We find a condition on the underlying graph of an Artin group that fully determines if it is subgroup separable. As consequence, separable and only can be obtained from groups ranks at most 2 via finite sequence free products direct with infinite cyclic group. This result generalizes Metaftsis-Raptis criterion for Right-Angled groups.

Equivalent conditions for the separability of the range of the subdifferential of a given convex Lipschitz function f defined on a separable Banach space are studied. The conditions are in terms of a majorization of f by a C-smooth function, separability of the boundary for f or an approximation of f by Fréchet smooth convex functions.

The original analysis of sponsored search auctions by Varian and independently by Aggarwal et al. did not take into account the notion of reserve prices, which are common across all major search engines. We investigate this further and show that the separability assumption derived by Aggarwal et al. is not sufficient for aligning the greedy allocation employed by GSP and the efficient allocatio...