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Formally, a strongly continuous semigroup is a representation of the semigroup ( R < sub >+</ sub >,+) on some Banach space X that is continuous in the strong operator topology.

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## Some Related Sentences

Formally and strongly

By 1912

**,****the**Social Democrats**,**with an explicitly anti-antisemitic program**,**were**the**largest party**in****the**German Reichstag**,**and**the**Progressives ran very__strongly__as well ...__Formally__**,**at least**,****the**Jews had been fully emancipated with**the**establishment**of****the**German Empire**,**although they were kept out**of**certain influential occupations**,**enjoyed extraordinary prosperity ... Germans intermarried with Jews:**in****the**1930s**some**50**,**000 Jews were living**in**mixed German-Jewish marriages**,**so at least 50**,**000 Germans**,**and presumably parts**of**their families**,**had familial contact with**the**Jews**.**

Formally and continuous

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 representation

__Formally__

**,**

**a**system

**is**said to be observable if

**,**for any possible sequence

**of**state and control vectors

**,**

**the**current state can be determined

**in**finite time using only

**the**outputs

**(**this definition

**is**slanted towards

**the**state

**space**

__representation__).

Formally and R

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__

**,**

**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:

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__**.**__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**ring

**is**an Abelian group

**(**

__R__

**,**+), together with

**a**second binary operation * such

**that**for all

**a**

**,**b and c

**in**

__R__

**,**

__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__designated as

**the**Manhattan Engineer District

**,**it refers specifically to

**the**period

**of**

**the**project from 1941 – 1946 under

**the**control

**of**

**the**U

**.**S

**.**Army Corps

**of**Engineers

**,**under

**the**administration

**of**General Leslie

__R__

**.**Groves

**.**

__Formally__

**,**if

__R__

**is**

**a**Noetherian ring and I

**is**

**a**principal

**,**proper ideal

**of**

__R__

**,**then I has height at most one

**.**

__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__

**,**let G be

**a**Coxeter group with reduced root system

__R__and k

**<**

**sub**> v </

**sub**>

**a**multiplicity function

**on**

__R__

**(**so k

**<**

**sub**> u </

**sub**> = k

**<**

**sub**> v </

**sub**> whenever

**the**reflections σ

**<**

**sub**> u </

**sub**> and σ

**<**

**sub**> v </

**sub**> corresponding to

**the**roots u and v are conjugate

**in**G ).

Formally and <

__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__

**,**

**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__

**,**

**the**ith row

**,**jth column element

**of**A

__<__sup > T </ sup >

**is**

**the**jth row

**,**ith column element

**of**A:

__Formally__

**,**

**the**outcomes Y

__<__

**sub**> i </

**sub**> are described as being Bernoulli-distributed data

**,**where each outcome

**is**determined by an unobserved probability p

__<__

**sub**> i </

**sub**>

**that**

**is**specific to

**the**outcome at hand

**,**but related to

**the**explanatory variables

**.**

__Formally__

**,**Minkowski

**space**

**is**

**a**four-dimensional real vector

**space**equipped with

**a**nondegenerate

**,**symmetric bilinear form with signature

__<__tt >(−,+,+,+)</ tt >

**(**Some may also prefer

**the**alternative signature

__<__tt >(+,−,−,−)</ tt >;

**in**general

**,**mathematicians and general relativists prefer

**the**former while particle physicists tend to use

**the**latter

**.**

__Formally__

**,**

**a**ringed

**space**

**(**

**X**

**,**O

__<__

**sub**>

**X**</

**sub**>)

**is**

**a**topological

**space**

**X**together with

**a**sheaf

**of**rings O

__<__

**sub**>

**X**</

**sub**>

**on**

**X**

**.**

__Formally__

**,**if we write F

__<__

**sub**> Δ </

**sub**>( x ) to mean

**the**f-polynomial

**of**Δ

**,**then

**the**h-polynomial

**of**Δ

**is**

__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**>).

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