2,713 research outputs found

    Computational Geometry Column 43

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    The concept of pointed pseudo-triangulations is defined and a few of its applications described.Comment: 3 pages, 1 figur

    Uniform Infinite Planar Triangulations

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    The existence of the weak limit as n --> infinity of the uniform measure on rooted triangulations of the sphere with n vertices is proved. Some properties of the limit are studied. In particular, the limit is a probability measure on random triangulations of the plane.Comment: 36 pages, 4 figures; Journal revised versio

    Liftings and stresses for planar periodic frameworks

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    We formulate and prove a periodic analog of Maxwell's theorem relating stressed planar frameworks and their liftings to polyhedral surfaces with spherical topology. We use our lifting theorem to prove deformation and rigidity-theoretic properties for planar periodic pseudo-triangulations, generalizing features known for their finite counterparts. These properties are then applied to questions originating in mathematical crystallography and materials science, concerning planar periodic auxetic structures and ultrarigid periodic frameworks.Comment: An extended abstract of this paper has appeared in Proc. 30th annual Symposium on Computational Geometry (SOCG'14), Kyoto, Japan, June 201

    BPS Graphs: From Spectral Networks to BPS Quivers

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    We define "BPS graphs" on punctured Riemann surfaces associated with AN1A_{N-1} theories of class S\mathcal{S}. BPS graphs provide a bridge between two powerful frameworks for studying the spectrum of BPS states: spectral networks and BPS quivers. They arise from degenerate spectral networks at maximal intersections of walls of marginal stability on the Coulomb branch. While the BPS spectrum is ill-defined at such intersections, a BPS graph captures a useful basis of elementary BPS states. The topology of a BPS graph encodes a BPS quiver, even for higher-rank theories and for theories with certain partial punctures. BPS graphs lead to a geometric realization of the combinatorics of Fock-Goncharov NN-triangulations and generalize them in several ways.Comment: 48 pages, 44 figure
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