Page 29 -

cardinality The number of elements in a set. products of a reaction is called autocatalysis.
Comment: Often denoted by S or |S|, where Catalysis brought about by a group on a reactant
S is any set. See also countable set, denumerably molecule itself is called intramolecular catalysis.
infinite set, finite set, infinite set, and uncountably The term catalysis is also often used when
infinite set. the substance is consumed in the reaction (for
example: base-catalyzed hydrolysis of esters).
carotenes Hydrocarbon carotenoids (a sub- Strictly, such a substance should be called an
class of tetraterpenes). activator.
Comment: Reactions are written at many
carotenoids Tetraterpenoids(C ),formally levels of resolution, and those reactions which
40
derived from the acylic parent ψ, ψ-carotene I detail how the catalyst functions will not meet
by hydrogenation, dehydrogenation, cyclization, this definition. But that is appropriate, since
oxidation, or combination of these processes. the molecule which functions catalytically in one
This class includes carotenes, xanthophylls, and reaction will be a substrate in the reactions which
certain compounds that arise from rearrangement describe the catalysis.
of the skeleton of I or by loss of part of this
structure. Retinoids are excluded. Cauchy-Goursat theorem Suppose that
f : D → Cis analytic on a disk D ⊂ C; then f
Cartan matrix For a semisimple Lie alge- has an antiderivative F on D, i.e., F (z) = f(z),

bra of rank l the Cartan matrix [A ] is defined which is analytic in D, and, if γ is any closed
ij
2(α i ,α j )
by A = (α i ,α i ) where α , ··· ,α is the system curve in D, then γ f = 0.
l
ij
1
of simple roots.
Cauchy integral formula Let f be analytic
Casimir Elements in the universal envelop- within and on a simple closed curve γ and z a
0
ing algebra of a Lie algebra that commute with point in the interior of γ . Then
all other elements of the Lie algebra are called
1 f(z)
Casimirs. f(z ) = dz.
0
2πi γ z − z 0
Casimir function On a Poisson manifold
Cauchy-Riemann equations (Cauchy-
(P, { , }), a function C that Poisson commutes
Riemann theorem) Let ; ⊂ C be an open set
with every function F is called a Casimir func-
and f : ; → C be a function f(z) = u(x, y)
tion, i.e., {C, F}= 0 for all F.

+iv(x, y). Then f (z ) exists if and only if f is
0
differentiable in the sense of real variables and,
Casorati-Weierstrass theorem Let f have
at z = (x ,y ), u, v satisfy the Cauchy-
an essential singularity at z and let w ∈ C. Then 0 0 0
0
there exists z ,z ,z ,... such that z → z and Riemann equations
0
n
3
2
1
f(z ) → w. ∂u ∂v ∂u ∂v
n
= , and =− .
∂x ∂y ∂y ∂x
catalyst A substance that increases the rate
of a reaction without modifying the overall stand- Cauchy-Riemann theorem See Cauchy-
ard Gibbs energy change in the reaction; the pro- Riemann equations.
cess is called catalysis. The catalyst is both a
reactant and product of the reaction. The words Cauchy-Schwarz inequality In any inner
catalyst and catalysis should not be used when product space (V, < , >) for any x, y ∈ V
the added substance reduces the rate of reaction
(see inhibitor). | <x, y > |≤ x y .
Catalysis can be classified as homogeneous
catalysis, in which only one phase is involved, Cauchy sequence A sequence {x } in a
n
and heterogeneous catalysis, in which the reac- metric space (M, d) such that for every "> 0
tion occurs at or near an interface between there exists an integer N such that d(x ,x )<"
m n
phases. Catalysis brought about by one of the whenever m>N and n>N.
© 2003 by CRC Press LLC
© 2003 by CRC Press LLC