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mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, 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 map In algebra, an additive map, Z-linear map or additive function is a function f that preserves the addition operation: f(x + y) = f(x) + f(y) for every pair of elements x and y in the domain of f. For example, any linear map is additive. When ...
s are special cases of subadditive functions.


Definitions

A subadditive function is a function f \colon A \to B, having a domain ''A'' and an ordered
codomain In mathematics, a codomain, counter-domain, or set of destination of a function is a set into which all of the output of the function is constrained to fall. It is the set in the notation . The term '' range'' is sometimes ambiguously used to ...
''B'' that are both closed under addition, with the following property: \forall x, y \in A, f(x+y)\leq f(x)+f(y). An example is the
square root In mathematics, a square root of a number is a number such that y^2 = x; in other words, a number whose ''square'' (the result of multiplying the number by itself, or y \cdot y) is . For example, 4 and −4 are square roots of 16 because 4 ...
function, having the
non-negative In mathematics, the sign of a real number is its property of being either positive, negative, or 0. Depending on local conventions, zero may be considered as having its own unique sign, having no sign, or having both positive and negative sign. ...
real number In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
s as domain and codomain: since \forall x, y \geq 0 we have: \sqrt\leq \sqrt+\sqrt. A
sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is cal ...
\left \_ is called subadditive if it satisfies the inequality a_\leq a_n+a_m 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. Note that while a concave sequence is subadditive, the converse is false. For example, arbitrarily assign a_1, a_2, ... with values in .5, 1/math>; then the sequence is subadditive but not concave.


Properties


Sequences

A useful result pertaining to subadditive sequences is the following lemma due to
Michael Fekete Michael (Mihály) Fekete (; 19 July 1886 – 13 May 1957) was a Hungarian-Israelis, Israeli mathematician. Biography Michael Fekete was born in Zenta, Austria-Hungary (today Senta, Serbia). He received his PhD in 1909 from the University ...
. The analogue of Fekete's lemma holds for superadditive sequences as well, that is: a_\geq a_n + a_m. (The limit then may be positive infinity: consider the sequence a_n = \log n!.) There are extensions of Fekete's lemma that do not require the inequality a_\le a_n + a_m to hold for all ''m'' and ''n'', but only for ''m'' and ''n'' such that \frac 1 2 \le \frac m n \le 2. Moreover, the condition a_\le a_n + a_m may be weakened as follows: a_\le a_n + a_m + \phi(n+m) provided that \phi is an increasing function such that the integral \int \phi(t) t^ \, dt 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. Besides, analogues of Fekete's lemma have been proved 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''(0) ≥ 0. To see this, take the inequality at the top. f(x) \ge f(x+y) - f(y). Hence f(0) \ge f(0+y) - f(y) = 0 A
concave function In mathematics, a concave function is one for which the function value at any convex combination of elements in the domain is greater than or equal to that convex combination of those domain elements. Equivalently, a concave function is any funct ...
f: superadditive.


Examples in various domains


Entropy

superadditivity">superadditive.


Examples in various domains


Entropy

Entropy plays a fundamental role in information theory">Entropy">superadditivity">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, 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 functions. It is, generally, a necessary and sufficient condition for the verification of a
natural monopoly A natural monopoly is a monopoly in an industry in which high infrastructural costs and other barriers to entry relative to the size of the market give the largest supplier in an industry, often the first supplier in a market, an overwhelming adv ...
. 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 In microeconomics, economies of scale are the cost advantages that enterprises obtain due to their scale of operation, and are typically measured by the amount of Productivity, output produced per unit of cost (production cost). A decrease in ...
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 Risk management is the identification, evaluation, and prioritization of risks, followed by the minimization, monitoring, and control of the impact or probability of those risks occurring. Risks can come from various sources (i.e, Threat (sec ...
. 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 of risk factors. The Gaussian VaR ensures subadditivity: for example, the Gaussian VaR of a two unitary long positions portfolio V at the confidence level 1-p is, assuming that the mean portfolio value variation is zero and the VaR is defined as a negative loss, \text_p \equiv z_\sigma_ = z_\sqrt where z_p is the inverse of the normal
cumulative distribution function In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x. Ever ...
at probability level p , \sigma_x^2,\sigma_y^2 are the individual positions returns variances and \rho_ is the linear correlation measure between the two individual positions returns. Since
variance In probability theory and statistics, variance is the expected value of the squared deviation from the mean of a random variable. The standard deviation (SD) is obtained as the square root of the variance. Variance is a measure of dispersion ...
is always positive, \sqrt \leq \sigma_x + \sigma_y Thus the Gaussian VaR is subadditive for any value of \rho_ \in 1,1 and, in particular, it equals the sum of the individual risk exposures when \rho_=1 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 Language is a structured system of communication that consists of grammar and vocabulary. It is the primary means by which humans convey meaning, both in spoken and signed language, signed forms, and may also be conveyed through writing syste ...
L is one where if a
word A word is a basic element of language that carries semantics, meaning, can be used on its own, and is uninterruptible. Despite the fact that language speakers often have an intuitive grasp of what a word is, there is no consensus among linguist ...
is in L, then all factors of that word are also in L. In combinatorics on words, a common problem is to determine the number A(n) of length-n words in a factorial language. Clearly A(m+n) \leq A(m)A(n), so \log A(n) is subadditive, and hence Fekete's lemma can be used to estimate the growth of A(n). For every k \geq 1, sample two strings of length n uniformly at random on the alphabet 1, 2, ..., k. The expected length of the longest common subsequence is a ''super''-additive function of n, and thus there exists a number \gamma_k \geq 0, such that the expected length grows as \sim \gamma_k n. By checking the case with n=1, we easily have \frac 1k < \gamma_k \leq 1. The exact value of even \gamma_2, however, is only known to be between 0.788 and 0.827.


See also

* * * *


Notes


References

* György Pólya and Gábor Szegő. '' Problems and Theorems in Analysis'', vol. 1. Springer-Verlag, New York (1976). . * Einar Hille.
Functional analysis and semi-groups
. American Mathematical Society, New York (1948). *N.H. Bingham, A.J. Ostaszewski. "Generic subadditive functions." Proceedings of American Mathematical Society, vol. 136, no. 12 (2008), pp. 4257–4266.


External links

{{PlanetMath attribution, id=4615, title=subadditivity Mathematical analysis Sequences and series Types of functions