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Formally, we define a bad field as a structure of the form ( K, T ), where K is an algebraically closed field and T is an infinite, proper, distinguished subgroup of K, such that ( K, T ) is of finite Morley rank in its full language.

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## Some Related Sentences

Formally and we

__Formally__

**,**

__we__start with

**a**category C with

**finite**products

**(**i

**.**e

**.**C has

**a**terminal object 1

**and**any two objects

**of**C have

**a**product ).

__Formally__it

**is**precisely

**in**allowing quantification over class variables α

**,**β

**,**etc.,

**that**

__we__assume

**a**range

**of**values for these variables to refer to

**.**

__Formally__

**,**

__we__are given

**a**set

**of**hypotheses

**and**

**a**set

**of**manifestations ; they are related by

**the**domain knowledge

**,**represented by

**a**function

**that**takes

**as**

**an**argument

**a**set

**of**hypotheses

**and**gives

**as**

**a**result

**the**corresponding set

**of**manifestations

**.**

__Formally__

**,**

__we__have for

**the**approximation to

**the**

**full**solution A

**,**

**a**series

**in**

**the**small parameter

**(**here called

**),**like

**the**following:

__Formally__

**,**if

__we__write F < sub > Δ </ sub >( x

**)**to mean

**the**f-polynomial

**of**Δ

**,**then

**the**h-polynomial

**of**Δ

**is**

__Formally__

**,**for

**a**countable set

**of**events A < sub > 1 </ sub >, A < sub > 2 </ sub >, A < sub > 3 </ sub >, ...,

__we__have

__Formally__

**,**

__we__begin by considering some family

**of**distributions for

**a**random variable X

**,**

**that**

**is**indexed by some θ

**.**

__Formally__

**,**let A be

**a**real matrix

**of**which

__we__want to compute

**the**eigenvalues

**,**

**and**let A < sub > 0 </ sub >:= A

**.**

__Formally__

**,**

**an**antihomomorphism between X

**and**Y

**is**

**a**homomorphism

**,**

**where**equals Y

**as**

**a**set

**,**but has multiplication reversed: denoting

**the**multiplication on Y

**as**

**and**

**the**multiplication on

**as**

**,**

__we__have

**.**

__Formally__

**,**

**the**definition only requires some invertibility

**,**so

__we__can substitute for Q any matrix M whose eigenvalues do not include − 1

**.**

__Formally__

**,**given

**a**

**finite**set X

**,**

**a**collection C

**of**subsets

**of**X

**,**all

**of**size n

**,**has Property B if

__we__can partition X into two disjoint subsets Y

**and**Z

**such**

**that**every set

**in**C meets both Y

**and**Z

**.**

__Formally__

**,**if

__we__denote

**the**set

**of**stable functions by S

**(**D

**)**

**and**

**the**stability radius by r

**(**f

**,**D

**),**then:

Formally and define

__Formally__

**,**

**the**issue

**is**

**that**interfertile " able to interbreed "

**is**not

**a**transitive relation – if A can breed with B

**,**

**and**B can breed with C

**,**it does not follow

**that**A can breed with C –

**and**thus does not

__define__

**an**equivalence relation

**.**

__Formally__

**,**

__define__

**the**set

**of**lines

**in**

**the**plane P

**as**L

**(**P ); then

**a**rigid motion

**of**

**the**plane takes lines to lines –

**the**group

**of**rigid motions acts on

**the**set

**of**lines –

**and**one may ask which lines are unchanged by

**an**action

**.**

__Formally__however they

__define__it

**as**any variable

**that**does not directly affect

**the**fundamentals

**of**

**the**economy

**.**

Formally and field

__Formally__organized vocational programs supported by federal funds allow high school students to gain experience

**in**

**a**

__field__

**of**work which

**is**likely to lead to

**a**full-time job on graduation

**.**

More rigorously

**,****the**divergence**of****a**vector__field__F at**a**point p**is**defined**as****the**limit**of****the**net flow**of**F across**the**smooth boundary**of****a**three dimensional region V divided by**the**volume**of**V**as**V shrinks to p**.**__Formally__**,**__Formally__

**,**

**an**inner product space

**is**

**a**vector space V over

**the**

__field__together with

**an**inner product

**,**i

**.**e., with

**a**map

In late 1900s

**the**day**that****of****the**City was founded**is**Jakin School**is****a**Private School**is**__Formally__Part**of**Jakin School Board was Grades One through Twelve there**is**no Football__field__on this school**.**In 1966 When Jakin School was Closed were Sent to Blakely-Union Elementary**,**Junior High**,****and**High School**(**which later Became Early County High**,**Middle**,****and**Elementary School ). while Carver school was built for African-American students grades One through Twelve**.**although late 1960s when Carver school was**closed**along with Kestler School was based**in**Damascus were sent to Washington High & Elementary at Blakely**.**In addition to small farm agriculture**,**Jakin's early economic growth resulted from turpentine**.**__Formally__

**,**

**an**analytic function ƒ

**(**z

**)**

**of**

**the**real or complex variables z < sub > 1 </ sub >,…, z < sub > n </ sub >

**is**transcendental if z < sub > 1 </ sub >, …, z < sub > n </ sub >, ƒ

**(**z

**)**are

**algebraically**independent

**,**i

**.**e., if ƒ

**is**transcendental over

**the**

__field__C

**(**z < sub > 1 </ sub >, …, z < sub > n </ sub >).

__Formally__

**,**

**a**coalgebra over

**a**

__field__

**K**

**is**

**a**vector space C over

**K**together with K-linear maps Δ: C → C ⊗ C

**and**ε: C →

**K**

**such**

**that**

__Formally__

**,**

**a**Hopf algebra

**is**

**a**

**(**associative

**and**coassociative

**)**bialgebra H over

**a**

__field__

**K**together with

**a**K-linear map S: H → H

**(**called

**the**antipode

**)**

**such**

**that**

**the**following diagram commutes:

__Formally__dedicated "

**as**

**a**memorial to

**the**boys

**that**were " on October 30

**,**1915

**,**Alumni Field

**and**

**its**distinctive " maroon goal-posts on

**a**

__field__

**of**green " were hailed

**in**

**that**evening's edition

**of**

**the**Boston Saturday Evening Transcript

**as**" one

**of**

**the**sights

**in**Boston

**.**

__Formally__

**,**

**an**algebraic function

**in**n variables over

**the**

__field__

**K**

**is**

**an**element

**of**

**the**algebraic closure

**of**

**the**

__field__

**of**rational functions

**K**

**(**x < sub > 1 </ sub >,..., x < sub > n </ sub >).

Formally and structure

__Formally__

**,**

**the**problem can be stated

**as**follows: given

**a**desired property

**,**expressed

**as**

**a**temporal logic formula p

**,**

**and**

**a**

__structure__M with initial state s

**,**decide if

**.**

__Formally__

**,**

**a**cumulativity predicate CUM can be defined

**as**follows

**,**

**where**capital X

**is**

**a**variable over sets

**,**U

**is**

**the**universe

**of**discourse

**,**p

**is**

**a**mereological part

__structure__on U

**,**

**and**

**is**

**the**mereological sum operation

**.**

__Formally__

**,**

**a**quantization predicate QUA can be defined

**as**follows

**,**

**where**

**is**

**the**universe

**of**discourse

**,**

**and**

**is**

**a**variable over sets

**,**

**and**

**is**

**a**mereological part

__structure__on with < math > < _p </ math >

**the**mereological part-of relation

**.**

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