homogeneous
Language: English
Part of speech: adj
Pronunciation: ipa: /ˌhɒm.ə(ʊ)ˈd͡ʒiː.nɪəs/
ipa: /ˌhəʊ.mə(ʊ)ˈd͡ʒiː.nɪəs/
ipa: /ˌhoʊ.moʊˈd͡ʒiː.njəs/
ipa: /ˌhoʊ.məˈd͡ʒiː.njəs/
ipa: /ˌhoʊ.moʊˈd͡ʒɛ.njəs/
ipa: /həˈmɑ.d͡ʒə.nəs/
Forms: form: homogenous
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).
Senses: glosses: Of the same kind; alike, similar.
examples: text: Their citizens were not of homogeneous origin, but were from all parts of Greece.
ref: 1946, Bertrand Russell, History of Western Philosophy, I.25:
text: All pseudolatex formulations were homogeneous and smooth in texture and elegant in appearance.
ref: 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:
glosses: Having the same composition throughout; of uniform make-up.
topics: chemistry
natural-sciences
physical-sciences
glosses: In the same state of matter.
topics: mathematics
sciences
glosses: 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.).
examples: The polynomial x²+5xy+y² is homogeneous of degree 2, because x², xy, and y² are all degree 2 monomials
topics: algebra
mathematics
sciences
glosses: 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.).
Of polynomials, functions, equations, systems of equations, or linear maps:
Such that all its nonzero terms have the same degree.
raw_tags: of a polynomial
topics: linear-algebra
mathematics
sciences
glosses: 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.).
Of polynomials, functions, equations, systems of equations, or linear maps:
Such that all the constant terms are zero.
qualifier: by specialization
raw_tags: of a system of linear equations
topics: mathematical-analysis
mathematics
sciences
glosses: 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.).
Of polynomials, functions, equations, systems of equations, or linear maps:
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(
qualifier: generalizing the case of polynomial functions
raw_tags: of a function f
glosses: 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;.
glosses: In ordinary differential equations (by analogy with the case for polynomial and functional homogeneity):
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.
raw_tags: of a first-order differential equation
glosses: In ordinary differential equations (by analogy with the case for polynomial and functional homogeneity):
Having its degree-zero term equal to zero; admitting the trivial solution.
raw_tags: of a linear differential equation
glosses: In ordinary differential equations (by analogy with the case for polynomial and functional homogeneity):
Homogeneous as a function of the dependent variable and its derivatives.
raw_tags: of a general differential equation
glosses: 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)
qualifier: ring theory
raw_tags: of an element of a graded ring
glosses: In abstract algebra and geometry:
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.
raw_tags: of a linear map f between vector spaces graded by a commutative monoid I
topics: geometry
mathematics
sciences
glosses: In abstract algebra and geometry:
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).
raw_tags: of a space equipped with a group action
topics: geometry
mathematics
sciences
glosses: In abstract algebra and geometry:
Of or relating to homogeneous coordinates.
topics: mathematics
probability-theory
sciences
glosses: In miscellaneous other senses:
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.
qualifier: Fourier analysis
raw_tags: of a distribution S on Euclidean n-space (or on ℝⁿmathbf 0)
topics: mathematics
order-theory
sciences
set-theory
glosses: In miscellaneous other senses:
Holding between a set and itself; being an endorelation.
raw_tags: of a relation
Synonyms: word: homogeneous
word: invariable
word: of a piece
word: onefold
word: samely
word: uniform
word: uniformal
word: unmixed
word: unvaried
word: unvarying
Hyponyms: word: connatural
word: monolithic
word: monotone
word: steady
word: unalloyed
Derived 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