The Bernstein problem for intrinsic graphs in Heisenberg groups and calibrations

In this paper we deal with some problems concerning minimal hypersurfaces in Carnot-Caratheodory (CC) structures. More precisely we will introduce a general calibration method in this setting and we will study the Bernstein problem for entire regular intrinsic minimal graphs in a meaningful and simpler class of CC spaces, i.e. the Heisenberg group H^n. In particular we will positively answer to the Bernstein problem in the case n=1 and we will provide counterexamples when n>=5

Boundary Triples for the Dirac Operator with Coulomb-Type Spherically Symmetric Perturbations

We determine explicitly a boundary triple for the Dirac operator $H:=-i\alpha\cdot \nabla + m\beta + \mathbb V(x)$ in $\mathbb R^3$, for $m\in\mathbb R$ and $\mathbb V(x)= |x|^{-1} ( \nu \mathbb{I}_4 +\mu \beta -i \lambda \alpha\cdot{x}/|x|\,\beta)$, with $\nu,\mu,\lambda \in \mathbb R$. Consequently we determine all the self-adjoint realizations of $H$ in terms of the behaviour of the functions of their domain in the origin. When $\sup_{x} |x| |\mathbb V(x)| \leq 1$, we discuss the problem of selecting the distinguished extension requiring that its domain is included in the domain of the appropriate quadratic form

Self-Adjoint Extensions for the Dirac Operator with Coulomb-Type Spherically Symmetric Potentials

We describe the self-adjoint realizations of the operator $H:=-i\alpha\cdot \nabla + m\beta + \mathbb V(x)$, for $m\in\mathbb R$, and $\mathbb V(x)= |x|^{-1} ( \nu \mathbb{I}_4 +\mu \beta -i \lambda \alpha\cdot{x}/{|x|}\,\beta)$, for $\nu,\mu,\lambda \in \mathbb R$. We characterize the self-adjointness in terms of the behaviour of the functions of the domain in the origin, exploiting Hardy-type estimates and trace lemmas. Finally, we describe the distinguished extension.Istituto Italiano di Alta Matematica "F. Severi

A Hardy-type inequality and some spectral characterizations for the Dirac-Coulomb operator

We prove a sharp Hardy-type inequality for the Dirac operator. We exploit this inequality to obtain spectral properties of the Dirac operator perturbed with Hermitian matrix-valued potentials V of Coulomb type: we characterise its eigenvalues in terms of the Birman–Schwinger principle and we bound its discrete spectrum from below, showing that the ground-state energy is reached if and only if V verifies some rigidity conditions. In the particular case of an electrostatic potential, these imply that V is the Coulomb potential

First evidence of diffuse ultra-steep-spectrum radio emission surrounding the cool core of a cluster

Diffuse synchrotron radio emission from cosmic-ray electrons is observed at the center of a number of galaxy clusters. These sources can be classified either as giant radio halos, which occur in merging clusters, or as mini halos, which are found only in cool-core clusters. In this paper, we present the first discovery of a cool-core cluster with an associated mini halo that also shows ultra-steep-spectrum emission extending well beyond the core that resembles radio halo emission. The large-scale component is discovered thanks to LOFAR observations at 144 MHz. We also analyse GMRT observations at 610 MHz to characterise the spectrum of the radio emission. An X-ray analysis reveals that the cluster is slightly disturbed, and we suggest that the steep-spectrum radio emission outside the core could be produced by a minor merger that powers electron re-acceleration without disrupting the cool core. This discovery suggests that, under particular circumstances, both a mini and giant halo could co-exist in a single cluster, opening new perspectives for particle acceleration mechanisms in galaxy clusters

A giant radio halo in the low luminosity X-ray cluster Abell 523

Radio halos are extended and diffuse non-thermal radio sources present at the cluster center, not obviously associated with any individual galaxy. A strong correlation has been found between the cluster X-ray luminosity and the halo radio power. We observe and analyze the diffuse radio emission present in the complex merging structure Abell 523, classified as a low luminosity X-ray cluster, to discuss its properties in the context of the halo total radio power versus X-ray luminosity correlation. We reduced VLA archive observations at 1.4 GHz to derive a deep radio image of the diffuse emission, and compared radio, optical, and X-ray data. Low-resolution VLA images detect a giant radio halo associated with a complex merging region. The properties of this new halo agree with those of radio halos in general discussed in the literature, but its radio power is about a factor of ten higher than expected on the basis of the cluster X-ray luminosity. Our study of this giant radio source demonstrates that radio halos can also be present in clusters with a low X-ray luminosity. Only a few similar cases have so far been found . This result suggests that this source represent a new class of objects, that cannot be explained by classical radio halo models. We suggest that the particle reacceleration related to merging processes is very efficient and/or the X-ray luminosity is not a good indicator of the past merging activity of a cluster.Comment: 5 pages, 6 figures, Astronomy and Astrophysics Letter in pres

New developments in dual cure epoxies

Direct write 3D printing provides an exciting new means of generating novel structures from thermosetting resins via an additive approach. Initial efforts along these lines demonstrated the preparation of homogenous, high quality parts. More recent work has shown that it is possible to modulate the properties of these parts locally through changes in orientation of high aspect ratio fillers. In this work, we present complementary developments related to resin chemistry that should provide practitioners of direct write 3D printing with additional options as far as the realization of functional gradations is concerned. This is accomplished via the creation of so-called “dual cure” resins. Here this term refers to materials that undergo thermal cure in a similar fashion to many conventional resins. Then, in a second step, the selective exposure of these materials to high energy radiation (gamma rays, electrons, etc.) results in further increases in local crosslink density, altering thermomechanical performance and providing a means of inducing arbitrary gradations in properties in a post-processing step. A number of different families of dual cure resins have been explored and are presented here, as well as data concerning changes in thermal and mechanical performance as a function of the details of the dual curing process. Preliminary efforts giving evidence of the generation of stable functional gradations in practice are also described. Ongoing and future efforts are focused on the optimization of these systems and the incorporation of their cure-dependent mechanical behavior into simulations in order to enable design optimization

The connection between radio halos and cluster mergers and the statistical properties of the radio halo population

We discuss the statistical properties of the radio halo population in galaxy clusters. Radio bi-modality is observed in galaxy clusters: a fraction of clusters host giant radio halos while the majority of clusters do not show evidence of diffuse cluster-scale radio emission. The radio bi-modality has a correspondence in terms of dynamical state of the hosting clusters showing that merging clusters host radio halos and follow the well known radio--X-ray correlation, while more relaxed clusters do not host radio halos and populate a region well separated from that correlation. These evidences can be understood in the framework of a scenario where merger-driven turbulence re-accelerate the radio emitting electrons. We discuss the main statistical expectations of this scenario underlining the important role of upcoming LOFAR surveys to test present models.Comment: 11 pages, 2 figures, Invited talk at the conference "Diffuse Relativistic Plasmas", Bangalore, 1-4 March 2011; accepted for publication in the Journal of Astrophysics and Astronom