9,809 research outputs found
Discrete Jordan Curve Theorems
AbstractDiscrete versions of the Jordan Curve Theorem are proved
Two-Dimensional Critical Percolation: The Full Scaling Limit
We use SLE(6) paths to construct a process of continuum nonsimple loops in
the plane and prove that this process coincides with the full continuum scaling
limit of 2D critical site percolation on the triangular lattice -- that is, the
scaling limit of the set of all interfaces between different clusters. Some
properties of the loop process, including conformal invariance, are also
proved.Comment: 45 pages, 12 figures. This is a revised version of math.PR/0504036
without the appendice
Discrete Jordan Curve Theorem: A proof formalized in Coq with hypermaps
This paper presents a formalized proof of a discrete form of the Jordan Curve
Theorem. It is based on a hypermap model of planar subdivisions, formal
specifications and proofs assisted by the Coq system. Fundamental properties
are proven by structural or noetherian induction: Genus Theorem, Euler's
Formula, constructive planarity criteria. A notion of ring of faces is
inductively defined and a Jordan Curve Theorem is stated and proven for any
planar hypermap
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