Homogeneous (mathematics): Difference between revisions

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In mathematics, homogeneous may refer to:

a homogeneous polynomial, in algebra
a homogeneous function
a homogeneous differential equation
a homogeneous system of linear equations, in linear algebra
homogeneous coordinates
a homogeneous number
a homogeneous space for a Lie group G, or more general transformation group
a homogeneous ideal in a graded ring

This disambiguation page lists mathematics articles associated with the title Homogeneous.
If an internal link led you here, you may wish to change the link to point directly to the intended article.

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-In [[mathematics]], '''homogeneous''' has a variety of meanings.
+In [[mathematics]], '''homogeneous''' may refer to:
+* a [[homogeneous polynomial]], in algebra
-*In [[algebra]], a [[homogeneous polynomial]] is a [[polynomial]] whose terms are [[monomial]]s all having the same total [[degree (mathematics)|degree]]; or are elements of the same dimension.
+* a [[homogeneous function]]
-* A [[homogeneous function]] is a function ''f'' satisfying ''f''(&alpha;''v'') = &alpha;<sup>''k''</sup>''f''(''v'') for some value of ''k''.
+* a homogeneous [[differential equation]]
-* 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'''.
+* a homogeneous [[system of linear equations]], in linear algebra
+* [[homogeneous coordinates]]
-* Homogeneous [[homogeneous coordinates | coordinates]] enable [[affine transformations]], such as spatial [[translation (geometry) | translations]], to be treated the same as [[linear transformations]] and thus represented by [[matrix (mathematics) | matrices]].
+* a homogeneous [[number]]
-* Homogeneous [[number]]s 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. Equivalently, a homogeneous space is a coset space G/H where H is a closed subgroup.
+* a [[homogeneous space]] for a Lie group G, or more general transformation group
+* a [[homogeneous ideal]] in a graded ring
-** 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 [[group action | acts]] transitively on it; this is true for [[connected space | connected]] manifolds without boundary.
-* Given a colouring of the edges of a [[complete graph]], the term homogeneous applies to a subset of vertices such that all edges connecting two of the subset have the same colour; and in much greater generality in [[Ramsey theory]] for colourings of k-element subsets.
-** This is also the notion of homogeneity used in (combinatorial) [[set theory]]: given a function ''f'':[X]<SUP>&alpha;</SUP>&rarr;''C'' colouring the set of subsets of ''X'' of order type &alpha; with colours from ''C'', a subset ''H'' of ''X'' is called homogeneous for ''f'' if all elements of [''H'']<SUP>&alpha;</SUP> get the same colour by ''f''.
-*In a [[graded ring]], there is a concept of [[homogeneous ideal]], important in [[algebraic geometry]]
-{{disambig-cleanup}}
+{{disambig}}
 [[Category:Mathematical terminology]]