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Cauchy theorem If a function f is analytic an end-group,a branch point or an otherwise-
in a region ; and γ is a closed curve in ; which designated characteristic feature of the macro-

is homotopic to a point in ;, then f = 0. See molecule.
γ
homotopy. Notes: (1) Except in linear single-strand
macromolecules, the definition of a chain may
Cayley transform Let A be a closed sym- be somewhat arbitrary.
(2)A cyclic macromolecule has no end groups
metric operator on a Hilbert space. Then (A −
iI)(A + iI) −1 is called the Cayley transform but may nevertheless be regarded as a chain.
of A . (3) Any number of branch points may be
present between the boundary units.
(4) Where appropriate, definitions relating to
Cea’s lemma Let a be a bilinear/sesqui-
macromolecule may also be applied to chain.
linear form on a real/complex Banach space V ,
which satisfies:
chain rule Let X, Y, Z be Banach spaces
Continuity:
and f : X → Y , g : Y → Z be differen-
|a(u, v)|≤ C u v , ∀u, v ∈ V, tiable of class C . Then g ◦ f : X → Z is of
k
V
V
V-ellipticity: class C and
k
2
|R{a(u, v)}| ≥ α u , ∀u ∈ V.
V
Suppose that V ⊂ V is a closed subspace of D(g ◦ f )(x) = Dg(f (x)) ◦ Df (x).
h

V .For f ∈ V , let u ∈ V, u ∈ V stand for
h h See derivative.
solutions of the variational problems
characteristic classes Chern classes
a(u, v) = f(v) ∀v ∈ V,
c ,...,c are defined for a complex vector
1
k
bundle of dimension k (or equivalently for a
a(u ,v ) = f(v ) ∀v ∈ V . 2i
h
h
h
h
h
GL(k, C) principal bundle) c ∈ H (M).
i
These are unique according to the Bramble- Pontrjagin classes p ,...,p are defined for
1
j
Hilbert lemma. Then Cea’s lemma asserts that a real vector bundle of dimension k (or equiv-
alently for a GL(k, R) principal bundle) p ∈
i
C H (M).
4i
u − u ≤ inf v h ∈V h u − v .
h V
h
α Stiefel-Whitney classes w ,...,w k are
1
defined for a real vector bundle of dimension
If a is symmetric/Hermitian the constant C can
α k (or equivalently for a GL(k, R) principal
be replaced with 1. A generalization of Cea’s bundle). They are Z characteristic classes
2
lemma to sesqui-linear forms that satisfy an inf- w ∈ H (M; Z ).
i
2
i
sup condition is possible. Assume that instead
of being V-elliptic the bilinear form a satisfies, characteristic cone The principal sym-
α
c (x)ξ
with α> 0, bol P (x, ξ) = |α|=m α of a (lin-
m
ear partial) differential operator P(x, D) =
|a(u ,v )| α
c (x)D is homogeneous of degree m
h
h
sup ≥ α u ∀u ∈ V . |α|≤m α
v h ∈V h h V h h
v in ξ . The set
h V
n
Then, provided that a unique continuous solution C (x) ={ξ ∈ R | P (x, ξ) = 0}
P m m
u ∈ V of the above variational problem exists,
is called the characteristic cone of P at x.
there holds
−1 characteristic equation For an n×n matrix
u − u ≤ 1 + Cα inf u − v .
h V
h V
A the equation det(A − λI) = 0.
v h ∈V h
chain (in polymers) The whole or part of characteristic function The characteristic
a macromolecule,an oligomer molecule or a function of a set A is defined by
block, comprising a linear or branched sequence
of constitutional units between two boundary 1if x ∈ A
χ (x) =
A
constitutional units, each of which may be either 0if x ∈ A
© 2003 by CRC Press LLC
© 2003 by CRC Press LLC