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Formally, an elliptic function is a function meromorphic on for which there exist two non-zero complex numbers and with ( in other words, not parallel ), such that and for all.

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## Some Related Sentences

Formally and function

__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__

**,**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__

**,**the discrete cosine transform

**is**

**a**linear

**,**invertible

__function__

**(**where denotes the set of real

**numbers**

**),**or equivalently

**an**invertible N × N square matrix

**.**

__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__

**,**

**a**

__function__ƒ

**is**real analytic

**on**

**an**open set D

**in**the real line if

**for**any x < sub > 0 </ sub >

**in**D one can write

__Formally__

**,**if

**is**

**an**open subset of the

**complex**plane

**,**

**a**point of

**,**

**and**

**is**

**a**holomorphic

__function__

**,**then

**is**called

**a**removable singularity

**for**if

**there**exists

**a**holomorphic

__function__

**which**coincides

**with**

**on**

**.**

__Formally__

**,**the problem of supervised pattern recognition can be stated as follows: Given

**an**unknown

__function__

**(**the ground truth )

**that**maps input instances to output labels

**,**along

**with**training data assumed to represent accurate examples of the mapping

**,**produce

**a**

__function__

**that**approximates as closely as possible the correct mapping

**.**

__Formally__

**,**

**a**statistic s

**is**

**a**measurable

__function__of X ; thus

**,**

**a**statistic s

**is**evaluated

**on**

**a**random variable X

**,**taking the value s

**(**X

**),**

**which**

**is**itself

**a**random variable

**.**

__Formally__

**,**the discrete sine transform

**is**

**a**linear

**,**invertible

__function__F: R < sup > N </ sup > < tt >-></ tt > R < sup > N </ sup >

**(**where R denotes the set of real

**numbers**

**),**or equivalently

**an**N × N square matrix

**.**

__Formally__

**,**the discrete Hartley transform

**is**

**a**linear

**,**invertible

__function__H: R < sup > n </ sup > < tt >-></ tt > R < sup > n </ sup >

**(**where R denotes the set of real

**numbers**).

__Formally__

**,**

**a**cardinal κ

**is**defined to be weakly compact if it

**is**uncountable

**and**

**for**every

__function__f: < sup > 2 </ sup > →

__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__

**,**this means

**that**

**,**

**for**some

__function__f

**,**the image f

**(**D ) of

**a**directed set D

**(**i

**.**e

**.**the set of the images of each element of D )

**is**again directed

**and**has as

**a**least upper bound the image of the least upper bound of D

**.**One could also say

**that**f preserves directed suprema

**.**

__Formally__

**,**let be

**a**stochastic process

**and**let represent the cumulative distribution

__function__of the joint distribution of at times

**.**

__Formally__

**,**

**an**ultrametric space

**is**

**a**set of points

**with**

**an**associated distance

__function__

**(**also called

**a**metric )

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__

**,**oxidation state

__is__the hypothetical charge

**that**

**an**atom would have if

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

**.**

__Formally__

**,**

**an**inner product space

__is__

**a**vector space V over the field together

**with**

**an**inner product

**,**i

**.**e.,

**with**

**a**map

*

__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**.**__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

**.**

__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 and on

__Formally__speaking

**,**

**a**collation method typically defines

**a**total order

__on__

**a**set of possible identifiers

**,**called sort keys

**,**

**which**consequently produces

**a**total preorder

__on__the set of items of information

**(**items

**with**the same identifier are

**not**placed

**in**any defined order ).

*

**,**see On__Formally__Undecidable Propositions of Principia Mathematica**and**Related Systems**for**details__on__English translations**.**__Formally__

**,**

**in**the finite-dimensional case

**,**if the linear map

**is**represented as

**a**multiplication by

**a**matrix A

**and**the translation as the addition of

**a**vector

**,**

**an**affine map acting

__on__

**a**vector can be represented as

__Formally__

**,**

**a**detailed balanced process

**is**the one

__on__

**which**the probability to be

**in**the state x

**and**transit to x ' must be the same as to be

**in**the state x '

**and**transit to x

**.**

__Formally__

**,**the appointment remains

**a**royal one made

__on__the advice of the Lord Chancellor

**,**but he / she no longer comments

__on__individual applications

**,**

**and**merely supervises the process

**and**reviews the panel's recommendations

**in**general terms

**(**satisfying himself / herself

**that**the process as operated

**is**fair

**and**efficient ).

Informally

**,**G has the above presentation if it**is**the " freest group " generated by S subject only to the relations R**.**__Formally__**,**the group G**is**said to have the above presentation if it**is**isomorphic to the quotient of**a**free group__on__S by the normal subgroup generated by the relations R**.**
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__

**,**

**a**center of gravity

**is**

**an**application point of the resultant gravitational force

__on__the body

**.**

__Formally__named The King's College of Our Lady

**and**Saint Nicholas

**in**Cambridge

**,**the college lies

__on__the River Cam

**and**faces out onto King's Parade

**in**the centre of the city

**.**

__Formally__educated African Americans

**,**

__on__the

**other**hand

**,**either disregarded black minstrelsy or openly disdained it

**.**

__Formally__

**,**the motion merely calls

__on__parliament to thank the monarch or viceroy

**for**the speech via

**an**Address

**in**Reply

**.**

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