pinecone/core-2020-05-10-deduplication huggingface dataset:
Riemannian L p Averaging on Lie Group of Nonzero Quaternions

This paper discusses quaternion $L^p$ geometric weighting averaging working on the multiplicative Lie group of nonzero quaternions $\mathbb{H}^{*}$, endowed with its natural bi-invariant Riemannian metric. Algorithms for computing the Riemannian $L^p$ center of mass of a set of points, with $1 \leq p \leq \infty$ (i.e., median, mean, $L^p$ barycenter and minimax center), are particularized to the case of $\mathbb{H}^{*}$.Two different approaches are considered. The first formulation is based on computing the logarithm of quaternions which maps them to the Euclidean tangent space at the identity $\mathbf{1}$, associated to the Lie algebra of $\mathbb{H}^{*}$. In the tangent space, Euclidean algorithms for $L^p$ center of mass can be naturally applied. The second formulation is a family of methods based on gradient descent algorithms aiming at minimizing the sum of quaternion geodesic distances raised to power $p$. These algorithms converges to the quaternion Fr\'{e}chet-Karcher barycenter ($p=2$), the quaternion Fermat-Weber point ($p=1$) and the quaternion Riemannian 1-center ($p=+\infty$). Besides giving explicit forms of these algorithms, their application for quaternion image processing is shown by introducing the notion of quaternion bilateral filtering
Robust Pose Control of Robot Manipulators Using Conformal Geometric Algebra

A controller, based on sliding mode control, is proposed for the n-link robotic manipulator pose tracking problem. The point pair (a geometric entity expressed in geometric algebra) is used to represent position and orientation of the end-effector of a manipulator. This permits us to express the direct and differential kinematics of the endeffector of the manipulator in a simple and compact way. For the control, a sliding mode controller is designed with the following properties: robustness against perturbations and parameter variations, finite time convergence, and easy implementation. Finally, the application, of the proposed controller in a 6 DOF robotic manipulator is presented via simulation
Connections on Cahen-Wallach spaces

We systematically discuss connections on the spinor bundle of Cahen-Wallach
symmetric spaces. A large class of these connections is closely connected to a
quadratic relation on Clifford algebras. This relation in turn is associated to
the symmetric linear map that defines the underlying space. We present various
solutions of this relation. Moreover, we show that the solutions we present
provide a complete list with respect to a particular algebraic condition on the
parameters that enter into the construction.Comment: 30 pages, v3 coincides with published version. Advances in Applied
  Clifford Algebras, 201
A Clifford Bundle Approach to the Differential Geometry of Branes

The Clifford bundle formalism (CBF) of differential forms and the theory of
extensors acting on $\mathcal{C\ell}(M,g)$ is first used for a fomulation of
the intrinsic geometry of a differential manifold $M$ equipped with a metric
field $\boldsymbol{g}$ of signature $(p,q)$ and an arbitrary metric compatible
connection $\nabla$ introducing the torsion (2-1)-extensor field $\tau$, the
curvature $(2-2)$ extensor field $\mathfrak{R}$ and (once fixing a gauge) the
connection $(1-2)$-extensor $\omega$ and the Ricci operator
$\boldsymbol{\partial}\wedge\boldsymbol{\partial}$ (where
$\boldsymbol{\partial}$ is the Dirac operator acting on sections of
$\mathcal{C\ell}(M,g)$) which plays an important role in this paper. Next,
using the CBF we give a thoughtful presentation the Riemann or the Lorentzian
geometry of an orientable submanifold $M$ ($\dim M=m$) living in a manifold
$\mathring{M}$ (such that $\mathring{M}\simeq\mathbb{R}^{n}$ is equipped with a
semi-Riemannian metric $\boldsymbol{\mathring{g}}$ with signature
$(\mathring{p},\mathring{q})$ and \ $\mathring{p}+\mathring{q}=n$ and its
Levi-Civita connection $\mathring{D}$) and where there is defined a metric
$\boldsymbol{g=i}^{\ast}\mathring{g}$, where $\boldsymbol{i}:$ $M\rightarrow
\mathring{M}$ is the inclusion map. We prove several equivalent forms for the
curvature operator $\mathfrak{R}$ of $M$. It is shown that the Ricci operator
of $M$ is the (negative) square of the shape operator $\mathbf{S}$ of $M$. Also
we disclose the relationship between the connection (1-2%)-extensor $\omega$
and the shape biform $\mathcal{S}$ (an object related to $\mathbf{S}$). We hope
that our presentation will be useful for differential geometers and theoretical
physists interested, e.g, in string and brane theories and relativity theory.Comment: Version published in Advances in Applied Clifford Algebras. Advances
  In Applied Clifford Algebras (2014
Absence of Differential Correlations Between the Wave Equations for
  Upper-Lower One-Index Twistor Fields Borne by the Infeld-van der Waerden
  Spinor Formalisms for General Relativity

It is pointed out that the wave equations for any upper-lower one-index
twistor fields which take place in the frameworks of the Infeld-van der Waerden
{\gamma}{\epsilon}-formalisms must be formally the same. The only reason for
the occurrence of this result seems to be directly related to the fact that the
spinor translation of the traditional conformal Killing equation yields twistor
equations of the same form. It thus appears that the conventional torsionless
devices for keeping track in the {\gamma}-formalism of valences of spinor
differential configurations turn out not to be useful for sorting out the
typical patterns of the equations at issue.Comment: About 7 pages, 18 reference
Split-Quaternions and the Dirac Equation

We show that Dirac 4-spinors admit an entirely equivalent formulation in
terms of 2-spinors defined over the split-quaternions. In this formalism, a
Lorentz transformation is represented as a $2 \times 2$ unitary matrix over the
split-quaternions. The corresponding Dirac equation is then derived in terms of
these 2-spinors. In this framework the $SO(3,2; {\bf R})$ symmetry of the
Lorentz invariant scalar $\overline{\psi}\psi$ is manifest.Comment: 17 pages; corrected misprint in eqn (70); corrected sign for Weyl
  basis in Section 5; minor improvements in layou
Representation of distributions by harmonic and monogenic potentials in Euclidean space

In the framework of Clifford analysis, a chain of harmonic and monogenic potentials in the upper half of Euclidean space $R^{m+1}_+$ was recently constructed, including a higher dimensional analogue of the logarithmic function in the complex plane, and their distributional boundary values were computed. In this paper we determine these potentials in lower half-space $R^{m+1}_-$ and investigate whether they can be extended through the boundary $R^m$. This is a stepping stone to the representation of a doubly infinite sequence of distributions in $R^m$, consisting of positive and negative integer powers of the Dirac and the Hilbert-Dirac operator, as the jump across $R^m$ of monogenic functions in the upper and lower half-spaces, in this way providing a sequence of interesting examples of Clifford hyperfunctions
Some properties of the spinor fourier transform

In this paper, the theory of the spinor Fourier transform introduced in [Batard T, Berthier M, Saint-Jean C, Clifford-Fourier Transform for Color Image Processing, Geometric Algebra Computing for Engineering and Computer Science (E. Bayro-Corrochano and G. Scheuermann Eds.), Springer, London, 2010, pp. 135–161] is further developed. While in the original paper, the transform was determined for vector-valued functions only, it now will be extended to functions taking values in the entire Clifford algebra. Next, two bases are determined under which this Fourier transform is diagonalizable. A main stumbling block for further applications, in particular concerning filter design in the Fourier domain, is the lack of a proper convolution theorem. This problem will be tackled in the final section of this paper
Matrix mechanics of the relativistic point particle and string in
  Clifford space

We resolve the space-time canonical variables of the relativistic point
particle into inner products of Weyl spinors with components in a Clifford
algebra and find that these spinors themselves form a canonical system with
generalized Poisson brackets. For N particles, the inner products of their
Clifford coordinates and momenta form two NxN Hermitian matrices X and P which
transform under a U(N) symmetry in the generating algebra. This is used as a
starting point for defining matrix mechanics for a point particle in Clifford
space. Next we consider the string. The Lorentz metric induces a metric and a
scalar on the world sheet which we represent by a Jackiw-Teitelboim term in the
action. The string is described by a polymomenta canonical system and we find
the wave solutions to the classical equations of motion for a flat world sheet.
Finally, we show that the SL(2.C) charge and space-time momentum of the
quantized string satisfy the Poincare algebra.Comment: v2: improvement of section 7, results unchanged. arXiv admin note:
  substantial text overlap with arXiv:1304.402
Some remarks concerning invariant quantities in scalar-tensor gravity

The aim of the current paper is to clarify some aspects of the formalism used
for describing the scalar-tensor gravity characterized by four arbitrary local
functionals of the scalar field. We recall the objects that are invariant with
respect to a spacetime point under the local Weyl rescaling of the metric and
under the scalar field redefinition. We phrase and prove a theorem that allows
to link such an object to each quantity in a theory where two out of the four
arbitrary local functionals of the scalar field are specified in a suitable
manner. Based on these results we phrase and reason the existence of the so
called translation rules.Comment: 16 pages, Advances in Applied Clifford Algebras 201
Nonassociatuve generalization of supersymmetry

A nonassociative generalization of supersymmetry is studied, where
supersymmetry generators are considered to be the nonassociative ones.
Associators for the product of three and four multipliers are defined. Using a
special choice of the parameters, it is shown that the associator of the
product of four supersymmetry generators is connected with the angular momentum
operator. The connection of operator decomposition to the hidden variables
theory and alternative quantum mechanics is discussed.Comment: final versio
Clifford algebra is the natural framework for root systems and Coxeter
  groups. Group theory: Coxeter, conformal and modular groups

In this paper, we make the case that Clifford algebra is the natural
framework for root systems and reflection groups, as well as related groups
such as the conformal and modular groups: The metric that exists on these
spaces can always be used to construct the corresponding Clifford algebra. Via
the Cartan-Dieudonn\'e theorem all the transformations of interest can be
written as products of reflections and thus via `sandwiching' with Clifford
algebra multivectors. These multivector groups can be used to perform concrete
calculations in different groups, e.g. the various types of polyhedral groups,
and we treat the example of the tetrahedral group $A_3$ in detail. As an aside,
this gives a constructive result that induces from every 3D root system a root
system in dimension four, which hinges on the facts that the group of spinors
provides a double cover of the rotations, the space of 3D spinors has a 4D
euclidean inner product, and with respect to this inner product the group of
spinors can be shown to be closed under reflections. In particular the 4D root
systems/Coxeter groups induced in this way are precisely the exceptional ones,
with the 3D spinorial point of view also explaining their unusual automorphism
groups. This construction simplifies Arnold's trinities and puts the McKay
correspondence into a wider framework. We finally discuss extending the
conformal geometric algebra approach to the 2D conformal and modular groups,
which could have interesting novel applications in conformal field theory,
string theory and modular form theory.Comment: 14 pages, 1 figure, 5 table
Some results on the lattice parameters of quaternionic Gabor frames

Gabor frames play a vital role not only modern harmonic analysis but also in several fields of applied mathematics, for instances, detection of chirps, or image processing. In this work we present a non-trivial generalization of Gabor frames to the quaternionic case and give new density results. The key tool is the two-sided windowed quaternionic Fourier transform (WQFT). As in the complex case, we want to write the WQFT as an inner product between a quaternion-valued signal and shifts and modulates of a real-valued window function. We demonstrate a Heisenberg uncertainty principle and for the results regarding the density, we employ the quaternionic Zak transform to obtain necessary and sufficient conditions to ensure that a quaternionic Gabor system is a quaternionic Gabor frame. We conclude with a proof that the Gabor conjecture do not hold true in the quaternionic case
A Clifford Bundle Approach to the Wave Equation of a Spin 1/2 Fermion in
  the de Sitter Manifold

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
$\boldsymbol{g:=-i}^{\ast}\boldsymbol{\mathring{g}%}$ 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 invariants $C_{1}$
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 \textbf{DHESS1,}which in previous
studies by other authors was simply postulated.$.$Next we derive a wave
equation (called \textbf{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 that \textbf{DHESS1}and
\textbf{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 usualy used by
physicists.Comment: This version contains a new appendix and improves the presentation of
  section
Commutative $C^*$-algebras generated by Toeplitz operators on the super
  unit ball

We extend known results about commutative $C^*$-algebras generated Toeplitz
operators over the unit ball to the supermanifold setup. This is obtained by
constructing commutative $C^*$-algebras of super Toeplitz operators over the
super ball $\mathbb{B}^{p|q}$ and the super Siegel domain $\mathbb{U}^{p|q}$
that naturally generalize the previous results for the unit ball and the Siegel
domain. In particular, we obtain one such commutative $C^*$-algebra for each
even maximal Abelian subgroup of automorphisms of the super ball.Comment: To appear in Advances in Applied Clifford Algebra
Three-term recurrence relations for systems of Clifford algebra-valued orthogonal polynomials

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.This work was supported by Portuguese funds through the CIDMA - Center for Research and Development in Mathematics and Applications of the University of Aveiro, the CMAT - Research Centre of Mathematics of the University of Minho and the FCT - Portuguese Foundation for Science and Technology (“Fundação para a Ciˆencia e a Tecnologia”), within projects PEst-OE/MAT/UI4106/2014 and PEst-OE/MAT/UI0013/2014.info:eu-repo/semantics/publishedVersio
An Approach by Representation of Algebras for Decoherence-Free Subspaces

The aim of this paper is to present a general algebraic formulation for the
Decoherence-Free Subspaces (DFSs). For this purpose, we initially generalize
some results of Pauli and Artin about semisimple algebras. Then we derive
orthogonality theorems for algebras analogous to finite groups. In order to
build the DFSs we consider the tensor product of Clifford algebras and left
minimal ideals. Furthermore, we show that standard applications of group theory
in quantum chemistry can be obtained in our formalism. Advantages and some
perspectives are also discussed.Comment: 11 page
The eight Cayley-Dickson doubling product

The purpose of this paper is to identify all eight of the basic
Cayley-Dickson doubling products. A Cayley-Dickson algebra $\cda{N+1}$ of
dimension $2^{N+1}$ consists of all ordered pairs of elements of a
Cayley-Dickson algebra $\cda{N}$ of dimension $2^N$ where the product
$(a,b)(c,d)$ of elements of $\cda{N+1}$ is defined in terms of a pair of second
degree binomials $\left(f(a,b,c,d),g(a,b,c,d)\right)$ satisfying certain
properties. The polynomial pair$(f,g)$ is called a `doubling product.' While
$\cda{0}$ may denote any ring, here it is taken to be the set $\mathbb{R}$ of
real numbers. The binomials $f$ and $g$ should be devised such that
$\cda{1}=\mathbb{C}$ the complex numbers, $\cda{2}=\mathbb{H}$ the quaternions,
and $\cda{3}=\mathbb{O}$ the octonions. Historically, various researchers have
used different yet equivalent doubling products
Coordinate free integrals in Geometric Calculus

We introduce a method for evaluating integrals in geometric calculus without
introducing coordinates, based on using the fundamental theorem of calculus
repeatedly and cutting the resulting manifolds so as to create a boundary and
allow for the existence of an antiderivative at each step. The method is a
direct generalization of the usual method of integration on function of a real
variable. It may lead to both practical applications and help unveil new
connections to various fields of mathematics.Comment: 20 pages, 1 figure. Add details on orientation of subspaces, and
  mention a more systematic metho
Clifford Multivector Toolbox (for MATLAB)

matlab ® is a numerical computing environment oriented towards manipulation of matrices and vectors (in the linear algebra sense, that is arrays of numbers). Until now, there was no comprehensive toolbox (software library) for matlab to compute with Clifford algebras and matrices of multivectors. We present in the paper an account of such a toolbox, which has been developed since 2013, and released publically for the first time in 2015. The paper describes the major design decisions made in implementing the toolbox, gives implementation details, and demonstrates some of its capabilities, up to and including the LU decomposition of a matrix of Clifford multivectors
M\"obius transformation for left-derivative quaternion holomorphic
  functions

Holomorphic quaternion functions only admit affine functions; thus, the
M\"obius transformation for these functions, which we call quaternionic
holomorphic transformation (QHT), only comprises similarity transformations. We
determine a general group $\mathsf{X}$ which has the group $\mathsf{G}$ of QHT
as a particular case. Furthermore, we observe that the M\"obius group and the
Heisenberg group may be obtained by making $\mathsf{X}$ more symmetric. We
provide matrix representations for the group $\mathsf{X}$ and for its algebra
$\mathfrak{x}$. The Lie algebra is neither simple nor semi-simple, and so it is
not classified among the classical Lie algebras. They prove that the group
$\mathsf{G}$ comprises $\mathsf{SU}(2,\mathbb{C})$ rotations, dilations and
translations. The only fixed point of the QHT is located at infinity, and the
QHT does not admit a cross-ratio. Physical applications are addressed at the
conclusion
The $E_8$ geometry from a Clifford perspective

This paper considers the geometry of $E_8$ from a Clifford point of view in
three complementary ways. Firstly, in earlier work, I had shown how to
construct the four-dimensional exceptional root systems from the 3D root
systems using Clifford techniques, by constructing them in the 4D even
subalgebra of the 3D Clifford algebra; for instance the icosahedral root system
$H_3$ gives rise to the largest (and therefore exceptional)
non-crystallographic root system $H_4$. Arnold's trinities and the McKay
correspondence then hint that there might be an indirect connection between the
icosahedron and $E_8$. Secondly, in a related construction, I have now made
this connection explicit for the first time: in the 8D Clifford algebra of 3D
space the $120$ elements of the icosahedral group $H_3$ are doubly covered by
$240$ 8-component objects, which endowed with a `reduced inner product' are
exactly the $E_8$ root system. It was previously known that $E_8$ splits into
$H_4$-invariant subspaces, and we discuss the folding construction relating the
two pictures. This folding is a partial version of the one used for the
construction of the Coxeter plane, so thirdly we discuss the geometry of the
Coxeter plane in a Clifford algebra framework. We advocate the complete
factorisation of the Coxeter versor in the Clifford algebra into exponentials
of bivectors describing rotations in orthogonal planes with the rotation angle
giving the correct exponents, which gives much more geometric insight than the
usual approach of complexification and search for complex eigenvalues. In
particular, we explicitly find these factorisations for the 2D, 3D and 4D root
systems, $D_6$ as well as $E_8$, whose Coxeter versor factorises as
$W=\exp(\frac{\pi}{30}B_C)\exp(\frac{11\pi}{30}B_2)\exp(\frac{7\pi}{30}B_3)\exp(\frac{13\pi}{30}B_4)$.Comment: Distinction in the David Hestenes Prize 2015 19 pages, 9 figures, 1
  table. arXiv admin note: text overlap with arXiv:1602.06800, arXiv:1602.0598
Sparse Representations of Clifford and Tensor algebras in Maxima

Clifford algebras have broad applications in science and engineering. The use
of Clifford algebras can be further promoted in these fields by availability of
computational tools that automate tedious routine calculations. We offer an
extensive demonstration of the applications of Clifford algebras in
electromagnetism using the geometric algebra G3 = Cl(3,0) as a computational
model in the Maxima computer algebra system. We compare the geometric
algebra-based approach with conventional symbolic tensor calculations supported
by Maxima, based on the itensor package. The Clifford algebra functionality of
Maxima is distributed as two new packages called clifford - for basic
simplification of Clifford products, outer products, scalar products and
inverses; and cliffordan - for applications of geometric calculus.Comment: 23 pages, 2 figures; accepted for publication in Advances in Applied
  Clifford Algebras, special issue AGACSE 201