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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:

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## Some Related Sentences

Formally and algebra

__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

**is**reflexive if it

**is**equal to

**the**

__algebra__of bounded operators which leave invariant each subspace left invariant by every operator in A.

__Formally__

**,**this

**is**described in algebraic notation like this

**:**

**(**19 + 1

**)**+

**(**15 − 1

**)**= x

**,**but even

**a**young student might use this technique without calling it

__algebra__.

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 H

__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__

**,**

**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__

**,**

**a**rigged Hilbert space consists of

**a**Hilbert space

__H__

**,**

**together**

**with**

**a**subspace Φ which carries

**a**finer topology

**,**

**that**

**is**one for which

**the**natural inclusion

__Formally__

**,**

**a**frame on

**a**homogeneous space G /

__H__consists of

**a**point in

**the**tautological bundle G

**→**G /

__H__.

__Formally__

**,**

**the**bounded Borel functional calculus of

**a**self adjoint operator T on Hilbert space

__H__

**is**

**a**mapping defined on

**the**space of bounded complex-valued Borel functions f on

**the**real line

**,**

Formally and over

__Formally__introduced in September 1979

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

**,**

**with**production chips available

__over__

**the**counter in November.

__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**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__

**,**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.

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