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  • Sheaf-Theoretic Stratification Learning from Geometric and Topological Perspectives
    Discret. Comput. Geom. (IF 0.741) Pub Date : 2020-05-29
    Adam Brown, Bei Wang

    We investigate a sheaf-theoretic interpretation of stratification learning from geometric and topological perspectives. Our main result is the construction of stratification learning algorithms framed in terms of a sheaf on a partially ordered set with the Alexandroff topology. We prove that the resulting decomposition is the unique minimal stratification for which the strata are homogeneous and the

    更新日期:2020-05-29
  • On the Number of Monochromatic Lines in $$\pmb {\mathbb {R}}^d$$Rd
    Discret. Comput. Geom. (IF 0.741) Pub Date : 2020-05-27
    Mario Huicochea

    Let X be a nonempty finite subset of \({\mathbb {R}}^d\) and \(X=\bigcup _{i=1}^m X_i\) a coloring with \(m

    更新日期:2020-05-27
  • Symmetric Non-Negative Forms and Sums of Squares
    Discret. Comput. Geom. (IF 0.741) Pub Date : 2020-05-21
    Grigoriy Blekherman, Cordian Riener

    We study symmetric non-negative forms and their relationship with symmetric sums of squares. For a fixed number of variables n and degree 2d, symmetric non-negative forms and symmetric sums of squares form closed, convex cones in the vector space of n-variate symmetric forms of degree 2d. Using representation theory of the symmetric group we characterize both cones in a uniform way. Further, we investigate

    更新日期:2020-05-21
  • Admissible Complexes for the Projective X-ray Transform over a Finite Field
    Discret. Comput. Geom. (IF 0.741) Pub Date : 2020-05-09
    David V. Feldman, Eric L. Grinberg

    We consider the X-ray transform in a projective space over a finite field. It is well known (after Bolker) that this transform is injective. We formulate an analog of Gelfand’s admissibility problem for the Radon transform, which asks for a classification of all minimal sets of lines for which the restricted Radon transform is injective. The solution involves doubly ruled quadric surfaces.

    更新日期:2020-05-09
  • Simple Realizability of Complete Abstract Topological Graphs Simplified
    Discret. Comput. Geom. (IF 0.741) Pub Date : 2020-05-04
    Jan Kynčl

    An abstract topological graph (briefly an AT-graph) is a pair \(A=(G,{\mathcal {X}})\) where \(G=(V,E)\) is a graph and \({\mathcal {X}}\subseteq {E \atopwithdelims ()2}\) is a set of pairs of its edges. The AT-graph A is simply realizable if G can be drawn in the plane so that each pair of edges from \({\mathcal {X}}\) crosses exactly once and no other pair crosses. We show that simply realizable

    更新日期:2020-05-04
  • On open and closed convex codes.
    Discret. Comput. Geom. Pub Date : 2018-12-13
    Joshua Cruz,Chad Giusti,Vladimir Itskov,Bill Kronholm

    Neural codes serve as a language for neurons in the brain. Open (or closed) convex codes, which arise from the pattern of intersections of collections of open (or closed) convex sets in Euclidean space, are of particular relevance to neuroscience. Not every code is open or closed convex, however, and the combinatorial properties of a code that determine its realization by such sets are still poorly

    更新日期:2019-11-01
  • On Sets Defining Few Ordinary Circles.
    Discret. Comput. Geom. Pub Date : 2017-12-30
    Aaron Lin,Mehdi Makhul,Hossein Nassajian Mojarrad,Josef Schicho,Konrad Swanepoel,Frank de Zeeuw

    An ordinary circle of a set P of n points in the plane is defined as a circle that contains exactly three points of P. We show that if P is not contained in a line or a circle, then P spans at least [Formula: see text] ordinary circles. Moreover, we determine the exact minimum number of ordinary circles for all sufficiently large n and describe all point sets that come close to this minimum. We also

    更新日期:2019-11-01
  • On the Densest Packing of Polycylinders in Any Dimension.
    Discret. Comput. Geom. Pub Date : 2016-04-14
    Wöden Kusner

    Using transversality and a dimension reduction argument, a result of Bezdek and Kuperberg is applied to polycylinders, showing that the optimal packing density of [Formula: see text] equals [Formula: see text] for all natural numbers n.

    更新日期:2019-11-01
  • Liftings and stresses for planar periodic frameworks.
    Discret. Comput. Geom. Pub Date : 2016-03-15
    Ciprian Borcea,Ileana Streinu

    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

    更新日期:2019-11-01
  • Gain-Sparsity and Symmetry-Forced Rigidity in the Plane.
    Discret. Comput. Geom. Pub Date : 2016-02-24
    Tibor Jordán,Viktória E Kaszanitzky,Shin-Ichi Tanigawa

    We consider planar bar-and-joint frameworks with discrete point group symmetry in which the joint positions are as generic as possible subject to the symmetry constraint. We provide combinatorial characterizations for symmetry-forced rigidity of such structures with rotation symmetry or dihedral symmetry of order 2k with odd k, unifying and extending previous work on this subject. We also explore the

    更新日期:2019-11-01
  • Coloring [Formula: see text]-Embeddable [Formula: see text]-Uniform Hypergraphs.
    Discret. Comput. Geom. Pub Date : 2014-11-25
    Carl Georg Heise,Konstantinos Panagiotou,Oleg Pikhurko,Anusch Taraz

    This paper extends the scenario of the Four Color Theorem in the following way. Let [Formula: see text] be the set of all [Formula: see text]-uniform hypergraphs that can be (linearly) embedded into [Formula: see text]. We investigate lower and upper bounds on the maximum (weak) chromatic number of hypergraphs in [Formula: see text]. For example, we can prove that for [Formula: see text] there are

    更新日期:2019-11-01
  • Reconstruction Using Witness Complexes.
    Discret. Comput. Geom. Pub Date : 2008-10-01
    Leonidas J Guibas,Steve Y Oudot

    We present a novel reconstruction algorithm that, given an input point set sampled from an object S, builds a one-parameter family of complexes that approximate S at different scales. At a high level, our method is very similar in spirit to Chew's surface meshing algorithm, with one notable difference though: the restricted Delaunay triangulation is replaced by the witness complex, which makes our

    更新日期:2019-11-01
  • Poisson-Delaunay Mosaics of Order k.
    Discret. Comput. Geom. Pub Date : null
    Herbert Edelsbrunner,Anton Nikitenko

    The order-k Voronoi tessellation of a locally finite set X ⊆ R n decomposes R n into convex domains whose points have the same k nearest neighbors in X. Assuming X is a stationary Poisson point process, we give explicit formulas for the expected number and total area of faces of a given dimension per unit volume of space. We also develop a relaxed version of discrete Morse theory and generalize by

    更新日期:2019-11-01
  • Barcodes of Towers and a Streaming Algorithm for Persistent Homology.
    Discret. Comput. Geom. Pub Date : null
    Michael Kerber,Hannah Schreiber

    A tower is a sequence of simplicial complexes connected by simplicial maps. We show how to compute a filtration, a sequence of nested simplicial complexes, with the same persistent barcode as the tower. Our approach is based on the coning strategy by Dey et al. (SoCG, 2014). We show that a variant of this approach yields a filtration that is asymptotically only marginally larger than the tower and

    更新日期:2019-11-01
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