homogeneous🔊

Of the same kind; alike, similar.

Having the same composition throughout; of uniform make-up.

(chemistry) In the same state of matter.

(mathematics) Uniform; scalable; having its behavior or form determined by, or the same as, its behavior on or form at a smaller component (of its domain of definition, of itself, etc.).

(algebra, of a polynomial) Such that all its nonzero terms have the same degree.

(linear algebra) Such that all the constant terms are zero.

(mathematical analysis) Such that if each of f's inputs are multiplied by the same scalar, f's output is multiplied by the same scalar to some fixed power (called the degree of homogeneity or degree of f}). (Formally and more generally, of a partial function f between vector spaces whose domain is a linear cone) Satisfying the equality f ( s x ) = s k f ( x ) for some integer k and for all x in the domain and s scalars.

(of a first-order differential equation) Capable of being written in the form f ( x , y ) d y = g ( x , y ) d x where f and g are homogeneous functions of the same degree as each other.

(of a linear differential equation) Having its degree-zero term equal to zero; admitting the trivial solution.

(of a general differential equation) Homogeneous as a function of the dependent variable and its derivatives.

(ring theory, of an element of a graded ring) Belonging to one of the summands of the grading.

(of a linear map f between vector spaces graded by a commutative monoid I) Which respects the grading of its domain and codomain. Formally: Satisfying f ( V j ) ⊆ W i + j for fixed i (called the degree or grade of f), V j the jth component of the grading of f {\displaystyle f} 's domain, Wk the kth component of the grading of f's codomain, and + representing the monoid operation in I.

(geometry, of a space equipped with a group action) Informally: Everywhere the same, uniform, in the sense that any point can be moved to any other (via the group action) while respecting the structure of the space. Formally: Such that the group action is transitively and acts by automorphisms on the space (some authors also require that the action be faithful).

(geometry) Of or relating to homogeneous coordinates.

(probability theory, Fourier analysis, of a distribution S on Euclidean n-space (or on R n ∖ { 0 })) Informally: Determined by its restriction to the unit sphere. Formally: Such that, for all real t > 0 and test functions ϕ ( x ), the equality S [ t − n ϕ ( x / t ) ] = t m S [ ϕ ( x ) ] holds for some fixed real or complex m.

(set theory, order theory, of a relation) Holding between a set and itself; being an endorelation.