147 research outputs found
Boundary Value Problems on Planar Graphs and Flat Surfaces with integer cone singularities, II: The mixed Dirichlet-Neumann Problem
In this paper we continue the study started in part I (posted). We consider a
planar, bounded, -connected region , and let \bord\Omega be its
boundary. Let 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 where is a special type of a (possibly immersed)
genus singular flat surface, tiled by rectangles and is an energy
preserving mapping from onto . In part I the solution
of a Dirichlet problem defined on 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
Consider a planar, bounded, -connected region , and let
\bord\Omega be its boundary. Let 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 where is a genus singular flat surface tiled
by rectangles and is an energy preserving mapping from
onto .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
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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