Support polygon - Revision history

en>Yobot: WP:CHECKWIKI error fixes + other fixes using AWB (10065)

2014-03-28T23:54:01Z

WP:CHECKWIKI error fixes + other fixes using AWB (10065)

← Older revision		Revision as of 01:54, 29 March 2014
Line 1:		Line 1:
	~~In [[mathematics]], '''Maclaurin's inequality''', named after [[Colin Maclaurin]], is a refinement~~ of the ~~[[inequality of arithmetic~~ and ~~geometric means]]~~.		The title of the author is Luther. Delaware has usually been my living location and will never move. Managing individuals is how I make money and it's something I truly enjoy. The favorite pastime for him and his children is to drive and now he is attempting to earn money with it.<br><br>My web site; extended auto warranty ([http://www.pr-boot-camp.eu/index.php?mod=users&action=view&id=11672 Recommended Looking at])

	~~Let~~ '~~'a''<sub>1</sub>, ''a''<sub>2</sub>, ..., ''a''<sub>''n''</sub> be [[positive number\|positive]] [[real number]]~~s~~, and for ''k'' = 1, 2, .~~.~~., ''n'' define the averages ''S''<sub>''k''</sub> as follows:~~

	~~:<math> S_k = \frac{\displaystyle \sum_{ 1\leq i_1 < \cdots < i_k \leq n}a_{i_1} a_{i_2} \cdots a_{i_k}}{\displaystyle {n \choose k}}.</math>~~

	The ~~numerator of this fraction~~ is ~~the [[elementary symmetric polynomial]] of degree ''k'' in the ''n'' variables ''a''<sub>1</sub>, ''a''<sub>2</sub>, ..., ''a''<sub>''n''</sub>, that~~ is~~, the sum of all products of ''k'' of the numbers ''a''<sub>1</sub>, ''a''<sub>2</sub>, ..., ''a''<sub>''n''</sub>~~ with ~~the indices in increasing order~~. ~~The denominator is the number of terms in the numerator, the [[binomial coefficient]]~~ <~~math~~>~~\scriptstyle {n\choose k}.~~<~~/math~~>

	~~Maclaurin's inequality is the following chain of~~ [~~[inequalities]]~~:

	~~:<math> S_1 \geq \sqrt{S_2} \geq \sqrt[3]{S_3} \geq \cdots \geq \sqrt[n]{S_n}<~~/~~math>~~

	~~with equality if and only if all the ''a''<sub>''i''<~~/~~sub> are equal~~.

	~~For ''n'' = 2, this gives the usual inequality of arithmetic and geometric means of two numbers. Maclaurin's inequality is well illustrated by the case ''n'' = 4:~~

	~~: <math>~~
	~~\begin{align}~~
	~~& {} \quad \frac{a_1+a_2+a_3+a_4}{4} \\[8pt]~~
	~~& {} \ge \sqrt{\frac{a_1a_2+a_1a_3+a_1a_4+a_2a_3+a_2a_4+a_3a_4}{6}} \\[8pt]~~
	~~& {} \ge \sqrt[3]{\frac{a_1a_2a_3+a_1a_2a_4+a_1a_3a_4+a_2a_3a_4}{4}} \\[8pt]~~
	~~& {} \ge \sqrt[4]{a_1a_2a_3a_4}~~.
	~~\end{align}~~
	</~~math>~~

	~~Maclaurin's inequality can be proved using the [[Newton's inequalities]]~~.

	==~~See also==~~

	* [[Newton's inequalities]]
	* [[Muirhead's inequality]]
	* [[Generalized mean inequality]]

	~~==References==~~

	*{{cite book
	~~\| last = Biler~~
	~~\| first = Piotr~~
	~~\| coauthors = Witkowski, Alfred~~
	~~\| title = Problems in mathematical analysis~~
	~~\| publisher = New York, N.Y.: M. Dekker~~
	~~\| date = 1990~~
	~~\| pages =~~
	~~\| isbn = 0-8247-8312-3~~
	}}

	~~{{PlanetMath attribution\|~~id=~~3835\|title=MacLaurin's Inequality}}~~

	~~[[Category:Real analysis]]~~
	~~[[Category:Inequalities]]~~
	~~[[Category:Symmetric functions]~~]

en>Addbot: Bot: Migrating 1 interwiki links, now provided by Wikidata on d:q3395587

2013-03-16T17:39:53Z

Bot: Migrating 1 interwiki links, now provided by Wikidata on d:q3395587

← Older revision		Revision as of 19:39, 16 March 2013
Line 1:		Line 1:
	~~Hello friend~~. Let ~~me introduce myself~~. ~~I am Ron but I don't like when people use my complete title~~. ~~Delaware has always been my living location and will by no means move~~. ~~I am~~ a ~~manufacturing~~ and ~~distribution officer~~. The ~~favorite hobby for him and his children~~ is ~~to generate and now he~~ is ~~attempting to earn money~~ with it.<br><br>~~Take a look at my web site~~ :: [~~http:~~//~~www~~.~~Chillister~~.~~com/blogs/post/27388 http~~:/~~/www~~.~~Chillister~~.~~com~~]		In [[mathematics]], '''Maclaurin's inequality''', named after [[Colin Maclaurin]], is a refinement of the [[inequality of arithmetic and geometric means]].

			Let ''a''<sub>1</sub>, ''a''<sub>2</sub>, ..., ''a''<sub>''n''</sub> be [[positive number\|positive]] [[real number]]s, and for ''k'' = 1, 2, ..., ''n'' define the averages ''S''<sub>''k''</sub> as follows:

			:<math> S_k = \frac{\displaystyle \sum_{ 1\leq i_1 < \cdots < i_k \leq n}a_{i_1} a_{i_2} \cdots a_{i_k}}{\displaystyle {n \choose k}}.</math>

			The numerator of this fraction is the [[elementary symmetric polynomial]] of degree ''k'' in the ''n'' variables ''a''<sub>1</sub>, ''a''<sub>2</sub>, ..., ''a''<sub>''n''</sub>, that is, the sum of all products of ''k'' of the numbers ''a''<sub>1</sub>, ''a''<sub>2</sub>, ..., ''a''<sub>''n''</sub> with the indices in increasing order. The denominator is the number of terms in the numerator, the [[binomial coefficient]] <math>\scriptstyle {n\choose k}.</math>

			Maclaurin's inequality is the following chain of [[inequalities]]:

			:<math> S_1 \geq \sqrt{S_2} \geq \sqrt[3]{S_3} \geq \cdots \geq \sqrt[n]{S_n}</math>

			with equality if and only if all the ''a''<sub>''i''</sub> are equal.

			For ''n'' = 2, this gives the usual inequality of arithmetic and geometric means of two numbers. Maclaurin's inequality is well illustrated by the case ''n'' = 4:

			: <math>
			\begin{align}
			& {} \quad \frac{a_1+a_2+a_3+a_4}{4} \\[8pt]
			& {} \ge \sqrt{\frac{a_1a_2+a_1a_3+a_1a_4+a_2a_3+a_2a_4+a_3a_4}{6}} \\[8pt]
			& {} \ge \sqrt[3]{\frac{a_1a_2a_3+a_1a_2a_4+a_1a_3a_4+a_2a_3a_4}{4}} \\[8pt]
			& {} \ge \sqrt[4]{a_1a_2a_3a_4}.
			\end{align}
			</math>

			Maclaurin's inequality can be proved using the [[Newton's inequalities]].

			==See also==

			* [[Newton's inequalities]]
			* [[Muirhead's inequality]]
			* [[Generalized mean inequality]]

			==References==

			*{{cite book
			\| last = Biler
			\| first = Piotr
			\| coauthors = Witkowski, Alfred
			\| title = Problems in mathematical analysis
			\| publisher = New York, N.Y.: M. Dekker
			\| date = 1990
			\| pages =
			\| isbn = 0-8247-8312-3
			}}

			{{PlanetMath attribution\|id=3835\|title=MacLaurin's Inequality}}

			[[Category:Real analysis]]
			[[Category:Inequalities]]
			[[Category:Symmetric functions]]

en>Ptbotgourou: r2.6.5) (robot Adding: fr:Polygone de sustentation

2011-07-18T10:41:22Z

r2.6.5) (robot Adding: fr:Polygone de sustentation

New page

Hello friend. Let me introduce myself. I am Ron but I don't like when people use my complete title. Delaware has always been my living location and will by no means move. I am a manufacturing and distribution officer. The favorite hobby for him and his children is to generate and now he is attempting to earn money with it.<br><br>Take a look at my web site :: [http://www.Chillister.com/blogs/post/27388 http://www.Chillister.com]