5,379 research outputs found
The Scalar Curvature Equation on
We give existence results for solutions of the prescribed scalar curvature
equation on , when the curvature function is a positive Morse function and
satisfies an index-count condition.Comment: 23 pages, 1 figur
Perturbation results of critical elliptic equations of Caffarelli-Kohn-Nirenberg type
We find for small positive solutions to the equation
in ,
which branch off from the manifold of minimizers in the class of radial
functions of the corresponding Caffarelli-Kohn-Nirenberg type inequality.
Moreover, our analysis highlights the symmetry-breaking phenomenon in these
inequalities, namely the existence of non-radial minimizers.Comment: 15 pages, 4 figure
Aperture Multipole Moments from Weak Gravitational Lensing
The projected mass of a gravitational lens inside (circular) apertures can be
derived from the measured shear inside an annulus which is caused by the tidal
field of the deflecting mass distribution. Here we show that also the
multipoles of the two-dimensional mass distribution can be derived from the
shear in annuli. We derive several expressions for these mass multipole moments
in terms of the shear, which allow large flexibility in the choice of a radial
weight function. In contrast to determining multipole moments from weak-lensing
mass reconstructions, this approach allows to quantify the signal-to-noise
ratio of the multipole moments directly from the observed galaxy ellipticities,
and thus to estimate the significance of the multipole detection. Radial weight
functions can therefore be chosen such as to optimize the significance of the
detection given an assumed radial mass profile. Application of our formulae to
numerically simulated clusters demonstrates that the quadrupole moment of
realistic cluster models can be detected with high signal-to-noise ratio S/N;
in about 85 per cent of the simulated cluster fields S/N >~ 3. We also show
that the shear inside a circular annulus determines multipole moments inside
and outside the annulus. This is relevant for clusters whose central region is
too bright to allow the observation of the shear of background galaxies, or
which extend beyond the CCD. We also generalize the aperture mass equation to
the case of `radial' weight functions which are constant on arbitrarily-shaped
curves which are not necessarily self-similar.Comment: 14 pages including 3 figures; submitted to MNRAS; replaced to improve
printing on non-A4 pape
- …