Characterization (mathematics)

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

  • "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.)

Wikimedia Foundation. 2010.

Игры ⚽ Нужно решить контрольную?

Look at other dictionaries:

  • Characterization (disambiguation) — Characterization or characterisation may refer to: Characterisation of a character in a narrative or dramatic work of art Characterisation (conflict), a procedure to resolve a lawsuit Characterization (mathematics), a concept in mathematics… …   Wikipedia

  • mathematics — /math euh mat iks/, n. 1. (used with a sing. v.) the systematic treatment of magnitude, relationships between figures and forms, and relations between quantities expressed symbolically. 2. (used with a sing. or pl. v.) mathematical procedures,… …   Universalium

  • List of mathematics articles (C) — NOTOC C C closed subgroup C minimal theory C normal subgroup C number C semiring C space C symmetry C* algebra C0 semigroup CA group Cabal (set theory) Cabibbo Kobayashi Maskawa matrix Cabinet projection Cable knot Cabri Geometry Cabtaxi number… …   Wikipedia

  • Outline of discrete mathematics — The following outline is presented as an overview of and topical guide to discrete mathematics: Discrete mathematics – study of mathematical structures that are fundamentally discrete rather than continuous. In contrast to real numbers that have… …   Wikipedia

  • List of basic discrete mathematics topics — Discrete mathematics, also called finite mathematics, is the study of mathematical structures that are fundamentally , in the sense of not supporting or requiring the notion of continuity. Most, if not all, of the objects studied in finite… …   Wikipedia

  • Distribution (mathematics) — This article is about generalized functions in mathematical analysis. For the probability meaning, see Probability distribution. For other uses, see Distribution (disambiguation). In mathematical analysis, distributions (or generalized functions) …   Wikipedia

  • List of mathematics articles (F) — NOTOC F F₄ F algebra F coalgebra F distribution F divergence Fσ set F space F test F theory F. and M. Riesz theorem F1 Score Faà di Bruno s formula Face (geometry) Face configuration Face diagonal Facet (mathematics) Facetting… …   Wikipedia

  • Space (mathematics) — This article is about mathematical structures called spaces. For space as a geometric concept, see Euclidean space. For all other uses, see space (disambiguation). A hierarchy of mathematical spaces: The inner product induces a norm. The norm… …   Wikipedia

  • Matrix (mathematics) — Specific elements of a matrix are often denoted by a variable with two subscripts. For instance, a2,1 represents the element at the second row and first column of a matrix A. In mathematics, a matrix (plural matrices, or less commonly matrixes)… …   Wikipedia

  • Duality (mathematics) — In mathematics, a duality, generally speaking, translates concepts, theorems or mathematical structures into other concepts, theorems or structures, in a one to one fashion, often (but not always) by means of an involution operation: if the dual… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”