OPTICS algorithm - Revision history

94.216.214.38: Undid revision 638967681 by ClueBot NG (talk)

2014-12-21T08:27:54Z

Undid revision 638967681 by ClueBot NG (talk)

← Older revision		Revision as of 10:27, 21 December 2014
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	~~[[File:MobiusF.PNG\|297px\|right\|thumb\|A Möbius band is a non-orientable~~ I~~-bundle. The dark line is the base for a set of transversal lines that are [[homeomorphic]] to the fiber~~ and ~~that each touch the edge of the band twice.]]~~		I am Jeffery and was born on 2 November 1988. My hobbies are Computer programming and Volleyball.<br><br>Feel free to surf to my web page: [http://mediaspydr.com/members/margemcdougal/activity/519553/ вы можете посмотреть здесь]
	~~[[File:Hopf_band_wikipedia.png\|right\|thumb\|An annulus is an orientable I-bundle. This example is embedded in 3-space with an even number of twists\|200px]]~~

	~~[[File:MxS1.PNG\|right\|thumb\|This image represents the twisted I-bundle over the~~ 2~~-torus, which is also fibered as a Möbius strip times the circle~~. ~~So, this space is also a [[circle bundle]]\|200px]]~~

	~~In mathematics, an '''I-bundle''' is a [[fiber bundle]] whose fiber is an [[interval (mathematics)\|interval]]~~ and ~~whose base is a [[manifold]]~~. ~~Any kind of interval, open, closed, semi-open, semi-closed, open-bounded, compact, even [[Line (mathematics)#Ray\|ray]]s, can be the fiber.~~

	~~Two simple examples of '''I-bundles''' are the [[Annulus (mathematics)\|annulus]] and the [[Möbius band]], the only two possible '''I-bundles''' over the circle~~ <~~math~~>~~\scriptstyle S^1~~<~~/math~~>~~. The annulus is a trivial or untwisted bundle because it corresponds~~ to the [[Cartesian product]] <math>\scriptstyle S^1\times I</math>, and the Möbius band is a non-trivial or twisted bundle. Both bundles are [[2-manifold]]s, but the annulus is an [[orientable manifold]] while the Möbius band is a [[non-orientable manifold]].

	~~Curiously, there are only two kinds of '''I-bundles''' when the base manifold is any [[surface]] but the [[Klein bottle]] <math>\scriptstyle K</math>. That surface has three I-bundles~~: ~~the trivial bundle <math>\scriptstyle K\times I</math> and two twisted bundles.~~

	Together with the [[Seifert fiber space]]s, '''I-bundles''' are fundamental elementary building blocks for the description of three dimensional spaces. These observations are simple well known facts on elementary [[3-manifold]]s.

	[[Line bundle]]s are both '''I-bundles''' and [[vector bundle]]s of rank one. When considering '''I-bundles''', one is interested mostly in their [[topological property\|topological properties]] and not their possible vector properties, as we might be for [[line bundle]]s.

	~~==References==~~
	* Scott, Peter, "The geometries of 3-manifolds". ''Bulletin of the London Mathematical Society'' 15 (1983), number 5, 401–487.
	* Hempel, John, "3-manifolds", ''Annals of Mathematics Studies'', number 86, Princeton University Press (1976).

	~~==External links==~~
	* [http://~~www~~.~~math.lsu.edu~~/~~~kasten~~/~~LSUTalk.pdf Example of use of I-bundles], nice pdf-slide presentation by Jeff Boerner at Dept. of Math, University of Iowa.~~
	* [http://~~www.m-hikari.com~~/imf-password2008/5-8-2008/casaliIMF5-8-2008.pdf I-bundles over the Klein-Bottle], "elementary" work on the orientable I-bundle over ''K'', by Maria Rita Casali, Dipartimento di Matematica Pura e Applicata, Università di Modena e Reggio Emilia.

	~~[[Category:Fiber bundles]]~~
	~~[[Category:Geometric topology]]~~
	~~[[Category:3-manifolds]~~]

2602:306:CD4A:D330:582A:1FD:FD8A:2064: /* Basic idea */

2013-10-31T23:26:30Z

Basic idea

← Older revision		Revision as of 01:26, 1 November 2013
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	~~My name~~ is ~~Omer Gardin but everybody calls me Omer~~. I'~~m from Canada~~. I'~~m studying at~~ the ~~high school~~ (~~final year~~) and I ~~play~~ the ~~Trumpet for 9 years~~. ~~Usually~~ I ~~choose songs from~~ the ~~famous films~~ :D. <br>I ~~have~~ two ~~brothers~~. I ~~like Amateur radio~~, ~~watching TV~~ (~~Doctor Who~~) ~~and Crocheting~~.~~<br><br>Also visit my page~~ [http://~~oumma~~.com/~~profil~~/vs-~~20 похудеть за дня~~]		[[File:MobiusF.PNG\|297px\|right\|thumb\|A Möbius band is a non-orientable I-bundle. The dark line is the base for a set of transversal lines that are [[homeomorphic]] to the fiber and that each touch the edge of the band twice.]]
			[[File:Hopf_band_wikipedia.png\|right\|thumb\|An annulus is an orientable I-bundle. This example is embedded in 3-space with an even number of twists\|200px]]

			[[File:MxS1.PNG\|right\|thumb\|This image represents the twisted I-bundle over the 2-torus, which is also fibered as a Möbius strip times the circle. So, this space is also a [[circle bundle]]\|200px]]

			In mathematics, an '''I-bundle''' is a [[fiber bundle]] whose fiber is an [[interval (mathematics)\|interval]] and whose base is a [[manifold]]. Any kind of interval, open, closed, semi-open, semi-closed, open-bounded, compact, even [[Line (mathematics)#Ray\|ray]]s, can be the fiber.

			Two simple examples of '''I-bundles''' are the [[Annulus (mathematics)\|annulus]] and the [[Möbius band]], the only two possible '''I-bundles''' over the circle <math>\scriptstyle S^1</math>. The annulus is a trivial or untwisted bundle because it corresponds to the [[Cartesian product]] <math>\scriptstyle S^1\times I</math>, and the Möbius band is a non-trivial or twisted bundle. Both bundles are [[2-manifold]]s, but the annulus is an [[orientable manifold]] while the Möbius band is a [[non-orientable manifold]].

			Curiously, there are only two kinds of '''I-bundles''' when the base manifold is any [[surface]] but the [[Klein bottle]] <math>\scriptstyle K</math>. That surface has three I-bundles: the trivial bundle <math>\scriptstyle K\times I</math> and two twisted bundles.

			Together with the [[Seifert fiber space]]s, '''I-bundles''' are fundamental elementary building blocks for the description of three dimensional spaces. These observations are simple well known facts on elementary [[3-manifold]]s.

			[[Line bundle]]s are both '''I-bundles''' and [[vector bundle]]s of rank one. When considering '''I-bundles''', one is interested mostly in their [[topological property\|topological properties]] and not their possible vector properties, as we might be for [[line bundle]]s.

			==References==
			* Scott, Peter, "The geometries of 3-manifolds". ''Bulletin of the London Mathematical Society'' 15 (1983), number 5, 401–487.
			* Hempel, John, "3-manifolds", ''Annals of Mathematics Studies'', number 86, Princeton University Press (1976).

			==External links==
			* [http://www.math.lsu.edu/~kasten/LSUTalk.pdf Example of use of I-bundles], nice pdf-slide presentation by Jeff Boerner at Dept. of Math, University of Iowa.
			* [http://www.m-hikari.com/imf-password2008/5-8-2008/casaliIMF5-8-2008.pdf I-bundles over the Klein-Bottle], "elementary" work on the orientable I-bundle over ''K'', by Maria Rita Casali, Dipartimento di Matematica Pura e Applicata, Università di Modena e Reggio Emilia.

			[[Category:Fiber bundles]]
			[[Category:Geometric topology]]
			[[Category:3-manifolds]]

en>Hirsutism at 14:41, 14 May 2012

2012-05-14T14:41:26Z

New page

My name is Omer Gardin but everybody calls me Omer. I'm from Canada. I'm studying at the high school (final year) and I play the Trumpet for 9 years. Usually I choose songs from the famous films :D. <br>I have two brothers. I like Amateur radio, watching TV (Doctor Who) and Crocheting.<br><br>Also visit my page [http://oumma.com/profil/vs-20 похудеть за дня]