Homogeneous

Homogeneous is an adjective that has several meanings.

In mathematics, it means an expression consisting of terms that are sums of monomials of the same total degree; or of elements of the same dimension.

A function f mapping a vector space V over a field F to another vector space W over F is said to be homogeneous of degree k if the equation f(a·v) = a^k·f(v) holds for a in F and v in V. For a function f(x) = f(x₁, ..., x_n) that is homogeneous of degree k Euler's homogeneous function theorem holds:

\sum _{i=1}^{n}x_{i}{\frac {\partial f}{\partial x_{i}}}(x)=kf(x)

More generally, a function f is said to be homogeneous if the equation f(a v) = g(a) f(v) holds for some strictly increasing positive function g.
A homogeneous differential equation is usually one of the form Lf = 0, where L is a differential operator, the corresponding inhomogeneous equation being Lf = g with g a given function; the word homogeneous is also used of equations in the form Dy = f(y/x).
In linear algebra a homogeneous system is a one of the form Ax=0.
Homogeneous numbers share identical prime factors (may be repeated).
A homogeneous space for a Lie group G , or more general transformation group, is a space X on which G acts transitively and continuously - so equivalently a coset space G/H where H is a closed subgroup.
As a special case of the previous meaning, a manifold is said to be homogeneous for its homeomorphism group, or diffeomorphism group, if that group acts transitively on it; this is true for connected manifolds.
Given a colouring of the edges of a complete graph, the term homogeneous applies to a subset of vertices such that all edge connecting two of the subset have the same colour; and in much greater generality in Ramsey theory for colourings of k-element subsets.

Homogeneity has a precise meaning in physics.

In biology homogeneous has a meaning similar to its meaning in mathematics. Generally it means "the same" or "of the same quality or general property", such as a homogeneous sample, homogeneous population, etc.

Homogenous (without the second e) has a similar meaning of being the same throughout, and is perhaps more common in everyday speech. For example homogenised milk is milk which has been processed so that the cream is mixed throughout the milk, rather than being left to settle at the top.

See also