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In mathematics, subadditivity is a property of a function that states, roughly, that evaluating the function for a sum of two numbers always returns something less than or equal to the sum of the function's values at each number. Image File history File links Broom_icon. ...

In mathematics, a subadditive function is a function , having a domain A and a codomain B that are both closed under addition, with the following property: Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... Graph of example function, The mathematical concept of a function expresses the intuitive idea of deterministic dependence between two quantities, one of which is viewed as primary (the independent variable, argument of the function, or its input) and the other as secondary (the value of the function, or output). A... In mathematics, the domain of a function is the set of all input values to the function. ... A codomain in mathematics is the set of output values associated with (or mapped to) the domain of inputs in a function. ... In mathematics, the closure C(X) of an object X is defined to be the smallest object that both includes X as a subset and possesses some given property. ...

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An example is the square root function, having the non-negative real numbers as domain and codomain, since we have: In mathematics, a square root of a number x is a number r such that , or in words, a number r whose square (the result of multiplying the number by itself) is x. ... A negative number is a number that is less than zero, such as −3. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ...

A sequence , is called subadditive if it satisfies the inequality For other senses of this word, see sequence (disambiguation). ... This article is about inequalities in mathematics. ...

for all m and n. The major reason for use of subadditive sequences is the following lemma due to M. Fekete.^[1] In mathematics, a lemma is a proven proposition which is used as a stepping stone to a larger result rather than an independent statement, in and of itself. ...

Lemma: For every subadditive sequence , the limit exists and is equal to . (The limit may be .)

The analogue of Fekete's lemma holds for superadditive functions as well, that is: (The limit then may be positive infinity: consider the sequence a_n = logn!.)

There are extensions of Fekete's lemma that do not require equation (1) 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.^[2] A sequence { an }, n ≥ 1, is called superadditive if it satisfies the inequality for all m and n. ...

See also

• Triangle inequality

In mathematics, triangle inequality is the theorem stating that for any triangle, the measure of a given side must be less than the sum of the other two sides but greater than the difference between the two sides. ...

References

1. ^ Fekete, M. "Uber die Verteilung der Wurzeln bei gewissen algebraischen Gleichungen mit. ganzzahligen Koeffizienten." Mathematische Zeitschrift 17 (1923), pp. 228–249.
2. ^ Michael J. Steele. "Probability theory and combinatorial optimization". SIAM, Philadelphia (1997). ISBN 0-89871-380-3.

• György Pólya and Gábor Szegö. "Problems and theorems in analysis, volume 1". Springer-Verlag, New York (1976). ISBN 0-387-05672-6.

George Pólya ca 1973 George Pólya (December 13, 1887 – September 7, 1985, in Hungarian Pólya György) was a Hungarian mathematician. ... Gábor Szegő (January 20, 1895 - August 7, 1985) was a Hungarian mathematician. ...

External link

This article incorporates material from subadditivity on PlanetMath, which is licensed under the GFDL. PlanetMath is a free, collaborative, online mathematics encyclopedia. ...