homogeneous

UK /ˌhɒm.ə(ʊ)ˈd͡ʒiː.nɪəs/ · US /ˌhoʊ.moʊˈd͡ʒi.ni.əs/ adjective

Forms homogenous

1. adjective Of the same kind; alike, similar.
2. adjective Having the same composition throughout; of uniform make-up.
3. adjective (chemistry) In the same state of matter.
4. adjective (mathematics) In any of several technical senses uniform; scalable; having its behavior or form determined by, or the same as, its behavior on or form at a smaller component (of its domain of definition, of itself, etc.).
5. adjective (algebra, of a polynomial) Such that all its nonzero terms have the same degree.
6. adjective (linear algebra, by specialization, of a system of linear equations) Such that all the constant terms are zero.
7. adjective (mathematical analysis, generalizing the case of polynomial functions, of a function f) Such that if each of f 's inputs are multiplied by the same scalar, f 's output is multiplied by the same scalar to some fixed power (called the degree of homogeneity or degree of f). (Formally and more generally, of a partial function f between vector spaces whose domain is a linear cone) Satisfying the equality f(s mathbf x)=sᵏᶠ(
8. adjective The function f(x,y)#61;x²#43;x²ʸ#43;y² is not homogeneous on all of #92;mathbb#123;R#125;² because f(2,2)#61;16#92;neq 2ᵏ#42;3#61;2ᵏf(1,1) for any k, but f is homogeneous on the subspace of #92;mathbb#123;R#125;² spanned by (1,0) because f(#92;alphax,#92;alphay)#61;#92;alphax²#61;#92;alpha²f(x,y) for all (x,y)#92;in#92;operatorname#123;Span#125;#92;#123;(1,0)#92;#125;.

Sources: Wiktionary · WordNet 3.0 · Webster’s 1913 — from the Wiktionary entry “homogeneous”, CC BY-SA 4.0. Details