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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.

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## Some Related Sentences

Formally and discrete

__Formally__stated

**,**

**the**FFT

**is**

**a**method for computing

**the**

__discrete__Fourier

**transform**

**of**

**a**sampled signal

**.**

__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**

__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**frieze group

**is**

**a**class

**of**infinite

__discrete__symmetry groups for patterns on

**a**strip

**(**infinitely wide rectangle

**),**hence

**a**class

**of**groups

**of**isometries

**of**

**the**plane

**,**

**or**

**of**

**a**strip

**.**

Formally and transform

__Formally__

**,**this follows from

**the**convolution theorem in mathematics

**,**which relates

**the**Fourier

__transform__

**of**

**the**power spectrum

**(**

**the**intensity

**of**each frequency ) to its autocorrelation

**.**

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 linear

__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

__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__

**,**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 function

__Formally__

**,**we are given

**a**

**set**

**of**hypotheses and

**a**

**set**

**of**manifestations ; they are related by

**the**domain knowledge

**,**represented by

**a**

__function__that takes as

**an**argument

**a**

**set**

**of**hypotheses and gives as

**a**result

**the**corresponding

**set**

**of**manifestations

**.**

__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__

**,**

**an**elliptic

__function__

**is**

**a**

__function__meromorphic on for which there exist two non-zero complex

**numbers**and with

**(**in other words

**,**not parallel

**),**such that and for all

**.**

__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__

**,**

**the**problem

**of**supervised pattern recognition can be stated as follows

**:**Given

**an**unknown

__function__

**(**

**the**ground truth ) that maps input instances to output labels

**,**along with training data assumed to represent accurate examples

**of**

**the**mapping

**,**produce

**a**

__function__that approximates as closely as possible

**the**correct mapping

**.**

__Formally__

**,**

**a**statistic s

**is**

**a**measurable

__function__

**of**X ; thus

**,**

**a**statistic s

**is**evaluated on

**a**random variable X

**,**taking

**the**value s

**(**X

**),**which

**is**itself

**a**random variable

**.**

__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 >).

__Formally__

**,**this means that

**,**for some

__function__f

**,**

**the**image f

**(**D )

**of**

**a**directed

**set**D

**(**i

**.**e

**.**

**the**

**set**

**of**

**the**images

**of**each element

**of**D )

**is**again directed and has as

**a**least upper bound

**the**image

**of**

**the**least upper bound

**of**D

**.**One could also say that f preserves directed suprema

**.**

__Formally__

**,**let be

**a**stochastic process and let represent

**the**cumulative distribution

__function__

**of**

**the**joint distribution

**of**at times

**.**

__Formally__

**,**

**an**ultrametric space

**is**

**a**

**set**

**of**points with

**an**associated distance

__function__

**(**also called

**a**metric )

0.296 seconds.