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stable mathematical program Roughly, steady state (stationary state) In a kinetic
one whose solution does not change much under analysisofacomplexreactioninvolvingunstable
perturbation. For the inequality case, we have intermediates in low concentration, the rate of
the following stability conditions: change of each such intermediate is set equal to
(i.) {x ∈ X : g(x) ≤ b} is bounded for some zero, so that the rate equation can be expressed
b> 0. as a function of the concentrations of chemical
species present in macroscopic amounts. For
(ii.) cl{x ∈ X : g(x) < 0}={x ∈ X : g(x)
example, assume that X is an unstable interme-
≤ 0}.
diate in the reaction sequence:
The first stability condition pertains to upper
k 1
semicontinuity and the second, called the closure A + BA −→ X
←−
k −1
condition, pertains to lower semicontinuity.
k 2
The conditions are not only sufficient to X + C −→ D.
ensure the respective semicontinuity, but they are
Conservation of mass requires that:
necessary when:
(i.) {x ∈ X : g(x) ≤ 0} is bounded, [A] + [X] + [D] = [A] 0
(ii.) {x ∈ X : g(x) < 0} is not empty.
which, since [A] is constant, implies:
0
standard deviation, s s s The positive square
root of the sum of the squares of the deviations −d[X]/dt = d[A]/dt + d[D]/dt.
between the observations and the mean of the
Since [X] is negligibly small, the rate of forma-
series, divided by one less than the total num-
ber in the series. The standard deviation is the tion of D is essentially equal to the rate of dis-
appearance of A, and the rate of change of [X]
positive square root of the variance, a more fun-
can be set equal to zero. Applying the steady
damental statistical quantity.
state approximation (d[X]/dt = 0) allows the
state A state of a system, s ∈ S, where elimination of [X] from the kinetic equations,
j
S is the set of states of the system, is a distinct whereupon the rate of reaction is expressed:
observable or derivable variable.
Comment: The definition is meant to cover k k [A][C]
1 2
d[D]/dt =−d[A]/dt =
both the intensive states of thermodynamics and k −1 + k [C]
2
the states of computations and computational
devices. By these definitions Post production Notes: (1) The steady state approximation
systems have neither memory nor states; instead, does not imply that [X] is even approximately
the set of constructs C formed during a derivation constant, only that its absolute rate of change
is discussed. is very much smaller than that of [A] and
[D]. Since according to the reaction scheme
stationary point Usually this is used to d[D]/dt = k [X][C], the asusmption that [X]
2
mean a Kuhn-Tucker point, which specializes to is constant would lead, for the case in which C
one for which gradf(x) = 0 if the mathematical is in large excess, to the absurd conclusion that
program is unconstrained. In the context of an formation of the product D will continue at a
algorithm, it is a fixed point. constant rate even after the reactant A has been
consumed.
stationary policy In a dynamic program, a
(2) In a stirred flow reactor a steady state
∗
policy that is independent of time, i.e., x (s, t) =
implies a regime in which all concentrations are
T(s) (some function of state, but not of time, t).
independent of time.
statistical genetics Genetics is a stochastic
process. Statistical genetics studies the genetics steel beam assortment problem A steel
by using the concepts and methods from the the- corporation manufactures structured beams of a
ory of probability and statistics. See population standard length, but a variety of strengths. There
genetics. is a known demand of each type of strength, but a

© 2003 by CRC Press LLC
© 2003 by CRC Press LLC