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en>Trappist the monk: /* References */replace mr template with mr parameter in CS1 templates; using AWB

2014-09-24T00:18:45Z

References: replace mr template with mr parameter in CS1 templates; using AWB

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	~~In [[mathematical analysis]],~~ the '~~''Hardy–Littlewood inequality'''~~, ~~named after [[G~~. ~~H. Hardy]]~~ and ~~[[John Edensor Littlewood]]~~, ~~states that if ''f'' and ''g'' are nonnegative [[measurable function\|measurable]] [[real functions]] vanishing~~ at ~~[[infinity]] that are defined on ''n''-[[dimension]]al [[Euclidean space]] '''R'''~~<~~sup~~>~~''n''<~~/~~sup> then~~		Although I take the driver's license of space for more than seven years, you can not open the several vehicles.<br>Some people feel that the replica handbags are a fake and cheap imitation of the original designer handbags, people from top brass of the society look at the replica handbags as something meant for the low income groups.<br>http://peterlongogolfshow.com/coach/?key=coach-factory-outlet-online-sale-invitation-9 <br /> http://peterlongogolfshow.com/coach/?key=coach-factory-outlet-store-sale-27 <br /> http://peterlongogolfshow.com/coach/?key=coach-outlet-online-factory-sale-7 <br /> http://peterlongogolfshow.com/coach/?key=coach-factory-store-sales-33 <br /> http://peterlongogolfshow.com/coach/?key=coach-outlet-factory-online-sale-23 <br /><br><br>In case you have almost any questions about wherever along with the way to make use of [http://www.restaurantcalcuta.com/outlet/ugg.asp Ugg Outlet Store], you are able to e-mail us at our web site.

	~~:<math>\int_{\mathbb{R}^n} f(x)g(x) \, dx \leq \int_{\mathbb{R}^n} f^(x)g^(x) \, dx~~</~~math~~>

	~~where~~ ~~''f''<sup><~~/~~sup> and ''g''<sup><~~/~~sup> are the [[symmetric decreasing rearrangement]]s of ''f''(''x'') and ''g''(''x''), respectively~~.~~<ref name~~=~~liebloss>{{cite book \| author=Lieb, Elliott H., & Loss, Michael \| title=Analysis \| edition=Second \| publisher=American Mathematical Society \| location=Providence, RI \| year=2001 \| isbn=0~~-~~8218~~-~~2783~~-~~9 }}~~
	</~~ref~~>~~<ref name=burchard>{{cite book\|title=A Short Course on Rearrangement Inequalities\|first=Almut\|last=Burchard\|url=~~http://~~www.math~~.~~toronto.edu~~/~~almut~~/~~rearrange.pdf}}~~</~~ref~~>

	~~==Proof==~~
	~~From [[layer cake representation]] we have~~:~~<ref name=liebloss~~/~~><ref name=burchard~~/>
	~~:<math>f(x)~~= ~~\int_0^\infty \chi_{f(x)>r} \, dr~~</~~math~~>
	:~~<math>g(x)~~= ~~\int_0^\infty \chi_{g(x)>s} \, ds~~</~~math~~>

	~~where~~ <~~math>\chi_{f(x)~~>r}<~~/math~~> ~~denotes~~ the [~~[indicator function]] of the subset ''E''<sub> ''f''</sub> given by~~

	:~~<math>E_f=\left\{x\in X: f(x)>r\right\} \, <~~/~~math>~~

	~~Analogously, <math>\chi_{g(x)>s}<~~/~~math> denotes the indicator function of the subset ''E''<sub> ''g''<~~/~~sub> given by~~

	~~:<math>E_g=\left\{x\in X: g(x)>s\right\} \, <~~/~~math>~~

	~~:<math>\begin{align}~~
	~~\int_{\mathbb{R}^n} f(x)g(x) \, dx &= \displaystyle\int_{\mathbb{R}^n}\int_0^\infty \int_0^\infty \chi_{f(x)>r}\chi_{g(x)>s} \, dr \, ds \, dx \\[8pt~~]
	~~&= \int_0^\infty \int_0^\infty \int_{\mathbb{R}^n}\chi_{f(x)>r\cap g(x)>s} \~~, ~~dx \, dr \, ds \\[8pt]~~
	~~&= \int_0^\infty \int_0^\infty \mu\left(\left\{f(x)>r\right\}\cap\left\{ g(x)>s\right\}\right) \, dr \, ds\\[8pt]~~
	~~&\leq \int_0^\infty \int_0^\infty \min\left(\mu\left(f(x)>r\right);\mu\left(g(x)>s\right)\right) \, dr \, ds\\[8pt]~~
	~~&= \int_0^\infty \int_0^\infty \min\left(\mu\left(f^(x)>r\right);\mu\left(g^(x)>s\right)\right) \, dr \, ds\\[8pt]~~
	~~&= \int_0^\infty \int_0^\infty \mu\left(\left\{f^\ast(x)>r\right\}\cap\left\{ g^\ast(x)>s\right\}\right) \, dr \, ds\\[8pt]~~
	~~&= \int_{\mathbb{R}^n} f^(x)g^(x) \, dx~~
	~~\end{align}~~
	~~</math>~~

	~~==See also==~~
	* [[Rearrangement inequality]]
	* [[Chebyshev's sum inequality]]

	~~==References==~~
	~~<references/>~~

	~~{{DEFAULTSORT:Hardy~~-~~Littlewood inequality}}~~
	~~[[Category:Inequalities]]~~
	~~[[Category:Articles containing proofs]]~~

en>Giftlite: −sp

2012-12-18T18:14:34Z

−sp

← Older revision		Revision as of 20:14, 18 December 2012
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	~~Sales however have dropped in~~ the ~~past couple of quarters~~.~~<br>And she may be into her career~~, ~~but what she~~'~~s really saying is or possibly but which one is it? If you have any kind of concerns concerning where~~ and ~~ways to use~~ [~~http://www.restaurantcalcuta.com/outlet/ugg.asp Cheap Uggs For Sale~~]~~, you could call us~~ at ~~our internet site. 60% of all human communication is non verbal, so~~ that ~~means 90% of what you~~'~~re saying ain~~'~~t coming out of your mouth. Of course she~~'~~s going to lie to you! She doesn~~'~~t want to hurt your feelings! What else she going to say? She doesn~~'~~t even know you.~~<br>~~http~~:~~//www.bendtrapclub.com/cheap/ugg.asp?p=453~~ <br /> ~~http:~~//~~www~~.~~bendtrapclub~~.~~com/cheap/ugg.asp?p~~=~~317~~ <br /> http://www.~~bendtrapclub~~.~~com~~/~~cheap~~/~~ugg~~.~~asp?p=312~~ <br /> ~~http~~://~~www.bendtrapclub.com/cheap/ugg.asp?p~~=~~457~~ <br /> ~~http~~://~~www.bendtrapclub.com~~/~~cheap/ugg.asp?p~~=81 <br /> <br>~~http:~~//~~atlantaka.org/?document_srl~~=~~925954~~<br>~~http~~://~~wikiweedia.org/index.php?title=Unbiased_Report_Exposes_The_Unanswered_Questions_on_Bags_Outlet~~		In [[mathematical analysis]], the '''Hardy–Littlewood inequality''', named after [[G. H. Hardy]] and [[John Edensor Littlewood]], states that if ''f'' and ''g'' are nonnegative [[measurable function\|measurable]] [[real functions]] vanishing at [[infinity]] that are defined on ''n''-[[dimension]]al [[Euclidean space]] '''R'''<sup>''n''</sup> then

			:<math>\int_{\mathbb{R}^n} f(x)g(x) \, dx \leq \int_{\mathbb{R}^n} f^(x)g^(x) \, dx</math>

			where ''f''<sup></sup> and ''g''<sup></sup> are the [[symmetric decreasing rearrangement]]s of ''f''(''x'') and ''g''(''x''), respectively.<ref name=liebloss>{{cite book \| author=Lieb, Elliott H., & Loss, Michael \| title=Analysis \| edition=Second \| publisher=American Mathematical Society \| location=Providence, RI \| year=2001 \| isbn=0-8218-2783-9 }}
			</ref><ref name=burchard>{{cite book\|title=A Short Course on Rearrangement Inequalities\|first=Almut\|last=Burchard\|url=http://www.math.toronto.edu/almut/rearrange.pdf}}</ref>

			==Proof==
			From [[layer cake representation]] we have:<ref name=liebloss/><ref name=burchard/>
			:<math>f(x)= \int_0^\infty \chi_{f(x)>r} \, dr</math>
			:<math>g(x)= \int_0^\infty \chi_{g(x)>s} \, ds</math>

			where <math>\chi_{f(x)>r}</math> denotes the [[indicator function]] of the subset ''E''<sub> ''f''</sub> given by

			:<math>E_f=\left\{x\in X: f(x)>r\right\} \, </math>

			Analogously, <math>\chi_{g(x)>s}</math> denotes the indicator function of the subset ''E''<sub> ''g''</sub> given by

			:<math>E_g=\left\{x\in X: g(x)>s\right\} \, </math>

			:<math>\begin{align}
			\int_{\mathbb{R}^n} f(x)g(x) \, dx &= \displaystyle\int_{\mathbb{R}^n}\int_0^\infty \int_0^\infty \chi_{f(x)>r}\chi_{g(x)>s} \, dr \, ds \, dx \\[8pt]
			&= \int_0^\infty \int_0^\infty \int_{\mathbb{R}^n}\chi_{f(x)>r\cap g(x)>s} \, dx \, dr \, ds \\[8pt]
			&= \int_0^\infty \int_0^\infty \mu\left(\left\{f(x)>r\right\}\cap\left\{ g(x)>s\right\}\right) \, dr \, ds\\[8pt]
			&\leq \int_0^\infty \int_0^\infty \min\left(\mu\left(f(x)>r\right);\mu\left(g(x)>s\right)\right) \, dr \, ds\\[8pt]
			&= \int_0^\infty \int_0^\infty \min\left(\mu\left(f^(x)>r\right);\mu\left(g^(x)>s\right)\right) \, dr \, ds\\[8pt]
			&= \int_0^\infty \int_0^\infty \mu\left(\left\{f^\ast(x)>r\right\}\cap\left\{ g^\ast(x)>s\right\}\right) \, dr \, ds\\[8pt]
			&= \int_{\mathbb{R}^n} f^(x)g^(x) \, dx
			\end{align}
			</math>

			==See also==
			* [[Rearrangement inequality]]
			* [[Chebyshev's sum inequality]]

			==References==
			<references/>

			{{DEFAULTSORT:Hardy-Littlewood inequality}}
			[[Category:Inequalities]]
			[[Category:Articles containing proofs]]

en>Katharineamy: added Category:Algebra; removed {{uncategorized}} using HotCat

2012-08-27T00:36:45Z

added Category:Algebra; removed {{uncategorized}} using HotCat

New page

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