Homogeneous

English

Alternative forms

• homogenous (may be considered incorrect; see usage note at § Usage notes)

Etymology

From Template:Der, from Template:Der, from Template:M + Template:M. Compare Template:M and Template:M.

Pronunciation

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Adjective

1. Of the same kind; alike, similar.
2. Having the same composition throughout; of uniform make-up.
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3. Template:Lb In the same state of matter.
4. Template:Lb 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.).
   1. Template:Non-gloss
      1. Template:Lb Such that all its nonzero terms have the same degree.

         The polynomial $x^2+5xy+y^2$ is homogeneous of degree 2, because $x^2$, $xy$Template:, and $y^2$ are all degree 2 monomials

      2. Template:Lb Such that all the constant terms are zero.
      3. Template:Lb 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 $f$). Template:Lb Satisfying the equality $f(s𝐱)=s^{k}f(𝐱)$ for some integer $k$ and for all $𝐱$ in the domain and $s$ scalars.

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      4. Template:Non-gloss
         1. Template:Lb Capable of being written in the form $f(x,y)dy=g(x,y)dx$ where $f$ and $g$ are homogeneous functions of the same degree as each other.
         2. Template:Lb Having its degree-zero term equal to zero; admitting the trivial solution.
         3. Template:Lb Homogeneous as a function of the dependent variable and its derivatives.
   2. Template:Non-gloss
      1. Template:CTemplate:Lb 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)
      2. Template:Lb Which respects the grading of its domain and codomain. Formally: Satisfying $f(V_j)\subseteq W_{i+j}$ for fixed $i$ (called the degree or grade of $f$), $V_j$ the $j$th component of the grading of $f$ 's domain, $W_k$ the $k$th component of the grading of $f$ 's codomain, and + representing the monoid operation in $I$.
      3. Template:Lb 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).
      4. Template:Lb Of or relating to homogeneous coordinates.
   3. Template:Non-gloss
      1. Template:Lb Informally: Determined by its restriction to the unit sphere. Formally: Such that, for all real $t>0$ and test functions $\varphi (𝐱)$, the equality $S[t^{-n}\varphi (𝐱/t)]=t^{m}S[\varphi (𝐱)]$ holds for some fixed real or complex $m$.
      2. Template:Lb Holding between a set and itself; being an endorelation.

Antonyms

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Translations

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Further reading

• Template:R:Webster 1913
• Template:R:Century 1911
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