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Formally, an algebraic function in n variables over the field K is an element of the algebraic closure of the field of rational functions K ( x < sub > 1 </ sub >,..., x < sub > n </ sub >).

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## Some Related Sentences

Formally and algebraic

__Formally__

**,**Liouville numbers and elementary numbers contain

**the**

__algebraic__numbers

**,**and they include some but not all transcendental numbers.

__Formally__

**,**this

**is**described

**in**

__algebraic__notation like this:

**(**19 +

**1**) +

**(**15 −

**1**) =

**x**

**,**but even a young student might use this technique without calling it algebra.

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 n

__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__

**,**

**the**singular value decomposition

**of**

**an**m ×

__n__real or complex matrix M

**is**a factorization

**of**

**the**form

__Formally__

**,**Aff

**(**V )

**is**naturally isomorphic to a subgroup

**of**

**,**with V embedded as

**the**affine plane

**,**namely

**the**stabilizer

**of**this affine plane ;

**the**above matrix formulation

**is**

**the**

**(**transpose

**of**)

**the**realization

**of**this

**,**with

**the**

**(**

__n__×

__n__and

**1**×

**1**) blocks corresponding to

**the**direct sum decomposition.

__Formally__

**,**a complex projective space

**is**

**the**space

**of**complex lines through

**the**origin

**of**

**an**

**(**

__n__+

**1**)- dimensional complex vector space.

__Formally__

**,**given a finite set X

**,**a collection C

**of**subsets

**of**X

**,**all

**of**size

__n__

**,**has Property B if we can partition X into two disjoint subsets Y and Z such that every set

**in**C meets both Y and Z.

__Formally__

**,**a composite number

__n__= d · 2

**<**sup

**>**s

**</**sup

**>**+

**1**with d being odd

**is**called a strong pseudoprime to a relatively prime base a when one

**of**

**the**following conditions hold:

__Formally__

**,**

**the**use

**of**a reduction

**is**

**the**

**function**that sends each natural number

__n__to

**the**largest natural number m whose membership

**in**

**the**set B was queried by

**the**reduction while determining

**the**membership

**of**

__n__

**in**A.

__Formally__

**,**P

**is**a symmetric polynomial

**,**if for any permutation σ

**of**

**the**subscripts

**1**

**,**2

**,**...,

__n__one has P

**(**X

**<**

**sub**

**>**σ

**(**

**1**)</

**sub**>, X

**<**

**sub**

**>**σ

**(**2 )</

**sub**>, …, X

**<**

**sub**

**>**σ

**(**

__n__)</

**sub**>) = P

**(**X

**<**

**sub**

**>**

**1**

**</**

**sub**>, X

**<**

**sub**

**>**2

**</**

**sub**>, …, X

**<**

**sub**

**>**

__n__

**</**

**sub**

**>).**

Formally and variables

__Formally__

**,**collective noun forms such as “ a group

**of**people ” are represented by second-order

__variables__

**,**or by first-order

__variables__standing for sets

**(**which are well-defined objects

**in**mathematics and logic ).

__Formally__it

**is**precisely

**in**allowing quantification

**over**class

__variables__α

**,**β

**,**etc., that we assume a range

**of**values for these

__variables__to refer to.

__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.

__Formally__

**,**they are partial derivatives

**of**

**the**option price with respect to

**the**independent

__variables__

**(**technically

**,**one Greek

**,**gamma

**,**

**is**a partial derivative

**of**another Greek

**,**called delta ).

__Formally__

**,**dependence refers to any situation

**in**which random

__variables__do not satisfy a mathematical condition

**of**probabilistic independence.

__Formally__

**,**a constraint satisfaction problem

**is**defined as a triple

**,**where

**is**a set

**of**

__variables__

**,**

**is**a domain

**of**values

**,**and

**is**a set

**of**constraints.

__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__

**,**

**the**algorithm's performance will be a random variable determined by

**the**random bits ; thus either

**the**running time

**,**or

**the**output

**(**or both ) are random

__variables__.

__Formally__

**,**propositional models can be represented by sets

**of**propositional

__variables__; namely

**,**each model

**is**represented by

**the**set

**of**propositional

__variables__it assigns to true.

__Formally__

**,**

**the**extension

**of**circumscription that incorporate varying and fixed

__variables__

**is**as follows

**,**where

**is**

**the**set

**of**

__variables__to minimize

**,**

**the**fixed

__variables__

**,**and

**the**varying

__variables__are those not

**in**:

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