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col (saddle point) A mountain-pass in a commutative Referring to a set A with a
potential-energy surface is known as a col or sad- binary operation * satisfying
dle point. It is a point at which the gradient is
zero along all coordinates, and the curvature is a ∗ b = b ∗ a
positive along all but one coordinate, which is the
reaction coordinate, along which the curvature is for all a, b ∈ A. See also Abelian group.
negative.
commutator (in an associative algebra A)
colligation The formation of a covalent The binary operation given by
bondbythecombinationorrecombinationoftwo
radicals (the reverse of unimolecular homolysis). [A, B] = A · B − B · A.
˙
˙
For example, OH + H C → CH OH.
3
3
For example, this is the definition for the com-
colloid A short synonym for colloidal sys- mutator of the Lie algebra GL(n, R) of n × n
tem. matrices. Notice that it satisfies the Jacobi iden-
tities
colloidal The term refers to a state of sub-
division, implying that the molecules or poly-
[[A, B],C] + [[B, C],A] + [[C, A],B] = 0.
molecular particles dispersed in a medium have
at least in one direction a dimension roughly
The commutator of two vector fields X =
between 1 nm and 1 µm, or that in a system dis- X ∂ , Y = Y ∂ is defined by
µ
µ
µ
µ
continuities are found at distances of that order.
µ
µ
ν
ν
[X, Y] = (X ∂ Y − Y ∂ X )∂ .
ν
µ
µ
color (1) (in mathematics) A nonlabel token
for an edge or node of a graph. A coloring of
More generally any binary operation defining
the graph G is the set of colors assigned to the
a Lie algebra, namely, the Lie-product obeying
nodes, such that no two adjacent nodes have the Jacobi identities. See also Lie algebra.
same color. An edge coloring is the set of colors
assigned to edges; a proper edge coloring is the
compact (1)A topological space (X, τ(X))
set of colors such that no two adjacent edges have
is compact if from any open covering of X one
the same color.
can always extract a finite subcovering.If X is a
Comment: These need not be physical colors,
topological subspace of a metric space, “com-
though it is perhaps easiest to think of them that
pact” is equivalent to “closed and bounded.”
way. Note that colorings are by definition proper,
Thence closed intervals are compact in R; closed
whereas edge colorings need not be. The differ- m
balls are compact subsets of R (as well as in any
ence between definitions of node and edge color-
metric space).
ings may have been motivated in the beginning
(2) (locally) A topological space X is locally
by the four color map problem.
compact if every point p ∈ X has a compact
(2) (in quantum QCD) Gluons and quarks
neighborhood.
have an additional type of polarization (degree
of freedom) not related to geometry. “The idiots
compact operator Let X and Y be Banach
physicists, unable to come up with any wonder-
spaces and T : X → Y a bounded linear oper-
ful Greek words anymore, called this type of
ator. T is called compact or completely continu-
polarization by the unfortunate name of color.”
ous if T maps bounded sets in X into precompact
(R.P. Feynman in QED: The Strange Theory of
sets (the closure is compact) in Y. Equivalently,
Light and Matter, Princeton University Press,
n
Princeton, NJ 1985). for every bounded sequence {x }⊂ X , {Tx }
n
has a subsequence convergent in Y.
commutation relations If a classical
Hamiltonian H(q ,p ) is quantized, one con- compact set A subset M of a topological
i
i
siders q and p as operators on a Hilbert space X is called compact if every system of
i
i
space satisfying the commutation relations open sets of X which covers M contains a finite
[q ,p ] = δ and [q ,q ] = [p ,p ] = 0. subsystem also covering M.
i j ij i j i j
© 2003 by CRC Press LLC
© 2003 by CRC Press LLC