abstract or concrete. The objects can be considered the extensions of the world, while the features are expression of the intensions. A continuing program of characterization of various objects leads to their categorization. Examples A rational number, generally defined as a ratio of two integers, can be characterized as a number with finite or repeating decimal expansion. A parallelogram is a quadrilateral whose opposing sides are parallel. One of its characterizations is that its diagonals bisect each other. This means that the diagonals in all parallelograms bisect each other, and conversely, that any quadrilateral whose diagonals bisect each other must be a parallelogram. The latter statement is only true if inclusive definitions of quadrilaterals are used (so that, for example, rectangles count as parallelograms), which is the dominant way of defining objects in mathematics nowadays. "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, provided that the distribution is continuous as defined above (see Characterization of probability distributions for more). "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. 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. See also Characterization of probability distributions Characterizations of the category of topological spaces Characterizations of the exponential function Characteristic (algebra) Characteristic (exponent notation) Classification theorem Euler characteristic Character (mathematics) References Mathematical terminology Equivalence (mathematics)