Corrigendum to ``Determining a sound-soft polyhedral scatterer by a single far-field measurement''

In the paper, G. Alessandrini and L. Rondi, ``Determining a sound-soft polyhedral scatterer by a single far-field measurement'', Proc. Amer. Math. Soc. 133 (2005), pp. 1685-1691, on the determination of a sound-soft polyhedral scatterer by a single far-field measurement, the proof of Proposition 3.2 is incomplete. In this corrigendum we provide a new proof of the same proposition which fills the previous gap.Comment: 3 page

Approximation of the Helfrich's functional via Diffuse Interfaces

We give a rigorous proof of the approximability of the so-called Helfrich's functional via diffuse interfaces, under a constraint on the ratio between the bending rigidity and the Gauss-rigidity

Spontaneous symmetry breaking and collapse in bosonic Josephson junctions

We investigate an attractive atomic Bose-Einstein condensate (BEC) trapped by a double-well potential in the axial direction and by a harmonic potential in the transverse directions. We obtain numerically, for the first time, a quantum phase diagram which includes all the three relevant phases of the system: Josephson, spontaneous symmetry breaking (SSB), and collapse. We consider also the coherent dynamics of the BEC and calculate the frequency of population-imbalance mode in the Josephson phase and in the SSB phase up to the collapse. We show that these phases can be observed by using ultracold vapors of 7Li atoms in a magneto-optical trap.Comment: 5 pages, 4 figures, to be published in Phys. Rev.

Load optimization in a planar network

We analyze the asymptotic properties of a Euclidean optimization problem on the plane. Specifically, we consider a network with three bins and $n$ objects spatially uniformly distributed, each object being allocated to a bin at a cost depending on its position. Two allocations are considered: the allocation minimizing the bin loads and the allocation allocating each object to its less costly bin. We analyze the asymptotic properties of these allocations as the number of objects grows to infinity. Using the symmetries of the problem, we derive a law of large numbers, a central limit theorem and a large deviation principle for both loads with explicit expressions. In particular, we prove that the two allocations satisfy the same law of large numbers, but they do not have the same asymptotic fluctuations and rate functions.Comment: Annals of Applied Probability 2010, Vol. 20, No. 6, 2040-2085 Published in at http://dx.doi.org/10.1214/09-AAP676 the Annals of Applied Probability by the Institute of Mathematical Statistics (http://www.imstat.org) 10.1214/09-AAP67