Home > Homogeneous (mathematics)

• 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:

• 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 homeomorphismThis word should not be confused with homomorphism. In topology, two geometrical objects (or "spaces") are called homeomorphic if, roughly speaking, the first can be deformed into the second by stretching and bending; cutting is also allowed, but only if group, or diffeomorphismIn mathematics, a diffeomorphism is a kind of isomorphism of smooth manifolds. Here is definition Given two differentiable manifolds M and N a bijective map from M to N is called a diffeomorphism if both and its inverse are smooth. Two manifolds M and N a group, if that group acts transitively on it; this is true for connected manifolds.
• Given a colouring of the edges of a complete graphGraphs Graph theory A complete graph is a simple graph where an edge connects every pair of vertices. A complete graph on n vertices has n vertices and n ''n minus;1)/2 edges, and is indicated by the notation K''. It is a regular graph of degree n minus;1, 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 theoryRamsey theory Ramsey theory named for Frank P. Ramsey, is a branch of mathematics that studies the conditions under which order must appear. Problems in Ramsey theory typically ask a question of the form: how many elements of some structure must there be for colourings of k-element subsets.