured shoulder joint data.

Keywords: Quaternion, Isosurface, Quaternion volume, Joint limit, Motion capture

Julien Flamant, Pierre Chainais, Nicolas Le Bihan, A complete framework for linear filtering of bivariate signals, Submitted 7 Feb 2018 to Signal Processing [eess.SP], Published 8 Feb 2018, URL: https://scirate.com/arxiv/1802.02469, http://arxiv.org/abs/1802.02469, PDF: http://arxiv.org/pdf/1802.02469.pdf, Author comments: 11 pages, 3 figures

Abstract: A complete framework for the linear time-invariant (LTI) filtering theory of bivariate signals is proposed based on a tailored quaternion Fourier transform. This framework features a direct description of LTI filters in terms of their eigenproperties enabling compact calculus and physically interpretable filtering relations in the frequency domain. The design of filters exhibiting fondamental properties of polarization optics (birefringence, diattenuation) is straightforward. It yields an efficient spectral synthesis method and new insights on Wiener filtering for bivariate signals with prescribed frequency-dependent polarization properties. This generic framework facilitates original descriptions of bivariate signals in two components with specific geometric or statistical properties. Numerical experiments support our theoretical analysis and illustrate the relevance of the approach on synthetic data.

Jouko Mickelsson, 3D Current Algebra and Twisted K Theory, (Submitted to arxiv on 8 Feb 2018), URL: https://arxiv.org/abs/1802.02806, PDF: https://arxiv.org/pdf/1802.02806

Abstract: Equivariant twisted K theory classes on compact Lie groups G can be realized as families of Fredholm operators acting in a tensor product of a fermionic Fock space and a representation space of a central extension of the loop algebra LG using a supersymmetric Wess-Zumino-Witten model. The aim of the present article is to extend the construction to higher loop algebras using an abelian extension of a 3D current algebra. We have only partial success: Instead of true Fredholm operators we have formal algebraic expressions in terms of the generators of the current algebra and an infinite dimensional Clifford algebra. These give rise to sesquilinear forms in a Hilbert bundle which transform in the expected way with respect to 3D gauge transformations but do not define true Hilbert space operators.

Source: Emails from Nek Valous, nek.valous_AT_nct-heidelberg.de, 06/02/2018 04:45, 10/02/2018 05:12, https://www.sciencedirect.com/science/article/pii/S0960077917304812, https://www.sciencedirect.com/science/article/pii/S0393044017302735, https://link.springer.com/article/10.1007/s11785-017-0664-6, https://link.springer.com/article/10.1007/s11785-017-0715-z, https://www.sciencedirect.com/science/article/pii/S1746809417301611, https://scirate.com/arxiv/1802.02469, https://arxiv.org/abs/1802.02806, accessed: 12 Feb. 2018

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