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* Formally, when working over the reals, as here, this is accomplished by considering the limit as ε → 0 ; but the " infinitesimal " language generalizes directly to Lie groups over general rings.

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## Some Related Sentences

Formally and when

__Formally__

**the**Prince-Electors elected a King of

**the**Romans

**,**who was elected in Germany

**but**became Holy Roman Emperor only

__when__crowned

**by**

**the**Pope

**.**

__Formally__

**,**these failed

__when__they were rejected

**by**

**the**Church of England's General Synod in 1972

**;**conversations and co-operation continued

**,**however

**,**leading in 2003

**to**

**the**signing of a covenant between

**the**two churches

**.**

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__

**,**knowing

**the**step response of a dynamical system gives information on

**the**stability of such a system

**,**and on its ability

**to**reach one stationary state

__when__starting from another

**.**

__Formally__

**,**a composite number n = d · 2 < sup > s </ sup > + 1 with d being odd

**is**called a strong pseudoprime

**to**a relatively prime base a

__when__one of

**the**following conditions hold:

__Formally__

**,**a graph

**is**symmetric with respect

**to**

**the**origin if it

**is**unchanged

__when__reflected across both

**the**x-axis and y-axis

**.**

__Formally__

**,**these failed

__when__they were rejected

**by**

**the**Church of England's General Synod in 1972

**,**however conversations and co-operation continued leading in 2003

**to**

**the**signing of a covenant between

**the**two churches

**.**

Formally and working

Following a merger with

**the**Furnishing Industry Trust (__Formally__The Furnishing Trades Benevolent Association ) in 2011**,****the**Company has become**the**most significant charity in**the**sector**,**supporting**the**needy and promoting welfare and education of those entering and__working__within**the**industry**,****as**well**as****the**wider support of**the**charitable work of**the**Mayoralty and**the**Corporation of**the**City of London**.**

Formally and over

__Formally__

**,**an inner product space

**is**a vector space V

__over__

**the**field together with an inner product

**,**i

**.**e., with a map

__Formally__introduced in September 1979

**,**Initial samples were released in February 1980

**,**with production chips available

__over__

**the**counter in November

**.**

__Formally__

**,**a frame

**is**defined

**to**be a lattice L in which finite meets distribute

__over__arbitrary joins

**,**i

**.**e

**.**every ( even infinite ) subset

__Formally__

**the**

**"**head of state

**"**can also personally be

**the**

**"**head of government

**"**( ex officio or

**by**ad hoc cumulation such

**as**an absolute monarch nominating himself )

**but**otherwise has formal precedence

__over__

**the**Head of Government and other ministers

**,**whether he

**is**their actual political superior ( absolute monarch

**,**executive president ) or rather theoretical or ceremonial in character

**.**

__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**

**.**

If R

**is**a ring**,**let R denote**the**ring of polynomials in**the**indeterminate X__over__R**.**Hilbert proved that if R**is****"**not too large ", in**the**sense that if R**is**Noetherian**,****the**same must be true for R**.**__Formally__**,**__Formally__

**,**

**the**statement that

**"**value decreases

__over__time

**"**

**is**given

**by**defining

**the**linear differential operator

**as**:

__Formally__

**,**

**this**effect

**is**referred

**to**

**as**and

**is**a process whereby

**the**capture frame rate of

**the**camera changes

__over__time

**.**

__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

**Lie**superalgebra

**is**a ( nonassociative ) Z < sub > 2 </ sub >- graded algebra

**,**or superalgebra

**,**

__over__a commutative ring ( typically R or C ) whose product

**,**called

**the**

**Lie**superbracket or supercommutator

**,**satisfies

**the**two conditions ( analogs of

**the**usual

**Lie**algebra axioms

**,**with grading ):

__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__

**,**an unlabelled state transition system

**is**a tuple ( S

**,**→) where S

**is**a set ( of states ) and

**→**⊆ S × S

**is**a binary relation

__over__S ( of transitions ).

__Formally__known

**as**

**the**Inverurie Parish Church

**,**a split in Church of Scotland

__over__

**the**appointment of ministers in 1843 ( The Disruption ) led

**to**

**the**creation of

**the**West Parish Church ( known locally

**as**

**"**The West Kirk ") which was founded

**as**a Free Church ( a church free from Edinburgh control ) thus causing

**the**Inverurie Parish Church

**to**change its name

**.**

__Formally__established during

**the**Han Dynasty

**,**zhou continued

**to**exist until

**the**establishment of

**the**Republic of China — a period of

__over__2000 years

**.**

__Formally__trained in Rome and Paris

**,**he exhibited in venues from Paris

**to**New York

**to**Beirut

__over__his career

**.**

__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__

**,**if there

**is**a utility function that describes preferences

__over__L commodities

**,**

**the**expenditure function

__Formally__established in 1922

**,**

__over__

**the**decades Woodcraft Rangers has modified Seton ’ s original emphasis on outdoor life

**to**incorporate activities that meet

**the**needs of an increasingly urban population

**,**

**but**

**the**goal of changing behavior and encouraging positive outcomes through interaction and education remains central

**to**its mission today

**.**

Formally and here

__Formally__

**,**we have for

**the**approximation

**to**

**the**full solution A

**,**a series in

**the**small parameter (

__here__called ), like

**the**following:

Formally and is

__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

**.**

__Formally__

**,**

**the**set of all context-free languages

__is__identical

**to**

**the**set of languages accepted

**by**pushdown automata ( PDA ).

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__

**,**

**the**base

__is__known

**as**Naval Support Facility Diego Garcia (

**the**US activity ) or Permanent Joint Operating Base ( PJOB ) Diego Garcia (

**the**UK's term ).

__Formally__

**,**there

__is__a clear distinction:

**"**DFT

**"**refers

**to**a mathematical transformation or function

**,**regardless of how it

__is__computed

**,**whereas

**"**FFT

**"**refers

**to**a specific family of algorithms for computing DFTs

**.**

__Formally__

**,**oxidation state

__is__

**the**hypothetical charge that an atom would have if all bonds

**to**atoms of different elements were 100 % ionic

**.**

__Formally__

**,**if M

__is__a set

**,**

**the**identity function f on M

__is__defined

**to**be that function with domain and codomain M which satisfies

__Formally__

**,**a profinite group

__is__a Hausdorff

**,**compact

**,**and totally disconnected topological group: that

__is__

**,**a topological group that

__is__also a Stone space

**.**

__Formally__

**,**

**this**sharing of dynamics

__is__referred

**to**

**as**universality

**,**and systems with precisely

**the**same critical exponents are said

**to**belong

**to**

**the**same universality class

**.**

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