Birkhoff interpolation - Revision history

en>Yobot: WP:CHECKWIKI error fixes / Tagging using AWB (10691)

2015-01-06T21:56:41Z

WP:CHECKWIKI error fixes / Tagging using AWB (10691)

← Older revision		Revision as of 23:56, 6 January 2015
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	~~In mathematics~~, ~~a [[sequence]] { ''a<sub>n</sub>'' }~~, ~~''n'' ≥ 1, is called '''superadditive''' if it satisfies the [[inequality (mathematics)\|inequality]]~~		Microbiologist Verderber from Clifford, spends time with interests which includes fast, new launch property singapore and bee keeping. In recent time took some time to make an expedition to Swartkrans.<br><br>Here is my web blog [http://maritademaine.soup.io/ The Skywoods title]

	~~: <math>a_{n+m} \geq a_n+a_m\,</math>~~

	~~for all ''m''~~ and ~~''n''~~. ~~The major reason for the use of superadditive sequences is the following [[lemma (mathematics)\|lemma]] due to [[Michael Fekete]].~~

	~~'''Lemma:''' (Fekete) For every superadditive sequence { ''a<sub>n</sub>'' }, ''n'' ≥ 1, the limit lim ''a<sub>n</sub>''/''n'' exists and equal~~ to ~~sup ''a<sub>n</sub>''/''n''.~~ ~~(The limit may be positive infinity, for instance, for the sequence ''a<sub>n</sub>'' = log ''n''<nowiki>!</nowiki>.)~~

	~~Similarly, a [[function (mathematics)\|function]] ''f''(''x'') is ''superadditive'' if~~

	~~: <math>f(x+y) \geq f(x)+f(y)\,</math>~~

	~~for all ''x'' and ''y'' in the [[domain (mathematics)\|domain]] of ''f''~~.

	~~For example,~~ <~~math~~>~~f(x)=x^2~~<~~/math~~> is a superadditive function for nonnegative real numbers because the square of <math>(x+y)</math> is always greater than or equal to the square of <math>x</math> plus the square of <math>y</math>, for nonnegative real numbers <math>x</math> and <math>y</math>.

	~~The analogue of Fekete's lemma holds for subadditive functions as well.~~
	There are extensions of Fekete's lemma that do not require the definition of superadditivity above to hold for all ''m'' and ''n''. There are also results that allow one to deduce the rate of convergence to the limit whose existence is stated in Fekete's lemma if some kind of both superadditivity and [[subadditivity]] is present. A good exposition of this topic may be found in Steele (1997).<ref>{{cite book\|author=Michael J. Steele\|title=Probability theory and combinatorial optimization\|publisher=SIAM, Philadelphia\|year=1997\|isbn=0-89871-380-3}}</ref><ref>{{cite video\|author=Michael J. Steele\|title=CBMS Lectures on Probability Theory and Combinatorial Optimization\|publisher=University of Cambridge\|year=2011\|url=http://~~sms.cam~~.ac.~~uk/collection/1189351}}<~~/~~ref>~~

	If ''f'' is a superadditive function, and if 0 is in its domain, then ''f''(0) ≤ 0. To see this, take the inequality at the top. <math>f(x) \le f(x+y) - f(y)</math>. Hence <math>f(0) \le f(0+y) - f(y) = 0</math>

	The ~~negative of a superadditive function is [[subadditivity\|subadditive]].~~

	~~== Examples of superadditive functions ==~~
	* [[Mutual information]]
	* Horst Alzer proved <ref>{{cite book\|author=Horst Alzer\|title=A superadditive property of Hadamard’s gamma function\| publisher=Springer\|year=2009\| doi=10.1007/s12188-008-0009-5}}</ref> that [[Hadamard’s gamma function]] H(''x'') is superadditive for all real numbers ''x'',''y'' with ''x'', ''y'' ≥ 1.5031.

	~~==See also==~~
	*[[Subadditivity]]
	*[[Choquet integral]]

	~~== References ==~~
	~~{{reflist}}~~
	~~;Notes~~
	*{{cite book\|author=György Polya and Gábor Szegö.\|title=Problems and theorems in analysis, volume 1\|publisher=Springer-Verlag, New York\|year=1976\|isbn=0-387-05672-6}}

	~~{{PlanetMath attribution\|id=4616\|title=Superadditivity}}~~

	~~[[Category:Mathematical analysis]]~~
	~~[[Category:Sequences and series]~~]

en>Khaianne: Joined a hanging sentence to its counterpart to make a cohesive sentence.

2013-09-18T14:59:37Z

Joined a hanging sentence to its counterpart to make a cohesive sentence.

← Older revision		Revision as of 16:59, 18 September 2013
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	~~Hello. Let me introduce~~ the ~~writer~~. ~~Her title~~ ~~healthy food delivery~~ is ~~Emilia Shroyer but~~ ~~healthy food delivery it~~'~~s not the most female title out there~~. [~~http~~://~~Healthyeater.com~~/~~diet-food-delivery-services Minnesota] has~~ always ~~been his house but his spouse desires them~~ to ~~transfer~~. To do ~~aerobics is a thing that I~~'m ~~totally addicted~~ to~~. Since she was 18 she~~'s ~~been operating as a receptionist but her~~ [http://~~Www~~.~~Deliverlean~~.~~com~~/ ~~promotion] never comes~~.<br><br>~~Here~~ is ~~my web page :: meal meal delivery service delivery service;~~ [~~http:~~//s.~~Argucracy~~.~~de/diettogoreviews74620 try what he says~~],		In mathematics, a [[sequence]] { ''a<sub>n</sub>'' }, ''n'' ≥ 1, is called '''superadditive''' if it satisfies the [[inequality (mathematics)\|inequality]]

			: <math>a_{n+m} \geq a_n+a_m\,</math>

			for all ''m'' and ''n''. The major reason for the use of superadditive sequences is the following [[lemma (mathematics)\|lemma]] due to [[Michael Fekete]].

			'''Lemma:''' (Fekete) For every superadditive sequence { ''a<sub>n</sub>'' }, ''n'' ≥ 1, the limit lim ''a<sub>n</sub>''/''n'' exists and equal to sup ''a<sub>n</sub>''/''n''. (The limit may be positive infinity, for instance, for the sequence ''a<sub>n</sub>'' = log ''n''<nowiki>!</nowiki>.)

			Similarly, a [[function (mathematics)\|function]] ''f''(''x'') is ''superadditive'' if

			: <math>f(x+y) \geq f(x)+f(y)\,</math>

			for all ''x'' and ''y'' in the [[domain (mathematics)\|domain]] of ''f''.

			For example, <math>f(x)=x^2</math> is a superadditive function for nonnegative real numbers because the square of <math>(x+y)</math> is always greater than or equal to the square of <math>x</math> plus the square of <math>y</math>, for nonnegative real numbers <math>x</math> and <math>y</math>.

			The analogue of Fekete's lemma holds for subadditive functions as well.
			There are extensions of Fekete's lemma that do not require the definition of superadditivity above to hold for all ''m'' and ''n''. There are also results that allow one to deduce the rate of convergence to the limit whose existence is stated in Fekete's lemma if some kind of both superadditivity and [[subadditivity]] is present. A good exposition of this topic may be found in Steele (1997).<ref>{{cite book\|author=Michael J. Steele\|title=Probability theory and combinatorial optimization\|publisher=SIAM, Philadelphia\|year=1997\|isbn=0-89871-380-3}}</ref><ref>{{cite video\|author=Michael J. Steele\|title=CBMS Lectures on Probability Theory and Combinatorial Optimization\|publisher=University of Cambridge\|year=2011\|url=http://sms.cam.ac.uk/collection/1189351}}</ref>

			If ''f'' is a superadditive function, and if 0 is in its domain, then ''f''(0) ≤ 0. To see this, take the inequality at the top. <math>f(x) \le f(x+y) - f(y)</math>. Hence <math>f(0) \le f(0+y) - f(y) = 0</math>

			The negative of a superadditive function is [[subadditivity\|subadditive]].

			== Examples of superadditive functions ==
			* [[Mutual information]]
			* Horst Alzer proved <ref>{{cite book\|author=Horst Alzer\|title=A superadditive property of Hadamard’s gamma function\| publisher=Springer\|year=2009\| doi=10.1007/s12188-008-0009-5}}</ref> that [[Hadamard’s gamma function]] H(''x'') is superadditive for all real numbers ''x'',''y'' with ''x'', ''y'' ≥ 1.5031.

			==See also==
			*[[Subadditivity]]
			*[[Choquet integral]]

			== References ==
			{{reflist}}
			;Notes
			*{{cite book\|author=György Polya and Gábor Szegö.\|title=Problems and theorems in analysis, volume 1\|publisher=Springer-Verlag, New York\|year=1976\|isbn=0-387-05672-6}}

			{{PlanetMath attribution\|id=4616\|title=Superadditivity}}

			[[Category:Mathematical analysis]]
			[[Category:Sequences and series]]

en>Headbomb: Various citation & identifier cleanup, plus AWB genfixes. using AWB

2011-09-10T05:01:07Z

Various citation & identifier cleanup, plus AWB genfixes. using AWB

New page

Hello. Let me introduce the writer. Her title healthy food delivery is Emilia Shroyer but healthy food delivery it's not the most female title out there. [http://Healthyeater.com/diet-food-delivery-services Minnesota] has always been his house but his spouse desires them to transfer. To do aerobics is a thing that I'm totally addicted to. Since she was 18 she's been operating as a receptionist but her [http://Www.Deliverlean.com/ promotion] never comes.<br><br>Here is my web page :: meal meal delivery service delivery service; [http://s.Argucracy.de/diettogoreviews74620 try what he says],