Cross-flow filtration - Revision history

en>Discospinster: Reverted edits by 117.223.88.38 (talk) to last revision by Ndemarco (HG)

2014-12-14T20:18:07Z

Reverted edits by 117.223.88.38 (talk) to last revision by Ndemarco (HG)

← Older revision		Revision as of 22:18, 14 December 2014
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	~~Wilber Berryhill is what his spouse loves to call~~ him ~~and he completely enjoys this title. To perform lacross is some thing he would never give up~~. ~~Invoicing~~ is what I do for ~~[http://hknews~~.~~classicmall.com.hk/groups/some-simple-tips-for-personal-development-progress/ real psychics]~~ a ~~living but~~ I'~~ve always wanted my own business~~. Her family life in ~~Ohio~~ but ~~her husband wants them~~ to move.<br><br>~~Also visit~~ my ~~web blog :: love [http://test.jeka-nn.ru/node/129 live psychic reading] readings~~; [http://~~www~~.~~weddingwall.com.au~~/~~groups~~/~~easy-advice-for-successful-personal-development-today~~/ mouse click the up coming ~~internet~~ site],		Friends contact him Royal. Interviewing is what I do for a residing but I strategy on altering it. Playing crochet is a factor that I'm completely addicted to. Her family life in Delaware but she requirements to move simply because of her family.<br><br>my website car warranty; [http://nauri.biz/xe/test/937675 mouse click the up coming web site],

en>Monkbot: Fix CS1 deprecated date parameter errors

2014-02-04T12:29:05Z

Fix CS1 deprecated date parameter errors

← Older revision		Revision as of 14:29, 4 February 2014
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	~~In [[mathematics]], the '''Brunn–Minkowski theorem''' (or '''Brunn–Minkowski inequality''')~~ is ~~an inequality relating the volumes (or more generally [[Lebesgue measure]]s) of [[compact space\|compact]] [[subset]]s of [[Euclidean space]]~~. ~~The original version of the Brunn–Minkowski theorem ([[Hermann Brunn]] 1887; [[Hermann Minkowski]] 1896) applied to convex sets; the generalization to compact nonconvex sets stated here~~ is ~~due to [[Lazar Lyusternik\|L~~. ~~A. Lyusternik]] (1935).~~		Wilber Berryhill is what his spouse loves to call him and he completely enjoys this title. To perform lacross is some thing he would never give up. Invoicing is what I do for [http://hknews.classicmall.com.hk/groups/some-simple-tips-for-personal-development-progress/ real psychics] a living but I've always wanted my own business. Her family life in Ohio but her husband wants them to move.<br><br>Also visit my web blog :: love [http://test.jeka-nn.ru/node/129 live psychic reading] readings; [http://www.weddingwall.com.au/groups/easy-advice-for-successful-personal-development-today/ mouse click the up coming internet site],

	~~==Statement of the theorem==~~
	~~Let ''n'' ≥ 1 and let ''μ'' denote the~~ [~~[Lebesgue measure]] on '''R'''<sup>''n''<~~/~~sup>~~. ~~Let ''A'' and ''B'' be two nonempty compact subsets of '''R'''<sup>''n''</sup>~~. ~~Then the following [[inequality (mathematics)\|inequality]] holds:~~

	~~:<math>[ \mu (A + B) ]^{1~~/~~n} \geq [\mu (A)]^{1~~/~~n} + [\mu (B)]^{1/n},<~~/~~math>~~

	~~where ''A'' + ''B'' denotes the [[Minkowski sum]~~]:

	~~:<math>A + B := \{\,~~ a ~~+ b \~~in ~~\mathbb{R}^{n} \mid a \in A,\ b \in B \,\}~~.<~~/math~~>

	~~==Remarks==~~
	~~The proof of the Brunn–Minkowski theorem establishes that the function~~

	:<~~math~~>~~A \mapsto~~ [~~\mu (A)]^{1~~/~~n}<~~/~~math>~~

	~~is [[concave function\|concave]] in the sense that, for every pair of nonempty compact subsets ''A'' and ''B'' of '''R'''<sup>''n''<~~/~~sup> and every 0 ≤ ''t'' ≤ 1,~~

	~~:<math>\left[ \mu (t A + (1 - t) B ) \right]^{1~~/~~n} \geq t [ \mu (A)~~ ]~~^{1/n} + (1 - t)~~ [ ~~\mu (B) ]^{1~~/~~n}.<~~/~~math>~~

	~~For [[convex set\|convex]] sets ''A'' and ''B'', the inequality in the theorem is strict~~
	~~for 0 < ''t'' < 1 unless ''A'' and ''B'' are [[homothetic]], i~~.e. ~~are equal up to [[translation (geometry)\|translation]] and [[Scaling (geometry)\|dilation]]~~.

	~~==See also==~~
	* [[Isoperimetric inequality]]
	* [[Milman's reverse Brunn–Minkowski inequality]]
	* [[Minkowski–Steiner formula]]
	* [[Prékopa–Leindler inequality]]
	* [[Vitale's random Brunn–Minkowski inequality]]

	~~==References==~~
	* {{cite journal \| author=Brunn, H. \| author-~~link=Hermann Brunn \| title=Über Ovale und Eiflächen \| year = 1887 \| version=Inaugural Dissertation, München}}~~
	*{{cite book
	~~\| last=Fenchel~~
	~~\| first=Werner~~
	~~\| author~~-~~link = Werner Fenchel~~
	~~\| coauthors=Bonnesen, Tommy~~
	~~\| title=Theorie der konvexen Körper~~
	~~\| series=Ergebnisse der Mathematik und ihrer Grenzgebiete~~
	~~\| volume=3~~
	~~\| publisher=1. Verlag von Julius Springer~~
	~~\| location=Berlin~~
	~~\| year=1934~~
	}}
	*{{cite book
	~~\| last=Fenchel~~
	~~\| first=Werner~~
	~~\| author-link=Werner Fenchel~~
	~~\| coauthors=Bonnesen, Tommy~~
	~~\| title=Theory of convex bodies~~
	~~\| publisher=L. Boron, C. Christenson and B. Smith. BCS Associates~~
	~~\| location=Moscow, Idaho~~
	~~\| year=1987~~
	}}
	* {{cite book \| last=Dacorogna \| first=Bernard \| title=Introduction to the Calculus of Variations \| publisher=Imperial College Press \| location=London \| year=2004 \| isbn=1-86094-508-~~2 \| unused_data=\|ISBN status=May be invalid – please double check }}~~
	* [[Heinrich Guggenheimer]] (1977) ''Applicable Geometry'', page 146, Krieger, Huntington ISBN 0-~~88275~~-~~368~~-~~1 .~~
	* {{cite journal \| last=Lyusternik \| first=Lazar A. \| authorlink=Lazar Lyusternik \| title=Die Brunn–Minkowskische Ungleichnung für beliebige messbare Mengen \| journal = Comptes Rendus (Doklady) de l'académie des Sciences de l'uRSS (Nouvelle Série) \| volume = III \| year = 1935 \| pages = 55–58}}
	* {{cite book \| last=Minkowski \| first=Hermann \| authorlink=Hermann Minkowski \| title = Geometrie der Zahlen \| location = Leipzig \| publisher = Teubner \| year = 1896}}
	* {{cite article\|last=Ruzsa\|first=Imre Z.\|authorlink=Imre Z. Ruzsa\|title=The Brunn–Minkowski inequality and nonconvex sets\|journal=Geometriae Dedicata\|volume=67\|doi=10.1023/A:1004958110076 \|doi=10.1023/~~A:1004958110076\|year=1997\|number=3\|pages=337–348\|mr=1475877}}~~
	* Rolf Schneider, ''Convex bodies: the ~~Brunn–Minkowski theory~~,~~'' Cambridge University Press, Cambridge, 1993.~~

	~~{{DEFAULTSORT:Brunn-Minkowski theorem}}~~
	~~[[Category:Theorems in measure theory]]~~
	~~[[Category:Theorems in convex geometry]]~~
	~~[[Category:Calculus of variations]]~~
	~~[[Category:Geometric inequalities]]~~
	~~[[Category:Sumsets]]~~

en>Peter in s: /* Benefits over conventional filtration */

2013-09-09T08:42:44Z

Benefits over conventional filtration

← Older revision		Revision as of 10:42, 9 September 2013
Line 1:		Line 1:
	~~Wilber Berryhill~~ is the ~~title his parents gave him~~ and ~~he completely digs that title~~. ~~For a while I~~'~~ve been in Mississippi but now I~~'~~m considering other choices. My day job is an info officer but I~~'~~ve currently applied for another one. It~~'~~s not a typical thing but what she likes doing is to perform domino but she doesn~~'~~t online psychic reading -~~ [~~http~~://1.~~234.36.240~~/~~fxac~~/~~m001_2~~/~~7330 36.240~~], ~~have the time lately.~~<br><br>~~Also visit my web blog~~ :~~: good~~ [~~http:~~//~~fashionlinked~~.~~com~~/~~index~~.~~php?do~~=~~/profile-13453/info/ best psychic~~], ~~[http://brazil~~.~~amor~~-~~amore~~.~~com/irboothe amor~~-~~amore~~.~~com~~],		In [[mathematics]], the '''Brunn–Minkowski theorem''' (or '''Brunn–Minkowski inequality''') is an inequality relating the volumes (or more generally [[Lebesgue measure]]s) of [[compact space\|compact]] [[subset]]s of [[Euclidean space]]. The original version of the Brunn–Minkowski theorem ([[Hermann Brunn]] 1887; [[Hermann Minkowski]] 1896) applied to convex sets; the generalization to compact nonconvex sets stated here is due to [[Lazar Lyusternik\|L. A. Lyusternik]] (1935).

			==Statement of the theorem==
			Let ''n'' ≥ 1 and let ''μ'' denote the [[Lebesgue measure]] on '''R'''<sup>''n''</sup>. Let ''A'' and ''B'' be two nonempty compact subsets of '''R'''<sup>''n''</sup>. Then the following [[inequality (mathematics)\|inequality]] holds:

			:<math>[ \mu (A + B) ]^{1/n} \geq [\mu (A)]^{1/n} + [\mu (B)]^{1/n},</math>

			where ''A'' + ''B'' denotes the [[Minkowski sum]]:

			:<math>A + B := \{\, a + b \in \mathbb{R}^{n} \mid a \in A,\ b \in B \,\}.</math>

			==Remarks==
			The proof of the Brunn–Minkowski theorem establishes that the function

			:<math>A \mapsto [\mu (A)]^{1/n}</math>

			is [[concave function\|concave]] in the sense that, for every pair of nonempty compact subsets ''A'' and ''B'' of '''R'''<sup>''n''</sup> and every 0 ≤ ''t'' ≤ 1,

			:<math>\left[ \mu (t A + (1 - t) B ) \right]^{1/n} \geq t [ \mu (A) ]^{1/n} + (1 - t) [ \mu (B) ]^{1/n}.</math>

			For [[convex set\|convex]] sets ''A'' and ''B'', the inequality in the theorem is strict
			for 0 < ''t'' < 1 unless ''A'' and ''B'' are [[homothetic]], i.e. are equal up to [[translation (geometry)\|translation]] and [[Scaling (geometry)\|dilation]].

			==See also==
			* [[Isoperimetric inequality]]
			* [[Milman's reverse Brunn–Minkowski inequality]]
			* [[Minkowski–Steiner formula]]
			* [[Prékopa–Leindler inequality]]
			* [[Vitale's random Brunn–Minkowski inequality]]

			==References==
			* {{cite journal \| author=Brunn, H. \| author-link=Hermann Brunn \| title=Über Ovale und Eiflächen \| year = 1887 \| version=Inaugural Dissertation, München}}
			*{{cite book
			\| last=Fenchel
			\| first=Werner
			\| author-link = Werner Fenchel
			\| coauthors=Bonnesen, Tommy
			\| title=Theorie der konvexen Körper
			\| series=Ergebnisse der Mathematik und ihrer Grenzgebiete
			\| volume=3
			\| publisher=1. Verlag von Julius Springer
			\| location=Berlin
			\| year=1934
			}}
			*{{cite book
			\| last=Fenchel
			\| first=Werner
			\| author-link=Werner Fenchel
			\| coauthors=Bonnesen, Tommy
			\| title=Theory of convex bodies
			\| publisher=L. Boron, C. Christenson and B. Smith. BCS Associates
			\| location=Moscow, Idaho
			\| year=1987
			}}
			* {{cite book \| last=Dacorogna \| first=Bernard \| title=Introduction to the Calculus of Variations \| publisher=Imperial College Press \| location=London \| year=2004 \| isbn=1-86094-508-2 \| unused_data=\|ISBN status=May be invalid – please double check }}
			* [[Heinrich Guggenheimer]] (1977) ''Applicable Geometry'', page 146, Krieger, Huntington ISBN 0-88275-368-1 .
			* {{cite journal \| last=Lyusternik \| first=Lazar A. \| authorlink=Lazar Lyusternik \| title=Die Brunn–Minkowskische Ungleichnung für beliebige messbare Mengen \| journal = Comptes Rendus (Doklady) de l'académie des Sciences de l'uRSS (Nouvelle Série) \| volume = III \| year = 1935 \| pages = 55–58}}
			* {{cite book \| last=Minkowski \| first=Hermann \| authorlink=Hermann Minkowski \| title = Geometrie der Zahlen \| location = Leipzig \| publisher = Teubner \| year = 1896}}
			* {{cite article\|last=Ruzsa\|first=Imre Z.\|authorlink=Imre Z. Ruzsa\|title=The Brunn–Minkowski inequality and nonconvex sets\|journal=Geometriae Dedicata\|volume=67\|doi=10.1023/A:1004958110076 \|doi=10.1023/A:1004958110076\|year=1997\|number=3\|pages=337–348\|mr=1475877}}
			* Rolf Schneider, ''Convex bodies: the Brunn–Minkowski theory,'' Cambridge University Press, Cambridge, 1993.

			{{DEFAULTSORT:Brunn-Minkowski theorem}}
			[[Category:Theorems in measure theory]]
			[[Category:Theorems in convex geometry]]
			[[Category:Calculus of variations]]
			[[Category:Geometric inequalities]]
			[[Category:Sumsets]]

en>Wavelength: revising letter case—MOS:HEAD

2012-06-19T15:46:29Z

revising letter case—MOS:HEAD

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