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n
Molecular entity is used here as a general term Monge-Ampere equation Let U ⊂ R be
for singular entities, irrespective of their nature, open and u : U × R → R. The Monge-Ampere
while chemical species stands for sets or ensem- equation for u is
blesofmolecularentities. Notethatthenameofa
2
compound may refer to the respective molecular det(D u) = f,
entity or to the chemical species; e.g., methane,
may mean a single molecule of CH (molecu- where Du = D u = (u ,...,u ) denotes the
x
4
x n
x 1
lar entity) or a molar amount, specified or not gradient of u with respect to the spatial variable
(chemical species), participating in a reaction. x = (x ,...,x ).
1
n
The degree of precision necessary to describe
a molecular entity depends on the context. For monomer A substance composed of mono-
example “hydrogen molecule” is an adequate mer molecules.
definition of a certain molecular entity for some
purposes, whereas for others it is necessary to monomer molecule A molecule which can
distinguish the electronic state and/or vibrational undergo polymerization thereby contributing
state and/or nuclear spin, etc. of the hydrogen constitutional units to the essential structure of a
molecule. macromolecule.
molecularity The number of reactant
monomeric unit (nomoner unit, mer) The
molecular entities that are involved in the
largest constitutional unit contributed by a single
microscopic chemical event constituting an
monomer molecule to the structure of a macro-
elementary reaction. (For reactions in solution
molecule or oligomer molecule.
this number is always taken to exclude molecular
Note: The largest constitutional unit con-
entities that form part of the medium and which
are involved solely by virtue of their solvation tributed by a single monomer molecule to
the structure of a macromolecule or oligomer
of solutes.) A reaction with a molecularity
molecule may be described either as monomeric
of one is called “unimolecular,” one with a
or by monomer used adjectivally.
molecularity of two “bimolecular,” and of three
“termolecular.”
monomorphism An injective morphism
moment of a force, M M M, about a point The between objects of a category. For example, a
vector product of the radius vector from this point monomorphism of vector spaces is a linear injec-
to a point on the line of action of the force and tive map; a monomorphism of groups is an injec-
the force, M = r × r × F. tive group homomorphism;a monomorphism of
manifolds is an injective differentiable map. See
momentum, p p p Vector quantity equal to the also bundle morphisms.
product of mass and velocity.
motif For any collection of objects X =
momentum map Let a Lie group G act on a
{x ,x ,...,x }, x is a motif if there exists an
i
n
2
1
Poisson manifold P and let g be the Lie algebra
x ∈ X, i = j such that f(x ) = x .If x i
j
i
j
of G.Any ξ ∈ g generates a vector field ξ (the
P occurs in X at least twice, that is, if x = x for
infinitesimal generator of the action) on P by i j
1 ≤ i, j ≤ n, i = j, itisan exact motif or an
d exact match.
ξ (x) = [exp(tξ) · x]| t=0 .
P
dt Comment: The relationship f is a transfor-
mation between x and x . For nucleic acid and
j
i
Suppose there is a map J : g → C (P ) such
∞
protein sequences, the most common motives are
that X = ξ , for all ξ ∈ g. The map J :
J(ξ) P the percent identity and percent similarity rela-
P → g defined by
∗
tionships, which are usually set to some thresh-
old value such that the identity (similarity) of the
/J(x), ξ0= J(ξ)(x) , ξ ∈ g,x ∈ P
resulting string is at least that threshold value.
is called the momentum map of the action. When X is a set of graphs, then f is either the
© 2003 by CRC Press LLC