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discrete spectrum The discrete spectrum or divergence theorem Also called Gauss’s
3
point spectrum σ (A) of a linear operator A is theorem. Let ; be a region in R and ∂; the
p
the set of all λ for which (λI − A) is not one- oriented surface that bounds ;, and denote by n
to-one, i.e., σ (A) is the set of all eigenvalues the unit outward normal vector to ∂;. Let X be
p
of A. a vector field defined on ;. Then

discrete topology In the discrete topology (divX)dV = (X · n)dS.
on a set S every subset of S is an open set. ; ∂;
divergent sequence A sequence that is not
disjoint Two collections, especially sets convergent.
such as A and B, are said to be disjoint if
dividing surface A surface, usually taken
A ∩ B =∅.
to be a hyperplane, constructed at right angles
dissipative A linear operator T : H → H to the minimum-energy path on a potential-
on a Hilbert space (H, < , >) is called dissi- energy surface. In conventional transition-state
pative if Re <T u, u>≤ 0 for all u ∈ H. theory it passes through the highest point on the
minimum-energy path. In generalized versions
dissociation (1) The separation of a molecu- of transition-state theory the dividing surface can
lar entity into two or more molecular entities be at other positions; in variational transition-
(or any similar separation within a polyatomic state theory the position of the dividing surface
molecular entity). Examples include unimolecu- is varied so as to get a better estimate of the rate
lar heterolysis and homolysis, and the separation constant.
of the constituents of an ion pair into free ions.
DNA See deoxyribonucleic acids.
(2) The separation of the constituents of any
aggregate of molecular entities. DNA supercoil A DNA molecule has a dou-
In both senses dissociation is the reverse of ble helical structure (see double helix). Con-
association. sider the axis of the double helix as a space
curve; it is known experimentally that the curve
distribution A distribution on an open set can have non-planar geometry. For example, it
n
U ⊂ R is a continuous linear functional on
can be solenoidal itself, thus the name supercoil.
∞
C (U) (smooth functions with compact sup- DNA from some organisms have their helical
c
port).
axis being a closed space curve with topological
linking numbers, i.e., knot. (cf., W.R. Bauer,
distribution (of subspaces) A family of
F.H.C. Crick, and J.H. White, Sci. Am., 243,
subspaces x ⊂ T M one for each x ∈ M.
x
When the dimension of the subspaces is con- 118, 1980).
x
stant with respect to x such a dimension k is domain In computer science, a domain is a
called the rank of the distribution. One should discipline, anareaofphysicalreality, orathought
add some regularity condition in order not to modeled by a representation; in essence, its sub-
allow too odd dependence of the subspace on ject. In mathematics, a domain is the set on
the point x. Usually one asks that locally there whose members a relation operates.
exist k local vector fields, called generators of , Comment: For further comments on the
spanning at each point of an open set U ⊂ M.
x mathematical sense of domain, see relation. See
also image and range.
divergence Let M be a manifold with vol-
ume ; and X a vector field on M. Then the domain model A formal model of a particu-
unique function div X ∈ C (M) such that lar domain (in the computational sense).
∞
;
the Lie derivative L ; = (div X); is called Comment: It is this that a database or artificial
;
X
3
the divergence of X.If M = R and X = intelligence system, or indeed any abstract model
(f ,f ,f ) then of a phenomenon, reifies. It is distinguished
1
3
2
from a data model, which is an implementation
∂f 1 ∂f 2 ∂f 3
divX = + + . method (such as relational, object-oriented, or
∂x ∂y ∂z declarative database).
© 2003 by CRC Press LLC