homogeneous (mathematics)

In mathematics, homogeneous has a variety of meanings.

In algebra 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) = k f(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.

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