Transparency (graphic) - Revision history

en>Jarble at 18:41, 6 March 2014

2014-03-06T18:41:16Z

← Older revision		Revision as of 20:41, 6 March 2014
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	~~[[File:Venn A subset B~~.~~svg\|150px\|thumb\|right\|''A'' is a subset of ''B''~~, ~~and ''B'' is a superset of ''A''~~.]]		Om , kan ni inte segrar. Människor verkar bedöma sin gällande lotto, lotterier, hästkapplöpning, poker, för att omnämna . Människor har spelat av antiken, Kina samt Rom, att saken där aktuell arbetsdagen. dagens ålder från burken, spelande äger tagit någon tillägg led befinna on-line. Du beskåda spelande överallt du . Metoden av onlinespel innerligt populärt men det kan innehava faror.<br><br>
	~~In [[mathematics]]~~, if ~~<math>A</math> is a [[subset]] of <math>B</math>~~, ~~then the '''inclusion map''' (also '''inclusion function'''~~, ~~'''insertion''', or '''canonical injection''')~~ ~~<ref>{{citation\| first1 = S~~. ~~\| last1 = Mac Lane \| first2 = G~~. \| last2 = Birkhoff \| title = Algebra \| year = 1967}}, page 5</ref> is the [[function (mathematics)\|function]] <math>\iota</math> that sends each element, <math>x</math> of <math>A</math> to <math>x</math>, treated as an element of <~~math~~>B<~~/math~~>:

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	~~A "hooked arrow"~~ <~~math~~>~~\hookrightarrow~~<~~/math~~> ~~is sometimes used~~ in ~~place of the function arrow above to denote an inclusion map~~.

	~~This and other analogous~~ [~~[injective~~]~~] functions~~ <~~ref~~>~~{{citation\| first = C. \| last = Chevalley \| title = Fundamental Concepts of Algebra \| year = 1956}}, page 1~~
	<~~/ref~~> ~~from [[substructure]]s are sometimes called '''''natural injections'''''~~.

	~~Given any [[morphism]] between [[object (category theory)\|objects]] ''X'' and ''Y'', if there is an inclusion map into the [[domain (mathematics)\|domain]]~~ <~~math~~>~~\iota : A\rightarrow X~~<~~/math~~>~~, then one can form the [[Function~~ (~~mathematics~~)~~#Restrictions and extensions\|restriction]] ''fi'' of ''f''~~. ~~In many instances~~, ~~one can also construct a canonical inclusion into the [[codomain]] ''R''→''Y'' known as the [[range (mathematics)\|range]] of ''f''~~.

	~~== Applications of inclusion maps ==~~
	~~Inclusion maps tend to be [[homomorphism]]s of [[algebraic structure]]s; thus~~, ~~such inclusion maps are [[embedding]]s~~. ~~More precisely, given a sub-structure closed under some operations, the inclusion map will be an embedding for tautological reasons, given the very definition by restriction of what one checks~~. ~~For example, for a binary operation~~ <~~math~~>~~\star~~<~~/math~~>, ~~to require that~~

	:<~~math~~>~~\iota(x\star y)=\iota(x)\star \iota(y)~~<~~/math~~>

	~~is simply to say that~~ <~~math~~>~~\star~~<~~/math~~> ~~is consistently computed in the sub-structure and the large structure. The case of a [[unary operation]] is similar; but one should also look at [[nullary]] operations~~, ~~which pick out a ''constant'' element~~. ~~Here the point is that [[closure (mathematics)\|closure]] means such constants must already be given in the substructure~~.

	~~Inclusion maps are seen in algebraic topology where if ''A'' is a [[strong deformation retract]] of ''X''~~, ~~the inclusion map yields an isomorphism between all homotopy groups (i~~.e. ~~is a [[Homotopy\|homotopy equivalence]])~~

	Inclusion maps in [[geometry]] come in different kinds: for example [[embedding]]s of [[submanifold]]s. [[Covariance and contravariance of vectors\|Contravariant]] objects such as [[differential form]]s ''restrict'' to submanifolds, ~~giving a mapping in the ''other direction''~~. ~~Another example, more sophisticated, is that of [[affine scheme]]s, for which the inclusions~~

	~~:''Spec(R/I)'' → ''Spec(R)''~~

	~~and~~

	~~:''Spec(R/I~~<~~sup~~>2<~~/sup~~>~~)'' → ''Spec(R)''~~

	~~may be different [[morphism]]s, where ''R'' is a [[commutative ring]] and ''I'' an [[ideal (ring theory)\|ideal]]~~.

	~~== See also ==~~
	*[[Cofibration]]
	*[[Identity function]]

	~~==Notes==~~
	~~{{reflist}}~~

	~~== References ==~~
	* {{citation\| first = C. ~~\| last = Chevalley \| title = Fundamental Concepts of Algebra \| publisher = Academic Press, New York \| year = 1956\| isbn = 0~~-12-~~172050-0}}~~.
	* {{citation\| first1 = S. ~~\| last1 = Mac Lane \| first2 = G~~. ~~\| last2 = Birkhoff~~ ~~\| title = Algebra \| publisher = AMS Chelsea Publishing~~, ~~Providence, Rhode Island \| year = 1967\| isbn = 0-8218-1646-2}}~~.

	~~{{DEFAULTSORT:Inclusion Map}}~~
	~~[[Category:Functions~~ and ~~mappings]]~~
	[~~[Category~~:~~Basic concepts in set theory]~~]

en>Kashmiri: Not a proper noun

2013-05-07T14:29:04Z

Not a proper noun

← Older revision		Revision as of 16:29, 7 May 2013
Line 1:		Line 1:
	~~Raul~~ is ~~the name his mothers~~ and ~~fathers gave him but individuals usually misspell it. As~~ a ~~man what he really likes is kayaking but he can~~'~~t make it his~~ [~~http:~~//~~Answers~~.~~yahoo~~.~~com/search/search_result?p~~=~~occupation&submit-go~~=~~Search+Y!+Answers occupation~~]~~. She's always loved living in Maryland. He is a manufacturing and planning officer and his salary has been truly satisfying. I'm not good at webdesign but you may want to check my website: http:~~//~~eu4wiki.de~~/~~index.php?title=Never_Lose_Your_Nya_Online_Svenska_Casinon_Again~~<br><br>		[[File:Venn A subset B.svg\|150px\|thumb\|right\|''A'' is a subset of ''B'', and ''B'' is a superset of ''A''.]]
			In [[mathematics]], if <math>A</math> is a [[subset]] of <math>B</math>, then the '''inclusion map''' (also '''inclusion function''', '''insertion''', or '''canonical injection''') <ref>{{citation\| first1 = S. \| last1 = Mac Lane \| first2 = G. \| last2 = Birkhoff \| title = Algebra \| year = 1967}}, page 5</ref> is the [[function (mathematics)\|function]] <math>\iota</math> that sends each element, <math>x</math> of <math>A</math> to <math>x</math>, treated as an element of <math>B</math>:

	~~Also visit my blog post~~: [~~http~~://~~eu4wiki~~.de/~~index~~.~~php?~~title=~~Never_Lose_Your_Nya_Online_Svenska_Casinon_Again nya online casinon på nätet~~]		:<math>\iota: A\rightarrow B, \qquad \iota(x)=x.</math>

			A "hooked arrow" <math>\hookrightarrow</math> is sometimes used in place of the function arrow above to denote an inclusion map.

			This and other analogous [[injective]] functions <ref>{{citation\| first = C. \| last = Chevalley \| title = Fundamental Concepts of Algebra \| year = 1956}}, page 1
			</ref> from [[substructure]]s are sometimes called '''''natural injections'''''.

			Given any [[morphism]] between [[object (category theory)\|objects]] ''X'' and ''Y'', if there is an inclusion map into the [[domain (mathematics)\|domain]] <math>\iota : A\rightarrow X</math>, then one can form the [[Function (mathematics)#Restrictions and extensions\|restriction]] ''fi'' of ''f''. In many instances, one can also construct a canonical inclusion into the [[codomain]] ''R''→''Y'' known as the [[range (mathematics)\|range]] of ''f''.

			== Applications of inclusion maps ==
			Inclusion maps tend to be [[homomorphism]]s of [[algebraic structure]]s; thus, such inclusion maps are [[embedding]]s. More precisely, given a sub-structure closed under some operations, the inclusion map will be an embedding for tautological reasons, given the very definition by restriction of what one checks. For example, for a binary operation <math>\star</math>, to require that

			:<math>\iota(x\star y)=\iota(x)\star \iota(y)</math>

			is simply to say that <math>\star</math> is consistently computed in the sub-structure and the large structure. The case of a [[unary operation]] is similar; but one should also look at [[nullary]] operations, which pick out a ''constant'' element. Here the point is that [[closure (mathematics)\|closure]] means such constants must already be given in the substructure.

			Inclusion maps are seen in algebraic topology where if ''A'' is a [[strong deformation retract]] of ''X'', the inclusion map yields an isomorphism between all homotopy groups (i.e. is a [[Homotopy\|homotopy equivalence]])

			Inclusion maps in [[geometry]] come in different kinds: for example [[embedding]]s of [[submanifold]]s. [[Covariance and contravariance of vectors\|Contravariant]] objects such as [[differential form]]s ''restrict'' to submanifolds, giving a mapping in the ''other direction''. Another example, more sophisticated, is that of [[affine scheme]]s, for which the inclusions

			:''Spec(R/I)'' → ''Spec(R)''

			and

			:''Spec(R/I<sup>2</sup>)'' → ''Spec(R)''

			may be different [[morphism]]s, where ''R'' is a [[commutative ring]] and ''I'' an [[ideal (ring theory)\|ideal]].

			== See also ==
			*[[Cofibration]]
			*[[Identity function]]

			==Notes==
			{{reflist}}

			== References ==
			* {{citation\| first = C. \| last = Chevalley \| title = Fundamental Concepts of Algebra \| publisher = Academic Press, New York \| year = 1956\| isbn = 0-12-172050-0}}.
			* {{citation\| first1 = S. \| last1 = Mac Lane \| first2 = G. \| last2 = Birkhoff \| title = Algebra \| publisher = AMS Chelsea Publishing, Providence, Rhode Island \| year = 1967\| isbn = 0-8218-1646-2}}.

			{{DEFAULTSORT:Inclusion Map}}
			[[Category:Functions and mappings]]
			[[Category:Basic concepts in set theory]]

en>Mblumber: Disambiguated: anti-aliasing → Spatial anti-aliasing (2) using Dab solver

2012-08-21T02:23:25Z

Disambiguated: anti-aliasing → Spatial anti-aliasing (2) using Dab solver

New page

Raul is the name his mothers and fathers gave him but individuals usually misspell it. As a man what he really likes is kayaking but he can't make it his [http://Answers.yahoo.com/search/search_result?p=occupation&submit-go=Search+Y!+Answers occupation]. She's always loved living in Maryland. He is a manufacturing and planning officer and his salary has been truly satisfying. I'm not good at webdesign but you may want to check my website: http://eu4wiki.de/index.php?title=Never_Lose_Your_Nya_Online_Svenska_Casinon_Again<br><br>

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