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Given a field ordering ≤ as in Def 1, the elements such that x ≥ 0 forms a positive cone of F. Conversely, given a positive cone P of F as in Def 2, one can associate a total ordering ≤< sub > P </ sub > by setting x ≤ y to mean y − x ∈ P. This total ordering ≤< sub > P </ sub > satisfies the properties of Def 1.

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## Some Related Sentences

Given and field

__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__

**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__**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__

**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**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__

**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__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 >).

__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

__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

**.**

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