89 research outputs found

    A Note on the Complexity of One-Sided Crossing Minimization of Trees

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    In 2011, Harrigan and Healy published a polynomial-time algorithm for one-sided crossing minimization for trees. We point out a counterexample to that algorithm, and show that one-sided crossing minimization is NP-hard for trees.Comment: 3 pages, 2 figure

    Turbocharging Heuristics for Weak Coloring Numbers

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    Bounded expansion and nowhere-dense classes of graphs capture the theoretical tractability for several important algorithmic problems. These classes of graphs can be characterized by the so-called weak coloring numbers of graphs, which generalize the well-known graph invariant degeneracy (also called k-core number). Being NP-hard, weak-coloring numbers were previously computed on real-world graphs mainly via incremental heuristics. We study whether it is feasible to augment such heuristics with exponential-time subprocedures that kick in when a desired upper bound on the weak coloring number is breached. We provide hardness and tractability results on the corresponding computational subproblems. We implemented several of the resulting algorithms and show them to be competitive with previous approaches on a previously studied set of benchmark instances containing 86 graphs with up to 183831 edges. We obtain improved weak coloring numbers for over half of the instances

    LinSets.zip: Compressing Linear Set Diagrams

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    Linear diagrams are used to visualize set systems by depicting set memberships as horizontal line segments in a matrix, where each set is represented as a row and each element as a column. Each such line segment of a set is shown in a contiguous horizontal range of cells of the matrix indicating that the corresponding elements in the columns belong to the set. As each set occupies its own row in the matrix, the total height of the resulting visualization is as large as the number of sets in the instance. Such a linear diagram can be visually sparse and intersecting sets containing the same element might be represented by distant rows. To alleviate such undesirable effects, we present LinSets.zip, a new approach that achieves a more space-efficient representation of linear diagrams. First, we minimize the total number of gaps in the horizontal segments by reordering columns, a criterion that has been shown to increase readability in linear diagrams. The main difference of LinSets.zip to linear diagrams is that multiple non-intersecting sets can be positioned in the same row of the matrix. Furthermore, we present several different rendering variations for a matrix-based representation that utilize the proposed row compression. We implemented the different steps of our approach in a visualization pipeline using integer-linear programming, and suitable heuristics aiming at sufficiently fast computations in practice. We conducted both a quantitative evaluation and a small-scale user experiment to compare the effects of compressing linear diagrams.Comment: To be presented at PacificVis 202

    The Complexity of Cluster Vertex Splitting and Company

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    Clustering a graph when the clusters can overlap can be seen from three different angles: We may look for cliques that cover the edges of the graph, we may look to add or delete few edges to uncover the cluster structure, or we may split vertices to separate the clusters from each other. Splitting a vertex vv means to remove it and to add two new copies of vv and to make each previous neighbor of vv adjacent with at least one of the copies. In this work, we study the underlying computational problems regarding the three angles to overlapping clusterings, in particular when the overlap is small. We show that the above-mentioned covering problem, which also has been independently studied in different contexts,is NP-complete. Based on a previous so-called critical-clique lemma, we leverage our hardness result to show that Cluster Editing with Vertex Splitting is also NP-complete, resolving an open question by Abu-Khzam et al. [ISCO 2018]. We notice, however, that the proof of the critical-clique lemma is flawed and we give a counterexample. Our hardness result also holds under a version of the critical-clique lemma to which we currently do not have a counterexample. On the positive side, we show that Cluster Vertex Splitting admits a vertex-linear problem kernel with respect to the number of splits.Comment: 30 pages, 9 figure

    Fermi Bubbles in the Milky Way: the closest AGN feedback laboratory courtesy of Sgr A*?

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    Deposition of a massive (10410^4 to 10^5 \msun) giant molecular cloud (GMC) into the inner parsec of the Galaxy is widely believed to explain the origin of over a hundred unusually massive young stars born there 6\sim 6 Myr ago. An unknown fraction of that gas could have been accreted by Sgr A*, the supermassive black hole (SMBH) of the Milky Way. It has been recently suggested that two observed γ\gamma-ray-emitting bubbles emanating from the very center of our Galaxy were inflated by this putative activity of Sgr A*. We run a suite of numerical simulations to test whether the observed morphology of the bubbles could be due to the collimation of a wide angle outflow from Sgr A* by the disc-like Central Molecular Zone (CMZ), a well known massive repository of molecular gas in the central 200\sim 200 pc. We find that an Eddington-limited outburst of Sgr A* lasting 1\simeq 1 Myr is required to reproduce the morphology of the {\it Fermi} bubbles, suggesting that the GMC mass was \sim 10^5 \msun and it was mainly accreted by Sgr A* rather than used to make stars. We also find that the outflow from Sgr A* enforces strong angular momentum mixing in the CMZ disc, robustly sculpting it into a much narrower structure -- a ring -- perhaps synonymous with the recently reported "Herschel ring". In addition, we find that Sgr A* outflow is likely to have induced formation of massive star-forming GMCs in the CMZ. In this scenario, the Arches and Quintuplet clusters, the two observed young star clusters in the central tens of parsecs of the Galaxy, and also GMCs such as Sgr B2, owe their existence to the recent Sgr A* activity.Comment: 18 pages, 10 figures, submitted to MNRA

    Turbulence induced collisional velocities and density enhancements: large inertial range results from shell models

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    To understand the earliest stages of planet formation, it is crucial to be able to predict the rate and the outcome of dust grains collisions, be it sticking and growth, bouncing, or fragmentation. The outcome of such collisions depends on the collision speed, so we need a solid understanding of the rate and velocity distribution of turbulence-induced dust grain collisions. The rate of the collisions depends both on the speed of the collisions and the degree of clustering experienced by the dust grains, which is a known outcome of turbulence. We evolve the motion of dust grains in simulated turbulence, an approach that allows a large turbulent inertial range making it possible to investigate the effect of turbulence on meso-scale grains (millimeter and centimeter). We find three populations of dust grains: one highly clustered, cold and collisionless; one warm; and the third "hot". Our results can be fit by a simple formula, and predict both significantly slower typical collisional velocities for a given turbulent strength than previously considered, and modest effective clustering of the collisional populations, easing difficulties associated with bouncing and fragmentation barriers to dust grain growth. Nonetheless, the rate of high velocity collisions falls off merely exponentially with relative velocity so some mid- or high-velocity collisions will still occur, promising some fragmentation.Comment: 14 pages, 8 figures, 4 tables, Accepted, MNRA

    LIGHT-RAILS MODEL IN TEMPE, ARIZONA: STRATEGIES TO REDUCE AIR POLLUTION

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    To rethink growth is important to keep up with an increase demand of transport in order to decrease greenhouse gas emissions. It is clear that transportation generates a loss in vegetation then leads to an increase of carbon dioxide, a prime contributor to global warming, in the atmosphere. However, new alternatives are available to show an attempt to solve congestion that ultimately promotes a further increase of pollutants in the atmosphere.  From this insight, we find solutions to resolve traffic problem in the megalopolis Phoenix, in particular the city of Tempe. Although this research we highlight the factors that make up a good ridership of the Phoenix light rail looking at similar low density cities like Phoenix, we searched for solutions that would prevent low ridership in future light rail extensions. It was found that a more pedestrian friendly environment was created with businesses and residential buildings close together, especially in Tempe City around the Arizona State University (ASU), extremely benefited by the project

    The Chang-Refsdal Lens Revisited

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    This paper provides a complete theoretical treatment of the point-mass lens perturbed by constant external shear, often called the Chang-Refsdal lens. We show that simple invariants exist for the products of the (complex) positions of the four images, as well as moment sums of their signed magnifications. The image topographies and equations of the caustics and critical curves are also studied. We derive the fully analytic expressions for precaustics, which are the loci of non-critical points that map to the caustics under the lens mapping. They constitute boundaries of the region in the image domain that maps onto the interior of the caustics. The areas under the critical curves, caustics and precaustics are all evaluated, which enables us to calculate the mean magnification of the source within the caustics. Additionally, the exact analytic expression for the magnification distribution for the source in the triangular caustics is derived, as well as a useful approximate expression. Finally, we find that the Chang-Refsdal lens with the convergence greater than unity can exhibit third-order critical behaviour, if the reduced shear is exactly equal to \sqrt{3}/2, and that the number of images for N-point masses with non-zero constant shear cannot be greater than 5N-1.Comment: to appear in MNRAS (including 6 figures, 3 appendices; v2 - minor update with corrected typos etc.

    From Lab to Pilot Scale: Commissioning of an Integrated Device for the Generation of Crystals

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    Fast time-to-market, increased efficiency, and flexibility of production processes are major motivators for the development of integrated, continuous apparatuses with short changeover times. Following this trend, the modular belt crystallizer was developed and characterized in lab scale with the model system sucrose-water. Based on the promising results, the plant concept was upscaled and commissioned in industrial environment. The results are presented within the scope of this work. Starting from small seed crystals in solution, it was possible to grow, separate, and dry product particles. Further, the conducted experiments demonstrated that it is feasible to transfer the results from laboratory to pilot scale, which in turn enables accelerated process design as well as development

    High accretion rates in magnetised Keplerian discs mediated by a Parker instability driven dynamo

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    Hydromagnetic stresses in accretion discs have been the subject of intense theoretical research over the past one and a half decades. Most of the disc simulations have assumed a small initial magnetic field and studied the turbulence that arises from the magnetorotational instability. However, gaseous discs in galactic nuclei and in some binary systems are likely to have significant initial magnetisation. Motivated by this, we performed ideal magnetohydrodynamic simulations of strongly magnetised, vertically stratified discs in a Keplerian potential. Our initial equilibrium configuration, which has an azimuthal magnetic field in equipartion with thermal pressure, is unstable to the Parker instability. This leads to the expelling of magnetic field arcs, anchored in the midplane of the disc, to around five scale heights from the midplane. Transition to turbulence happens primarily through magnetorotational instability in the resulting vertical fields, although magnetorotational shear instability in the unperturbed azimuthal field plays a significant role as well, especially in the midplane where buoyancy is weak. High magnetic and hydrodynamical stresses arise, yielding an effective α\alpha-value of around 0.1 in our highest resolution run. Azimuthal magnetic field expelled by magnetic buoyancy from the disc is continuously replenished by the stretching of a radial field created as gas parcels slide in the linear gravity field along inclined magnetic field lines. This dynamo process, where the bending of field lines by the Parker instability leads to re-creation of the azimuthal field, implies that highly magnetised discs are astrophysically viable and that they have high accretion rates.Comment: 14 pages, 14 figures, accepted for publication in A&
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