We consider a natural generalization of the eigenvalue problem for the Laplacian with homogeneous Dirichlet boundary conditions. This corresponds to look for the critical values of the Dirichlet integral, constrained to the unit Lq sphere. We collect some results, present some counter-examples and compile a list of open problems.

The optimal Orlicz target space and the optimal rearrangement- invariant target space are exhibited for embeddings of fractional-order Orlicz- Sobolev spaces W^{s,A}(R^n). Related Hardy type inequalities are proposed as well. Versions for fractional Orlicz-Sobolev seminorms of the Bourgain-Brezis-Mironescu theorem on the limit as s->1^- and of the Maz'ya-Shaposhnikova theorem on the limit as s ->0^+ are established. This is a joint work with Andrea Cianchi, Lubos Pick and Lenka Slavikova.

We numerically study by lattice Boltzmann simulations the rheological properties of an active emulsion made of a suspension of an active polar gel embedded in an isotropic passive background. We find that the hexatic equilibrium configuration of polar droplets is highly sensitive to both active injection and external forcing and may either lead to asymmetric unidirectional states which break top-bottom symmetry or symmetric ones. In this latter case, for large enough activity, the system develops a shear thickening regime at low shear rates.

We discuss the well-posedness of the "transient eddy current" magneto-quasi-static approximation of Maxwell's initial value problem with bounded and measurable conductivity, with sources, on a domain. We prove the existence and uniqueness of weak solutions, and we provide global Hölder estimates for the magnetic part.

MicroRNAs play a fundamental role in retinal development and function. To characterise the miRNome of the human retina, we carried out deep sequencing analysis on sixteen individuals. We established the catalogue of retina-expressed miRNAs, determined their relative abundance and found that a small number of miRNAs accounts for almost 90% of the retina miRNome. We discovered more than 3000 miRNA variants (isomiRs), encompassing a wide range of sequence variations, which include seed modifications that are predicted to have an impact on miRNA action.

Unsupervised learning approaches are frequently employed to stratify patients into clinically relevant subgroups and to identify biomarkers such as disease-associated genes.

The long time behaviour of the solutions of the equation u(t) = Delta F(u) in exterior domains is studied.

Mass spectrometry is a widely applied technology with a strong impact in the proteomics field. MALDI-TOF is a combined technology in mass spectrometry with many applications in characterizing biological samples from different sources, such as the identification of cancer biomarkers, the detection of food frauds, the identification of doping substances in athletes' fluids, and so on. The massive quantity of data, in the form of mass spectra, are often biased and altered by different sources of noise.

Background: Systems Medicine is a novel approach to medicine, i.e. an interdisciplinary field that considers the human body as a system, composed of multiple parts and of complex relationships at multiple levels, and further integrated into an environment. Exploring Systems Medicine implies understanding and combining concepts coming from diametral different fields, including medicine, biology, statistics, modelling and simulation, and data science. Such heterogeneity leads to semantic issues, which may slow down implementation and fruitful interaction between these highly diverse fields.

We present a simple model describing the chemical aggression undergone by calcium carbonate rocks in presence of acid atmosphere. A large literature is available on the deterioration processes of building stones, in particular in connection with problems concerning historical buildings in the field of Cultural Heritage. It is well known that the greatest aggression is caused by sulfur dioxide and nitrate. In this paper we consider the corrosion caused by sulphur dioxide, which, reacting with calcium carbonate, produces gypsum.