Ricerc@Sapienza

Random plane wave is conjectured to be a universal model for high-energy eigenfunctions of the Laplace operator on generic compact Riemannian manifolds. This is known to be true on average. In the present paper we discuss one of important geometric observable: critical points. We first compute one-point function for the critical point process, in particular we compute the expected number of critical points inside any open set. After that we compute the short-range asymptotic behaviour of the two-point function.

We study the behaviour of the point process of critical points of isotropic stationary
Gaussian fields. We compute the main term in the asymptotic expansion of the
two-point correlation function near the diagonal. Our main result implies that for a
‘generic’ field the critical points neither repel nor attract each other. Our analysis
also allows to study how the short-range behaviour of critical points depends on their
index.

We study here the random fluctuations in the number of critical points with values in an interval I⊂R for Gaussian spherical eigenfunctions fℓ, in the high energy regime where ℓ→∞. We show that these fluctuations are asymptotically equivalent to the centred L2-norm of fℓ times the integral of a (simple and fully explicit) function over the interval under consideration. We discuss also the relationships between these results and the asymptotic behaviour of other geometric functionals on the excursion sets of random spherical harmonics.

We prove some existence results for the following Schrodinger-Maxwell system of elliptic equations:
{-div(M(x)del u) + A phi vertical bar u vertical bar(r-2) u = f, u is an element of W-0(1,2) (Omega), -div(M(x)del phi) = vertical bar u vertical bar(r), phi is an element of W-0(1,2) (Omega).
In particular, we prove the existence of a finite energy solution (u, phi) if r > 2* and f does not belong to the "dual space" L2N/N+2 (Omega).

In the Intelligent Transportation Systems, integration of different components of the classical driver-vehicleinfrastructure system is supported by advances in technology and communications. This study presents a general road safety analysis framework that exploits different types of data on traffic, geometry, and accidents to develop a Road Safety Analysis Center and an on-board Road Safety Driver Advisory. The Road Safety Analysis Center considers different sources of data: accident inventories, road geometry, and floating car data, which reveal drivers' behavior.