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Electromagnetic Vector Wave and Polarization Descriptors
Published in Jong-Sen Lee, Eric Pottier, Polarimetric Radar Imaging, 2017
where the matrices verify σi−1=σiT* and |det(σi)|=1. These matrices are a representation of the quaternion group with the following multiplicative table [15]:
Elementary Algebra
Published in Dan Zwillinger, CRC Standard Mathematical Tables and Formulas, 2018
The division ring of quaternions is the ring ({a + bi + cj + dk | a, b, c, d ∈ R}, +, ·), where operations are carried out using the rules for polynomial addi- tion and multiplication and the defining relations for the quaternion group Q.
Normal Bipolar Soft Subgroups
Published in Fuzzy Information and Engineering, 2021
Faruk Karaaslan, Aman Ullah, Imtiaz Ahmad
Assume that is a universal set and , quaternion group and , subgroup of G, be the subset of parameter set G. We define the BS-group by and we define a BS-set by It is clear that is a NBS-subgroup of BS-group over U.
Modelling and control of a spherical pendulum via a non–minimal state representation
Published in Mathematical and Computer Modelling of Dynamical Systems, 2021
Ricardo Campa, Israel Soto, Omar Martínez
The proof that the quaternion multiplication (6) is closed and associative relies in using the unit norm constraint (5) for each operand, and some of the above properties of the skew–symmetric operator. The identity element of the unit quaternion group is , and the inverse element of is , so that
Hopfions, heliknotons, skyrmions, torons and both abelian and nonabelian vortices in chiral liquid crystals
Published in Liquid Crystals Reviews, 2022
Jin-Sheng Wu, Ivan I. Smalyukh
Besides dislocations composed of disclinations, the elastic energy of individual disclinations can be numerically computed using tensorial order parameter given by Equation (14). (In the cases of half-integer defects, directors cannot be vectorized without introducing additional singularities like wall defects connecting them.) Under parameters for common LCs like 5CB, τ lines have higher elastic energies compared with χ lines, with λ lines being most stable disclinations due to the absence of singularity in the molecular field. Furthermore, since these disclinations are represented by different elements in the same homotopy group (the quaternion group Q8), their interaction with and transformation into each other are thoroughly characterized by the multiplication rules of the group [23,64,65,108,117,118] (Figure 6c). This provides an explanation for the lack of experimental observation of an isolated τ line, which is expected to quickly transform into pairs of other disclinations due to its high energetic costs. However, as far as we are aware, there have not been clear experimental demonstrations of these multiplication rules involving cholesteric disclinations beyond the representation of defects by homotopy group elements. By employing director analysis of dislocations and disclinations in CLCs, we can see that the different types of dislocation cores in CLCs effectively comprise distinct arrangements of λ and τ lines embedded within helical quasilayers, or equivalently, pairs of half-integer defects within uniform far-field in terms of χ(r) directors. The quaternion interpretation also provides deep insights into the geometries and energetics of the CLC defect lines. Interestingly, the classification of CLC disclinations as λ, τ and χ lines was introduced by Kleman and Friedel [7] well before the quaternion interpretation and homotopy (also co-invented by Kleman) theory classification of CLC defects was introduced and appreciated [21,22,65,102], but it naturally became an important part of it.