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closed orbit An orbit γ (t) of a vector field coboundary operator See cochain com-
X is called closed if there is a τ > 0 such that plex.
γ (t + τ) = γ (t) for all t.
cochain complex A cochain complex con-
closed set The complement of an open set in sists of a sequence of modules and homomor-
a topological space (X, τ(X)). The class χ(X) phisms
of closed sets has the following properties:
q
··· → C q−1 → C → C q+1 → ···
(i.) The empty set ∅ and the whole space X
are elements in χ(X); such that at each stage the image of a given
(ii.) The union of a finite number of elements homomorphism is contained in the kernel of the
q
q
in χ(X) is still in χ(X); and next. The homomorphism d : C → C q+1 is
2
called coboundary operator.We have d = 0.
(iii.) The intersection of a (possibly infinite)
q
q
The kernel Z = ker d is the module of qth
family of elements in χ(X) is still in χ(X).
q
degree cocycles and the image B = im d q−1
closure For a subset A of a topological is the module of qth degree coboundaries. The
q q
space, the smallest closed set containing A. qth cohomology module H is defined as H =
q
q
Denoted A. Z /B .
¯
coacervation The separation into two liquid cocycle See cochain complex
phases in colloidal systems. The phase that is
codifferential On a Riemannian manifold
more concentrated in colloid component is the
k
(M, g) the codifferential δ : ; (M) →
coacervate, and the other phase is the equilibrium
; k−1 (M) is defined by
solution.
k
δα = (−1) n(n+k)+1+Ind(g) ∗ d ∗ α, α ∈ ; (M),
coadjoint action The coadjoint action Ad ∗
of a Lie group G on the dual of its Lie algebra where ∗ is the Hodge operator and d the exterior
∗
g is defined as the dual of the adjoint action of 2
derivative.We have δ = 0.
G on g. It is given by
codomain See relation. See also domain,
∗
< Ad α, ξ >=< α, Ad ξ >,
g g image, range.
∗
for g ∈ G, α ∈ g , ξ ∈ g, where <, >
∗ coercivity Let V be a Banach space. A
is the pairing between g and g and Ad (ξ) =
g
T (R −1 ◦ L )ξ. sesqui-linear form a : V × V → C is coer-
g
g
e
cive on V , if there exists a constant α> 0 and a

coadjoint orbit The orbits of the coadjoint compact operator K : V → V such that
∗ ∗
action, i.e., O = {Ad α | g ∈ G, α ∈ g }⊂ 2
α
g
g . Coadjoint orbits carry a natural symplectic |R{a(u, u)}+ (K(u), u)|≥ c u V ∀u ∈ V.
∗
structure, the induced Poisson bracket is the Lie-
cohomology See deRham cohomology
Poisson bracket given by
group.

δF δH
{F, H}(α) =− α, , . coisotropic (sub)manifold (in a symplectic
δα δα
manifold [P, ω]) A submanifold M ⊂ P of
coadjoint representation The coadjoint a symplectic manifold (P, ω) such that at any
representation of a Lie group G on the dual of its point p ∈ M the tangent space is contained in its
Lie algebra g is given by the coadjoint action symplectic polar, i.e.,
∗
Ad by o
∗
T M ⊂ (T M) .
p p
∗
∗
∗
Ad : G → GL(g , g ),
The dimension of a coisotropic manifold M is at
∗
∗
Ad −1 = (T (R ◦ L −1)) .
g e g g least half of the dimension of P .
© 2003 by CRC Press LLC
© 2003 by CRC Press LLC