A new acquisition system for remote control of wall paintings has been realized and tested in the field. The system measures temperature and atmospheric pressure in an archeological site where a fresco has been put under control. The measuring chain has been designed to be used in unfavorable environments where neither electric power nor telecommunication infrastructures are available. The environmental parameters obtained from the local monitoring are then transferred remotely allowing an easier management by experts in the field of conservation of cultural heritage. The local acquisition system uses an electronic card based on microcontrollers and sends the data to a central unit realized with a Raspberry-Pi. The latter manages a high quality camera to pick up pictures of the fresco. Finally, to realize the remote control at a site not reached by internet signals, a WiMAX connection based on different communication technologies such as WiMAX, Ethernet, GPRS and Satellite, has been set up.

In the context of a general lattice Γ in R n and a strictly expanding map M which preserves the lattice, we characterize all the wavelet families, all the MSF wavelets, all the multiwavelets associated with a Multiresolution Analysis (MRA) of multiplicity d ≥ 1, and all the scaling functions. Moreover , we give several examples: in particular, we construct a single, MRA and C ∞ (R n) wavelet, which is nonseparable and with compactly supported Fourier transform.

The purpose of this paper is to bring a new light on the state-dependent HamiltonJacobi equation and its connection with the Hopf-Lax formula in the framework of a Carnot group (G, •). The equation we shall consider is of the formwhere X 1 , . . . , X m are a basis of the first layer of the Lie algebra of the group G, and Ψ : R m → R is a superlinear, convex function. The main result shows that the unique viscosity solution of the Hamilton-Jacobi equation can be given by the Hopf-Lax formula u(x, t) = infwhere Ψ G : G → R is the G-Legendre-Fenchel transform of Ψ, defined by a control theoretical approach. We recover, as special cases some known results: the classical Hopf-Lax formula in the Euclidean spaces R n showing that Ψ R n is the Legendre-Fenchel trasnsform Ψ * of Ψ; moreover, we recover the result by Manfredi and Stroffolini in the Heisenberg group for particular Hamiltonian function Ψ. In this paper we follow an optimal control problem approach and we obtain several properties for the value functions u and Ψ G : in particular we prove a precise estimate for the horizontal gradient of the solution u, two existence results of the optimal control for the optimal problems and we show that Ψ G is convex.

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