pinecone/core-2020-05-10-deduplication huggingface dataset:
On Cayley's factorization of 4D rotations and applications

The final publication is available at link.springer.comA 4D rotation can be decomposed into a left- and a right-isoclinic rotation. This decomposition, known as Cayley’s factorization of 4D rotations, can be performed using the Elfrinkhof–Rosen method. In this paper, we present a more straightforward alternative approach using the corresponding orthogonal subspaces, for which orthogonal bases can be defined. This yields easy formulations, both in the space of 4×44×4 real orthogonal matrices representing 4D rotations and in the Clifford algebra C4,0,0C4,0,0. Cayley’s factorization has many important applications. It can be used to easily transform rotations represented using matrix algebra to different Clifford algebras. As a practical application of the proposed method, it is shown how Cayley’s factorization can be used to efficiently compute the screw parameters of 3D rigid-body transformations.Peer ReviewedPostprint (author's final draft
Motor Estimation using Heterogeneous Sets of Objects in Conformal Geometric Algebra

In this paper we present a novel method for nonlinear rigid body motion estimation from noisy data using heterogeneous sets of objects of the conformal model in geometric algebra. The rigid body motions are represented by motors. We employ state-of-the-art nonlinear optimization tools and compute gradients and Jacobian matrices using forward-mode automatic differentiation based on dual numbers. The use of automatic differentiation enables us to employ a wide range of cost functions in the estimation process. This includes cost functions for motor estimation using points, lines and planes. Moreover, we explain how these cost functions make it possible to use other geometric objects in the conformal model in the motor estimation process, e.g., spheres, circles and tangent vectors. Experimental results show that we are able to successfully estimate rigid body motions from synthetic datasets of heterogeneous sets of conformal objects including a combination of points, lines and planes.This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made
Geometric Algebra as a Unifying Language for Physics and Engineering and Its Use in the Study of Gravity

Geometric Algebra (GA) is a mathematical language that aids a unified approach and understanding in topics across mathematics, physics and engineering. In this contribution, we introduce the Space-time Algebra (STA), and discuss some of its applications in electromagnetism, quantum mechanics and acoustic physics. Then we examine a gauge theory approach to gravity that employs GA to provide a coordinate free formulation of General Relativity, and discuss what a suitable Lagrangian for gravity might look like in two dimensions. Finally the extension of the gauge theory approach to include scale invariance is briefly introduced, and attention drawn to the interesting properties with respect to the cosmological constant of the type of Lagrangians which are favoured in this approach. The intention throughout is to provide a survey accessible to anyone, equipped only with an introductory knowledge of GA, whether in maths, physics or engineering.This is the final version of the article. It first appeared from Springer via http://dx.doi.org/10.1007/s00006-016-0700-
Remarks on the spectrum of the Dirac operator of pseudo-Riemannian spin
  manifolds

We study the spectrum of the Dirac operator $D$ on pseudo-Riemannian spin
manifolds of signature $(p,q)$, considered as an unbounded operator in the
Hilbert space $L^2_\xi(S)$. The definition of $L^2_\xi(S)$ involves the choice
of a $p$-dimensional time-like subbundle $\xi\subset TM$. We establish a
sufficient criterion for the spectra of $D$ induced by two maximal time-like
subbundles $\xi_1,\xi_2\subset TM$ to be equal. If the base manifold $M$ is
compact, the spectrum does not depend on $\xi$ at all. We then proceed by
explicitely computing the full spectrum of $D$ for $\mathbb R^{p,q}$, the flat
torus $\mathbb T^{p,q}$ and products of the form $\mathbb T^{1,1}\times F$ with
$F$ being an arbitrary compact, even-dimensional Riemannian spin manifold
Dolbeault-Dirac operators, quantum Clifford algebras and the
  Parthasarathy formula

We consider Dolbeault-Dirac operators on quantized irreducible flag manifolds
as defined by Kr\"ahmer and Tucker-Simmons. We show that, in general, these
operators do not satisfy a formula of Parthasarathy-type. This is a consequence
of two results that we prove here: we always have quadratic commutation
relations for the relevant quantum root vectors, up to terms in the quantized
Levi factor; there are examples of quantum Clifford algebras where the
commutation relations are not of quadratic-constant type.Comment: 23 page
Doing euclidean plane geometry using projective geometric algebra

The article presents a new approach to euclidean plane geometry based on
projective geometric algebra (PGA). It is designed for anyone with an interest
in plane geometry, or who wishes to familiarize themselves with PGA. After a
brief review of PGA, the article focuses on $\mathbf{P}(\mathbb{R}^*_{2,0,1})$,
the PGA for euclidean plane geometry. It first explores the geometric product
involving pairs and triples of basic elements (points and lines), establishing
a wealth of fundamental metric and non-metric properties. It then applies the
algebra to a variety of familiar topics in plane euclidean geometry and shows
that it compares favorably with other approaches in regard to completeness,
compactness, practicality, and elegance. The seamless integration of euclidean
and ideal (aka infinite) elements forms an essential and novel feature of the
treatment. Numerous figures accompany the text. For readers with the requisite
mathematical background, a self-contained coordinate-free introduction to the
algebra is provided in an appendix.Comment: 30 pages, 20 figure
Free fermions on a piecewise linear four-manifold. I: Exotic chain
  complex

Recently, an algebraic realization of the four-dimensional Pachner move 3--3
was found in terms of Grassmann--Gaussian exponentials, and a remarkable
nonlinear parameterization for it, going in terms of a $\mathbb C$-valued
2-cocycle. Here we define, for a given triangulated four-dimensional manifold
and a 2-cocycle on it, an `exotic' chain complex intimately related to the
mentioned parameterization, thus providing a basis for algebraic realizations
of all four-dimensional Pachner moves.Comment: 23 pages. v2: new and better `gauge' for operators proposed, and many
  small improvement
The Non-Metricity Formulation of General Relativity

After recalling the differential geometry of non-metric connections in the
formalism of differential forms, we introduce the idea of a Non-Metricity (NM)
connection, whose connection $1$--forms coincides with the non-metricity
$1$--forms for a class of cobase fields. Then we formulate a theory of
gravitation (equivalent to General Relativity (GR)) which admits a geometrical
interpretation in a flat torsionless space where the gravitational field is
completely manifest in the non-metricity of a NM connection. We define and then
apply the non-metricity gauge to a gravitational Lagrangian density discovered
by Wallner and which is equivalent to the Einstein-Hilbert Lagrangian density.
The Einstein equations coupled to the matter currents $\left(
\mathcal{J}_{\alpha}\right) $ thus becomes $\delta
dg_{\alpha}=\mathcal{T}_{\alpha}+\mathcal{J}_{\alpha}$, where $\left(
\mathcal{T}_{\alpha}\right) $ is identified as the gravitational
energy-momentum currents, to which we shall find a relatively simple and
physically appealing form. It is also shown that in the gravitational analogue
of the Lorenz gauge, our field equations can be written as a system of Proca
equations, which may be of interest in the study of propagation of
gravitational-electromagnetic waves.Comment: 35 pages, no figures; improved version in Adv. Appl. Clifford
  Algebras 201
Quaternionic Wiener Algebras, Factorization and Applications

We define an almost periodic extension of the Wiener algebras in the
quaternionic setting and prove a Wiener-Levy type theorem for it, as well as
extending the theorem to the matrix-valued case. We prove a Wiener-Hopf
factorization theorem for the quaternionic matrix-valued Wiener algebras
(discrete and continuous) and explore the connection to the Riemann-Hilbert
problem in that setting. As applications, we characterize solvability of two
classes of quaternionic functional equations and give an explicit formula for
the canonical factorization of quaternionic rational matrix functions via
realization.Comment: 34 pages, accepted for publication in AACA (Advances in Applied
  Clifford Algebras). The final publication will be available at Springer via
  http://dx.doi.org/10.1007/s00006-016-0750-
Mass, zero mass and ... nophysics

In this paper we demonstrate that massless particles cannot be considered as
limiting case of massive particles. Instead, the usual symmetry structure based
on semisimple groups like $U(1)$, $SU(2)$ and $SU(3)$ has to be replaced by
less usual solvable groups like the minimal nonabelian group ${\rm sol}_2$.
Starting from the proper orthochronous Lorentz group ${\rm Lor}_{1,3}$ we
extend Wigner's little group by an additional generator, obtaining the maximal
solvable or Borel subgroup ${\rm Bor}_{1,3}$ which is equivalent to the
Kronecker sum of two copies of ${\rm sol}_2$, telling something about the
helicity of particle and antiparticle states.Comment: 44 pages, no figures, published versio
Transgression field theory at the interface of topological insulators

Topological phases of matter can be classified by using Clifford algebras
through Bott periodicity. We consider effective topological field theories of
quantum Hall systems and topological insulators that are Chern-Simons and BF
field theories. The edge states of these systems are related to the gauge
invariance of the effective actions. For the edge states at the interface of
two topological insulators, transgression field theory is proposed as a gauge
invariant effective action. Transgression actions of Chern-Simons theories for
(2+1)D and (4+1)D and BF theories for (3+1)D are constructed. By using
transgression actions, the edge states are written in terms of the bulk
connections of effective Chern-Simons and BF theories.Comment: 7 pages, title changed, new section, discussions and references
  added, published versio
The theory of local mass dimension one fermions of spin one half

About a decade ago the present author in collaboration with Daniel Grumiller
presented an `unexpected theoretical discovery' of spin one-half fermions with
mass dimension one [JCAP 2005, PRD 2005]. In the decade that followed a
significant number of groups explored intriguing mathematical and physical
properties of the new construct. However, the formalism suffered from two
troubling features, that of non-locality and a subtle violation of Lorentz
symmetry. Here, we trace the origin of both of these issues to a hidden freedom
in the definition of duals of spinors and the associated field adjoints. In the
process, for the first time, we provide a quantum theory of spin one-half
fermions that is free from all the mentioned issues. The interactions of the
new fermions are restricted to dimension-four quartic self interaction, and
also to a dimension-four coupling with the Higgs. A generalised Yukawa coupling
of the new fermions with neutrinos provides an hitherto unsuspected source of
lepton-number violation. The new fermions thus present a first-principle dark
matter partner to Dirac fermions of the standard model of high energy physics
with contrasting mass dimensions -- that of three halves for the latter versus
one of the former without mutating the statistics from fermionic to bosonic.Comment: 41 pages. Much of the discussion made more pedagogic. References
  updated and enlarged. Accepted for publication by Advances in Applied
  Clifford Algebra
The integration of angular velocity

A common problem in physics and engineering is determination of the
orientation of an object given its angular velocity. When the direction of the
angular velocity changes in time, this is a nontrivial problem involving
coupled differential equations. Several possible approaches are examined, along
with various improvements over previous efforts. These are then evaluated
numerically by comparison to a complicated but analytically known rotation that
is motivated by the important astrophysical problem of precessing black-hole
binaries. It is shown that a straightforward solution directly using
quaternions is most efficient and accurate, and that the norm of the quaternion
is irrelevant. Integration of the generator of the rotation can also be made
roughly as efficient as integration of the rotation. Both methods will
typically be twice as efficient as naive vector- or matrix-based methods.
Implementation by means of standard general-purpose numerical integrators is
stable and efficient, so that such problems can be readily solved as part of a
larger system of differential equations. Possible generalization to integration
in other Lie groups is also discussed.Comment: Final version in press with Advances in Applied Clifford Algebra
Free fermions on a piecewise linear four-manifold. II: Pachner moves

This is the second in a series of papers where we construct an invariant of a
four-dimensional piecewise linear manifold $M$ with a given middle cohomology
class $h\in H^2(M,\mathbb C)$. This invariant is the square root of the torsion
of unusual chain complex introduced in Part I (arXiv:1605.06498) of our work,
multiplied by a correcting factor. Here we find this factor by studying the
behavior of our construction under all four-dimensional Pachner moves, and show
that it can be represented in a multiplicative form: a product of same-type
multipliers over all 2-faces, multiplied by a product of same-type multipliers
over all pentachora.Comment: 20 pages. v2: two of three Experimental Results now proven, and many
  small improvement
General Dynamics of Spinors

In this paper, we consider a general twisted-curved space-time hosting Dirac
spinors and we take into account the Lorentz covariant polar decomposition of
the Dirac spinor field: the corresponding decomposition of the Dirac spinor
field equation leads to a set of field equations that are real and where
spinorial components have disappeared while still maintaining Lorentz
covariance. We will see that the Dirac spinor will contain two real scalar
degrees of freedom, the module and the so-called Yvon-Takabayashi angle, and we
will display their field equations. This will permit us to study the coupling
of curvature and torsion respectively to the module and the YT angle.Comment: 9 page
Spinor Fields, Singular Structures, Charge Conjugation, ELKO and
  Neutrino Masses

In this paper, we consider the most general treatment of spinor fields, their
kinematic classification and the ensuing dynamic polar reduction, for both
classes of regular and singular spinors; specifying onto the singular class, we
discuss features of the corresponding field equations, taking into special
account the sub-classes of Weyl and Majorana spinors; for the latter case, we
study the condition of charge-conjugation, presenting a detailed introduction
to a newly-defined type of spinor, that is the so-called ELKO spinor: at the
end of our investigation, we will assess how all elements will concur to lay
the bases for a simple proposal of neutrino mass generation.Comment: 8 page
Resolution of the $k$-Dirac operator

This is the second part in a series of two papers. The $k$-Dirac complex is a
complex of differential operators which are natural to a particular
$|2|$-graded parabolic geometry. In this paper we will consider the $k$-Dirac
complex over a homogeneous space of the parabolic geometry and as a first
result, we will prove that the $k$-Dirac complex is exact with formal power
series at any fixed point. Then we will show that the $k$-Dirac complex
descends from an affine subset of the homogeneous space to a complex of linear,
constant coefficient differential operators and that the first operator in the
descended complex is the $k$-Dirac operator studied in Clifford analysis. The
main result of this paper is that the descended complex is locally exact and
thus it forms a resolution of the $k$-Dirac operator
Classical field theories from Hamiltonian constraint: Local symmetries
  and static gauge fields

We consider the Hamiltonian constraint formulation of classical field
theories, which treats spacetime and the space of fields symmetrically, and
utilizes the concept of momentum multivector. The gauge field is introduced to
compensate for non-invariance of the Hamiltonian under local transformations.
It is a position-dependent linear mapping, which couples to the Hamiltonian by
acting on the momentum multivector. We investigate symmetries of the ensuing
gauged Hamiltonian, and propose a generic form of the gauge field strength. In
examples we show how a generic gauge field can be specialized in order to
realize gravitational and/or Yang-Mills interaction. Gauge field dynamics is
not discussed in this article. Throughout, we employ the mathematical language
of geometric algebra and calculus.Comment: 24 page
The Standard Model Algebra - Leptons, Quarks, and Gauge from the Complex
  Clifford Algebra Cl6

A simple geometric algebra is shown to contain automatically the leptons and
quarks of a family of the Standard Model, and the electroweak and color gauge
symmetries, without predicting extra particles and symmetries. The algebra is
already naturally present in the Standard Model, in two instances of the
Clifford algebra $\mathbb{C}\ell_6$, one being algebraically generated by the
Dirac algebra and the weak symmetry generators, and the other by a complex
three-dimensional representation of the color symmetry, which generates a Witt
decomposition which leads to the decomposition of the algebra into ideals
representing leptons and quarks. The two instances being isomorphic, the
minimal approach is to identify them, resulting in the model proposed here. The
Dirac and Lorentz algebras appear naturally as subalgebras acting on the ideals
representing leptons and quarks. The resulting representations on the ideals
are invariant to the electromagnetic and color symmetries, which are generated
by the bivectors of the algebra. The electroweak symmetry is also present, and
it is already broken by the geometry of the algebra. The model predicts a bare
Weinberg angle $\theta_W$ given by $\sin^2\theta_W=0.25$. The model shares
common ideas with previously known models, particularly with Chisholm and
Farwell, 1996, Trayling and Baylis, 2004, and Furey, 2016
Diffusive Logistic Equations with Degenerate Boundary Conditions

The purpose of this paper is to provide a careful and accessible exposition of static bifurcation theory for a class of degenerate boundary value problems for diffusive logistic equations with indefinite weights that model population dynamics in environments with spatial heterogeneity. We discuss the changes that occur in the structure of the positive solutions as a parameter varies near the first eigenvalue of the linearized problem, and prove that the most favorable situations will occur if there is a relatively large favorable region (with good resources and without crowding effects) located some distance away from the boundary of the environment
Christoffel transforms and Hermitian linear functionals

8 pages, no figures.-- MSC2000 codes: Primary 42C05; Secondary 15A23.MR#: MR2192525 (2006h:42044)Zbl#: Zbl 1121.42017In this contribution we are focused on some spectral transformations of Hermitian linear functionals. They are the analogues of the Christoffel transform for linear functionals, i. e. for Jacobi matrices which has been deeply studied in the past. We consider Hermitian linear functionals associated with a probability measure supported on the unit circle. In such a case we compare the Hessenberg matrices associated with such a probability measure and its Christoffel transform. In this way, almost unitary matrices appear. We obtain the deviation to the unit matrix both for principal submatrices and the complete matrices respectively.The work of the first author (F. Marcellán) was supported by Dirección General de Investigación, Ministerio de Educación y Ciencia of Spain, under grant BFM 2003-06335-C03-02, and INTAS Research Network NeCCA INTAS 03-51-6637. The work of the second author (J. Hernández) was supported by Fundación Universidad Carlos III de Madrid.Publicad
Infinitely many securities and the fundamental theorem of asset pricing

Several authors have pointed out the possible absence of martingale
measures for static arbitrage free markets with an infinite number of available
securities. Accordingly, the literature constructs martingale measures by generalizing
the concept of arbitrage (free lunch, free lunch with bounded risk,
etc.) or introducing the theory of large financial markets. This paper does
not modify the definition of arbitrage and addresses the caveat by drawing
on projective systems of probability measures. Thus we analyze those situations
for which one can provide a projective system of σ−additive measures
whose projective limit may be interpreted as a risk-neutral probability of an
arbitrage free market. Hence the Fundamental Theorem of Asset Pricing is
extended so that it can apply for models with infinitely many assets.Partially funded by the Spanish Ministry of Science and Education (ref: BEC2003 − 09067 −
C04 − 03) and Comunidad Autónoma de Madrid (ref: s − 0505/tic/000230).Publicad
New structures on the tangent bundles and tangent sphere bundles

In this paper we study a Riemanian metric on the tangent bundle $T(M)$ of a
Riemannian manifold $M$ which generalizes Sasaki metric and Cheeger Gromoll
metric and a compatible almost complex structure which together with the metric
confers to $T(M)$ a structure of locally conformal almost K\"ahlerian manifold.
This is the natural generalization of the well known almost K\"ahlerian
structure on $T(M)$. We found conditions under which $T(M)$ is almost
K\"ahlerian, locally conformal K\"ahlerian or K\"ahlerian or when $T(M)$ has
constant sectional curvature or constant scalar curvature. Then we will
restrict to the unit tangent bundle and we find an isometry with the tangent
sphere bundle (not necessary unitary) endowed with the restriction of the
Sasaki metric from $T(M)$. Moreover, we found that this map preserves also the
natural almost contact structures obtained from the almost Hermitian ambient
structures on the unit tangent bundle and the tangent sphere bundle,
respectively.Comment: 20 pages, LaTeX2e 4.1
Degenerate Elliptic Eigenvalue Problems with Indefinite Weights

The purpose of this paper is to provide a careful and accessible exposition of the Kreĭn and Rutman Theory of degenerate elliptic eigenvalue problems with indefinite weights that model population dynamics in environments with spatial heterogeneity. We prove that the first eigenvalue of our problem is algebraically simple and its corresponding eigenfunction may be chosen to be positive everywhere. Here the approach is distinguished by the extensive use of the ideas and techniques characteristic of the recent developments in the theory of partial differential equations. The results extend an earlier theorem due to Manes and Micheletti to the degenerate case