A 1-parameter family of spherical CR uniformizations of the figure eight knot complement

We describe a simple fundamental domain for the holonomy group of the boundary unipotent spherical CR uniformization of the figure eight knot complement, and deduce that small deformations of that holonomy group (such that the boundary holonomy remains parabolic) also give a uniformization of the figure eight knot complement. Finally, we construct an explicit 1-parameter family of deformations of the boundary unipotent holonomy group such that the boundary holonomy is twist-parabolic. For small values of the twist of these parabolic elements, this produces a 1-parameter family of pairwise non-conjugate spherical CR uniformizations of the figure eight knot complement

On the Nature of the Radial Velocity Variability of Aldebaran: A Search for Spectral Line Bisector Variations

The shape of the Ti I 6303.8 A spectral line of Aldebaran as measured by the line bisector was investigated using high signal-to-noise, high resolution data. The goal of this study was to understand the nature of the 643-day period in the radial velocity for this star reported by Hatzes and Cochran. Variations in the line bisector with the radial velocity period would provide strong evidence in support of rotational modulation or stellar pulsations as the cause of the 643-day period. A lack of any bisector variability at this period would support the planet hypothesis. Variations in the line asymmetries are found with a period of 49.93 days. These variations are uncorrelated with 643-day period found previously in the radial velocity measurements. It is demonstrated that this 50-day period is consistent with an m=4 nonradial sectoral g-mode oscillation. The lack of spectral variability with the radial velocity period of 643 days may provide strong evidence in support of the hypothesis that this variability stems from the reflex motion of the central star due to a planetary companion having a mass of 11 Jupiter masses. However, this long-period variability may still be due to a low order (m=2) pulsation mode since these would cause bisector variations less than the error measurement.Comment: LaTeX, 8 pages, 10 figures. Accepted in Monthly Notices of the Royal Astronomical Societ

A new method of measuring center-of-mass velocities of radially pulsating stars from high-resolution spectroscopy

We present a radial velocity analysis of 20 solar neighborhood RR Lyrae and 3 Population II Cepheids variables. We obtained high-resolution, moderate-to-high signal-to-noise ratio spectra for most stars and obtained spectra were covering different pulsation phases for each star. To estimate the gamma (center-of-mass) velocities of the program stars, we use two independent methods. The first, `classic' method is based on RR Lyrae radial velocity curve templates. The second method is based on the analysis of absorption line profile asymmetry to determine both the pulsational and the gamma velocities. This second method is based on the Least Squares Deconvolution (LSD) technique applied to analyze the line asymmetry that occurs in the spectra. We obtain measurements of the pulsation component of the radial velocity with an accuracy of $\pm$ 3.5 km s$^{-1}$. The gamma velocity was determined with an accuracy $\pm$ 10 km s$^{-1}$, even for those stars having a small number of spectra. The main advantage of this method is the possibility to get the estimation of gamma velocity even from one spectroscopic observation with uncertain pulsation phase. A detailed investigation of the LSD profile asymmetry shows that the projection factor $p$ varies as a function of the pulsation phase -- this is a key parameter which converts observed spectral line radial velocity variations into photospheric pulsation velocities. As a byproduct of our study, we present 41 densely-spaced synthetic grids of LSD profile bisectors that are based on atmospheric models of RR Lyr covering all pulsation phases.Comment: 17 pages, 16 figures, accepted for publication in MNRAS; doi:10.1093/mnras/stx294

On the Complexity of Randomly Weighted Voronoi Diagrams

In this paper, we provide an $O(n \mathrm{polylog} n)$ bound on the expected complexity of the randomly weighted Voronoi diagram of a set of $n$ sites in the plane, where the sites can be either points, interior-disjoint convex sets, or other more general objects. Here the randomness is on the weight of the sites, not their location. This compares favorably with the worst case complexity of these diagrams, which is quadratic. As a consequence we get an alternative proof to that of Agarwal etal [AHKS13] of the near linear complexity of the union of randomly expanded disjoint segments or convex sets (with an improved bound on the latter). The technique we develop is elegant and should be applicable to other problems

Computing bisectors in a dynamic geometry environment

In this note, an approach combining dynamic geometry and automated deduction techniques is used to study the bisectors between points and curves. Usual teacher constructions for bisectors are discussed, showing that inherent limitations in dynamic geometry software impede their thorough study. We show that the interactive sketching of bisectors and an automatic treatment of the algebraic problem involved can give a reasonable knowledge about them. Since some cases are currently out of computational scope, despite the simplicity of the bisector problem, we sketch an alternative method for dealing with them