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Characterization (mathematics)

In [mathematics] , the statement that "Property * P * characterizes object * X * " means, not simply that * X * has property * P * , but that * X * is the * only * thing that has property * P * . It is also common to find statements such as "Property * Q * characterises * Y * [up to] [isomorphism] ". The first type of statement says in different words that the [extension] of * P * is a [singleton] set. The second says that the extension of * Q * is a single [equivalence class] (for isomorphism, in the given example — depending on how * up to * is being used, some other [equivalence relation] might be involved).

** Examples **

- "Among [probability distribution] s on the interval from 0 to ∞ on the real line, [memorylessness] characterizes the [exponential distribution] s." This statement means that the exponential distributions are the only probability distributions that are memoryless.

- "According to [Bohr–Mollerup theorem] , among all functions * f * such that * f * (1) = 1 and * x f * ( * x * ) = * f * ( * x * 1) for * x * > 0, log-convexity characterizes the [gamma function] ." This means that among all such functions, the gamma function is the * only * one that is log-convex. (A function * f * is * log-convex * [iff] log( * f * ) is a [convex function] . The base of the logarithm does not matter as long as it is more than 1, but conventionally mathematicians take "log" with no subscript to mean the [natural logarithm] , whose base is * e * .)

- The circle is characterized as a [manifold] by being one-dimensional, [compact] and [connected] ; here the characterization, as a smooth manifold, is [up to] [diffeomorphism] .

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