homogeneous

adjective homogenous

1. not comparable Of the same kind; alike, similar.
2. not comparable Having the same composition throughout; of uniform make-up. Their citizens were not of homogeneous origin, but were from all parts of Greece.
3. not comparable In the same state of matter.
4. not comparable 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. not comparable, 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
6. not comparable, of a system of linear equations Such that all the constant terms are zero.
7. not comparable, 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(
8. not comparable 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. not comparable, 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.
10. not comparable, of a linear differential equation Having its degree-zero term equal to zero; admitting the trivial solution.
11. not comparable, of a general differential equation Homogeneous as a function of the dependent variable and its derivatives.
12. not comparable, 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)
13. not comparable, of a linear map f between vector spaces graded by a commutative monoid I 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.
14. not comparable, 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).
15. not comparable Of or relating to homogeneous coordinates.
16. not comparable, of a distribution S on Euclidean n space (or on ℝⁿmathbf 0) 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.
17. not comparable, of a relation Holding between a set and itself; being an endorelation.

Similar: homogeneous, invariable, of a piece, onefold, samely, uniform, uniformal, unmixed, unvaried, unvarying · Opposite: heterogeneous, inhomogeneous