homogeneous

adj

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).

1. Of the same kind; alike, similar.
2. Having the same composition throughout; of uniform make-up.

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

   “All pseudolatex formulations were homogeneous and smooth in texture and elegant in appearance.”

3. In the same state of matter.
4. 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. 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.).

   “The polynomial x²+5xy+y² is homogeneous of degree 2, because x², xy, and y² are all degree 2 monomials”

6. 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.).
7. 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.).
8. 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;.
9. In ordinary differential equations (by analogy with the case for polynomial and functional homogeneity):
10. In ordinary differential equations (by analogy with the case for polynomial and functional homogeneity):
11. In ordinary differential equations (by analogy with the case for polynomial and functional homogeneity):
12. In abstract algebra and geometry:
13. In abstract algebra and geometry:
14. In abstract algebra and geometry:
15. In abstract algebra and geometry:
16. In miscellaneous other senses:
17. In miscellaneous other senses: