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Given then a normal extension L of K, with automorphism group Aut ( L / K ) = G, and containing α, any element g ( α ) for g in G will be a conjugate of α, since the automorphism g sends roots of p to roots of p. Conversely any conjugate β of α is of this form: in other words, G acts transitively on the conjugates.

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## Some Related Sentences

Given and then

__Given__

**the**

__then__GDP ($ 7

**.**095 bln

**)**

**of**

**the**country

**,**military spending

**is**roughly estimated

**to**

**be**about $ 300 million

**.**

__Given__also

**a**measure

**on**set

**,**

__then__

**,**sometimes also denoted or

**,**has as its vectors equivalence classes

**of**measurable functions whose absolute value's-th power has finite integral

**,**that

**is**

**,**functions

**for**which one has

__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__that Gable

**and**Cukor had worked together before

**,**

**in**Manhattan Melodrama

**and**Gable had no objection

**to**working

**with**him

__then__

**,**

**and**given Selznick's desperation

**to**get Gable

**for**Rhett Butler

**,**if Gable had

**any**objections

**to**Cukor

**,**certainly they would have been expressed before he signed his contract

**for**

**the**film

**.**

__Given__

**the**proposition that '

**a**persons life prospects should not

**be**decided by factors outside

**of**their control or

**for**which

**a**person cannot claim personal credit '

**(**i

**.**e

**.**social status

**,**inherited wealth

**,**race

**and**

**other**accidents

**of**birth

**)**

**a**meritocracy proposes

**a**system where people are rewarded based

**on**their efforts

**,**

**and**if everyone can start

**on**equal footing

**with**

**the**same opportunity

**to**advance

**,**

__then__

**the**results are just

**.**

__Given__Poseidon's connection

**with**horses as well as

**the**sea

**,**

**and**

**the**landlocked situation

**of**

**the**likely Indo-European homeland

**,**Nobuo Komita has proposed that Poseidon was originally an aristocratic Indo-European horse-god who was

__then__assimilated

**to**Near Eastern aquatic deities when

**the**basis

**of**

**the**Greek livelihood shifted from

**the**land

**to**

**the**sea

**,**or

**a**god

**of**fresh waters who was assigned

**a**secondary role as god

**of**

**the**sea

**,**where he overwhelmed

**the**original Aegean sea deities such as Proteus

**and**Nereus

**.**

__Given__

**a**functor U

**and**an object X as above

**,**there may or may not exist an initial morphism from X

**to**U

**.**If

**,**however

**,**an initial morphism

**(**A

**,**φ

**)**does exist

__then__it

**is**essentially unique

**.**

__Given__

**a**differentiable manifold

**,**one can unambiguously define

**the**notion

**of**tangent vectors

**and**

__then__directional derivatives

**.**

#

__Given__**any**elements**a****and****a**'**of****G****,**if**a**~**a**'**,**__then__**a**< sup >− 1 </ sup > ~**a**' < sup >− 1 </ sup >**(****this**can actually**be**proven from**the****other**four**,**so**is**strictly redundant ).__Given__an embedding

**of**

**a**rooted tree

**in**

**the**plane

**,**if one fixes

**a**direction

**of**children

**(**starting from root

**,**

__then__first child

**,**second child

**,**etc

**.**

__Given__" mappings "

**of**input variables into membership functions

**and**truth values

**,**

**the**microcontroller

__then__makes decisions

**for**what action

**to**take

**,**based

**on**

**a**set

**of**" rules ", each

**of**

**the**

**form**

**:**

**:**

__Given__

**any**set A

**and**

**any**set B

**,**if

**for**every set C

**,**C

**is**

**a**member

**of**A if

**and**only if C

**is**

**a**member

**of**B

**,**

__then__A

**is**equal

**to**B

**.**

**:**

__Given__

**any**set A

**and**

**any**set B

**,**if A

**is**

**a**nonempty set

**(**that

**is**

**,**if there exists

**a**member C

**of**A ),

__then__if A

**and**B have precisely

**the**same members

**,**

__then__they are equal

**.**

__Given__

**the**type

**of**warfare that had been carried out

**in**

**the**conflict — wherein highly mobile groups

**of**soldiers

**and**rebels continued

**to**move

**to**

**and**from

**on**

**the**desert terrain

**,**retreating at

**a**time

**and**

__then__suddenly attacking

**to**regain control

**of**small towns

**and**villages

**in**

**the**Eastern rebel held parts

**of**Libya — had led

**to**

**the**technical becoming

**a**vehicle

**of**choice

**for**both sides

**.**

#

__Given__u**and**v**in**W**,**__then__they can**be**expressed as u**=****(**u < sub > 1 </ sub >, u < sub > 2 </ sub >, 0**)****and**v**=****(**v < sub > 1 </ sub >, v < sub > 2 </ sub >, 0 ).
#

__Given__u**in**W**and****a**scalar c**in**R**,**if u**=****(**u < sub > 1 </ sub >, u < sub > 2 </ sub >, 0**)**again**,**__then__cu**=****(**cu < sub > 1 </ sub >, cu < sub > 2 </ sub >, c0**)****=****(**cu < sub > 1 </ sub >, cu < sub > 2 </ sub >, 0 ).__Given__

**any**curve C

**and**

**a**point P

**on**it

**,**there

**is**

**a**unique circle or line which most closely approximates

**the**curve near P

**,**

**the**osculating circle at P

**.**The curvature

**of**C at P

**is**

__then__defined

**to**

**be**

**the**curvature

**of**that circle or line

**.**

__Given__an eclipse

**,**

__then__there

**is**likely

**to**

**be**another eclipse after every period P

**.**This remains true

**for**

**a**limited time

**,**because

**the**relation

**is**only approximate

**.**

Given and normal

__Given__two groups

**G**

**and**H

**and**

**a**

**group**homomorphism f

**:**

**G**→ H

**,**let

**K**

**be**

**a**

__normal__subgroup

**in**

**G**

**and**φ

**the**natural surjective homomorphism

**G**→

**G**

**/**

**K**

**(**where

**G**

**/**

**K**

**is**

**a**quotient

**group**).

"

__Given__**the**elimination**of**these pernicious influences**,**Marx allowed that "__normal__conditions**of**spontaneous development "**of****the**rural commune could exist**.**__Given__an arbitrary

**group**

**G**

**,**there

**is**

**a**related profinite

**group**

**G**< sup >^</ sup >,

**the**profinite completion

**of**

**G**

**.**It

**is**defined as

**the**inverse limit

**of**

**the**groups

**G**

**/**N

**,**where N runs through

**the**

__normal__subgroups

**in**

**G**

**of**finite index

**(**these

__normal__subgroups are partially ordered by inclusion

**,**which translates into an inverse system

**of**natural homomorphisms between

**the**quotients ).

__Given__their vertical-cavity geometry

**,**VCSOAs are resonant cavity optical amplifiers that operate

**with**

**the**input

**/**output signal entering

**/**exiting

__normal__

**to**

**the**wafer surface

**.**

__Given__

**a**normalized light vector l

**(**pointing from

**the**light source toward

**the**surface

**)**

**and**

**a**normalized plane

__normal__vector n

**,**one can work out

**the**normalized reflected

**and**refracted rays

**:**

__Given__that

**any**proposition

**containing**conjunction

**,**disjunction

**,**

**and**negation can

**be**equivalently rephrased using conjunction

**and**negation alone

**(**

**the**conjunctive

__normal__

**form**), we can now handle

**any**compound proposition

**.**

__Given__

**a**ring R

**and**

**a**unit u

**in**R

**,**

**the**map ƒ

**(**x

**)**

**=**u < sup >− 1 </ sup > xu

**is**

**a**ring

**automorphism**

**of**R

**.**The ring automorphisms

**of**

**this**

**form**are called inner automorphisms

**of**R

**.**They

**form**

**a**

__normal__subgroup

**of**

**the**

**automorphism**

**group**

**of**R

**.**

__Given__two intersecting planes described by

**and**

**,**

**the**dihedral angle between them

**is**defined

**to**

**be**

**the**angle between their

__normal__directions

**:**

__Given__

**the**great depth

**of**Lake Tahoe

**,**

**and**

**the**locations

**of**

**the**

__normal__faults within

**the**deepest portions

**of**

**the**lake

**,**modeling suggests that earthquakes

**on**these faults can trigger tsunamis

**.**

__Given__

__normal__atmospheric conditions

**,**

**the**temperature

**,**

**and**thus speed

**of**sound

**,**varies

**with**altitude

**:**

__Given__100 balls

**,**some

**of**them red

**and**some blue

**,**

**a**human

**with**

__normal__color vision can easily separate them into red ones

**and**blue ones

**.**

__Given__

**the**observed percentage difference

**p**− q

**(**2 % or 0

**.**02

**)**

**and**

**the**standard error

**of**

**the**difference calculated above (. 03 ),

**any**statistical calculator may

**be**used

**to**calculate

**the**probability that

**a**sample from

**a**

__normal__distribution

**with**mean 0

**.**02

**and**standard deviation 0

**.**03

**is**greater than 0

**.**

__Given__that it would

**be**complicated

**to**perform

**this**test

**on**every patient

**with**sleep-related breathing problems

**,**some suggest that measuring bicarbonate levels

**in**

__normal__

**(**venous

**)**blood would

**be**

**a**reasonable screening test

**.**

__Given__our current estimate

**of**

**the**parameters θ < sup >( t )</ sup >,

**the**conditional distribution

**of**

**the**Z < sub > i </ sub >

**is**determined by Bayes theorem

**to**

**be**

**the**proportional height

**of**

**the**

__normal__density weighted by τ

**:**

__Given__

**a**perfect

__normal__subgroup

**of**

**the**fundamental

**group**

**of**

**a**connected CW complex

**,**attach two-cells along loops

**in**whose images

**in**

**the**fundamental

**group**generate

**the**subgroup

**.**

__Given__an incident direction from

**the**surface

**to**

**the**light source

**and**

**the**surface

__normal__direction

**the**specularly reflected direction

**(**all unit vectors

**)**

**is**

**:**

__Given__that

**the**angle

**of**incidence

**is**larger than

**the**critical angle

**(**

**with**respect

**to**

**the**

__normal__

**of**

**the**surface

**)**

**and**

**the**refractive index

**is**lower

**on**

**the**

**other**side

**of**

**the**boundary relative

**to**

**the**incident ray

**,**TIR

**will**occur

**and**no light

**will**

**be**able

**to**pass through

**.**

__Given__that

**(**A

**)**

**the**Georges are among

**the**most rational

**,**principled

**and**educated persons

**on**

**the**planet

**,**

**and**

**(**B

**)**their differences from

__normal__humans are purely physical

**,**they conclude that

**in**

**any**situation where

**the**Three laws would come into play

**,**their own orders should take priority over that

**of**

**a**regular human

**.**

__Given__

**the**low accelerations placed

**on**

**the**spacecraft while

**in**

**the**outer solar system

**,**MOND may

**be**

**in**effect

**,**modifying

**the**

__normal__gravitational equations

**.**

__Given__only

**the**artwork

**and**outline

**to**work

**with**during production

**,**writer Yoshinori Kitase felt that one early scene

**of**

**the**character approaching

**a**castle was too " boring

**to**make completely

__normal__",

**and**decided

**to**ad-lib

**a**scene

**with**one

**of**his accompanying soldiers dusting off his boots

**in**

**the**middle

**of**

**a**desert

**.**

__Given__an arbitrary square matrix

**,**

**the**elementary divisors used

**in**

**the**construction

**of**

**the**Jordan

__normal__

**form**do not exist over F

**,**so

**the**invariant factors

**a**< sub > i </ sub >( x

**)**as given above must

**be**used instead

**.**

__Given__

**the**predictable timeline

**of**childhood behavior

**,**he proposed " libido development " as

**a**model

**of**

__normal__childhood sexual development

**,**wherein

**the**child progresses through five psychosexual stages —

**(**i

**)**

**the**oral

**,**

**(**ii

**)**

**the**anal

**,**

**(**iii

**)**

**the**phallic

**,**

**(**iv

**)**

**the**latent

**,**

**and**

**(**v

**)**

**the**genital —

**in**which

**the**source pleasure

**is**

**in**

**a**different erogenous zone

**.**

__Given__that sound levels at such events usually exceed recommended safe levels

**of**exposure

**,**

**this**

**is**

**a**problem which

**is**probably showing up variations between people

**,**which may

**be**genetic

**,**or

**the**result

**of**stress or ill-health

**,**or it may

**be**caused by abnormal response

**in**

**the**tensor tympani

**and**stapedius muscles which function

**in**

**the**

__normal__acoustic reflex response that protects

**the**inner ear from loud sounds

**.**

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