History of string theory - Revision history

en>Tom.Reding: Gen fixes (page/s, endash, , et al., unicodify, and/or concising wikilinks, etc.), & ref cleanup using AWB

2015-01-04T06:22:33Z

Gen fixes (page/s, endash, &nbsp, et al., unicodify, and/or concising wikilinks, etc.), & ref cleanup using AWB

← Older revision		Revision as of 08:22, 4 January 2015
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	~~In the [[mathematics\|mathematical]] field of [[topology]], a~~ ''~~'regular homotopy''' refers to a special kind of [[homotopy]] between [[immersion (mathematics)\|immersion]]~~s ~~of one [[manifold]]~~ in ~~another~~. ~~The homotopy must be a 1-parameter family of immersions.~~		Friends contact him Royal Seyler. Camping is some thing that I've carried out for years. For years she's been living in Kansas. Bookkeeping is what she does.<br><br>My page :: [http://enjoythisworld.com/UserProfile/tabid/464/userId/33651/Default.aspx extended auto warranty]

	~~Similar to [[homotopy class]]es, one defines two immersions to be in the same regular homotopy class if there exists a regular homotopy between them. Regular homotopy for immersions~~ is ~~similar to [[Homotopy#Isotopy\|isotopy]] of embeddings: they are both restricted types of homotopies~~. ~~Stated another way, two continuous functions~~ <~~math~~>~~f,g : M \to N~~<~~/math~~> ~~are homotopic if they represent points in the same path-components of the mapping space <math>C(M,N)</math>, given the~~ [~~[compact-open topology]]. The '''space of immersions''' is the subspace of <math>C(M,N)</math> consisting of immersions, denote it by <math>Imm(M,N)</math>. Two immersions <math>f,g~~:~~M \to N<~~/~~math> are '''regularly homotopic''' if they represent points in the same path-component of <math>Imm(M,N)<~~/~~math>.~~

	~~== Examples ==~~
	~~[[File:Winding Number Around Point~~.~~svg\|thumb\|300px\|This curve has [[total curvature]] 6''π'', and [[turning number]] 3.]]~~
	The '''Whitney–Graustein theorem''' {{anchor\|Whitney–Graustein theorem}} classifies the regular homotopy classes of a circle into the plane; two immersions are regularly homotopic if and only if they have the same [[turning number]] – equivalently, [[total curvature]]; equivalently, if and only if their [[Gauss map]]s have the same degree/~~[[winding number]].~~

	~~[[File:MorinSurfaceFromTheTop.PNG\|thumb\|Smale's classification of immersions of spheres shows that [[sphere eversion]]s exist, which can be realized via this [[Morin surface]].]]~~
	~~[[Stephen Smale]] classified the regular homotopy classes of a ''k''-sphere immersed in <math>\mathbb R^n<~~/math> – they are classified by homotopy groups of [[Stiefel manifold]]s, which is a generalization of the Gauss map, with here ''k'' partial derivatives not vanishing. A corollary of his work is that there is only one regular homotopy class of a ''2''-sphere immersed in <math>\mathbb R^3</math>. In particular, this means that [[sphere eversion]]s exist, i.e. one can turn the 2-sphere "inside-out".

	~~Both of these examples consist of reducing regular homotopy to homotopy; this has subsequently been substantially generalized in the [[homotopy principle]] (or ''h''-principle) approach.~~

	~~==References==~~
	*[[Hassler Whitney]], ''[http://~~www.numdam.org~~/~~numdam-bin~~/~~item?id=CM_1937__4__276_0 On regular closed curves in the plane]''~~. ~~[[Compositio Mathematica]], 4 (1937), p. 276–284~~
	*Stephen Smale, ''A classification of immersions of the two-sphere''. [[Transactions of the American Mathematical Society\|Trans. Amer. Math. Soc.]] 90 1958 281–290.
	*Stephen Smale, ''The classification of immersions of spheres in Euclidean spaces''. [[Annals of Mathematics\|Ann. of Math.]] (2) 69 1959 327–344.

	~~[[Category:Differential topology]]~~
	~~[[Category:Algebraic topology]~~]

en>Omnipaedista: provide a lead section

2013-10-09T06:08:13Z

provide a lead section

← Older revision		Revision as of 08:08, 9 October 2013
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	~~Nice~~ to ~~meet you~~, ~~my title~~ is ~~Ling~~ and ~~I totally dig~~ that ~~title~~. ~~Bookkeeping has been his working day occupation for~~ a ~~whilst. What she enjoys performing~~ is ~~to perform croquet but she hasn't produced~~ a ~~dime~~ with it. ~~Kansas~~ is ~~exactly where her home~~ is ~~but she needs to move because~~ of ~~her family.~~<br><br>~~Feel free~~ to ~~surf to my web page ::~~ [http://~~109~~.~~230~~.~~254.127~~/~~index.php~~?~~mod~~=~~users~~&~~action=view&id=29690 extended car warranty~~]		In the [[mathematics\|mathematical]] field of [[topology]], a '''regular homotopy''' refers to a special kind of [[homotopy]] between [[immersion (mathematics)\|immersion]]s of one [[manifold]] in another. The homotopy must be a 1-parameter family of immersions.

			Similar to [[homotopy class]]es, one defines two immersions to be in the same regular homotopy class if there exists a regular homotopy between them. Regular homotopy for immersions is similar to [[Homotopy#Isotopy\|isotopy]] of embeddings: they are both restricted types of homotopies. Stated another way, two continuous functions <math>f,g : M \to N</math> are homotopic if they represent points in the same path-components of the mapping space <math>C(M,N)</math>, given the [[compact-open topology]]. The '''space of immersions''' is the subspace of <math>C(M,N)</math> consisting of immersions, denote it by <math>Imm(M,N)</math>. Two immersions <math>f,g:M \to N</math> are '''regularly homotopic''' if they represent points in the same path-component of <math>Imm(M,N)</math>.

			== Examples ==
			[[File:Winding Number Around Point.svg\|thumb\|300px\|This curve has [[total curvature]] 6''π'', and [[turning number]] 3.]]
			The '''Whitney–Graustein theorem''' {{anchor\|Whitney–Graustein theorem}} classifies the regular homotopy classes of a circle into the plane; two immersions are regularly homotopic if and only if they have the same [[turning number]] – equivalently, [[total curvature]]; equivalently, if and only if their [[Gauss map]]s have the same degree/[[winding number]].

			[[File:MorinSurfaceFromTheTop.PNG\|thumb\|Smale's classification of immersions of spheres shows that [[sphere eversion]]s exist, which can be realized via this [[Morin surface]].]]
			[[Stephen Smale]] classified the regular homotopy classes of a ''k''-sphere immersed in <math>\mathbb R^n</math> – they are classified by homotopy groups of [[Stiefel manifold]]s, which is a generalization of the Gauss map, with here ''k'' partial derivatives not vanishing. A corollary of his work is that there is only one regular homotopy class of a ''2''-sphere immersed in <math>\mathbb R^3</math>. In particular, this means that [[sphere eversion]]s exist, i.e. one can turn the 2-sphere "inside-out".

			Both of these examples consist of reducing regular homotopy to homotopy; this has subsequently been substantially generalized in the [[homotopy principle]] (or ''h''-principle) approach.

			==References==
			*[[Hassler Whitney]], ''[http://www.numdam.org/numdam-bin/item?id=CM_1937__4__276_0 On regular closed curves in the plane]''. [[Compositio Mathematica]], 4 (1937), p. 276–284
			*Stephen Smale, ''A classification of immersions of the two-sphere''. [[Transactions of the American Mathematical Society\|Trans. Amer. Math. Soc.]] 90 1958 281–290.
			*Stephen Smale, ''The classification of immersions of spheres in Euclidean spaces''. [[Annals of Mathematics\|Ann. of Math.]] (2) 69 1959 327–344.

			[[Category:Differential topology]]
			[[Category:Algebraic topology]]

en>Lambiam: /* 1968-1974: Dual resonance model */ wikilink

2012-07-21T20:16:34Z

1968-1974: Dual resonance model: wikilink

New page

Nice to meet you, my title is Ling and I totally dig that title. Bookkeeping has been his working day occupation for a whilst. What she enjoys performing is to perform croquet but she hasn't produced a dime with it. Kansas is exactly where her home is but she needs to move because of her family.<br><br>Feel free to surf to my web page :: [http://109.230.254.127/index.php?mod=users&action=view&id=29690 extended car warranty]