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The Jordan curve theorem is non-trivial
Journal of Mathematics and the Arts, 2011The 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 noth...
William T Ross
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On Selfsimilar Jordan Curves on the Plane
Siberian Mathematical Journal, 2003Summary: We study the attractors of a finite system of planar contraction similarities \(S_j\) \((j=1,\dots ,n)\) satisfying the coupling condition: for a set \(\{x_0,\dots ,x_n\}\) of points and a binary vector \((s_1,\dots , s_n)\), called the signature, the mapping \(S_j\) takes the pair \(\{x_0, x_n\}\) either into the pair \(\{ x_{j-1}, x_j ...
Aseev, V. V. +2 more
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Acta Mathematica Hungarica, 1988
The aim of this paper is to define the notion of the linking Jordan curve and to show that under certain conditions we can find such a curve. The subject is based on the axiomatization of linking theory which is described by the author in Dokl. Akad. Nauk SSSR 203, 986-988 (1972; Zbl 0261.57010) and Acta Math. Hung. 49, 3-28 (1987; Zbl 0616.55007).
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The aim of this paper is to define the notion of the linking Jordan curve and to show that under certain conditions we can find such a curve. The subject is based on the axiomatization of linking theory which is described by the author in Dokl. Akad. Nauk SSSR 203, 986-988 (1972; Zbl 0261.57010) and Acta Math. Hung. 49, 3-28 (1987; Zbl 0616.55007).
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A Conformal Proof of a Jordan Curve Problem
Canadian Mathematical Bulletin, 1969The following theorem appears in [1].Let R be a closed simply connected region of the inversive plane bounded by a Jordan curve J, and let J be divided into three closed arcs A1, A2, A3. Then there exists a circle contained in R and having points in common with all three arcs.
Spoar, G., Lane, N. D.
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RENDICONTO DELL’ACCADEMIA DELLE SCIENZE FISICHE E MATEMATICHE SERIE IV - VOL. XCI - ANNO CLXIII (2024)
After some background, at first we prove the equivalence between two definitions of “Jordan’s curve”, defined as simple and closed plane curve but known also as subset of R2 homeomorphic to a circle. Using the ideas of Maehara ([6]) who gave a proof of Jordan’s classical theorem, for a given Jordan’s curve Γ we introduce the notions of central point ...
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After some background, at first we prove the equivalence between two definitions of “Jordan’s curve”, defined as simple and closed plane curve but known also as subset of R2 homeomorphic to a circle. Using the ideas of Maehara ([6]) who gave a proof of Jordan’s classical theorem, for a given Jordan’s curve Γ we introduce the notions of central point ...
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Strong Szegő Theorem on a Jordan Curve
2022We 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. This also corresponds to studying a planar Coulomb gas on the curve at inverse temperature $β=2$.
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A Bound for Reflections Across Jordan Curves
gmj, 2003Abstract The properties of logarithmic derivative of the Riemann mapping function of a quasidisk are applied to quantitative estimation of quasiconformal reflections across its boundary.
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The Jordan Curve Theorem for Piecewise Smooth Curves
The American Mathematical Monthly, 1969(1969). The Jordan Curve Theorem for Piecewise Smooth Curves. The American Mathematical Monthly: Vol. 76, No. 6, pp. 605-610.
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