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sample (in analytical chemistry) A portion
of material selected from a larger quantity of
material. The term needs to be qualified, e.g.,
S bulk sample, representative sample, primary
sample, bulked sample, or test sample.
The term “sample” implies the existence of
a sampling error, i.e., the results obtained on
the portions taken are only estimates of the
saddle point Let f : X × Y −→ R. Then,
concentration of a constituent or the quantity of
(x ,y ) is a saddle point of f if x minimizes
∗
∗
∗
a property present in the parent material. If there
f(x, y ) on X, and y maximizes f(x ,y) on Y.
∗
∗
∗
is no or negligible sampling error, the portion
Equivalently,
removed is a test portion, aliquot, or specimen.
The term “specimen” is used to denote a portion
∗
∗
∗
∗
f(x ,y) ≤ f(x ,y ) ≤ f(x, y )
taken under conditions such that the sampling
variability cannot be assessed (usually because
for all x in X, y in Y. the population is changing), and is assumed for
convenience, to be zero. The manner of selection
von Neumann (1928) proved this equivalent to: of the sample should be prescribed in a sampling
plan.
Inf [Sup{f (x, y) : y ∈ Y} : x ∈ X]
sample unit The discrete identifiable por-
tion suitable for taking as a sample or as a portion
= Sup[Inf{f (x, y) : x in X} : y ∈ Y]
of a sample. These units may be different at dif-
ferent stages of sampling.
∗
∗
= f(x ,y ).
satisfiability problem Find a truth assign-
saddle point problem On Banach spaces ment to logical propositions such that a (given)
V, W considerthe functional J : V ×W → R.A collection of clauses is true (or ascertain that
saddlepointproblemseeksapair(u, p) ∈ V ×W at least one clause must be false in every truth
such that assignment). This fundamental problem in com-
putational logic forms the foundation for NP-
completeness
J(u, p) = inf v∈V sup q∈W J(v, q).
scalar Given a vector space V , a member
Necessary conditions for this kind of stationary
of the field from which scalar multiplication of
point often lead to symmetric linear variational
vectors in V is defined.
problems with saddle point structure: seek u ∈
V, p ∈ W such that
scalar product See inner product.
a(u, v) + b(v, p) = f(v) ∀v ∈ V,
scaling Changing the units of measurement,
usually for the numerical stability of an algo-

b(u, q) = g(q) ∀q ∈ W, rithm. The variables are transformed as x = Sx,
where S = diag(s ). The diagonal elements are
j
where a : V × V → C,b : V × W → C thescalevalues, whicharepositive: s ,...,s >
n
1
are sesqui-linear forms, and f, g stand for lin- 0. Constraint function values can also be scaled.
ear forms on V and W, respectively. Specimens For example, in an LP, the constraints Ax = b,
of variational saddle point problems are encoun- can be scaled by RAx = Rb, where R = diag(r )
i
tered in the case of mixed variational formula- such that r> 0. (This affects the dual values.)
tions, the Stokes problem of fluid mechanics, and Some LP scaling methods simply scale each col-
whenever a linear constraint is taken into account umn of A by dividing by its greatest magnitude
by a Lagrangian multiplier. (null columns are identified and removed).
c
© 2003 by CRC Press LLC
© 2003 by CRC Press LLC