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 equation 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.

See homogeneity for the expression in the physics field.

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