Symmetrization in Geometry

The concept of an $i$-symmetrization is introduced, which provides a convenient framework for most of the familiar symmetrization processes on convex sets. Various properties of $i$-symmetrizations are introduced and the relations between them investigated. New expressions are provided for the Steiner and Minkowski symmetrals of a compact convex set which exhibit a dual relationship between them. Characterizations of Steiner, Minkowski and central symmetrization, in terms of natural properties that they enjoy, are given and examples are provided to show that none of the assumptions made can be dropped or significantly weakened. Other familiar symmetrizations, such as Schwarz symmetrization, are discussed and several new ones introduced.Comment: A chacterization of central symmetrization has been added and several typos have been corrected. This version has been accepted for publication on Advances in Mathematic

UV spectral analysis of very hot H-deficient [WCE]-type central stars of planetary nebulae: NGC 2867, NGC 5189, NGC 6905, Pb 6, and Sand 3

We analysed UV FUSE, IUE, and HST/STIS spectra of five of the hottest [WCE]-type central stars of planetary nebulae: NGC 2867, NGC 5189, NGC 6905, Pb 6, and Sand 3. The analysis leveraged on our grid of CMFGEN synthetic spectra, which covers the parameter regime of hydrogen deficient central stars of planetary nebulae and allows a uniform and systematic study of the stellar spectra. The stellar atmosphere models calculated by us include many elements and ionic species neglected in previous analyses, which allowed us to improve the fits to the observed spectra considerably and provided an additional diagnostic line: the Ne VII $\lambda$ 973 $\mathrm{\AA}$, which had not been modelled in [WCE] spectra and which presents, in these stars, a strong P-Cygni profile. We report newly derived photospheric and wind parameters and elemental abundances. The central stars of NGC 2867, NGC 5189, and Pb 6 had their temperatures revised upward in comparison with previous investigations and we found the carbon to helium mass ratio of the sample objects to span a wide range of values, 0.42$\leq$C:He$\leq$1.96. Modelling of the Ne VII $\lambda$ 973 $\mathrm{\AA}$ P-Cygni profile indicated strong neon overabundances for the central stars of NGC 2867, NGC 5189, NGC 6905, and Pb 6, with Ne mass fractions between 0.01 and 0.04. Nitrogen abundances derived by us for the central stars of NGC 5189, Pb 6, and Sand 3 are higher than previous determinations by factors of 3, 10, and 14, respectively.Comment: Accepted on MNRA

Cooperative Online Learning: Keeping your Neighbors Updated

We study an asynchronous online learning setting with a network of agents. At each time step, some of the agents are activated, requested to make a prediction, and pay the corresponding loss. The loss function is then revealed to these agents and also to their neighbors in the network. Our results characterize how much knowing the network structure affects the regret as a function of the model of agent activations. When activations are stochastic, the optimal regret (up to constant factors) is shown to be of order $\sqrt{\alpha T}$, where $T$ is the horizon and $\alpha$ is the independence number of the network. We prove that the upper bound is achieved even when agents have no information about the network structure. When activations are adversarial the situation changes dramatically: if agents ignore the network structure, a $\Omega(T)$ lower bound on the regret can be proven, showing that learning is impossible. However, when agents can choose to ignore some of their neighbors based on the knowledge of the network structure, we prove a $O(\sqrt{\overline{\chi} T})$ sublinear regret bound, where $\overline{\chi} \ge \alpha$ is the clique-covering number of the network

Holographic tracking and sizing of optically trapped microprobes in diamond anvil cells

We demonstrate that Digital Holographic Microscopy can be used for accurate 3D tracking and sizing of a colloidal probe trapped in a diamond anvil cell (DAC). Polystyrene beads were optically trapped in water up to Gigapascal pressures while simultaneously recording in-line holograms at 1 KHz frame rate. Using Lorenz-Mie scattering theory to fit interference patterns, we detected a 10% shrinking in the bead’s radius due to the high applied pressure. Accurate bead sizing is crucial for obtaining reliable viscosity measurements and provides a convenient optical tool for the determination of the bulk modulus of probe material. Our technique may provide a new method for pressure measurements inside a DAC

Polar features in the flagellar propulsion of E. coli bacteria

E. coli bacteria swim following a run and tumble pattern. In the run state all flagella join in a single helical bundle that propels the cell body along approximately straight paths. When one or more flagellar motors reverse direction the bundle unwinds and the cell randomizes its orientation. This basic picture represents an idealization of a much more complex dynamical problem. Although it has been shown that bundle formation can occur at either pole of the cell, it is still unclear whether this two run states correspond to asymmetric propulsion features. Using holographic microscopy we record the 3D motions of individual bacteria swimming in optical traps. We find that most cells possess two run states characterised by different propulsion forces, total torque and bundle conformations. We analyse the statistical properties of bundle reversal and compare the hydrodynamic features of forward and backward running states. Our method is naturally multi-particle and opens up the way towards controlled hydrodynamic studies of interacting swimming cells

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

Massive higher spins and holography

We review recent progress towards the understanding of higher spin gauge symmetry breaking in AdS space from a holographic vantage point. According to the AdS/CFT correspondence, N=4 SYM theory at vanishing coupling constant should be dual to a theory in AdS which exhibits higher spin gauge symmetry enhancement. When the SYM coupling is non-zero, all but a handful of HS currents are violated by anomalies, and correspondingly local higher spin symmetry in the bulk gets spontaneously broken. In agreement with previous results and holographic expectations, we find that, barring one notable exception (spin 1 eating spin 0), the Goldstone modes responsible for HS symmetry breaking in AdS have non-vanishing mass even in the limit in which the gauge symmetry is restored. We show that spontaneous breaking a' la Stueckelberg implies that the mass of the relevant spin s'=s-1 Goldstone field is exactly the one predicted by the correspondence.Comment: 8 pages, talk presented by M.B. at the "Fourth Meeting on Constrained Dynamics and Quantum gravity" held in Cala Gonone (Sardinia, Italy), September 12-16, 200

Discovery Potential of SUSY and UED in ATLAS

This poster presents an evaluation of the discovery potential of Supersymmetry and Universal Extra Dimensions for channels with jets, leptons and missing transverse energy. The LHC running scenario at a centre-of-mass energy of 10 TeV, delivering an integrated luminosity of 200 inverse pb for the 2009-2010 run is investigated

Phacoemulsification Cataract Surgery without Viscoelastic Substance: Bianchi’s Method

Life expectancy of the population increase and cataract development will affect all the people with aging. Cataract surgery, a worldwide performed procedure, evolves and progresses. However, different techniques exist, which could be selected for different cases. Any ideal technique should be safe, simple, fast, and easy to learn with good clinical outcome. This chapter will describe one technique to operate cataracts with those characteristics and to perform phacoemulsification cataract surgery without viscoelastic substance. Some advantages of this technique are related to avoiding viscoelastic potential problems, as postoperative intraocular pressure elevation or anterior chamber inflammation associated with viscoelastic. Moreover, a fundamental factor to remark is the difference between work into the anterior chamber with negative pressure or positive pressure. Because the anterior chamber is maintained by the balanced salt solution with the continuous irrigation without viscoelastic. Performing the capsulorhexis is easier. Other advantages are shorten surgical time, fewer economical cost, and potentially fewer complications. Some limitations are as follows: intraocular lens must be one piece foldable, and principally, patients with corneal endothelial pathology must be excluded. Tips, step-by-step surgery, recommendations, and evolution of the technique will be described, with the wish that many surgeons will try to perform Bianchi’s method (bimanual, microincision phacoemulsification cataract surgery without viscoelastic substance) for your next patient