Oneparameter

Oneparameter, typically written as one-parameter or one parameter, is a term used in mathematics and related disciplines to describe an object that depends on a single scalar parameter. A one-parameter family is a collection of objects indexed by a parameter t in an interval I ⊆ R, with t ↦ O_t forming a smooth (or continuous) map in the appropriate context. The central idea is to study how objects change as the parameter varies along one dimension.

In geometry and analysis, a common example is a one-parameter family of functions f_t, where f_t(x) = f(x, t) and t ∈ I. The family traces out a surface or a trajectory in the space of functions. In differential geometry and Lie theory, a one-parameter subgroup of a Lie group G is a homomorphism φ: R → G with φ(s + t) = φ(s)φ(t); its derivative at zero yields a Lie algebra element that generates the subgroup.

In physics and applied sciences, time is frequently treated as a single parameter driving evolution, yielding a one-parameter family of states or solutions to equations of motion. In statistics, a one-parameter family of probability distributions is defined by a single parameter—such as the Poisson distribution with rate λ or the Bernoulli distribution with success probability p. One-parameter exponential families also arise, with densities of the form f(x; θ) = h(x) exp(η(θ)T(x) − A(θ)).

The term emphasizes variation along one degree of freedom; more complex models may involve multiple parameters. See also deformation, parametric family, one-parameter subgroup, and parametric curve.