# Homogeneous

**is an adjective that has several meanings.**

*Homogeneous*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*(*x*_{1}, ...,*x*_{n}) that is homogeneous of degree*k*Euler's homogeneous function theorem holds:

- 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 A**x**=**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.

**has a precise meaning in physics.**

*Homogeneity*
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.

## See also

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