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We present the analysis of the stationary equilibria and their stability in case of an opinion formation process in presence of binary opposite opinions evolving according to majority-like rules on social networks. The starting point is a kinetic Boltzmann-type model derived from microscopic interactions rules for the opinion exchange among individuals holding a certain degree of connectivity. The key idea is to derive from the kinetic model an Allen-Cahn type equation for the fraction of individuals holding one of the two opinions. The latter can be studied by means of a linear stability analysis and by exploiting integral operator analysis. While this is true for ternary interactions, for binary interactions the derived equation of interest is a linear scattering equation, that can be studied by means of General Relative Entropy tools and integral operators.

The aim of this paper is to discuss the mathematical modeling of Brownian active particle systems, a recently popular paradigmatic system for self-propelled particles. We present four microscopic models with different types of repulsive interactions between particles and their associated macroscopic models, which are formally obtained using different coarse-graining methods. The macroscopic limits are integro-differential equations for the density in phase space (positions and orientations) of the particles and may include nonlinearities in both the diffusive and advective components. In contrast to passive particles, systems of active particles can undergo phase separation without any attractive interactions, a mechanism known as motility-induced phase separation (MIPS). We explore the onset of such a transition for each model in the parameter space of occupied volume fraction and P\'eclet number via a linear stability analysis and numerical simulations at both the microscopic and macroscopic levels. We establish that one of the models, namely the mean-field model which assumes long-range repulsive interactions, cannot explain the emergence of MIPS. In contrast, MIPS is observed for the remaining three models that assume short-range interactions that localize the interaction terms in space.

This paper is devoted to the investigation of inverse problems related to stationary drift-diffusion equations modeling semiconductor devices. In this context we analyze several identification problems corresponding to different types of measurements, where the parameter to be reconstructed is an inhomogeneity in the PDE model (doping profile). For a particular type of measurement (related to the voltage-current map) we consider special cases of drift-diffusion equations, where the inverse problems reduces to a classical inverse conductivity problem. A numerical experiment is presented for one of these special situations (linearized unipolar case)., Comment: 37 pages, 7 figures

Cellular micromotion - a tiny movement of cell membranes on the nanometer-micrometer scale - has been proposed as a pathway for inter-cellular signal transduction and as a label-free proxy signal to neural activity. Here we harness several recent approaches of signal processing to detect such micromotion in video recordings of unlabeled cells. Our survey includes spectral filtering of the video signal, matched filtering, as well as 1D and 3D convolutional neural networks acting on pixel-wise time-domain data and a whole recording respectively.

In this paper we analyse a class of nonlinear cross-diffusion systems for two species with local repulsive interactions that exhibit a formal gradient flow structure with respect to the Wasserstein metric. We show that systems where the population pressure is given by a function of the total population are critical with respect to cross-diffusion perturbations. This criticality is showcased by proving that adding an extra cross-diffusion term that breaks the symmetry of the population pressure in the system leads to completely different behaviours, namely segregation or mixing, depending on the sign of the perturbation. We show these results at the level of the minimisers of the associated free energy functionals. We also analyse certain implications of these results for the gradient flow systems of PDEs associated to these functionals and we present a numerical exploration of the time evolution of these phenomena.

We study the Weyl chamber length boundary both of the Hitchin and of the maximal character varieties and determine therein an open set of discontinuity for the action of the mapping class group. This result is obtained as consequence of a canonical decomposition of a geodesic current on a surface of finite type arising from a topological decomposition of the surface along special geodesics. We show that each component either is associated to a measured lamination or has positive systole. For a current with positive systole, we show that the intersection function on the set of closed curves is bi-Lipschitz equivalent to the length function with respect to a hyperbolic metric. The results of this paper on currents generalise the ones in arXiv:1710.07060v1 to the case of surfaces of finite area with geodesic boundary. Concerning the Weyl chamber boundary we improve on the results in arXiv:1710.07060v1 by showing that for higher rank groups, said open set of discontinuity is not empty. We give also explicit examples in the case of the $SL(3,\mathbb R)$-Hitchin component.

We study the existence and uniqueness of nontrivial stationary solutions to a nonlocal aggregation equation with quadratic diffusion arising in many contexts in population dynamics. The equation is the Wasserstein gradient flow generated by the energy E, which is the sum of a quadratic free energy and the interaction energy. The interaction kernel is taken radial and attractive, nonnegative and integrable, with further technical smoothness assumptions. The existence vs. nonexistence of such solutions is ruled by a threshold phenomenon, namely nontrivial steady states exist if and only if the diffusivity constant is strictly smaller than the total mass of the interaction kernel. In the one dimensional case we prove that steady states are unique up to translations and mass constraint. The strategy is based on a strong version of the Krein-Rutman theorem. The steady states are symmetric with respect to their center of mass x0, compactly supported on sets of the form [x0 -L, x0+L], C^2 on their support, strictly decreasing on (x0, x0+L). Moreover, they are global minimizers of the energy functional E. The results are complemented by numerical simulations., Comment: 26 pages, 8 figures

We study isometric actions of tree automorphism groups on the infinite-dimensional hyperbolic spaces. On the one hand, we exhibit a general one-parameter family of such representations and analyse the corresponding equivariant embeddings of the trees, showing that they are convex-cocompact and asymptotically isometric. On the other hand, focusing on the case of sufficiently transitive groups of automorphisms of locally finite trees, we classify completely all irreducible representations by isometries of hyperbolic spaces. It turns out that in this case our one-parameter family exhausts all non-elementary representations., Comment: Added two references