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Formally, an absolute coequalizer of a pair in a category C is a coequalizer as defined above but with the added property that given any functor F ( Q ) together with F ( q ) is the coequalizer of F ( f ) and F ( g ) in the category D. Split coequalizers are examples of absolute coequalizers.

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## Some Related Sentences

Formally and absolute

__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__

**,**

**the**theorem

**is**stated

**as**follows: There exist unique integers

**q**

**and**r such

**that**

**a**= qd + r

**and**0 ≤ r < | d |, where | d | denotes

**the**

__absolute__value

**of**d

**.**

Formally and pair

__Formally__

**,**

**a**hypergraph

**is**

**a**

__pair__where

**is**

**a**set

**of**elements called nodes or vertices

**,**

**and**

**is**

**a**set

**of**non-empty subsets

**of**called hyperedges or edges

**.**

__Formally__

**,**each

**of**

**the**following definitions defines

**a**concrete

**category**

**,**

**and**every

__pair__

**of**these categories can be shown to be concretely isomorphic

**.**

__Formally__

**,**

**a**signed graph Σ

**is**

**a**

__pair__

**(**G

**,**σ

**)**

**that**consists

**of**

**a**graph G =

**(**V

**,**E

**)**

**and**

**a**sign mapping or signature σ from E to

**the**sign group

__Formally__

**,**

**a**biased graph Ω

**is**

**a**

__pair__

**(**G

**,**B

**)**where B

**is**

**a**linear class

**of**circles ; this by definition

**is**

**a**class

**of**circles

**that**satisfies

**the**theta-graph

**property**mentioned

**above**

**.**

Formally and category

Limits

**and**colimits**in****a**__category__**C****are****defined**by means**of**diagrams**in****C****.**__Formally__**,****a**diagram**of**type J**in****C****is****a****functor**from J to**C**:__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__

**,**complexification

**is**

**a**

**functor**Vect < sub > R </ sup > → Vect < sub >

**C**</ sup >, from

**the**

__category__

**of**real vector spaces to

**the**

__category__

**of**complex vector spaces

**.**

Formally and C

__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__

**,**

**given**two categories

__C__

**and**

**D**

**,**

**an**equivalence

**of**categories consists

**of**

**a**

**functor**

**F**:

__C__→

**D**

**,**

**a**

**functor**G:

**D**→

__C__

**,**

**and**two natural isomorphisms ε: FG → I < sub >

**D**</ sub >

**and**η: I < sub >

__C__</ sub >→ GF

**.**

__Formally__

**,**

**a**vertex cover

**of**

**a**graph G

**is**

**a**set

__C__

**of**vertices such

**that**each edge

**of**G

**is**incident to at least one vertex

**in**

__C__

**.**The set

__C__

**is**said to cover

**the**edges

**of**G

**.**The following figure shows

**examples**

**of**vertex covers

**in**two graphs

**(**

**and**

**the**set

__C__

**is**marked

**with**red ).

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__

**,**

**an**inner product space

__is__

**a**vector space V over

**the**field

**together**

**with**

**an**inner product

**,**i

**.**e.,

**with**

**a**map

__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__**,**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 defined

__Formally__

**,**

**as**per

**the**2002 Memorandum

**of**Understanding between

**the**BSI

**and**

**the**United Kingdom Government

**,**British Standards

**are**

__defined__

**as**:

__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 ).

__Formally__

**,**these reside

**in**

**a**complex separable Hilbert space-variously called

**the**" state space " or

**the**" associated Hilbert space "

**of**

**the**system-that

**is**well

__defined__up to

**a**complex number

**of**norm 1

**(**

**the**phase factor ).

__Formally__

**,**

**the**convex hull may be

__defined__

**as**

**the**intersection

**of**all convex sets containing X or

**as**

**the**set

**of**all convex combinations

**of**points

**in**X

**.**

__Formally__

**,**

**a**unique factorization domain

**is**

__defined__to be

**an**integral domain R

**in**which every non-zero

**and**non-unit x

**of**R can be written

**as**

**a**product

**(**including

**an**empty product

**)**

**of**irreducible elements p < sub > i </ sub >

**of**R

**and**

**a**unit u:

__Formally__

**,**

**a**transductive support vector machine

**is**

__defined__by

**the**following primal optimization problem:

__Formally__

**,**bending modulus

**is**

__defined__

**as**

**the**energy required to deform

**a**membrane from its intrinsic curvature to some other curvature

**.**

__Formally__

**,**

**a**constraint satisfaction problem

**is**

__defined__

**as**

**a**triple

**,**where

**is**

**a**set

**of**variables

**,**

**is**

**a**domain

**of**values

**,**

**and**

**is**

**a**set

**of**constraints

**.**

__Formally__

**,**

**the**case where only

**a**subset

**of**parameters

**is**

__defined__

**is**still

**a**composite hypothesis ; nonetheless

**,**

**the**term point hypothesis

**is**often applied

**in**such cases

**,**particularly where

**the**hypothesis test can be structured

**in**such

**a**way

**that**

**the**distribution

**of**

**the**test statistic

**(**

**the**distribution under

**the**null hypothesis

**)**does not depend on

**the**parameters whose values have not been specified under

**the**point null hypothesis

**.**

__Formally__

**,**

**the**index

**of**H

**in**G

**is**

__defined__

**as**

**the**number

**of**cosets

**of**H

**in**G

**.**

**(**The number

**of**left cosets

**of**H

**in**G

**is**always equal to

**the**number

**of**right cosets

**.**

__Formally__

**,**it

**is**

**a**norm

__defined__on

**the**space

**of**bounded linear operators between two

**given**normed vector spaces

**.**

__Formally__

**,**

**a**cardinal κ

**is**

__defined__to be weakly compact if it

**is**uncountable

**and**for every function

**f**: < sup > 2 </ sup > →

__Formally__

**,**

**the**sets

**of**free

**and**bound names

**of**

**a**process

**in**π – calculus

**are**

__defined__inductively

**as**follows

**.**

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