Journal Articles

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Browsing Journal Articles by Subject "0101 Pure Mathematics"

Now showing 1 - 3 of 3
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    Constructing reparameterization invariant metrics on spaces of plane curves
    (Elsevier, 9/05/2014) Bauer M; Bruveris M; Marsland SR; Michor P
    Metrics on shape spaces are used to describe deformations that take one shape to another, and to define a distance between shapes. We study a family of metrics on the space of curves, which includes several recently proposed metrics, for which the metrics are characterised by mappings into vector spaces where geodesics can be easily computed. This family consists of Sobolev-type Riemannian metrics of order one on the space Imm(S1,R2)Imm(S1,R2) of parameterized plane curves and the quotient space Imm(S1,R2)/Diff(S1)Imm(S1,R2)/Diff(S1) of unparameterized curves. For the space of open parameterized curves we find an explicit formula for the geodesic distance and show that the sectional curvatures vanish on the space of parameterized open curves and are non-negative on the space of unparameterized open curves. For one particular metric we provide a numerical algorithm that computes geodesics between unparameterized, closed curves, making use of a constrained formulation that is implemented numerically using the RATTLE algorithm. We illustrate the algorithm with some numerical tests between shapes.
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    Counting the spanning trees of the 3-cube using edge slides
    (2012-10) Tuffley CP
    We give a direct combinatorial proof of the known fact that the 3-cube has 384 spanning trees, using an "edge slide" operation on spanning trees. This gives an answer in the case n=3 to a question implicitly raised by Stanley. Our argument gives a bijective proof of the n=3 case of a weighted count of the spanning trees of the n-cube due to Martin and Reiner.
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