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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

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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__

**,**

**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 ƒ

__Formally__

**,**

**in**Euclidean space

**,**

**the**wave front

**set**of

__ƒ__

**is**defined as

**the**complement of

**the**

**set**of all pairs (

**x**

**<**

**sub**

**>**

**0**

**</**

**sub**>, v ) such that there exists

**a**test

**function**with φ (

**x**

**<**

**sub**

**>**

**0**

**</**

**sub**>) ≠

**0**and

**an**

**open**cone Γ containing v such that

**the**estimate

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__**,**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 real

__Formally__

**,**

**the**singular value decomposition of

**an**m × n

__real__or complex matrix M

**is**

**a**factorization of

**the**form

__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__

**,**let A be

**a**

__real__matrix of which we want to compute

**the**eigenvalues

**,**and let A

**<**

**sub**

**>**

**0**

**</**

**sub**>:= A.

__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__

**,**

**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**

**,**

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