A note on the convexity number for complementary prisms

In the geodetic convexity, a set of vertices $S$ of a graph $G$ is $\textit{convex}$ if all vertices belonging to any shortest path between two vertices of $S$ lie in $S$. The cardinality $con(G)$ of a maximum proper convex set $S$ of $G$ is the $\textit{convexity number}$ of $G$. The $\textit{complementary prism}$ $G\overline{G}$ of a graph $G$ arises from the disjoint union of the graph $G$ and $\overline{G}$ by adding the edges of a perfect matching between the corresponding vertices of $G$ and $\overline{G}$. In this work, we we prove that the decision problem related to the convexity number is NP-complete even restricted to complementary prisms, we determine $con(G\overline{G})$ when $G$ is disconnected or $G$ is a cograph, and we present a lower bound when $diam(G) \neq 3$.Comment: 10 pages, 2 figure

Renormalization in a Lorentz-violating model and higher-order operators

The renormalization in a Lorentz-breaking scalar-spinor higher-derivative model involving $\phi^4$ self-interaction and the Yukawa-like coupling is studied. We explicitly de- monstrate that the convergence is improved in comparison with the usual scalar-spinor model, so, the theory is super-renormalizable, with no divergences beyond four loops. We compute the one-loop corrections to the propagators for the scalar and fermionic fields and show that in the presence of higher-order Lorentz invariance violation, the poles that dominate the physical theory, are driven away from the standard on-shell pole mass due to radiatively induced lower dimensional operators. The new operators change the standard gamma-matrix structure of the two-point functions, introduce large Lorentz-breaking corrections and lead to modifications in the renormalization conditions of the theory. We found the physical pole mass in each sector of our model.Comment: 20 pages, 5 figures. New version with modifications in the renormalized Lagrangian. To be published in EPJ

Some Extended Classes of Distributions: Characterizations and Properties

Based on a simple relationship between two truncated moments and certain functions of the th order statistic, we characterize some extended classes of distributions recently proposed in the statistical literature, videlicet Beta-G, Gamma-G, Kumaraswamy-G and McDonald-G. Several properties of these extended classes and some special cases are discussed. We compare these classes in terms of goodness-of-fit criteria using some baseline distributions by means of two real data sets

On the duality in four-dimensional Lorentz-breaking field theories

We consider new issues of duality in four-dimensional Lorentz-breaking field theories. In particular, we demonstrate that the arising of the aether-like Lorentz-breaking term is necessary in order for the 4D models to display the duality analog between the MCS and self-dual models in 3D. We further study the dispersion relations in both theories and discuss the physical contents of the models involved in this new dualilty.Comment: 16 page

Single and Double Photoionization and Photodissociation of Toluene by Soft X-rays in Circumstellar Environment

The formation of polycyclic aromatic hydrocarbons (PAHs) and their methyl derivatives occurs mainly in the dust shells of asymptotic giant branch (AGB) stars. The bands at 3.3 and 3.4 $\mu$m, observed in infrared emission spectra of several objects, are attributed C-H vibrational modes in aromatic and aliphatic structures, respectively. In general, the feature at 3.3 $\mu$m is more intense than the 3.4 $\mu$m. Photoionization and photodissociation processes of toluene, the precursor of methylated PAHs, were studied using synchrotron radiation at soft X-ray energies around the carbon K edge with time-of-flight mass spectrometry. Partial ion yields of a large number of ionic fragments were extracted from single and 2D-spectra, where electron-ion coincidences have revealed the doubly charged parent-molecule and several doubly charged fragments containing seven carbon atoms with considerable abundance. \textit{Ab initio} calculations based on density functional theory were performed to elucidate the chemical structure of these stable dicationic species. The survival of the dications subjected to hard inner shell ionization suggests that they could be observed in the interstellar medium, especially in regions where PAHs are detected. The ionization and destruction of toluene induced by X-rays were examined in the T Dra conditions, a carbon-rich AGB star. In this context, a minimum photodissociation radius and the half-life of toluene subjected to the incidence of the soft X-ray flux emitted from a companion white dwarf star were determined.Comment: 11 pages, 4 figures, accept for publication in Ap

Dynamical Lorentz and CPT symmetry breaking in a 4D four-fermion model

In a 4D chiral Thirring model we analyse the possibility that radiative corrections may produce spontaneous breaking of Lorentz and CPT symmetry. By studying the effective potential, we verified that the chiral current $\bar\psi\gamma^{\mu} \gamma_5 \psi$ may assume a nonzero vacuum expectation value which triggers the Lorentz and CPT violations. Furthermore, by making fluctuations on the minimum of the potential we dynamically induce a bumblebee like model containing a Chern-Simons term.Comment: Small modifications in the text and new references added, 12 pages, 4 figures, revtex4. To appear in Phys. Rev.