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Wikipedia / Subadditivity

In mathematics, subadditivity is a property of a function that states, roughly, that evaluating the function for the sum of two elements of the domain always returns something less than or equal to the sum of the function's values at each element. There are numerous examples of subadditive functions in various areas of mathematics, particularly norms and square roots. Additive maps are special cases of subadditive functions.

Definitions

A subadditive function is a function, having a domain A and an ordered codomain B that are both closed

An example is the square root function, having the non-negative real numbers

A sequence is called subadditive if it satisfies the inequality for all m and n. This is a special case of subadditive function, if a sequence is interpreted as a function on the set of natural numbers.

; then the sequence is subadditive but not concave.

Properties

Sequences

A useful result pertaining to subadditive sequences is Fekete's lemma.

.)

to hold for all m and n, but only for m and n

is an increasing function converges (near the infinity).

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.

Additionally, analogues of Fekete's lemma have been proven for subadditive real maps (with additional assumptions) from finite subsets of an amenable group, and further, of a cancellative left-amenable semigroup.

Functions

If f is a subadditive function, and if 0 is in its domain, then f

A concave function, will finally verify that f is subadditive.

The negative of a subadditive function is superadditive.

Examples in various domains

Entropy

Entropy plays a fundamental role in information theory and statistical physics, as well as in quantum mechanics in a generalized formulation due to von Neumann. Entropy appears always as a subadditive quantity in all of its formulations, meaning the entropy of a supersystem or a set union of random variables is always less or equal than the sum of the entropies of its individual components. Additionally, entropy in physics satisfies several more strict inequalities such as the Strong Subadditivity of Entropy in classical statistical mechanics and its quantum analog.

Economics

Subadditivity is an essential property of some particular cost functionss. It is, generally, a necessary and sufficient condition for the verification of a natural monopoly. It implies that production from only one firm is socially less expensive (in terms of average costs) than production of a fraction of the original quantity by an equal number of firms.

Economies of scale are represented by subadditive average cost functions.

Except in the case of complementary goods, the price of goods (as a function of quantity) must be subadditive. Otherwise, if the sum of the cost of two items is cheaper than the cost of the bundle of two of them together, then nobody would ever buy the bundle, effectively causing the price of the bundle to "become" the sum of the prices of the two separate items. Thus proving that it is not a sufficient condition for a natural monopoly; since the unit of exchange may not be the actual cost of an item. This situation is familiar to everyone in the political arena where some minority asserts that the loss of some particular freedom at some particular level of government means that many governments are better; whereas the majority assert that there is some other correct unit of cost.

Finance

Subadditivity is one of the desirable properties of coherent risk measures in risk management. The economic intuition behind risk measure subadditivity is that a portfolio risk exposure should, at worst, simply equal the sum of the risk exposures of the individual positions that compose the portfolio. The lack of subadditivity is one of the main critiques of VaR models which do not rely on the assumption of normality is the inverse of the normal cumulative distribution function is the linear correlation measure between the two individual positions returns. Since variance which is the case of no diversification effects on portfolio risk.

Thermodynamics

Subadditivity occurs in the thermodynamic properties of non-ideal solutions and mixtures like the excess molar volume and heat of mixing or excess enthalpy.

Combinatorics on words

A factorial language is one where if a word, then all factors. In combinatorics on words.

. The expected length of the longest common subsequence is a super, however, is only known to be between 0.788 and 0.827.

See also

Notes

References