Latest Results for Advances in Applied Clifford Algebras
Geometric Data Manipulation with Clifford Algebras and Möbius Transforms
2015-09-21 03:00:00 AM
Abstract
We propose the usage of Möbius transformations, defined in the context of Clifford algebras, for geometrically manipulating a point cloud data lying in a vector space of arbitrary dimension. We present this method as an application to signal classification in a dimensionality reduction framework. We first discuss a general situation where data analysis problems arise in signal processing. In this context, we introduce the construction of special Möbius transformations on vector spaces \({\mathbb{R}^n}\) , customized for a classification setting. A computational experiment is presented indicating the potential and shortcomings of this framework.Canal Surfaces with Quaternions
2015-09-19 03:00:00 AM
Abstract
Quaternions are more usable than three Euler angles in the three dimensional Euclidean space. Thus, many laws in different fields can be given by the quaternions. In this study, we show that canal surfaces and tube surfaces can be obtained by the quaternion product and by the matrix representation. Also, we show that the equation of canal surface given by the different frames of its spine curve can be obtained by the same unit quaternion. In addition, these surfaces are obtained by the homothetic motion. Then, we give some results.Expansions for the Dirac Operator and Related Operators in Super Spinor Space
2015-09-19 03:00:00 AM
Abstract
In this paper, we study expansions for the Dirac operator D, the modified Dirac operator \({D-\lambda,}\) and the polynomial Dirac operator P(D) in super spinor space. These expansions are a meaningful generalization of the classical Almansi expansion in polyharmonic functions theory. As an application of the expansions, the generalized Riquier problem in super spinor space is investigated.Using the Symmetries of the Cube and Tetrahedron as a Vehicle for Introducing Some Clifford Algebra to High School Students and Undergraduate College Students
2015-09-16 03:00:00 AM
Abstract
This paper outlines how Clifford Algebra can be presented to high school or college students to deal with the symmetries of the Platonic solids. I concentrate on the cube but all five solids are discussed.Two-Sided Hypergenic Functions
2015-09-12 03:00:00 AM
Abstract
In this paper we present an analogous of the class of two-sided axial monogenic functions to the case of axial \({\kappa}\) -hypermonogenic functions. In order to do that we will solve a Vekua-type system in terms of Bessel functions.The $${h(x)}$$ h ( x ) -Fibonacci Quaternion Polynomials: Some Combinatorial Properties
2015-09-12 03:00:00 AM
Abstract
In this paper we consider the \({h(x)}\) -Fibonacci quaternion polynomials and present some properties involving these polynomials, including the exponential and Poisson generating functions.On Rotation About Lightlike Axis in Three-Dimensional Minkowski Space
2015-09-10 03:00:00 AM
Abstract
We obtain matrix of the rotation about arbitrary lightlike axis in three-dimensional Minkowski space by deriving the Rodrigues’ rotation formula and using the corresponding Cayley map. We prove that a unit timelike split quaternion q with a lightlike vector part determines rotation R q about lightlike axis and show that a split quaternion product of two unit timelike split quaternions with null vector parts determines the rotation about a spacelike, a timelike or a lightlike axis. Finally, we give some examples.A Clifford Bundle Approach to the Wave Equation of a Spin 1/2 Fermion in the de Sitter Manifold
2015-09-09 03:00:00 AM
Abstract
In this paper we give a Clifford bundle motivated approach to the wave equation of a free spin 1/2 fermion in the de Sitter manifold, a brane with topology \({M=\mathrm {S0}(4,1)/\mathrm {S0}(3,1)}\) living in the bulk spacetime \({{\mathbb{R}^{4,1}}=(\mathring{M}=\mathbb{R}^5,\boldsymbol{\mathring{g}})}\) and equipped with a metric field \({\boldsymbol{g}:\boldsymbol{=}-\boldsymbol{i}^{\ast} \boldsymbol{\mathring{g}}}\) with\({\boldsymbol{i}:M\rightarrow \mathring{M}}\) being the inclusion map. To obtain the analog of Dirac equation in Minkowski spacetime in the structure \({\mathring{M}}\) we appropriately factorize the two Casimir invariantsC 1 and C 2 of the Lie algebra of the de Sitter group using the constraint given in the linearization of C 2 as input to linearize C 1. In this way we obtain an equation that we called DHESS1, which in previous studies by other authors was simply postulated. Next we derive a wave equation (called DHESS2) for a free spin 1/2 fermion in the de Sitter manifold using a heuristic argument which is an obvious generalization of a heuristic argument (described in detail in Appendix D) permitting a derivation of the Dirac equation in Minkowski spacetime and which shows that such famous equation express nothing more than the fact that the momentum of a free particle is a constant vector field over timelike integral curves of a given velocity field. It is a remarkable fact thatDHESS1 and DHESS2 coincide. One of the main ingredients in our paper is the use of the concept of Dirac-Hestenes spinor fields. Appendices B and C recall this concept and its relation with covariant Dirac spinor fields usually used by physicists.Point Particle with Extrinsic Curvature as a Boundary of a Nambu–Goto String: Classical and Quantum Model
2015-09-09 03:00:00 AM
Abstract
It is shown how a string living in a higher dimensional space can be approximated as a point particle with squared extrinsic curvature. We consider a generalized Howe–Tucker action for such a “rigid particle” and consider its classical equations of motion and constraints. We find that the algebra of the Dirac brackets between the dynamical variables associated with velocity and acceleration contains the spin tensor. After quantization, the corresponding operators can be represented by the Dirac matrices, projected onto the hypersurface that is orthogonal to the direction of momentum. A condition for the consistency of such a representation is that the states must satisfy the Dirac equation with a suitable effective mass. The Pauli–Lubanski vector composed with such projected Dirac matrices is equal to the Pauli–Lubanski vector composed with the usual, non projected, Dirac matrices, and its eigenvalues thus correspond to spin one half states.Three-Term Recurrence Relations for Systems of Clifford Algebra-Valued Orthogonal Polynomials
2015-09-08 03:00:00 AM
Abstract
Recently, systems of Clifford algebra-valued orthogonal polynomials have been studied from different points of view. We prove in this paper that for their building blocks there exist some three-term recurrence relations, similar to that for orthogonal polynomials of one real variable. As a surprising byproduct of own interest we found out that the whole construction process of Clifford algebra-valued orthogonal polynomials via Gelfand–Tsetlin basis or otherwise relies only on one and the same basic Appell sequence of polynomials.One-Parameter Homothetic Motions and Euler-Savary Formula in Generalized Complex Number Plane $${\mathbb{C}_{J}}$$ C J
2015-09-04 03:00:00 AM
Abstract
In this paper, we introduce one-parameter homothetic motions in the generalized complex number plane ( \({\mathfrak{p}}\) -complex plane) $$\mathbb{C}_{J}=\left\{x+Jy:\,\,\, x,y \in \mathbb{R},\quad J^2=\mathfrak{p},\quad \mathfrak{p} \in \{-1,0,1\} \right\} \subset \mathbb{C}_\mathfrak{p}$$ where$$\mathbb{C}_\mathfrak{p}=\{x+Jy:\,\,\, x,y \in \mathbb{R}, \quad J^2=\mathfrak{p}\}$$ such that \({-\infty < \mathfrak{p} < \infty}\) . The velocities, accelerations and pole points of the motion are analysed. Moreover, three generalized complex number planes, of which two are moving and the other one is fixed, are considered and a canonical relative system for one-parameter planar homothetic motion in \({\mathbb{C}_{J}}\) is defined. Euler-Savary formula, which gives the relationship between the curvatures of trajectory curves, during the one-parameter homothetic motions, is obtained with the aim of this canonical relative system.The Fermi-Walker Derivative on the Spherical Indicatrix of a Space Curve
2015-09-03 03:00:00 AM
Abstract
In this paper Fermi-Walker derivative and Fermi-Walker parallelism and non-rotating frame concepts are given along the spherical indicatrix of a curve in E 3. First, we consider a curve in Euclid space and investigate the Fermi-Walker derivative along the tangent. The concepts which Fermi-Walker derivative are analyzed along its tangent. Then, the Fermi-Walker derivative and its theorems are analyzed along the principal normal indicatrix and the binormal indicatrix of any curve in E 3.Extended Lorentz Transformations in Clifford Space Relativity Theory
2015-09-01 03:00:00 AM
Abstract
Some novel physical consequences of the Extended Relativity Theory in C-spaces (Clifford spaces) were explored recently. In particular, generalized photon dispersion relations allowed for energy-dependent speeds of propagation while still retaining the Lorentz symmetry in ordinary spacetimes, but breaking the extended Lorentz symmetry in C-spaces. In this work we analyze in further detail the extended Lorentz transformations in Clifford Space and their physical implications. Based on the notion of “extended events” one finds a very different physical explanation of the phenomenon of “relativity of locality” than the one described by the Doubly Special Relativity (DSR) framework. A generalized Weyl-Heisenberg algebra, involving polyvector-valued coordinates and momenta operators, furnishes a realization of an extended Poincare algebra in C-spaces. In addition to the Planck constantħ, one finds that the commutator of the Clifford scalar components of the Weyl-Heisenberg algebra requires the introduction of a dimensionless parameter which is expressed in terms of the ratio of two length scales : the Planck and Hubble scales. We finalize by discussing the concept of “photons”, null intervals, effective temporal variables and the addition/subtraction laws of generalized velocities in C-space.Fundamental Solutions for Second Order Elliptic Operators in Clifford-Type Algebras
2015-09-01 03:00:00 AM
Abstract
We show fundamental solutions for some second order elliptic operators in the framework of parameter-depending Clifford-type algebras. Then we show integral representation formulae for sufficiently smooth functions defined in a domain of \({\mathbb{R}^{n+1}}\) . This leads to integral formulae for solutions of certain partial differential equations of higher order.Bicomplex Linear Operators on Bicomplex Hilbert Spaces and Littlewood’s Subordination Theorem
2015-09-01 03:00:00 AM
Abstract
In this paper we study some basic properties of bicomplex linear operators on bicomplex Hilbert spaces. Further, we introduce the notion of bicomplex quotient modules and annihilators of submodules of a bicomplex module. We also introduce and discuss some bicomplex holomorphic function spaces and prove Littlewood’s Subordination Principle for bicomplex Hardy space.Symplectic, Orthogonal and Linear Lie Groups in Clifford Algebra
2015-09-01 03:00:00 AM
Abstract
In this paper we prove isomorphisms between 5 Lie groups (of arbitrary dimension and fixed signatures) in Clifford algebra and classical matrix Lie groups - symplectic, orthogonal and linear groups. Also we obtain isomorphisms of corresponding Lie algebras.The Column and Row Immanants Over A Split Quaternion Algebra
2015-09-01 03:00:00 AM
Abstract
The theory of the column-row determinants has been considered for matrices over a non-split quaternion algebra. In this paper the concepts of column-row determinants are extending to a split quaternion algebra. New definitions of the column and row immanants (permanents) for matrices over a split quaternion algebra are introduced, and their basic properties are investigated. The key theorem about the column and row immanants of a Hermitian matrix over a split quaternion algebra is proved. Based on this theorem an immanant (permanent, determinant) of a Hermitian matrix over a split quaternion algebra is introduced.Best Pair of Two Skew Lines over the Octonions
2015-09-01 03:00:00 AM
Abstract
This is a work on an application of octonions to Analytic Geometry. In the octonionic context, the orthogonal projection of a point onto a straight line is presented. Further, the best approximation pair of points of two skew lines over the octonions is studied.Polynomial Dirac Operators in Superspace
2015-09-01 03:00:00 AM
Abstract
In this paper, we construct fundamental solutions for the polynomial operators \({{(\partial_{x}-\lambda)^l}, \prod \limits_{k=1}^{l} (\partial_{x}- \lambda_{k})^{n_k},\, \partial^{l}_{x}+ \sum \limits_{j=1}^{l} b_j \partial^{l-j}_{x}}\) , where \({{\partial_{x}}}\) is the Dirac operator in superspace, and \({{\lambda, \lambda_{k}, b_j}}\) are complex numbers. Applying the fundamental solutions, we investigate Cauchy-Pompeiu formulas for the operators \({{{(\partial_{x}- \lambda)^l}, \partial^{l}_{x} + \sum \limits_{j=1}^{l} b_j \partial^{l-j}_{x}}}\) . In addition, we obtain a Cauchy-Pompeiu formula for the operator \({\prod \limits_{k=1}^{l} (\partial_{x}- \lambda_{k})^{n_k}}\) and a Cauchy formula for the operator \({\prod \limits_{k=1}^{l} (\partial_{x}- \lambda_{k})}\) by another way.Split Fibonacci and Lucas Octonions
2015-09-01 03:00:00 AM
Abstract
In this paper, split Fibonacci and Lucas octonions are proposed and their some properties and relations are obtained.
