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2014-11-16T07:36:45Z

top: Added to introduction to make it a more adequate summary of the article to comply with WP:LEAD

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	In [~~[topology]~~], ~~a branch~~ of ~~mathematics~~, ~~'''collapse''' is~~ a ~~concept due~~ to [[J. ~~H. C. Whitehead]~~].<~~ref name="Whitehead"~~>~~Whitehead~~, J.H.C. ~~(1938) ''Simplical spaces~~, ~~nuclei~~ and ~~m-groups''~~, ~~Proceedings~~ of the ~~London Mathematical Society 45, pp 243–327~~</~~ref>~~		Do you expeгience feeling juѕt like you might be lethargic the entire [http://otorongo.com.au/bba/uncategorized/vigrx-plus-in-uae-pharmacy-steps-to-reaching-the-correct-nutritious-needs/ vigrx plus growth chart] day? If so, it can ƅe what you will be having. It's important to offer the corrеct vitamins and minerals which means you haνe poաer.<br><br>Careful гeading through of food tags is vital. Decreased excess fat merсhandisе can cеrtainly still include great amounts of sea salt, sweets or other ƅad preserѵatives. Very processed fߋodѕ are not best for weight lоss. 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	== ~~Definition~~ ==

	~~Let~~ <~~math~~>K<~~/math~~> be an [~~[abstract simplicial complex]~~].

	~~Suppose that~~ <~~math~~> ~~\tau, \sigma \in K~~<~~/math~~> ~~such that~~ the ~~following two conditions are satisfied:~~

	~~(i)~~ <~~math~~> ~~\tau \subset \sigma~~<~~/math~~>, in ~~particular <math> \dim \tau <\dim~~ ~~\sigma<~~/~~math>;~~

	~~(ii) <math> \sigma<~~/~~math> is a maximal face of K~~ and ~~no other maximal face of K contains~~ <~~math~~> ~~\tau~~ <~~/math>,~~

	~~then <math> \tau </math~~> ~~is called a '''free face~~'''.

	~~A simplicial '''collapse'''~~ of K is ~~the removal of all simplices~~ <~~math~~>~~\gamma~~<~~/math~~> ~~such~~ that <~~math~~> ~~\tau \subseteq\gamma \subseteq \sigma~~<~~/math~~>, ~~provided that~~ <~~math~~>~~\tau~~<~~/math~~> ~~is a free face. If additionally we have dim τ = dim σ-1~~, ~~then this is called an '''elementary collapse'''.~~

	~~A simplicial complex that has a collapse~~ to a ~~point is called '''collapsible''~~'. ~~Every collapsible complex is contractible~~, ~~but~~ the ~~converse~~ is not ~~true.~~

	~~This definition can be extended~~ to ~~[[CW-complex]]es and is the basis for the concept of [[simple-homotopy equivalence]]~~.<~~ref~~>~~Cohen~~, M.M. ~~(1973) ''A Course in Simple-Homotopy Theory'', Springer-Verlag New York<~~/~~ref>~~

	== ~~Examples~~==
	* Complexes that do not have a free face cannot be collapsible. ~~Two such interesting examples are [[RH Bing\|Bing]]'s [[house with two rooms]]~~ and ~~Zeeman's [[Dunce hat (topology)\|dunce hat]]; they are [[contractible]] (homotopy equivalent to a point)~~, ~~but not collapsible~~.
	* Any ''n''-dimensional [[PL manifold]] that is collapsible is in fact piecewise-linearly isomorphic to ~~an ''n''-ball~~.~~<ref name="Whitehead" />~~

	~~== References ==~~
	~~{{reflist}}~~

	~~==See also==~~
	*[[Shelling (topology)]]
	~~[[Category:Algebraic topology]]~~


	~~{{topology-stub}}~~

209.101.171.115: /* Advantages & disadvantages */ rm worthless link

2013-02-20T22:30:22Z

Advantages & disadvantages: rm worthless link

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Juice frequently has added sweeteners~~, ~~whilst fresh fruit has [http://Dict~~.~~leo.org/?search=all~~-~~natural all-natural~~] ~~sugar. The amount of sugarѕ in many fruit drinks~~ is ~~beуond the amount of sweets~~ in ~~sߋft drinks~~. ~~Βesidеs getting naturallу wondeгful, fruits offers [http:~~/~~/www.dict.cc/?s~~=~~beneficial beneficial~~] nutrients like vitamin supplements, vitamins and minerals, and fіber. These could decrease the chance of seveгe ailments liҝe cardiovascular dіseɑse.<br><br>Mentioned prеviously in the beginning of this aɡen vigrx plus di indonesia - [~~http~~:~~//www.Clickkar.in/author/jau47/ http://www.clickkar.in~~] - guiԁe, the guidelines stated earlіer will assist you in imρroving your way of living. This article must have given you much more cߋmprehensіon of thе field of nutrients. Think around this article's guidance and try out theѕe recommendations that ѕeem specifically appropriate for you.		In [[topology]], a branch of mathematics, '''collapse''' is a concept due to [[J. H. C. Whitehead]].<ref name="Whitehead">Whitehead, J.H.C. (1938) ''Simplical spaces, nuclei and m-groups'', Proceedings of the London Mathematical Society 45, pp 243–327</ref>

			== Definition ==

			Let <math>K</math> be an [[abstract simplicial complex]].

			Suppose that <math> \tau, \sigma \in K</math> such that the following two conditions are satisfied:

			(i) <math> \tau \subset \sigma</math>, in particular <math> \dim \tau <\dim \sigma</math>;

			(ii) <math> \sigma</math> is a maximal face of K and no other maximal face of K contains <math> \tau </math>,

			then <math> \tau </math> is called a '''free face'''.

			A simplicial '''collapse''' of K is the removal of all simplices <math>\gamma</math> such that <math> \tau \subseteq\gamma \subseteq \sigma</math>, provided that <math>\tau</math> is a free face. If additionally we have dim τ = dim σ-1, then this is called an '''elementary collapse'''.

			A simplicial complex that has a collapse to a point is called '''collapsible'''. Every collapsible complex is contractible, but the converse is not true.

			This definition can be extended to [[CW-complex]]es and is the basis for the concept of [[simple-homotopy equivalence]].<ref>Cohen, M.M. (1973) ''A Course in Simple-Homotopy Theory'', Springer-Verlag New York</ref>

			== Examples==
			* Complexes that do not have a free face cannot be collapsible. Two such interesting examples are [[RH Bing\|Bing]]'s [[house with two rooms]] and Zeeman's [[Dunce hat (topology)\|dunce hat]]; they are [[contractible]] (homotopy equivalent to a point), but not collapsible.
			* Any ''n''-dimensional [[PL manifold]] that is collapsible is in fact piecewise-linearly isomorphic to an ''n''-ball.<ref name="Whitehead" />

			== References ==
			{{reflist}}

			==See also==
			*[[Shelling (topology)]]
			[[Category:Algebraic topology]]


			{{topology-stub}}

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