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en>C.vessiaris: Add the link of Google Image Labeler to Wikipedia

2014-11-02T01:10:26Z

Add the link of Google Image Labeler to Wikipedia

← Older revision		Revision as of 03:10, 2 November 2014
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	In mathematics, a '''Schur-convex function''', also known as '''S-convex''', '''isotonic function''' and '''order-preserving function''' is a [[function (mathematics)\|function]] <math>f: \mathbb{R}^d\rightarrow \mathbb{R}</math>, for which if <math>\forall x,y\in \mathbb{R}^d </math> where <math>x</math> is ~~[[majorization\|majorized]] by <math>y</math>, then <math>f(x)\le f(y)</math>. Named after [[Issai Schur]], Schur-convex functions are used in~~ the ~~study of [[majorization]]. Every function that is [[Convex function\|convex]] and [[Symmetric function\|symmetric]] is also Schur-convex. The opposite implication is not true,~~ but ~~all Schur-convex functions are symmetric (under permutations of the arguments).~~		Alyson Meagher is the title her parents gave her but she doesn't like when individuals use her full name. He is an purchase clerk and it's some thing he truly enjoy. I've always cherished residing in Mississippi. To climb is something I really enjoy doing.<br><br>Here is my blog: [http://modenpeople.co.kr/modn/qna/292291 psychic solutions by lynne]

	~~== Schur-concave function ==~~
	~~A function <math>f</math> is~~ '~~Schur-concave' if its negative,<math>-f</math>, is Schur-convex~~.

	~~==A simple criterion==~~

	~~If <math>f</math>~~ is ~~Schur-convex~~ and ~~all first partial derivatives exist, then the following holds, where <math> f_{(i)}(x) </math> denotes the partial derivative with respect to <math> x_i </math>:~~
	~~:<math> (x_1 - x_2)(f_{(1)}(x) - f_{(2)}(x)) \ge 0~~
	~~</math> for all <math> x </math>~~. ~~Since <math> f </math> is a symmetric function, the above condition implies all the similar conditions for the remaining indexes!~~

	~~== Examples ==~~
	* <math> f(x)=\min(x) </math> is Schur-concave while <math> f(x)=\max(x) </math> is Schur-convex. This can be seen directly from the definition.

	* The [[Shannon entropy]] function <math>\sum_{i=1}^d{P_i \cdot \log_2{\frac{1}{P_i}}}</math> is Schur-concave.

	* The [[Rényi entropy]] function is ~~also Schur-concave~~.

	* <~~math~~> ~~\sum_{i=1}^d{x_i^k},k \ge 1~~ <~~/math~~> is ~~Schur-convex.~~

	* The function <math> f(x) = \prod_{i=1}^n x_i </math> is Schur-concave, when we assume all <math> x_i > 0 </math>. In the same way, all the [~~[Elementary symmetric polynomial\|Elementary symmetric function]]s are Schur-concave, when <math> x_i > 0 </math>.~~

	* A natural interpretation of [[majorization]] is that if <math> x \succ y </~~math> then <math> x <~~/~~math> is more spread out than <math> y </math>~~. ~~So it is natural to ask if statistical measures of variability are Schur-convex~~. ~~The [[variance]] and [[standard deviation]] are Schur-convex functions, while the [[Median absolute deviation]] is not.~~

	* If <math> g </~~math> is a convex function defined on a real interval, then <math> \sum_{i=1}^n g(x_i) <~~/~~math> is Schur-convex.~~

	* Some probability examples: If <math> X_1, \dots, X_n </~~math> are exchangeable random variables, then the function~~
	~~:<math> \text{E} \prod_{j=1}^n X_j^{a_j} </math>~~
	~~is Schur-convex as a function of <math> a=(a_1, \dots, a_n) </math>, assuming that the expectations exist.~~

	* The [[Gini coefficient]] is strictly Schur concave.

	~~==See also==~~
	* [[majorization]]
	* [[Quasiconvex function]]
	* [[Convex function]]

	~~[[Category:Convex analysis]]~~
	~~[[Category:Inequalities]~~]


	~~{{mathanalysis-stub}}~~

en>Andyhowlett: /* Cheating */

2013-10-31T18:16:04Z

Cheating

← Older revision		Revision as of 20:16, 31 October 2013
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	~~The author~~ is ~~recognized~~ by the ~~name~~ of ~~Figures Wunder. For many years I've been operating as a payroll clerk~~. ~~Doing ceramics~~ is ~~what her family members~~ and ~~her appreciate~~. ~~Puerto Rico~~ is ~~where he's usually been residing~~ but ~~she needs to move simply because~~ of ~~her family members~~.<br><br>~~Feel free~~ to ~~visit my site; [http~~://~~torontocartridge~~.~~com~~/~~uncategorized~~/the-~~ideal~~-way-to-~~battle~~-a-~~yeast~~-~~infection~~/ ~~home std test~~]		In mathematics, a '''Schur-convex function''', also known as '''S-convex''', '''isotonic function''' and '''order-preserving function''' is a [[function (mathematics)\|function]] <math>f: \mathbb{R}^d\rightarrow \mathbb{R}</math>, for which if <math>\forall x,y\in \mathbb{R}^d </math> where <math>x</math> is [[majorization\|majorized]] by <math>y</math>, then <math>f(x)\le f(y)</math>. Named after [[Issai Schur]], Schur-convex functions are used in the study of [[majorization]]. Every function that is [[Convex function\|convex]] and [[Symmetric function\|symmetric]] is also Schur-convex. The opposite implication is not true, but all Schur-convex functions are symmetric (under permutations of the arguments).

			== Schur-concave function ==
			A function <math>f</math> is 'Schur-concave' if its negative,<math>-f</math>, is Schur-convex.

			==A simple criterion==

			If <math>f</math> is Schur-convex and all first partial derivatives exist, then the following holds, where <math> f_{(i)}(x) </math> denotes the partial derivative with respect to <math> x_i </math>:
			:<math> (x_1 - x_2)(f_{(1)}(x) - f_{(2)}(x)) \ge 0
			</math> for all <math> x </math>. Since <math> f </math> is a symmetric function, the above condition implies all the similar conditions for the remaining indexes!

			== Examples ==
			* <math> f(x)=\min(x) </math> is Schur-concave while <math> f(x)=\max(x) </math> is Schur-convex. This can be seen directly from the definition.

			* The [[Shannon entropy]] function <math>\sum_{i=1}^d{P_i \cdot \log_2{\frac{1}{P_i}}}</math> is Schur-concave.

			* The [[Rényi entropy]] function is also Schur-concave.

			* <math> \sum_{i=1}^d{x_i^k},k \ge 1 </math> is Schur-convex.

			* The function <math> f(x) = \prod_{i=1}^n x_i </math> is Schur-concave, when we assume all <math> x_i > 0 </math>. In the same way, all the [[Elementary symmetric polynomial\|Elementary symmetric function]]s are Schur-concave, when <math> x_i > 0 </math>.

			* A natural interpretation of [[majorization]] is that if <math> x \succ y </math> then <math> x </math> is more spread out than <math> y </math>. So it is natural to ask if statistical measures of variability are Schur-convex. The [[variance]] and [[standard deviation]] are Schur-convex functions, while the [[Median absolute deviation]] is not.

			* If <math> g </math> is a convex function defined on a real interval, then <math> \sum_{i=1}^n g(x_i) </math> is Schur-convex.

			* Some probability examples: If <math> X_1, \dots, X_n </math> are exchangeable random variables, then the function
			:<math> \text{E} \prod_{j=1}^n X_j^{a_j} </math>
			is Schur-convex as a function of <math> a=(a_1, \dots, a_n) </math>, assuming that the expectations exist.

			* The [[Gini coefficient]] is strictly Schur concave.

			==See also==
			* [[majorization]]
			* [[Quasiconvex function]]
			* [[Convex function]]

			[[Category:Convex analysis]]
			[[Category:Inequalities]]


			{{mathanalysis-stub}}

en>Hmainsbot1: /* Idea */AWB general fixes and delink dates per :WP:DATELINK, :WP:YEARLINK and :MOS:UNLINKYEARS using AWB (8097)

2012-08-04T05:17:26Z

Idea: AWB general fixes and delink dates per WP:DATELINK, WP:YEARLINK and MOS:UNLINKYEARS using AWB (8097)

New page

The author is recognized by the name of Figures Wunder. For many years I've been operating as a payroll clerk. Doing ceramics is what her family members and her appreciate. Puerto Rico is where he's usually been residing but she needs to move simply because of her family members.<br><br>Feel free to visit my site; [http://torontocartridge.com/uncategorized/the-ideal-way-to-battle-a-yeast-infection/ home std test]