# homogeneous

Pronunciation: US say "hoh-moh-JEE-nyuhs" /ˌhoʊ.moʊˈd͡ʒiː.njəs/; UK say "hom-uh-JEE-nee-uhs" /ˌhɒm.ə(ʊ)ˈd͡ʒiː.nɪəs/

## Adjective

1. Consistent in composition throughout, so that every part is made up the same way.
   - Simple: The same all the way through, with no different parts.
   - _Blend the ingredients until the dough is homogeneous and no dry flour remains._
   - _Under the microscope the sample proved homogeneous, with no visible layers or grains._
2. Of the same kind; alike or similar to one another.
   - Simple: All of the same kind, or very much alike.
   - _The early settlers formed a homogeneous community, sharing one language and one church._
   - _The committee was criticized as homogeneous, with every member drawn from the same industry._
3. In chemistry, existing in or forming a single phase, as when a catalyst is dissolved in the same liquid or gas as the reactants.
   - Simple: In chemistry, forming one single phase with no boundaries inside, such as a gas mixture or a liquid solution.
   - _The chemists chose a homogeneous catalyst that dissolves in the same solvent as the reactants._
   - _Because both reactants are gases that mix freely, the reaction between them is homogeneous._
4. Of a space with a group action, the same everywhere: the group acts transitively by automorphisms, carrying any point to any other while preserving the structure (some authors also require a faithful action).
   - Simple: In geometry, looking the same from every point, since a symmetry can move any point to any other.
   - _The sphere is homogeneous: a rotation can carry any point on it to any other point._
   - _Euclidean space is a homogeneous space under the group of rigid motions._
6. Of a function (or, by extension, a distribution), such that scaling all its inputs by one factor multiplies the output by a fixed power of that factor, called the degree.
   - Simple: In maths, describes a function where scaling all inputs by a number scales the output by a fixed power of it.
   - _The function f(x, y) = x² + y² is homogeneous of degree 2, since f(tx, ty) = t²f(x, y)._
   - _A production function that is homogeneous of degree 1 shows constant returns to scale._
5. Of a polynomial, having every nonzero term of the same degree; more generally, of an element of a graded ring, lying wholly in one graded component.
   - Simple: In maths, describes a polynomial whose terms all have the same total degree, like x² + xy + y².
   - _The polynomial x³ + 2xy² − y³ is homogeneous of degree 3._
   - _The polynomial x² + 3xy + y² is homogeneous because every term has degree two._
7. Of a differential equation, homogeneous in the dependent variable and its derivatives: scaling them all by one factor scales the expression by a fixed power of that factor.
   - Simple: Said of a differential equation in which scaling the unknown and its derivatives by a number scales every term alike.
   - _Since every term of y·y″ − 2(y′)² = 0 has degree two in y and its derivatives, the equation is homogeneous._
   - _Because y·y″ + (y′)² = 0 is homogeneous, the substitution v = y′/y reduces it to a first-order equation._
8. Of a first-order differential equation, able to be written as f(x, y) dy = g(x, y) dx, where f and g are homogeneous functions of the same degree.
   - Simple: Of a first-order differential equation that can be rewritten so that x and y appear only as the ratio y/x.
   - _The equation dy/dx = (x + y)/x is homogeneous, so the substitution y = vx turns it into a separable equation._
   - _Homogeneous equations like (x² + y²) dx = 2xy dy can be solved by setting y = vx._

## Other meanings

- (adjective) In geometry, relating to homogeneous coordinates.
- (adjective) Of a linear equation, system of linear equations, or linear differential equation, having every term free of the unknowns equal to zero, so that the zero solution always satisfies it.
- (adjective) Of a relation, holding between a set and itself, so that it relates members of one set to members of the same set; being an endorelation.
- (adjective) Of a linear map between graded vector spaces, respecting the gradings: it sends the degree-j component into the degree-(i + j) component, for a fixed i, the map's degree.

## Origin

Greek homós ("same") and génos ("kind") combined in Ancient Greek homogenḗs, meaning "of the same kind, family or race," which Medieval Latin took over as homogeneus. The word can be compared with homo- ("same") and the adjective suffix -ous.

Source: DictionaryAI, https://dictionaryai.com/define/homogeneous (written from Wiktionary data, CC BY-SA 4.0: https://creativecommons.org/licenses/by-sa/4.0/).