Jackson network - Revision history

en>Gareth Jones: wlink Jean Walrand

2014-04-25T16:19:02Z

wlink Jean Walrand

← Older revision		Revision as of 18:19, 25 April 2014
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128.101.152.60: supplying a conspicuously missing comma

2014-02-19T16:22:28Z

supplying a conspicuously missing comma

Show changes

en>Monkbot: Fix CS1 deprecated date parameter errors

2014-01-22T15:31:03Z

Fix CS1 deprecated date parameter errors

← Older revision		Revision as of 17:31, 22 January 2014
Line 1:		Line 1:
	~~Let me first begin by introducing myself. My name~~ is [~~http~~:/~~/www~~.~~gaysphere~~.~~net~~/~~blog/212030 std testing at home~~] ~~Boyd Butts although it~~ is ~~not~~ the ~~over~~ the ~~counter std test name on my beginning certification~~. ~~California is exactly where~~ I'~~ve always been residing and I love each day residing here~~. ~~I utilized~~ to be ~~unemployed but now I am~~ a [~~http:~~//~~Www~~.~~marieclaire.com~~/~~health~~-~~fitness/weight~~-~~loss~~/at-~~home~~-~~sexually-transmitted-disease-testing librarian] and~~ the ~~salary has been truly std testing at home satisfying~~. ~~One of the things~~ [~~http~~://~~www~~.~~pathwayschico.org~~/~~blogs~~/~~post~~/~~216965 home std test kit~~] ~~she loves most is to do aerobics~~ and ~~now she is attempting to make money~~ with it.<br><br>~~Feel at home [http://www~~.~~wbli~~.~~com~~/~~gallery~~/~~entertainment/morning~~-~~show/celebrities~~-~~who~~-~~have~~-~~admitted~~-~~having~~-~~stds/gCF9q/ std test] free to surf to my blog post~~ - ~~home std test~~ ([~~http~~:~~//www.hotporn123.com/blog/154369 Additional Info~~])		In [[mathematics]], the '''resolvent formalism''' is a technique for applying concepts from [[complex analysis]] to the study of the [[spectrum (functional analysis)\|spectrum]] of [[Operator (mathematics)\|operator]]s on [[Hilbert space]]s and more general spaces.

			The '''resolvent''' captures the spectral properties of an operator in the analytic structure of the resolvent. Given an operator ''A'', the resolvent may be defined as

			: <math> R(z;A)= (A-zI)^{-1}. </math>

			Among other uses, the resolvent may be used to solve the inhomogeneous [[Fredholm integral equation]]s; a commonly used approach is a series solution, the [[Liouville-Neumann series]].

			The resolvent of ''A'' can be used to directly obtain information about the spectral decomposition
			of ''A''. For example, suppose <math> \lambda </math> is an isolated [[eigenvalue]] in the
			[[spectrum]] of ''A''. That is, suppose there exists a simple closed curve <math> C_\lambda </math>
			in the complex plane
			that separates <math> \lambda </math> from the rest of the spectrum of ''A''.
			Then the [[residue (complex analysis)\|residue]]

			: <math> \frac{-1}{2\pi i} \oint_{C_\lambda} (A- z I)^{-1}~ dz </math>

			defines a [[projection operator]] onto the <math> \lambda </math> [[eigenspace]] of ''A''.

			The [[Hille-Yosida theorem]] relates the resolvent to an integral over the one-parameter [[group (mathematics)\|group]] of transformations generated by ''A''. Thus, for example, if ''A'' is [[Hermitian operator\|Hermitian]], then <math>U(t)=\exp(itA)</math> is a one-parameter group of unitary operators. The resolvent can be expressed as the integral

			: <math>R(z;A)= \int_0^\infty e^{-zt}U(t) dt.</math>

			==History==
			The first major use of the resolvent operator was by [[Ivar Fredholm]], in a landmark 1903 paper in ''Acta Mathematica'' that helped establish modern [[operator theory]]. The name ''resolvent'' was given by [[David Hilbert]].

			==Resolvent identity==

			For all <math>z, w</math> in <math>\rho(A)</math>, the [[resolvent set]] of an operator <math>A</math>, we have that the '''resolvent identity''' (also called Hilbert's identity) holds:<ref>Dunford and Schwartz, Vol I, Lemma 6, p568.</ref>
			:<math>R(z; A) - R(w; A) = (w-z) R(z;A) R(w;A)\,</math>
			(Note that Dunford and Schwartz define the resolvent as <math>(zI-A)^{-1}</math> so that the formula above is slightly different from theirs.)

			==Compact resolvent==
			When studying an [[unbounded operator]] <math>A:H\to H</math> on a [[Hilbert space]] <math>H</math>, if there exists <math>z\in\rho(A)</math> such that <math>R(z;A)</math> is a [[compact operator]], we say that <math>A</math> has compact resolvent. The spectrum <math>\sigma(A)</math> of such <math>A</math> is a discrete subset of <math>\mathbb{C}</math>. If furthermore <math>A</math> is [[self-adjoint]], then <math>\sigma(A)\subset\mathbb{R}</math> and there exists an orthonormal basis <math>\{v_i\}_{i\in\mathbb{N}}</math> of eigenvectors of <math>A</math> with eigenvalues <math>\{\lambda_i\}_{i\in\mathbb{N}}</math> respectively. Also, <math>\{\lambda_i\}</math> has no finite accumulation point.<ref>Taylor, p515.</ref>

			==See also==
			* [[Resolvent set]]
			* [[Stone's theorem]]
			* [[Holomorphic functional calculus]]
			* [[Spectral theory]]
			* [[Compact operator]]
			* [[Unbounded operator]]

			==References==
			<references/>
			* [[Nelson Dunford]], [[Jacob T. Schwartz]] ''Linear Operators'', Part I General Theory ISBN 0-471-60848-3, Part II Spectral Theory, Self Adjoint Operators in Hilbert Space ISBN 0-471-60847-5, Part III Spectral Operators ISBN 0-471-60846-7
			* E.I. Fredholm, "Sur une classe d'equations fonctionnelles", ''Acta Mathematica'', '''27''' (1903) pp. 365–390.
			* {{Citation
			\| last = Kato
			\| first = Tosio
			\| title = Perturbation Theory for Linear Operators
			\| edition = 2nd
			\| publisher = Spinger-Verlag
			\| location = New York, NY
			\| year = 1980
			\| isbn = 0-387-07558-5 }}.
			*{{Citation
			\| last = Taylor
			\| first = Michael E. \| authorlink = Michael E. Taylor
			\| title = Partial Differential Equations I
			\| publisher = Spinger-Verlag
			\| location = New York, NY
			\| year = 1996
			\| isbn = 7-5062-4252-4 }}

			[[Category:Fredholm theory]]
			[[Category:Formalism (deductive)]]

en>Sun Creator: A to An typo fixing, replaced: of a open → of an open using AWB

2012-08-26T18:19:40Z

A to An typo fixing, replaced: of a open → of an open using AWB

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