Diffusion dynamics on multiplex networks

We study the time scales associated with diffusion processes that take place on multiplex networks, i.e., on a set of networks linked through interconnected layers. To this end, we propose the construction of a supra-Laplacian matrix, which consists of a dimensional lifting of the Laplacian matrix of each layer of the multiplex network. We use perturbative analysis to reveal analytically the structure of eigenvectors and eigenvalues of the complete network in terms of the spectral properties of the individual layers. The spectrum of the supra-Laplacian allows us to understand the physics of diffusionlike processes on top of multiplex networks.

Application of ultrasound in bone surgery: two case reports

The present study describes some of the applications of ultrasound in bone surgery, based on the presentation of two clinical cases. The Piezosurgery® ultrasound device was used (Tecnología Mectron Medical, Carasco, Italy). In one case the instrument was used to harvest a chin bone graft for placement in a bone defect at level 1.2, while in the other case a bony window osteotomy was made in the external wall of the maxillary sinus, in the context of a sinus membrane lift procedure. The Piezosurgery® device produces specific ultrasound frequency modulation (25-29 kHz), and has been designed to secure increased precision in application to bone surgery. This instrument produces selective sectioning of the mineralized bone structures, and causes less intra- and postoperative bleeding. One of the advantages of the Piezosurgery® device is that it can be used for maxillary sinus lift procedures in dental implant placement. In this context it considerably lessens the risk of sinus mucosa laceration by preparing the bony window in the external wall of the upper maxilla, and can be used to complete the lifting maneuver. The use of ultrasound in application to hard tissues can be regarded as a slow technique compared with the conventional rotary instruments, since it requires special surgical skill and involves a certain learning curve.

A Cartan-Eilenberg approach to Homotopical Algebra

In this paper we propose an approach to homotopical algebra where the basic ingredient is a category with two classes of distinguished morphisms: strong and weak equivalences. These data determine the cofibrant objects by an extension property analogous to the classical lifting property of projective modules. We define a Cartan-Eilenberg category as a category with strong and weak equivalences such that there is an equivalence of categories between its localisation with respect to weak equivalences and the relative localisation of the subcategory of cofibrant objects with respect to strong equivalences. This equivalence of categories allows us to extend the classical theory of derived additive functors to this non additive setting. The main examples include Quillen model categories and categories of functors defined on a category endowed with a cotriple (comonad) and taking values on a category of complexes of an abelian category. In the latter case there are examples in which the class of strong equivalences is not determined by a homotopy relation. Among other applications of our theory, we establish a very general acyclic models theorem.