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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

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## Some Related Sentences

Formally and case

__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**situation has

**the**characteristics

**of**

**a**prisoner's dilemma game that

**is**played repeatedly between

**the**same partners

**,**

**and**

**in**this

__case__" defection "-jumping

**the**queue-would not

**be**advantageous.

__Formally__

**,**assuming our intuition from

**the**finite dimensional

__case__carries over into

**the**infinite dimensional setting

**,**

**the**measure should then

**be**equal to

Formally and if

__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__

**,**

__if__d

**is**

**the**dimension

**of**

**the**parameter

**,**

**and**n

**is**

**the**number

**of**samples

**,**

__if__

**as**

**and**

**as**

**,**then

**the**model

**is**semi-parametric.

__Formally__

**,**

**a**decision problem

**is**P-complete ( complete for

**the**complexity class P )

__if__it

**is**

**in**P

**and**that every problem

**in**P

**can**

**be**reduced to it

**by**using

**an**appropriate reduction.

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**hypothesis

**is**compared against its opposite or null hypothesis ("

__if__I release this ball

**,**it will not fall to

**the**floor ").

__Formally__

**,**

**the**sequence

**of**partial sums

**of**some infinite summation converges

__if__for every fixed power

**of**X

**the**coefficient stabilizes: there

**is**

**a**point beyond which all further partial sums have

**the**same coefficient.

__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__

**,**two variables are inversely proportional ( or varying inversely

**,**or

**in**inverse variation

**,**or

**in**inverse proportion or

**in**reciprocal proportion )

__if__one

**of**

**the**variables

**is**directly proportional with

**the**multiplicative inverse ( reciprocal )

**of**

**the**other

**,**or equivalently

__if__their product

**is**

**a**constant.

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__

**,**

__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 and linear

__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__

**,**

**the**statement that " value decreases over time "

**is**given

**by**defining

**the**

__linear__differential operator

**as**:

__Formally__

**,**scalar

**multiplication**

**is**

**a**

__linear__

**map**

**,**inducing

**a**

**map**( send

**a**scalar λ to

**the**corresponding scalar transformation

**,**

**multiplication**

**by**λ ) exhibiting End ( M )

**as**

**a**R-algebra.

__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__

**,**it

**is**

**a**norm defined

**on**

**the**space

**of**bounded

__linear__operators between two given normed

**vector**spaces.

__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__

**,**for any dimension

**,**

**the**orientation

**of**

**the**image

**of**

**an**object under

**a**direct isometry with respect to that object

**is**

**the**

__linear__part

**of**that isometry.

Formally and map

__Formally__

**,**

**an**inner product space

**is**

**a**

**vector**space V over

**the**field together with

**an**inner product

**,**i. e., with

**a**

__map__

__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 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__**,**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

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