Définitions
1.AdjectiveOf the same kind; alike, similar.
2.AdjectiveHaving the same composition throughout; of uniform make-up.
3.AdjectiveIn the same state of matter.
4.AdjectiveIn 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."The polynomial
x
2
+
5
x
y
+
y
2
{\displaystyle x^{2}+5xy+y^{2}}
is homogeneous of degree 2, because
x
2
{\displaystyle x^{2}}
,
x
y
{\displaystyle xy}
, and
y
2
{\displaystyle y^{2}}
are all degree 2 monomials; The function
f
(
x
,
y
)
=
x
2
+
x
2
y
+
y
2
{\displaystyle f(x,y)=x^{2}+x^{2}y+y^{2}}
is not homogeneous on all of
R
2
{\displaystyle \mathbb {R} ^{2}}
because
f
(
2
,
2
)
=
16
≠
2
k
∗
3
=
2
k
f
(
1
,
1
)
{\displaystyle f(2,2)=16\neq 2^{k}*3=2^{k}f(1,1)}
for any
k
{\displaystyle k}
, but
f
{\displaystyle f}
is homogeneous on the subspace of
R
2
{\displaystyle \mathbb {R} ^{2}}
spanned by
(
1
,
0
)
{\displaystyle (1,0)}
because
f
(
α
x
,
α
y
)
=
α
x
2
=
α
2
f
(
x
,
y
)
{\displaystyle f(\alpha x,\alpha y)=\alpha x^{2}=\alpha ^{2}f(x,y)}
for all
(
x
,
y
)
∈
Span
{
(
1
,
0
)
}
{\displaystyle (x,y)\in \operatorname {Span} \{(1,0)\}}
."
5.AdjectiveOf polynomials, functions, equations, systems of equations, or linear maps:
Such that all its nonzero terms have the same degree."The polynomial
x
2
+
5
x
y
+
y
2
{\displaystyle x^{2}+5xy+y^{2}}
is homogeneous of degree 2, because
x
2
{\displaystyle x^{2}}
,
x
y
{\displaystyle xy}
, and
y
2
{\displaystyle y^{2}}
are all degree 2 monomials; The function
f
(
x
,
y
)
=
x
2
+
x
2
y
+
y
2
{\displaystyle f(x,y)=x^{2}+x^{2}y+y^{2}}
is not homogeneous on all of
R
2
{\displaystyle \mathbb {R} ^{2}}
because
f
(
2
,
2
)
=
16
≠
2
k
∗
3
=
2
k
f
(
1
,
1
)
{\displaystyle f(2,2)=16\neq 2^{k}*3=2^{k}f(1,1)}
for any
k
{\displaystyle k}
, but
f
{\displaystyle f}
is homogeneous on the subspace of
R
2
{\displaystyle \mathbb {R} ^{2}}
spanned by
(
1
,
0
)
{\displaystyle (1,0)}
because
f
(
α
x
,
α
y
)
=
α
x
2
=
α
2
f
(
x
,
y
)
{\displaystyle f(\alpha x,\alpha y)=\alpha x^{2}=\alpha ^{2}f(x,y)}
for all
(
x
,
y
)
∈
Span
{
(
1
,
0
)
}
{\displaystyle (x,y)\in \operatorname {Span} \{(1,0)\}}
."
6.AdjectiveSuch that all its nonzero terms have the same degree."The polynomial
x
2
+
5
x
y
+
y
2
{\displaystyle x^{2}+5xy+y^{2}}
is homogeneous of degree 2, because
x
2
{\displaystyle x^{2}}
,
x
y
{\displaystyle xy}
, and
y
2
{\displaystyle y^{2}}
are all degree 2 monomials"
7.AdjectiveSuch that all the constant terms are zero.
8.AdjectiveSuch that if each of
f
{\displaystyle f}
's inputs are multiplied by the same scalar,
f
{\displaystyle f}
's output is multiplied by the same scalar to some fixed power (called the degree of homogeneity or degree of
f
{\displaystyle f}
). Satisfying the equality
f
(
s
x
)
=
s
k
f
(
x
)
{\displaystyle f(s\mathbf {x} )=s^{k}f(\mathbf {x} )}
for some integer
k
{\displaystyle k}
and for all
x
{\displaystyle \mathbf {x} }
in the domain and
s
{\displaystyle s}
scalars."The function
f
(
x
,
y
)
=
x
2
+
x
2
y
+
y
2
{\displaystyle f(x,y)=x^{2}+x^{2}y+y^{2}}
is not homogeneous on all of
R
2
{\displaystyle \mathbb {R} ^{2}}
because
f
(
2
,
2
)
=
16
≠
2
k
∗
3
=
2
k
f
(
1
,
1
)
{\displaystyle f(2,2)=16\neq 2^{k}*3=2^{k}f(1,1)}
for any
k
{\displaystyle k}
, but
f
{\displaystyle f}
is homogeneous on the subspace of
R
2
{\displaystyle \mathbb {R} ^{2}}
spanned by
(
1
,
0
)
{\displaystyle (1,0)}
because
f
(
α
x
,
α
y
)
=
α
x
2
=
α
2
f
(
x
,
y
)
{\displaystyle f(\alpha x,\alpha y)=\alpha x^{2}=\alpha ^{2}f(x,y)}
for all
(
x
,
y
)
∈
Span
{
(
1
,
0
)
}
{\displaystyle (x,y)\in \operatorname {Span} \{(1,0)\}}
."
9.AdjectiveIn ordinary differential equations (by analogy with the case for polynomial and functional homogeneity):
Capable of being written in the form
f
(
x
,
y
)
d
y
=
g
(
x
,
y
)
d
x
{\displaystyle f(x,y)\mathop {dy} =g(x,y)\mathop {dx} }
where
f
{\displaystyle f}
and
g
{\displaystyle g}
are homogeneous functions of the same degree as each other.
Having its degree-zero term equal to zero; admitting the trivial solution.
Homogeneous as a function of the dependent variable and its derivatives.
10.AdjectiveCapable of being written in the form
f
(
x
,
y
)
d
y
=
g
(
x
,
y
)
d
x
{\displaystyle f(x,y)\mathop {dy} =g(x,y)\mathop {dx} }
where
f
{\displaystyle f}
and
g
{\displaystyle g}
are homogeneous functions of the same degree as each other.
11.AdjectiveHaving its degree-zero term equal to zero; admitting the trivial solution.
12.AdjectiveHomogeneous as a function of the dependent variable and its derivatives.
13.AdjectiveIn abstract algebra and geometry:
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)
Which respects the grading of its domain and codomain. Formally: Satisfying
f
(
V
j
)
⊆
W
i
+
j
{\displaystyle f(V_{j})\subseteq W_{i+j}}
for fixed
i
{\displaystyle i}
(called the degree or grade of
f
{\displaystyle f}
),
V
j
{\displaystyle V_{j}}
the
j
{\displaystyle j}
th component of the grading of
f
{\displaystyle f}
's domain,
W
k
{\displaystyle W_{k}}
the
k
{\displaystyle k}
th component of the grading of
f
{\displaystyle f}
's codomain, and + representing the monoid operation in
I
{\displaystyle I}
.
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).
Of or relating to homogeneous coordinates.
14.AdjectiveBelonging 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)
15.AdjectiveWhich respects the grading of its domain and codomain. Formally: Satisfying
f
(
V
j
)
⊆
W
i
+
j
{\displaystyle f(V_{j})\subseteq W_{i+j}}
for fixed
i
{\displaystyle i}
(called the degree or grade of
f
{\displaystyle f}
),
V
j
{\displaystyle V_{j}}
the
j
{\displaystyle j}
th component of the grading of
f
{\displaystyle f}
's domain,
W
k
{\displaystyle W_{k}}
the
k
{\displaystyle k}
th component of the grading of
f
{\displaystyle f}
's codomain, and + representing the monoid operation in
I
{\displaystyle I}
.
16.AdjectiveInformally: 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).
17.AdjectiveOf or relating to homogeneous coordinates.
18.AdjectiveIn miscellaneous other senses:
Informally: Determined by its restriction to the unit sphere. Formally: Such that, for all real 0}">
t
>
0
{\displaystyle t>0}
0}"> and test functions
ϕ
(
x
)
{\displaystyle \phi (\mathbf {x} )}
, the equality
S
[
t
−
n
ϕ
(
x
/
t
)
]
=
t
m
S
[
ϕ
(
x
)
]
{\displaystyle S[t^{-n}\phi (\mathbf {x} /t)]=t^{m}S[\phi (\mathbf {x} )]}
holds for some fixed real or complex
m
{\displaystyle m}
.
Holding between a set and itself; being an endorelation.
19.AdjectiveInformally: Determined by its restriction to the unit sphere. Formally: Such that, for all real 0}">
t
>
0
{\displaystyle t>0}
0}"> and test functions
ϕ
(
x
)
{\displaystyle \phi (\mathbf {x} )}
, the equality
S
[
t
−
n
ϕ
(
x
/
t
)
]
=
t
m
S
[
ϕ
(
x
)
]
{\displaystyle S[t^{-n}\phi (\mathbf {x} /t)]=t^{m}S[\phi (\mathbf {x} )]}
holds for some fixed real or complex
m
{\displaystyle m}
.
20.AdjectiveHolding between a set and itself; being an endorelation.