arXiv subjects

Daniel Williams

Publications and source records attributed to Daniel Williams.

21 records · Page 2Linked to original sources

arXiv2017

Constraints On Short, Hard Gamma-Ray Burst Beaming Angles From Gravitational Wave Observations

The first detection of a binary neutron star merger, GW170817, and an associated short gamma-ray burst confirmed that neutron star mergers are responsible for at least some of these bursts. The prompt gamma ray emission from these events is thought to be highly relativistically beamed. We present a method for inferring limits on the extent of this beaming by comparing the number of short gamma-ray bursts observed electromagnetically to the number of neutron star binary mergers detected in gravitational waves. We demonstrate that an observing run comparable to the expected Advanced LIGO 2016--2017 run would be capable of placing limits on the beaming angle of approximately $θ\in (2.88^\circ,14.15^\circ)$, given one binary neutron star detection. We anticipate that after a year of observations with Advanced LIGO at design sensitivity in 2020 these constraints would improve to $θ\in (8.10^\circ,14.95^\circ)$.

astro-ph.HE↗

arXiv2014

Global solutions to the homogeneous and inhomogeneous Navier-Stokes equations

In this paper we take a new approach to a proof of existence and uniqueness of solutions for the 3D-Navier-Stokes equations, which leads to essentially the same proof for both bounded and unbounded domains and for homogeneous or inhomogeneous incompressible fluids. Our approach is to construct the largest separable Hilbert space ${\bf{SD}}^2[\R^3]$, for which the Leray-Hopf (type) solutions in $L^2[{\mathbb R}^3]$ are strong solutions in ${\bf{SD}}^2[\R^3]$. We say Leray-Hopf type because our solutions are weak in the spatial sense but not in time.

math-ph↗

arXiv2012

Note on the Spectral Theorem

In this note, we show that the spectral theorem, has two representations; the Stone-von Neumann representation and one based on the polar decomposition of linear operators, which we call the deformed representation. The deformed representation has the advantage that it provides an easy extension to all closed densely defined linear operators on Hilbert space. Furthermore, the deformed representation can also be extended all separable reflexive Banach spaces and has a limited extension to non-reflexive Banach spaces.

math-ph↗