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outside U is zero everywhere. On the contrary, valid subsequences, but /d, c0 is not a subse-
α
the functions f can be required to satisfy less quence of S. See also bag, empty collection, list,
α
∞
stringent regularity conditions such as C . sequence, set, and superset.
Partitions of unity are often used to prove
existence of a global object once local objects
are known to exist in M. For example, let M be path An alternating sequence of nodes and
a (paracompact) C ∞ manifold and {(U ,x )} α∈I edgesdrawnfromagraphG(V, E), suchthatthey
α
α

an atlas of M. This means that U are diffeo- form a connected subgraph, G (V , E ), where
α

m
morphic (via the maps x : U → R )toan V ⊆ V and E ⊆ E.
α α
m
open set x (U ) ⊂ R (m = dim(M)). Con- Comment: A path through a graph is simply
α
α
sequently, there exist local (strictly) Riemann- a connected subgraph whose nodes and edges
ian metrics g induced on U (via the maps appear in sequence: just begin at the beginning
α
α
m
x α : U α → R ) by the standard metric and walk along the path to the end. If one permits
m
δ αβ on R . Notice that metrics are second oneself to let the nodes of the path be implicitly
rank, nondegenerate, positive definite tensors. represented by the edges, the path becomes a
If {(U ,f )} is a partition of unity relative to sequence of edges. See also pathway, sequence,
α
α
the open covering {U }, we can then define the and terminal nodes.
α
second-order tensors f g , one for each α ∈ I.
α α
They are global tensors but they identically van-
pathway A sequence of biochemical reac-
ish outside U so that they are not suitable to
α
define a Riemannian metric over M. Let us, how- tions and their compounds whose nodes and
edges form a path and have historically been con-
ever, consider the combination g = f g
α∈I α α
(which exists since {U } is locally finite). Then sidered by biochemists to be a pathway.
α
g is a global Riemannian metric over M, since Comment: This definition seems a little
linear combinations with positive coefficients of circular, but in fact what we define as biochem-
positive definite tensors are still positive definite. ical pathways is largely determined by the his-
The same argument cannot be applied in gen- tory and results of the experiments involved in
eral to arbitrary signature and in particular to the their discovery. Definitions of particular path-
Lorentzian case. In those cases, in fact, further ways vary slightly among different sources. For
topological conditions have to be satisfied for a example, some authors include phosphorylation
global metric to exist. of d-glucopyranose as part of glycolysis; others
do not. Still others refer to the next step as the
parts of collections For any bag, list, first “committed step” in glycolysis. See also
sequence, or set, a subpart (subbag, sublist, sub- path, sequence, and terminal nodes.
sequence, or subset) is a portion of the original
collection. If the part is less than the original
pathwise connected The property of a topo-
collection, it is a proper subbag, sublist, subse-
logical space (X, τ(X)) that any two points can
quence, or subset, and we denote the relationship
be joined by a curve. See connected. Pathwise
between the part and the collection by part ⊂ col-
connectedness implies connectedness. The con-
lection. Otherwise, therelationshipisdenotedby
verse is false.
⊆ to indicate the part may be equal to the whole.
2
Example: The subset in R given by the union
Comment: A set, bag, list, or sequence may
of the set L ={(0,x) : x ∈ [0, 1]} and the set
contain another of its type. For example, {a, b}⊂ 1
S ={(x, sin( )) :0 ≤ x} is connected but not
{a, b, c} and /a, b0⊂/a, b, c0. x
pathwise connected.
In forming parts of collections, bear in
mind that the part must satisfy the same
properties as the whole. For example, if we
pattern See motif.
have S =/a, b, c, d, e0, then any derived
subsequence must have the same precedence
relations: {/a, b0, /b, c, d0, /d, e0} is a set of pendant node See singleton node.
© 2003 by CRC Press LLC