Transfer-matrix method - Revision history

en>Rcrumpf: /* References */

2014-02-28T15:40:56Z

References

← Older revision		Revision as of 17:40, 28 February 2014
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	~~In [[mathematics]], a '''coercive function''' is a function that "grows rapidly" at the extremes of the space on which it is defined~~. ~~Depending on the context~~		Friends call him Royal Seyler. The factor she adores most is to play handball but she can't make it her occupation. He presently life in Idaho and his mothers and fathers live nearby. His day job is a cashier and his salary has been truly fulfilling.<br><br>my weblog - [http://Ktva-Online.com/index.php?mod=users&action=view&id=12078 http://Ktva-Online.com/index.php?mod=users&action=view&id=12078]
	~~different exact definitions of this idea are in use.~~

	~~==Coercive vector fields ==~~
	~~A vector field ''f'' : '''R'''<sup>''n''</sup> → '''R'''<sup>''n''</sup>~~ is ~~called '''coercive''' if~~

	~~:<math>\frac{f(x) \cdot x}{\\| x \\|} \~~to ~~+ \infty \mbox{ as } \\| x \\| \to + \infty,</math>~~

	~~where "<math>\cdot</math>" denotes the usual [[dot product]] and <math>\\|x\\|</math> denotes the usual Euclidean [[norm (mathematics)\|norm]] of the vector ''x'~~'.

	~~A coercive vector field is~~ in ~~particular norm-coercive since~~
	~~<math>\\|f(x)\\| \geq (f(x) \cdot x) / \\| x \\|</math> for~~
	~~<math>x \in \mathbb{R}^n \setminus \{0\} </math>, by~~
	~~[[Cauchy–Schwarz inequality\|Cauchy Schwarz inequality]].~~
	~~However a norm-coercive mapping~~
	~~''f'' : '''R'''<sup>''n''</sup> → '''R'''<sup>''n''</sup>~~
	~~is not necessarily a coercive vector field. For instance~~
	~~the rotation~~
	~~''f'' : '''R'''<sup>''2''</sup> → '''R'''<sup>''2''</sup>, ''f(x) = (-x<sub>2</sub>, x<sub>1</sub>)''~~
	~~by 90° is a norm-coercive mapping which fails to be a coercive vector field since~~
	~~<math>f(x) \cdot x = 0</math> for every <math>x \in \mathbb{R}^2</math>.~~

	~~==Coercive operators~~ and ~~forms==~~
	~~A [[self-adjoint operator]] <math>A:H\to H,</math> where <math>H</math> is a real [[Hilbert space]], is called '''coercive''' if there exists a constant <math>c>0</math> such that~~

	~~: <math>\langle Ax, x\rangle \ge c\\|x\\|^2</math>~~

	~~for all <math>x</math> in <math>H~~.~~</math>~~

	~~A [[bilinear form]] <math>a:H\times H\to \mathbb R</math>~~ is ~~called '''coercive''' if there exists a constant <math>c>0</math> such that~~

	~~:<math>a(x, x)\ge c\\|x\\|^2</math>~~

	~~for all <math>x</math> in <math>H.</math>~~

	~~It follows from the [[Riesz representation theorem]] that any symmetric (<math>~~a~~(x, y)=a(y, x)</math> for all <math>x, y</math> in <math>H</math>), continuous (<math>\|a(x, y)\|\le K\\|x\\|\,\\|y\\|</math> for all <math>x, y</math> in <math>H</math> and some constant <math>K>0</math>)~~ and ~~coercive bilinear form <math>a</math>~~ has ~~the representation~~

	~~: <math>a(x, y)=\langle Ax, y\rangle</math>~~

	~~for some self-adjoint operator <math>A:H\to H,</math> which then turns out to be a coercive operator~~. ~~Also, given a coercive operator self-adjoint operator <math>A,</math> the bilinear form~~ <~~math~~>a<~~/math~~> ~~defined as above is coercive.~~

	~~One can also show that any self~~-~~adjoint operator <math>A:H\to H</math> is a coercive operator if and only if it is a coercive mapping~~
	~~(in the sense of coercivity of a vector field, where one has to replace the dot product with the more general inner product).~~
	~~The definitions of coercivity for vector fields, operators, and bilinear forms are closely related and compatible.~~

	~~==Norm-coercive mappings==~~
	~~A mapping~~
	~~<math>f~~ : ~~X \to X' </math> between two normed vectorspaces~~
	~~<math>(X, \\| \cdot \\|)<~~/~~math> and <math>(X', \\| \cdot \\|')<~~/~~math>~~
	~~is called '''norm~~-~~coercive''' iff~~

	~~:<math> \\|f(x)\\|' \to + \infty \mbox{ as } \\|x\\| \to +\infty </math>~~.

	~~More generally, a function <math>f : X \to X' </math> between two [[topological space]]s <math>X</math> and <math>X'<~~/~~math> is called '''coercive''' if for every [[compact space\|compact subset]] <math>K'</math> of <math>X'</math> there exists a compact subset <math>K</math> of <math>X</math> such that~~

	~~:<math>f (X \setminus K) \subseteq X' \setminus K'.</math>~~

	~~The [[function composition\|composition]] of a [[bijection\|bijective]] [[proper map]] followed by a coercive map is coercive~~.

	==~~(Extended valued) coercive functions~~==
	~~An (extended valued) function~~
	~~<math>f~~:~~\mathbb{R}^n \to \mathbb{R} \cup \{- \infty, + \infty\}<~~/~~math>~~
	~~is called '''coercive''' iff~~
	~~:<math> f(x) \to + \infty \mbox{ as } \\| x \\| \to + \infty.<~~/~~math>~~
	~~A realvalued coercive function <math>f:\mathbb{R}^n \to \mathbb{R} </math>~~
	~~is in particular norm-coercive. However a norm~~-~~coercive function~~
	~~<math>f:\mathbb{R}^n \to \mathbb{R} </math> is not necessarily coercive~~.
	~~For instance the identity function on <math> \mathbb{R} <~~/~~math> is norm-coercive~~
	~~but not coercive~~.

	==~~References==~~
	* {{cite book\| author=Renardy, Michael and Rogers, Robert C. \| title=An introduction to partial differential equations \| edition=Second edition \| publisher=Springer-Verlag \| location=New York, NY \| year=2004 \| pages=xiv+434 \| isbn=0-387-00444-0 }}
	* {{cite book
	~~\| last = Bashirov~~
	~~\| first = Agamirza E~~
	~~\| title = Partially observable linear systems under dependent noises~~
	~~\| publisher = Basel; Boston: Birkhäuser Verlag~~
	~~\| year = 2003~~
	~~\| pages =~~
	~~\| isbn = 0-8176-6999-X~~
	}}
	*{{cite book
	~~\| author2-link=Neil Trudinger~~
	~~\| first1=D.~~
	~~\| last1=Gilbarg~~
	~~\| first2=N.~~
	~~\| last2=Trudinger~~
	~~\| title = Elliptic partial differential equations of second order, 2nd ed~~
	~~\| publisher = Berlin; New York: Springer~~
	~~\| year = 2001~~
	~~\| pages =~~
	~~\| isbn = 3-540-41160-7~~
	}}

	~~{{PlanetMath attribution\|~~id=~~7154\|title=Coercive Function}}~~

	~~[[Category:Functional analysis]]~~
	~~[[Category:General topology]]~~
	~~[[Category:Types of functions]~~]

en>Apolisi4 at 17:07, 2 February 2014

2014-02-02T17:07:37Z

← Older revision		Revision as of 19:07, 2 February 2014
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	~~The writer~~ is ~~called Araceli Gulledge~~. ~~He currently life~~ in ~~Idaho~~ and ~~his mothers and fathers live nearby~~. ~~Bookkeeping~~ is ~~what she does~~. ~~Playing croquet~~ is ~~something I will never give up~~.<br><br>~~my web blog~~; [~~http~~://~~www~~.~~visionassetmanagement~~.~~com~~/~~ActivityFeed~~/~~MyProfile~~/~~tabid~~/60/~~userId~~/~~2039~~/~~Default~~.~~aspx http~~://~~www~~.~~visionassetmanagement~~.~~com~~]		In [[mathematics]], a '''coercive function''' is a function that "grows rapidly" at the extremes of the space on which it is defined. Depending on the context
			different exact definitions of this idea are in use.

			==Coercive vector fields ==
			A vector field ''f'' : '''R'''<sup>''n''</sup> → '''R'''<sup>''n''</sup> is called '''coercive''' if

			:<math>\frac{f(x) \cdot x}{\\| x \\|} \to + \infty \mbox{ as } \\| x \\| \to + \infty,</math>

			where "<math>\cdot</math>" denotes the usual [[dot product]] and <math>\\|x\\|</math> denotes the usual Euclidean [[norm (mathematics)\|norm]] of the vector ''x''.

			A coercive vector field is in particular norm-coercive since
			<math>\\|f(x)\\| \geq (f(x) \cdot x) / \\| x \\|</math> for
			<math>x \in \mathbb{R}^n \setminus \{0\} </math>, by
			[[Cauchy–Schwarz inequality\|Cauchy Schwarz inequality]].
			However a norm-coercive mapping
			''f'' : '''R'''<sup>''n''</sup> → '''R'''<sup>''n''</sup>
			is not necessarily a coercive vector field. For instance
			the rotation
			''f'' : '''R'''<sup>''2''</sup> → '''R'''<sup>''2''</sup>, ''f(x) = (-x<sub>2</sub>, x<sub>1</sub>)''
			by 90° is a norm-coercive mapping which fails to be a coercive vector field since
			<math>f(x) \cdot x = 0</math> for every <math>x \in \mathbb{R}^2</math>.

			==Coercive operators and forms==
			A [[self-adjoint operator]] <math>A:H\to H,</math> where <math>H</math> is a real [[Hilbert space]], is called '''coercive''' if there exists a constant <math>c>0</math> such that

			: <math>\langle Ax, x\rangle \ge c\\|x\\|^2</math>

			for all <math>x</math> in <math>H.</math>

			A [[bilinear form]] <math>a:H\times H\to \mathbb R</math> is called '''coercive''' if there exists a constant <math>c>0</math> such that

			:<math>a(x, x)\ge c\\|x\\|^2</math>

			for all <math>x</math> in <math>H.</math>

			It follows from the [[Riesz representation theorem]] that any symmetric (<math>a(x, y)=a(y, x)</math> for all <math>x, y</math> in <math>H</math>), continuous (<math>\|a(x, y)\|\le K\\|x\\|\,\\|y\\|</math> for all <math>x, y</math> in <math>H</math> and some constant <math>K>0</math>) and coercive bilinear form <math>a</math> has the representation

			: <math>a(x, y)=\langle Ax, y\rangle</math>

			for some self-adjoint operator <math>A:H\to H,</math> which then turns out to be a coercive operator. Also, given a coercive operator self-adjoint operator <math>A,</math> the bilinear form <math>a</math> defined as above is coercive.

			One can also show that any self-adjoint operator <math>A:H\to H</math> is a coercive operator if and only if it is a coercive mapping
			(in the sense of coercivity of a vector field, where one has to replace the dot product with the more general inner product).
			The definitions of coercivity for vector fields, operators, and bilinear forms are closely related and compatible.

			==Norm-coercive mappings==
			A mapping
			<math>f : X \to X' </math> between two normed vectorspaces
			<math>(X, \\| \cdot \\|)</math> and <math>(X', \\| \cdot \\|')</math>
			is called '''norm-coercive''' iff

			:<math> \\|f(x)\\|' \to + \infty \mbox{ as } \\|x\\| \to +\infty </math>.

			More generally, a function <math>f : X \to X' </math> between two [[topological space]]s <math>X</math> and <math>X'</math> is called '''coercive''' if for every [[compact space\|compact subset]] <math>K'</math> of <math>X'</math> there exists a compact subset <math>K</math> of <math>X</math> such that

			:<math>f (X \setminus K) \subseteq X' \setminus K'.</math>

			The [[function composition\|composition]] of a [[bijection\|bijective]] [[proper map]] followed by a coercive map is coercive.

			==(Extended valued) coercive functions==
			An (extended valued) function
			<math>f:\mathbb{R}^n \to \mathbb{R} \cup \{- \infty, + \infty\}</math>
			is called '''coercive''' iff
			:<math> f(x) \to + \infty \mbox{ as } \\| x \\| \to + \infty.</math>
			A realvalued coercive function <math>f:\mathbb{R}^n \to \mathbb{R} </math>
			is in particular norm-coercive. However a norm-coercive function
			<math>f:\mathbb{R}^n \to \mathbb{R} </math> is not necessarily coercive.
			For instance the identity function on <math> \mathbb{R} </math> is norm-coercive
			but not coercive.

			==References==
			* {{cite book\| author=Renardy, Michael and Rogers, Robert C. \| title=An introduction to partial differential equations \| edition=Second edition \| publisher=Springer-Verlag \| location=New York, NY \| year=2004 \| pages=xiv+434 \| isbn=0-387-00444-0 }}
			* {{cite book
			\| last = Bashirov
			\| first = Agamirza E
			\| title = Partially observable linear systems under dependent noises
			\| publisher = Basel; Boston: Birkhäuser Verlag
			\| year = 2003
			\| pages =
			\| isbn = 0-8176-6999-X
			}}
			*{{cite book
			\| author2-link=Neil Trudinger
			\| first1=D.
			\| last1=Gilbarg
			\| first2=N.
			\| last2=Trudinger
			\| title = Elliptic partial differential equations of second order, 2nd ed
			\| publisher = Berlin; New York: Springer
			\| year = 2001
			\| pages =
			\| isbn = 3-540-41160-7
			}}

			{{PlanetMath attribution\|id=7154\|title=Coercive Function}}

			[[Category:Functional analysis]]
			[[Category:General topology]]
			[[Category:Types of functions]]

130.102.158.16 at 22:17, 22 July 2012

2012-07-22T22:17:19Z

New page

The writer is called Araceli Gulledge. He currently life in Idaho and his mothers and fathers live nearby. Bookkeeping is what she does. Playing croquet is something I will never give up.<br><br>my web blog; [http://www.visionassetmanagement.com/ActivityFeed/MyProfile/tabid/60/userId/2039/Default.aspx http://www.visionassetmanagement.com]