2,048 research outputs found
Numerical aspects of nonlinear Schrodinger equations in the presence of caustics
The aim of this text is to develop on the asymptotics of some 1-D nonlinear
Schrodinger equations from both the theoretical and the numerical perspectives,
when a caustic is formed. We review rigorous results in the field and give some
heuristics in cases where justification is still needed. The scattering
operator theory is recalled. Numerical experiments are carried out on the focus
point singularity for which several results have been proven rigorously.
Furthermore, the scattering operator is numerically studied. Finally,
experiments on the cusp caustic are displayed, and similarities with the focus
point are discussed.Comment: 20 pages. To appear in Math. Mod. Meth. Appl. Sc
Scattering for nonlinear Schrodinger equation under partial harmonic confinement
We consider the nonlinear Schrodinger equation under a partial quadratic
confinement. We show that the global dispersion corresponding to the
direction(s) with no potential is enough to prove global in time Strichartz
estimates, from which we infer the existence of wave operators thanks to
suitable vector-fields. Conversely, given an initial Cauchy datum, the solution
is global in time and asymptotically free, provided that confinement affects
one spatial direction only. This stems from anisotropic Morawetz estimates,
involving a marginal of the position density.Comment: 26 pages. Some typos fixed, especially in Section
Extended depth-of-field imaging and ranging in a snapshot
Traditional approaches to imaging require that an increase in depth of field is associated with a reduction in
numerical aperture, and hence with a reduction in resolution and optical throughput. In their seminal
work, Dowski and Cathey reported how the asymmetric point-spread function generated by a cubic-phase
aberration encodes the detected image such that digital recovery can yield images with an extended depth of
field without sacrificing resolution [Appl. Opt. 34, 1859 (1995)]. Unfortunately recovered images are
generally visibly degraded by artifacts arising from subtle variations in point-spread functions with defocus.
We report a technique that involves determination of the spatially variant translation of image components
that accompanies defocus to enable determination of spatially variant defocus. This in turn enables recovery
of artifact-free, extended depth-of-field images together with a two-dimensional defocus and range map
of the imaged scene. We demonstrate the technique for high-quality macroscopic and microscopic imaging
of scenes presenting an extended defocus of up to two waves, and for generation of defocus maps with an
uncertainty of 0.036 waves
Video-rate computational super-resolution and integral imaging at longwave-infrared wavelengths
We report the first computational super-resolved, multi-camera integral
imaging at long-wave infrared (LWIR) wavelengths. A synchronized array of FLIR
Lepton cameras was assembled, and computational super-resolution and
integral-imaging reconstruction employed to generate video with light-field
imaging capabilities, such as 3D imaging and recognition of partially obscured
objects, while also providing a four-fold increase in effective pixel count.
This approach to high-resolution imaging enables a fundamental reduction in the
track length and volume of an imaging system, while also enabling use of
low-cost lens materials.Comment: Supplementary multimedia material in
http://dx.doi.org/10.6084/m9.figshare.530302
Super-resolution imaging using a camera array
The angular resolution of many commercial imaging systems is limited, not by diffraction or optical aberrations, but by pixilation effects. Multiaperture imaging has previously demonstrated the potential for super-resolution (SR) imaging using a lenslet array and single detector array. We describe the practical demonstration of SR imaging using an array of 25 independent commercial-off-the-shelf cameras. This technique demonstrates the potential for increasing the angular resolution toward the diffraction limit, but without the limit on angular resolution imposed by the use of a single detector array
Computational localization microscopy with extended axial range
A new single-aperture 3D particle-localization and tracking technique is presented that demonstrates an increase in depth range by more than an order of magnitude without compromising optical resolution and throughput. We exploit the extended depth range and depth-dependent translation of an Airy-beam PSF for 3D localization over an extended volume in a single snapshot. The technique is applicable to all bright-field and fluorescence modalities for particle localization and tracking, ranging from super-resolution microscopy through to the tracking of fluorescent beads and endogenous particles within cells. We demonstrate and validate its application to real-time 3D velocity imaging of fluid flow in capillaries using fluorescent tracer beads. An axial localization precision of 50 nm was obtained over a depth range of 120μm using a 0.4NA, 20× microscope objective. We believe this to be the highest ratio of axial range-to-precision reported to date
Definable orthogonality classes in accessible categories are small
We lower substantially the strength of the assumptions needed for the
validity of certain results in category theory and homotopy theory which were
known to follow from Vopenka's principle. We prove that the necessary
large-cardinal hypotheses depend on the complexity of the formulas defining the
given classes, in the sense of the Levy hierarchy. For example, the statement
that, for a class S of morphisms in a locally presentable category C of
structures, the orthogonal class of objects is a small-orthogonality class
(hence reflective) is provable in ZFC if S is \Sigma_1, while it follows from
the existence of a proper class of supercompact cardinals if S is \Sigma_2, and
from the existence of a proper class of what we call C(n)-extendible cardinals
if S is \Sigma_{n+2} for n bigger than or equal to 1. These cardinals form a
new hierarchy, and we show that Vopenka's principle is equivalent to the
existence of C(n)-extendible cardinals for all n. As a consequence, we prove
that the existence of cohomological localizations of simplicial sets, a
long-standing open problem in algebraic topology, is implied by the existence
of arbitrarily large supercompact cardinals. This result follows from the fact
that cohomology equivalences are \Sigma_2. In contrast with this fact, homology
equivalences are \Sigma_1, from which it follows (as is well known) that the
existence of homological localizations is provable in ZFC.Comment: 38 pages; some results have been improved and former inaccuracies
have been correcte
Efficient implementation of finite volume methods in Numerical Relativity
Centered finite volume methods are considered in the context of Numerical
Relativity. A specific formulation is presented, in which third-order space
accuracy is reached by using a piecewise-linear reconstruction. This
formulation can be interpreted as an 'adaptive viscosity' modification of
centered finite difference algorithms. These points are fully confirmed by 1D
black-hole simulations. In the 3D case, evidence is found that the use of a
conformal decomposition is a key ingredient for the robustness of black hole
numerical codes.Comment: Revised version, 10 pages, 6 figures. To appear in Phys. Rev.
Linear vs. nonlinear effects for nonlinear Schrodinger equations with potential
We review some recent results on nonlinear Schrodinger equations with
potential, with emphasis on the case where the potential is a second order
polynomial, for which the interaction between the linear dynamics caused by the
potential, and the nonlinear effects, can be described quite precisely. This
includes semi-classical regimes, as well as finite time blow-up and scattering
issues. We present the tools used for these problems, as well as their
limitations, and outline the arguments of the proofs.Comment: 20 pages; survey of previous result
Perturbation of the sierpinski antenna to allocate the operating bands
A scheme for modifying the spacing between the bands of the Sierpinski antenna is introduced. Experimental results of two novel designs of fractal antennas suggest that the fractal structure can be perturbed to enable the log-period to be changed while still maintaining the multiband behaviour of the antenna.Peer ReviewedPostprint (published version
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