homogeneous
Meanings
adj
- Of the same kind; alike, similar.
- Having the same composition throughout; of uniform make-up.
- In the same state of matter.
- 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.
- Such that all the constant terms are zero.
- 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(
- 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 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.
- Having its degree-zero term equal to zero; admitting the trivial solution.
Pronunciation
Word forms
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).
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