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- valuation
. - May be
read at http://www.harcourt.com/dictionary /browse/19/
.
- valuation domain
. - May be
read at http://www.harcourt.com/dictionary /browse/19/
.
- valuation ring
. - May be
read at http://www.harcourt.com/dictionary /browse/19/
.
- Vandermonde matrix
. - May be
read at http://www.harcourt.com/dictionary /browse/19/
.
- vanish
. - May be
read at http://www.harcourt.com/dictionary /browse/19/
.
- vanishing point
. - May be
read at http://www.harcourt.com/dictionary /browse/19/
.
- variable
- Misnomer for input of
function (PL). As noted by Bertrand Russell, a "variable" does not vary! Properly
named functand (PL).
- variation of parameters
. - May be
read at http://www.harcourt.com/dictionary /browse/19/
.
- variety
. - May be
read at http://www.harcourt.com/dictionary /browse/19/
.
- vector
. - The purpose of the vector
format is to obtain an extension of language, as when the scalar (0-vector)
of speed can "speak of direction" by passing to a vector for velocity.
Both name and notion are creations of the great Irish mathematician, William Rowan Hamilton
(1805-1865), to explicate a complex number as a vector of real numbers. German
mathematician, Felix Klein (1849-1929), gave us the best definition of a vector: the
standard 2-D or 3-D coordinates, (x,y), (x,y,z) constitute
components of a vector and are invariant under a transformation of coordinates
(PL), x2 + y2
x'2
+ y'2 or x2 + y2 + z2
x'2 + y'2 + z'2. Hence, any
n-tuple is a vector iff it is invariant under the same type of
transformation. Under rotation of coordinate axes, PL occurs the point
transformation: [x, y, z] [x', y', z']. Since
the vector models as a single point (PL vector by coordinate specification), the Klein
criterion provides an effective test. Thus, v(x, y, z) transforms
into the vector v'(x' y', z') under rotation of the coordinate
axes, PL: v'x = vxcosq +
vysinq + vz, v'x = -vxsin
q + vycosq + vz, v'
z = vz. This is the definitive test for vector (and, by extension,
tensor) status. For example, the structure, [x, -y]
transforms as x' = x cosq, y' = ycosq, hence, does not represent a vector. PL orthogonality. (To see how
the vector format allows extension from the declarative mood to other moods and from
the aleatic mode to other modes of language, PL vectorlogic.)
- vector addition
. - Geometrically, by
placing the arrow of one vector at the tail of the other and closing the triangle, from tail of
first to arrow of second, as the vector sum. Algebraically, by adding corresponding coordinates:
[x1, y1, z1] + [x2, y2,
z2] = [x1 + x2, y1 + y2, z1 +
z2].
- vector analysis
. - PL vector. The
standard Gibbs-Heaviside vector algebra-analysis is hybrid. defective, cumbersome
compared to multivector algebra-analysis (PL) which includes the Gibbs-Heaviside system
as a special case.
- vector by coordinate specification
. - PL
vector. Geometrically, a vector can be designated by its intial and terminal points
, which is simplified by placing initial point at origin of coordinate system,
and speficying only it terminal ("arrow") point. Thus, v = [x,
y, z] = [x, 0, 0] + [0, y, 0] + [0, 0, z]. Such a vector is equivalent to any line-segment of equal length
along the line containing the above specification. Its magnitude is
|r| = (x2 + y2 + z2)½. The unit vectors
can be written as [i, j, k] = [1, 0, 0] + [0, 1, 0] + [0, 0, 1]. A
general vector can be written in terms of these: g = ig
x + jgy + kgz, g = 0 gx = g
y = gz = 0, with magnitude, (gx
2 + gy2 + gz2)½. (PL
vector by direction cosine specification.)
- vector by direction cosine specification
.
- PL vector, vector by coordinate specification. Given, the vector coordinate
specification, v = [x1, y1, z1]
, find its projection upon the three coordinate axes, forming three right triangles.
Label as a the origin-angle of the
right triangle in the X-Y plane; as b
the origin-angle of the right triangle in the X-Z plane; as g the origin-angle of the right triangle in the Y-Z
plane. Now, the adjacent side of each triangle is the projection of the specified vector
on each axis with cosine specfications, x1 = r cos a, y1 = r cos b, z1 = r cos
g. These are the direction cosines, providing another
vector specification: v = [x1, y1, z
1] = [r cos a, r cos b, r cos g].
- vector calculus
. - PL vector.
- vector component
. - PL vector.
- vector equation
. - An equation containing
one or more "unknowns" with vector (PL) structure.
- vector field
. - May be
read at http://www.harcourt.com/dictionary /browse/19/
.
- vector function
. - A function with a vector
domain (PL) and a vector or scalar codomain (PL).
.
- vector logic
. - Standard logic concerns
only simple declarative statements, which can be assigned the truth value of "T"
for "true" and "F" for "false", as well as complex or compound statements constructed via
connectives conserving that declarative aspect. These are:
- conjunction (and-ing) of statements, which is true only if all components are true;
- disjunction (or-ing) of statements, which is false only if all components are false;
- conditional (if one statement, then another), which is false only if first component is true and
the second is false;
- biconditional (iff), which is true only when both components argree in truth-value (both true or
both false)l
- negation, which reverses the truth value of a statment (T -> F, or F -> T).
This severely
limits the language. (Thus, the "because" we so frequently use is not truth functional. And
the assignment of computer programming is imperative, not declarative.)
This limitation is bypassed by forming a two component vector in which the second component
is the usual declarative of standard logic or scalar logic, while the second component
declares that the sign user or speaker uses a particular mood of speech -- declarative, interrogative,
subjunctive, petitive, etc. Thus, the declarative aspect is conserved by shifting from scalar to
vector logic, just as some physical property is conserved by shifting from, say, scalar "speed" to
vector "velocity". This approach can be extended to deal with modes of expression, such as deontic.
- vector space
. - A vector space V over a field F
(PL) is (traditionally) defined as a set of elements (vectors)
such that their sum is defined, as is the multiplication of any element
by a scalar, providing an Abelian group under addition. (PL linear
space.) This limited definition is due to the Gibbs-Heaviside definition of vector
(see above), whose vector product is anticommutative and nonassociative, so
that vectors are noninversive, hence, the system cannot contain an Abelian
group under multiplication. However, in multivector theory (PL), every vector has an inverse
as has a vast class of higher multivectors, so that the definition of vector space could be
reformulated and extended. Only widespread bias prevents this.
- vector space associated with a graph
.
- May be read at http://www.harcourt.com/dictionary /browse/19/
.
- Venn diagram
. - An extension by John Venn of the Euler diagram,
used to explicate logical or set-theoretic relations. (Thus, a Venn diagram can be used to prove the set-theoretic
distributive law.) Given three statements or sets (PL) -- it is difficult to diagram more than three -- their
interrelationship is symbolized by three overlapping circles or rectangles. Venn enclosed these in a larger circle or
rectangle to denote the universe of discourse. The diagram can be used to apply the inclusion-exclusion algorithm
(PL) for acccounting membership in subsets. (Example). The Venn Diagram
applies only to cases wherein each type or kind exists only without increase in order or degree. To deal with the latter,
PL the hays diagram.)
- versine
. - May be
read at http://www.harcourt.com/dictionary /browse/19/
.
- vertex
. - May be
read at http://www.harcourt.com/dictionary /browse/19/
.
- vertex cut
. - May be
read at http://www.harcourt.com/dictionary /browse/19/
.
- vertex-induced subgraph
. - May be
read at http://www.harcourt.com/dictionary /browse/19/
.
- vertical angles
. - May be
read at http://www.harcourt.com/dictionary /browse/19/
.
- Vizing's theorem
. - May be
read at http://www.harcourt.com/dictionary /browse/19/
.
- void set
. - May be
read at http://www.harcourt.com/dictionary /browse/19/
.
- volume
. - May be
read at http://www.harcourt.com/dictionary /browse/19/
.
- volume form
. - May be
read at http://www.harcourt.com/dictionary /browse/19/
.
- volume integral
. - May be
read at http://www.harcourt.com/dictionary /browse/19/
.