Metric Dimension for Gabriel Unit Disk Graphs is NP-Complete

We show that finding a minimal number of landmark nodes for a unique virtual addressing by hop-distances in wireless ad-hoc sensor networks is NP-complete even if the networks are unit disk graphs that contain only Gabriel edges. This problem is equivalent to Metric Dimension for Gabriel unit disk graphs. The Gabriel edges of a unit disc graph induce a planar O(\sqrt{n}) distance and an optimal energy spanner. This is one of the most interesting restrictions of Metric Dimension in the context of wireless multi-hop networks.Comment: A brief announcement of this result has been published in the proceedings of ALGOSENSORS 201

Quadratic Maps in Two Variables on Arbitrary Fields

Let $\mathbb{F}$ be a field of characteristic different from $2$ and $3$, and let $V$ be a vector space of dimension $2$ over $\mathbb{F}$. The generic classification of homogeneous quadratic maps $f\colon V\to V$ under the action of the linear group of $V$, is given and efficient computational criteria to recognize equivalence are provided.Comment: 12 pages, no figure

The little-studied cluster Berkeley 90. II. The foreground ISM

Context: Nearly one century after their discovery, the carrier(s) of Diffuse Interstellar Bands is/are still unknown and there are few sightlines studied in detail for a large number of DIBs. Aims: We want to study the ISM sightlines towards LS III +46 11 and LS III +46 12, two early-O-type stellar systems, and LS III +46 11 B, a mid-B-type star. The three targets are located in the stellar cluster Berkeley 90 and have a high extinction. Methods: We use the multi-epoch high-S/N optical spectra presented in paper I (Ma\'iz Apell\'aniz et al. 2015), the extinction results derived there, and additional spectra. Results: We have measured equivalent widths, velocities, and FWHMs for a large number of absorption lines in the rich ISM spectrum in front of Berkeley 90. The absorbing ISM has at least two clouds at different velocities, one with a lower column density (thinner) in the K I lines located away from Berkeley 90 and another one with a higher column density (thicker) associated with the cluster. The first cloud has similar properties for both O-star sightlines but the second one is thicker for LS III +46 11. The comparison between species indicate that the cloud with a higher column density has a denser core, allowing us to classify the DIBs in a sigma-zeta scale, some of them for the first time. The LS III +46 12 sightline also has a high-velocity redshifted component.Comment: Accepted for publication in A&

Ediacaran Obduction of a Fore-Arc Ophiolite in SW Iberia: A Turning Point in the Evolving Geodynamic Setting of Peri- Gondwana

The Calzadilla Ophiolite is an ensemble of mafic and ultramafic rocks that represents the transition between lower crust and upper mantle of a Cadomian (peri-Gondwanan) fore arc. Mapping and structural analysis of the ophiolite demonstrates that it was obducted in latest Ediacaran times, because the Ediacaran-Early Cambrian sedimentary series (Malcocinado Formation) discordantly covers it. The ophiolite and emplacement-related structures are affected by Variscan deformation (Devonian-Carboniferous), which includes SW verging overturned folds (D1) and thrusts (D2), upright folds (D3), extensional faults (D4), and later faults (D5). These phases of deformation are explained in the context of Variscan tectonics as the result of the progressive collision between Gondwana and Laurussia. Qualitative unstraining of Variscan deformation reveals the primary geometry of Ediacaran-Cambrian structures and uncovers the generation of east verging thrusts as responsible for the primary obduction of the Calzadilla Ophiolite. Restoration of planar and linear structures associated with this event indicates an Ediacaran, east directed obduction of the ophiolite, that is, emplacement of the Cadomian fore arc onto inner sections of the northern margin of Gondwana. According to regional data, the obduction separates two extension-dominated stages in the tectonic evolution of the African margin of northern Gondwana preserved in southern Europe. Preobduction extension brought about the onset and widening of fore-arc and back-arc basins in the external part of the continent, while postobduction extension facilitated the formation of extensional migmatitic domes, an oceanward migration of back-arc spreading centers across peri-Gondwana, and the eventual opening of a major basin such as the Rheic Ocean