Eikonal Equations for Null Radial Geodesics in the Schwarzschild Metric

We study the eikonal function φ corresponding to outgoing and ingoing radial null geodesics (light rays in the short wave length limit) in the Schwarzschild spacetime. Contrary to the behavior of the expansion scalar θ at the singularities (past and future), φ turns out to be finite at r = 0 (except for light travelling along the horizons) and inversely proportional to M, the mass of the black hole, and so proportional to the Hawking temperature.

Co-location of the downdip end of seismic locking and the continental shelf break

<p>Along subduction margins, the morphology of the near shore domain records the combined action of erosion from ocean waves and permanent tectonic deformation from the convergence of plates. We observe that at subduction margins around the globe, the edge of continental shelves tends to be located above the downdip end of seismic coupling on the megathrust (locking depth). Coastlines lie farther landward at variable distances. This observation stems from a compilation of well-resolved coseismic and interseismic<span>  </span>coupling datasets. The permanent interseismic uplift component of the total tectonic deformation can explain the localization of the shelf break. It contributes a short wave-length gradient in vertical deformation on top of the structural and isostatic deformation of the margin. This places a hinge line between seaward subsidence and landward uplift above the locking depth. Landward of the hinge line, rocks are uplifted in the domain of wave-base erosion and a shelf is maintained by the competition of rock uplift and wave erosion. Wave erosion then sets the coastline back from the tectonically meaningful shelf break. We combine a wave erosion model with an elastic deformation model to show how the locking depth pins the location of the shelf break. In areas where the shelf is wide, onshore geodetic constraints on seismic coupling is limited and could be advantageously complemented by considering the location of the shelf break. Subduction margin morphology integrates hundreds of seismic cycles and could inform seismic coupling stability through time.</p>

Short and Long Wave Peristaltic Flow: Modeling and Mathematical Analysis

In this paper, we present a mathematical model for the peristaltic flow of a Newtonian fluid in an axisymmetric channel with small aspect ratio. In particular, we study the effects of the wave length of the wall oscillation distinguishing between long wave length (same order of the vessel's length) and short wave length (same order of the vessel's radius). We prove that the oscillation produces flow even in the absence of a pressure gradient in case of long wave. In case of short wave length, peristalsis does not affect the flow. We also prove that, in both cases, the tube resistance increases as the oscillation amplitude increases.

Optical Characterization of Chrysotile for the Real Time Detection

We measured chrominance of chrysotile powder using a colorimeter and a luminance meter under different light sources with seven color filters to test feasibility for the real time asbestos detector development. Chrysotile powder was prepared by heating asbestos fabric at 150 °C for 1hr and hydrochloric acid treatment was followed to remove other elements. Refractive index liquid was used as a color changing analyte to observe the change in chrominance of chrysotile. From the measurements, it revealed that a colorimeter was more suitable for the chrysotile detection by the maximum change in chrominance, especially in the short wave length range of blue regions. A luminance meter was not able to separate the difference in chrominance depending on different color and light sources regardless dyeing. It might be due to that reflected light from chrysotile particle was relatively weaker than that from illuminated area. It is our suggestion that the real time monitoring of asbestos in indoor air can be possible using a color sensor, specified to the unique wavelength of dyed chrysotile based on our chrominance data.