homogeneous

English

Alternative forms

• homogenous (may be considered incorrect; see usage note at §§ Usage notes)

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).

Pronunciation

• (UK) IPA^(key): /ˌhɒ.mə(ʊ)ˈdʒiː.nɪəs/, /ˌhəʊ.mə(ʊ)ˈdʒiː.nɪəs/
• (US) IPA^(key): /ˌhoʊ.moʊˈd͡ʒiː.njəs/, /ˌhoʊ.məˈd͡ʒiː.njəs/, /ˌhoʊ.moʊˈd͡ʒɛ.njəs/, /həˈmɑ.d͡ʒə.nəs/

• Audio (UK)

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Adjective

homogeneous (not comparable)

1. Of the same kind; alike, similar.
2. Having the same composition throughout; of uniform make-up.
   • 1946, Bertrand Russell, History of Western Philosophy, I.25:

     Their citizens were not of homogeneous origin, but were from all parts of Greece.

3. (chemistry) In the same state of matter.
4. (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.).
   1. Of polynomials, functions, equations, systems of equations, or linear maps:
      1. (algebra, of a polynomial) Such that all its nonzero terms have the same degree.

         The polynomial ${\displaystyle x^{2}+5xy+y^{2}}$ is homogeneous of degree 2, because ${\displaystyle x^{2}}$, ${\displaystyle xy}$, and ${\displaystyle y^{2}}$ are all degree 2 monomials

      2. (linear algebra, by specialization, of a system of linear equations) Such that all the constant terms are zero.
      3. (mathematical analysis, generalizing the case of polynomial functions, of a function ${\displaystyle f}$) Such that if each of ${\displaystyle f}$ 's inputs are multiplied by the same scalar, ${\displaystyle f}$ 's output is multiplied by the same scalar to some fixed power (called the degree of homogeneity or degree of ${\displaystyle f}$). (Formally and more generally, of a partial function ${\displaystyle f}$ between vector spaces whose domain is a linear cone) Satisfying the equality ${\displaystyle f(s\mathbf {x} )=s^{k}f(\mathbf {x} )}$ for some integer ${\displaystyle k}$ and for all ${\displaystyle \mathbf {x} }$ in the domain and ${\displaystyle s}$ scalars.

         The function ${\displaystyle f(x,y)=x^{2}+x^{2}y+y^{2}}$ is not homogeneous on all of ${\displaystyle \mathbb {R} ^{2}}$ because ${\displaystyle f(2,2)=16\neq 2^{k}*3=2^{k}f(1,1)}$ for any ${\displaystyle k}$, but ${\displaystyle f}$ is homogeneous on the subspace of ${\displaystyle \mathbb {R} ^{2}}$ spanned by ${\displaystyle (1,0)}$ because ${\displaystyle f(\alpha x,\alpha y)=\alpha x^{2}=\alpha ^{2}f(x,y)}$ for all ${\displaystyle (x,y)\in \operatorname {Span} \{(1,0)\}}$.

      4. In ordinary differential equations (by analogy with the case for polynomial and functional homogeneity):
         1. (of a first-order differential equation) Capable of being written in the form ${\displaystyle f(x,y)\mathop {dy} =g(x,y)\mathop {dx} }$ where ${\displaystyle f}$ and ${\displaystyle g}$ are homogeneous functions of the same degree as each other.
         2. (of a linear differential equation) Having its degree-zero term equal to zero; admitting the trivial solution.
         3. (of a general differential equation) Homogeneous as a function of the dependent variable and its derivatives.
   2. In abstract algebra and geometry:
      1. (ring theory, of an element of a graded ring) 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)
      2. (of a linear map ${\displaystyle f}$ between vector spaces graded by a commutative monoid ${\displaystyle I}$) Which respects the grading of its domain and codomain. Formally: Satisfying ${\displaystyle f(V_{j})\subseteq W_{i+j}}$ for fixed ${\displaystyle i}$ (called the degree or grade of ${\displaystyle f}$), ${\displaystyle V_{j}}$ the ${\displaystyle j}$th component of the grading of ${\displaystyle f}$ 's domain, ${\displaystyle W_{k}}$ the ${\displaystyle k}$th component of the grading of ${\displaystyle f}$ 's codomain, and + representing the monoid operation in ${\displaystyle I}$.
      3. (geometry, of a space equipped with a group action) 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).
      4. (geometry) Of or relating to homogeneous coordinates.
   3. In miscellaneous other senses:
      1. (probability theory, Fourier analysis, of a distribution ${\displaystyle S}$ on Euclidean n-space (or on ${\displaystyle \mathbb {R} ^{n}\backslash \{\mathbf {0} \}}$)) Informally: Determined by its restriction to the unit sphere. Formally: Such that, for all real ${\displaystyle t>0}$ and test functions ${\displaystyle \phi (\mathbf {x} )}$, the equality ${\displaystyle S=t^{m}S}$ holds for some fixed real or complex ${\displaystyle m}$.
      2. (set theory, order theory, of a relation) Holding between a set and itself; being an endorelation.

Antonyms

• heterogeneous
• inhomogeneous

Derived terms

Related terms

Translations

Further reading

• “homogeneous”, in Webster’s Revised Unabridged Dictionary, Springfield, Mass.: G. & C. Merriam, 1913, →OCLC.
• “homogeneous”, in The Century Dictionary , New York, N.Y.: The Century Co., 1911, →OCLC.
• “homogeneous”, in OneLook Dictionary Search.