Anderson Localization in Disordered Vibrating Rods

We study, both experimentally and numerically, the Anderson localization phenomenon in torsional waves of a disordered elastic rod, which consists of a cylinder with randomly spaced notches. We find that the normal-mode wave amplitudes are exponentially localized as occurs in disordered solids. The localization length is measured using these wave amplitudes and it is shown to decrease as a function of frequency. The normal-mode spectrum is also measured as well as computed, so its level statistics can be analyzed. Fitting the nearest-neighbor spacing distribution a level repulsion parameter is defined that also varies with frequency. The localization length can then be expressed as a function of the repulsion parameter. There exists a range in which the localization length is a linear function of the repulsion parameter, which is consistent with Random Matrix Theory. However, at low values of the repulsion parameter the linear dependence does not hold.Comment: 10 pages, 6 figure

Discrete model for laser driven etching and microstructuring of metallic surfaces

We present a unidimensional discrete solid-on-solid model evolving in time using a kinetic Monte Carlo method to simulate micro-structuring of kerfs on metallic surfaces by means of laser-induced jet-chemical etching. The precise control of the passivation layer achieved by this technique is responsible for the high resolution of the structures. However, within a certain range of experimental parameters, the microstructuring of kerfs on stainless steel surfaces with a solution of $\mathrm{H}_3\mathrm{PO}_4$ shows periodic ripples, which are considered to originate from an intrinsic dynamics. The model mimics a few of the various physical and chemical processes involved and within certain parameter ranges reproduces some morphological aspects of the structures, in particular ripple regimes. We analyze the range of values of laser beam power for the appearance of ripples in both experimental and simulated kerfs. The discrete model is an extension of one that has been used previously in the context of ion sputtering and is related to a noisy version of the Kuramoto-Sivashinsky equation used extensively in the field of pattern formation.Comment: Revised version. Etching probability distribution and new simulations adde

Confinement-induced resonances for a two-component ultracold atom gas in arbitrary quasi-one-dimensional traps

We solve the two-particle s-wave scattering problem for ultracold atom gases confined in arbitrary quasi-one-dimensional trapping potentials, allowing for two different atom species. As a consequence, the center-of-mass and relative degrees of freedom do not factorize. We derive bound-state solutions and obtain the general scattering solution, which exhibits several resonances in the 1D scattering length induced by the confinement. We apply our formalism to two experimentally relevant cases: (i) interspecies scattering in a two-species mixture, and (ii) the two-body problem for a single species in a non-parabolic trap.Comment: 22 pages, 3 figure

On the criticality of inferred models

Advanced inference techniques allow one to reconstruct the pattern of interaction from high dimensional data sets. We focus here on the statistical properties of inferred models and argue that inference procedures are likely to yield models which are close to a phase transition. On one side, we show that the reparameterization invariant metrics in the space of probability distributions of these models (the Fisher Information) is directly related to the model's susceptibility. As a result, distinguishable models tend to accumulate close to critical points, where the susceptibility diverges in infinite systems. On the other, this region is the one where the estimate of inferred parameters is most stable. In order to illustrate these points, we discuss inference of interacting point processes with application to financial data and show that sensible choices of observation time-scales naturally yield models which are close to criticality.Comment: 6 pages, 2 figures, version to appear in JSTA

Class of PPT bound entangled states associated to almost any set of pure entangled states

We analyze a class of entangled states for bipartite $d \otimes d$ systems, with $d$ non-prime. The entanglement of such states is revealed by the construction of canonically associated entanglement witnesses. The structure of the states is very simple and similar to the one of isotropic states: they are a mixture of a separable and a pure entangled state whose supports are orthogonal. Despite such simple structure, in an opportune interval of the mixing parameter their entanglement is not revealed by partial transposition nor by the realignment criterion, i.e. by any permutational criterion in the bipartite setting. In the range in which the states are Positive under Partial Transposition (PPT), they are not distillable; on the other hand, the states in the considered class are provably distillable as soon as they are Nonpositive under Partial Transposition (NPT). The states are associated to any set of more than two pure states. The analysis is extended to the multipartite setting. By an opportune selection of the set of multipartite pure states, it is possible to construct mixed states which are PPT with respect to any choice of bipartite cuts and nevertheless exhibit genuine multipartite entanglement. Finally, we show that every $k$-positive but not completely positive map is associated to a family of nondecomposable maps.Comment: 12 pages, 3 figures. To appear in Phys. Rev.

A second order minimality condition for the Mumford-Shah functional

A new necessary minimality condition for the Mumford-Shah functional is derived by means of second order variations. It is expressed in terms of a sign condition for a nonlocal quadratic form on $H^1_0(\Gamma)$, $\Gamma$ being a submanifold of the regular part of the discontinuity set of the critical point. Two equivalent formulations are provided: one in terms of the first eigenvalue of a suitable compact operator, the other involving a sort of nonlocal capacity of $\Gamma$. A sufficient condition for minimality is also deduced. Finally, an explicit example is discussed, where a complete characterization of the domains where the second variation is nonnegative can be given.Comment: 30 page