Integer solutions to bankruptcy problems with non-integer claims

In this paper, we consider a bankruptcy problem with the additional constraint that the estate has to be assigned in integer unities; differently from the approaches by other authors, we allow for non-integer claims. This situation poses new questions on the existence and the uniqueness of the solution. We propose a new solution method that leads to the “box solution”, for which uniqueness may be guaranteed introducing a ranking on the claimants. Some examples complete the paper.     

Cahn–Hilliard and thin film equations with nonlinear mobility as gradient flows in weighted-Wasserstein metrics

In this paper, we establish a novel approach to proving existence of non-negative weak solutions for degenerate parabolic equations of fourth order, like the Cahn–Hilliard and certain thin film equations. The considered evolution equations are in the form of a gradient flow for a perturbed Dirichlet energy with respect to a Wasserstein-like transport metric, and weak solutions are obtained as curves of maximal slope. Our main assumption is that the mobility of the particles is a concave function of their spatial density. A qualitative difference of our approach to previous ones is that essential properties of the solution – non-negativity, conservation of the total mass and dissipation of the energy – are automatically guaranteed by the construction from minimizing movements in the energy landscape.     

Materials and technological aspects of gilded buckles from a North Eastern Medieval Italian context

This study concerns the archaeometric characterization of two artefacts from the medieval site of San Rocco (Castelfranco Veneto, Italy). Both of them belong to a larger set of metallic objects, some being part of tomb ornaments belonging to two very well distinct periods of frequentation of the site. Both items are buckles and they have been selected as representative of two very well-known and established typologies. The older one, at the end of the sixth—beginning of the seventh century AD, is made of silver, although relicts of gold have been found. A number of relevant counterparts of this item have been found in different sites of the Longobard Italy. The main body of the second artefact (second half of the fourteenth century AD) consists of a bent strip of a copper rich alloy, coated on one side with a nearly continuous, decorated, gold layer. This is a so-called lyre-buckle, with a widespread diffusion not only in Italian, but also North European contexts. From the results of the analyses carried out using low-vacuum scanning electron microscopy, energy dispersive X-ray spectroscopy, and X-ray diffractometry, information on manufacturing, gilding technology and materials have been obtained that will be an useful benchmark for the archaeometric characterization of similar items from selected collections. In this way, the already established typological affinity of these two classes of items will be extended to technological and materials aspects, also.