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chromosome A self-replicating structure classical symbol A symbol a(x, ξ) of a
consisting of DNA complexes with various pro- pseudodifferential operator (or Fourier integral
teins and involved in the storage and transmission operator) of order m is called classical if there
ofgeneticinformation; thephysicalstructurethat exist C ∞ functions a (x, ξ), positively homoge-
j
contains genes (cf. plasmid). Eukaryotic cells neous of degree m − j in ξ (i.e., a(x, τξ) =
have a characteristic number of chromosomes τ m−j a (x, ξ), τ > 0) such that asymptotically
j
per cell (cf. ploidy) and contain DNA as linear
∞

duplexes. The chromosomes of bacteria consist a(x, ξ) ∼ a (x, ξ).
j
of double-standed, circular DNA molecules. j=0
Clifford algebra Clifford algebra is a for-
circulation In the theory of Markov pro-
mulation of algebra which unifies and extends
cesses, the probability flux in a stationary
complex numbers and vector algebra. It is based
state which can be achieved by two types of
on the Clifford product of two vectors a and b
balance: detailed balance and circular balance.
which is written ab. The product has two parts,
For most Markov processes, the sufficient and
a scalar part and a bivector part. The scalar part
necessary condition for zero circulation is time-
is symmetric and corresponds with the usual dot
reversibility. 1
product a · b = (ab + ba). The bivector part is
2
antisymmetric and can be thought of as a directed
k k k
n
m
class C C C A function f : U ⊂ R → R area, defining a plane a ∧b = (ab −ba). Then
1
k
2
is differentiable of class C ,0 ≤ k ≤∞, if all the Clifford product can be written as ab = a ·
α
partial derivatives ∂ f α ,0 ≤|α|≤ k,of f up to
∂x b + a ∧ b.
order k exist and are continuous.
closed curve A closed curve (or closed path)
classical Fourier integral operator A in a space M is a curve γ :[a, b] → M such that

Fourier integral operator Au = e φ(x,ξ) γ(a) = γ(b).
a(x, ξ)u(y)dy is called classical if it has a clas-
closed form An exterior form ω on a mani-
sical symbol a(x, ξ).
fold having vanishing exterior differential dω =
0. Exact forms are closed. The converse is not
classical groups The matrix groups GL(n),
true in general. It is true on contractible mani-
SL(n), U(n), SU(n), O(n), SO(n), Sp(2n) are m
folds (e.g., on star-shaped open sets of R by
called classical Lie groups.
Poincar´e’s lemma). De Rham cohomology stud-
ies the topological properties of manifolds by
classical limit In quantum mechanics when classifying closed forms that are not exact.
2 −1
the Planck constant h ¯→0 classical mechanics is Example: The form ω = (x + y )
2
recovered. (xdy − ydx) is a closed (but not exact) form
2
defined on R −{0}; in fact, it locally reduces to
2
2
classical mechanics Classical mechanics dϕ on the unit circle x + y = 1, butitisnot
vis-`a-vis quantum mechanics and relativistic an exact form since ϕ cannot extend to a single-
mechanics. The equations of motion in classical valued coordinate function on the whole of the
mechanics are Hamilton’s equations or equiva- circle.
lently the Euler-Lagrange equations, formulated
closed graph theorem If X and Y are
on a finite dimensional symplectic manifold.
Banach spaces and T : X → Y is a closed linear
operator defined on all of X then T is bounded
classical path A distinction between clas-
(i.e., continuous).
sical and quantum mechanics. In classical
mechanics a particle takes only one path to go closedoperator AlinearoperatorT : X →
from a point q to a point q , while all paths Y, between two Banach spaces X and Y and

contribute in quantum mechanics. The three having a dense domain D(T ) ⊂ X is closed if its
colors R, G, and B belong to the representation graph N(T ) ={(x, T x) : x ∈ D(T )} is closed
of SU(3). in X × Y.
© 2003 by CRC Press LLC
© 2003 by CRC Press LLC