45,816 research outputs found
Approximate Approximations from scattered data
The aim of this paper is to extend the approximate quasi-interpolation on a
uniform grid by dilated shifts of a smooth and rapidly decaying function on a
uniform grid to scattered data quasi-interpolation. It is shown that high order
approximation of smooth functions up to some prescribed accuracy is possible,
if the basis functions, which are centered at the scattered nodes, are
multiplied by suitable polynomials such that their sum is an approximate
partition of unity. For Gaussian functions we propose a method to construct the
approximate partition of unity and describe the application of the new
quasi-interpolation approach to the cubature of multi-dimensional integral
operators.Comment: 29 pages, 17 figure
Computation of volume potentials over bounded domains via approximate approximations
We obtain cubature formulas of volume potentials over bounded domains
combining the basis functions introduced in the theory of approximate
approximations with their integration over the tangential-halfspace. Then the
computation is reduced to the quadrature of one dimensional integrals over the
halfline. We conclude the paper providing numerical tests which show that these
formulas give very accurate approximations and confirm the predicted order of
convergence.Comment: 18 page
Compression for Smooth Shape Analysis
Most 3D shape analysis methods use triangular meshes to discretize both the
shape and functions on it as piecewise linear functions. With this
representation, shape analysis requires fine meshes to represent smooth shapes
and geometric operators like normals, curvatures, or Laplace-Beltrami
eigenfunctions at large computational and memory costs.
We avoid this bottleneck with a compression technique that represents a
smooth shape as subdivision surfaces and exploits the subdivision scheme to
parametrize smooth functions on that shape with a few control parameters. This
compression does not affect the accuracy of the Laplace-Beltrami operator and
its eigenfunctions and allow us to compute shape descriptors and shape
matchings at an accuracy comparable to triangular meshes but a fraction of the
computational cost.
Our framework can also compress surfaces represented by point clouds to do
shape analysis of 3D scanning data
Topological Phases of Sound and Light
Topological states of matter are particularly robust, since they exploit
global features insensitive to local perturbations. In this work, we describe
how to create a Chern insulator of phonons in the solid state. The proposed
implementation is based on a simple setting, a dielectric slab with a suitable
pattern of holes. Its topological properties can be wholly tuned in-situ by
adjusting the amplitude and frequency of a driving laser that controls the
optomechanical interaction between light and sound. The resulting chiral,
topologically protected phonon transport along the edges can be probed
completely optically. Moreover, we identify a regime of strong mixing between
photon and phonon excitations, which gives rise to a large set of different
topological phases. This would be an example of a Chern insulator produced from
the interaction between two physically very different particle species, photons
and phonons
Phonon-phonon interactions due to non-linear effects in a linear ion trap
We examine in detail the theory of the intrinsic non-linearities in the
dynamics of trapped ions due to the Coulomb interaction. In particular the
possibility of mode-mode coupling, which can be a source of decoherence in
trapped ion quantum computation, or, alternatively, can be exploited for
parametric down-conversion of phonons, is discussed and conditions under which
such coupling is possible are derived.Comment: 25 pages, 4 figure
Geometry-induced localization of thermal fluctuations in ultrathin superconducting structures
Thermal fluctuations of the order parameter in an ultrathin triangular shaped
superconducting structure are studied near , in zero applied field. We
find that the order parameter is prone to much larger fluctuations in the
corners of the structure as compared to its interior. This geometry-induced
localization of thermal fluctuations is attributed to the fact that condensate
confinement in the corners is characterised by a lower effective
dimensionality, which favors stronger fluctuations.Comment: 9 pages, 5 figure
Optomechanical creation of magnetic fields for photons on a lattice
We propose using the optomechanical interaction to create artificial magnetic
fields for photons on a lattice. The ingredients required are an optomechanical
crystal, i.e. a piece of dielectric with the right pattern of holes, and two
laser beams with the right pattern of phases. One of the two proposed schemes
is based on optomechanical modulation of the links between optical modes, while
the other is an lattice extension of optomechanical wavelength-conversion
setups. We illustrate the resulting optical spectrum, photon transport in the
presence of an artificial Lorentz force, edge states, and the photonic
Aharonov-Bohm effect. Moreover, wWe also briefly describe the gauge fields
acting on the synthetic dimension related to the phonon/photon degree of
freedom. These can be generated using a single laser beam impinging on an
optomechanical array
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