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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

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## Some Related Sentences

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__**,**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__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**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 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__**,**
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__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__

**,**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.

Formally and K

* 2004

**:**Newsboys**,**Relient__K__**,**Christafari**,**The Tribe (__Formally__The World Wide Message Tribe ), Pillar**,**Paul Colman Trio**,**Magnify**,**Darlene Zschech**,**Hillsong United

Formally and is

__Formally__

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

__is__identical to the set of languages accepted by pushdown automata ( PDA ).

__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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