Abstract We prove that a flow of cold collisionless particles from all directions in and out of a region necessarily forms a caustic. A corollary is that, in cold dark matter cosmology, galactic halos have inner caustics in addition to the more obvious outer caustics. The outer caustics are fold catastrophes located on topological spheres surrounding the galaxy. To obtain the catastrophe struct...

We study quantum caustics (i.e., the quantum analogue of the classical singularity in the Dirichlet boundary problem) in d-dimensional systems with quadratic Lagrangians of the form L = 1 2 Pij(t) ẋ ẋ +Qij(t)x ẋ + 1 2 Rij(t)x x + Si(t) x . Based on Schulman’s procedure in the path-integral we derive the transition amplitude on caustics in a closed form for generic multiplicity f , and thereby c...

Realistic rendering of underwater scenes is one of the most anticipated research topics in computer graphics. Caustics are the important component enhancing the realism of this kind of scenes. Unfortunately, rendering caustics is a time consuming task. As a result, most existing algorithms cannot handle this at interactive rate. In recent years, volumetric texture based rendering algorithms hav...

In this paper we present an improved technique for interactive rendering of caustics using programmable graphics hardware. Previous real-time methods have used simple prisms for the caustic volumes and a constant intensity approximation at the receiver. Our approach uses interpolated caustic volumes to render smooth high-quality caustics. We have derived a simple formula for evaluating the dens...

The late infall of cold dark matter onto an isolated galaxy such as our own produces flows with definite velocity vectors at any physical point in the galactic halo. It also produces caustics which are places where the dark matter density is very large. The outer caustics are topological spheres whereas the inner caustics are rings. The self-similar model of galactic halo formation predicts tha...

In this article, we attempt to study the possible link between the dynamics of a circle map and the caustics of its iterations. The attention is on a geometrically defined off-center reflections, which, coincidentally, is also a generalization of standard perturbation of rotation. Some results of the caustics and dynamics of the family of maps are given. The method of symplectic topology is par...

The caustics of Fourier integral operators are defined as caustics of the corresponding Schwartz kernels (Lagrangian distributions on X × Y). The caustic set Σ(C) of the canonical relation C is characterized as the set of points where the rank of the projection π : C → X × Y is smaller than its maximal value, dim(X × Y) − 1. We derive the L p (Y) → L q (X) estimates on Fourier integral operator...

Caustics are curves with the property that a billiard trajectory, once tangent to it, stays tangent after every reflection at the boundary of the billiard table. When the billiard table is an ellipse, any nonsingular billiard trajectory has a caustic, which can be either a confocal ellipse or a confocal hyperbola. Resonant caustics —the ones whose tangent trajectories are closed polygons— are d...

Caustics are a generic feature of the nonlinear growth of structure in the dark matter distribution. If the dark matter were absolutely cold, its mass density would diverge at caustics, and the integrated annihilation probability would also diverge for individual particles participating in them. For realistic dark matter candidates, this behaviour is regularised by small but non-zero initial th...

background: caustics produce one of the most serious complications of poisonings. this research was aimed to study demographical, clinical and endoscopic findings in patients with corrosive injury. methods: in a retrospective survey, all records of the patients who had been admitted to razi and 17th shahrivar hospital in rasht city due to the ingestion of caustic agents during 2002-2009 and end...