Mouse keys - Revision history

76.113.36.83: the description of the key command to activate Mouse keys was not specific. the key command only works with the left shift bar, not just any shift bar.

2014-11-10T16:11:50Z

the description of the key command to activate Mouse keys was not specific. the key command only works with the left shift bar, not just any shift bar.

← Older revision		Revision as of 18:11, 10 November 2014
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	~~In [[mathematical analysis]] (in particular [[convex analysis]]) and [[optimization (mathematics)\|optimization]], a '''proper convex function'''~~ is ~~a [[convex function]]~~ ''~~f'' taking values in the [[extended real number line]]~~ such that		Catrina Le is what's posted on her birth card though she doesn't extremely like being called such as that. Software achieving is where her primary income comes from remember, though , soon her husband and her will start your own business. What the woman loves doing is to visit to karaoke but she's thinking on starting a new generation. For years she's been living in Vermont. She is running and verifying tire pressures regularly a blog here: http://[http://Www.Twitpic.com/tag/prometeu.net prometeu.net]<br><br>My weblog - [http://prometeu.net hack clash of clans iphone]

	~~:<math>f(x) < +\infty</math>~~

	~~for at least one ''x'' and~~

	~~:<math>f(x) > -\infty</math>~~

	~~for every ''x''~~. ~~That~~ is, ~~a convex function is ''proper'' if its [[effective domain]] is nonempty~~ and ~~it never attains <math>-\infty</math>~~.~~<ref name="AB">{{cite book\|last1=Aliprantis\|first1=C.D.\|last2=Border\|first2=K.C.\|title=Infinite Dimensional Analysis: A Hitchhiker~~'s ~~Guide\|edition=3\|publisher=Springer\|year=2007\|isbn=978-3-540-32696-0\|doi=10~~.~~1007/3-540-29587-9\|page=254}}</ref> Convex functions that are not proper are called~~ '~~'improper convex functions''.<ref>{{cite book\|author=[[Rockafellar, R~~. ~~Tyrrell]]\|title=Convex Analysis\|publisher=Princeton University Press\|location=Princeton, NJ\|year=1997\|origyear=1970\|isbn=978-0-691-01586-6\|page=24}}</ref>~~

	~~A ''proper concave function'' is any function ''g'' such that <math>f = -g</math>~~ is a ~~proper convex function.~~

	~~== Properties ==~~

	~~For every proper convex function ''f'' on '''R'''<sup>n<~~/~~sup> there exist some ''b'' in '''R'''<sup>n<~~/~~sup> and β in '''R''' such that~~

	:~~<math>f(x) \ge x \cdot b - \beta<~~/~~math>~~

	~~for every ''x''~~.

	~~The sum of two proper convex functions is not necessarily proper or convex~~. ~~For instance if the sets <math>A \subset X<~~/~~math> and <math>B \subset X<~~/~~math> are [[convex set~~]~~]s in the [[vector space]] ''X'', then the [[Characteristic function (convex analysis)\|indicator function]]s~~ <~~math~~>~~I_A~~<~~/math~~> ~~and <math>I_B</math> are proper convex functions, but <math>I_A + I_B</math> is not convex (unless <math>A \cup B</math> is convex), and is identically equal to <math>+\infty<~~/~~math> if <math>A \cap B = \emptyset<~~/~~math>.~~

	~~The [[infimal convolute\|infimal convolution]] of two proper convex functions is convex but not necessarily proper convex~~.~~{{Citation needed\|date=October 2011}}~~

	~~== References ==~~
	~~{{Reflist}}~~

	~~[[Category:Convex analysis]]~~
	~~[[Category:Types~~ of ~~functions]~~]

63.92.247.41 at 20:59, 14 January 2014

2014-01-14T20:59:53Z

← Older revision		Revision as of 22:59, 14 January 2014
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	~~Person who wrote~~ the [~~http~~://~~Www~~.~~Bing~~.~~com/search?q~~=~~article&form~~=~~MSNNWS&mkt~~=en-~~us&pq~~=~~article article] is~~ called ~~Roberto Ledbetter and his~~ [~~http:~~//~~browse~~.~~deviantart.com~~/~~?q=wife+doesn%27t wife doesn~~'~~t] like it at all of. His wife~~ and ~~your guy live~~ in ~~Idaho but he needs to glide because~~ of ~~his family. In his professional life he~~ is ~~a filing assistant together with his salary has been recently really fulfilling~~. ~~To go to karaoke is~~ the ~~thing he loves most of nearly all. He~~'s ~~been working on any website for some spare time now. Check it available here: http:~~//~~circuspartypanama.com~~<br><br>~~Also visit my weblog :~~: [~~http~~:~~//circuspartypanama.com clash~~ of ~~clans unlimited gems~~]		In [[mathematical analysis]] (in particular [[convex analysis]]) and [[optimization (mathematics)\|optimization]], a '''proper convex function''' is a [[convex function]] ''f'' taking values in the [[extended real number line]] such that

			:<math>f(x) < +\infty</math>

			for at least one ''x'' and

			:<math>f(x) > -\infty</math>

			for every ''x''. That is, a convex function is ''proper'' if its [[effective domain]] is nonempty and it never attains <math>-\infty</math>.<ref name="AB">{{cite book\|last1=Aliprantis\|first1=C.D.\|last2=Border\|first2=K.C.\|title=Infinite Dimensional Analysis: A Hitchhiker's Guide\|edition=3\|publisher=Springer\|year=2007\|isbn=978-3-540-32696-0\|doi=10.1007/3-540-29587-9\|page=254}}</ref> Convex functions that are not proper are called ''improper convex functions''.<ref>{{cite book\|author=[[Rockafellar, R. Tyrrell]]\|title=Convex Analysis\|publisher=Princeton University Press\|location=Princeton, NJ\|year=1997\|origyear=1970\|isbn=978-0-691-01586-6\|page=24}}</ref>

			A ''proper concave function'' is any function ''g'' such that <math>f = -g</math> is a proper convex function.

			== Properties ==

			For every proper convex function ''f'' on '''R'''<sup>n</sup> there exist some ''b'' in '''R'''<sup>n</sup> and β in '''R''' such that

			:<math>f(x) \ge x \cdot b - \beta</math>

			for every ''x''.

			The sum of two proper convex functions is not necessarily proper or convex. For instance if the sets <math>A \subset X</math> and <math>B \subset X</math> are [[convex set]]s in the [[vector space]] ''X'', then the [[Characteristic function (convex analysis)\|indicator function]]s <math>I_A</math> and <math>I_B</math> are proper convex functions, but <math>I_A + I_B</math> is not convex (unless <math>A \cup B</math> is convex), and is identically equal to <math>+\infty</math> if <math>A \cap B = \emptyset</math>.

			The [[infimal convolute\|infimal convolution]] of two proper convex functions is convex but not necessarily proper convex.{{Citation needed\|date=October 2011}}

			== References ==
			{{Reflist}}

			[[Category:Convex analysis]]
			[[Category:Types of functions]]

en>Grandfatherclok: override erroneous editor application of case

2012-08-10T04:38:08Z

override erroneous editor application of case

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