93,908 research outputs found
Lifting smooth curves over invariants for representations of compact Lie groups, III
Any sufficiently often differentiable curve in the orbit space of a
real finite-dimensional orthogonal representation of a finite
group admits a differentiable lift into the representation space with
locally bounded derivative. As a consequence any sufficiently often
differentiable curve in the orbit space can be lifted twice
differentiably. These results can be generalized to arbitrary polar
representations. Finite reflection groups and finite rotation groups in
dimensions two and three are discussed in detail.Comment: 19 pages, Late
Supersymmetric Kaluza-Klein reductions of M-waves and MKK-monopoles
We investigate the Kaluza-Klein reductions to ten dimensions of the purely
gravitational half-BPS M-theory backgrounds: the M-wave and the Kaluza-Klein
monopole. We determine the moduli space of smooth (supersymmetric) Kaluza-Klein
reductions by classifying the freely-acting spacelike Killing vectors which
preserve some Killing spinor. As a consequence we find a wealth of new
supersymmetric IIA configurations involving composite and/or bound-state
configurations of waves, D0 and D6-branes, Kaluza-Klein monopoles in type IIA
and flux/nullbranes, and some other new configurations. Some new features
raised by the geometry of the Taub-NUT space are discussed, namely the
existence of reductions with no continuous moduli. We also propose an
interpretation of the flux 5-brane in terms of the local description (close to
the branes) of a bound state of D6-branes and ten-dimensional Kaluza-Klein
monopoles.Comment: 36 pages (v2: Reference added, "draft" mode disabled; v3: two
singular reductions discarded, appendix on spin structures added, references
updated
Representation and Characterization of Non-Stationary Processes by Dilation Operators and Induced Shape Space Manifolds
We have introduce a new vision of stochastic processes through the geometry
induced by the dilation. The dilation matrices of a given processes are
obtained by a composition of rotations matrices, contain the measure
information in a condensed way. Particularly interesting is the fact that the
obtention of dilation matrices is regardless of the stationarity of the
underlying process. When the process is stationary, it coincides with the
Naimark Dilation and only one rotation matrix is computed, when the process is
non-stationary, a set of rotation matrices are computed. In particular, the
periodicity of the correlation function that may appear in some classes of
signal is transmitted to the set of dilation matrices. These rotation matrices,
which can be arbitrarily close to each other depending on the sampling or the
rescaling of the signal are seen as a distinctive feature of the signal. In
order to study this sequence of matrices, and guided by the possibility to
rescale the signal, the correct geometrical framework to use with the
dilation's theoretic results is the space of curves on manifolds, that is the
set of all curve that lies on a base manifold. To give a complete sight about
the space of curve, a metric and the derived geodesic equation are provided.
The general results are adapted to the more specific case where the base
manifold is the Lie group of rotation matrices. The notion of the shape of a
curve can be formalized as the set of equivalence classes of curves given by
the quotient space of the space of curves and the increasing diffeomorphisms.
The metric in the space of curve naturally extent to the space of shapes and
enable comparison between shapes.Comment: 19 pages, draft pape
A Spin-Statistics Theorem for Certain Topological Geons
We review the mechanism in quantum gravity whereby topological geons,
particles made from non-trivial spatial topology, are endowed with nontrivial
spin and statistics. In a theory without topology change there is no
obstruction to ``anomalous'' spin-statistics pairings for geons. However, in a
sum-over-histories formulation including topology change, we show that
non-chiral abelian geons do satisfy a spin-statistics correlation if they are
described by a wave function which is given by a functional integral over
metrics on a particular four-manifold. This manifold describes a topology
changing process which creates a pair of geons from .Comment: 21 pages, Plain TeX with harvmac, 3 figures included via eps
Importance and effectiveness of representing the shapes of Cosserat rods and framed curves as paths in the special Euclidean algebra
We discuss how the shape of a special Cosserat rod can be represented as a
path in the special Euclidean algebra. By shape we mean all those geometric
features that are invariant under isometries of the three-dimensional ambient
space. The representation of the shape as a path in the special Euclidean
algebra is intrinsic to the description of the mechanical properties of a rod,
since it is given directly in terms of the strain fields that stimulate the
elastic response of special Cosserat rods. Moreover, such a representation
leads naturally to discretization schemes that avoid the need for the expensive
reconstruction of the strains from the discretized placement and for
interpolation procedures which introduce some arbitrariness in popular
numerical schemes. Given the shape of a rod and the positioning of one of its
cross sections, the full placement in the ambient space can be uniquely
reconstructed and described by means of a base curve endowed with a material
frame. By viewing a geometric curve as a rod with degenerate point-like cross
sections, we highlight the essential difference between rods and framed curves,
and clarify why the family of relatively parallel adapted frames is not
suitable for describing the mechanics of rods but is the appropriate tool for
dealing with the geometry of curves.Comment: Revised version; 25 pages; 7 figure
Left-invariant evolutions of wavelet transforms on the Similitude Group
Enhancement of multiple-scale elongated structures in noisy image data is
relevant for many biomedical applications but commonly used PDE-based
enhancement techniques often fail at crossings in an image. To get an overview
of how an image is composed of local multiple-scale elongated structures we
construct a multiple scale orientation score, which is a continuous wavelet
transform on the similitude group, SIM(2). Our unitary transform maps the space
of images onto a reproducing kernel space defined on SIM(2), allowing us to
robustly relate Euclidean (and scaling) invariant operators on images to
left-invariant operators on the corresponding continuous wavelet transform.
Rather than often used wavelet (soft-)thresholding techniques, we employ the
group structure in the wavelet domain to arrive at left-invariant evolutions
and flows (diffusion), for contextual crossing preserving enhancement of
multiple scale elongated structures in noisy images. We present experiments
that display benefits of our work compared to recent PDE techniques acting
directly on the images and to our previous work on left-invariant diffusions on
orientation scores defined on Euclidean motion group.Comment: 40 page
Integrable flows and Backlund transformations on extended Stiefel varieties with application to the Euler top on the Lie group SO(3)
We show that the -dimensional Euler--Manakov top on can be
represented as a Poisson reduction of an integrable Hamiltonian system on a
symplectic extended Stiefel variety , and present its Lax
representation with a rational parameter.
We also describe an integrable two-valued symplectic map on the
4-dimensional variety . The map admits two different reductions,
namely, to the Lie group SO(3) and to the coalgebra .
The first reduction provides a discretization of the motion of the classical
Euler top in space and has a transparent geometric interpretation, which can be
regarded as a discrete version of the celebrated Poinsot model of motion and
which inherits some properties of another discrete system, the elliptic
billiard.
The reduction of to gives a new explicit discretization of
the Euler top in the angular momentum space, which preserves first integrals of
the continuous system.Comment: 18 pages, 1 Figur
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