For G open bounded subset of R^2 with C^1 boundary, we study the regularity of the variational solution u in H^1_0(G) to the quasilinear elliptic equation of Leray-Lions type: -div A(x,Du)=f , when f belongs to the Zygmund space L(log L)^{\delta}, \delta>0. As an interpolation between known results for \delta=1/2 and \delta=1 of [Stampacchia] and [Alberico-Ferone], we prove that |Du| belongs to the Lorentz space L^{2, 1/\delta}(G) for \delta in [1/2, 1].

Expansions in noninteger bases often appear in number theory and probability theory, and they are closely connected to ergodic theory, measure theory and topology. For two-letter alphabets the golden ratio plays a special role: in smaller bases only trivial expansions are unique, whereas in greater bases there exist nontrivial unique expansions. In this paper we determine the corresponding critical bases for all three-letter alphabets and we establish the fractal nature of these bases in function of the alphabets.

We report a search for new gravitational physics phenomena based on Riemann-Cartan theory of general relativity including spacetime torsion. Starting from the parametrized torsion framework of Mao, Tegmark, Guth, and Cabi, we analyze the motion of test bodies in the presence of torsion, and, in particular, we compute the corrections to the perihelion advance and to the orbital geodetic precession of a satellite. We consider the motion of a test body in a spherically symmetric field, and the motion of a satellite in the gravitational field of the Sun and the Earth. We describe the torsion field by means of three parameters, and we make use of the autoparallel trajectories, which in general differ from geodesics when torsion is present. We derive the specific approximate expression of the corresponding system of ordinary differential equations, which are then solved with methods of celestial mechanics. We calculate the secular variations of the longitudes of the node and of the pericenter of the satellite. The computed secular variations show how the corrections to the perihelion advance and to the orbital de Sitter effect depend on the torsion parameters. All computations are performed under the assumptions of weak field and slow motion. To test our predictions, we use the measurements of the Moon's geodetic precession from lunar laser ranging data, and the measurements of Mercury's perihelion advance from planetary radar ranging data. These measurements are then used to constrain suitable linear combinations of the torsion parameters.

We consider an integro-differential model for evolutionary game theory which describes the evolution of a population adopting mixed strategies. Using a reformulation based on the first moments of the solution, we prove some analytical properties of the model and global estimates. The asymptotic behavior and the stability of solutions in the case of two strategies is analyzed in details. Numerical schemes for two and three strategies which are able to capture the correct equilibrium states are also proposed together with several numerical examples.

The flow in the stern region of a fully appended hull is analyzed by both computational and experimental fluid dynamics. The study is focused on the velocity field induced by the rotating propellers. Measurements have been performed by laser Doppler velocimetry (LDV) on the vertical midplane of the rudder and in two transversal planes behind the propeller and behind the rudder. In the numerical approach, the real geometry of the propeller has been considered. To this purpose, a dynamic overlapping grids method has been used, which is implemented in the unsteady Reynolds averaged Navier-Stokes equations (URANSE) solver developed at INSEAN. Uncertainty analysis has been performed on both data sets and the results from the two approaches are compared. The agreement between the two data sets is found to be good, the deviation in the velocity and vorticity fields lying within the evaluated uncertainties. Numerical data allowed the analysis of further details of the flow that could not be measured, like load conditions of the single blades, interaction of the propeller wake with the rudder, and pressure oscillations induced by the propeller on the vault of the stern.