Double Bubbles Minimize

The classical isoperimetric inequality in R^3 states that the surface of smallest area enclosing a given volume is a sphere. We show that the least area surface enclosing two equal volumes is a double bubble, a surface made of two pieces of round spheres separated by a flat disk, meeting along a single circle at an angle of 120 degrees.Comment: 57 pages, 32 figures. Includes the complete code for a C++ program as described in the article. You can obtain this code by viewing the source of this articl

The Real Estate Investment Trust: State Law Problems

We present a map of the three-dimensional (3D) distribution of dust in the Orion complex. Orion is the closest site of high-mass star formation, making it an excellent laboratory for studying the interstellar medium and star formation. We used data from the Gaia-TGAS catalogue combined with photometry from 2MASS and WISE to get the distances and extinctions of individual stars in the vicinity of the Orion complex. We use a Gaussian process and adopt a non-parametric method to infer the probability distribution function of the dust densities at arbitrary points throughout the region. We map the dust distribution towards different parts of the Orion complex. We find that the distance and depth of the cloud are compatible with other recent works, which show that the method can be applicable to local molecular clouds to map their 3D dust distribution. We also demonstrate the danger of only using colours of stars to derive their extinctions without considering further physical constraints, such as the colour-magnitude diagram (CMD)

An Empirical Relation between Sodium Absorption and Dust Extinction

Dust extinction and reddening are ubiquitous in astronomical observations and are often a major source of systematic uncertainty. We present here a study of the correlation between extinction in the Milky Way and the equivalent width of the NaI D absorption doublet. Our sample includes more than 100 high resolution spectra from the KECK telescopes and nearly a million low resolution spectra from the Sloan Digital Sky Survey (SDSS). We measure the correlation to unprecedented precision, constrain its shape, and derive an empirical relation between these quantities with a dispersion of order 0.15 magnitude in E(B-V). From the shape of the curve of growth we further show that a typical sight line through the Galaxy, as seen within the SDSS footprint, crosses about three dust clouds. We provide a brief guide on how to best estimate extinction to extragalactic sources such as supernovae, using the NaI D absorption feature, under a variety of circumstances.Comment: MNRAS accepte