The lognormal distribution represents the probability of finding randomly distributed particles in a micro canonical ensemble with high entropy. To a first approximation, a modified form of this distribution with a truncated termination may represent an isolated galactic disc, and this disc density distribution model was therefore run to give the best fit to the observational rotation curves for 37 representative galaxies. The resultant curves closely matched the observational data for a wide range of velocity profiles and galaxy types with rising, flat or descending curves in agreement with Verheijen's classification of ‘R’, ‘F’ and ‘D’ type curves, and the corresponding theoretical total disc masses could be fitted to a baryonic Tully–Fisher relation. Nine of the galaxies were matched to galaxies with previously published masses, suggesting a mean excess dynamic disc mass of dex 0.61 ± 0.26 over the baryonic masses. Although questionable with regard to other measurements of the shape of disc galaxy gravitational potentials, this model can accommodate a scenario in which the gravitational mass distribution, as measured via the rotation curve, is confined to a thin plane without requiring a dark matter halo or the use of modified Newtonian dynamics.

The relaxed motion of stars and gas in galactic discs is well approximated by a rotational velocity that is a function of radial position only, implying that individual components have lost any information about their prior states. Thermodynamically, such an equilibrium state is a microcanonical ensemble with maximum entropy, characterised by a lognormal probability distribution. Assuming this for the surface density distribution yields rotation curves that closely match observational data across a wide range of disc masses and galaxy types and provides a useful tool for modelling the theoretical density distribution in the disc. A universal disc spin parameter emerges from the model, giving a tight virial mass estimator with strong correlation between angular momentum and disc mass, suggesting a mechanism by which the proto-disc developed by dumping excess mass to the core or excess angular momentum to a satellite galaxy. The baryonic-to-dynamic mass ratio for the model approaches unity for high mass galaxies, but is generally <1 for low mass discs, and this discrepancy appears to follow a similar relationship to that shown in recent work on the Radial Acceleration Relation (RAR). Although this may support Modified Newtonian Dynamics (MOND) in preference to a Dark Matter (DM) halo, it does not exclude undetected baryonic mass or a gravitational DM component in the disc.

By postulating that the majority of the mass and angular momentum of a disc galaxy is confined to the disc with a lognormal surface density distribution, and that galactic discs are substantially, if not fully, self-gravitating, it may be shown that the resultant rotation curves (RCs) display a good overall fit to observational data for a wide range of galaxy types. With this hypothesis, the total angular momentum J, and total energy |E| of 38 disc galaxies was computed and plotted against the derived disc masses, with best fit slopes for J of 1.683 ± 0.018 and |E| of 1.643 ± 0.038, and a universal disc spin parameter λ = 0.423 ± 0.014. Using the disc parameters V_max and R_max as surrogates for the virial velocity and radius, a virial mass estimator $M_{{\rm disc}} \propto R_{{\rm max}} V_{{\rm max}}^2 $ was generated, with a log–log slope of 1.024 ± 0.014 for the 38 galaxies, and a proportionality constant λ* = 1.47 ± 0.20 × 10⁵ M_⊙kpc^{− 1} km^{− 2} s². This relationship has less scatter than M∝(V_rot)^α, and may provide an alternative to the Tully–Fisher relation in determining virial disc masses.

Galaxy number counts in the K-, H-, I-, R-, B- and U-bands from the Durham Extragalactic Astronomy and Cosmology catalogue could be well-fitted over their whole range using luminosity function (LF) parameters derived from the SDSS at the bright region and required only modest luminosity evolution with the steepening of the LF slope ($\alpha$), except for a sudden steep increase in the B-band and a less steep increase in the U-band at faint magnitudes that required a starburst evolutionary model to account for the excess faint number counts. A cosmological model treating Hubble expansion as an Einstein curvature required less correction at faint magnitudes than a standard $\mathrm{\Lambda }$CDM model, without requiring dark matter or dark energy. Data from DR17 of the SDSS in the g, i, r, u and z bands over two areas of the sky centred on the North Galactic Cap (NGC) and above the South Galactic Cap (SGC), with areas of 5954 and 859 sq. deg., respectively, and a combined count of 622,121 galaxies, were used to construct bright galaxy number counts and galaxy redshift/density plots within the limits of redshift $\le 0.4$ and mag $\le 20$. Their comparative densities confirmed an extensive void in the Southern sky with a deficit of 26% out to a redshift z ≤ 0.15. Although not included in the number count data set because of its incompleteness at fainter magnitudes, extending the SDSS redshift-number count survey to fainter and more distant galaxies with redshift ≤ 1.20 showed a secondary peak in the number counts with many QSOs, bright X-ray and radio sources, and evolving irregular galaxies with rapid star formation rates. This sub-population at redshifts of 0.45–0.65 may account for the excess counts observed in the B-band. Recent observations from the HST and James Webb Space Telescope (JWST) have also begun to reveal a high density of massive galaxies at high redshifts ($z>7$) with high UV and X-ray emissions, and future observations by the JWST may reveal the assembly of galaxies in the early universe going back to the first light in the universe.

After their initial formation, disk galaxies are observed to be rotationally stable over periods of >6 Gyr, implying that any large velocity disturbances of stars and gas clouds are damped rapidly on the timescale of their rotation. However, it is also known that despite this damping, there must be a degree of random local motion to stabilize the orbits against degenerate collapse. A mechanism for such damping is proposed by a combination of inter-stellar gravitational interactions, and interactions with the Oort clouds and exo-Oort objects associated with each star. These mechanisms may produce rapid damping of large perturbations within a time period that is short on the scale of observational look-back time, but long on the scale of the disk rotational period for stars with small perturbations. This mechanism may also account for the locally observed mean perturbations in the Milky Way of 8–15 km/s for younger stars and 20–30 km/s for older stars.

Excoriated pruritus can be an intolerable symptom in patients with cancer where Type 2 inflammation and its associated cytokines IL-4 and IL-13 play major roles in the pruritus. Dupilumab, an antibody blocking IL-4 and IL-13, is approved for treating moderate to severe atopic dermatitis (AD) where itching is a significant symptom. We present a case report of intractable malignancy-associated AD and pruritus with eosinophilia in a patient with stage IV malignant melanoma who was treated with dupilumab. Biweekly treatment with dupilumab led to an immediate improvement in itching and resolution of the AD, which subsided after a few doses and without significant adverse effects. Routine radiologic monitoring of the malignant melanoma showed concomitant resolution of secondary nodules in the lung, liver, and pleura. It was concluded that dupilumab may be a safe and effective treatment for intractable malignancy-associated AD with pruritus and may have potential for moderating metastatic malignant melanoma.