On the reconstruction of planar lattice-convex sets from the covariogram

A finite subset $K$ of $\mathbb{Z}^d$ is said to be lattice-convex if $K$ is the intersection of $\mathbb{Z}^d$ with a convex set. The covariogram $g_K$ of $K\subseteq \mathbb{Z}^d$ is the function associating to each u \in \integer^d the cardinality of $K\cap (K+u)$. Daurat, G\'erard, and Nivat and independently Gardner, Gronchi, and Zong raised the problem on the reconstruction of lattice-convex sets $K$ from $g_K$. We provide a partial positive answer to this problem by showing that for $d=2$ and under mild extra assumptions, $g_K$ determines $K$ up to translations and reflections. As a complement to the theorem on reconstruction we also extend the known counterexamples (i.e., planar lattice-convex sets which are not reconstructible, up to translations and reflections) to an infinite family of counterexamples.Comment: accepted in Discrete and Computational Geometr

Effective Classification using a small Training Set based on Discretization and Statistical Analysis

This work deals with the problem of producing a fast and accurate data classification, learning it from a possibly small set of records that are already classified. The proposed approach is based on the framework of the so-called Logical Analysis of Data (LAD), but enriched with information obtained from statistical considerations on the data. A number of discrete optimization problems are solved in the different steps of the procedure, but their computational demand can be controlled. The accuracy of the proposed approach is compared to that of the standard LAD algorithm, of Support Vector Machines and of Label Propagation algorithm on publicly available datasets of the UCI repository. Encouraging results are obtained and discusse

A Chandra view of the clumpy reflector at the heart of the Circinus galaxy

We present a spectral and imaging analysis of the X-ray reflecting structure at the heart of the Circinus galaxy, investigating the innermost regions surrounding the central black hole. By studying an archival 200 ks Chandra ACIS-S observation, we are able to image the extended clumpy structure responsible for both cold reflection of the primary radiation and neutral iron Ka line emission. We measure an excess of the equivalent width of the iron Ka line which follows an axisymmetric geometry around the nucleus on a hundred pc scale. Spectra extracted from different regions confirm a scenario in which the dominant mechanism is the reflection of the nuclear radiation from Compton-thick gas. Significant differences in the equivalent width of the iron Ka emission line (up to a factor of 2) are found. It is argued that these differences are due to different scattering angles with respect to the line of sight rather than to different iron abundances.Comment: 6 pages, 4 figures, accepted for publication on MNRA

Planar Duality in SU(2)SU(2) WZW Models

We show how to generalize the $SU(2)$ WZW models to allow for open and unoriented sectors. The construction exhibits some novel patterns of Chan-Paton charge assignments and projected spectra that reflect the underlying current algebra.Comment: 13 pages, latex, 1 figure

String amplitudes in the Hpp-wave limit of AdS3xS3

We compute string amplitudes on pp-waves supported by NS-NS 3-form fluxes and arising in the Penrose limit of AdS3xS3xM. We clarify the role of the non-chiral accidental SU(2) symmetry of the background. We comment on the extension of our results to the superstring and propose a holographic formula in the BMN limit of the AdS3/CFT2 correspondence valid for any correlator.Comment: Latex,no figures, 47 p

X-ray reprocessing in Seyfert Galaxies: simultaneous XMM-Newton/BeppoSAX observations

We selected a sample of eight bright unobscured (at least at the iron line energy) Seyfert Galaxies observed simultaneously by XMM-Newton and BeppoSAX, taking advantage of the complementary characteristics of the two missions. The main results of our analysis can be summarized as follows: narrow neutral iron lines are confirmed to be an ubiquitous component in Seyfert spectra; none of the analyzed sources shows unambiguously a broad relativistic iron line; all the sources of our sample (with a single exception) show the presence of a Compton reflection component; emission lines from ionized iron are observed in some sources; peculiar weak features around 5-6 keV (possibly arising from rotating spots on the accretion disk) are detected in two sources. The scenario emerging from these results strongly requires some corrections for the classical model of reprocessing from the accretion disk. As for materials farther away from the Black Hole, our results represent a positive test for the Unification Model, suggesting the presence of the torus in (almost) all sources, even if unobscured.Comment: Accepted for publication in A&