We present a catalogue of over 7000 sources from the GLEAM survey which have significant structure on sub-arcsecond scales at 162 MHz. The compact nature of these sources was detected and quantified via their Interplanetary Scintillation (IPS) signature, measured in interferometric images from the Murchison Widefield Array. The advantage of this approach is that all sufficiently compact sources across the survey area are included down to a well-defined flux density limit. The survey is based on ${\sim}250\times 10\hbox{-}\mathrm{min}$ observations, and the area covered is somewhat irregular, but the area within $1\,\mathrm{h}<\mathrm{RA}<11\,\mathrm{h}$ ; $-10^\circ<\mathrm{Decl.}<+20^\circ$ is covered entirely, and over 85% of this area has a detection limit for compact structure below 0.2 Jy. 7839 sources clearly showing IPS were detected ( ${>}5\sigma$ confidence), with a further 5550 tentative ( ${>}2\sigma$ confidence) detections. Normalised Scintillation Indices (NSI; a measure of the fraction of flux density coming from a compact component) are reported for these sources. Robust and informative upper limits on the NSI are reported for a further 31081 sources. This represents the largest survey of compact sources at radio frequencies ever undertaken.

While unobscured and radio-quiet active galactic nuclei are regularly being found at redshifts $z > 6$ , their obscured and radio-loud counterparts remain elusive. We build upon our successful pilot study, presenting a new sample of low-frequency-selected candidate high-redshift radio galaxies (HzRGs) over a sky area 20 times larger. We have refined our selection technique, in which we select sources with curved radio spectra between 72–231 MHz from the GaLactic and Extragalactic All-sky Murchison Widefield Array (GLEAM) survey. In combination with the requirements that our GLEAM-selected HzRG candidates have compact radio morphologies and be undetected in near-infrared $K_{\rm s}$ -band imaging from the Visible and Infrared Survey Telescope for Astronomy Kilo-degree Infrared Galaxy (VIKING) survey, we find 51 new candidate HzRGs over a sky area of approximately $1200\ \mathrm{deg}^2$ . Our sample also includes two sources from the pilot study: the second-most distant radio galaxy currently known, at $z=5.55$ , with another source potentially at $z \sim 8$ . We present our refined selection technique and analyse the properties of the sample. We model the broadband radio spectra between 74 MHz and 9 GHz by supplementing the GLEAM data with both publicly available data and new observations from the Australia Telescope Compact Array at 5.5 and 9 GHz. In addition, deep $K_{\rm s}$ -band imaging from the High-Acuity Widefield K-band Imager (HAWK-I) on the Very Large Telescope and from the Southern Herschel Astrophysical Terahertz Large Area Survey Regions $K_{\rm s}$ -band Survey (SHARKS) is presented for five sources. We discuss the prospects of finding very distant radio galaxies in our sample, potentially within the epoch of reionisation at $z \gtrsim 6.5$ .

The GaLactic and Extragalactic All-sky Murchison Widefield Array (GLEAM) is a radio continuum survey at 76–227 MHz of the entire southern sky (Declination $<\!{+}30^{\circ}$ ) with an angular resolution of ${\approx}2$ arcmin. In this paper, we combine GLEAM data with optical spectroscopy from the 6dF Galaxy Survey to construct a sample of 1 590 local (median $z \approx 0.064$ ) radio sources with $S_{200\,\mathrm{MHz}} > 55$ mJy across an area of ${\approx}16\,700\,\mathrm{deg}^{2}$ . From the optical spectra, we identify the dominant physical process responsible for the radio emission from each galaxy: 73% are fuelled by an active galactic nucleus (AGN) and 27% by star formation. We present the local radio luminosity function for AGN and star-forming (SF) galaxies at 200 MHz and characterise the typical radio spectra of these two populations between 76 MHz and ${\sim}1$ GHz. For the AGN, the median spectral index between 200 MHz and ${\sim}1$ GHz, $\alpha_{\mathrm{high}}$ , is $-0.600 \pm 0.010$ (where $S \propto \nu^{\alpha}$ ) and the median spectral index within the GLEAM band, $\alpha_{\mathrm{low}}$ , is $-0.704 \pm 0.011$ . For the SF galaxies, the median value of $\alpha_{\mathrm{high}}$ is $-0.650 \pm 0.010$ and the median value of $\alpha_{\mathrm{low}}$ is $-0.596 \pm 0.015$ . Among the AGN population, flat-spectrum sources are more common at lower radio luminosity, suggesting the existence of a significant population of weak radio AGN that remain core-dominated even at low frequencies. However, around 4% of local radio AGN have ultra-steep radio spectra at low frequencies ( $\alpha_{\mathrm{low}} < -1.2$ ). These ultra-steep-spectrum sources span a wide range in radio luminosity, and further work is needed to clarify their nature.

The Murchison Widefield Array (MWA) has observed the entire southern sky (Declination, $\delta< 30^{\circ}$ ) at low radio frequencies, over the range 72–231MHz. These observations constitute the GaLactic and Extragalactic All-sky MWA (GLEAM) Survey, and we use the extragalactic catalogue (EGC) (Galactic latitude, $|b| >10^{\circ}$ ) to define the GLEAM 4-Jy (G4Jy) Sample. This is a complete sample of the ‘brightest’ radio sources ( $S_{\textrm{151\,MHz}}>4\,\text{Jy}$ ), the majority of which are active galactic nuclei with powerful radio jets. Crucially, low-frequency observations allow the selection of such sources in an orientation-independent way (i.e. minimising the bias caused by Doppler boosting, inherent in high-frequency surveys). We then use higher-resolution radio images, and information at other wavelengths, to morphologically classify the brightest components in GLEAM. We also conduct cross-checks against the literature and perform internal matching, in order to improve sample completeness (which is estimated to be $>95.5$ %). This results in a catalogue of 1863 sources, making the G4Jy Sample over 10 times larger than that of the revised Third Cambridge Catalogue of Radio Sources (3CRR; $S_{\textrm{178\,MHz}}>10.9\,\text{Jy}$ ). Of these G4Jy sources, 78 are resolved by the MWA (Phase-I) synthesised beam ( $\sim2$ arcmin at 200MHz), and we label 67% of the sample as ‘single’, 26% as ‘double’, 4% as ‘triple’, and 3% as having ‘complex’ morphology at $\sim1\,\text{GHz}$ (45 arcsec resolution). We characterise the spectral behaviour of these objects in the radio and find that the median spectral index is $\alpha=-0.740 \pm 0.012$ between 151 and 843MHz, and $\alpha=-0.786 \pm 0.006$ between 151MHz and 1400MHz (assuming a power-law description, $S_{\nu} \propto \nu^{\alpha}$ ), compared to $\alpha=-0.829 \pm 0.006$ within the GLEAM band. Alongside this, our value-added catalogue provides mid-infrared source associations (subject to 6” resolution at 3.4 $\mu$ m) for the radio emission, as identified through visual inspection and thorough checks against the literature. As such, the G4Jy Sample can be used as a reliable training set for cross-identification via machine-learning algorithms. We also estimate the angular size of the sources, based on their associated components at $\sim1\,\text{GHz}$ , and perform a flux density comparison for 67 G4Jy sources that overlap with 3CRR. Analysis of multi-wavelength data, and spectral curvature between 72MHz and 20GHz, will be presented in subsequent papers, and details for accessing all G4Jy overlays are provided at https://github.com/svw26/G4Jy.

The entire southern sky (Declination, $\delta< 30^{\circ}$ ) has been observed using the Murchison Widefield Array (MWA), which provides radio imaging of $\sim$ 2 arcmin resolution at low frequencies (72–231 MHz). This is the GaLactic and Extragalactic All-sky MWA (GLEAM) Survey, and we have previously used a combination of visual inspection, cross-checks against the literature, and internal matching to identify the ‘brightest’ radio-sources ( $S_{\mathrm{151\,MHz}}>4$ Jy) in the extragalactic catalogue (Galactic latitude, $|b| >10^{\circ}$ ). We refer to these 1 863 sources as the GLEAM 4-Jy (G4Jy) Sample, and use radio images (of ${\leq}45$ arcsec resolution), and multi-wavelength information, to assess their morphology and identify the galaxy that is hosting the radio emission (where appropriate). Details of how to access all of the overlays used for this work are available at https://github.com/svw26/G4Jy. Alongside this we conduct further checks against the literature, which we document here for individual sources. Whilst the vast majority of the G4Jy Sample are active galactic nuclei with powerful radio-jets, we highlight that it also contains a nebula, two nearby, star-forming galaxies, a cluster relic, and a cluster halo. There are also three extended sources for which we are unable to infer the mechanism that gives rise to the low-frequency emission. In the G4Jy catalogue we provide mid-infrared identifications for 86% of the sources, and flag the remainder as: having an uncertain identification (129 sources), having a faint/uncharacterised mid-infrared host (126 sources), or it being inappropriate to specify a host (2 sources). For the subset of 129 sources, there is ambiguity concerning candidate host-galaxies, and this includes four sources (B0424–728, B0703–451, 3C 198, and 3C 403.1) where we question the existing identification.

We describe the parameters of a low-frequency all-sky survey of compact radio sources using Interplanetary Scintillation, undertaken with the Murchison Widefield Array. While this survey gives important complementary information to low-resolution survey, providing information on the sub-arsecond structure of every source, a survey of this kind has not been attempted in the era of low-frequency imaging arrays such as the Murchison Widefield Array and LOw Frequency Array. Here we set out the capabilities of such a survey, describing the limitations imposed by the heliocentric observing geometry and by the instrument itself. We demonstrate the potential for Interplanetary Scintillation measurements at any point on the celestial sphere and we show that at 160 MHz, reasonable results can be obtained within 30° of the ecliptic (2π str: half the sky). We also suggest some observational strategies and describe the first such survey, the Murchison Widefield Array Phase I Interplanetary Scintillation survey. Finally we analyse the potential of the recently upgraded Murchison Widefield Array and discuss the potential of the Square Kilometre Array-low to use Interplanetary Scintillation to probe sub-mJy flux density levels at sub-arcsecond angular resolution.

We consider a quasilinear elliptic problem of the form

where λ > 0 is a parameter, 1 < p < 2 and Ω is a strictly convex bounded domain in ℝN, N > p, with C2 boundary ∂Ω. The nonlinearity f : [0, ∞) → ℝ is a continuous function that is semipositone (f(0) < 0) and p-superlinear at infinity. Using degree theory, combined with a rescaling argument and uniform L∞a priori bound, we establish the existence of a positive solution for λ small. Moreover, we show that there exists a connected component of positive solutions bifurcating from infinity at λ = 0. We also extend our study to systems.

This paper motivates and introduces a number of new concepts of positive dependence among sets of random variables. In particular, setwise positive upper (and lower) orthant dependence, setwise association, and other related concepts are studied and their relationships are explored. These new concepts are applied to various multivariate normal distributions, some of which have both positive and negative covariances. A variety of new and interesting inequalities are also obtained.