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 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; it is also used of equations in the form Dy = f(y/x).
- In linear algebra it is a system in 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 an isomorphic meaning 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.
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