multiplicity

Multiplicity is a term used across disciplines to denote the number of occurrences or instances of a given object, value, or category in a context.

In mathematics, multiplicity has several precise senses. The algebraic multiplicity of a root r of a polynomial is the exponent of (x - r) in the polynomial's factorization; the root may occur multiple times. In linear algebra, the algebraic multiplicity of an eigenvalue is its multiplicity as a root of the characteristic polynomial, while the geometric multiplicity is the dimension of its eigenspace. These concepts relate to degeneracy of an eigenvalue.

In combinatorics, multiplicity can refer to the number of occurrences of a given element in a multiset; counting multiplicities is essential for generating functions and counting problems.

In chemistry and physics, multiplicity has different uses. Spin multiplicity, defined as 2S+1, describes the number of possible spin states of a system with total spin S. In atomic and molecular spectroscopy, degeneracy or multiplicity reflects the number of energy states sharing the same energy.

In astronomy, multiplicity describes the number of gravitationally bound stars in a system, such as binaries, triples, or higher-order multiples, affecting statistics of star formation and dynamics.

In information science and software design, multiplicity constrains relationships between entities; for example in UML diagrams, an association end may have a multiplicity indicating how many objects may participate.