order3

Order3 is a term used across several disciplines to denote third-order concepts or, in algebra, elements of order three. Its exact meaning depends on context but generally signals a level, degree, or cycle that is threefold.

In algebra, an element g has order 3 if g cubed equals the identity and no smaller positive power returns the identity. In a group such as the cyclic group of order 3, generated by g, every non-identity element has order 3. In permutation groups, a 3-cycle is a permutation with order 3.

In tensor analysis and multilinear algebra, a third-order (order-3) tensor is a multilinear map with three indices, written as Tijk. Such tensors describe relationships among triplets of vector components and appear in physics, computer graphics, and machine learning. Operations on third-order tensors include contractions that reduce their order or combine them with other tensors to yield matrices or vectors.

In calculus and differential equations, the third derivative of a function, denoted f'''(x), is related to third-order rates of change. Third-order differential equations involve the third derivative and arise in various physical and engineering models.

In asymptotic analysis and computer science, order-3 commonly appears in Big-O notation, as in O(n^3), indicating cubic growth in time or space with respect to input size.

In statistics, third-order moments capture skewness, describing asymmetry in a distribution, and are part of higher-order moment analysis alongside variance (second-order) and kurtosis (fourth-order).

See also: order, order of an element, third-order tensor, cubic.