We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
We examine the energy distribution of the fast radio burst (FRB) population using a well-defined sample of 63 FRBs from the Australian Square Kilometre Array Pathfinder (ASKAP) radio telescope, 28 of which are localised to a host galaxy. We apply the luminosity-volume (
$V/V_{\mathrm{max}}$) test to examine the distribution of these transient sources, accounting for cosmological and instrumental effects, and determine the energy distribution for the sampled population over the redshift range 
$0.01 \lesssim z \lesssim 1.02$. We find the distribution between 
$10^{23}$ and 
$10^{26}$ J Hz
$^{-1}$ to be consistent with both a pure power-law with differential slope 
$\gamma=-1.96 \pm 0.15$, and a Schechter function with 
$\gamma = -1.82 \pm 0.12$ and downturn energy 
$E_\mathrm{max} \sim 6.3 \, \times 10^{25}$ J Hz
$^{-1}$. We identify systematic effects which currently limit our ability to probe the luminosity function outside this range and give a prescription for their treatment. Finally, we find that with the current dataset, we are unable to distinguish between the evolutionary and spectral models considered in this work.
Liquid flowing down a fibre readily destabilises into a train of beads, commonly called a bead-on-fibre pattern. Bead formation results from capillary-driven instability and gives rise to patterns with constant velocity and time-invariant bead frequency 
$f$ whenever the instability is absolute. In this study, we develop a scaling law for 
$f$ that relates the Strouhal number 
$St$ and capillary number 
$Ca$ for Ostwaldian power-law liquids with Newtonian liquids recovered as a limiting case. We validate our proposed scaling law by comparing it with prior experimental data and new experimental data using xanthan gum solutions to produce a low capillary number 
$Ca \leq 10^{-2}$ regime. The experimental data encompasses both Ostwaldian and Newtonian flow, as well as symmetric and asymmetric patterns, and we find the data collapses along the predicted trend across seven orders of magnitude in 
$Ca$. Our proposed scaling law is a powerful tool for studying and applying bead-on-fibre flows where 
$f$ is critical, such as heat and mass transfer systems.
We prove convergence in norm and pointwise almost everywhere on 
$L^p$, 
$p\in (1,\infty )$, for certain multi-parameter polynomial ergodic averages by establishing the corresponding multi-parameter maximal and oscillation inequalities. Our result, in particular, gives an affirmative answer to a multi-parameter variant of the Bellow–Furstenberg problem. This paper is also the first systematic treatment of multi-parameter oscillation semi-norms which allows an efficient handling of multi-parameter pointwise convergence problems with arithmetic features. The methods of proof of our main result develop estimates for multi-parameter exponential sums, as well as introduce new ideas from the so-called multi-parameter circle method in the context of the geometry of backwards Newton diagrams that are dictated by the shape of the polynomials defining our ergodic averages.
