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	Jordan curve theorem - Wikipedia, the free encyclopedia
		In topology, the Jordan curve theorem states that every non-self-intersecting loop in the plane divides the plane into an "inside" and an "outside".
		A rigorous 200,000-line formal proof of the Jordan curve theorem was eventually produced in 2005 by an international team of of mathematicians using the Mizar system.
		There is a generalisation of the Jordan curve theorem called the Jordan-Schönflies theorem which states that any Jordan curve in the plane can be extended to a homeomorphism of the plane.
	en.wikipedia.org /wiki/Jordan_curve_theorem (389 words)

	Jordan (Site not responding. Last check: 2007-11-06)
		Jordan was examined on 14 January 1861 by Duhamel, Serret and Puiseux.
		Jordan married Marie-Isabelle Munet, the daughter of the deputy mayor of Lyon, in 1862.
		Jordan's use of the group concept in geometry in 1869 was motivated by studies of crystal structure.
	www-groups.dcs.st-and.ac.uk /~history/Mathematicians/Jordan.html (2043 words)

	PlanetMath: Jordan curve theorem
		Informally, the Jordan curve theorem states that every Jordan curve divides the Euclidean plane into an ``outside'' and an ``inside''.
		The two connected components mentioned in each formulation are, of course, the inside and the outside the Jordan curve, although only in the first formulation is there a clear way to say what is out and what is in.
		This is version 6 of Jordan curve theorem, born on 2002-11-11, modified 2004-02-16.
	planetmath.org /encyclopedia/JordanCurveTheorem.html (179 words)

	Colorful Mathematics: Part II
		Proofs of theorems about coloring problems in the plane, like the four-color problem, must use properties of the geometry/topology of the plane.
		The Jordan Curve Theorem is surprisingly hard to prove, precisely because it is so basic that one has to rely directly on properties of the plane itself, properties that distinguish the plane from other surfaces such as a torus (donut).
		It is somewhat easier to prove the polygonal version of the theorem which states that a simple closed polygon in the plane divides the points into three sets.
	www.ams.org /featurecolumn/archive/color2.html (1230 words)

	The Jordan Curve Theorem
		Any simple closed curve C divides the points of the plane not on C into two distinct domains (with no points in common) of which C is the common boundary.
		To give a concrete example, it happens that this theorem has a simple proof for the case of polygons, which are a class of simple curves that occurs very often in most important problems.
		Therefore, proving the theorem for this particular case (a much simpler task than proving it for the case of general simple closed curves) may be of great use in solving a lot of other important problems.
	www-cgrl.cs.mcgill.ca /~godfried/teaching/cg-projects/97/Octavian/compgeom.html (1554 words)

	Search Results for Jordan
		Jordan was aware that his work was at a level that would be somewhat inappropriate for engineering students for he once said to Lebesgue that he called it "Ecole Polytechnique analysis course" since:-.
		A converse to the Jordan curve theorem, proved by Schonflies, states that a subset of the 2-sphere is a simple closed curve if it has two complementary domains, is the boundary of each of them, and is accessible from each of these domains.
		Jordan seems to have succeeded in a way that Sylow did not, for Jordan made Lie realise how important group theory was for the study of geometry.
	www-groups.dcs.st-and.ac.uk /history/Search/historysearch.cgi?SUGGESTION=Jordan&CONTEXT=1 (3236 words)

	Edouard Goursat - Wikipedia, the free encyclopedia
		At that time the topological foundations of complex analysis were still not clarified, with the Jordan curve theorem considered a challenge to mathematical rigour (as it would remain until L.
		Hardy, to be exemplary in facing up to the difficulties inherent in stating the fundamental Cauchy integral theorem properly.
		For that reason it is sometimes called the Cauchy-Goursat theorem.
	en.wikipedia.org /wiki/Edouard_Goursat (187 words)

	Formalization of Jordan Curve Theorem
		The proof of Jordan curve theorem for special polygons is the first part of the formalization of general Jordan curve theorem for simple closed curves:
		Jordan curve theorem - the theorem that states that every simple closed curve divides a plane into two parts and is the common boundary between them (see The American Heritage Dictionary of the English Language).
		This theorem seems quite obvious, however it is common knowledge that it is very difficult to it prove rigorously.
	mizar.uwb.edu.pl /jordan (414 words)

	Jordan Curve Theorem
		Although this theorem is obvious, it has proven to be extremely important to topologists and provides recreational math enthusiasts with answers to many classic problems.
		If we shade the simple closed curve that passes through house 1, water, house 2, and electricity, we find that house 3 is inside this curve and that it has not yet been joined to gas, which is outside the curve.
		According to the Jordan Curve Theorem, it is impossible to connect them without crossing the curve.
	britton.disted.camosun.bc.ca /jbjordan.htm (671 words)

	JCT 8/29/91 (Site not responding. Last check: 2007-11-06)
		An Elementary Proof Of The Jordan Curve Theorem
		The Jordan Curve Theorem states that every simple closed curve in the Euclidean plane
		C and on the fact that simple closed curves are not retracts of
	math.uc.edu /~bellh/JCT (1331 words)

	Regular polyhedra
		For a polygon it means that every closed curve in its interior can be continuously shrunk into a point while the deformation is being carried entirely inside the polygon.
		The Euler's Theorem, also known as the Euler's formula, deals with the relative number of faces, edges and vertices that a polyhedron (or polygon) may have.
		It actually follows from the proof of Theorem 1 that the Theorem applies to what might be (and is) called maps and to which I referred to as networks.
	www.cut-the-knot.org /do_you_know/polyhedra.shtml (1138 words)

	Jordan Curve Theorem
		I selected a theorem that is non-trivial and from the twentieth century (well, maybe it is late nineteenth century, but at least it's fairly recent.) The Jordan curve theorem.
		I think theorem proving tends to make this problem worse: when the computer produced something, the chance that the human doesn't understand it is bigger than when the human produced it.
		What I would like to see is a system that's kept rather simple because it has "factored out" the theorem proving and computer algebra parts: moreover it would be nice to have this in such a modular way that there could be multiple variants of "plug-ins" (theorem proving plug-ins, computer algebra plug-ins) to choose from.
	www.cs.ru.nl /~freek/jordan (4021 words)

	[No title]
		It is a celebrated theorem of Kuratowski (1929) that a graph is planar iff it contains no subgraph homeomorphic to the "forbidden subgraphs" K[3,3] or K[5], the 5-vertex complete graph.
		This theorem was also proven earlier in 1927-28 by Pontryagin, and later by Frink and Smith in 1930; see the paper by Kennedy and Quintas for the history (a bibliography is below).
		Thomassen has constructed a proof of the Jordan curve theorem based upon this (see his 1992 Monthly paper), as well as short graph-theoretical proofs of the the Jordan-Schonflies theorem, Hurwitz's theorem, etc., see his 1994 survey paper.
	www.math.niu.edu /~rusin/known-math/96/kuratowski (848 words)

	Digital Jordan Curve Theorems (ResearchIndex)
		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.
		We present a new, short proof of this theorem using induction on the Euclidean length of the curve.
		We also prove that the theorem holds with another topology on the digital plane but then only for a restricted class of Jordan curves.
	citeseer.ist.psu.edu /411748.html (242 words)

	March 2002 solution
		Specifically, a Jordan curve is described by the set of points (x(t), y(t)) where x(t) and y(t) are continuous real-valued functions on the unit interval (with t running from 0 to 1) such that
		The Jordan curve theorem (published by Camille Jordan in 1893) says that such a curve separates the plane into two regions, the interior (which is bounded by the curve) and the exterior (which is unbounded).
		Of course, if one imagines the Jordan curve to be the boundary of a pancake, such a result is obvious - the interior is the part you eat.
	mathcentral.uregina.ca /MP/previous2001/apr02sol.html (1179 words)

	Citebase - On the theorem converse to Jordan's curve theorem
		Theorem converse to Jordan's curve theorem says that \it if a compact set K has two complementary domains in R
		We show that the requirement of this theorem that \it all points of K were accessible from \it both complementary domains is surplus and prove one generalization of this theorem.
		Citation coverage and analysis is incomplete and hit coverage and analysis is both incomplete and noisy.
	citebase.eprints.org /cgi-bin/citations?id=oai:arXiv.org:math/0009164 (272 words)

	Earliest Known Uses of Some of the Words of Mathematics (J) (Site not responding. Last check: 2007-11-06)
		JORDAN CURVE appears in W. Osgood, "On the Existence of the Green's Function for the Most General Simply Connected Plane Region," Transactions of the American Mathematical Society, Vol.
		(July 1900): "By a Jordan curve is meant a curve of the general class of continuous curves without multiple points, considered by Jordan, Cours d'Analyse, vol.
		Jordan curve-theorem is found in D. Woodard, "On two-dimensional analysis situs with special reference to the Jordan curve-theorem," Fundamenta (1929).
	members.aol.com /jeff570/j.html (416 words)

	[No title] (Site not responding. Last check: 2007-11-06)
		The theorem says that a closed (f(a)=f(b)) continuous curve f > : [a,b] -> R2, with no double points, that is f : (a,b) -> R2 is > injective, will divide the plane into two connected components, one > which is unbounded and another which is bounded.
		> > Books on algebraic topology typically prove the generalized Jordan Curve > theorem: if you remove from the n-sphere S^n a subset C homeomorphic to > S^(n-1), what remains has two components whose common boundary is C. > The proof depends on homology theory.
		It is constructive in that the proof shows how to determine if a point in in the bounded or unbounded regions, and it show how to contruct a path joing two points that are in the same component of the complement.
	www.math.niu.edu /~rusin/papers/known-math/99/jordan_crv (419 words)

	Euler's Formula
		Otherwise, it is a Jordan curve and separates two faces; remove it and reduce F and E by one.
		Assume the formula holds for a smaller than F number of faces and consider a surface with number of faces equal to F. Pick two vertices on the boundary (left by the removed face) of the surface and connect them by a chain of internal edges.
		As usual, the Jordan curve theorem is involved, in the fact that a lake doubling back on itself creates an island.
	www.ics.uci.edu /~eppstein/junkyard/euler/all.html (5087 words)

	gustavolacerda: my philosophy about expressivity (Site not responding. Last check: 2007-11-06)
		I'll be the first to admit that if I refuse to prove the Jordan Curve Theorem, then I should choose a set of higher-level axioms that will "easily" prove not just the JCT, but also similar problems.
		I think that if the JCT was not provable, then the counterexamples to it would show where our intuition was "wrong".
		That happens to be the case (it's a theorem of Brouwer, I think), and that does not sound so different from having the possibility of a closed curve that divides the plane in n parts with n different from two.
	www.livejournal.com /users/gustavolacerda/156135.html (1856 words)

	Table of contents for Library of Congress control number 00048799 (Site not responding. Last check: 2007-11-06)
		Planar graphs and the Jordan Curve Theorem 2.2.
		The Jordan Curve Theorem and Kuratowski's Theorem in general topological spaces Chapter 3.
		Tree-width and the excluded grid theorem 205 7.2.
	www.loc.gov /catdir/toc/fy044/00048799.html (228 words)

	Citebase - A nonstandard proof of the Jordan curve theorem
		A nonstandard proof of the Jordan curve theorem
		A new elementary nonstandard proof of the Jordan curve theorem is given.
		The proof (the technical part consists of 4 pages) is self-contained, except for the Jordan theorem for polygons taken for granted.
	citebase.eprints.org /cgi-bin/citations?id=oai:arXiv.org:math/9608204 (327 words)

	Glossary
		Jordan Curve Theorem The Jordan Curve theorem states the following:
		Any simple closed curve in the plane partitions the plane into three disjoint connected sets such that the set that is in the curve is the boundary of both the other sets.
		This theorem is important for proving properties about polygons because they are simple closed curves.
	cgm.cs.mcgill.ca /~athens/cs507/Projects/2000/MS/diameter/node19.html (648 words)

	jordan curve theorem - OneLook Dictionary Search
		Jordan curve theorem : The American Heritage® Dictionary of the English Language [home, info]
		Jordan curve theorem : Infoplease Dictionary [home, info]
		Jordan curve theorem : PlanetMath Encyclopedia [home, info]
	www.onelook.com /?w=jordan+curve+theorem&ls=a (150 words)

	Classifications of Curve-Curve Intersections from the CAD/CAM Viewpoint (Site not responding. Last check: 2007-11-06)
		Intersection points of curves are the basic elements for many algorithms in the CAD/CAM area.
		In most circumstances, the key property of an intersection point is the behavior of the curves in a neighborhood of that intersection point.
		The movement is evaluated in the neighborhood of an intersection point under the constraint that the particle moves along one curve with respect to the other.
	csdl2.computer.org /persagen/DLAbsToc.jsp?resourcePath=/dl/proceedings/&toc=comp/proceedings/cgi/1996/7518/00/7518toc.xml&DOI=10.1109/CGI.1996.511865 (191 words)

	MSN Encarta - Dictionary - Jordan curve definition
		MSN Encarta - Dictionary - Jordan curve definition
		Search for "Jordan curve" in all of MSN Encarta
		closed curve: in mathematics, any simple closed curve, e.g.
	encarta.msn.com /encnet/features/dictionary/DictionaryResults.aspx?refid=1861623293 (65 words)

	Topics: J
		Idea: Every simple closed curve divides the plane into exactly two components.
		Jordan Normal Form of a Matrix > see matrices.
		Jordan Theory > see higher-dimensional gravity; kaluza-klein; scalar-tensor.
	www.phy.olemiss.edu /~luca/Topics/j.html (237 words)

	[No title]
		A useful (and hard!) theorem involves Jordan curves, which are continuous non-self-intersecting ("simple") closed curves in the plane.
		Find a cycle C in G -- in the drawing C must be a simple closed curve Now try to add other vertices and edges of G, creating more cycles, all with "insides" and "outsides".
		Find some edge that is forced to go from the inside of one cycle to its outside.
	www.skidmore.edu /~adean/MC3020409/Slides/MC302041130.ppt (166 words)

	physics - Simple polygon (Site not responding. Last check: 2007-11-06)
		A simple polygon is a polygon which does not intersect itself anywhere.
		These are also called Jordan polygons, because the Jordan curve theorem can be used to prove that such a polygon divides the plane into two regions, the region inside it and the region outside it.
		A polygon that is not simple is a complex polygon, and does not always have a well-defined inside and outside.
	www.physicsdaily.com /physics/Simple_polygon (243 words)

	Dudley W. Woodard, Mathematician of the African Diaspora
		Woodard retired in 1947 and died July 1, 1965 in his home in Cleveland Ohio.
		The first research paper published in an acredited mathematics journal by an african american is the first of two papers by Dudley Weldon Woodard, On two dimensional analysis situs with special reference to the Jordan Curve Theorem.
		Woodard, D. On two dimensional analysis situs with special reference to the Jordan Curve Theorem, Fundamenta Mathematicae 13 (1929), 121-145.
	www.math.buffalo.edu /mad/PEEPS/woodard_dudleyw.html (697 words)

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