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implicit function theorem Let V, W, Z be
Banach spaces, U ⊂ V, U ⊂ W open and
1 2
I U ,y ∈ U assume D f(x ,y ) : W → Z is
f : U × U → Z differentiable. For some x ∈
1
2
0
0
2
0
0
1
2
an isomorphism. Then there are neighborhoods
U of x and Z of f(x ,y ) and a unique differ-
0
0
0
0
0
entiable map g : U × Z → U such that for all
0
0
2
ideal (of an algebra A) A left ideal is a vector (x, z) ∈ U × Z 0
0
subspace I ⊂ A such that ∀a ∈ A, i ∈ I, a ·i ∈
f (x, g(x, z)) = z.
I. Analogously, a right ideal is a vector subspace
I ⊂ A such that ∀a ∈ A, i ∈ I, i · a ∈ I.A
incidence relation See edge.
bilateral ideal is left ideal which is also a right
ideal.
incident A node v is incident to an edge
i
The quotient of an algebra by a bilateral ideal
(v ,v ), since it is an end point of the edge.
i
j
is an algebra. In the category of algebras with
unity, proper ideals are not subalgebras. In fact, index See Fredholm operator, Atiyah-
if the unit 1 ∈ I belongs to the ideal, then I = A. Singer index theorem.
On the contrary, ideals of Lie algebras are always
subalgebras. inf-sup condition A sesqui-linear form a :
V × V → C on a Hilbert space V satisfies an
identity (1) An element e of a set X with a inf-sup condition if, for some α> 0,
binary operation * satisfying
|a(u, v)|
sup v∈V ≥ α u V ∀u ∈ V,
a ∗ e = e ∗ a = a v V
for all a ∈ X. sup v∈V |a(u, v)| > 0 ∀u ∈ V.
(2) A true mathematical equation. v V
This is necessary and sufficient for the variational
image Let r be a relation with domain A and problem
codomain B, and let a ∈ A, r(a) ∈ B. Then the
u ∈ V : a(u, v) = f(v) ∀v ∈ V
image of a under r, r(a) ∈ B, is produced by
applying r to a.
to possess a unique solution for any f ∈ V . The
Comment: See the comment on relation for −1
solution satisfies u ≤ α f .
V
V

more details. See also codomain, domain, range,
For a linear symmetric variational saddle
and relation.
point problem with sesqui-linear forms a : V ×
V → C and b : V × W → C the suitable inf-
immersional wetting A process in which a sup conditions claim the existence of constants
solid or liquid, β, is covered with a liquid, α, α, β > 0 such that
both of which were initially in contact with a gas
2
or liquid, δ, without changing the area of the αδ- R{a(u, u)}≥ α u V ∀u ∈ Ker(B),
interface.
|b(v, p)|
sup v∈V ≥ β p V ∀p ∈ V
v
immunoglobulin (Ig) A protein of the V
globulin-type found in serum or other body fluids where
that possesses antibody activity. An individual Ig
Ker(B) := {v ∈ V : b(v, q) = 0 ∀q ∈ W} .
molecule is built up from two light (L) and two
heavy (H) polypeptide chains linked together by Then the variational saddle point problem, which
disulfide bonds. Igs are divided into five classes seeks u ∈ V, p ∈ W such that
based on antigenic and structural differences in
the H chains. a(u, v) + b(v, p) = f(v) ∀v ∈ V
c
© 2003 by CRC Press LLC
© 2003 by CRC Press LLC