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2533212 | 10.1007/s00006-008-0092-9 | The second-order equation in the (1/2,0)+(0,1/2) representation of the
Lorentz group has been proposed by A. Barut in the 70s. It permits to explain
the mass splitting of leptons (e,mu,tau). Recently, the interest has grown to
this model (see, for instance, the papers by S. Kruglov and J. P. Vigier et
al). We continue the research deriving the equation from the first principles,
finding dynamical invariants for this model, investigating the influence of
potential interactions.Comment: 8 pages, no figures. The talk given at the Int. Conf. on Clifford
Algebras and Applications (ICCA7), Toulouse, France, May 19-29, 2005. To be
published in the Proceeding | The Barut Second-Order Equation: Lagrangian, Dynamical Invariants and
Interactions | the barut second-order equation: lagrangian, dynamical invariants and interactions | lorentz barut permits splitting leptons grown papers kruglov vigier continue deriving principles invariants investigating pages figures. talk int. conf. clifford algebras icca toulouse proceeding | non_dup | [] |
2557174 | 10.1007/s00006-008-0109-4 | We show that diffeomorphism invariance of the Maxwell and the Dirac-Hestenes
equations implies the equivalence among different universe models such that if
one has a linear connection with non-null torsion and/or curvature the others
have also. On the other hand local Lorentz invariance implies the surprising
equivalence among different universe models that have in general different
G-connections with different curvature and torsion tensors.Comment: 19 pages, Revtex, Plenary Talk presented at VII International
Conference on Clifford Algebras and their Applications, Universite Paul
Sabatier UFR MIG, Toulouse (FRANCE), to appear in "Clifford Algebras,
Applications to Mathematics, Physics and Engineering", Progress in Math.
Phys., Birkhauser, Berlin 200 | Diffeomorphism Invariance and Local Lorentz Invariance | diffeomorphism invariance and local lorentz invariance | diffeomorphism invariance maxwell dirac hestenes equivalence universe connection torsion curvature also. lorentz invariance surprising equivalence universe connections curvature torsion pages revtex plenary talk clifford algebras universite paul sabatier toulouse clifford algebras mathematics progress math. phys. birkhauser berlin | non_dup | [] |
9995785 | 10.1007/s00006-008-0128-1 | We present a new polar representation of quaternions inspired by the Cayley-Dickson representation. In this new polar representation, a quaternion is represented by a pair of complex numbers as in the Cayley-Dickson form, but here these two complex numbers are a complex 'modulus' and a complex 'argument'. As in the Cayley-Dickson form, the two complex numbers are in the same complex plane (using the same complex root of - 1), but the complex phase is multiplied by a different complex root of - 1 in the exponential function. We show how to calculate the 'modulus' and 'argument' from an arbitrary quaternion in Cartesian form. © 2008 Birkhäuser Verlag Basel/Switzerland | Quaternion polar representation with a complex modulus and complex argument inspired by the Cayley-Dickson form | quaternion polar representation with a complex modulus and complex argument inspired by the cayley-dickson form | polar quaternions inspired cayley dickson representation. polar quaternion cayley dickson modulus argument cayley dickson multiplied exponential function. modulus argument quaternion cartesian form. birkhäuser verlag basel switzerland | non_dup | [] |
2019259 | 10.1007/s00006-010-0203-2 | We investigate the SL(2,R) invariant geodesic curves with the as- sociated
invariant distance function in parabolic geometry. Parabolic geom- etry
naturally occurs in the study of SL(2,R) and is placed in between the elliptic
and the hyperbolic (also known as the Lobachevsky half-plane and 2- dimensional
Minkowski half-plane space-time) geometries. Initially we attempt to use
standard methods of finding geodesics but they lead to degeneracy in this
setup. Instead, by studying closely the two related elliptic and hyperbolic
geometries we discover a unified approach to a more exotic and less obvious
parabolic case. With aid of common invariants we describe the possible dis-
tance functions that turn out to have some unexpected, interesting properties.Comment: LaTeX, 10 pages, 9 EPS figure | Isometric action of SL(2,R) on homogeneous spaces | isometric action of sl(2,r) on homogeneous spaces | geodesic sociated parabolic geometry. parabolic geom etry naturally placed elliptic hyperbolic lobachevsky minkowski geometries. initially attempt geodesics degeneracy setup. studying closely elliptic hyperbolic geometries discover unified exotic obvious parabolic case. invariants tance unexpected latex pages | non_dup | [] |
2138198 | 10.1007/s00006-010-0243-7 | A model of Yang-Mills interactions and gravity in terms of the Clifford
algebra Cl(0,6) is presented. The gravity and Yang-Mills actions are formulated
as different order terms in a generalized action. The feebleness of gravity as
well as the smallness of the cosmological constant and theta terms are
discussed at the classical level. The invariance groups, including the de
Sitter and the Pati-Salam SU(4) subgroups, consist of gauge transformations
from either side of an algebraic spinor. Upon symmetry breaking via the Higgs
fields, the remaining symmetries are the Lorentz SO(1,3), color SU(3),
electromagnetic U(1)_EM, and an additional U(1). The first generation leptons
and quarks are identified with even and odd parts of spinor idempotent
projections. There are still several shortcomings with the current model.
Further research is needed to fully recover the standard model results.Comment: 20 pages, to appear in Advances in Applied Clifford Algebra | Yang-Mills Interactions and Gravity in Terms of Clifford Algebra | yang-mills interactions and gravity in terms of clifford algebra | mills clifford presented. mills formulated action. feebleness smallness cosmological theta level. invariance sitter pati salam subgroups consist transformations algebraic spinor. breaking symmetries lorentz electromagnetic leptons quarks spinor idempotent projections. shortcomings model. recover pages advances clifford | non_dup | [] |
2074316 | 10.1007/s00006-010-0249-1 | Conventional descriptions of transverse waves in an elastic solid are limited
by an assumption of infinitesimally small gradients of rotation. By assuming a
linear response to variations in orientation, we derive an exact description of
a restricted class of rotational waves in an ideal isotropic elastic solid. The
result is a nonlinear equation expressed in terms of Dirac bispinors. This
result provides a simple classical interpretation of relativistic quantum
mechanical dynamics. We construct a Lagrangian of the form L=-E+U+K=0, where E
is the total energy, U is the potential energy, and K is the kinetic energy.Comment: 9 pages; Added references in revisio | Exact Description of Rotational Waves in an Elastic Solid | exact description of rotational waves in an elastic solid | descriptions elastic infinitesimally gradients rotation. derive restricted rotational ideal isotropic elastic solid. dirac bispinors. relativistic dynamics. lagrangian pages revisio | non_dup | [] |
2098738 | 10.1007/s00006-010-0255-3 | In \cite{Mul} one-parameter planar motion was first introduced and the
relations between absolute, relative, sliding velocities (and accelerations) in
the Euclidean plane $\mathbb{E}^2$ were obtained. Moreover, the relations
between the Complex velocities one-parameter motion in the Complex plane were
provided by \cite{Mul}. One-parameter planar homothetic motion was defined in
the Complex plane, \cite{Kur}. In this paper, analogous to homothetic motion in
the Complex plane given by \cite{Kur}, one-parameter planar homothetic motion
is defined in the Hyperbolic plane. Some characteristic properties about the
velocity vectors, the acceleration vectors and the pole curves are given.
Moreover, in the case of homothetic scale $h$ identically equal to 1, the
results given in \cite{Yuc} are obtained as a special case. In addition, three
hyperbolic planes, of which two are moving and the other one is fixed, are
taken into consideration and a canonical relative system for one-parameter
planar hyperbolic homothetic motion is defined. Euler-Savary formula, which
gives the relationship between the curvatures of trajectory curves, is obtained
with the help of this relative system | One-Parameter Homothetic Motion in the Hyperbolic Plane and Euler-Savary
Formula | one-parameter homothetic motion in the hyperbolic plane and euler-savary formula | cite planar sliding velocities accelerations euclidean mathbb obtained. velocities cite planar homothetic cite analogous homothetic cite planar homothetic hyperbolic plane. acceleration pole given. homothetic identically cite case. hyperbolic planes moving consideration canonical planar hyperbolic homothetic defined. euler savary curvatures trajectory | non_dup | [] |
9995786 | 10.1007/s00006-010-0263-3 | The fundamental properties of biquaternions (complexified quaternions) are presented including several different representations, some of them new, and definitions of fundamental operations such as the scalar and vector parts, conjugates, semi-norms, polar forms, and inner products. The notation is consistent throughout, even between representations, providing a clear account of the many ways in which the component parts of a biquaternion may be manipulated algebraically. © 2010 Springer Basel AG | Fundamental Representations and Algebraic Properties of Biquaternions or Complexified Quaternions | fundamental representations and algebraic properties of biquaternions or complexified quaternions | biquaternions complexified quaternions representations definitions operations conjugates norms polar products. notation representations ways biquaternion manipulated algebraically. springer basel | non_dup | [] |
2128663 | 10.1007/s00006-010-0269-x | We investigate the utility of geometric (Clifford) algebras (GA) methods in
two specific applications to quantum information science. First, using the
multiparticle spacetime algebra (MSTA, the geometric algebra of a relativistic
configuration space), we present an explicit algebraic description of one and
two-qubit quantum states together with a MSTA characterization of one and
two-qubit quantum computational gates. Second, using the above mentioned
characterization and the GA description of the Lie algebras SO(3) and SU(2)
based on the rotor group Spin+(3, 0) formalism, we reexamine Boykin's proof of
universality of quantum gates. We conclude that the MSTA approach does lead to
a useful conceptual unification where the complex qubit space and the complex
space of unitary operators acting on them become united, with both being made
just by multivectors in real space. Finally, the GA approach to rotations based
on the rotor group does bring conceptual and computational advantages compared
to standard vectorial and matricial approaches.Comment: 18 pages; accepted for publication in Adv. Appl. Clifford Alg. (2010 | A Geometric Algebra Perspective On Quantum Computational Gates And
Universality In Quantum Computing | a geometric algebra perspective on quantum computational gates and universality in quantum computing | utility geometric clifford algebras science. multiparticle spacetime msta geometric relativistic algebraic qubit msta qubit gates. algebras rotor formalism reexamine boykin universality gates. msta conceptual unification qubit unitary acting multivectors space. rotations rotor bring conceptual advantages vectorial matricial pages publication adv. appl. clifford alg. | non_dup | [] |
2142779 | 10.1007/s00006-011-0285-5 | Using the standard Cayley transform and elementary tools it is reiterated
that the conformal compactification of the Minkowski space involves not only
the "cone at infinity" but also the 2-sphere that is at the base of this cone.
We represent this 2-sphere by two additionally marked points on the Penrose
diagram for the compactified Minkowski space. Lacks and omissions in the
existing literature are described, Penrose diagrams are derived for both,
simple compactification and its double covering space, which is discussed in
some detail using both the U(2) approach and the exterior and Clifford algebra
methods. Using the Hodge * operator twistors (i.e. vectors of the
pseudo-Hermitian space H_{2,2}) are realized as spinors (i.e., vectors of a
faithful irreducible representation of the even Clifford algebra) for the
conformal group SO(4,2)/Z_2. Killing vector fields corresponding to the left
action of U(2) on itself are explicitly calculated. Isotropic cones and
corresponding projective quadrics in H_{p,q} are also discussed. Applications
to flat conformal structures, including the normal Cartan connection and
conformal development has been discussed in some detail.Comment: 38 pages, 8 figures, late | On Conformal Infinity and Compactifications of the Minkowski Space | on conformal infinity and compactifications of the minkowski space | cayley transform elementary reiterated conformal compactification minkowski involves cone infinity sphere cone. sphere additionally marked penrose compactified minkowski space. lacks omissions penrose diagrams compactification covering exterior clifford methods. hodge twistors i.e. pseudo hermitian realized spinors i.e. faithful irreducible clifford conformal killing explicitly calculated. isotropic cones projective quadrics discussed. conformal cartan connection conformal pages | non_dup | [] |
2428226 | 10.1007/s00006-011-0303-7 | Does relativistic gravity provide arguments against the existence of a
preferred frame? Our answer is negative. We define a viable theory of gravity
with preferred frame. In this theory, the EEP holds exactly, and the Einstein
equations of GR limit are obtained in a natural limit. Despite some remarkable
differences (stable "frozen stars" instead of black holes, a "big bounce"
instead of the big bang, exclusion of nontrivial topologies and closed causal
loops, and a preference for a flat universe) the theory is viable.
The equations of the theory are derived from simple axioms about some
fundamental condensed matter (the generalized Lorentz ether), so that, in
particular, the EEP is not postulated but derived.
The theory is compatible with the condensed matter interpretation for the
fermions and gauge fields of the standard model.Comment: Some changes in the presentatio | A generalization of the Lorentz ether to gravity with
general-relativistic limit | a generalization of the lorentz ether to gravity with general-relativistic limit | relativistic arguments preferred answer negative. viable preferred frame. einstein limit. remarkable frozen holes bounce bang exclusion nontrivial topologies causal loops preference universe viable. axioms condensed lorentz ether postulated derived. compatible condensed fermions presentatio | non_dup | [] |
2128208 | 10.1007/s00006-011-0316-2 | We investigate the properties of the Extended Fock Basis (EFB) of Clifford
algebras introduced in [1]. We show that a Clifford algebra can be seen as a
direct sum of multiple spinor subspaces that are characterized as being left
eigenvectors of \Gamma. We also show that a simple spinor, expressed in Fock
basis, can have a maximum number of non zero coordinates that equals the size
of the maximal totally null plane (with the notable exception of vectorial
spaces with 6 dimensions).Comment: Minimal corrections to the published versio | The Extended Fock Basis of Clifford Algebra | the extended fock basis of clifford algebra | fock clifford algebras clifford spinor subspaces eigenvectors gamma. spinor fock equals maximal totally notable exception vectorial .comment versio | non_dup | [] |
80915823 | 10.1007/s00006-012-0333-9 | A Clifford A-algebra of a quadratic A-module (E, q) is an associative
and unital A-algebra (i.e. sheaf of A-algebras) associated with
the quadratic ShSetX-morphism q, and satisfying a certain universal
property. By introducing sheaves of sets of orthogonal bases (or simply
sheaves of orthogonal bases), we show that with every Riemannian quadratic
free A-module of finite rank, say, n, one can associate a Clifford
free A-algebra of rank 2n. This “main” result is stated in Theorem 3.2.http://link.springer.com/journal/6hb2016Mathematics and Applied Mathematic | Clifford A-algebras of quadratic A-modules | clifford a-algebras of quadratic a-modules | clifford quadratic module associative unital i.e. sheaf algebras quadratic shsetx morphism satisfying universal property. introducing sheaves orthogonal bases sheaves orthogonal bases riemannian quadratic module associate clifford “main” stated mathematic | non_dup | [] |
9325076 | 10.1007/s00006-012-0350-8 | In this paper we discuss generalized properties of non-associativity in
Clifford bundles on the 7-sphere S7. Novel and prominent properties inherited
from the non-associative structure of the Clifford bundle on S7 are
demonstrated. They naturally lead to general transformations of the spinor
fields on S7 and have dramatic consequences for the associated Kac-Moody
current algebras. All additional properties concerning the non-associative
structure in the Clifford bundle on S7 are considered. We further discuss and
explore their applications.Comment: 16 page | Non-Associativity in the Clifford Bundle on the Parallelizable Torsion
7-Sphere | non-associativity in the clifford bundle on the parallelizable torsion 7-sphere | associativity clifford bundles sphere prominent inherited associative clifford bundle demonstrated. naturally transformations spinor dramatic consequences moody algebras. concerning associative clifford bundle considered. explore | non_dup | [] |
19597622 | 10.1007/s00006-012-0371-3 | Quaternionic representations of Coxeter (reflection) groups of ranks 3 and 4, as well as those of E8, have been used extensively in the literature. The present paper analyses such Coxeter groups in the Clifford Geometric Algebra framework, which affords a simple way of performing reflections and rotations whilst exposing more clearly the underlying geometry. The Clifford approach shows that the quaternionic representations in fact have very simple geometric interpretations. The representations of the groups A1 × A1 × A1, A3, B3 and H3 of rank 3 in terms of pure quaternions are shown to be simply the Hodge dualised root vectors, which determine the reflection planes of the Coxeter groups. Two successive reflections result in a rotation, described by the geometric product of the two reflection vectors, giving a Clifford spinor. The spinors for the rank-3 groups A1 × A1 × A1, A3, B3 and H3 yield a new simple construction of binary polyhedral groups. These in turn generate the groups A1 × A1 × A1 × A1, D4, F4 and H4 of rank 4 and their widely used quaternionic representations are shown to be spinors in disguise. Therefore, the Clifford geometric product in fact induces the rank-4 groups from the rank-3 groups. In particular, the groups D4, F4 and H4 are exceptional structures, which our study sheds new light on. IPPP/12/26, DCPT/12/5 | Clifford algebra unveils a surprising geometric significance of quaternionic root systems of Coxeter groups. | clifford algebra unveils a surprising geometric significance of quaternionic root systems of coxeter groups. | quaternionic representations coxeter reflection ranks extensively literature. coxeter clifford geometric affords performing reflections rotations whilst exposing geometry. clifford quaternionic representations geometric interpretations. representations quaternions hodge dualised reflection planes coxeter groups. successive reflections geometric reflection giving clifford spinor. spinors polyhedral groups. widely quaternionic representations spinors disguise. clifford geometric induces groups. exceptional sheds ippp dcpt | non_dup | [] |
24935020 | 10.1007/s00006-013-0378-4 | We survey the development of Clifford's geometric algebra and some of its
engineering applications during the last 15 years. Several recently developed
applications and their merits are discussed in some detail. We thus hope to
clearly demonstrate the benefit of developing problem solutions in a unified
framework for algebra and geometry with the widest possible scope: from quantum
computing and electromagnetism to satellite navigation, from neural computing
to camera geometry, image processing, robotics and beyond.Comment: 26 pages, 91 reference | Applications of Clifford's Geometric Algebra | applications of clifford's geometric algebra | clifford geometric years. merits detail. hope benefit unified widest scope electromagnetism satellite navigation camera robotics pages | non_dup | [] |
24938968 | 10.1007/s00006-013-0404-6 | This paper explains how, following the representation of 3D crystallographic
space groups in Clifford's geometric algebra, it is further possible to
similarly represent the 162 so called subperiodic groups of crystallography in
Clifford's geometric algebra. A new compact geometric algebra group
representation symbol is constructed, which allows to read off the complete set
of geometric algebra generators. For clarity moreover the chosen generators are
stated explicitly. The group symbols are based on the representation of point
groups in geometric algebra by versors (Clifford monomials, Lipschitz
elements).
Keywords: Subperiodic groups, Clifford's geometric algebra, versor
representation, frieze groups, rod groups, layer groups .Comment: 17 pages, 6 figures, 11 tables. arXiv admin note: substantial text
overlap with arXiv:1306.128 | Representation of Crystallographic Subperiodic Groups in Clifford's
Geometric Algebra | representation of crystallographic subperiodic groups in clifford's geometric algebra | explains crystallographic clifford geometric subperiodic crystallography clifford geometric algebra. geometric symbol read geometric generators. clarity generators stated explicitly. symbols geometric versors clifford monomials lipschitz keywords subperiodic clifford geometric versor frieze .comment pages tables. admin substantial overlap | non_dup | [] |
24945960 | 10.1007/s00006-013-0421-5 | The formulae of the relativistic products are found S=1
Barut-Muzinich-Williams matrices. They are analogs of the well-known
Chisholm-Caianiello-Fubini identities. The obtained results can be useful in
the higher-order calculations of the high-energy processes with S=1 particles
in the framework of the 2(2S+1) Weinberg formalism, which recently attracted
attention again. PACS numbers: 02.90.+p, 11.90.+t, 12.20.DsComment: 5pp. This is the modernized version of the EFUAZ FT-95-14 (never
submitted and published) preprint of the second author. The modifications are
due to the Thesis of the first autho | Chisholm-Caianiello-Fubini Identities for S=1 Barut-Muzinich-Williams
Matrices | chisholm-caianiello-fubini identities for s=1 barut-muzinich-williams matrices | formulae relativistic barut muzinich williams matrices. analogs chisholm caianiello fubini identities. weinberg formalism attracted again. pacs .dscomment modernized efuaz never submitted preprint author. modifications thesis autho | non_dup | [] |
19597619 | 10.1007/s00006-013-0422-4 | Real physical systems with reflective and rotational symmetries such as viruses, fullerenes and quasicrystals have recently been modeled successfully in terms of three-dimensional (affine) Coxeter groups. Motivated by this progress, we explore here the benefits of performing the relevant computations in a Geometric Algebra framework, which is particularly suited to describing reflections. Starting from the Coxeter generators of the reflections, we describe how the relevant chiral (rotational), full (Coxeter) and binary polyhedral groups can be easily generated and treated in a unified way in a versor formalism. In particular, this yields a simple construction of the binary polyhedral groups as discrete spinor groups. These in turn are known to generate Lie and Coxeter groups in dimension four, notably the exceptional groups D4, F4 and H4. A Clifford algebra approach thus reveals an unexpected connection between Coxeter groups of ranks 3 and 4. We discuss how to extend these considerations and computations to the Conformal Geometric Algebra setup, in particular for the non-crystallographic groups, and construct root systems and quasicrystalline point arrays. We finally show how a Clifford versor framework sheds light on the geometry of the Coxeter element and the Coxeter plane for the examples of the twodimensional non-crystallographic Coxeter groups I2(n) and the threedimensional groups A3, B3, as well as the icosahedral group H3. IPPP/12/49, DCPT/12/9 | A Clifford algebraic framework for Coxeter group theoretic computations. | a clifford algebraic framework for coxeter group theoretic computations. | reflective rotational symmetries viruses fullerenes quasicrystals modeled successfully affine coxeter groups. motivated progress explore benefits performing computations geometric suited describing reflections. coxeter generators reflections chiral rotational coxeter polyhedral unified versor formalism. polyhedral spinor groups. coxeter notably exceptional clifford reveals unexpected connection coxeter ranks extend considerations computations conformal geometric setup crystallographic quasicrystalline arrays. clifford versor sheds coxeter coxeter twodimensional crystallographic coxeter threedimensional icosahedral ippp dcpt | non_dup | [] |
51224322 | 10.1007/s00006-013-0432-2 | 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 | Riemannian L p Averaging on Lie Group of Nonzero Quaternions | riemannian l p averaging on lie group of nonzero quaternions | discusses quaternion geometric weighting averaging multiplicative nonzero quaternions mathbb endowed riemannian metric. riemannian infty i.e. barycenter minimax particularized mathbb .two considered. formulation logarithm quaternions euclidean tangent mathbf mathbb tangent euclidean naturally applied. formulation descent aiming minimizing quaternion geodesic distances raised converges quaternion chet karcher barycenter quaternion fermat weber quaternion riemannian infty besides giving quaternion introducing notion quaternion bilateral filtering | non_dup | [] |
47248035 | 10.1007/s00006-014-0448-2 | 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 | Robust Pose Control of Robot Manipulators Using Conformal Geometric Algebra | robust pose control of robot manipulators using conformal geometric algebra | controller sliding robotic manipulator pose tracking problem. geometric entity geometric effector manipulator. permits express kinematics endeffector manipulator way. sliding controller robustness perturbations implementation. controller robotic manipulator | non_dup | [] |
24958486 | 10.1007/s00006-014-0451-7 | 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 | Connections on Cahen-Wallach spaces | connections on cahen-wallach spaces | systematically connections spinor bundle cahen wallach spaces. connections closely quadratic clifford algebras. defines space. relation. algebraic enter pages coincides version. advances clifford algebras | non_dup | [] |
24963102 | 10.1007/s00006-014-0452-6 | 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 | A Clifford Bundle Approach to the Differential Geometry of Branes | a clifford bundle approach to the differential geometry of branes | clifford bundle formalism extensors acting mathcal fomulation intrinsic manifold equipped boldsymbol signature compatible connection nabla introducing torsion extensor curvature extensor mathfrak fixing connection extensor omega ricci boldsymbol wedge boldsymbol boldsymbol dirac acting mathcal plays paper. thoughtful presentation riemann lorentzian orientable submanifold living manifold mathring mathring simeq mathbb equipped riemannian boldsymbol mathring signature mathring mathring mathring mathring levi civita connection mathring boldsymbol mathring boldsymbol rightarrow mathring inclusion map. curvature mathfrak ricci mathbf disclose connection extensor omega biform mathcal mathbf hope presentation geometers physists interested brane relativity advances clifford algebras. advances clifford algebras | non_dup | [] |
24981731 | 10.1007/s00006-014-0456-2 | 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 | 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 | 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 | pointed twistor frameworks infeld waerden gamma epsilon formalisms formally same. occurrence spinor translation traditional conformal killing twistor form. torsionless devices keeping track gamma formalism valences spinor configurations sorting pages | non_dup | [] |
25023178 | 10.1007/s00006-014-0475-z | 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 | Split-Quaternions and the Dirac Equation | split-quaternions and the dirac equation | dirac spinors admit entirely formulation spinors split quaternions. formalism lorentz unitary split quaternions. dirac spinors. lorentz overline pages corrected misprint corrected weyl minor improvements layou | non_dup | [] |
55709577 | 10.1007/s00006-014-0478-9 | 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 | Representation of distributions by harmonic and monogenic potentials in Euclidean space | representation of distributions by harmonic and monogenic potentials in euclidean space | clifford harmonic monogenic potentials euclidean analogue logarithmic distributional computed. potentials stepping stone doubly infinite consisting integer powers dirac hilbert dirac jump monogenic clifford hyperfunctions | non_dup | [] |
55800919 | 10.1007/s00006-015-0555-8 | 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 | Some properties of the spinor fourier transform | some properties of the spinor fourier transform | spinor fourier transform batard berthier saint jean clifford fourier transform geometric bayro corrochano scheuermann eds. springer developed. transform valued clifford algebra. bases fourier transform diagonalizable. stumbling concerning filter fourier proper convolution theorem. tackled | non_dup | [] |
25033796 | 10.1007/s00006-015-0557-6 | 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 | Matrix mechanics of the relativistic point particle and string in
Clifford space | matrix mechanics of the relativistic point particle and string in clifford space | resolve canonical relativistic weyl spinors clifford spinors canonical poisson brackets. clifford momenta hermitian transform generating algebra. defining mechanics clifford space. string. lorentz induces sheet jackiw teitelboim action. polymomenta canonical sheet. quantized satisfy poincare unchanged. admin substantial overlap | non_dup | [] |
29569355 | 10.1007/s00006-015-0567-4 | 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 | Some remarks concerning invariant quantities in scalar-tensor gravity | some remarks concerning invariant quantities in scalar-tensor gravity | clarify formalism describing functionals field. spacetime weyl rescaling redefinition. phrase quantity functionals specified manner. phrase translation pages advances clifford algebras | non_dup | [] |
24799006 | 10.1007/s00006-015-0580-7 | 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 | Nonassociatuve generalization of supersymmetry | nonassociatuve generalization of supersymmetry | nonassociative generalization supersymmetry supersymmetry generators nonassociative ones. associators multipliers defined. associator supersymmetry generators operator. connection decomposition hidden mechanics versio | non_dup | [] |
42683482 | 10.1007/s00006-015-0584-3 | 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 | Clifford algebra is the natural framework for root systems and Coxeter
groups. Group theory: Coxeter, conformal and modular groups | clifford algebra is the natural framework for root systems and coxeter groups. group theory: coxeter, conformal and modular groups | clifford reflection conformal modular clifford algebra. cartan dieudonn transformations reflections sandwiching clifford multivectors. multivector concrete e.g. polyhedral treat tetrahedral detail. aside constructive induces hinges facts spinors cover rotations spinors euclidean spinors reflections. coxeter precisely exceptional spinorial explaining unusual automorphism groups. simplifies arnold trinities puts mckay correspondence wider framework. extending conformal geometric conformal modular conformal modular pages | non_dup | [] |
32244237 | 10.1007/s00006-015-0587-0 | 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 | Some results on the lattice parameters of quaternionic Gabor frames | some results on the lattice parameters of quaternionic gabor frames | gabor frames vital modern harmonic mathematics instances chirps processing. trivial generalization gabor frames quaternionic results. sided windowed quaternionic fourier transform wqft want wqft quaternion valued shifts modulates valued window function. heisenberg employ quaternionic transform ensure quaternionic gabor quaternionic gabor frame. gabor conjecture hold quaternionic | non_dup | [] |
29512582 | 10.1007/s00006-015-0588-z | 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 | A Clifford Bundle Approach to the Wave Equation of a Spin 1/2 Fermion in
the de Sitter Manifold | a clifford bundle approach to the wave equation of a spin 1/2 fermion in the de sitter manifold | clifford bundle motivated fermion sitter manifold brane topology mathrm mathrm living spacetime mathbb mathring mathbb boldsymbol mathring equipped boldsymbol boldsymbol mathring boldsymbol rightarrow mathring inclusion map. analog dirac minkowski spacetime mathring appropriately factorize casimir invariants sitter linearization linearize textbf dhess postulated. derive textbf dhess fermion sitter manifold heuristic argument obvious generalization heuristic argument permitting derivation dirac minkowski spacetime famous express nothing timelike field. remarkable textbf dhess textbf dhess coincide. ingredients dirac hestenes spinor fields. appendices covariant dirac spinor usualy improves presentation | non_dup | [] |
29563725 | 10.1007/s00006-015-0593-2 | 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 | Commutative $C^*$-algebras generated by Toeplitz operators on the super
unit ball | commutative $c^*$-algebras generated by toeplitz operators on the super unit ball | extend commutative algebras toeplitz ball supermanifold setup. constructing commutative algebras super toeplitz super ball mathbb super siegel mathbb naturally generalize ball siegel domain. commutative maximal abelian subgroup automorphisms super advances clifford | non_dup | [] |
84049216 | 10.1007/s00006-015-0596-z | 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 | Three-term recurrence relations for systems of Clifford algebra-valued orthogonal polynomials | three-term recurrence relations for systems of clifford algebra-valued orthogonal polynomials | clifford valued orthogonal polynomials view. blocks recurrence orthogonal polynomials variable. surprising byproduct clifford valued orthogonal polynomials gelfand tsetlin relies appell polynomials.this portuguese funds cidma mathematics aveiro cmat mathematics minho portuguese foundation “fundação para ciˆencia tecnologia” projects pest pest .info repo semantics publishedversio | non_dup | [] |
25023602 | 10.1007/s00006-015-0623-0 | 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 | An Approach by Representation of Algebras for Decoherence-Free Subspaces | an approach by representation of algebras for decoherence-free subspaces | algebraic formulation decoherence subspaces dfss initially generalize pauli artin semisimple algebras. derive orthogonality theorems algebras analogous groups. build dfss clifford algebras ideals. formalism. advantages perspectives | non_dup | [] |
86414148 | 10.1007/s00006-015-0638-6 | 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 | The eight Cayley-Dickson doubling product | the eight cayley-dickson doubling product | eight cayley dickson doubling products. cayley dickson ordered cayley dickson binomials satisfying properties. doubling product. mathbb numbers. binomials devised mathbb mathbb quaternions mathbb octonions. historically researchers doubling | non_dup | [] |
42640090 | 10.1007/s00006-016-0655-0 | 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 | Coordinate free integrals in Geometric Calculus | coordinate free integrals in geometric calculus | evaluating integrals geometric calculus introducing calculus repeatedly cutting manifolds create antiderivative step. generalization usual variable. practical unveil connections pages figure. subspaces mention metho | non_dup | [] |
74373991 | 10.1007/s00006-016-0666-x | 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 | Clifford Multivector Toolbox (for MATLAB) | clifford multivector toolbox (for matlab) | matlab oriented manipulation arrays comprehensive toolbox library matlab clifford algebras multivectors. toolbox released publically describes decisions implementing toolbox demonstrates capabilities decomposition clifford multivectors | non_dup | [] |
29560824 | 10.1007/s00006-016-0673-y | 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 | M\"obius transformation for left-derivative quaternion holomorphic
functions | m\"obius transformation for left-derivative quaternion holomorphic functions | holomorphic quaternion admit affine obius call quaternionic holomorphic comprises similarity transformations. mathsf mathsf case. obius heisenberg mathsf symmetric. representations mathsf mathfrak neither classified algebras. mathsf comprises mathsf mathbb rotations dilations translations. infinity admit ratio. addressed | non_dup | [] |
42691484 | 10.1007/s00006-016-0675-9 | 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 | The $E_8$ geometry from a Clifford perspective | the $e_8$ geometry from a clifford perspective | considers clifford complementary ways. firstly exceptional clifford constructing subalgebra clifford icosahedral exceptional crystallographic arnold trinities mckay correspondence hint indirect connection icosahedron secondly connection clifford icosahedral doubly covered endowed system. splits subspaces folding relating pictures. folding coxeter thirdly coxeter clifford framework. advocate factorisation coxeter versor clifford exponentials bivectors describing rotations orthogonal planes giving exponents geometric insight usual complexification eigenvalues. explicitly factorisations coxeter versor factorises frac frac frac frac .comment distinction david hestenes prize pages table. admin overlap | non_dup | [] |
42712295 | 10.1007/s00006-016-0682-x | 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 | Sparse Representations of Clifford and Tensor algebras in Maxima | sparse representations of clifford and tensor algebras in maxima | clifford algebras broad engineering. clifford algebras promoted availability automate tedious routine calculations. offer extensive demonstration clifford algebras electromagnetism geometric maxima system. geometric symbolic maxima itensor package. clifford functionality maxima packages clifford simplification clifford outer inverses cliffordan geometric pages publication advances clifford algebras agacse | non_dup | [] |
148622160 | 10.1007/s00006-016-0683-9 | 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 | On Cayley's factorization of 4D rotations and applications | on cayley's factorization of 4d rotations and applications | publication decomposed isoclinic rotation. decomposition cayley’s factorization rotations elfrinkhof–rosen method. straightforward orthogonal subspaces orthogonal bases defined. formulations orthogonal representing rotations clifford cayley’s factorization applications. transform rotations clifford algebras. practical cayley’s factorization efficiently screw rigid transformations.peer reviewedpostprint draft | non_dup | [] |
52134889 | 10.1007/s00006-016-0692-8 | 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 | Motor Estimation using Heterogeneous Sets of Objects in Conformal Geometric Algebra | motor estimation using heterogeneous sets of objects in conformal geometric algebra | rigid noisy heterogeneous conformal geometric algebra. rigid motions motors. employ gradients jacobian automatic numbers. automatic enables employ process. motor planes. geometric conformal motor e.g. spheres circles tangent vectors. successfully rigid motions synthetic datasets heterogeneous conformal planes.this creative commons attribution permits unrestricted reproduction credit creative commons | non_dup | [] |
77413939 | 10.1007/s00006-016-0700-z | 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- | Geometric Algebra as a Unifying Language for Physics and Engineering and Its Use in the Study of Gravity | geometric algebra as a unifying language for physics and engineering and its use in the study of gravity | geometric mathematical aids unified topics mathematics engineering. electromagnetism mechanics acoustic physics. examine employs coordinate formulation relativity lagrangian look dimensions. invariance briefly drawn cosmological lagrangians favoured approach. intention accessible anyone equipped introductory maths engineering.this article. appeared springer | non_dup | [] |
42674604 | 10.1007/s00006-016-0725-3 | 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 | Remarks on the spectrum of the Dirac operator of pseudo-Riemannian spin
manifolds | remarks on the spectrum of the dirac operator of pseudo-riemannian spin manifolds | dirac pseudo riemannian manifolds signature unbounded hilbert involves subbundle establish criterion maximal subbundles equal. manifold all. proceed explicitely mathbb torus mathbb mathbb riemannian manifold | non_dup | [] |
42737161 | 10.1007/s00006-016-0730-6 | 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 | Dolbeault-Dirac operators, quantum Clifford algebras and the
Parthasarathy formula | dolbeault-dirac operators, quantum clifford algebras and the parthasarathy formula | dolbeault dirac quantized irreducible flag manifolds ahmer tucker simmons. satisfy parthasarathy type. quadratic commutation quantized levi clifford algebras commutation quadratic | non_dup | [] |
29505496 | 10.1007/s00006-016-0731-5 | 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 | Doing euclidean plane geometry using projective geometric algebra | doing euclidean plane geometry using projective geometric algebra | presents euclidean projective geometric anyone wishes familiarize pga. brief focuses mathbf mathbb euclidean geometry. explores geometric involving triples establishing wealth properties. applies familiar topics euclidean compares favorably regard completeness compactness practicality elegance. seamless euclidean ideal infinite treatment. numerous accompany text. readers requisite mathematical coordinate pages | non_dup | [] |
42727920 | 10.1007/s00006-016-0746-y | 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 | Free fermions on a piecewise linear four-manifold. I: Exotic chain
complex | free fermions on a piecewise linear four-manifold. i: exotic chain complex | algebraic realization pachner move grassmann exponentials remarkable parameterization going mathbb valued cocycle. triangulated manifold cocycle exotic intimately parameterization algebraic realizations pachner pages. | non_dup | [] |
25031703 | 10.1007/s00006-016-0749-8 | 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 | The Non-Metricity Formulation of General Relativity | the non-metricity formulation of general relativity | recalling connections formalism metricity connection connection coincides metricity cobase fields. formulate gravitation relativity admits geometrical torsionless gravitational manifest metricity connection. metricity gravitational lagrangian discovered wallner einstein hilbert lagrangian density. einstein currents mathcal alpha delta alpha mathcal alpha mathcal alpha mathcal alpha gravitational currents physically appealing form. gravitational analogue lorenz proca propagation gravitational electromagnetic pages adv. appl. clifford algebras | non_dup | [] |
42731122 | 10.1007/s00006-016-0750-2 | 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- | Quaternionic Wiener Algebras, Factorization and Applications | quaternionic wiener algebras, factorization and applications | wiener algebras quaternionic wiener levy extending valued case. wiener hopf factorization quaternionic valued wiener algebras explore connection riemann hilbert setting. characterize solvability quaternionic canonical factorization quaternionic rational pages publication aaca advances clifford algebras publication springer | non_dup | [] |
42742417 | 10.1007/s00006-017-0758-2 | 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 | Mass, zero mass and ... nophysics | mass, zero mass and ... nophysics | massless limiting massive particles. usual semisimple replaced usual solvable nonabelian proper orthochronous lorentz extend wigner generator obtaining maximal solvable borel subgroup kronecker copies telling something helicity antiparticle pages versio | non_dup | [] |
24977372 | 10.1007/s00006-017-0761-7 | 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 | Transgression field theory at the interface of topological insulators | transgression field theory at the interface of topological insulators | topological classified clifford algebras bott periodicity. topological hall topological insulators chern simons theories. invariance actions. topological insulators transgression action. transgression chern simons constructed. transgression connections chern simons pages title changed discussions versio | non_dup | [] |
42672416 | 10.1007/s00006-017-0775-1 | 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 theory of local mass dimension one fermions of spin one half | the theory of local mass dimension one fermions of spin one half | decade daniel grumiller unexpected discovery fermions jcap decade explored intriguing mathematical construct. formalism suffered troubling locality subtle violation lorentz symmetry. trace hidden freedom duals spinors adjoints. fermions issues. fermions restricted quartic higgs. generalised yukawa fermions neutrinos hitherto unsuspected lepton violation. fermions partner dirac fermions contrasting halves former mutating fermionic pages. pedagogic. updated enlarged. publication advances clifford | non_dup | [] |
42714482 | 10.1007/s00006-017-0793-z | 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 | The integration of angular velocity | the integration of angular velocity | velocity. nontrivial involving equations. improvements efforts. numerically complicated analytically motivated astrophysical precessing binaries. straightforward quaternions norm quaternion irrelevant. generator roughly rotation. twice naive methods. integrators readily solved equations. generalization advances clifford | non_dup | [] |
74203093 | 10.1007/s00006-017-0811-1 | 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 | Free fermions on a piecewise linear four-manifold. II: Pachner moves | free fermions on a piecewise linear four-manifold. ii: pachner moves | papers piecewise manifold cohomology mathbb torsion unusual multiplied correcting factor. studying pachner moves multiplicative multipliers faces multiplied multipliers pages. proven | non_dup | [] |
84329992 | 10.1007/s00006-017-0816-9 | 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 | General Dynamics of Spinors | general dynamics of spinors | twisted curved hosting dirac spinors lorentz covariant polar decomposition dirac spinor decomposition dirac spinor spinorial disappeared maintaining lorentz covariance. dirac spinor freedom module yvon takabayashi display equations. permit curvature torsion module | non_dup | [] |
129361811 | 10.1007/s00006-018-0821-7 | 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 | Spinor Fields, Singular Structures, Charge Conjugation, ELKO and
Neutrino Masses | spinor fields, singular structures, charge conjugation, elko and neutrino masses | spinor kinematic ensuing polar singular spinors specifying singular weyl majorana spinors conjugation presenting newly spinor elko spinor concur bases proposal | non_dup | [] |
83864166 | 10.1007/s00006-018-0830-6 | 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 | Resolution of the $k$-Dirac operator | resolution of the $k$-dirac operator | papers. dirac graded parabolic geometry. dirac homogeneous parabolic dirac formal point. dirac descends affine homogeneous descended dirac clifford analysis. descended locally dirac | non_dup | [] |
73410988 | 10.1007/s00006-018-0865-8 | 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 | Classical field theories from Hamiltonian constraint: Local symmetries
and static gauge fields | classical field theories from hamiltonian constraint: local symmetries and static gauge fields | formulation treats spacetime symmetrically utilizes multivector. compensate invariance transformations. couples acting multivector. symmetries ensuing gauged propose generic strength. generic specialized realize gravitational mills interaction. article. employ mathematical geometric | non_dup | [] |
78509732 | 10.1007/s00006-018-0869-4 | 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 | The Standard Model Algebra - Leptons, Quarks, and Gauge from the Complex
Clifford Algebra Cl6 | the standard model algebra - leptons, quarks, and gauge from the complex clifford algebra cl6 | geometric automatically leptons quarks electroweak symmetries predicting extra symmetries. naturally instances clifford mathbb algebraically dirac generators generates witt decomposition decomposition ideals representing leptons quarks. instances isomorphic here. dirac lorentz algebras naturally subalgebras acting ideals representing leptons quarks. representations ideals electromagnetic symmetries bivectors algebra. electroweak broken algebra. predicts bare weinberg theta theta shares ideas chisholm farwell trayling baylis furey | non_dup | [] |
56658344 | 10.1007/s00009-004-0018-2 | 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 | Diffusive Logistic Equations with Degenerate Boundary Conditions | diffusive logistic equations with degenerate boundary conditions | careful accessible exposition bifurcation degenerate diffusive logistic indefinite weights environments heterogeneity. varies eigenvalue linearized favorable situations favorable crowding away | non_dup | [] |
29400241 | 10.1007/s00009-005-0057-3 | 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 | Christoffel transforms and Hermitian linear functionals | christoffel transforms and hermitian linear functionals | pages figures. codes focused transformations hermitian functionals. analogues christoffel transform functionals jacobi deeply past. hermitian functionals circle. hessenberg christoffel transform. unitary appear. principal submatrices respectively.the marcellán dirección investigación ministerio educación ciencia spain intas necca intas hernández fundación universidad carlos madrid.publicad | non_dup | [] |
29402634 | 10.1007/s00009-007-0121-2 | 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 | Infinitely many securities and the fundamental theorem of asset pricing | infinitely many securities and the fundamental theorem of asset pricing | pointed martingale arbitrage markets infinite securities. accordingly constructs martingale generalizing arbitrage lunch lunch etc. introducing markets. modify arbitrage addresses caveat drawing projective measures. analyze situations projective σ−additive projective interpreted neutral arbitrage market. asset pricing infinitely assets.partially funded spanish ministry comunidad autónoma madrid .publicad | non_dup | [] |
2588681 | 10.1007/s00009-008-0135-4 | 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 | New structures on the tangent bundles and tangent sphere bundles | new structures on the tangent bundles and tangent sphere bundles | riemanian tangent bundle riemannian manifold generalizes sasaki cheeger gromoll compatible confers locally conformal ahlerian manifold. generalization ahlerian ahlerian locally conformal ahlerian ahlerian sectional curvature curvature. restrict tangent bundle isometry tangent sphere bundle unitary endowed restriction sasaki preserves hermitian ambient tangent bundle tangent sphere bundle pages latex | non_dup | [] |
56658350 | 10.1007/s00009-008-0140-7 | 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 | Degenerate Elliptic Eigenvalue Problems with Indefinite Weights | degenerate elliptic eigenvalue problems with indefinite weights | careful accessible exposition kreĭn rutman degenerate elliptic eigenvalue indefinite weights environments heterogeneity. eigenvalue algebraically eigenfunction everywhere. distinguished extensive ideas developments equations. extend manes micheletti degenerate | non_dup | [] |
29400202 | 10.1007/s00009-009-0008-5 | 17 pages, 2 figures.-- MSC2000 codes: Primary 42C05; Secondary 15A23.In this manuscript we analyze some linear spectral transformations of a Hermitian linear functional using the multiplication by some class of Laurent polynomials. We focus our attention in the behavior of the Verblunsky parameters of the perturbed linear functional. Some illustrative examples are pointed out.The work of the first author has been supported by a grant of Universidad Autónoma de Tamaulipas. The work of the second author has been supported by Dirección General de Investigación, Ministerio de Educación y Ciencia of Spain, grant MTM06-13000-C03-02. Both authors have been supported by project CCG07-UC3M/ESP-3339 with the financial
support of Comunidad de Madrid-Universidad Carlos III de Madrid.Publicad | Linear spectral transformations and Laurent polynomials | linear spectral transformations and laurent polynomials | pages figures. codes analyze transformations hermitian multiplication laurent polynomials. verblunsky perturbed functional. illustrative pointed out.the universidad autónoma tamaulipas. dirección investigación ministerio educación ciencia spain comunidad madrid universidad carlos madrid.publicad | non_dup | [] |
1941003 | 10.1007/s00009-009-0020-9 | A dynamical system on the total space of the fibre bundle of second order
accelerations, $T^2M$, is defined as a third order vector field $S$ on $T^2M$,
called semispray, which is mapped by the second order tangent structure into
one of the Liouville vector field. For a regular Lagrangian of second order we
prove that this semispray is uniquely determined by two associated
Cartan-Poincar\'e one-forms. To study the geometry of this semispray we
construct a nonlinear connection, which is a Lagrangian subbundle for the
presymplectic structure. Using this semispray and the associated nonlinear
connection we define covariant derivatives of first and second order. With
respect to this, the second order dynamical derivative of the Lagrangian metric
tensor vanishes | The geometry of systems of third order differential equations induced by
second order Lagrangians | the geometry of systems of third order differential equations induced by second order lagrangians | fibre bundle accelerations semispray mapped tangent liouville field. lagrangian semispray uniquely cartan poincar forms. semispray connection lagrangian subbundle presymplectic structure. semispray connection covariant derivatives order. lagrangian vanishes | non_dup | [] |
49299151 | 10.1007/s00009-009-0022-7 | 30 pages.International audienceWe generalize known results on transport equations associated to a Lipschitz field $\mathbf{F}$ on some subspace of $\mathbb{R}^N$ endowed with some general space measure $\mu$. We provide a new definition of both the transport operator and the trace measures over the incoming and outgoing parts of $\partial \Omega$ generalizing known results from the literature. We also prove the well-posedness of some suitable boundary-value transport problems and describe in full generality the generator of the transport semigroup with no-incoming boundary conditions | A new approach to transport equations associated to a regular field: trace results and well-posedness. | a new approach to transport equations associated to a regular field: trace results and well-posedness. | pages.international audiencewe generalize lipschitz mathbf subspace mathbb endowed trace incoming outgoing omega generalizing literature. posedness generality generator semigroup incoming | non_dup | [] |
2160390 | 10.1007/s00009-010-0108-2 | We consider spacelike surfaces in the four-dimensional Minkowski space and
introduce geometrically an invariant linear map of Weingarten-type in the
tangent plane at any point of the surface under consideration. This allows us
to introduce principal lines and an invariant moving frame field. Writing
derivative formulas of Frenet-type for this frame field, we obtain eight
invariant functions. We prove a fundamental theorem of Bonnet-type, stating
that these eight invariants under some natural conditions determine the surface
up to a motion. We show that the basic geometric classes of spacelike surfaces
in the four-dimensional Minkowski space, determined by conditions on their
invariants, can be interpreted in terms of the properties of the two geometric
figures: the tangent indicatrix, and the normal curvature ellipse. We apply our
theory to a class of spacelike general rotational surfaces.Comment: 23 pages; to appear in Mediterr. J. Math., Vol. 9 (2012 | An Invariant Theory of Spacelike Surfaces in the Four-dimensional
Minkowski Space | an invariant theory of spacelike surfaces in the four-dimensional minkowski space | spacelike minkowski geometrically weingarten tangent consideration. principal moving field. writing formulas frenet eight functions. bonnet stating eight invariants motion. geometric spacelike minkowski invariants interpreted geometric tangent indicatrix curvature ellipse. spacelike rotational pages mediterr. math. vol. | non_dup | [] |
29405063 | 10.1007/s00009-011-0153-5 | The paper introduces a new notion of vector-valued risk function, a crucial notion in Actuarial and Financial Mathematics. Both deviations and expectation bounded or coherent risk measures are defined and analyzed. The relationships with both scalar and vector risk functions of previous literature are discussed, and it is pointed out that this new approach seems to appropriately integrate several preceding points of view. The framework of the study is the general setting of Banach lattices and Bochner integrable vector-valued random variables. Sub-gradient linked representation theorems and practical examples are provided.This research was partially supported by “Ministerio de Ciencia en Innovación”
(Spain), Grant ECO2009 − 14457 − C04Publicad | Vector Risk Functions | vector risk functions | introduces notion valued crucial notion actuarial mathematics. deviations expectation coherent analyzed. pointed appropriately integrate preceding view. banach lattices bochner integrable valued variables. theorems practical provided.this partially “ministerio ciencia innovación” spain publicad | non_dup | [] |
56658357 | 10.1007/s00009-012-0212-6 | The purpose of this paper is to study a class of semilinear elliptic boundary value problems with degenerate boundary conditions which include as particular cases the Dirichlet and Robin problems. The approach here is distinguished by the extensive use of the ideas and techniques characteristic of the recent developments in the theory of partial differential equations. By making use of the Leray–Schauder degree, we prove very exact results on the number of solutions of our problem. The results here extend earlier theorems due to Berger–Podolak, Castro–Lazer and Ambrosetti–Mancini to the degenerate case | Multiple Solutions of Semilinear Elliptic Problems with Degenerate Boundary Conditions | multiple solutions of semilinear elliptic problems with degenerate boundary conditions | semilinear elliptic degenerate dirichlet robin problems. distinguished extensive ideas developments equations. leray–schauder problem. extend theorems berger–podolak castro–lazer ambrosetti–mancini degenerate | non_dup | [] |
24796339 | 10.1007/s00009-013-0365-y | In this paper we consider a sum of modified Bessel functions of the first
kind of which particular case is used in the study of Kanter's sharp modified
Bessel function bound for concentrations of some sums of independent symmetric
random vectors. We present some monotonicity and convexity properties for that
sum of modified Bessel functions of the first kind, as well as some Tur\'an
type inequalities, lower and upper bounds. Moreover, we point out an error in
Kanter's paper [Ka] and at the end of the paper we pose an open problem, which
may be of interest for further research.Comment: 8 page | On a sum of modified Bessel functions | on a sum of modified bessel functions | bessel kind kanter sharp bessel sums vectors. monotonicity convexity bessel kind inequalities bounds. kanter pose | non_dup | [] |
24977136 | 10.1007/s00009-014-0396-z | This paper focuses on the numerical solution of initial value problems for
fractional differential equations of linear type. The approach we propose
grounds on expressing the solution in terms of some integral weighted by a
generalized Mittag-Leffler function. Then suitable quadrature rules are devised
and order conditions of algebraic type are derived. Theoretical findings are
validated by means of numerical experiments and the effectiveness of the
proposed approach is illustrated by means of comparisons with other standard
methods | Exponential quadrature rules for linear fractional differential
equations | exponential quadrature rules for linear fractional differential equations | focuses fractional type. propose grounds expressing weighted mittag leffler function. quadrature devised algebraic derived. validated effectiveness illustrated comparisons | non_dup | [] |
24999134 | 10.1007/s00009-014-0431-0 | Filter convergence of vector lattice-valued measures is considered, in order
to deduce theorems of convergence for their decompositions. First the
$\sigma$-additive case is studied, without particular assumptions on the
filter; later the finitely additive case is faced, first assuming uniform
$s$-boundedness (without restrictions on the filter), then relaxing this
condition but imposing stronger properties on the filter. In order to obtain
the last results, a Schur-type convergence theorem is used.Comment: 18 page | Filter convergence and decompositions for vector lattice-valued measures | filter convergence and decompositions for vector lattice-valued measures | filter valued deduce theorems decompositions. sigma additive assumptions filter finitely additive faced boundedness restrictions filter relaxing imposing stronger filter. schur | non_dup | [] |
24973788 | 10.1007/s00009-014-0440-z | We mainly establish a monotonicity property between some special Riemann sums
of a convex function $f$ on $[a,b]$, which in particular yields that
$\frac{b-a}{n+1}\sum_{i=0}^n f\left(a+i\frac{b-a}{n}\right)$ is decreasing
while $\frac{b-a}{n-1}\sum_{i=1}^{n-1} f\left(a+i\frac{b-a}{n}\right)$ is an
increasing sequence. These give us a new refinement of the Hermitt-Hadamard
inequality.
Moreover, we give a refinement of the classical Alzer's inequality together
with a suitable converse to it. Applications regarding to some important convex
functions are also included | Some Monotonicity Properties of Convex Functions with Applications | some monotonicity properties of convex functions with applications | establish monotonicity riemann sums convex frac frac decreasing frac frac sequence. refinement hermitt hadamard inequality. refinement alzer inequality converse convex | non_dup | [] |
29500113 | 10.1007/s00009-014-0456-4 | The aim of this paper is to give an $s$-cobordism classification of
topological $4$-manifolds in terms of the standard invariants using the group
of homotopy self-equivalences. Hambleton and Kreck constructed a braid to study
the group of homotopy self-equivalences of $4$-manifolds. Using this braid
together with the modified surgery theory of Kreck, we give an $s$-cobordism
classification for certain $4$-manifolds with fundamental group $\pi$, such
that cd $\pi \leq 2$.Comment: appears in Mediterr. J. Math. (2015 | S-Cobordism Classification of $4$-Manifolds Through the Group of
Homotopy Self-Equivalences | s-cobordism classification of $4$-manifolds through the group of homotopy self-equivalences | cobordism topological manifolds invariants homotopy equivalences. hambleton kreck braid homotopy equivalences manifolds. braid kreck cobordism manifolds .comment mediterr. math. | non_dup | [] |
24939686 | 10.1007/s00009-014-0461-7 | In this work we investigate Ricci flows of almost Kaehler structures on Lie
algebroids when the fundamental geometric objects are completely determined by
(semi) Riemannian metrics, or effective) regular generating Lagrange/ Finsler,
functions. There are constructed canonical almost symplectic connections for
which the geometric flows can be represented as gradient ones and characterized
by nonholonomic deformations of Grigory Perelman's functionals. The first goal
of this paper is to define such thermodynamical type values and derive almost
K\"ahler - Ricci geometric evolution equations. The second goal is to study how
fixed Lie algebroid, i.e. Ricci soliton, configurations can be constructed for
Riemannian manifolds and/or (co) tangent bundles endowed with nonholonomic
distributions modelling (generalized) Einstein or Finsler - Cartan spaces.
Finally, there are provided some examples of generic off-diagonal solutions for
Lie algebroid type Ricci solitons and (effective) Einstein and Lagrange-Finsler
algebroids.Comment: This version is accepted by Mediterranian J. Math. and modified
following editor/referee's requests. File latex2e 11pt generates 29 page | Almost Kaehler Ricci Flows and Einstein and Lagrange-Finsler Structures
on Lie Algebroids | almost kaehler ricci flows and einstein and lagrange-finsler structures on lie algebroids | ricci flows kaehler algebroids geometric riemannian metrics generating lagrange finsler functions. canonical symplectic connections geometric flows nonholonomic deformations grigory perelman functionals. goal thermodynamical derive ahler ricci geometric equations. goal algebroid i.e. ricci soliton configurations riemannian manifolds tangent bundles endowed nonholonomic einstein finsler cartan spaces. generic diagonal algebroid ricci solitons einstein lagrange finsler mediterranian math. editor referee requests. file latex generates | non_dup | [] |
24992449 | 10.1007/s00009-014-0470-6 | We propose a method to construct first integrals of a dynamical system,
starting with a given set of independent infinitesimal symmetries. In the case
of two infinitesimal symmetries, a rank two Poisson structure on the ambient
space it is found, such that the vector field that generates the dynamical
system, becomes a Poisson vector field. Moreover, the symplectic leaves and the
Casimir functions of the associated Poisson manifold are characterized.
Explicit conditions that guarantee Hamilton-Poisson realizations of the
dynamical system are also given.Comment: 14 page | A method to generate first integrals from infinitesimal symmetries | a method to generate first integrals from infinitesimal symmetries | propose integrals infinitesimal symmetries. infinitesimal symmetries poisson ambient generates poisson field. symplectic leaves casimir poisson manifold characterized. guarantee hamilton poisson realizations | non_dup | [] |
25044294 | 10.1007/s00009-015-0521-7 | One of the classical problems concerns the class of analytic functions $f$ on
the open unit disk $|z|<1$ which have finite Dirichlet integral $\Delta(1,f)$,
where $$\Delta(r,f)=\iint_{|z|<r}|f'(z)|^2 \, dxdy \quad (0<r\leq 1). $$ The
class ${\mathcal S}^*(A,B)$ of normalized functions $f$ analytic in $|z|<1$ and
satisfies the subordination condition $zf'(z)/f(z)\prec (1+Az)/(1+Bz)$ in
$|z|<1$ and for some $-1\leq B\leq 0$, $A\in {\mathbb C}$ with $A\neq B$, has
been studied extensively. In this paper, we solve the extremal problem of
determining the value of $$\max_{f\in {\mathcal S}^*(A,B)}\Delta(r,z/f)$$ as a
function of $r$. This settles the question raised by Ponnusamy and Wirths in
[11]. One of the particular cases includes solution to a conjecture of
Yamashita which was settled recently by Obradovi\'{c} et. al [9].Comment: 16 pages, 8 figures, 3 table | Maximal area integral problem for certain class of univalent analytic
functions | maximal area integral problem for certain class of univalent analytic functions | concerns analytic dirichlet delta delta iint dxdy quad mathcal analytic satisfies subordination prec mathbb extensively. solve extremal determining mathcal delta settles raised ponnusamy wirths conjecture yamashita settled obradovi .comment pages | non_dup | [] |
25051471 | 10.1007/s00009-015-0559-6 | For a smooth manifold $M$, it was shown in \cite{BPH} that every affine
connection on the tangent bundle $TM$ naturally gives rise to covariant
differentiation of multivector fields (MVFs) and differential forms along MVFs.
In this paper, we generalize the covariant derivative of \cite{BPH} and
construct covariant derivatives along MVFs which are not induced by affine
connections on $TM$. We call this more general class of covariant derivatives
\textit{higher affine connections}. In addition, we also propose a framework
which gives rise to non-induced higher connections; this framework is obtained
by equipping the full exterior bundle $\wedge^\bullet TM$ with an associative
bilinear form $\eta$. Since the latter can be shown to be equivalent to a set
of differential forms of various degrees, this framework also provides a link
between higher connections and multisymplectic geometry.Comment: 37 pages; main definition and results generalized; substantial
changes after section | Higher Affine Connections | higher affine connections | manifold cite affine connection tangent bundle naturally covariant multivector mvfs mvfs. generalize covariant cite covariant derivatives mvfs affine connections call covariant derivatives textit affine connections propose connections equipping exterior bundle wedge bullet associative bilinear connections multisymplectic pages substantial | non_dup | [] |
61480427 | 10.1007/s00009-015-0607-2 | In this paper, we describe into real-linear isometries defined between (not necessarily unital) function algebras and show, based on an example, that this type of isometries behaves differently from surjective real-linear isometries and from classical linear isometries. Next we introduce jointly norm-additive mappings and apply our results on real-linear isometries to provide a complete description of these mappings when defined between function algebras which are not necessarily unital or uniformly closed.Research of J. J. Font was partially supported by Universitat Jaume I (Projecte P1-1B2014-35) | Real-Linear Isometries and Jointly Norm-Additive Maps on Function Algebras | real-linear isometries and jointly norm-additive maps on function algebras | isometries necessarily unital algebras isometries behaves differently surjective isometries isometries. jointly norm additive mappings isometries mappings algebras necessarily unital uniformly closed.research font partially universitat jaume projecte | non_dup | [] |
29503943 | 10.1007/s00009-015-0633-0 | We use methods of harmonic analysis and group representation theory to study
the spectral properties of the abstract parabolic operator $\mathscr L =
-d/dt+A$ in homogeneous function spaces. We provide sufficient conditions for
invertibility of such operators in terms of the spectral properties of the
operator $A$ and the semigroup generated by $A$. We introduce a homogeneous
space of functions with absolutely summable spectrum and prove a generalization
of the Gearhart-Pr\"uss Theorem for such spaces. We use the results to prove
existence and uniqueness of solutions of a certain class of non-linear
equations | Harmonic and Spectral Analysis of Abstract Parabolic Operators in
Homogeneous Function Spaces | harmonic and spectral analysis of abstract parabolic operators in homogeneous function spaces | harmonic parabolic mathscr homogeneous spaces. invertibility semigroup homogeneous absolutely summable generalization gearhart spaces. uniqueness | non_dup | [] |
29548960 | 10.1007/s00009-015-0650-z | It is well known that quasi-isometric embeddings of Gromov hyperbolic spaces
induce topological embeddings of their Gromov boundaries. A more general
question is to detect classes of functions between Gromov hyperbolic spaces
that induce continuous maps between their Gromov boundaries. In this paper we
introduce the class of visual functions $f$ that do induce continuous maps
$\tilde f$ between Gromov boundaries. Its subclass, the class of radial
functions, induces Hoelder maps between Gromov boundaries. Conversely, every
Hoelder map between Gromov boundaries of visual hyperbolic spaces induces a
radial function. We study the relationship between large scale properties of f
and small scale properties of $f$, especially related to the dimension theory.
In particular, we prove a form of the dimension raising theorem. We give a
natural example of a radial dimension raising map and we also give a general
class of radial functions that raise asymptotic dimension | Inducing maps between Gromov boundaries | inducing maps between gromov boundaries | quasi isometric embeddings gromov hyperbolic induce topological embeddings gromov boundaries. detect gromov hyperbolic induce gromov boundaries. induce tilde gromov boundaries. subclass induces hoelder gromov boundaries. conversely hoelder gromov boundaries hyperbolic induces function. theory. raising theorem. raising raise asymptotic | non_dup | [] |
54613170 | 10.1007/s00009-016-0685-9 | Classical elastic curves (elastica) are variational objects with many applications in physics and engineering. Elastica in real space forms are well understood, but in other ambient spaces there are few known explicit examples, except geodesics. In this work, we study elastica living in the total space of a Killing submersion focusing on those curves whose osculating plane forms a constant angle with the vertical foliation (slant elastica). First, we compute the Euler–Lagrange equations for elastica and construct new examples of slant elastica in Killing submersions. Then, we completely classify the two main families of slant elastica in Bianchi–Cartan–Vranceanu ambient spaces (giving also explicit parametrizations) | Elasticae in Killing submersions | elasticae in killing submersions | elastic elastica variational engineering. elastica understood ambient geodesics. elastica living killing submersion focusing osculating foliation slant elastica euler–lagrange elastica slant elastica killing submersions. classify families slant elastica bianchi–cartan–vranceanu ambient giving parametrizations | non_dup | [] |
42647205 | 10.1007/s00009-016-0705-9 | In this paper, we study some aspects of the irreducibility of
$\widetilde{M_g^{Pl}(G)}$ and its interrelation with the existence of "normal
forms", i.e. non-singular plane equations (depending on a set of parameters)
such that a specialization of the parameters gives a certain non-singular plane
model associated to the elements of $\widetilde{M_g^{Pl}(G)}$. In particular,
we introduce the concept of being equation strongly irreducible
(ES-Irreducible) for which the locus $\widetilde{M_g^{Pl}(G)}$ is represented
by a single "normal form". Henn, and Komiya-Kuribayashi, observed that
$\widetilde{M_3^{Pl}(G)}$ is ES-Irreducible. In this paper we prove that this
phenomena does not occur for any odd $d>4$. More precisely, let
$\mathbb{Z}/m\mathbb{Z}$ be the cyclic group of order $m$, we prove that
$\widetilde{M_g^{Pl}(\mathbb{Z}/(d-1)\mathbb{Z})}$ is not ES-Irreducible for
any odd integer $d\geq5$, and the number of its irreducible components is at
least two. Furthermore, we conclude the previous result when $d=6$ for the
locus $\widetilde{M_{10}^{Pl}(\mathbb{Z}/3\mathbb{Z})}$.
Lastly, we prove the analogy of these statements when $K$ is any
algebraically closed field of positive characteristic $p$ such that
$p>(d-1)(d-2)+1$.Comment: This paper is a recent version of chapter 1 of arXiv:1503.0114 | On the locus of smooth plane curves with a fixed automorphism group | on the locus of smooth plane curves with a fixed automorphism group | irreducibility widetilde interrelation i.e. singular specialization singular widetilde irreducible irreducible locus widetilde henn komiya kuribayashi widetilde irreducible. phenomena precisely mathbb mathbb cyclic widetilde mathbb mathbb irreducible integer irreducible two. locus widetilde mathbb mathbb lastly analogy statements algebraically .comment | non_dup | [] |
42684323 | 10.1007/s00009-016-0719-3 | Given a left Quillen presheaf of localized model structures, we study the
homotopy limit model structure on the associated category of sections. We focus
specifically on towers and fibered products (pullbacks) of model categories. As
applications we consider Postnikov towers of model categories, chromatic towers
of spectra and Bousfield arithmetic squares of spectra. For stable model
categories, we show that the homotopy fiber of a stable left Bousfield
localization is a stable right Bousfield localization.Comment: 20 pages. The paper "Bousfield localisations along Quillen bifunctors
and applications" (arXiv:1411.0500v1) has been divided into two parts:
"Towers and fibered products of model structures", which is this arXiv
submission, and "Bousfield localisations along Quillen bifunctors"
(arXiv:1411.0500v2 | Towers and fibered products of model structures | towers and fibered products of model structures | quillen presheaf localized homotopy sections. towers fibered pullbacks categories. postnikov towers categories chromatic towers bousfield arithmetic squares spectra. categories homotopy fiber bousfield localization bousfield pages. bousfield localisations quillen bifunctors divided towers fibered submission bousfield localisations quillen bifunctors | non_dup | [] |
87655095 | 10.1007/s00009-016-0735-3 | This paper is devoted to the construction and analysis of a Moser–Steffensen iterative scheme. The method has quadratic convergence without evaluating any derivative nor inverse operator. We present a complete study of the order of convergence for systems of equations, hypotheses ensuring the local convergence, and finally, we focus our attention to its numerical behavior. The conclusion is that the method improves the applicability of both Newton and Steffensen methods having the same order of convergence.Peer ReviewedPostprint (author's final draft | On a Moser-Steffensen type method for nonlinear systems of equations | on a moser-steffensen type method for nonlinear systems of equations | devoted moser–steffensen iterative scheme. quadratic evaluating operator. hypotheses ensuring behavior. improves applicability newton steffensen convergence.peer reviewedpostprint draft | non_dup | [] |
29571288 | 10.1007/s00009-016-0762-0 | In this paper we study the invariant metrizability and projective
metrizability problems for the special case of the geodesic spray associated to
the canonical connection of a Lie group. We prove that such canonical spray is
projectively Finsler metrizable if and only if it is Riemann metrizable. This
result means that this structure is rigid in the sense that considering
left-invariant metrics, the potentially much larger class of projective Finsler
metrizable canonical sprays, corresponding to Lie groups, coincides with the
class of Riemann metrizable canonical sprays. Generalisation of these results
for geodesic orbit spaces are given.Comment: final version, accepted by MJO | Invariant metrizability and projective metrizability on Lie groups and
homogeneous spaces | invariant metrizability and projective metrizability on lie groups and homogeneous spaces | metrizability projective metrizability geodesic spray canonical connection group. canonical spray projectively finsler metrizable riemann metrizable. rigid metrics potentially projective finsler metrizable canonical sprays coincides riemann metrizable canonical sprays. generalisation geodesic orbit | non_dup | [] |
42725014 | 10.1007/s00009-016-0787-4 | In the study, the collocation method based on exponential cubic B-spline
functions is proposed to solve one dimensional Boussinesq systems numerically.
Two initial boundary value problems for Regularized and Classical Boussinesq
systems modeling motion of traveling waves are considered. The accuracy of the
method is validated by measuring the error between the numerical and analytical
solutions. The numerical solutions obtained by various values of free parameter
$p$ are compared with some solutions in literature.Comment: 13 pages, 4 figure | Solitary wave simulations of the Boussinesq Systems | solitary wave simulations of the boussinesq systems | collocation exponential cubic spline solve boussinesq numerically. regularized boussinesq traveling considered. validated measuring solutions. pages | non_dup | [] |
25010095 | 10.1007/s00009-016-0812-7 | The tangent bundle $T^kM$ of order $k$, of a smooth Banach manifold $M$
consists of all equivalent classes of curves that agree up to their
accelerations of order $k$. For a Banach manifold $M$ and a natural number $k$
first we determine a smooth manifold structure on $T^kM$ which also offers a
fiber bundle structure for $(\pi_k,T^kM,M)$. Then we introduce a particular
lift of linear connections on $M$ to geometrize $T^kM$ as a vector bundle over
$M$. More precisely based on this lifted nonlinear connection we prove that
$T^kM$ admits a vector bundle structure over $M$ if and only if $M$ is endowed
with a linear connection. As a consequence applying this vector bundle
structure we lift Riemannian metrics and Lagrangians from $M$ to $T^kM$. Also,
using the projective limit techniques, we declare a generalized Fr\'echet
vector bundle structure for $T^\infty M$ over $M$ | Higher order tangent bundles | higher order tangent bundles | tangent bundle banach manifold agree accelerations banach manifold manifold offers fiber bundle lift connections geometrize bundle precisely lifted connection admits bundle endowed connection. bundle lift riemannian metrics lagrangians projective declare echet bundle infty | non_dup | [] |
73419167 | 10.1007/s00009-016-0814-5 | Using computational techniques we tabulate prime knots up to five crossings
in the solid torus and the infinite family of lens spaces $L(p,q)$. For these
knots we calculate the second and third skein module and establish which prime
knots in the solid torus are amphichiral. Most knots are distinguished by the
skein modules. For the handful of cases where the skein modules fail to detect
inequivalent knots, we calculate and compare the hyperbolic structures of the
knot complements. We were unable to resolve a handful of 5-crossing cases for
$p\geq 13$.Comment: To appear in Mediterr. J. Mat | Tabulation of prime knots in lens spaces | tabulation of prime knots in lens spaces | tabulate prime knots crossings torus infinite lens knots skein module establish prime knots torus amphichiral. knots distinguished skein modules. handful skein modules fail detect inequivalent knots hyperbolic knot complements. unable resolve handful crossing .comment mediterr. | non_dup | [] |
42734040 | 10.1007/s00009-017-0878-x | In the present paper we consider a special class of Lorentz surfaces in the
four-dimensional pseudo-Euclidean space with neutral metric which are
one-parameter systems of meridians of rotational hypersurfaces with timelike or
spacelike axis and call them meridian surfaces. We give the complete
classification of minimal and quasi-minimal meridian surfaces. We also classify
the meridian surfaces with non-zero constant mean curvature.Comment: 15 pages. arXiv admin note: text overlap with arXiv:1606.0004 | Meridian Surfaces with Constant Mean Curvature in Pseudo-Euclidean
4-space with Neutral Metric | meridian surfaces with constant mean curvature in pseudo-euclidean 4-space with neutral metric | lorentz pseudo euclidean neutral meridians rotational hypersurfaces timelike spacelike call meridian surfaces. quasi meridian surfaces. classify meridian pages. admin overlap | non_dup | [] |
84094677 | 10.1007/s00009-017-0907-9 | We show that for every sequence $(n_i)$, where each $n_i$ is either an
integer greater than 1 or is $\infty$, there exists a simply connected open
3-manifold $M$ with a countable dense set of ends $\{e_i\}$ so that, for every
$i$, the genus of end $e_i$ is equal to $n_i$. In addition, the genus of the
ends not in the dense set is shown to be less than or equal to 2. These simply
connected 3-manifolds are constructed as the complements of certain Cantor sets
in $S^3$. The methods used require careful analysis of the genera of ends and
new techniques for dealing with infinite genus | Simply Connected 3-Manifolds with a Dense Set of Ends of Specified Genus | simply connected 3-manifolds with a dense set of ends of specified genus | integer infty manifold countable dense ends genus genus ends dense manifolds complements cantor careful genera ends dealing infinite genus | non_dup | [] |
73390901 | 10.1007/s00009-017-0926-6 | In this paper, we obtain some properties of biconservative Lorentz
hypersurface $M_{1}^{n}$ in $E_{1}^{n+1}$ having shape operator with complex
eigen values. We prove that every biconservative Lorentz hypersurface
$M_{1}^{n}$ in $E_{1}^{n+1}$ whose shape operator has complex eigen values with
at most five distinct principal curvatures has constant mean curvature. Also,
we investigate such type of hypersurface with constant length of second
fundamental form having six distinct principal curvatures | On Biconservative Lorentz Hypersurface with non-diagonalizable shape
operator | on biconservative lorentz hypersurface with non-diagonalizable shape operator | biconservative lorentz hypersurface eigen values. biconservative lorentz hypersurface eigen principal curvatures curvature. hypersurface principal curvatures | non_dup | [] |
42747400 | 10.1007/s00009-017-0954-2 | The aim of this paper is to geometrize time dependent Lagrangian mechanics in
a way that the framework of second order tangent bundles plays an essential
role. To this end, we first introduce the concepts of time dependent
connections and time dependent semisprays on a manifold $M$ and their induced
vector bundle structures on the second order time dependent tangent bundle
$\R\times T^2M$. Then we turn our attention to regular time Lagrangians and
their interaction with $\R\times T^2M$ in different situations such as
mechanical systems with potential fields, external forces and holonomic
constraints. Finally we propose an examples to support our theory | Second order time dependent tangent bundles and their applications | second order time dependent tangent bundles and their applications | geometrize lagrangian mechanics tangent bundles plays role. concepts connections semisprays manifold bundle tangent bundle lagrangians situations forces holonomic constraints. propose | non_dup | [] |
83845895 | 10.1007/s00009-017-0991-x | An almost Golden Riemannian structure $(\varphi ,g)$ on a manifold is given
by a tensor field $\varphi $ of type (1,1) satisfying the Golden section
relation $\varphi ^{2}=\varphi +1$, and a pure Riemannian metric $g$, i.e., a
metric satisfying $g(\varphi X,Y)=g(X,\varphi Y)$. We study connections adapted
to such a structure, finding two of them, the first canonical and the well
adapted, which measure the integrability of $\varphi $ and the integrability of
the $G$-structure corresponding to $(\varphi ,g)$ | On the geometry of almost Golden Riemannian manifolds | on the geometry of almost golden riemannian manifolds | golden riemannian varphi manifold varphi satisfying golden varphi varphi riemannian i.e. satisfying varphi varphi connections adapted canonical adapted integrability varphi integrability varphi | non_dup | [] |
154274130 | 10.1007/s00009-017-1017-4 | The concept of the inverse along an element was introduced by Mary in 2011. Later, Zhu et al. introduced the one-sided inverse along an element. In this paper, we first give a new existence criterion for the one-sided inverse along a product and characterize the existence of Moore–Penrose inverse by means of one-sided invertibility of certain element in a ring. In addition, we show that a∈ S † ⋂ S # if and only if (a∗a)k is invertible along a if and only if (aa∗)k is invertible along a in a ∗ -monoid S, where k is an arbitrary given positive integer. Finally, we prove that the inverse of a along aa ∗ coincides with the core inverse of a under the condition a∈ S { 1 , 4 } in a ∗ -monoid S.FCT - Fuel Cell Technologies Program(CXLX13-072)This research was supported by the National Natural Science Foundation
of China (No. 11371089), the Specialized Research Fund for the Doctoral Program of
Higher Education (No. 20120092110020), the Natural Science Foundation of Jiangsu Province
(No. BK20141327) and the Foundation of Graduate Innovation Program of Jiangsu Province
(No. KYZZ15-0049).info:eu-repo/semantics/publishedVersio | The one-sided inverse along an element in semigroups and rings | the one-sided inverse along an element in semigroups and rings | mary sided element. criterion sided characterize moore–penrose sided invertibility ring. invertible invertible monoid integer. coincides monoid s.fct fuel technologies cxlx foundation specialized fund doctoral foundation jiangsu province foundation graduate innovation jiangsu province kyzz .info repo semantics publishedversio | non_dup | [] |
129360892 | 10.1007/s00009-017-1036-1 | We introduce \textcolor{red}{general} new techniques for computing the
geometric index of a link $L$ in the interior of a solid torus $T$. These
techniques simplify and unify previous ad hoc methods used to compute the
geometric index in specific examples \textcolor{red}{ and allow the simple
computation of geometric index for new examples where the index was not
previously known}. The geometric index measures the minimum number of times any
meridional disc of $T$ must intersect $L$. It is related to the algebraic index
in the sense that adding up signed intersections of an interior simple closed
curve $C$ in $T$ with a meridional disc gives $\pm$ the algebraic index of $C$
in $T$. One key idea is introducing the notion of geometric index for solid
chambers of the form $B^2\times I$ in $T$. After that we prove that if a solid
torus can be divided into solid chambers by meridional discs in a specific
\textcolor{red}{(and often easy to obtain)} way, then the geometric index can
be easily computed | New Techniques for Computing Geometric Index | new techniques for computing geometric index | textcolor geometric interior torus simplify unify geometric textcolor geometric geometric meridional disc intersect algebraic adding signed intersections interior meridional disc algebraic introducing notion geometric chambers torus divided chambers meridional discs textcolor geometric | non_dup | [] |
84330803 | 10.1007/s00009-017-1042-3 | Let $V_{n}$ denote the third order linear recursive sequence defined by the
initial values $V_{0}$, $V_{1}$ and $V_{2}$ and the recursion
$V_{n}=rV_{n-1}+sV_{n-2}+tV_{n-3}$ if $n\geq 3$, where $r$, $s$, and $t$ are
real constants. The $\{V_{n}\}_{n\geq0}$ are generalized Tribonacci numbers and
reduce to the usual Tribonacci numbers when $r=s=t=1$ and to the $3$-bonacci
numbers when $r=s=1$ and $t=0$. In this study, we introduced a quaternion
sequence which has not been introduced before. We show that the new quaternion
sequence that we introduced includes the previously introduced Tribonacci,
Padovan, Narayana and Third order Jacobsthal quaternion sequences. We obtained
the Binet formula, summation formula and the norm value for this new quaternion
sequence | On a Generalization for Tribonacci Quaternions | on a generalization for tribonacci quaternions | recursive recursion constants. tribonacci usual tribonacci bonacci quaternion before. quaternion tribonacci padovan narayana jacobsthal quaternion sequences. binet summation norm quaternion | non_dup | [] |