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The continuous injection of energy at a localized region in space in a stationary gas creates a shock wave that propagates radially outwards. We study the hydrodynamics of this disturbance using event driven molecular dynamics of a hard sphere gas in three dimensions, the numerical solution of the Euler equation with a virial equation of state for the gas, and the numerical solution of the Navier-Stokes equation. We show that the results from the Euler equation do not agree with the data from hard sphere simulations. Including dissipative terms through the Navier-Stokes equation results in reasonably good description of the data, when the coefficients of dissipation are chose parametrically.

We present the luminosity-halo mass relations of satellite (sLHMRs) galaxies in the SDSS redMaPPer cluster catalogue and the effects of the dense cluster environment on subhalo mass evolution. We use data from the Subaru Hyper Suprime-Cam survey Year-3 catalogue of galaxy shapes to measure the weak lensing signal around these satellites. This signal serves as a probe of the matter distribution around the satellites, thereby providing the masses of their associated subhalos. We bin our satellites based on physical observable quantities such as their luminosity or the host cluster's richness, combined with their cluster-centric radial separations. Our results indicate that although more luminous satellites tend to reside in more massive halos, the sLHMRs depend on the distance of the satellite from the cluster centre. Subhalos near the cluster centre (within $<0.3 h^{-1}Mpc$) are stripped of mass. Consequently, the ratio of subhalo mass to luminosity decreases near the cluster centre. For low luminosity galaxies ($L < 10^{10} h^{-2}L_{\odot}$), the lack of evidence of increasing subhalo masses with luminosity shows the impact of tidal stripping. We also present stellar-to-subhalo mass relations (sSHMRs) for our satellite sample evolving at different cluster-centric separations. Inferred sSHMRs in the outer radial bin appear to match that observed for the field galaxies. We show that the sSHMRs from the mock-redMaPPer run on galaxy catalogues generated by the empirical UniverseMachine galaxy formation model are in good agreement with our observational results. Satellites, when binned based on the host cluster's richness, show very little dependence of the subhalo mass on the richness.

This paper considers the scenario in which there are multiple institutions, each with a limited capacity for candidates, and candidates, each with preferences over the institutions. A central entity evaluates the utility of each candidate to the institutions, and the goal is to select candidates for each institution in a way that maximizes utility while also considering the candidates' preferences. The paper focuses on the setting in which candidates are divided into multiple groups and the observed utilities of candidates in some groups are biased--systematically lower than their true utilities. The first result is that, in these biased settings, prior algorithms can lead to selections with sub-optimal true utility and significant discrepancies in the fraction of candidates from each group that get their preferred choices. Subsequently, an algorithm is presented along with proof that it produces selections that achieve near-optimal group fairness with respect to preferences while also nearly maximizing the true utility under distributional assumptions. Further, extensive empirical validation of these results in real-world and synthetic settings, in which the distributional assumptions may not hold, are presented., Comment: The conference version of this paper appears in ICML 2024