Controlled Perturbation is a framework for perturbing geometric arrangements to make them more robust for further manipulation. The idea is to slightly perturb the arrangement to make it more robust for finite precision arithmetic. We present a Controlled Perturbation scheme for arrangements of line segments in IR (CPAL for short). CPAL perturbs the endpoints of the arrangement to eliminate deg...

For hydrologic applications, terrain models should have few local minima, and drainage lines should coincide with edges. We show that triangulating a set of points with elevations such that the number of local minima of the resulting terrain is minimized is NP-hard for degenerate point sets. The same result applies when there are no degeneracies for higher-order Delaunay triangulations. Two heu...

This paper shows that the mass-sheet degeneracy and other degeneracies in lensing have simple geometrical interpretations: they are mostly rescalings of the arrival-time surface. Different degeneracies appear in Local Group lensing and in cosmological lensing, because in the former the absolute magnification is measured but the image structure is not resolved, whereas in the latter the reverse ...

The accelerating expansion of the universe suggests that an unknown component with strongly negative pressure, called dark energy, currently dominates the dynamics of the universe. Such a component makes up ;70% of the energy density of the universe yet has not been predicted by the standard model of particle physics. The best method for exploring the nature of this dark energy is to map the re...

In a classical Symbolic Perturbation scheme, degeneracies are handled by substituting some polynomials in ε for the inputs of a predicate. Instead of a single perturbation, we propose to use a sequence of (simpler) perturbations. Moreover, we look at their effects geometrically instead of algebraically; this allows us to tackle cases that were not tractable with the classical algebraic approach...

We prove sharp L estimates for oscillatory integral and Fourier integral operators for which the associated canonical relation C ⊂ T ∗ΩL × T ∗ΩR projects to T ∗ΩL and to T ∗ΩR with corank one singularities of type ≤ 2. This includes two-sided cusp singularities. Applications are given to operators with one-sided swallowtail singularities such as restricted X-ray transforms for well-curved line ...

We show that in the worst case, (n d) sidedness queries are required to determine whether a set of n points in IR d is aanely degenerate, i.e., whether it contains d + 1 points on a common hyperplane. This matches known upper bounds. We give a straightforward adversary argument, based on the explicit construction of a point set containing (n d) \collapsible" simplices, any one of which can be m...

The spectrum of eigenenergies of a quantum integrable system whose hamiltonian depends on a single parameter shows degeneracies (crossings) when the parameter varies. We derive a semiclassical expression for the density of crossings in the plane energy-parameter, that is the number of crossings per unit of energy and unit of parameter, in terms of classical periodic orbits. We compare the resul...

We generalize to non-flat geometries the formalism of Simon et al. (2005) to reconstruct the dark energy potential. This formalismmakes use of quantities similar to the Horizon-flow parameters in inflation, can, in principle, be made non-parametric and is general enough to be applied outside the simple, single scalar field quintessence. Since presently available and forthcoming data do not allo...