2,653 research outputs found

    Covering orthogonal polygons with star polygons: The perfect graph approach

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    AbstractThis paper studies the combinatorial structure of visibility in orthogonal polygons. We show that the visibility graph for the problem of minimally covering simple orthogonal polygons with star polygons is perfect. A star polygon contains a point p, such that for every point q in the star polygon, there is an orthogonally convex polygon containing p and q. This perfectness property implies a polynomial algorithm for the above polygon covering problem. It further provides us with an interesting duality relationship. We first establish that a minimum clique cover of the visibility graph of a simple orthogonal polygon corresponds exactly to a minimum star cover of the polygon. In general, simple orthogonal polygons can have concavities (dents) with four possible orientations. In this case, we show that the visibility graph is weakly triangulated. We thus obtain an O(n8) algorithm. Since weakly triangulated graphs are perfect, we also obtain an interesting duality relationship. In the case where the polygon has at most three dent orientations, we show that the visibility graph is triangulated or chordal. This gives us an O(n3) algorithm

    Scanning Gate Microscopy of Quantum Contacts Under Parallel Magnetic Field: Beating Patterns Between Spin-Split Transmission Peaks or Channel Openings

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    We study the conductance gg of an electron interferometer created in a two dimensional electron gas between a nanostructured contact and the depletion region induced by the charged tip of a scanning gate microscope. Using non-interacting models, we study the beating pattern of interference fringes exhibited by the images giving gg as a function of the tip position when a parallel magnetic field is applied. The analytical solution of a simplified model allows us to distinguish between two cases: (i) If the field is applied everywhere, the beating of Fabry-P\'erot oscillations of opposite spins gives rise to interference rings which can be observed at low temperatures when the contact is open between spin-split transmission resonances. (ii) If the field acts only upon the contact, the interference rings cannot be observed at low temperatures, but only at temperatures of the order of the Zeeman energy. For a contact made of two sites in series, a model often used for describing an inversion-symmetric double-dot setup, a pseudo-spin degeneracy is broken by the inter-dot coupling and a similar beating effect can be observed without magnetic field at temperatures of the order of the interdot coupling. Eventually, numerical studies of a quantum point contact with quantized conductance plateaus confirm that a parallel magnetic field applied everywhere or only upon the contact gives rises to similar beating effects between spin-split channel openings.Comment: 11 pages, 17 figure

    Generalized kernels of polygons under rotation

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    Given a set O\mathcal{O} of kk orientations in the plane, two points inside a simple polygon PP O\mathcal{O}-see each other if there is an O\mathcal{O}-staircase contained in PP that connects them. The O\mathcal{O}-kernel of PP is the subset of points which O\mathcal{O}-see all the other points in PP. This work initiates the study of the computation and maintenance of the O\mathcal{O}-Kernel{\rm Kernel} of a polygon PP as we rotate the set O\mathcal{O} by an angle Ξ\theta, denoted O\mathcal{O}-KernelΞ(P){\rm Kernel}_{\theta}(P). In particular, we design efficient algorithms for (i) computing and maintaining {0o}\{0^{o}\}-KernelΞ(P){\rm Kernel}_{\theta}(P) while Ξ\theta varies in [−π2,π2)[-\frac{\pi}{2},\frac{\pi}{2}), obtaining the angular intervals where the {0o}\{0^{o}\}-KernelΞ(P){\rm Kernel}_{\theta}(P) is not empty and (ii) for orthogonal polygons PP, computing the orientation Ξ∈[0,π2)\theta\in[0, \frac{\pi}{2}) such that the area and/or the perimeter of the {0o,90o}\{0^{o},90^{o}\}-KernelΞ(P){\rm Kernel}_{\theta}(P) are maximum or minimum. These results extend previous works by Gewali, Palios, Rawlins, Schuierer, and Wood.Comment: 12 pages, 4 figures, a version omitting some proofs appeared at the 34th European Workshop on Computational Geometry (EuroCG 2018

    A multi-wavelength analysis for interferometric (sub-)mm observations of protoplanetary disks: radial constraints on the dust properties and the disk structure

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    Theoretical models of grain growth predict dust properties to change as a function of protoplanetary disk radius, mass, age and other physical conditions. We lay down the methodology for a multi-wavelength analysis of (sub-)mm and cm continuum interferometric observations to constrain self-consistently the disk structure and the radial variation of the dust properties. The computational architecture is massively parallel and highly modular. The analysis is based on the simultaneous fit in the uv-plane of observations at several wavelengths with a model for the disk thermal emission and for the dust opacity. The observed flux density at the different wavelengths is fitted by posing constraints on the disk structure and on the radial variation of the grain size distribution. We apply the analysis to observations of three protoplanetary disks (AS 209, FT Tau, DR Tau) for which a combination of spatially resolved observations in the range ~0.88mm to ~10mm is available (from SMA, CARMA, and VLA), finding evidence of a decreasing maximum dust grain size (a_max) with radius. We derive large a_max values up to 1 cm in the inner disk between 15 and 30 AU and smaller grains with a_max~1 mm in the outer disk (R > 80AU). In this paper we develop a multi-wavelength analysis that will allow this missing quantity to be constrained for statistically relevant samples of disks and to investigate possible correlations with disk or stellar parameters.Comment: 19 pages, 15 figures, accepted for publication in A&

    Epsilon-Unfolding Orthogonal Polyhedra

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    An unfolding of a polyhedron is produced by cutting the surface and flattening to a single, connected, planar piece without overlap (except possibly at boundary points). It is a long unsolved problem to determine whether every polyhedron may be unfolded. Here we prove, via an algorithm, that every orthogonal polyhedron (one whose faces meet at right angles) of genus zero may be unfolded. Our cuts are not necessarily along edges of the polyhedron, but they are always parallel to polyhedron edges. For a polyhedron of n vertices, portions of the unfolding will be rectangular strips which, in the worst case, may need to be as thin as epsilon = 1/2^{Omega(n)}.Comment: 23 pages, 20 figures, 7 references. Revised version improves language and figures, updates references, and sharpens the conclusio

    Higher-harmonic adaptation and the detection of squarewave gratings

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    Adaptation to a high contrast sinewave grating of 1 c/deg spatial frequency causes a large increase in the contrast threshold for a 1 c/deg test grating, but fails to raise the threshold for a squarewave grating of 0.33 c/deg, although the sensitivity of the “channel” tuned to both the third and fifth harmonic components of the squarewave test grating should be thoroughly suppressed. Following sequential adaptation to sinewave gratings of 1 and 3 c/deg spatial frequency, detection of squarewave gratings at 0.33 c/deg likewise remains unaffected. In contrast, after adaptation to a 0.33 c/deg squarewave grating with missing fundamental the contrast threshold for a squarewave test grating of the same frequency is increased by 0.25 log unit, although the higher harmonic component frequencies are less affected than by sequential sinewave adaptation. The results suggest that independent spatial frequency channels detecting harmonic components are not alone sufficient to account for the visibility of low frequency squarewaves
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