10,572 research outputs found
Stability of Curvature Measures
We address the problem of curvature estimation from sampled compact sets. The
main contribution is a stability result: we show that the gaussian, mean or
anisotropic curvature measures of the offset of a compact set K with positive
-reach can be estimated by the same curvature measures of the offset of a
compact set K' close to K in the Hausdorff sense. We show how these curvature
measures can be computed for finite unions of balls. The curvature measures of
the offset of a compact set with positive -reach can thus be approximated
by the curvature measures of the offset of a point-cloud sample. These results
can also be interpreted as a framework for an effective and robust notion of
curvature
Ultraviolet Finite Quantum Field Theory on Quantum Spacetime
We discuss a formulation of quantum field theory on quantum space time where
the perturbation expansion of the S-matrix is term by term ultraviolet finite.
The characteristic feature of our approach is a quantum version of the Wick
product at coinciding points: the differences of coordinates q_j - q_k are not
set equal to zero, which would violate the commutation relation between their
components. We show that the optimal degree of approximate coincidence can be
defined by the evaluation of a conditional expectation which replaces each
function of q_j - q_k by its expectation value in optimally localized states,
while leaving the mean coordinates (q_1 + ... + q_n)/n invariant.
The resulting procedure is to a large extent unique, and is invariant under
translations and rotations, but violates Lorentz invariance. Indeed, optimal
localization refers to a specific Lorentz frame, where the electric and
magnetic parts of the commutator of the coordinates have to coincide *).
Employing an adiabatic switching, we show that the S-matrix is term by term
finite. The matrix elements of the transfer matrix are determined, at each
order in the perturbative expansion, by kernels with Gaussian decay in the
Planck scale. The adiabatic limit and the large scale limit of this theory will
be studied elsewhere.
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*) S. Doplicher, K. Fredenhagen, and J.E.Roberts, Commun. Math. Phys. 172,
187 (1995) [arXiv:hep-th/0303037]Comment: LaTeX (using amsmath, amssymb), 23 pages, 1 figure. Dedicated to
Rudolf Haag on the occasion of his 80th birthday. See also: hep-th/0303037,
hep-th/0201222. Second version: minor changes in exposition, two references
added. To appear on Commun. Math. Phy
Minimal Surfaces from Monopoles
The geometry of minimal surfaces generated by charge 2 Bogomolny monopoles on
3-dimensional Euclidean space is described in terms of the moduli parameter k.
We find that the distribution of Gaussian curvature on the surface reflects the
monopole structure. This is elucidated by the behaviour of the Gauss maps of
the minimal surfaces.Comment: 23 pages, 2 figure
Gluon flux-tube distribution and linear confinement in baryons
We have observed the formation of gluon flux-tubes within baryons using
lattice QCD techniques. A high-statistics approach, based on translational and
rotational symmetries of the four-dimensional lattice, enables us to observe
correlations between vacuum action density and quark positions in a completely
gauge independent manner. This contrasts with earlier studies which used
gauge-dependent smoothing techniques. We used 200 O(a^2) improved quenched QCD
gauge-field configurations on a 16^3x32 lattice with a lattice spacing of 0.123
fm. In the presence of static quarks flux tubes representing the suppression of
gluon-field fluctuations are observed. We have analyzed 11 L-shapes and 8 T and
Y shapes of varying sizes in order to explore a variety of flux-tube
topologies, including the ground state. At large separations, Y-shape flux-tube
formation is observed. T-shaped paths are observed to relax towards a Y-shaped
topology, whereas L-shaped paths give rise to a large potential energy. We do
not find any evidence for the formation of a Delta-shaped flux-tube (empty
triangle) distribution. However, at small quark separations, we observe an
expulsion of gluon-field fluctuations in the shape of a filled triangle with
maximal expulsion at the centre of the triangle. Having identified the precise
geometry of the flux distribution, we are able to perform quantitative
comparison between the length of the flux-tube and the associated static quark
potential. For every source configuration considered we find a universal string
tension, and conclude that, for large quark separations, the ground state
potential is that which minimizes the length of the flux-tube. The flux tube
radius of the baryonic ground state potential is found to be 0.38 \pm 0.03 fm,
with vacuum fluctuations suppressed by 7.2 \pm 0.6 %.Comment: 16 pages, final version as accepted for publication in Physical
review D1. Abstract, text, references and some figures have been revise
GOGMA: Globally-Optimal Gaussian Mixture Alignment
Gaussian mixture alignment is a family of approaches that are frequently used
for robustly solving the point-set registration problem. However, since they
use local optimisation, they are susceptible to local minima and can only
guarantee local optimality. Consequently, their accuracy is strongly dependent
on the quality of the initialisation. This paper presents the first
globally-optimal solution to the 3D rigid Gaussian mixture alignment problem
under the L2 distance between mixtures. The algorithm, named GOGMA, employs a
branch-and-bound approach to search the space of 3D rigid motions SE(3),
guaranteeing global optimality regardless of the initialisation. The geometry
of SE(3) was used to find novel upper and lower bounds for the objective
function and local optimisation was integrated into the scheme to accelerate
convergence without voiding the optimality guarantee. The evaluation
empirically supported the optimality proof and showed that the method performed
much more robustly on two challenging datasets than an existing
globally-optimal registration solution.Comment: Manuscript in press 2016 IEEE Conference on Computer Vision and
Pattern Recognitio
Covariant Mappings for the Description of Measurement, Dissipation and Decoherence in Quantum Mechanics
The general formalism of quantum mechanics for the description of statistical
experiments is briefly reviewed, introducing in particular position and
momentum observables as POVM characterized by their covariance properties with
respect to the isochronous Galilei group. Mappings describing state
transformations both as a consequence of measurement and of dynamical evolution
for a closed or open system are considered with respect to the general
constraints they have to obey and their covariance properties with respect to
symmetry groups. In particular different master equations are analyzed in view
of the related symmetry group, recalling the general structure of mappings
covariant under the same group. This is done for damped harmonic oscillator,
two-level system and quantum Brownian motion. Special attention is devoted to
the general structure of translation-covariant master equations. Within this
framework a recently obtained quantum counterpart of the classical linear
Boltzmann equation is considered, as well as a general theoretical framework
for the description of different decoherence experiments, pointing to a
connection between different possible behaviours in the description of
decoherence and the characteristic functions of classical L\'evy processes.Comment: Comments: 38 pages, to appear in Lecture Notes in Physics,
Springer-Verla
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