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Dini’s theorem Let {f } be a sequence of Dirac operator A linear first-order partial
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continuous functions converging pointwise to differential operator with nonconstant coeffi-
a continuous function f . If {f } is monotone cients, defined between sections of the spin bun-
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increasing sequence, then the convergence is uni- dle. It is defined as covariant derivative followed
form. by Clifford product. Locally the Dirac operator
is given by
dipole–dipole interaction Intermolecular
δ
or intramolecular interaction between molecules Dψ = γ µ ψ
m
or groups having a permanent electric dipole µ δx u
moment. The strength of the interaction depends µ
where γ are the Dirac gamma matrices.
on the distance and relative orientation of the
dipoles. The term applies also to intramolecu-
Dirac spinors Elements in the full complex
lar interactions between bonds having permanent
spin representation as opposed to elements in the
dipole moments.
half spin representation which are called Weyl
spinors and elements in the real spin representa-
Dirac Paul Adrien Maurice Dirac (1902–
tion which are called Majorana spinors.
1984), Swiss/English theoretical physicist,
founder of relativistic quantum mechanics.
directed edge A sequence of two nodes
Nobel prize for physics 1933 (shared with
(ordered pair) in a graph; often denoted
Schr¨odinger).
/v , v i+1 0.
i
Comment: These are the edges that are drawn
Dirac delta function The generalized func-
tion δ(x − x ) defined by with arrows, indicating a direction of travel or
0 flow. See edge.

∞
f (x)δ(x − x )dx = f (x ). direction In a formal reaction equation, the
0
0
consumption of sinistralateral coreactants (the
−∞
Dirac delta measure The measure δ y “forward” direction); equally, the consumption
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located at some arbitrary, but fixed y ∈ R is of dextralateral coreactants (the “reverse” direc-
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defined for A ⊂ R as tion).
Comment: Chemical and biochemical reac-

1 if y ∈ A
δ (A) = tions can be thought of as a composite of two
y
0 if y ∈ A. reactions. One that consumes sinistralateral core-
actants, producing dextralateral ones or proceed-
Dirac equation The equation
ing left to right and the opposite. These two
(i ∂ − m)ψ = 0 directions are often represented as half reactions
which are formal equations that specify only
where ψ is a wave function describing a relativis- one of the two directions. The intrinsic rates of
µ
tic particle of mass m and ∂ := γ δ with γ µ the two directions, as measured by their respec-
µ
the Dirac gamma matrices. tive rate constants can differ very significantly.
See also dextralateral, dynamic equilibrium, for-
Dirac gamma matrices The 4 × 4 matrices
j mal reaction equation, microscopic reversibility,
I 0 0 σ
0 j product, rate constant, reversibility, sinistralat-
γ = , γ = j ,
0 −I −σ 0 eral, and substrate.
j = 1, 2, 3, where I is the 2 × 2 identity matrix

j 1 01 Dirichlet boundary condition Specifica-
and σ are the Pauli matrices σ = ,
10 tion of the value of the solution to a partial dif-

0 −i 10 ferential equation along a bounding surface.
2 1 2
σ = , σ = , i = −1.
i 0 0 −1
Dirichlet problem The problem of finding
Dirac Laplacian The square root of the solutions of the Laplace equation u = 0(u +
xx
2
Dirac operator, i.e., we have D ψ = ψ, the u = 0 in two dimensions) that take on given
yy
d’Alembert operator. boundary values.
© 2003 by CRC Press LLC