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| Characterization (mathematics) | 2/2 | https://en.wikipedia.org/wiki/Characterization_(mathematics) | reference | science, encyclopedia | 2026-05-05T07:23:28.881720+00:00 | kb-cron |
The least-upper-bound property The greatest-lower-bound property The nested interval property The Bolzano-Weierstrass theorem The convergence of Cauchy sequences A typical real analysis university course would begin with the first of these, the least-upper-bound property, as an axiomatic definition of the reals (sometimes called the "axiom of completeness" in texts), and gradually prove its way to the last, the convergence of Cauchy sequences. The proofs are quite nontrivial. Among these five characterizations, the Cauchy-sequence perspective turns out to be the easiest to generalize, and is chosen as the definition for the completeness of an abstract metric space. However, the least-upper-bound property is often the most useful to prove facts about real numbers themselves, such as the intermediate value theorem. Thus the most useful and most generalizable characterizations are at times different. Another example of this phenomenon is found in physics. Hamiltonian mechanics is a characterization of classical mechanics, being equivalent to Newton’s laws. However, it is much easier to generalize to quantum mechanics and statistical mechanics, which is its primary virtue. However, it is a different characterization, Lagrangian mechanics, that is often preferred for the study of classical mechanics itself. Since a characterization result is equivalent to the initial definition or axiom(s) of the object, it can be used as an equivalent definition, from which the original definition can be proved as a theorem. This leads to the question of which definition is “best” in a given situation, out of many possible options. There is no absolute answer, but the ones that are chosen by authors of books or papers is often a matter of aesthetic or pedagogical considerations, as well as convention, history, and tradition. For real numbers, the least-upper-bound property may have been chosen on the grounds of being easier to learn than Cauchy sequences. One of the most important results in complex analysis is a characterization result, namely the fact that all locally complex-differentiable functions are analytic (equal to their Taylor series). Characterisations are very common in abstract algebra, where they often take the form of “structure theorems,” expressing the structure of an object in a simple form. These results are often very difficult to prove. In the theory of matrices, the Jordan Canonical Form is a characterization, or structure theorem, for complex matrices, and the spectral theorem is likewise for symmetric matrices (if real) or Hermitian matrices (if complex). According to the spectral theorem, the real symmetric matrices are precisely the ones that have a basis of perpendicular eigenvectors (called principal axes in physics). In the theory of groups, there is a structure theorem for finite abelian groups, that states that every such group is a direct product of cyclic groups. As if-and-only-if statements, characterizations are, in a sense, the “strongest” type of mathematical theorem, which is in line with the difficulty of their proofs. Consider a generic mathematical theorem, that A implies B. If B does not imply A, the theorem may be said to be “underpowered”, as the proved statement B is “weaker” than the ingredient A, being not strong enough to prove A on its own. In a characterization, however, B must imply A also – the proved statement is as strong as the ingredient, and it can be no stronger. In a sense, such a result “uses” all of the structure in A in proving B.
== 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. "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 == Characterizations of the category of topological spaces – Multiple equivalent ways to define a topological spacePages displaying short descriptions of redirect targets Characterizations of the exponential function – Mathematical concept Characteristic (algebra) – Smallest integer n for which n equals 0 in a ring Characteristic (exponent notation) – Mathematical functionPages displaying short descriptions of redirect targets Classification theorem – Describes the objects of a given type, up to some equivalence Euler characteristic – Topological invariant in mathematics Character (mathematics)
== References ==