This paper deals with the variational analysis of topological singularities in two dimensions. We consider two canonical zero-temperature models: the core radius approach and the Ginzburg-Landau energy. Denoting by epsilon the length scale parameter in such models, we focus on the vertical bar log epsilon VERBAR; energy regime.

We consider low-energy configurations for the Heitmann-Radin sticky discs functional, in the limit of diverging number of discs. More precisely, we renormalize the Heitmann-Radin potential by subtracting the minimal energy per particle, i.e. the so-called kissing number.

We show that minimizers of the Heitmann-Radin energy (Heitmann and Radin in J Stat Phys 22(3): 281-287, 1980) are unique if and only if the particle number N belongs to an infinite sequence whose first thirty-five elements are 1, 2, 3, 4, 5, 7, 8, 10, 12, 14, 16, 19, 21, 24, 27, 30, 33, 37, 40, 44, 48, 52, 56, 61, 65, 70, 75, 80, 85, 91, 96, 102, 108, 114, 120 (see the paper for a closed-form description of this sequence).

We consider 2D discrete systems, described by scalar functions and governed by periodic interaction potentials. We focus on anisotropic nearest neighbors interactions in the hexagonal lattice and on isotropic long range interactions in the square lattice. In both these cases, we perform a complete Gamma-convergence analysis of the energy induced by a configuration of discrete topological singularities. This analysis allows to prove the existence of many metastable configurations of singularities in the hexagonal lattice.

We consider two-dimensional zero-temperature systems of N particles to which we associate an energy of the form E[V](X):=?1?i<j?NV(|X(i)-X(j)|),where X(j) ? R represents the position of the particle j and V(r) ? R is the pairwise interaction energy potential of two particles placed at distance r. We show that under suitable assumptions on the single-well potential V, the ground state energy per particle converges to an explicit constant E¯ [V] , which is the same as the energy per particle in the square lattice infinite configuration.

We consider a simple discrete model for screw dislocations in crystals. Using a variational discrete scheme we study the motion of a configuration of dislocations toward low energy configurations. We deduce an effective fully overdamped dynamics that follows the maximal dissipation criterion introduced in Cermelli and Gurtin (1999) and predicts motion along the glide directions of the crystal. (C) 2016 Elsevier Inc. All rights reserved.

We consider a simple discrete model for screw dislocations in crystals. Using a variational discrete scheme we study the motion of a configuration of dislocations toward low energy configurations. We deduce an effective fully overdamped dynamics that follows the maximal dissipation criterion introduced in Cermelli and Gurtin (1999) and predicts motion along the glide directions of the crystal. (C) 2016 Elsevier Ltd. All rights reserved.

In this paper, we use the Z-control approach to get further insight on the role of awareness in the management of epidemics that, just like Covid-19, display a high rate of overexposure because of the large number of asymptomatic people. We focus on a SEIR model including a overexposure mechanism and consider awareness as a time-dependent variable whose dynamics is not assigned a priori.

This paper aims at building a variational approach to the dynamics of discrete topological singularities in two dimensions, based on I"-convergence. We consider discrete systems, described by scalar functions defined on a square lattice and governed by periodic interaction potentials. Our main motivation comes from XY spin systems, described by the phase parameter, and screw dislocations, described by the displacement function. For these systems, we introduce a discrete notion of vorticity.

Starting from recent experimental observations of starlings and jackdaws, we propose a minimal agent-based mathematical model for bird flocks based on a system of second-order delayed stochastic differential equations with discontinuous (both in space and time) right-hand side. The model is specifically designed to reproduce self-organized spontaneous sudden changes of direction, not caused by external stimuli like predator's attacks. The main novelty of the model is that every bird is a potential turn initiator, thus leadership is formed in a group of indistinguishable agents.