All functions are (locally) ss-harmonic (up to a small error) - and applications

The classical and the fractional Laplacians exhibit a number of similarities, but also some rather striking, and sometimes surprising, structural differences. A quite important example of these differences is that any function (regardless of its shape) can be locally approximated by functions with locally vanishing fractional Laplacian, as it was recently proved by Serena Dipierro, Ovidiu Savin and myself. This informal note is an exposition of this result and of some of its consequences

Ground state solutions for nonlinear fractional Schr\"{o}dinger equations in RN\mathbb{R}^N

We construct solutions to a class of Schr\"{o}dinger equations involving the fractional laplacian. Our approach is variational in nature, and based on minimization on the Nehari manifold.Comment: 17 page

Data-Driven Epidemic Intelligence Strategies Based on Digital Proximity Tracing Technologies in the Fight against COVID-19 in Cities

In a modern pandemic outbreak, where collective threats require global strategies and local operational defence applications, data-driven solutions for infection tracing and forecasting epidemic trends are crucial to achieve sustainable and socially resilient cities. Indeed, the need for monitoring, containing, and mitigating the ongoing COVID-19 pandemic has generated a great deal of interest in Digital Proximity Tracing Technology (DPTT) on smartphones, as well as their function and effectiveness and insights of population acceptance. This paper introduces and compares different Data-Driven Epidemic Intelligence Strategies (DDEIS) developed on DPTTs. It aims to clarify to what extent DDEIS could be effective and both technologically and socially suitable in reaching the objective of a swift return to normality for cities, guaranteeing public health safety and minimizing the risk of epidemic resurgence. It assesses key advantages and limits in supporting both individual decision-making and policy-making, considering the role of human behaviour. Specifically, an online survey carried out in Italy revealed user preferences for DPTTs and provided preliminary data for an SEIR (Susceptible–Exposed–Infectious–Recovered) epidemiological model. This was developed to evaluate the impact of DDEIS on COVID-19 spread dynamics, and results are presented together with an evaluation of potential drawbacks

Chaotic Orbits for Systems of Nonlocal Equations

We consider a system of nonlocal equations driven by a perturbed periodic potential. We construct multibump solutions that connect one integer point to another one in a prescribed way. In particular, heteroclinic, homoclinic and chaotic trajectories are constructed. This is the first attempt to consider a nonlocal version of this type of dynamical systems in a variational setting and the first result regarding symbolic dynamics in a fractional framework

Spin-Orbit and Tensor Forces in Heavy-quark Light-quark Mesons: Implications of the New Ds state at 2.32 GeV

We consider the spectroscopy of heavy-quark light-quark mesons with a simple model based on the non-relativistic reduction of vector and scalar exchange between fermions. Four forces are induced: the spin-orbit forces on the light and heavy quark spins, the tensor force, and a spin-spin force. If the vector force is Coulombic, the spin-spin force is a contact interaction, and the tensor force and spin-orbit force on the heavy quark to order $1/m_1m_2$ are directly proportional. As a result, just two independent parameters characterize these perturbations. The measurement of the masses of three p-wave states suffices to predict the mass of the fourth. This technique is applied to the $D_s$ system, where the newly discovered state at 2.32 GeV provides the third measured level, and to the $D$ system. The mixing of the two $J^P=1^+$ p-wave states is reflected in their widths and provides additional constraints. The resulting picture is at odds with previous expectations and raises new puzzles.Comment: 6 pages, 1 figur

The Phillip island penguin parade (A mathematical treatment)

Penguins are flightless, so they are forced to walk while on land. In particular, they show rather specific behaviours in their homecoming, which are interesting to observe and to describe analytically. We observed that penguins have the tendency to waddle back and forth on the shore to create a sufficiently large group, and then walk home compactly together. The mathematical framework that we introduce describes this phenomenon, by taking into account "natural parameters", such as the eyesight of the penguins and their cruising speed. The model that we propose favours the formation of conglomerates of penguins that gather together, but, on the other hand, it also allows the possibility of isolated and exposed individuals. The model that we propose is based on a set of ordinary differential equations. Due to the discontinuous behaviour of the speed of the penguins, the mathematical treatment (to get existence and uniqueness of the solution) is based on a "stop-and-go" procedure. We use this setting to provide rigorous examples in which at least some penguins manage to safely return home (there are also cases in which some penguins remain isolated). To facilitate the intuition of the model, we also present some simple numerical simulations that can be compared with the actual movement of the penguin parade

Fractional-order operators: Boundary problems, heat equations

The first half of this work gives a survey of the fractional Laplacian (and related operators), its restricted Dirichlet realization on a bounded domain, and its nonhomogeneous local boundary conditions, as treated by pseudodifferential methods. The second half takes up the associated heat equation with homogeneous Dirichlet condition. Here we recall recently shown sharp results on interior regularity and on $L_p$-estimates up to the boundary, as well as recent H\"older estimates. This is supplied with new higher regularity estimates in $L_2$-spaces using a technique of Lions and Magenes, and higher $L_p$-regularity estimates (with arbitrarily high H\"older estimates in the time-parameter) based on a general result of Amann. Moreover, it is shown that an improvement to spatial $C^\infty$-regularity at the boundary is not in general possible.Comment: 29 pages, updated version, to appear in a Springer Proceedings in Mathematics and Statistics: "New Perspectives in Mathematical Analysis - Plenary Lectures, ISAAC 2017, Vaxjo Sweden