We use a large N-body simulation to study the characteristic scales in density gradient profiles and around halos with masses ranging from $10^{12}$ $10^{15} h^{-1}{\rm M_\odot}$. investigate separately along major (T_1) minor (T_3) axes of local tidal tensor how depend on halo mass, formation time, environment. find two kinds prominent features profiles, deep `valley' `peak'. Gaussian Process ...

We investigate the caustic structure of a lens composed by a discrete number of point–masses, having mutual distances smaller than the Einstein radius of the total mass of the system. Along with the main critical curve, it is known that the lens map is characterized by secondary critical curves producing small caustics far from the lens system. By exploiting perturbative methods, we derive the ...

This paper discusses numerical experiments of wave front propagation on a flat disk and related results. In particular, notions of extremal rays and wave front caustics are introduced to investigate the dynamic behaviors of wave fronts under repeated reflection. These results are related to earlier work on caustics of ray systems. It is shown that wave fronts require a modified notion of causti...

We show quantitatively how the collision rate of droplets of visible moisture in turbulent air increases very abruptly as the intensity of the turbulence passes a threshold, due to the formation of fold caustics in their velocity field. The formation of caustics is an activated process, in which a measure of the intensity of the turbulence, termed the Stokes number St, is analogous to temperatu...

Stochastic flows mimicking 2D turbulence in compressible media are considered. Particles driven by such can collide and we study the collision (caustic) frequency. Caustics occur when Jacobian of a flow vanishes. First, system nonlinear stochastic differential equations involving is derived reduced to smaller number unknowns. Then, for special cases forcing, upper lower bounds found mean causti...

A systematic investigation of the effect of the history force on particle advection is carried out in a paradigmatic model flow of chaotic advection, the von Kármán flow. All investigated properties turn out to heavily depend on the presence of memory when compared to previous studies neglecting this force. We find a weaker tendency for accumulation and for caustics formation. The Lyapunov expo...

We study experimentally how waves affect the distribution of particles that float on a liquid surface. We show that clustering of small particles in a standing wave is a nonlinear effect with the clustering time decreasing as the square of the wave amplitude. In a set of random waves, we show that small floaters concentrate on a multifractal set with caustics.

This work examines the small-scale E and B patterns of the Cosmic Microwave Background Radiation (CMBR) polarization anisotropy. Particularly, we address the topologi-cal natures and spatial symmetries of the E and B modes, and how one may make use of the local measurements of Stokes parameters to separate these two modes. We point out that the separation of these two modes in a local map of St...

A new uniform asymptotic approximation for the Wigner 6j-symbol is given in terms of Wigner rotation matrices (d-matrices). The approximation is uniform in the sense that it applies for all values of the quantum numbers, even those near caustics. The derivation of the new approximation is not given, but the geometrical ideas supporting it are discussed and numerical tests are presented, includi...

The first investigations of the response of the microlensing magnification pattern (at an optical depth of order unity) to the mass function of the microlenses found that the resulting statistics depend only on the mean microlens mass 〈m〉. In particular the mean microlensing caustic crossing rate was found to be proportional to