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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.
The Moment-SOS hierarchy, first introduced in optimization in 2000, is based on the theory of the S-moment problem and its dual counterpart: polynomials that are positive on S. It turns out that this methodology can also be used to solve problems with positivity constraints ‘f(x) ≥ 0 for all $\mathbf{x}\in S$
’ or linear constraints on Borel measures. Such problems can be viewed as specific instances of the generalized moment problem (GMP), whose list of important applications in various domains of science and engineering is almost endless. We describe this methodology in optimization and also in two other applications for illustration. Finally we also introduce the Christoffel function and reveal its links with the Moment-SOS hierarchy and positive polynomials.
