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One of the transition states of the two reaction Green’s theorem A special case of Stokes’
steps must (in general) have a higher standard theorem for the plane. Let P, Q be differentiable
2
Gibbs energy than the other, whatever the con- functions in a region ; ⊂ R , then
centrationofD inthesystem. However, thevalue

of that concentration will determine which of the ∂Q ∂P
− = Pdx + Qdy.
reaction steps is rate-limiting. If the particular ; ∂x ∂y ∂;
concentrations of interest, which may vary, are
chosen as the standard rate, then the rate-limiting grammar For any language L, the grammar
step is the one of highest Gibbs energy. X is the set of rules specifying the syntax of well-
formed constructs in L.
Gibbs energy of activation (standard free Comment: A synonym for the rules (and con-
‡ 0 fusingly, sentences formed by applying the rules)
energy of activation), G The standard
Gibbs energy difference between the transition is “productions.” Grammars have three func-
tions: to generate and to recognize constructs
state of a reaction (either an elementary reaction
in a language, and to transform one language
or a stepwise reaction) and the ground state of the
to another. The most familar example of a
reactants. It is calculated from the experimental
grammar comes from string grammars built from
rate constant k via the conventional form of the
natural languages, which specify the syntactic
absolute rate equation:
properties a sentence must fulfill for it to be
‡ “legal.” Many grammars, and the languages
G = RT [ln(k /h) − ln(k/T )]
B
they describe, fall into a hierarchy of increasing
where k B is the Boltzmann constant and h mathematical complexity first devised by Noam
the Planck constant (k /h = 2.08358 × Chomsky. A context-sensitive grammar, one of
B
−1 −1
10 10 K s ). The values of the rate constants, the more complex types, specifies that a token’s
and hence Gibbs energies of activation, depend output depends on its context. Examples in
upon the choice of concentration units (or of the English are a little contrived: perhaps the best
thermodynamic standard state). is “Dick and Jane went north and south, respec-
tively.” Here “respectively” signals a mapping
function, so that Dick went north and Jane south.
Gram-Schmidt orthogonalization A pro-
Grammars are commonly applied to recognize
cess to construct an orthonormal basis in a
features of DNA and protein sequence. In that
Hilbert space out of an arbitrary Hilbert basis.
context they are usually called string grammars.
They are also used to recognize and generate pat-
Green’s functions Auxiliary functions used
terns of chemical and biochemical structure and
to solve nonhomogeneous boundary value prob-
function. See graph grammar.
lems. Example: The general solution of the
boundary value problem
graph A graph G(V, E) consists of a set
of vertices V, V = ∅ and a set of edges

−y = f(x) , y(0) = 0,y(1) = 0
e(λ, v ,v ) ∈ E, E ≥∅, where λ ∈ L, L = ∅
i
j
can be written in the form is the type of relation the edge expresses, and
{v ,v }∈ V,i = j are the (possibly empty) ver-
j
i
1 tices associated with that edge.
y = φ(x) = G(x, s)f (s)ds
0 Comment: This definition has edges express-
ing relationships but allows them to be
where G(x, s) is the Green’s function defined by unbounded by vertices on either or both sides
(“free” edges). This latter feature is particularly

s(1 − x), 0 ≤ s ≤ x,
G(x, s) = useful in specifying certain types of graphs and
x(1 − s), x ≤ s ≤ 1 .
operators upon them. All the graphs considered
here are finite; have one and only one edge join-
ing any pair of nodes (are not multigraphs); and
© 2003 by CRC Press LLC