en>Bgwhite: WP:CHECKWIKI error fix #97. No content between TOC and first headline per WP:TOC and WP:LEAD. This is an accessibility issue for users of screen readers. Do general fixes and cleanup if needed. - using AWB (9991)

2014-03-22T07:38:17Z

WP:CHECKWIKI error fix #97. No content between TOC and first headline per WP:TOC and WP:LEAD. This is an accessibility issue for users of screen readers. Do general fixes and cleanup if needed. - using AWB (9991)

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en>Addbot: Bot: Migrating 8 interwiki links, now provided by Wikidata on d:q996052 (Report Errors)

2013-02-27T00:55:26Z

Bot: Migrating 8 interwiki links, now provided by Wikidata on d:q996052 (Report Errors)

← Older revision		Revision as of 02:55, 27 February 2013
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	~~Yoshiko~~ is ~~her car warranty name but she extended car warranty doesn't like when people use her full name~~. I'~~ve usually extended auto warranty loved residing~~ in ~~Idaho. One~~ of the ~~issues I love most~~ is ~~greeting card~~ [~~http~~://~~Www~~.~~Popularmechanics~~.~~com~~/~~cars~~/~~how-to~~/~~repair~~/~~how-to-get-a-used-car-warranty-and-not-get-screwed-6654348 gathering] but I don~~'t have the ~~time lately. The~~ [~~http:~~//~~Www~~.~~consumer.ftc.gov~~/~~articles~~/~~0054-auto-service-contracts-~~and-~~warranties occupation I've~~] ~~been occupying~~ for ~~years is a bookkeeper but I~~'~~ve currently utilized for another 1~~.<br><br>~~Here extended car warranty~~ is ~~my site~~ :: ~~[http:~~//~~Skyliner~~-~~Ehrengarde~~.de/~~index~~.~~php?mod~~=~~users&action~~=~~view&id~~=~~11990 http://Skyliner-Ehrengarde~~.~~de/index~~.~~php?mod~~=~~users&action~~=~~view&~~id=~~11990~~]		In [[mathematics]], '''Fredholm's theorems''' are a set of celebrated results of [[Ivar Fredholm]] in the [[Fredholm theory]] of [[integral equations]]. There are several closely related theorems, which may be stated in terms of integral equations, in terms of [[linear algebra]], or in terms of the [[Fredholm operator]] on [[Banach space]]s.

			The [[Fredholm alternative]] is one of the Fredholm theorems.

			==Linear algebra==
			Fredholm's theorem in linear algebra is as follows: if ''M'' is a [[Matrix (mathematics)\|matrix]], then the [[orthogonal complement]] of the [[row space]] of ''M'' is the [[null space]] of ''M'':

			:<math>(\operatorname{row } M)^\bot = \ker M.</math>

			Similarly, the orthogonal complement of the column space of ''M'' is the null space of the adjoint:

			:<math>(\operatorname{col } M)^\bot = \ker M^*.</math>

			==Integral equations==
			Fredholm's theorem for integral equations is expressed as follows. Let <math>K(x,y)</math> be an [[integral kernel]], and consider the [[Homogeneous polynomial\|homogeneous equation]]s

			:<math>\int_a^b K(x,y) \phi(y) \,dy = \lambda \phi(x)</math>

			and its complex adjoint

			:<math>\int_a^b \psi(x) \overline{K(x,y)} \, dx = \overline {\lambda}\psi(y).</math>

			Here, <math>\overline{\lambda}</math> denotes the [[complex conjugate]] of the [[complex number]] <math>\lambda</math>, and similarly for <math>\overline{K(x,y)}</math>. Then, Fredholm's theorem is that, for any fixed value of <math>\lambda</math>, these equations have either the trivial solution <math>\psi(x)=\phi(x)=0</math> or have the same number of [[linearly independent]] solutions <math>\phi_1(x),\cdots,\phi_n(x)</math>, <math>\psi_1(y),\cdots,\psi_n(y)</math>.

			A sufficient condition for this theorem to hold is for <math>K(x,y)</math> to be [[square integrable]] on the rectangle <math>[a,b]\times[a,b]</math> (where ''a'' and/or ''b'' may be minus or plus infinity).

			Here, the integral is expressed as a one-dimensional integral on the real number line. In [[Fredholm theory]], this result generalizes to [[integral operator]]s on multi-dimensional spaces, including, for example, [[Riemannian manifold]]s.

			==Existence of solutions==
			One of the Fredholm theorem's closely related to the [[Fredholm alternative]], concerns the existence of solutions to the inhomogeneous [[Fredholm equation]]

			:<math> \lambda \phi(x)-\int_a^b K(x,y) \phi(y) \,dy=f(x).</math>

			Solutions to this equation exist if and only if the function <math>f(x)</math> is [[orthogonal]] to the complete set of solutions <math>\{\psi_n(x)\}</math> of the corresponding homogeneous adjoint equation:

			:<math>\int_a^b \overline{\psi_n(x)} f(x) \,dx=0</math>

			where <math>\overline{\psi_n(x)}</math> is the complex conjugate of <math>\psi_n(x)</math> and the former is one of the complete set of solutions to

			:<math>\lambda\overline{\psi(y)} -\int_a^b \overline{\psi(x)} K(x,y) \,dx=0.</math>

			A sufficient condition for this theorem to hold is for <math>K(x,y)</math> to be [[square integrable]] on the rectangle <math>[a,b]\times[a,b]</math>.

			==References==
			* E.I. Fredholm, "Sur une classe d'equations fonctionnelles", ''Acta Math.'', '''27''' (1903) pp. 365–390.
			* {{mathworld\|urlname=FredholmsTheorem\|title=Fredholm's Theorem}}
			* {{springer\|id=f/f041470\|title=Fredholm theorems for integral operators\|author=B.V. Khvedelidze}}

			[[Category:Fredholm theory]]
			[[Category:Linear algebra]]
			[[Category:Theorems in functional analysis]]

en>JackieBot: r2.6.5) (Robot: Adding uk:STRIPS

2012-02-16T22:28:05Z

r2.6.5) (Robot: Adding uk:STRIPS

New page

Yoshiko is her car warranty name but she extended car warranty doesn't like when people use her full name. I've usually extended auto warranty loved residing in Idaho. One of the issues I love most is greeting card [http://Www.Popularmechanics.com/cars/how-to/repair/how-to-get-a-used-car-warranty-and-not-get-screwed-6654348 gathering] but I don't have the time lately. The [http://Www.consumer.ftc.gov/articles/0054-auto-service-contracts-and-warranties occupation I've] been occupying for years is a bookkeeper but I've currently utilized for another 1.<br><br>Here extended car warranty is my site :: [http://Skyliner-Ehrengarde.de/index.php?mod=users&action=view&id=11990 http://Skyliner-Ehrengarde.de/index.php?mod=users&action=view&id=11990]

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en>Bgwhite: WP:CHECKWIKI error fix #97. No content between TOC and first headline per WP:TOC and WP:LEAD. This is an accessibility issue for users of screen readers. Do general fixes and cleanup if needed. - using AWB (9991)

en>Addbot: Bot: Migrating 8 interwiki links, now provided by Wikidata on d:q996052 (Report Errors)

en>JackieBot: r2.6.5) (Robot: Adding uk:STRIPS