[permalink] [id link]
Some Related Sentences
Given and field
Given a field F, the assertion “
F is algebraically closed ” is equivalent
to other assertions:
Given any vector space V over
a field F, the dual space V * is defined
as the set
of all linear maps ( linear functionals ).

)
Given a smooth Φ < sup
> t
</ sup >, an autonomous vector
field can be derived from it
.
Given the difficulty
of finding exact solutions
, Einstein's
field equations are also solved frequently
by numerical integration on
a computer
, or
by considering small perturbations
of exact solutions
.
Given a field K
, the corresponding general linear groupoid GL <
sub >*</
sub >( K ) consists
of all invertible matrices whose entries range over K
. Matrix multiplication interprets composition
.
Given a vector space V over
the field R
of real numbers
, a function is called sublinear if

"
Given that science continually seeks
to adjust its theories structurally
to fit
the facts
, i
. e., adjusts its maps
to fit
the territory
, and thus advances more rapidly than any other
field, he believed
that the key
to understanding sanity would be found
in the study
of the methods
of science ( and
the study
of structure
as revealed
by science ).
Given the currently keen interest
in biotechnology and
the high levels
of funding
in that field, attempts
to exploit
the replicative ability
of existing cells are timely
, and may easily lead
to significant insights and advances
.
Given the union's commitment
to international solidarity
, its efforts and success
in the field come
as no surprise
.
Given two affine spaces and
, over
the same
field, a function is an affine map if and only if for every family
of weighted points
in such that
Given that this is
a plane wave
, each vector represents
the magnitude and direction
of the electric
field for an entire plane
that is perpendicular
to the axis
.
Given a bounded sequence
, there exists
a closed ball
that contains
the image
of ( is
a subset
of the scalar
field ).
Given a vector space V over
a field K
, the span
of a set S ( not necessarily finite ) is defined
to be
the intersection W
of all subspaces
of V which contain S
. W is referred
to as the subspace spanned
by S
, or
by the vectors
in S
. Conversely, S is called
a spanning set
of W
.
Given any
such interpretation
of a set
of points
as complex numbers
, the points constructible using valid compass and straightedge constructions alone are precisely
the elements of the smallest
field containing
the original set
of points and closed under
the complex conjugate and square root operations (
to avoid ambiguity
, we
can specify
the square root with complex argument less than π ).
Given a subset S
in R < sup
> n
</ sup >,
a vector
field is represented
by a vector-valued function V: S → R < sup
> n
</ sup
> in standard Cartesian coordinates (
x <
sub > 1 </ sub >, ...,
x <
sub > n
</ sub >).
Given a differentiable manifold M
, a vector
field on M is an assignment
of a tangent vector
to each point
in M
. More precisely
, a vector
field F is
a mapping from M into
the tangent bundle TM so
that is
the identity mapping
Given a core geometry
, the B
field needed for
a given force
can be calculated from (
2 ); if it comes out
to much more than
1. 6 T
, a larger core must be used
.
Given such a field, an absolute value
can be defined on it
.
Given a locally compact topological
field K
, an absolute value
can be defined
as follows
.
Given a grid point
field of geopotential height
, storm tracks
can be visualized
by contouring its average standard deviation
, after
the data has been band-pass filtered
.
Given an algebraically closed
field A containing K
, there is
a unique splitting
field L
of p between K and A
, generated
by the roots
of p
. If K is
a subfield
of the complex numbers
, the existence is automatic
.
Given a separable extension K ′
of K
, a Galois closure L
of K ′ is
a type
of splitting
field, and also
a Galois extension
of K containing K ′
that is minimal
, in an obvious sense
.
Given and ordering
Given the importance
of reason
, Clement stresses apatheia
as a reasonable
ordering of our passions
in order
to live within God's love
, which is seen
as a form
of truth
.
Given an-element array with an
ordering relation
as an input for
the sorting
, consider it
as a collection
of cards
, with
the ( unknown
in the beginning ) statistical
ordering of each element serving
as its index
.
Given that a preference
ordering allows deriving its associated knowledge base but also allows performing
a single step
of revision
, studies on iterated revision have been concentrated on how
a preference
ordering should be changed
in response
of a revision
.
Given this
ordering of A
, a subset B
of A is cofinal
in A if for every
a ∈ A there is
a b
∈ B
such that a ⊃ b
.
Given and ≤
Given the first n digits
of Ω and
a k
≤ n
, the algorithm enumerates
the domain
of F until enough
elements of the domain have been found so
that the probability they represent is within
2 < sup >-( k +
1 )</ sup
> of Ω
.
Given a group G
, a factor group G / N is abelian if and only if
≤ N
.
Given a set S with
a partial order ≤, an infinite descending chain is
a chain V
that is
a subset
of S upon which
≤ defines
a total order
such that V has no least element
, that is
, an element m
such that for all
elements n
in V it holds
that m
≤ n
.
Given constants C
, D and V
, there are only finitely many ( up
to diffeomorphism ) compact n-dimensional Riemannian manifolds with sectional curvature | K |
≤ C
, diameter
≤ D and volume
≥ V
.
Given constants C
, D and V
, there are only finitely many homotopy types
of compact n-dimensional Riemannian manifolds with sectional curvature K
≥ C
, diameter
≤ D and volume
≥ V
.
Given two states p and q
in S
, q simulates p
, written p
≤ q if there is
a simulation R
such that ( p
, q )
∈ R
. The relation
≤ is
a preorder
, and is usually called
the simulation preorder
.
Given two integers
a and b
, with b ≠
0, there exist unique integers q and r
such that a = bq + r and
0 ≤ r < | b |, where | b | denotes
the absolute value
of b
.
Given any r on this sphere
, and an angle – π <
a ≤ π
, the versor is on
the 3-sphere
of H
.
Given a set
of observations (
x <
sub > 1 </ sub >,
x <
sub > 2 </ sub >, …,
x <
sub > n
</ sub >), where each observation is
a d-dimensional real vector
, k-means clustering aims
to partition
the n observations into k sets ( k
≤ n ) S =
Given an 80-bit key k <
sub > 0 </ sub > ... k <
sub > 79
</ sub > and an l-bit IV v <
sub > 0 </ sub > ... v <
sub > l
− 1 </ sub > ( where
0 ≤ l
≤ 80 ), Trivium is initialized
as follows:
Given and 1

Idealists are skeptics about
the physical world
, maintaining either:
1 )
that nothing exists outside
the mind
, or
2 )
that we would have no access
to a mind-independent reality even if it may exist ;
the latter case often takes
the form
of a denial
of the idea
that we
can have unconceptualised experiences ( see Myth
of the Given ).

After Christians
in Ephesus first wrote
to their counterparts recommending Apollos
to them
, he went
to Achaia where Paul names him
as an apostle (
1 Cor 4: 6
, 9-13 )
Given that Paul only saw himself
as an apostle ' untimely born ' (
1 Cor 15: 8 ) it is certain
that Apollos became an apostle
in the regular way (
as a witness
to the risen Lord and commissioned
by Jesus-1 Cor 15: 5-9 ;
1 Cor 9:
1 ).< ref
> So
the Alexandrian recension ;
the text
in < sup
> 38
</ sup
> and Codex Bezae indicate
that Apollos went
to Corinth
.
Given a ket
of norm
1, the orthogonal projection onto
the subspace spanned
by is
Given points
P <
sub > 0 </ sub > and
P <
sub > 1 </ sub >,
a linear Bézier curve is simply
a straight line between those two points
.
Given that a father's age is
1 less than twice
that of his son
, and
that the digits AB making up
the father's age are reversed
in the son's age ( i
. e
. BA ), leads
to the equation 19B-8A
Given a trigonometric series f (
x ) with S
as its set
of zeros
, Cantor had discovered
a procedure
that produced another trigonometric series
that had S '
as its set
of zeros
, where S ' is
the set
of limit points
of S
. If p (
1 ) is
the set
of limit points
of S
, then he could construct
a trigonometric series whose zeros are p (
1 ).
Given f
∈ G (
x *
x < sup >-
1 </ sup >,
y *
y < sup >-1
</ sup >) and g
∈ G (
y *
y < sup >-1
</ sup >, z * z < sup >-1
</ sup >), their composite is defined
as g * f
∈ G (
x *
x < sup >-1
</ sup >, z * z < sup >-1
</ sup >).

:(
a )
Given player 2's strategy
, the best payoff possible for player
1 is V
, and
Given a function ƒ defined over
the reals
x, and its derivative ƒ < nowiki
> '</ nowiki >, we begin with
a first guess
x <
sub > 0 </ sub > for
a root
of the function f
. Provided
the function is reasonably well-behaved
a better approximation
x <
sub > 1 </ sub > is

# Minimisation operator:
Given a ( k +
1 )- ary
total function:
Given Ω microstates at
a particular energy
, the probability
of finding
the system
in a particular microstate is p =
1 / Ω
.
Given two points (
x <
sub > 1 </ sub >,
y <
sub > 1 </ sub >) and (
x <
sub > 2 </ sub >,
y <
sub > 2 </ sub >),
the change
in x from
one to the other is ( run ), while
the change
in y is ( rise ).
Given current trends
in the UMC — with overseas churches growing
, especially
in Africa
, and U
. S
. churches collectively losing about
1, 000 members
a week — it has been estimated
that Africans will make up at least 30 %
of the delegates at
the 2012 General Conference
, and it is also possible
that 40 %
of the delegates will be from outside
the U
. S
. One Congolese bishop has estimated
that typical Sunday attendance
of the UMC is higher
in his country than
in the entire United States
.
Given metric spaces ( X
, d <
sub > 1 </ sub >) and ( Y
, d <
sub > 2 </ sub >),
a function f: X → Y is called uniformly continuous if for every real number ε
> 0 there exists δ
> 0 such that for every
x, y ∈ X with d <
sub > 1 </ sub >(
x, y ) < δ
, we have
that d <
sub > 2 </ sub >( f (
x ), f (
y )) < ε
.
Given the low efficiency
of the laser amplification process ( about
1 to 1. 5 %), and
the losses
in generation ( steam-driven turbine systems are typically about 35 % efficient ), fusion gains would have
to be on
the order
of 350 just
to break even
.
0.688 seconds.