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products and exterior differentials of gener- the contraction. However, no other nonmathe-
ators). Accordingly, this includes also the 2- matical information is required for contrac-
ν
forms dω i =−dy i ∧ dx . tions of models of biological systems, such as
µ 1 ...µ k−1 µ 1 ...µ k−1 ν
k
∗
¯ ; (J B) is the ideal of all forms vanishing biochemical networks (unlike expansion). Such
c
on all holonomic sections, including all (m+h)- information may be beneficial, for example, in
forms with h> 0. suggesting appropriate approximations. An
expansion is the reverse operation.
continuity equation The law of conserva-
tion of mass of a fluid with density ρ(x, t) and contraction mapping theorem Let T :
velocity v is called continuity equation (M, ρ) → (M, ρ) be a contraction mapping of
a complete metric space (M, ρ) into itself. Then
∂ρ
+ div(ρv) = 0. T has a unique fixed point; i.e., there exists a
∂t unique x ∈ M such that Tx = x .
0
0
0
continuous function A function between
topological spaces such that the inverse image contravariant/covariant tensor A tensor T
of every open set is open. on a vector space E, contravariant of order r and
covariant of order s is a multilinear map
continuous spectrum The continuous spec-
∗
∗
T : E × ··· × E × E × ··· × E → R
trum σ (T ) of a linear operator T on a Hilbert
c
space H consists of all λ ∈ C such that T −λI is
∗
(r copies of E and s copies of E).
a one-to-one mapping of H onto a dense proper
subspace of H.
convergent sequence A sequence {x }
n
continuously differentiable A function f : having a limit L. That is, for every neighbor-
n
m
U ⊂ R → R whose partial derivatives ∂f i hood U of L,we have x ∈ U for all except
n
∂x j finitely many n.
exist and are continuous is called continuously
1
differentiable or of class C .
convex combinations See convex hull.
contractible A topological space X is con-
tractible when the identity map id : M → M convex hull The convex hull of a set A in a
(defined by id(x) = x)) and the constant map vector space X is the set of all convex combina-
c : M → M (defined by c (y) = x)) are homo- tions of elements of A, i.e., the set of all sums
x
x

topic. See homotopy. t x + ··· t x , with x ∈ A, t ≥ 0, t = 1,n
1 1
i
n n
i
i
arbitrary.
contraction (1) (in metric topology) A map
T : (M, ρ) → (M, ρ) of a metric space (M, ρ) convex set A set C ⊂ X is convex if for any
into itself such that there exists a constant λ, 0 < x, y ∈ C, tx +(1−t)y ∈ C for all t (0 <t < 1).
λ< 1 such that
convolution The convolution f ∗ g of two
ρ(T x, T y) ≤ λρ(x, y), for all x, y ∈ M.
functions f and g is given by
(2) (in networks) In a network (or graph),
the replacement of a larger subnetwork by a f ∗ g(x) = f(x − y)g(y)dy.
smaller using a sequence of mathematical oper-
ations. Network N is a contraction of network

N if N can be obtained from N by a sequence

of node, edge, parameter, or label combination coordinate See basis of a vector space.
operations.
Comment: The nature of the combination coordinate patch See chart.
operations and their operands can vary con-
siderably depending on the intended result of coordinate transition function See chart.
© 2003 by CRC Press LLC
© 2003 by CRC Press LLC