447 research outputs found

    Two-dimensional patterns with distinct differences; constructions, bounds, and maximal anticodes

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    A two-dimensional (2-D) grid with dots is called a configuration with distinct differences if any two lines which connect two dots are distinct either in their length or in their slope. These configurations are known to have many applications such as radar, sonar, physical alignment, and time-position synchronization. Rather than restricting dots to lie in a square or rectangle, as previously studied, we restrict the maximum distance between dots of the configuration; the motivation for this is a new application of such configurations to key distribution in wireless sensor networks. We consider configurations in the hexagonal grid as well as in the traditional square grid, with distances measured both in the Euclidean metric, and in the Manhattan or hexagonal metrics. We note that these configurations are confined inside maximal anticodes in the corresponding grid. We classify maximal anticodes for each diameter in each grid. We present upper bounds on the number of dots in a pattern with distinct differences contained in these maximal anticodes. Our bounds settle (in the negative) a question of Golomb and Taylor on the existence of honeycomb arrays of arbitrarily large size. We present constructions and lower bounds on the number of dots in configurations with distinct differences contained in various 2-D shapes (such as anticodes) by considering periodic configurations with distinct differences in the square grid

    Phase transitions for the Boolean model of continuum percolation for Cox point processes

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    We consider the Boolean model with random radii based on Cox point processes. Under a condition of stabilization for the random environment, we establish existence and non-existence of subcritical regimes for the size of the cluster at the origin in terms of volume, diameter and number of points. Further, we prove uniqueness of the infinite cluster for sufficiently connected environments.Comment: 22 pages, 2 figure

    Pattern Formation in Growing Sandpiles with Multiple Sources or Sinks

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    Adding sand grains at a single site in Abelian sandpile models produces beautiful but complex patterns. We study the effect of sink sites on such patterns. Sinks change the scaling of the diameter of the pattern with the number NN of sand grains added. For example, in two dimensions, in presence of a sink site, the diameter of the pattern grows as (N/log⁥N)\sqrt{(N/\log N)} for large NN, whereas it grows as N\sqrt{N} if there are no sink sites. In presence of a line of sink sites, this rate reduces to N1/3N^{1/3}. We determine the growth rates for these sink geometries along with the case when there are two lines of sink sites forming a wedge, and its generalization to higher dimensions. We characterize one such asymptotic patterns on the two-dimensional F-lattice with a single source adjacent to a line of sink sites, in terms of position of different spatial features in the pattern. For this lattice, we also provide an exact characterization of the pattern with two sources, when the line joining them is along one of the axes.Comment: 27 pages, 17 figures. Figures with better resolution is available at http://www.theory.tifr.res.in/~tridib/pss.htm

    Geometry-Induced Transport Properties of Two Dimensional Networks

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