823 research outputs found

    Computing tolerance parameters for fixturing and feeding

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    Fixtures and feeders are important components of automated assembly systems: fixtures accurately hold parts and feeders move parts into alignment. These components can fail when part shape varies. Parametric tolerance classes specify how much variation is allowable. In this paper we consider fixturing convex polygonal parts using right-angle brackets and feeding polygonal parts on conveyor belts using sequences of vertical fences. For both cases, we define new tolerance classes and give algorithms for computing the parameter specifications such that the fixture or feeder will work for all parts in the tolerance class. For fixturing we give an O(1) algorithm to compute the dimensions of rectangular tolerance zones. For feeding we give an O(n2) algorithm to compute the radius of the largest allowable tolerance zone around each vertex. For each, we give an O(n) time algorithm for testing if an n-sided part is in the tolerance class

    A spanning tree model for the Heegaard Floer homology of a branched double-cover

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    Given a diagram of a link K in S^3, we write down a Heegaard diagram for the branched-double cover Sigma(K). The generators of the associated Heegaard Floer chain complex correspond to Kauffman states of the link diagram. Using this model we make some computations of the homology \hat{HF}(Sigma(K)) as a graded group. We also conjecture the existence of a delta-grading on \hat{HF}(Sigma(K)) analogous to the delta-grading on knot Floer and Khovanov homology.Comment: 43 pages, 20 figure

    When the Cut Condition is Enough: A Complete Characterization for Multiflow Problems in Series-Parallel Networks

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    Let G=(V,E)G=(V,E) be a supply graph and H=(V,F)H=(V,F) a demand graph defined on the same set of vertices. An assignment of capacities to the edges of GG and demands to the edges of HH is said to satisfy the \emph{cut condition} if for any cut in the graph, the total demand crossing the cut is no more than the total capacity crossing it. The pair (G,H)(G,H) is called \emph{cut-sufficient} if for any assignment of capacities and demands that satisfy the cut condition, there is a multiflow routing the demands defined on HH within the network with capacities defined on GG. We prove a previous conjecture, which states that when the supply graph GG is series-parallel, the pair (G,H)(G,H) is cut-sufficient if and only if (G,H)(G,H) does not contain an \emph{odd spindle} as a minor; that is, if it is impossible to contract edges of GG and delete edges of GG and HH so that GG becomes the complete bipartite graph K2,pK_{2,p}, with p3p\geq 3 odd, and HH is composed of a cycle connecting the pp vertices of degree 2, and an edge connecting the two vertices of degree pp. We further prove that if the instance is \emph{Eulerian} --- that is, the demands and capacities are integers and the total of demands and capacities incident to each vertex is even --- then the multiflow problem has an integral solution. We provide a polynomial-time algorithm to find an integral solution in this case. In order to prove these results, we formulate properties of tight cuts (cuts for which the cut condition inequality is tight) in cut-sufficient pairs. We believe these properties might be useful in extending our results to planar graphs.Comment: An extended abstract of this paper will be published at the 44th Symposium on Theory of Computing (STOC 2012

    Orienting polyhedral parts by pushing

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    A common task in automated manufacturing processes is to orient parts prior to assembly. We consider sensorless orientation of an asymmetric polyhedral part by a sequence of push actions, and show that is it possible to move any such part from an unknown initial orientation into a known final orientation if these actions are performed by a jaw consisting of two orthogonal planes. We also show how to compute an orienting sequence of push actions.We propose a three-dimensional generalization of conveyor belts with fences consisting of a sequence of tilted plates with curved tips; each of the plates contains a sequence of fences. We show that it is possible to compute a set-up of plates and fences for any given asymmetric polyhedral part such that the part gets oriented on its descent along plates and fences

    Sandpiles and Dominos

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    We consider the subgroup of the abelian sandpile group of the grid graph consisting of configurations of sand that are symmetric with respect to central vertical and horizontal axes. We show that the size of this group is (i) the number of domino tilings of a corresponding weighted rectangular checkerboard; (ii) a product of special values of Chebyshev polynomials; and (iii) a double-product whose factors are sums of squares of values of trigonometric functions. We provide a new derivation of the formula due to Kasteleyn and to Temperley and Fisher for counting the number of domino tilings of a 2m x 2n rectangular checkerboard and a new way of counting the number of domino tilings of a 2m x 2n checkerboard on a M\"obius strip.Comment: 35 pages, 24 figure
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