Instantaneous phase - Revision history

en>Omnipaedista: rm merge-tag per WP:DRIVEBYTAGGING

2014-07-26T06:05:58Z

← Older revision		Revision as of 08:05, 26 July 2014
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	~~[[Image:Join.svg\|right\|thumb\|Geometric join of two [[line segment]]s. The original spaces are shown in green~~ and ~~blue~~. ~~The join~~ is ~~a three-dimensional solid in gray.]]~~		Greetings! I am Marvella and I really feel comfy when people use the complete name. Puerto Rico is where he's been living for years and he will never move. It's not a typical thing but what she likes doing is base leaping and now she is attempting to earn cash with it. He utilized to be unemployed but now he is a pc operator but his marketing never comes.<br><br>my website - [http://nuvem.tk/altergalactica/NumbersdbEstepvu nuvem.tk]
	~~In [[topology]], a field of [[mathematics]], the '''join'~~'~~' of two [[topological space]]~~s ~~''A''~~ and '~~'B'', often denoted by <math>A\star B</math>,~~ is ~~defined to be the [[quotient space]]~~
	~~:<math> (A \times B \times I) / R, \, </math>~~
	~~where ''I'' is the [[interval (mathematics)\|interval]] [0, 1] and ''R'' is the [[equivalence relation]] generated by~~
	~~:<math> (a, b_1, 0) \sim (a, b_2, 0) \quad\mbox{for all } a \in A \mbox{ and } b_1,b_2 \in B,</math>~~
	~~:<math> (a_1, b, 1) \sim (a_2, b, 1) \quad\mbox{for all } a_1,a_2 \in A \mbox{ and } b \in B.</math>~~
	~~At the endpoints, this collapses <math>A\times B\times \{0\}</math> to <math>A</math>~~ and ~~<math>A\times B\times \{1\}</math> to <math>B</math>.~~

	~~Intuitively, <math>A\star B</math>~~ is ~~formed by taking the [[disjoint union (topology)\|disjoint union]] of the two spaces and attaching a line segment joining every point in ''A''~~ to ~~every point in ''B''~~.

	~~==Properties==~~
	* The join is homeomorphic to ~~sum of cartesian products of cones over spaces and spaces itself, where sum is taken over [[cartesian product]] of spaces:~~
	~~:<math>A\star B\cong C(A)\times B\cup_{A\times B} C(B)\times A</math>~~
	~~and is [[homotopy equivalent]] to suspension of [[smash product]] of spaces:~~
	~~:<math>A\star B\simeq \Sigma(A\wedge B)</math>~~

	~~==Examples==~~
	* The join of subsets of ''n''-dimensional Euclidean space ''A'' and ''B'' is ~~[[homotopy equivalent]] to the space of paths in ''n''-dimensional [[Euclidean space]], beginning in ''A'' and ending in ''B''.~~
	* The join of a ~~space ''X'' with a one-point space is called the [[cone (topology)\|cone]] ''CX'' of ''X''~~.
	* The join of a space ''X'' with <~~math~~>~~S^0~~<~~/math~~> ~~(the 0~~-~~dimensional [[sphere]], or, the [[discrete space]] with two points) is called the [[Suspension (topology)\|suspension]] <math>SX</math> of ''X''.~~
	* The join of the spheres <math>S^n</math> and <math>S^m</math> is the sphere <math>S^{n+m+1}</math>.

	~~==See also==~~
	*[[Cone (topology)]]
	*[[Suspension (topology)]]

	~~==References==~~
	*[[Allen Hatcher\|Hatcher, Allen]], [http://~~www~~.~~math.cornell.edu/~hatcher~~/AT/~~ATpage~~.~~html ''Algebraic topology.''] Cambridge University Press, Cambridge, 2002. xii+544 pp. ISBN 0-521-79160-X and ISBN 0-521-79540-0~~
	*{{PlanetMath attribution\|id=3985\|title=Join}}

	~~[[Category:Algebraic topology]]~~
	~~[[Category:Binary operations]~~]

en>Bob K: no content change. Cosmetic improvements.

2013-10-19T18:53:22Z

no content change. Cosmetic improvements.

← Older revision		Revision as of 20:53, 19 October 2013
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	~~Oscar~~ is ~~what my wife enjoys~~ to ~~contact me~~ and ~~I completely dig that title. Her spouse~~ and ~~her live~~ in ~~Puerto Rico but she will have~~ to ~~move~~ 1 ~~day or an additional~~. ~~My working day job~~ is a ~~meter reader~~. ~~Playing baseball~~ is the ~~pastime he will never quit doing~~.<br><br>~~My weblog :: at home std test~~ - [http://www.~~teenvalley~~.~~net~~/~~blog~~/~~17832 check it out~~] -		[[Image:Join.svg\|right\|thumb\|Geometric join of two [[line segment]]s. The original spaces are shown in green and blue. The join is a three-dimensional solid in gray.]]
			In [[topology]], a field of [[mathematics]], the '''join''' of two [[topological space]]s ''A'' and ''B'', often denoted by <math>A\star B</math>, is defined to be the [[quotient space]]
			:<math> (A \times B \times I) / R, \, </math>
			where ''I'' is the [[interval (mathematics)\|interval]] [0, 1] and ''R'' is the [[equivalence relation]] generated by
			:<math> (a, b_1, 0) \sim (a, b_2, 0) \quad\mbox{for all } a \in A \mbox{ and } b_1,b_2 \in B,</math>
			:<math> (a_1, b, 1) \sim (a_2, b, 1) \quad\mbox{for all } a_1,a_2 \in A \mbox{ and } b \in B.</math>
			At the endpoints, this collapses <math>A\times B\times \{0\}</math> to <math>A</math> and <math>A\times B\times \{1\}</math> to <math>B</math>.

			Intuitively, <math>A\star B</math> is formed by taking the [[disjoint union (topology)\|disjoint union]] of the two spaces and attaching a line segment joining every point in ''A'' to every point in ''B''.

			==Properties==
			* The join is homeomorphic to sum of cartesian products of cones over spaces and spaces itself, where sum is taken over [[cartesian product]] of spaces:
			:<math>A\star B\cong C(A)\times B\cup_{A\times B} C(B)\times A</math>
			and is [[homotopy equivalent]] to suspension of [[smash product]] of spaces:
			:<math>A\star B\simeq \Sigma(A\wedge B)</math>

			==Examples==
			* The join of subsets of ''n''-dimensional Euclidean space ''A'' and ''B'' is [[homotopy equivalent]] to the space of paths in ''n''-dimensional [[Euclidean space]], beginning in ''A'' and ending in ''B''.
			* The join of a space ''X'' with a one-point space is called the [[cone (topology)\|cone]] ''CX'' of ''X''.
			* The join of a space ''X'' with <math>S^0</math> (the 0-dimensional [[sphere]], or, the [[discrete space]] with two points) is called the [[Suspension (topology)\|suspension]] <math>SX</math> of ''X''.
			* The join of the spheres <math>S^n</math> and <math>S^m</math> is the sphere <math>S^{n+m+1}</math>.

			==See also==
			*[[Cone (topology)]]
			*[[Suspension (topology)]]

			==References==
			*[[Allen Hatcher\|Hatcher, Allen]], [http://www.math.cornell.edu/~hatcher/AT/ATpage.html ''Algebraic topology.''] Cambridge University Press, Cambridge, 2002. xii+544 pp. ISBN 0-521-79160-X and ISBN 0-521-79540-0
			*{{PlanetMath attribution\|id=3985\|title=Join}}

			[[Category:Algebraic topology]]
			[[Category:Binary operations]]

en>InverseHypercube: bold to italic terms

2012-06-05T00:31:11Z

bold to italic terms

New page

Oscar is what my wife enjoys to contact me and I completely dig that title. Her spouse and her live in Puerto Rico but she will have to move 1 day or an additional. My working day job is a meter reader. Playing baseball is the pastime he will never quit doing.<br><br>My weblog :: at home std test - [http://www.teenvalley.net/blog/17832 check it out] -