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vector (1) A directed line segment in R or vertex cover Given a graph, G = [V, E], a
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R , determined by its magnitude and direction. vertex cover is a subset of V , say C, such that for
(2) An element of a vector space. each edge (u, v) in E, at least one of u and v is in
C. Given weights, {w(v)} for v in V , the weight
vector bundle A bundle (B,M,π; V) with of a vertex cover is the sum of weights of the
a vector space V as a standard fiber and such nodes in C. The minimum weight vertex cover
that the transition functions act on V by means problem is to find a vertex cover whose weight is
of linear transformations. Vector bundles always minimum.
allow global sections.
vertical automorphism A bundle mor-
phism ) : B → B on a bundle (B,M,π; F)
vector field A vector-valued function V ,
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defined in a region D (usually in R ). The vector which projects over the identity, i.e., such that
π ≡ id ◦ π = π ◦ ). Locally a vertical mor-
V(p), assigned to a point p ∈ D, is required to M
phism is of the following form
have its initial point at p.

x = x
vector product For two vectors x = (x , y = Y(x, y).

1
3
x ,x ) and y = (y ,y ,y ) in R , the vector
3
2
1
3
2
vertical vector field A vector field \ over
x ×y = (x y −x y ,x y −x y ,x y −x y ). a bundle (B,M,π; F) which projects over the
3 1
3 2
2 3
2 1
1 2
1 3
zero vector of M. Locally, it is expressed as
vectorspace Asetclosedunderadditionand i
\ = ξ (x, y)∂ i
scalar multiplication (by elements from a given
n
µ
i
field). One example is R , where addition is where (x ; y ) are fibered coordinates. The flow
the usual coordinate-wise addition, and scalar of a vertical vector field is formed by vertical
multiplication is t(x , ..., x ) = (tx , ..., tx ). automorphisms.
n
n
1
1
Another vector space is the set of all m × n
viscosity See dynamic viscosity.
matrices. If A and B are two matrices (of the
same size), so is A + B. Also, tA is a matrix for
voltage (in electroanalysis) The use of this
any scalar, t in R. Another vector space is the set term is discouraged, and the term applied poten-
n
of all functions with domain X and range in R .
tial should be used instead, for nonperiodic sig-
If f and g are two such functions, so are f + g
nals. However, it is retained here for sinusoidal
and tf for all t in R. Note that a vector space
and other periodic signals because no suitable
must have a zero since we can set t = 0. See
substitute for it has been proposed.
also module.
von Neumann machine (vNM) A device
vehicle routing problem (VRP) Find opti- that stores a program specifying an algo-
mal delivery routes from one or more depots to a rithm or heuristic in memory; includes separ-
set of geographically scattered points (e.g., popu- able arithmetic and logic processing units and
lation centers). A simple case is finding a route input/output facilities and executes a discrete
for snow removal, garbage collection, or street computation on the input data using the stored
sweeping (without complications, this is akin to program. It assumes a finite number of distinct
a shortest path problem). In its most complex internal states and operates on a finite number of
form, the VRP is a generalization of the TSP, as symbols K. It may or may not store the input
it can include additional time and capacity con- and output, as desired. For convenience allow
straints, precedence constraints, plus more. execution to be either sequential on a single pro-
cessor or sequential within each of a set of pro-
v,c
velocity, v,cv,c Vector quantity equal to the cessors joined together by various schemes for
derivative of the position vector with respect to message passing and storage. The machine is
time (symbols, u, v, w for components of c). a nonstochastic device executing deterministic
or nondeterministic algorithms. See universal
vertex See node. Turing machine and Post production system.
© 2003 by CRC Press LLC
© 2003 by CRC Press LLC