179,933 research outputs found
The Chang-Refsdal Lens Revisited
This paper provides a complete theoretical treatment of the point-mass lens
perturbed by constant external shear, often called the Chang-Refsdal lens. We
show that simple invariants exist for the products of the (complex) positions
of the four images, as well as moment sums of their signed magnifications. The
image topographies and equations of the caustics and critical curves are also
studied. We derive the fully analytic expressions for precaustics, which are
the loci of non-critical points that map to the caustics under the lens
mapping. They constitute boundaries of the region in the image domain that maps
onto the interior of the caustics. The areas under the critical curves,
caustics and precaustics are all evaluated, which enables us to calculate the
mean magnification of the source within the caustics. Additionally, the exact
analytic expression for the magnification distribution for the source in the
triangular caustics is derived, as well as a useful approximate expression.
Finally, we find that the Chang-Refsdal lens with the convergence greater than
unity can exhibit third-order critical behaviour, if the reduced shear is
exactly equal to \sqrt{3}/2, and that the number of images for N-point masses
with non-zero constant shear cannot be greater than 5N-1.Comment: to appear in MNRAS (including 6 figures, 3 appendices; v2 - minor
update with corrected typos etc.
Post-critical set and non existence of preserved meromorphic two-forms
We present a family of birational transformations in depending on
two, or three, parameters which does not, generically, preserve meromorphic
two-forms. With the introduction of the orbit of the critical set (vanishing
condition of the Jacobian), also called ``post-critical set'', we get some new
structures, some "non-analytic" two-form which reduce to meromorphic two-forms
for particular subvarieties in the parameter space. On these subvarieties, the
iterates of the critical set have a polynomial growth in the \emph{degrees of
the parameters}, while one has an exponential growth out of these subspaces.
The analysis of our birational transformation in is first carried out
using Diller-Favre criterion in order to find the complexity reduction of the
mapping. The integrable cases are found. The identification between the
complexity growth and the topological entropy is, one more time, verified. We
perform plots of the post-critical set, as well as calculations of Lyapunov
exponents for many orbits, confirming that generically no meromorphic two-form
can be preserved for this mapping. These birational transformations in ,
which, generically, do not preserve any meromorphic two-form, are extremely
similar to other birational transformations we previously studied, which do
preserve meromorphic two-forms. We note that these two sets of birational
transformations exhibit totally similar results as far as topological
complexity is concerned, but drastically different results as far as a more
``probabilistic'' approach of dynamical systems is concerned (Lyapunov
exponents). With these examples we see that the existence of a preserved
meromorphic two-form explains most of the (numerical) discrepancy between the
topological and probabilistic approach of dynamical systems.Comment: 34 pages, 7 figure
Total variation regularization for manifold-valued data
We consider total variation minimization for manifold valued data. We propose
a cyclic proximal point algorithm and a parallel proximal point algorithm to
minimize TV functionals with -type data terms in the manifold case.
These algorithms are based on iterative geodesic averaging which makes them
easily applicable to a large class of data manifolds. As an application, we
consider denoising images which take their values in a manifold. We apply our
algorithms to diffusion tensor images, interferometric SAR images as well as
sphere and cylinder valued images. For the class of Cartan-Hadamard manifolds
(which includes the data space in diffusion tensor imaging) we show the
convergence of the proposed TV minimizing algorithms to a global minimizer
Plus-minus construction leads to perfect invisibility
Recent theoretical advances applied to metamaterials have opened new avenues
to design a coating that hides objects from electromagnetic radiation and even
the sight. Here, we propose a new design of cloaking devices that creates
perfect invisibility in isotropic media. A combination of positive and negative
refractive indices, called plus-minus construction, is essential to achieve
perfect invisibility (i.e., no time delay and total absence of reflection).
Contrary to the common understanding that between two isotropic materials
having different refractive indices the electromagnetic reflection is
unavoidable, our method shows that surprisingly the reflection phenomena can be
completely eliminated. The invented method, different from the classical
impedance matching, may also find electromagnetic applications outside of
cloaking devices, wherever distortions are present arising from reflections.Comment: 24 pages, 10 figure
Analysis of the Strong Coupling Limit of the Richardson Hamiltonian using the Dyson Mapping
The Richardson Hamiltonian describes superconducting correlations in a
metallic nanograin. We do a perturbative analysis of this and related
Hamiltonians, around the strong pairing limit, without having to invoke Bethe
Ansatz solvability. Rather we make use of a boson expansion method known as the
Dyson mapping. Thus we uncover a selection rule that facilitates both
time-independent and time-dependent perturbation expansions. In principle the
model we analise is realised in a very small metalic grain of a very regular
shape. The results we obtain point to subtleties sometimes neglected when
thinking of the superconducting state as a Bose-Einstein condensate. An
appendix contains a general presentation of time-independent perturbation
theory for operators with degenerate spectra, with recursive formulas for
corrections of arbitrarily high orders.Comment: New final version accepted for publication in PRB. 17 two-column
pages, no figure
- …