Direct-ethanol fuel cell - Revision history

89.97.182.2: /* Advantages */

2014-05-06T09:16:36Z

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← Older revision		Revision as of 11:16, 6 May 2014
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en>BG19bot: WP:CHECKWIKI error fix for #61. Punctuation goes before References. Do general fixes if a problem exists. - using AWB (9916)

2014-02-07T02:21:35Z

WP:CHECKWIKI error fix for #61. Punctuation goes before References. Do general fixes if a problem exists. - using AWB (9916)

← Older revision		Revision as of 04:21, 7 February 2014
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	~~The~~ '~~''Sobolev conjugate''' of ''p'' for <math>1\leq p <n<~~/~~math>, where ''n'' is space dimensionality, is~~		Oscar is how he's known as and he [http://www.godlikeproductions.com/forum1/message696798/pg1 completely loves] this name. For years he's been residing in North Dakota and his family enjoys it. The factor she adores most is body developing and now she std [http://www.sddch.org/?document_srl=345265 home std test] test is trying to make cash with it. He utilized to [http://riddlesandpoetry.com/?q=node/19209 at home std test] home std testing be unemployed but now he is a meter reader.<br><br>Visit my [http://answers.Yahoo.com/question/index?qid=20061008193252AA7Bbdv web site] [http://facehack.ir/index.php?do=/blog/20/what-you-have-to-do-facing-candida/ http://facehack.ir/index.php?do=/blog/20/what-you-have-to-do-facing-candida]
	~~:<math> p^*=\frac{pn}{n-p}>p<~~/~~math>~~
	~~This is an important parameter in the [[Sobolev inequality\|Sobolev inequalities]~~].
	~~== Motivation ==~~
	~~A question arises whether~~ '~~'u'' from the~~ [~~[Sobolev space]] <math>W^{1,p}(R^n)<~~/~~math> belongs to <math>L^q(R^n)<~~/~~math> for some ''q''>''p''~~. ~~More specifically, when does <math>\\|Du\\|_{L^p(R^n)}<~~/~~math> control <math>\\|u\\|_{L^q(R^n)}</math>~~? It is ~~easy~~ to ~~check that~~
	~~the following inequality~~
	:~~<math>\\|u\\|_{L^~~q~~(R^n)}\leq C(p,q)\\|Du\\|_{L^p(R^n)}<~~/~~math> (*)~~
	~~can not~~ be ~~true for arbitrary ''q''~~. ~~Consider~~ <~~math~~>~~u(x)\in C^\infty_c(R^n)~~<~~/math~~>~~, infinitely differentiable function with compact support. Introduce <math>u_\lambda(x)~~:~~=u(\lambda x)<~~/~~math>~~. ~~We have that~~
	~~:<math>\\|u_\lambda\\|_{L^q(R^n)}^q~~=~~\int_{R^n}\|u(\lambda x)\|^qdx=\frac{1}{\lambda^n}\int_{R^n}\|u(y)\|^qdy=\lambda^{-n}\\|u\\|_{L^q(R^n)}^q</math>~~
	:~~<math>\\|Du_\lambda\\|_{L^p(R^n)}^p=\int_{R^n}\|\lambda Du(\lambda x)\|^pdx=\frac{\lambda^p}{\lambda^n}\int_{R^n}\|Du(y)\|^pdy=\lambda^{p-n}\\|Du\\|_{L^p(R^n)}^p<~~/~~math>~~
	~~The inequality (*) for <math>u_\lambda<~~/~~math> results in the following inequality for <math>u<~~/~~math>~~
	~~:<math>\\|u\\|_{L^q(R^n)}\leq \lambda^{1-n~~/~~p+n~~/~~q}C(p,q)\\|Du\\|_{L^p(R^n)}<~~/~~math>~~
	~~If <math>1~~-n/~~p+n~~/~~q\not = 0<~~/~~math>, then by letting <math>\lambda<~~/~~math> going to zero or infinity we obtain a contradiction~~. ~~Thus the inequality (*) could only be true for~~
	~~:<math>q~~=~~\frac{pn}{n~~-~~p}</math>,~~
	~~which is the Sobolev conjugate.~~

	~~==See also==~~
	*[[Sergei Lvovich Sobolev]]

	~~==References==~~
	* Lawrence C. Evans. Partial differential equations. Graduate studies in Mathematics, Vol 19. American Mathematical Society. 1998. ISBN 0-~~8218~~-~~0772~~-2

	~~[[Category:Sobolev spaces]~~]

en>Addbot: Bot: Migrating 4 interwiki links, now provided by Wikidata on d:q282799 (Report Errors)

2013-02-28T19:53:16Z

Bot: Migrating 4 interwiki links, now provided by Wikidata on d:q282799 (Report Errors)

← Older revision		Revision as of 21:53, 28 February 2013
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	~~Hello~~. ~~Let me introduce~~ the ~~author~~. ~~Her name~~ is ~~Refugia Shryock. He used~~ to be ~~unemployed but now he is a pc operator but his promotion never arrives~~. ~~Playing baseball is the hobby he will by no means stop doing~~. ~~Years ago we moved to North Dakota and I adore each working day residing here~~.<br><br>~~Here is my page~~ :: ~~[http:~~//~~dtekorea.com~~/xe/~~DS_030111~~/~~1073195 std home test~~]		The '''Sobolev conjugate''' of ''p'' for <math>1\leq p <n</math>, where ''n'' is space dimensionality, is
			:<math> p^*=\frac{pn}{n-p}>p</math>
			This is an important parameter in the [[Sobolev inequality\|Sobolev inequalities]].
			== Motivation ==
			A question arises whether ''u'' from the [[Sobolev space]] <math>W^{1,p}(R^n)</math> belongs to <math>L^q(R^n)</math> for some ''q''>''p''. More specifically, when does <math>\\|Du\\|_{L^p(R^n)}</math> control <math>\\|u\\|_{L^q(R^n)}</math>? It is easy to check that
			the following inequality
			:<math>\\|u\\|_{L^q(R^n)}\leq C(p,q)\\|Du\\|_{L^p(R^n)}</math> (*)
			can not be true for arbitrary ''q''. Consider <math>u(x)\in C^\infty_c(R^n)</math>, infinitely differentiable function with compact support. Introduce <math>u_\lambda(x):=u(\lambda x)</math>. We have that
			:<math>\\|u_\lambda\\|_{L^q(R^n)}^q=\int_{R^n}\|u(\lambda x)\|^qdx=\frac{1}{\lambda^n}\int_{R^n}\|u(y)\|^qdy=\lambda^{-n}\\|u\\|_{L^q(R^n)}^q</math>
			:<math>\\|Du_\lambda\\|_{L^p(R^n)}^p=\int_{R^n}\|\lambda Du(\lambda x)\|^pdx=\frac{\lambda^p}{\lambda^n}\int_{R^n}\|Du(y)\|^pdy=\lambda^{p-n}\\|Du\\|_{L^p(R^n)}^p</math>
			The inequality (*) for <math>u_\lambda</math> results in the following inequality for <math>u</math>
			:<math>\\|u\\|_{L^q(R^n)}\leq \lambda^{1-n/p+n/q}C(p,q)\\|Du\\|_{L^p(R^n)}</math>
			If <math>1-n/p+n/q\not = 0</math>, then by letting <math>\lambda</math> going to zero or infinity we obtain a contradiction. Thus the inequality (*) could only be true for
			:<math>q=\frac{pn}{n-p}</math>,
			which is the Sobolev conjugate.

			==See also==
			*[[Sergei Lvovich Sobolev]]

			==References==
			* Lawrence C. Evans. Partial differential equations. Graduate studies in Mathematics, Vol 19. American Mathematical Society. 1998. ISBN 0-8218-0772-2

			[[Category:Sobolev spaces]]

en>Edgeh72: /* Advantages */

2012-08-21T20:14:16Z

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Hello. Let me introduce the author. Her name is Refugia Shryock. He used to be unemployed but now he is a pc operator but his promotion never arrives. Playing baseball is the hobby he will by no means stop doing. Years ago we moved to North Dakota and I adore each working day residing here.<br><br>Here is my page :: [http://dtekorea.com/xe/DS_030111/1073195 std home test]