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Formally, the Cantor function c: → is defined as follows:

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## Some Related Sentences

Formally and function

__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__

**,**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__

**,**

**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__

**,**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__

**,**

**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 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 )

Formally and c

__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 and →

*

__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 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__

**,**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__

**,**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__

**,**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__

**,**a frame on a homogeneous space G / H consists of a point in

**the**tautological bundle G

__→__G / H.

__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__

**,**given two partially ordered sets ( S

**,**≤) and ( T

**,**≤), a

**function**f

**:**S

__→__T

**is**an order-embedding if f

**is**both order-preserving and order-reflecting

**,**i. e. for all x and y in S

**,**one has

__Formally__it can be seen just

**as**an ordinary

**function**from X to

**the**power set of Y

**,**written

**as**φ

**:**X

__→__2 < sup > Y </ sup >.

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__

**,**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__

**,**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

0.127 seconds.