A Directional Lipschitz Extension Lemma, with Applications to Uniqueness and Lagrangianity for the Continuity Equation

We prove a Lipschitz extension lemma in which the extension procedure simultaneously preserves the Lipschitz continuity for two non-equivalent distances. The two distances under consideration are the Euclidean distance and, roughly speaking, the geodesic distance along integral curves of a (possibly multi-valued) flow of a continuous vector field. The Lipschitz constant for the geodesic distance of the extension can be estimated in terms of the Lipschitz constant for the geodesic distance of the original function. This Lipschitz extension lemma allows us to remove the high integrability assumption on the solution needed for the uniqueness within the DiPerna-Lions theory of continuity equations in the case of vector fields in the Sobolev space W1,p, where p is larger than the space dimension, under the assumption that the so-called "forward-backward integral curves" associated to the vector field are trivial for almost every starting point. More precisely, for such vector fields we prove uniqueness and Lagrangianity for weak solutions of the continuity equation that are just locally integrable

A uniqueness result for the continuity equation in two dimensions

We consider certain properties of maps of class C 2 from Rd to Rd 121 that are strictly related to Sard\u2019s theorem, and show that some of them can be extended to Lipschitz maps, while others still require some additional regularity. We also give counterexamples showing that, in term of regularity, our results are optimal

How did the carrier shell Xenophora crispa (König, 1825) build its shell? Evidence from the Recent and fossil record

The genus Xenophora comprises species of marine gastropods (Cretaceous-Recent) able to add fragments of various origins to their shell surface. Agglutination potentials vary, from species lacking attachments to species completely covered by agglutinated materials, as in the Mediterranean species Xenophora crispa. Here, we analyse Recent and fossil specimens of Xenophora crispa from the Mediterranean area using SEM and XRD, to better understand their biomineralization patterns and the mechanisms leading to the agglutination of shells, bioclasts and lithoclasts, and their evolution in time. We also provide new data on poorly studied gastropod shell microstructures. We conclude that: (1) most of the Xenophora crispa shell consists of an aragonitic crossed lamellar fabric, but fibrous to spherulitic prismatic fabrics, seemingly of calcite, have been found in the columella and peripheral edge (the thickest parts of the shell); (2) attachment of objects is mediated by a prismatic microstructure, indicating that this may be the most functional fabric in attachment areas in molluscs; and (3) the functional purpose of the agglutination in Xenophora crispa may be related to a snowshoe strategy to successfully colonize muddy substrates, coupled with tactile and olfactory camouflage. Indeed, this species secretes in the columella and peripheral edge a less dense and a more organic rich calcitic fabric, possibly to lighten the shell thickest parts in order not to sink in soft sediments and to facilitate the shell raising from the substrate to create a protected feeding area. This behaviour seems to have been maintained by X. crispa over 2 My time span

On the Lp-differentiability of certain classes of functions

We prove the Lp-differentiability at almost every point for convolution products on \u211dd of the form K*\u3bc, where \u3bc is bounded measure and K is a homogeneous kernel of degree 1-d. From this result we derive the Lp-differentiability for vector fields on R d whose curl and divergence are measures, and also for vector fields with bounded deformation

Resolving vertical and east-west horizontal motion from differential interferometric synthetic aperture radar : The L'Aquila earthquake

Analysis of surface coseismic displacement has already been obtained for the 6 April 2009 L'Aquila (central Italy) earthquake from differential interferometric synthetic aperture radar (DInSAR) data. Working jointly on ascending and descending DInSAR data makes for a step forward with respect to published preliminary estimates: we process data in order to retrieve a continuous displacement pattern, both in the vertical and horizontal directions, the latter being limited to the eastward component because of the low sensibility of the SAR images used to resolve northward motion. Our analysis provides new insights on the horizontal component of displacement, obtaining a clear picture of eastward displacement patterns over the epicentral area. This result is noteworthy, as until now little information has been available on horizontal displacement following normal-fault events in the central Apennines (Umbria-Marche, 1997, and L'Aquila, 2009), given the lack of dense GPS networks, the only available source of horizontal displacement data in this area. Inverted fault characteristics from such data also show noteworthy differences compared to previous studies, localizing the Paganica fault as the causative fault for the earthquake

Leaf superposition property for integer rectifiable currents

We consider the class of integer rectifiable currents without boundary satisfying a positivity condition. We establish that these currents can be written as a linear superposition of graphs of finitely many functions with bounded variation.Comment: 13 page

Evolution of frustrated and stabilising contacts in reconstructed ancient proteins

Energetic properties of a protein are a major determinant of its evolutionary fitness. Using a reconstruction algorithm, dating the reconstructed proteins and calculating the interaction network between their amino acids through a coevolutionary approach, we studied how the interactions that stabilise 890 proteins, belonging to five families, evolved for billions of years. In particular, we focused our attention on the network of most strongly attractive contacts and on that of poorly optimised, frustrated contacts. Our results support the idea that the cluster of most attractive interactions extends its size along evolutionary time, but from the data, we cannot conclude that protein stability or that the degree of frustration tends always to decrease