Shortest common supersequence - Revision history

2001:638:403:5050:224:1DFF:FEDE:83A3: added reference

2015-01-02T08:44:48Z

added reference

← Older revision		Revision as of 10:44, 2 January 2015
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	~~In [[mathematics]]~~, ~~a '''Fredholm kernel'''~~ is a certain type of a [[kernel (integral operator)\|kernel]] on a [[Banach space]], associated with [[nuclear operator]]s on the Banach space. They are an abstraction of the idea of the [[Fredholm integral equation]] and the [[Fredholm operator]], and are one of the objects of study in [[Fredholm theory]]. Fredholm kernels are named in honour of [[Erik Ivar Fredholm]]. ~~Much of the abstract theory of Fredholm kernels was developed by [[Alexander Grothendieck]] and published in 1955.~~		Nice to meet you, my name is Refugia. Hiring is my occupation. The preferred pastime for my children and me is to play baseball but I haven't made a dime with it. Her spouse and her live in Puerto Rico but she will have to move 1 day or another.<br><br>Here is my weblog at home std testing ([http://Www.gaysphere.net/user/KJGI linked internet site])

	~~==Definition==~~
	~~Let ''B'' be an arbitrary [[Banach space]], and let ''B''<sup></sup> be its dual, that~~ is~~, the space of [[bounded linear functional]]s on ''B''~~. The ~~[[tensor product]] <math>B^\otimes B</math> has a [[complete space\|completion]] under the norm~~

	~~:<math>\Vert X \Vert_\pi =~~
	~~\inf \sum_{\{i\}} \Vert e^*_i\Vert \Vert e_i \Vert</math>~~

	~~where the [[infimum]] is taken over all finite representations~~

	~~:<math>X=\sum_{\{i\}} e^_i e_i \in B^\otimes B</math>~~

	~~The completion, under this norm, is often denoted as~~

	~~:<math>B^* \widehat{\,\otimes\,}_\pi B </math>~~

	and is ~~called the~~ '~~''[[topological tensor product\|projective topological tensor product]]'''. The elements of this space are called '''Fredholm kernels'''.~~

	~~==Properties==~~
	~~Every Fredholm kernel has~~ a ~~representation in the form~~

	~~:<math>X=\sum_{\{i\}} \lambda_i e^*_i \otimes e_i</math>~~

	with ~~<math>e_i \in B</math>~~ and ~~<math>e^_i \~~in ~~B^</math> such that <math>\Vert e_i \Vert = \Vert e^*_i \Vert =~~ 1~~</math> and~~

	~~:<math>\sum_{\{i\}} \vert \lambda_i \vert < \infty~~. ~~\, </math>~~

	~~Associated with each such kernel is a linear operator~~

	:<~~math~~>~~\mathcal {L}_X : B \to B~~<~~/math~~>

	~~which has the canonical representation~~

	~~:<math>\mathcal{L}_X f =\sum_{\{i\}} \lambda_i e^*_i(f) \otimes e_i. \, </math>~~

	~~Associated with every Fredholm kernel~~ is ~~a trace, defined as~~

	~~:<math>\mbox{tr} X = \sum_{\{i\}} \lambda_i e^*_i~~(~~e_i). \,</math>~~

	~~==''p''-summable kernels==~~
	~~A Fredholm kernel is said to be ''' ''p''-summable''' if~~

	~~:<math>\sum_{\{i\}} \vert \lambda_i \vert^p < \infty</math>~~

	~~A Fredholm kernel is said to be of '''order q''' if ''q'' is the~~ [~~[infimum]] of all <math>0<p\le 1</math> for all ''p'' for which it is ''p''-summable.~~

	~~==Nuclear operators on Banach spaces==~~
	~~An operator <math>\mathcal{L}~~:~~B \to B</math> is said to be a [[nuclear operator]] if there exists an~~
	<math>X\in B^* \widehat{\,\otimes\,}_\pi B</math> such that <math>\mathcal{L} = \mathcal{L}_X</math>. Such an operator is said to be ''p''-summable and of order ''q'' if ''X'' is. In general, there may be more than one ''X'' associated with such a nuclear operator, and so the trace is not uniquely defined. However, if the order <math>q \le 2/3</~~math>, then there is a unique trace, as given by a theorem of Grothendieck.~~

	~~==Grothendieck's theorem==~~
	~~If <math>\mathcal{L}:B\to B</math> is an operator of order <math>q \le 2/3</math> then a trace may be defined, with~~

	~~:<math>\mbox{Tr} \mathcal {L} = \sum_{\{i\}} \rho_i</math>~~

	~~where <math>\rho_i</math> are the [[eigenvalue]]s of <math>\mathcal{L}</math>. Furthermore, the [[Fredholm determinant]]~~

	~~:<math>\det \left( 1-z\mathcal{L}\right)=~~
	~~\prod_i \left(1-\rho_i z \right)</math>~~

	~~is an [[entire function]] of ''z''~~. ~~The formula~~

	~~:<math>\det \left( 1-z\mathcal{L}\right)=~~
	~~\exp \mbox{Tr} \log\left( 1-z\mathcal{L}\right) </math>~~

	~~holds as well~~. ~~Finally, if <math>\mathcal{L}<~~/~~math> is parameterized by some [[complex number\|complex]]-valued parameter ''w'', that is, <math>\mathcal{L}=\mathcal{L}_w<~~/~~math>, and the parameterization is [[holomorphic]~~] ~~on some domain, then~~

	~~:<math>\det \left( 1-z\mathcal{L}_w\right~~)~~</math>~~

	~~is holomorphic on the same domain.~~

	~~==Examples==~~
	An important example is the Banach space of holomorphic functions over a domain <math>D\subset \mathbb{C}^k</math>. In this space, every nuclear operator is of order zero, and is thus of [[trace-class]].

	~~==Nuclear spaces==~~
	~~The idea of a nuclear operator can be adapted to [[Fréchet space]]s. A '''[[nuclear space]]''' is a Fréchet space where every bounded map of the space to an arbitrary Banach space is nuclear.~~

	~~==References==~~
	*{{cite journal \|author=Grothendieck A \|title=Produits tensoriels topologiques et espaces nucléaires \|journal=Mem. Am. Math.Soc. \|volume=16 \|year=1955}}
	*{{cite journal \|author=Grothendieck A \|title=La théorie de Fredholm \|journal=Bull. Soc. Math. France \|volume=84 \|pages=319–84 \|year=1956}}
	* {{springer\|id=f/f041440\|title=Fredholm kernel\|author=B.V. Khvedelidze, G.L. Litvinov}}
	*{{cite journal \|author=Fréchet M \|title=On the Behavior of the nth Iterate of a Fredholm Kernel as n Becomes Infinite \|journal=Proc. Natl. Acad. Sci. U.S.A. \|volume=18 \|issue=11 \|pages=671–3 \|date=November 1932 \|pmid=16577494 \|pmc=1076308 \|doi=10.1073/pnas.18.11.671 }}

	~~[[Category:Fredholm theory]]~~
	~~[[Category:Banach spaces]]~~
	~~[[Category:Topology of function spaces]]~~

en>Materialscientist: Reverted 1 edit by 116.203.252.89 identified as test/vandalism using STiki

2013-10-21T23:13:58Z

Reverted 1 edit by 116.203.252.89 identified as test/vandalism using STiki

← Older revision		Revision as of 01:13, 22 October 2013
Line 1:		Line 1:
	They ~~call me Emilia~~. ~~Bookkeeping~~ is ~~her working day occupation now~~. He is ~~truly fond~~ of ~~performing ceramics but he~~ is ~~having difficulties~~ to ~~find time~~ for it~~. Minnesota~~ is ~~where he~~'~~s been residing for years~~.<br><br>~~My site~~ [~~http:~~//~~livestudentjobs~~.co.uk/~~groups~~/~~solid~~-~~advice~~-~~for~~-~~handling~~-a-~~candida~~/ ~~std home test~~]		In [[mathematics]], a '''Fredholm kernel''' is a certain type of a [[kernel (integral operator)\|kernel]] on a [[Banach space]], associated with [[nuclear operator]]s on the Banach space. They are an abstraction of the idea of the [[Fredholm integral equation]] and the [[Fredholm operator]], and are one of the objects of study in [[Fredholm theory]]. Fredholm kernels are named in honour of [[Erik Ivar Fredholm]]. Much of the abstract theory of Fredholm kernels was developed by [[Alexander Grothendieck]] and published in 1955.

			==Definition==
			Let ''B'' be an arbitrary [[Banach space]], and let ''B''<sup></sup> be its dual, that is, the space of [[bounded linear functional]]s on ''B''. The [[tensor product]] <math>B^\otimes B</math> has a [[complete space\|completion]] under the norm

			:<math>\Vert X \Vert_\pi =
			\inf \sum_{\{i\}} \Vert e^*_i\Vert \Vert e_i \Vert</math>

			where the [[infimum]] is taken over all finite representations

			:<math>X=\sum_{\{i\}} e^_i e_i \in B^\otimes B</math>

			The completion, under this norm, is often denoted as

			:<math>B^* \widehat{\,\otimes\,}_\pi B </math>

			and is called the '''[[topological tensor product\|projective topological tensor product]]'''. The elements of this space are called '''Fredholm kernels'''.

			==Properties==
			Every Fredholm kernel has a representation in the form

			:<math>X=\sum_{\{i\}} \lambda_i e^*_i \otimes e_i</math>

			with <math>e_i \in B</math> and <math>e^_i \in B^</math> such that <math>\Vert e_i \Vert = \Vert e^*_i \Vert = 1</math> and

			:<math>\sum_{\{i\}} \vert \lambda_i \vert < \infty. \, </math>

			Associated with each such kernel is a linear operator

			:<math>\mathcal {L}_X : B \to B</math>

			which has the canonical representation

			:<math>\mathcal{L}_X f =\sum_{\{i\}} \lambda_i e^*_i(f) \otimes e_i. \, </math>

			Associated with every Fredholm kernel is a trace, defined as

			:<math>\mbox{tr} X = \sum_{\{i\}} \lambda_i e^*_i(e_i). \,</math>

			==''p''-summable kernels==
			A Fredholm kernel is said to be ''' ''p''-summable''' if

			:<math>\sum_{\{i\}} \vert \lambda_i \vert^p < \infty</math>

			A Fredholm kernel is said to be of '''order q''' if ''q'' is the [[infimum]] of all <math>0<p\le 1</math> for all ''p'' for which it is ''p''-summable.

			==Nuclear operators on Banach spaces==
			An operator <math>\mathcal{L}:B \to B</math> is said to be a [[nuclear operator]] if there exists an
			<math>X\in B^* \widehat{\,\otimes\,}_\pi B</math> such that <math>\mathcal{L} = \mathcal{L}_X</math>. Such an operator is said to be ''p''-summable and of order ''q'' if ''X'' is. In general, there may be more than one ''X'' associated with such a nuclear operator, and so the trace is not uniquely defined. However, if the order <math>q \le 2/3</math>, then there is a unique trace, as given by a theorem of Grothendieck.

			==Grothendieck's theorem==
			If <math>\mathcal{L}:B\to B</math> is an operator of order <math>q \le 2/3</math> then a trace may be defined, with

			:<math>\mbox{Tr} \mathcal {L} = \sum_{\{i\}} \rho_i</math>

			where <math>\rho_i</math> are the [[eigenvalue]]s of <math>\mathcal{L}</math>. Furthermore, the [[Fredholm determinant]]

			:<math>\det \left( 1-z\mathcal{L}\right)=
			\prod_i \left(1-\rho_i z \right)</math>

			is an [[entire function]] of ''z''. The formula

			:<math>\det \left( 1-z\mathcal{L}\right)=
			\exp \mbox{Tr} \log\left( 1-z\mathcal{L}\right) </math>

			holds as well. Finally, if <math>\mathcal{L}</math> is parameterized by some [[complex number\|complex]]-valued parameter ''w'', that is, <math>\mathcal{L}=\mathcal{L}_w</math>, and the parameterization is [[holomorphic]] on some domain, then

			:<math>\det \left( 1-z\mathcal{L}_w\right)</math>

			is holomorphic on the same domain.

			==Examples==
			An important example is the Banach space of holomorphic functions over a domain <math>D\subset \mathbb{C}^k</math>. In this space, every nuclear operator is of order zero, and is thus of [[trace-class]].

			==Nuclear spaces==
			The idea of a nuclear operator can be adapted to [[Fréchet space]]s. A '''[[nuclear space]]''' is a Fréchet space where every bounded map of the space to an arbitrary Banach space is nuclear.

			==References==
			*{{cite journal \|author=Grothendieck A \|title=Produits tensoriels topologiques et espaces nucléaires \|journal=Mem. Am. Math.Soc. \|volume=16 \|year=1955}}
			*{{cite journal \|author=Grothendieck A \|title=La théorie de Fredholm \|journal=Bull. Soc. Math. France \|volume=84 \|pages=319–84 \|year=1956}}
			* {{springer\|id=f/f041440\|title=Fredholm kernel\|author=B.V. Khvedelidze, G.L. Litvinov}}
			*{{cite journal \|author=Fréchet M \|title=On the Behavior of the nth Iterate of a Fredholm Kernel as n Becomes Infinite \|journal=Proc. Natl. Acad. Sci. U.S.A. \|volume=18 \|issue=11 \|pages=671–3 \|date=November 1932 \|pmid=16577494 \|pmc=1076308 \|doi=10.1073/pnas.18.11.671 }}

			[[Category:Fredholm theory]]
			[[Category:Banach spaces]]
			[[Category:Topology of function spaces]]

en>ChrisGualtieri: General fixes / CHECKWIKI fixes using AWB

2012-08-30T23:30:13Z

General fixes / CHECKWIKI fixes using AWB

New page

They call me Emilia. Bookkeeping is her working day occupation now. He is truly fond of performing ceramics but he is having difficulties to find time for it. Minnesota is where he's been residing for years.<br><br>My site [http://livestudentjobs.co.uk/groups/solid-advice-for-handling-a-candida/ std home test]