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The Jordan curve theorem is non-trivial
Journal of Mathematics and the Arts, 2011 The formal mathematical definition of a Jordan curve (a non-self-intersecting continuous loop in the plane) is so simple that one is often lead to the unimaginative view that a Jordan curve is nothing more than a circle or an ellipse.
William T Ross
exaly +4 more sources
The Jordan curve theorem in the Khalimsky plane [PDF]
Applied General Topology, 2008 The connectivity in Alexandroff topological spaces is equivalent to the path connectivity. This fact gets some specific properties to Z2, equipped with the Khalimsky topology.
Ezzeddine Bouassida
doaj +6 more sources
The Jordan Curve Theorem, Formally and Informally [PDF]
American Mathematical Monthly, 2007 curve separates the plane into a bounded interior region and an unbounded exterior. One hundred years ago, Oswald Veblen declared that this theorem is “justly regarded as a most important step in the direction of a perfectly rigorous mathematics ” [13, p.
Thomas C Hales
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A combinatorial analog of the Jordan Curve Theorem
Journal of Combinatorial Theory Series B, 1983 The concept of the genus of a pair of permutations is defined in the same manner as was done by Jacques. The integrality of the genus is proven in a new way by applying a technique developed by Walkup for the reduction of products of permutations.
Saul Stahl
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Digital Jordan curve theorems [PDF]
, 2000 . Efim Khalimsky’s digital Jordan curve theorem states that the complement of a Jordan curve in the digital plane equipped with the Khalimsky topology has exactly two connectivity components.
Christer O. Kiselman
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Discrete Jordan Curve Theorems
Journal of Combinatorial Theory, Series B, 1989 Discrete versions of the Jordan Curve Theorem are ...
Vince, Andrew, Little, C.H.C
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The Complexity of Hex and the Jordan Curve Theorem [PDF]
, 2016 The Jordan curve theorem and Brouwer's fixed-point theorem are fundamental problems in topology. We study their computational relationship, showing that a stylized computational version of Jordan’s theorem is PPAD-complete, and therefore in a sense ...
Adler, Aviv +2 more
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A Jordan curve theorem for 2-dimensional tilings [PDF]
Topology and Its Applications, 2021 The classical Jordan curve theorem for digital curves asserts that the Jordan curve theorem remains valid in the Khalimsky plane. Since the Khalimsky plane is a quotient space of $\mathbb R^2$ induced by a tiling of squares, it is natural to ask for which other tilings of the plane it is possible to obtain a similar result.
Natalia Jonard-Pérez
exaly +3 more sources
A digital analogue of the Jordan curve theorem
Discrete Applied Mathematics, 2003 We study certain closure operations on Z2, with the aim of showing that they can provide a suitable framework for solving problems of digital topology. The Khalimsky topology on Z2, which is commonly used as a basic structure in digital topology nowadays,
Šlapal, J, J Šlapal
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Strong Szegő Theorem on a Jordan Curve
, 2022 We consider certain determinants with respect to a sufficiently regular Jordan curve γ in the complex plane that generalize Toeplitz determinants which are obtained when the curve is the circle.
Johansson, Kurt,
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