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covering{U } suchthatforanyx ∈ M thereis
α α∈I
an open neighborhood U which intersects just a
x
finite number of elements U of the covering. On
O a paracompact manifold M every open covering
α
contains a good covering, i.e., an open covering
{U } such that for any finite set of indices
α α∈I
α ,α ,...,α ∈ I the intersection U α 1 α 2 ...α k =
k
2
1
oligomer A substance composed of " k
i=0 U (which is an open set by definition) is
α i
oligomer molecules.An oligomer obtained by topologically trivial, i.e., it is contractible.
telomerization is often termed a telomer.
open neighborhood (of a point x of a topo-
oligomer molecule A molecule of inter- logical space [X, τ(X)]) An open subset
mediate relative molecular mass, the structure U ⊂ X containing x.
of which essentially comprises a small plurality
of units derived, actually or conceptually, from open set A subset U ⊂ X of a topological
molecules of lower relative molecular mass. spaceX whichisanelementofitstopologyτ(X).
n
Notes: (1) A molecule is regarded as having In R , with the standard metric topology, a
an intermediate relative molecular mass if it has subset U ⊂ R is open if any point x ∈ U is con-
n
properties which do vary significantly with the tained in U together with an open n-ball centered
removal of one or a few of the units. at x of radius r, i.e., B ={y ∈ R : |y−x| <r}.
r
n
x
(2) If a part or the whole of the molecule
has an intermediate relative molecular mass orbit of an action See left action and right
and essentially comprises a small plurality of action.
units derived, actually or conceptually, from
molecules of lower relative molecular mass, it order of reaction, n n n If the macroscopic
may be described as oligomeric, or by oligomer (observed, empirical, or phenomenological) rate
used adjectivally. of reaction (v) for any reaction can be expressed
by an empirical differential rate equation (or
one-dimensional Ising model One of the rate law) which contains a factor of the form
β
α
most important models in statistical mechanics. k[A] [B] ... (expressing in full the depen-
It consists of a set of identical spins {..., S , dence of the rate of reaction on the concentrations
−2
S ,S ,S ,S , ...} with nearest neighbor inter- [A],[B]...) where α, β are constant exponents
2
−1
0
1
action energy JS S . Each spin can take values (independent of concentration and time) and k
i i+1
±1. An infinite one-dimensional Ising system is independent of [A] and [B], etc. (rate con-
can exhibit phase transition behavior. stant, rate coefficient), then the reaction is said
to be of order α with respect to A, of order β
onto A mapping or transformation of a set with respect to B,..., and of (total or overall) order
X which transforms points of X to those in Y is n = α + β + .... The exponents α, β, ... can
a mapping of X onto Y if each point of Y is the be positive or negative integral or rational non-
image of at least one point of X. For example, integral numbers. They are the reaction orders
y = 3x + 2 is amapping ofthe real numbers onto with respect to A,B, ... and are sometimes called
2
the real numbers; y = x is a mapping of the real “partial orders of reaction.” Orders of reaction
numbers into the real numbers, or onto the non- deduced from the dependence of initial rates of
negative real numbers. See also into, bijection, reaction on concentration are called “orders of
and injection. reaction with respect to concentration”; orders
of reaction deduced from the dependence of the
opencovering(ofatopologicalspace[X,τ(X)]) rate of reaction on time of reaction are called
A family {U } of open sets in X such that “orders of reaction with respect to time.”
α α∈I
!
U = X where I is any set of indices. On The concept of order of reaction is also applic-
α∈I α
a paracompact manifold M every open covering able to chemical rate processes occurring in
contains a locally finite covering, i.e., an open systems for which concentration changes (and

c

© 2003 by CRC Press LLC
© 2003 by CRC Press LLC