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denumerably infinite set A set which is provided that this limit exists, f is called dif-
equivalent to the set of natural numbers, N. ferentiable at the point x with derivative f (x).

(There is a bijection between the set and N.) More generally, if X, Y are Banach spaces U ⊂
See also cardinality, finite set, infinite set, and X open, and f : U ⊂ X → Y, then the Frechet
uncountably infinite set. derivative of f is map Df : U → L(X, Y),
where L(X, Y) is the vector space of bounded
deoxyribonucleic acids (DNA) High- linear operators for X to Y defined by
molecular-weight linear polymers, composed of
nucleotides containing deoxyribose and linked f(x + th) − f(x)
Df (x)h = lim .
by phosphodiester bonds; DNA contain the t→0 t
genetic information of organisms. The double-
If this limit exists f is called differentiable at x
stranded form consists of a double helix of
with (total or Frechet) derivative Df (x).
two complementary chains that run in opposite
directions and are held together by hydrogen
bonds between pairs of the complementary deterministic computation A deterministic
nucleotides and Hoogsteen (stacking) forces. computation C specifies a computation C : K
d
I
→ K such that the cardinalities (denoted S,
O
deRham cohomology group The quotient where S is any set) of the sets of presented inputs
groups of closed forms by exact forms on a and produced outputs (K and K , respec-

O
I
manifold M are called the deRham cohomol- tively) are 1 (K I = K O = 1); each compu-
ogy groups of M. The kth deRham cohomology tational step is a one-to-one mapping between
group of M is the presented input and the produced output (∀c ,
i
c ∈ C is one-to-one); and the probability that
i
d
k
k
H (M) = ker d /range d k−1 , each symbol in the sets of presented inputs and
produced outputs exists is 1 (∀σ i,I , σ i,I ∈ K ,
I
k
where d is the exterior derivative on k forms. ∀σ , σ ∈ K , P (σ ) = P (σ ) = 1).
i,O i,O O e i,I e i,O
Comment: As with the definitions of compu-
derivation (of an R-algebra A) Given R tation, nondeterministic, and stochastic compu-
a ring,an R-linear map D : A → A (i.e., tations, the goal here is to place the usual theory
D(λx + µy) = λD(x) + µD(y)) such that the of computing within a framework that accom-
Leibniz rule holds, i.e., ∀x, y ∈ AD(xy) = modates molecular computers. Like any other
D(x)y + xD(y). A derivation on a manifold system of chemical reactions, a molecular com-
M is a derivation of the function algebra A = puterissurroundedbymanycopiesofitspossible
C (M). inputs and outputs. These copies of symbols
∞
Example: let F(R) be the R-algebra of real are populations of symbols, each type of sym-
valued (differentiable) functions f : R → R bol occuring at some frequency in its respective
over R. The ordinary derivative D : F(R) → population (P ). P (σ ) for any particular σ can
i
i
e
e
F(R) is a derivation. vary depending on the constitution of K ,K ,
I O
Notice that A is not required to be an associ- and the properties of a particular c ; so there
i
ative algebra. In particular the definition applies exists a probability density function over K.For
to Lie algebras where the Leibniz rule reads as computation over a number of steps, a vector P e
D([x, y]) = [D(x), y] + [x, D(y)]. of probabilities for each step to each σ would
i
Example: let L be a Lie algebra. For all x ∈ be assigned. Notice that a unit cardinality does
L, the map ad : L → L defined by ad : y → not imply that the length of the input and output
x
x
[x, y]isa derivation. tokens is one. See also nondeterministic and
stochastic computations.
derivative The derivative of a function f :

R → R at a point x is the function f defined by
device Asubnetworkofthebiochemicalnet-
f(x + h) − f(x) work with distinct dynamical (and perhaps bio-

f (x) = lim chemical) properties.
h→0 h
© 2003 by CRC Press LLC