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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] |
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| : <math>a_{n+m} \geq a_n+a_m\,</math>
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| for all ''m'' and ''n''. The major reason for the use of superadditive sequences is the following [[lemma (mathematics)|lemma]] due to [[Michael Fekete]].
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| '''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>.)
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| Similarly, a [[function (mathematics)|function]] ''f''(''x'') is ''superadditive'' if
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| : <math>f(x+y) \geq f(x)+f(y)\,</math>
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| for all ''x'' and ''y'' in the [[domain (mathematics)|domain]] of ''f''.
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| 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>.
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| The analogue of Fekete's lemma holds for subadditive functions as well.
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| 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>
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| 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>
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| The negative of a superadditive function is [[subadditivity|subadditive]]. | |
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| == Examples of superadditive functions ==
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| * [[Mutual information]]
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| * 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.
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| ==See also==
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| *[[Subadditivity]]
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| *[[Choquet integral]]
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| == References ==
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| {{reflist}}
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| ;Notes
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| *{{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}}
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| {{PlanetMath attribution|id=4616|title=Superadditivity}}
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| [[Category:Mathematical analysis]]
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| [[Category:Sequences and series]]
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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.
Here is my web blog The Skywoods title