2,955 research outputs found
An improved bound on the number of point-surface incidences in three dimensions
We show that points and smooth algebraic surfaces of bounded degree
in satisfying suitable nondegeneracy conditions can have at most
incidences, provided that any
collection of points have at most O(1) surfaces passing through all of
them, for some . In the case where the surfaces are spheres and no
three spheres meet in a common circle, this implies there are point-sphere incidences. This is a slight improvement over the previous
bound of for an (explicit) very
slowly growing function. We obtain this bound by using the discrete polynomial
ham sandwich theorem to cut into open cells adapted to the set
of points, and within each cell of the decomposition we apply a Turan-type
theorem to obtain crude control on the number of point-surface incidences. We
then perform a second polynomial ham sandwich decomposition on the irreducible
components of the variety defined by the first decomposition. As an
application, we obtain a new bound on the maximum number of unit distances
amongst points in .Comment: 17 pages, revised based on referee comment
Greedy Search for Descriptive Spatial Face Features
Facial expression recognition methods use a combination of geometric and
appearance-based features. Spatial features are derived from displacements of
facial landmarks, and carry geometric information. These features are either
selected based on prior knowledge, or dimension-reduced from a large pool. In
this study, we produce a large number of potential spatial features using two
combinations of facial landmarks. Among these, we search for a descriptive
subset of features using sequential forward selection. The chosen feature
subset is used to classify facial expressions in the extended Cohn-Kanade
dataset (CK+), and delivered 88.7% recognition accuracy without using any
appearance-based features.Comment: International Conference on Acoustics, Speech and Signal Processing
(ICASSP), 201
Casimir effect for parallel plates in de Sitter spacetime
The Wightman function and the vacuum expectation values of the field squared
and of the energy-momentum tensor are obtained, for a massive scalar field with
an arbitrary curvature coupling parameter, in the region between two infinite
parallel plates, on the background of de Sitter spacetime. The field is
prepared in the Bunch-Davies vacuum state and is constrained to satisfy Robin
boundary conditions on the plates. For the calculation, a mode-summation method
is used, supplemented with a variant of the generalized Abel-Plana formula.
This allows to explicitly extract the contributions to the expectation values
which come from each single boundary, and to expand the second-plate-induced
part in terms of exponentially convergent integrals. Several limiting cases of
interest are then studied. Moreover, the Casimir forces acting on the plates
are evaluated, and it is shown that the curvature of the background spacetime
decisively influences the behavior of these forces at separations larger than
the curvature scale of de Sitter spacetime. In terms of the curvature coupling
parameter and the mass of the field, two very different regimes are realized,
which exhibit monotonic and oscillatory behavior of the vacuum expectation
values, respectively. The decay of the Casimir force at large plate separation
is shown to be power-law (monotonic or oscillating), with independence of the
value of the field mass.Comment: 22 pages, 4 figures, added figures for a massless field, added
reference, added discussions and comments on thermal effect
Fermionic Casimir effect for parallel plates in the presence of compact dimensions with applications to nanotubes
We evaluate the Casimir energy and force for a massive fermionic field in the
geometry of two parallel plates on background of Minkowski spacetime with an
arbitrary number of toroidally compactified spatial dimensions. The bag
boundary conditions are imposed on the plates and periodicity conditions with
arbitrary phases are considered along the compact dimensions. The Casimir
energy is decomposed into purely topological, single plate and interaction
parts. With independence of the lengths of the compact dimensions and the
phases in the periodicity conditions, the interaction part of the Casimir
energy is always negative. In order to obtain the resulting force, the
contributions from both sides of the plates must be taken into account. Then,
the forces coming from the topological parts of the vacuum energy cancel out
and only the interaction term contributes to the Casimir force. Applications of
the general formulae to Kaluza-Klein type models and carbon nanotubes are
given. In particular, we show that for finite length metallic nanotubes the
Casimir forces acting on the tube edges are always attractive, whereas for
semiconducting-type ones they are attractive for small lengths of the nanotube
and repulsive for large lengths.Comment: 20 pages, 3 figure
Fermionic Casimir densities in anti-de Sitter spacetime
The fermionic condensate and vacuum expectation value of the energy-momentum
tensor, for a massive fermionic field on the background of anti-de Sitter
spacetime, in the geometry of two parallel boundaries with bag boundary
conditions, are investigated. Vacuum expectation values, expressed as series
involving the eigenvalues of the radial quantum number, are neatly decomposed
into boundary-free, single-boundary-induced, and second-boundary-induced parts,
with the help of the generalized Abel-Plana summation formula. In this way, the
renormalization procedure is very conveniently reduced to the one corresponding
to boundary-free AdS spacetime. The boundary-induced contributions to the
fermionic condensate and to the vacuum expectation value of the energy density
are proven to be everywhere negative. The vacuum expectation values are
exponentially suppressed at distances from the boundaries much larger than the
curvature radius of the AdS space. Near the boundaries, effects related with
the curvature of the background spacetime are shown to be subdominant and, to
leading order, all known results for boundaries in the Minkowski bulk are
recovered. Zeta function techniques are successfully used for the evaluation of
the total vacuum energy in the region between the boundaries. It is proven that
the resulting interaction forces between them are attractive and that, for
large separations, they also decay exponentially. Finally, our results are
extended and explicitly translated to fermionic Casimir densities in braneworld
scenarios of Randall-Sundrum type.Comment: 22 pages, 2 figure
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