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Examples

• "Among probability distributions on the interval from 0 to ∞ on the real line, memorylessness characterizes the exponential distributions." 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. 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 manifoldIn mathematics, a manifold ''M is a type of space, characterised in one of two equivalent ways: near every point of the space, we have a coordinate system; or near every point, the environment is like that in Euclidean space of a given dimension. Therefor by being one-dimensional, compactSeveral specialized usages of the terms compact and compactness exist. Multiple definitions of the term "compact" are found in mathematics: The most common usage relates to topology, where one considers compact spaces . This article also includes the clos and connected; here the characterization, as a smooth manifold, is up to diffeomorphismIn mathematics, a diffeomorphism is a kind of isomorphism of smooth manifolds. Here is definition Given two differentiable manifolds M and N a bijective map from M to N is called a diffeomorphism if both and its inverse are smooth. Two manifolds M and N a.