Brocard's problem - Revision history

en>K9re11: removed Category:Number theory; added Category:Diophantine equations using HotCat

2014-11-13T13:38:15Z

removed Category:Number theory; added Category:Diophantine equations using HotCat

← Older revision		Revision as of 15:38, 13 November 2014
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	~~In [[knot theory]], each [[link (knot theory)\|link]]~~ and ~~[[knot (mathematics)\|knot]] can have an assigned '''knot thickness'''. Each realization of a link or knot has a thickness assigned to it~~. ~~The thickness τ of a link allows us to introduce a scale with respect to which we can then define the [[ropelength]] of a link.~~		Andrew Simcox is the name his mothers and fathers gave him and he completely enjoys this name. To climb is some thing I truly appreciate doing. Mississippi is where her house is but her husband desires them to move. I am currently a travel agent.<br><br>Feel free to surf to my web-site ... free psychic reading ([http://www.herandkingscounty.com/content/information-and-facts-you-must-know-about-hobbies click the up coming document])

	~~==Definition==~~
	~~There exist several possible definitions of thickness that coincide for smooth enough curves.~~

	~~=== Global radius of curvature ===~~
	~~The thickness~~ is ~~defined using the simpler concept of the local thickness τ(''x'')~~. ~~The local thickness at a point ''x'' on the link~~ is ~~defined as~~
	~~:<math> \tau(x)=\inf r(x,y,z),\, </math>~~
	where ''x'', ''y'', and ''z'' are points on the link, all distinct, and ''r''(''x'', ''y'', ''z'') is the radius of the circle that passes through all three points (''x'', ''y'', ''z''). From this definition we can deduce that the local thickness is ~~at most equal~~ to ~~the local radius of curvature~~.

	~~The thickness of~~ a ~~link is defined as~~
	~~:<math>\tau(L) = \inf \tau(x)~~.<~~/math~~><~~ref~~>~~[http://lcvmwww~~.~~epfl~~.~~ch/~lcvm/articles/43/info~~.~~html O. Gonzalez, J.H. Maddocks, "Global Curvature, Thickness and the Ideal Shapes of Knots", Proc. National Academy of Sciences of the USA 96~~ (~~1999) 4769-4773]</ref>~~

	~~=== Injectivity radius ===~~
	~~This definition ensures that a [[normal tube]] to the link with radius equal to τ(''L'') will not self intersect, and so we arrive at a "real world" knot made out of a thick string.<ref>~~[http://~~george~~.~~math~~.~~stthomas.edu~~/~~rawdon~~/~~Preprints/thickness.pdf Thickness of knots]~~
	~~R. A. Litherland, J. Simon, O. Durumeric,~~ and ~~E. Rawdon~~
	~~Topology Appl., 91(3): 233~~-~~244, 1999.~~]~~</ref>~~

	~~==References==~~
	~~<references/>~~

	~~{{DEFAULTSORT:Knot Thickness}}~~
	~~[[Category:Knot theory]]~~

	~~{{knottheory-stub}}~~

24.87.60.131: Added link to avoid confusion.

2013-08-04T02:17:13Z

Added link to avoid confusion.

← Older revision		Revision as of 04:17, 4 August 2013
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	~~Hello~~, ~~my name is Andrew~~ and ~~my wife doesn~~'~~t like~~ it at all. He is ~~an purchase clerk and it's some thing he truly appreciate~~. ~~It's not~~ a ~~typical thing but what I like performing~~ is ~~to climb but I don't have~~ the ~~time recently~~. ~~For~~ a ~~while I~~'~~ve been in Mississippi but now I~~'~~m contemplating other options~~.<~~br><br~~>~~Also visit my web blog~~ [http://~~chorokdeul~~.co.kr/~~index~~.~~php?document_srl~~=~~324263&mid~~=~~customer21 clairvoyants~~]		In [[knot theory]], each [[link (knot theory)\|link]] and [[knot (mathematics)\|knot]] can have an assigned '''knot thickness'''. Each realization of a link or knot has a thickness assigned to it. The thickness τ of a link allows us to introduce a scale with respect to which we can then define the [[ropelength]] of a link.

			==Definition==
			There exist several possible definitions of thickness that coincide for smooth enough curves.

			=== Global radius of curvature ===
			The thickness is defined using the simpler concept of the local thickness τ(''x''). The local thickness at a point ''x'' on the link is defined as
			:<math> \tau(x)=\inf r(x,y,z),\, </math>
			where ''x'', ''y'', and ''z'' are points on the link, all distinct, and ''r''(''x'', ''y'', ''z'') is the radius of the circle that passes through all three points (''x'', ''y'', ''z''). From this definition we can deduce that the local thickness is at most equal to the local radius of curvature.

			The thickness of a link is defined as
			:<math>\tau(L) = \inf \tau(x).</math><ref>[http://lcvmwww.epfl.ch/~lcvm/articles/43/info.html O. Gonzalez, J.H. Maddocks, "Global Curvature, Thickness and the Ideal Shapes of Knots", Proc. National Academy of Sciences of the USA 96 (1999) 4769-4773]</ref>

			=== Injectivity radius ===
			This definition ensures that a [[normal tube]] to the link with radius equal to τ(''L'') will not self intersect, and so we arrive at a "real world" knot made out of a thick string.<ref>[http://george.math.stthomas.edu/rawdon/Preprints/thickness.pdf Thickness of knots]
			R. A. Litherland, J. Simon, O. Durumeric, and E. Rawdon
			Topology Appl., 91(3): 233-244, 1999.]</ref>

			==References==
			<references/>

			{{DEFAULTSORT:Knot Thickness}}
			[[Category:Knot theory]]

			{{knottheory-stub}}

en>Helpful Pixie Bot: ISBNs (Build KE)

2012-05-08T04:35:13Z

ISBNs (Build KE)

New page

Hello, my name is Andrew and my wife doesn't like it at all. He is an purchase clerk and it's some thing he truly appreciate. It's not a typical thing but what I like performing is to climb but I don't have the time recently. For a while I've been in Mississippi but now I'm contemplating other options.<br><br>Also visit my web blog [http://chorokdeul.co.kr/index.php?document_srl=324263&mid=customer21 clairvoyants]