A comprehensive approach to Sobolev-type embeddings, involving arbitrary rearrangement- invariant norms on the entire Euclidean space R^n, is offered. In particular, the optimal target space in any such embedding is exhibited.

The topic of controlled drug release has received much attention in recent years, for example in the design of tablets and in local drug delivery devices such as stents, transdermal patches, therapeutic contact lenses and orthopaedic implants. In recent years, we have developed a series of models for such devices to describe drug release from a polymeric platform, drug transport in surrounding biological tissues and fully coupled models of them.

In this paper we describe a theoretical mathematical model of dual drug delivery from a durable polymer coated medical device. We demonstrate how the release rate of each drug may in principle be controlled by altering the initial loading configuration of the two drugs. By varying the underlying microstructure of polymer coating, further control may be obtained, providing the opportunity to tailor the release profile of each drug for the given application.

We discuss the problem of partitioning a macroscopic system into a collection of independent subsystems. The partitioning of a system into replica-like subsystems is nowadays a subject of major interest in several fields of theoretical and applied physics. The thermodynamic approach currently favoured by practitioners is based on a phenomenological definition of an interface energy associated with the partition, due to a lack of easily computable expressions for a microscopic (i.e. particle-based) interface energy.

In this article we propose a mathematical model for the onset and progression of Alzheimer's disease based on transport and diffusion equations. We regard brain neurons as a continuous medium and structure them by their degree of malfunctioning. Two different mechanisms are assumed to be relevant for the temporal evolution of the disease: i) diffusion and agglomeration of soluble polymers of amyloid, produced by damaged neurons and ii) neuron-to-neuron prion-like transmission.

In the first part of this paper we review a mathematical model for the onset and progression of Alzheimer's disease (AD) that was developed in subsequent steps over several years. The model is meant to describe the evolution of AD in vivo. In Achdou et al (2013 J. Math. Biol. 67 1369-92) we treated the problem at a microscopic scale, where the typical length scale is a multiple of the size of the soma of a single neuron. Subsequently, in Bertsch et al (2017 Math. Med. Biol.

The Voronoi diagrams are an important tool having theoretical and practical applications in a large number of fields. We present a new procedure, implemented as a set of CUDA kernels, which detects, in a general and efficient way, topological changes in case of dynamic Voronoi diagrams whose generating points move in time. The solution that we provide has been originally developed to identify plastic events during simulations of soft-glassy materials based on a lattice Boltzmann model with frustrated-short range attractive and mid/long-range repulsive-interactions.

The existence of solutions to a one-dimensional problem arising in magneto-viscoelasticity is here considered. Specifically, a non-linear system of integro-differential equations is analysed; it is obtained coupling an integro-differential equation modelling the viscoelastic behaviour, in which the kernel represents the relaxation function, with the non-linear partial differential equations modelling the presence of a magnetic field.

We present a novel application of the Lattice Boltzmann Method to the study of pulsed reactive flows in transitional Knudsen number regimes, namely 0.1 < Kn < 1. We characterize the conversion efficiency of catalytic particles for different geometries and configurations, including single catalytic particle and nanoporous gold (npAu) spheres, within pulsed-flow reactors.