homogeneous
adjUK /ˌhɒm.ə(ʊ)ˈd͡ʒiː.nɪəs/ · US /ˌhoʊ.moʊˈd͡ʒi.ni.əs/
Etymology From Medieval Latin homogeneus, from Ancient Greek ὁμογενής (homogenḗs, “of the same race, family or kind”), from ὁμός (homós, “same”) + γένος (génos, “kind”). Compare homo- (“same”) and -ous (adjectival suffix).
17 senses
- 1Of the same kind; alike, similar.
- 2Having the same composition throughout; of uniform make-up.
- 3In the same state of matter.
- 4In any of several technical senses uniform; scalable…
- 5Such that all its nonzero terms have the same degree.
- 6Such that all the constant terms are zero.
- 7Such that if each of f 's inputs are multiplied by the…
- 8The function f(x,y)#61;x²#43;x²ʸ#43;y² is not…
- 9Capable of being written in the form f(x,y) mathop…
- 10Having its degree-zero term equal to zero; admitting the…
- 11Homogeneous as a function of the dependent variable and…
- 12Belonging to one of the summands of the grading (if the…
- 13Which respects the grading of its domain and codomain.…
- 14Informally: Everywhere the same, uniform, in the sense…
- 15Of or relating to homogeneous coordinates.
- 16Informally: Determined by its restriction to the unit…
- 17Holding between a set and itself; being an endorelation.
Of the same kind; alike, similar.
Having the same composition throughout; of uniform make-up.
- Their citizens were not of homogeneous origin, but were from all parts of Greece.1946, Bertrand Russell, History of Western Philosophy, I.25:
- All pseudolatex formulations were homogeneous and smooth in texture and elegant in appearance.2013 October 14, Jirapornchai Suksaeree, Laksana Charoenchai, Chaowalit Monton, Tun Chusut, Apirak Sakunpak, Wiwat Pichayakorn and Prapaporn Boonme, “Preparation of a Pseudolatex-Membrane for Ketoprofen Transdermal Drug Delivery Systems”, in Industrial & Engineering Chemistry Research, volume 52, number 45, →DOI:
Chemistry, Natural sciences, Physical sciences In the same state of matter.
Mathematics, Sciences 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.).
› In any of several technical senses uniform… › Of polynomials, functions, equations…
Algebra, Mathematics, Sciences Of a polynomial Such that all its nonzero terms have the same degree.
- The polynomial x²+5xy+y² is homogeneous of degree 2, because x², xy, and y² are all degree 2 monomials
Linear algebra, Mathematics, Sciences Such that all the constant terms are zero.
By specialization Of a system of linear equations
Mathematical analysis, Mathematics, Sciences 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(
Generalizing the case of polynomial functions
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;.
› In ordinary differential equations (by…
Of a first-order differential equation Capable of being written in the form f(x,y) mathop dy=g(x,y) mathop dx where f and g are homogeneous functions of the same degree as each other.
Of a linear differential equation Having its degree-zero term equal to zero; admitting the trivial solution.
Of a general differential equation Homogeneous as a function of the dependent variable and its derivatives.
› In abstract algebra and geometry:
Belonging to one of the summands of the grading (if the ring is graded over the natural numbers and the element is in the kth summand, it is said to be homogeneous of degree k; if the ring is graded over a commutative monoid I, and the element is an element of the ith summand, it is said to be of grade i)
Of an element of a graded ring
Which respects the grading of its domain and codomain. Formally: Satisfying f(V_j)⊆W_i+j for fixed i (called the degree or grade of f), V_j the jth component of the grading of f 's domain, W_k the kth component of the grading of f 's codomain, and + representing the monoid operation in I.
Of a linear map f between vector spaces graded by a commutative monoid I
Geometry, Mathematics, Sciences Informally: Everywhere the same, uniform, in the sense that any point can be moved to any other (via the group action) while respecting the structure of the space. Formally: Such that the group action is transitively and acts by automorphisms on the space (some authors also require that the action be faithful).
Of a space equipped with a group action
Geometry, Mathematics, Sciences Of or relating to homogeneous coordinates.
› In miscellaneous other senses:
Mathematics, Probability theory, Sciences Informally: Determined by its restriction to the unit sphere. Formally: Such that, for all real t>0 and test functions ϕ( mathbf x), the equality S[t⁻ⁿϕ( mathbf x/t)]=t^(mS)[ϕ( mathbf x)] holds for some fixed real or complex m.
Fourier analysis Of a distribution S on Euclidean n-space (or on ℝⁿmathbf 0)
Mathematics, Order theory, Sciences, Set theory Of a relation Holding between a set and itself; being an endorelation.
- Synonyms
- homogeneous, invariable, of a piece, onefold, samely, uniform, uniformal, unmixed, unvaried, unvarying
- Antonyms
- heterogeneous, inhomogeneous
- Derived terms
27 terms
bihomogeneous, dishomogeneous, homogeneous broadening, homogeneous catalysis, homogeneous coordinate, homogeneous equilibrium, homogeneous function, homogeneous ideal, homogeneously, homogeneous mixture, homogeneousness, homogeneous number, homogeneous polynomial, homogeneous radiation, homogeneous society, homogeneous space, homogeneous system, homogeneous width, inhomogeneous, nonhomogeneous, prehomogeneous, quasihomogeneous, subhomogeneous, superhomogeneous, time-homogeneous Markovian type, ultrahomogeneous, unhomogeneous- Related terms
- homogeneity, homogenesis, homogenise, homogenize, homogeny
- Hyponyms
- connatural, monolithic, monotone, steady, unalloyed