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dextralateral The set of obligatorily co- diameter For any bounded domain K ⊂ R n
reacting species arbitrarily written on the right- we define its diameter by
hand side of a formal reaction equation.
diam(K) := sup{|x − y|, x, y ∈ K}.
Comment: Formal reaction equations repre-
sent two sets of molecular species, one written on
diffeomorphism A diffeomorphism of class
the left and the other on the right-hand side of the k k
C is a C differentiable map f which is invert-
equation. The placement of a set on a side of the −1 k
ible such that f is also of class C .
equation is completely arbitrary. The equation is
understood to be symmetric in that the reaction’s
differentiable See derivative.
chemistry proceeds in both the “forward” (left
to right) and “backward” (right to left) simultan- differentiable manifold A topological space
p
eously, until dynamic equilibrium is achieved. M is called differentiable manifold of class C if
For the (bio)chemistry to proceed, each member each point x ∈ M has a neighborhood U(x) ⊂
of a set of coreacting species must be present. For M (called local chart or coordinate patch) which
each, the higher its concentration, the easier it is is homeomorphic to an open set in a vector
to observe the reaction and the faster the reaction space (R ), and such that the change of coor-
n
will occur. dinates (coordinate transition functions) are dif-
p
Since the reactions are reversible, the equa- ferentiable maps of class C . See chart.
tions are symmetric, and the placement of sets
of coreacting species in the equation is arbitrary, differential equation An equation involv-
the sets of species are designated sinistralateral ing a function and some of its derivatives, e.g.,

anddextralateral.Thesedesignationsdistinguish f (x) + f(x) = 0.
them from the sets of substrates and products:
differential form A differential form of
these terms indicate the role a molecule plays in
degree r (or an r-form) on an open set U in a
the reaction. See also direction, dynamic equi- r
vector space X is a smooth map ω : U → L X ∗
librium, formal reaction equation, microscopic
from U into the rth alternating product of X .
∗
reversibility, product, rate constant, reversibil-
ity, sinistralateral, and substrate.
differential operator Let E and F be vector
∞
bundles over a manifold M and C (E), C (F)
∞
diagonalizable A linear transformation T : the spaces of smooth sections. A differential
V → V on a vector space V is called diagonal- operator is a linear map L : C (E) → C (F)
∞
∞
izable (or semi-simple) if there exists a basis of
such that L(fg) = (Lf )g +(Lg)f for all f, g ∈
V consisting entirely of eigenvectors of T . See C (E).
∞
linear.
diffusion equation The diffusion equation
diagram level (in x-ray spectroscopy) A or heat equation is of the following form, for
level described by the removal of one electron u = u(x, t), x ∈ R n
from the configuration of the neutral ground
(∂ − )u = 0 .
state. These levels form a spectrum similar to t
that of a one-electron or hydrogen-like atom but, ∂ u(x,t) ∂u(x,t)
2
e.g., in two dimensions 2 = .
beingsingle-valencylevels, havetheenergyscale ∂x ∂t
reversed relative to that of single-electron levels. diffusion-driven instability This is a bifur-
Diagram levels may be divided into valence cation phenomenon in nonlinear diffusion-
levels and core levels according to the nature of reaction equations. If when all the diffusion
the electron vacancy. Diagram levels with orbital terms are absent, the remaining nonlinear ODE
angular momentum different from zero occur in has a stable fixed point, but when a diffusion term
pairs and form spin doublets. is present, the spatially homogeneous solution
corresponding to the stable fixed point is unsta-
diagram line (in x-ray spectroscopy) See ble, we say the system has a diffusion-driven
characteristic x-ray emission and x-ray satellite. instability.

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