Quantum cascade laser - Revision history

en>Chandraknpatel: /* External links */

2015-01-09T22:39:56Z

External links

← Older revision		Revision as of 00:39, 10 January 2015
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	~~Ed is what people call me and my spouse doesn't like it at all~~. ~~To climb is some thing she would by no means give up. Alaska is the only place~~ I've ~~been residing~~ in but now I'm ~~considering~~ other ~~options~~. My working ~~day job is~~ an ~~invoicing~~ officer but I've ~~already applied for an additional 1~~.<br><br>~~Feel free to surf to~~ my page: [http://~~koreanyelp~~.~~com~~/index.php?document_srl=~~1798~~&mid=~~SchoolNews email psychic~~ readings]		Hi there, I am Andrew Berryhill. I've usually cherished living in Kentucky but now I'm contemplating other choices. Since he was 18 he's been working as an info officer but he ideas on altering it. What me and my family adore is performing ballet but I've been taking on new issues recently.<br><br>my page: [http://c045.danah.co.kr/home/index.php?document_srl=1356970&mid=qna tarot readings]

en>Hike395: Disambiguated: underdetermined → underdetermined system using Dab solver

2014-02-11T08:08:32Z

Disambiguated: underdetermined → underdetermined system using Dab solver

← Older revision		Revision as of 10:08, 11 February 2014
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	~~In [[mathematics]], and specifically in [[algebraic geometry]], the concept of '''irreducible component'''~~ is ~~used to make formal the idea that a set such as defined by the equation~~		Ed is what people call me and my spouse doesn't like it at all. To climb is some thing she would by no means give up. Alaska is the only place I've been residing in but now I'm considering other options. My working day job is an invoicing officer but I've already applied for an additional 1.<br><br>Feel free to surf to my page: [http://koreanyelp.com/index.php?document_srl=1798&mid=SchoolNews email psychic readings]

	~~:''XY'' = 0~~

	~~is the union of the two lines~~

	~~:''X'' = 0~~

	and

	:'~~'Y'' = 0~~.
	~~Thus an [[algebraic set]]~~ is ~~'''irreducible''' if it is not the union of two proper algebraic subsets~~. ~~It is a fundamental theorem of classical algebraic geometry that every algebraic set~~ is the union of a finite number of irreducible algebraic subsets (varieties) and that this decomposition is unique if one removes those subsets that are contained in another one. The elements of this unique decomposition are called '~~''irreducible components'''.~~

	~~This notion may be reformulated~~ in [[topology\|topological]] terms, using [[Zariski topology]], for which the closed sets are the subvarieties: an algebraic set is irreducible if it is not the union of two proper subsets that are closed for Zariski topology. This allows a generalization in topology, and, through it, to general [[scheme (mathematics)\|scheme]]s for which the above property of finite decomposition is not necessarily true.

	~~== In topology ==~~
	~~A [[Topological spaces\|topological space]] ''X'' is '''reducible''~~' ~~if it can be written as a union <math>X = X_1 \cup X_2</math> of two non-empty [[Closed set\|closed]] [[Subset\|proper subsets]] <math>X_1</math>, <math>X_2</math> of <math>X</math>~~.
	~~A topological space~~ is '''irreducible''' (or '''[[hyperconnected space\|hyperconnected]]''') if it is not reducible. Equivalently, all non empty [[open set\|open]] subsets of ''X'' are [[dense set\|dense]] or any two nonempty open sets have nonempty [[intersection (set theory)\|intersection]].

	A subset ''F'' of a topological space ''X'' is called irreducible or reducible, if ''F'' considered as a topological space via the [[subspace topology]] has the corresponding property in the above sense. ~~That is, <math>F</math> is reducible if it can be written as a union <math>F = (G_1\cap F)\cup(G_2\cap F)</math> where~~ <~~math~~>~~G_1,G_2~~<~~/math~~> ~~are closed subsets of <math>X</math>, neither of which contains <math>F</math>.~~

	An '''irreducible component''' of a [[topological space]] is a [[maximal element\|maximal]] [[Reduction (mathematics)\|irreducible]] [[proper subset\|subset]]. If a subset is irreducible, its [[closure (topology)\|closure]] is, so irreducible components are [[topological space\|closed]].

	~~== In algebraic geometry ==~~

	Every [[affine variety\|affine]] or [[projective variety\|projective algebraic set]] is defined as the set of the zeros of an [[ideal (ring theory)\|ideal]] in a [[polynomial ring]]. In this case, the irreducible components are the varieties associated to ~~the minimal primes over the ideal. This is this identification that allows~~ to ~~prove the uniqueness and the finiteness of the decomposition. This decomposition is strongly related with the [[primary decomposition]] of the ideal.~~

	In general [[scheme theory]], every scheme is the union of its irreducible components, but the number of components is not necessary finite. However, in most cases occurring in "practice", namely for all [[noetherian scheme]]s, there are finitely many irreducible components.

	~~== Examples ==~~
	~~The irreducibility depends much on actual topology on some set. For example, possibly contradicting the intuition, the real numbers (with their usual topology) are reducible~~: ~~for example the open interval (−1, 1) is not dense, its closure is the closed interval~~ [~~−1, 1].~~

	~~However, the notion is fundamental and more meaningful in [[algebraic geometry]]~~: ~~consider the variety~~

	~~:''X'' := {''x'' · ''y'' = 0}~~

	(a subset of the affine plane, ''x'' and ''y'' are the variables) endowed with the ''[[Zariski topology]]''. It is reducible, its irreducible components are its closed subsets {''x = 0''} and {''y'' = 0}.

	~~This can also be read off the coordinate ring ''k''[''x'', ''y'']~~/~~(''xy'') (if the variety is defined over a [[field (mathematics)\|field]] ''k''), whose minimal prime ideals are (''x'') and (''y'')~~.

	~~{{PlanetMath attribution\|id~~=~~1109\|title=irreducible}}~~
	~~{{PlanetMath attribution\|id=3107\|title~~=~~Irreducible component}}~~

	~~{{DEFAULTSORT:Irreducible Component}}~~
	~~[[Category:Algebraic geometry]]~~
	~~[[Category:General topology]]~~
	~~[[Category:Algebraic varieties]]~~

	~~[[fr:Dimension_de_Krull#Composantes_irr.C3.A9ductibles]~~]

en>Monkbot: Fix CS1 deprecated date parameter errors

2014-01-29T00:53:23Z

Fix CS1 deprecated date parameter errors

← Older revision		Revision as of 02:53, 29 January 2014
Line 1:		Line 1:
	Ed is ~~what people contact me~~ and ~~my wife doesn~~'~~t like~~ it ~~at all~~. ~~Invoicing~~ is ~~what I do. Her family members life~~ in ~~Alaska but her spouse wants them to move~~. The ~~preferred pastime~~ for ~~him~~ and ~~his kids~~ is ~~fashion and he~~'ll be ~~beginning something else along with~~ it.<br><br>~~Here~~ is ~~my blog psychics online~~ (~~[http:~~//~~clothingcarearchworth~~.~~com/index~~.~~php?document_srl~~=~~441551~~&~~mid~~=~~customer_review mouse click~~ the ~~following article~~])		In [[mathematics]], and specifically in [[algebraic geometry]], the concept of '''irreducible component''' is used to make formal the idea that a set such as defined by the equation

			:''XY'' = 0

			is the union of the two lines

			:''X'' = 0

			and

			:''Y'' = 0.
			Thus an [[algebraic set]] is '''irreducible''' if it is not the union of two proper algebraic subsets. It is a fundamental theorem of classical algebraic geometry that every algebraic set is the union of a finite number of irreducible algebraic subsets (varieties) and that this decomposition is unique if one removes those subsets that are contained in another one. The elements of this unique decomposition are called '''irreducible components'''.

			This notion may be reformulated in [[topology\|topological]] terms, using [[Zariski topology]], for which the closed sets are the subvarieties: an algebraic set is irreducible if it is not the union of two proper subsets that are closed for Zariski topology. This allows a generalization in topology, and, through it, to general [[scheme (mathematics)\|scheme]]s for which the above property of finite decomposition is not necessarily true.

			== In topology ==
			A [[Topological spaces\|topological space]] ''X'' is '''reducible''' if it can be written as a union <math>X = X_1 \cup X_2</math> of two non-empty [[Closed set\|closed]] [[Subset\|proper subsets]] <math>X_1</math>, <math>X_2</math> of <math>X</math>.
			A topological space is '''irreducible''' (or '''[[hyperconnected space\|hyperconnected]]''') if it is not reducible. Equivalently, all non empty [[open set\|open]] subsets of ''X'' are [[dense set\|dense]] or any two nonempty open sets have nonempty [[intersection (set theory)\|intersection]].

			A subset ''F'' of a topological space ''X'' is called irreducible or reducible, if ''F'' considered as a topological space via the [[subspace topology]] has the corresponding property in the above sense. That is, <math>F</math> is reducible if it can be written as a union <math>F = (G_1\cap F)\cup(G_2\cap F)</math> where <math>G_1,G_2</math> are closed subsets of <math>X</math>, neither of which contains <math>F</math>.

			An '''irreducible component''' of a [[topological space]] is a [[maximal element\|maximal]] [[Reduction (mathematics)\|irreducible]] [[proper subset\|subset]]. If a subset is irreducible, its [[closure (topology)\|closure]] is, so irreducible components are [[topological space\|closed]].

			== In algebraic geometry ==

			Every [[affine variety\|affine]] or [[projective variety\|projective algebraic set]] is defined as the set of the zeros of an [[ideal (ring theory)\|ideal]] in a [[polynomial ring]]. In this case, the irreducible components are the varieties associated to the minimal primes over the ideal. This is this identification that allows to prove the uniqueness and the finiteness of the decomposition. This decomposition is strongly related with the [[primary decomposition]] of the ideal.

			In general [[scheme theory]], every scheme is the union of its irreducible components, but the number of components is not necessary finite. However, in most cases occurring in "practice", namely for all [[noetherian scheme]]s, there are finitely many irreducible components.

			== Examples ==
			The irreducibility depends much on actual topology on some set. For example, possibly contradicting the intuition, the real numbers (with their usual topology) are reducible: for example the open interval (−1, 1) is not dense, its closure is the closed interval [−1, 1].

			However, the notion is fundamental and more meaningful in [[algebraic geometry]]: consider the variety

			:''X'' := {''x'' · ''y'' = 0}

			(a subset of the affine plane, ''x'' and ''y'' are the variables) endowed with the ''[[Zariski topology]]''. It is reducible, its irreducible components are its closed subsets {''x = 0''} and {''y'' = 0}.

			This can also be read off the coordinate ring ''k''[''x'', ''y'']/(''xy'') (if the variety is defined over a [[field (mathematics)\|field]] ''k''), whose minimal prime ideals are (''x'') and (''y'').

			{{PlanetMath attribution\|id=1109\|title=irreducible}}
			{{PlanetMath attribution\|id=3107\|title=Irreducible component}}

			{{DEFAULTSORT:Irreducible Component}}
			[[Category:Algebraic geometry]]
			[[Category:General topology]]
			[[Category:Algebraic varieties]]

			[[fr:Dimension_de_Krull#Composantes_irr.C3.A9ductibles]]

en>Khazar2: /* Growth */clean up, typos fixed: , → , using AWB

2012-05-16T23:12:47Z

Growth: clean up, typos fixed: , → , using AWB

New page

Ed is what people contact me and my wife doesn't like it at all. Invoicing is what I do. Her family members life in Alaska but her spouse wants them to move. The preferred pastime for him and his kids is fashion and he'll be beginning something else along with it.<br><br>Here is my blog psychics online ([http://clothingcarearchworth.com/index.php?document_srl=441551&mid=customer_review mouse click the following article])