15,699 research outputs found
The Geometry of Axisymmetric Ideal Fluid Flows with Swirl
The sectional curvature of the volume preserving diffeomorphism group of a
Riemannian manifold can give information about the stability of inviscid,
incompressible fluid flows on . We demonstrate that the submanifold of the
volumorphism group of the solid flat torus generated by axisymmetric fluid
flows with swirl, denoted by , has positive sectional
curvature in every section containing the field iff
. This is in sharp contrast to the situation on
, where only Killing fields have nonnegative
sectional curvature in all sections containing it. We also show that this
criterion guarantees the existence of conjugate points on
along the geodesic defined by .Comment: 8 page
An elementary proof of Franks' lemma for geodesic flows
Given a Riemannian manifold and a geodesic , the
perpendicular part of the derivative of the geodesic flow along is a linear symplectic map. We give an
elementary proof of the following Franks' lemma, originally found in [G.
Contreras and G. Paternain, 2002] and [G. Contreras, 2010]: this map can be
perturbed freely within a neighborhood in by a -small perturbation
of the metric that keeps a geodesic for the new metric. Moreover,
the size of these perturbations is uniform over fixed length geodesics on the
manifold. When , the original metric must belong to a
--open and dense subset of metrics
Efficient MRF Energy Propagation for Video Segmentation via Bilateral Filters
Segmentation of an object from a video is a challenging task in multimedia
applications. Depending on the application, automatic or interactive methods
are desired; however, regardless of the application type, efficient computation
of video object segmentation is crucial for time-critical applications;
specifically, mobile and interactive applications require near real-time
efficiencies. In this paper, we address the problem of video segmentation from
the perspective of efficiency. We initially redefine the problem of video
object segmentation as the propagation of MRF energies along the temporal
domain. For this purpose, a novel and efficient method is proposed to propagate
MRF energies throughout the frames via bilateral filters without using any
global texture, color or shape model. Recently presented bi-exponential filter
is utilized for efficiency, whereas a novel technique is also developed to
dynamically solve graph-cuts for varying, non-lattice graphs in general linear
filtering scenario. These improvements are experimented for both automatic and
interactive video segmentation scenarios. Moreover, in addition to the
efficiency, segmentation quality is also tested both quantitatively and
qualitatively. Indeed, for some challenging examples, significant time
efficiency is observed without loss of segmentation quality.Comment: Multimedia, IEEE Transactions on (Volume:16, Issue: 5, Aug. 2014
Geometric Aspects of Holographic Bit Threads
We revisit the recent reformulation of the holographic prescription to
compute entanglement entropy in terms of a convex optimization problem,
introduced by Freedman and Headrick. According to it, the holographic
entanglement entropy associated to a boundary region is given by the maximum
flux of a bounded, divergenceless vector field, through the corresponding
region. Our work leads to two main results: (i) We present a general algorithm
that allows the construction of explicit thread configurations in cases where
the minimal surface is known. We illustrate the method with simple examples:
spheres and strips in vacuum AdS, and strips in a black brane geometry.
Studying more generic bulk metrics, we uncover a sufficient set of conditions
on the geometry and matter fields that must hold to be able to use our
prescription. (ii) Based on the nesting property of holographic entanglement
entropy, we develop a method to construct bit threads that maximize the flux
through a given bulk region. As a byproduct, we are able to construct more
general thread configurations by combining (i) and (ii) in multiple patches. We
apply our methods to study bit threads which simultaneously compute the
entanglement entropy and the entanglement of purification of mixed states and
comment on their interpretation in terms of entanglement distillation. We also
consider the case of disjoint regions for which we can explicitly construct the
so-called multi-commodity flows and show that the monogamy property of mutual
information can be easily illustrated from our constructions.Comment: 48 pages, multiple figures. v3: matches published versio
The volume entropy of a surface decreases along the Ricci flow
The volume entropy, h(g), of a compact Riemannian manifold (M,g) measures the growth rate of the volume of a ball of radius R in its universal cover. Under the Ricci flow, g evolves along a certain path that improves its curvature properties. For a compact surface of variable negative curvature we use a Katok–Knieper–Weiss formula to show that h(gt) is strictly decreasing
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