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sparsity The fraction of zeros in a matrix. If spectrum of a matrix The set of eigenval-
A is m by n, and A(i, j) = 0 for k of its elements, ues of A.
its sparsity is k/mn. Large linear programs tend
to be very sparse, increasing as the dimensions
stability region The set of parameter values
get large. For example, consider the standard
for which an optimal solution remains optimal.
transportation problem with s sources and d des- This arises naturally in combinatorial optimiza-
tinations. This has m = (s + d) constraints and
tion, where a solution is often a subgraph, such
n = sd variables. Each column, however, has
as a tree, and the question is for what range
exactly two nonzeros since A is the incidence
of arc weights is this subgraph optimal (such
matrix of the network, so its sparsity is 2n/mn,
as a spanning tree that is minimum for given
or simply 2/m, which decreases as the number
of sources and/or destinations grows large. The weights). More generally, x could be a solu-
tion generated by some algorithm, A, from an
sparsity of a simple graph (or network) is the 0
initial value x . Then, suppose the feasibility
sparsity of its adjacency matrix. More generally,
region F(p) depends on the parameter p and
the sparsity of a multigraph refers to the average
the objective f(x p) also depends on p. Let
degree of its nodes. j
0
X(p, A, x ) denote the generated solution from
0
specially ordered set (SOS) Certain sets of algorithm A, starting at x , with parameter value
nonnegative variables that are required to sum to p. Let the parameter set be P (which includes
1. For computational efficiency, it is sometimes p ). The stability region of x = X(p ,A,x )
0
∗
∗
∗
0
better to define these sets by some marking data is {p ∈ P : x = X(p, A, x )}. The algorithm
∗
structure, rather than include them along with may be a heuristic,so x need not be optimal.
∗
other equality constraints. There are two types For example, one could use an n-opt heuristic
of SOSs, distinguished by what they represent. for the traveling salesman problem, so x repre-
A Type 1 SOS is one in which each variable is sents a tour. The parameters could be the costs,
binary, so the constraint is one of multiple choice. or they could be the location of each point in a
A Type 2 SOS is one in which a restricted basis
euclidean TSP. The stability region is the set of
entry rule is used, as in the lambda-form of sep- costs, or coordinates in the plane, for which the
arable programming.
tour generated by n-opt is the same.
specifically labeled An isotopically labeled
compound is designated as specifically labeled stable As applied to chemical species, the
when a unique isotopically substituted com- term expresses a thermodynamic property, which
pound is formally added to the analogous isotopi- is quantitatively measured by relative molar
cally unmodified compound. In such a case, both standard Gibbs energies. A chemical species A
position(s) and number of each labeling nuclide is more stable than its isomer B if G > 0 for
0
r
are defined. the (real or hypothetical) reaction A → B, under
standard conditions. If for the two reactions:
spectral radius (of a matrix, A) The radius 0
P → X + Y( G )
r
1
of the following disk that contains the spectrum: 0
Q → X + Z( G )
r
r(A) = Max{|y| : y is an eigenvalue of A}. 2
0
0
G > G , P is more stable relative to
r
r
2
1
spectral responsivity function See respon-
the product Y than is Q relative to Z. Both in
sivity.
qualitative and quantitative usage the term stable
spectrum Let T be a linear operator on a is therefore always used in reference to some
Banach space V . A complex number λ is said explicitly stated or implicitly assumed standard.
to be in the resolvent set ρ(T ) of T if λI − T The term should not be used as a synonym
for unreactive or “less reactive” since this con-
is a bijection with bounded inverse. R (T ) =
λ
(λI − T) −1 is called the resolvent of T at λ.If fuses thermodynamics and kinetics. A relatively
λ ∈ ρ(T ), then λ is said to be in the spectrum more stable chemical species may be more reac-
σ(T ) of T . The set of all eigenvalues of T is tive than some reference species toward a given
called the point spectrum of T . reaction partner.
© 2003 by CRC Press LLC
© 2003 by CRC Press LLC