6,658 research outputs found
Global properties of gravitational lens maps in a Lorentzian manifold setting
In a general-relativistic spacetime (Lorentzian manifold), gravitational
lensing can be characterized by a lens map, in analogy to the lens map of the
quasi-Newtonian approximation formalism. The lens map is defined on the
celestial sphere of the observer (or on part of it) and it takes values in a
two-dimensional manifold representing a two-parameter family of worldlines. In
this article we use methods from differential topology to characterize global
properties of the lens map. Among other things, we use the mapping degree (also
known as Brouwer degree) of the lens map as a tool for characterizing the
number of images in gravitational lensing situations. Finally, we illustrate
the general results with gravitational lensing (a) by a static string, (b) by a
spherically symmetric body, (c) in asymptotically simple and empty spacetimes,
and (d) in weakly perturbed Robertson-Walker spacetimes.Comment: 26 page
Improved analysis of algorithms based on supporting halfspaces and quadratic programming for the convex intersection and feasibility problems
This paper improves the algorithms based on supporting halfspaces and
quadratic programming for convex set intersection problems in our earlier paper
in several directions. First, we give conditions so that much smaller quadratic
programs (QPs) and approximate projections arising from partially solving the
QPs are sufficient for multiple-term superlinear convergence for nonsmooth
problems. Second, we identify additional regularity, which we call the second
order supporting hyperplane property (SOSH), that gives multiple-term quadratic
convergence. Third, we show that these fast convergence results carry over for
the convex inequality problem. Fourth, we show that infeasibility can be
detected in finitely many operations. Lastly, we explain how we can use the
dual active set QP algorithm of Goldfarb and Idnani to get useful iterates by
solving the QPs partially, overcoming the problem of solving large QPs in our
algorithms.Comment: 27 pages, 2 figure
Conical defects in growing sheets
A growing or shrinking disc will adopt a conical shape, its intrinsic
geometry characterized by a surplus angle at the apex. If growth is slow,
the cone will find its equilibrium. Whereas this is trivial if , the
disc can fold into one of a discrete infinite number of states if is
positive. We construct these states in the regime where bending dominates,
determine their energies and how stress is distributed in them. For each state
a critical value of is identified beyond which the cone touches itself.
Before this occurs, all states are stable; the ground state has two-fold
symmetry.Comment: 4 pages, 4 figures, LaTeX, RevTeX style. New version corresponds to
the one published in PR
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