147 research outputs found

    Boundary Value Problems on Planar Graphs and Flat Surfaces with integer cone singularities, II: The mixed Dirichlet-Neumann Problem

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    In this paper we continue the study started in part I (posted). We consider a planar, bounded, mm-connected region Ω\Omega, and let \bord\Omega be its boundary. Let T\mathcal{T} be a cellular decomposition of \Omega\cup\bord\Omega, where each 2-cell is either a triangle or a quadrilateral. From these data and a conductance function we construct a canonical pair (S,f)(S,f) where SS is a special type of a (possibly immersed) genus (m−1)(m-1) singular flat surface, tiled by rectangles and ff is an energy preserving mapping from T(1){\mathcal T}^{(1)} onto SS. In part I the solution of a Dirichlet problem defined on T(0){\mathcal T}^{(0)} was utilized, in this paper we employ the solution of a mixed Dirichlet-Neumann problem.Comment: 26 pages, 16 figures (color

    Boundary Value Problems on Planar Graphs and Flat Surfaces with integer cone singularities, I: The Dirichlet Problem

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    Consider a planar, bounded, mm-connected region Ω\Omega, and let \bord\Omega be its boundary. Let T\mathcal{T} be a cellular decomposition of \Omega\cup\bord\Omega, where each 2-cell is either a triangle or a quadrilateral. From these data and a conductance function we construct a canonical pair (S,f)(S,f) where SS is a genus (m−1)(m-1) singular flat surface tiled by rectangles and ff is an energy preserving mapping from T(1){\mathcal T}^{(1)} onto SS.Comment: 27 pages, 11 figures; v2 - revised definition (now denoted by the flux-gradient metric (1.9)) in section 1 and minor modifications of proofs; corrected typo

    Electrical networks and Stephenson's conjecture

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    In this paper, we consider a planar annulus, i.e., a bounded, two-connected, Jordan domain, endowed with a sequence of triangulations exhausting it. We then construct a corresponding sequence of maps which converge uniformly on compact subsets of the domain, to a conformal homeomorphism onto the interior of a Euclidean annulus bounded by two concentric circles. As an application, we will affirm a conjecture raised by Ken Stephenson in the 90's which predicts that the Riemann mapping can be approximated by a sequence of electrical networks.Comment: Comments are welcome
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