Hamilton-Jacobi Equations and Distance Functions on Riemannian Manifolds

The paper is concerned with the properties of the distance function from a closed subset of a Riemannian manifold, with particular attention to the set of singularities

The special growth history of central galaxies in groups and clusters

Central galaxies (CGs) in galaxy groups and clusters are believed to form and assemble a good portion of their stellar mass at early times, but they also accrete significant mass at late times via galactic cannibalism, that is merging with companion group or cluster galaxies that experience dynamical friction against the common host dark-matter halo. The effect of these mergers on the structure and kinematics of the CG depends not only on the properties of the accreted satellites, but also on the orbital parameters of the encounters. Here we present the results of numerical simulations aimed at estimating the distribution of merging orbital parameters of satellites cannibalized by the CGs in groups and clusters. As a consequence of dynamical friction, the satellites' orbits evolve losing energy and angular momentum, with no clear trend towards orbit circularization. The distributions of the orbital parameters of the central-satellite encounters are markedly different from the distributions found for halo-halo mergers in cosmological simulations. The orbits of satellites accreted by the CGs are on average less bound and less eccentric than those of cosmological halo-halo encounters. We provide fits to the distributions of the central-satellite merging orbital parameters that can be used to study the merger-driven evolution of the scaling relations of CGs.Comment: 14 pages, 11 figures, accepted for publication in MNRAS. Minor changes with respect to previous versio

Lorentzian Connes Distance, Spectral Graph Distance and Loop Gravity

Connes' formula defines a distance in loop quantum gravity, via the spinfoam Dirac operator. A simple notion of spectral distance on a graph can be extended do the discrete Lorentzian context, providing a physically natural example of Lorentzian spectral geometry, with a neat space of Dirac operators. The Hilbert structure of the fermion space is Lorentz covariant rather than invariant.Comment: 4 page