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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.

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## Some Related Sentences

Formally and problem

__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

**.**

__Formally__

**,**

**a**transductive support vector machine is defined by

**the**following primal optimization

__problem__

**:**

__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**

__problem__

**can**

**be**

**stated**

**as**

**follows**

**:**given

**a**desired property

**,**expressed

**as**

**a**temporal logic formula p

**,**and

**a**structure M

**with**initial state s

**,**decide if

**.**

Formally and pattern

__Formally__speaking

**,**

**the**strategy

__pattern__defines

**a**family

**of**algorithms

**,**encapsulates each one

**,**and makes them interchangeable

**.**

__Formally__

**,**

**a**graph rewriting system usually consists

**of**

**a**set

**of**graph rewrite rules

**of**

**the**form

**,**

**with**being called

__pattern__graph

**(**or left-hand side

**)**and being called replacement graph

**(**or right-hand side

**of**

**the**rule ).

Formally and can

__Formally__

**,**arbitrage transactions have negative skew – prices

__can__get

**a**small amount closer

**(**but often no closer than 0 ), while they

__can__get very far apart

**.**

__Formally__

**the**" head

**of**state "

__can__also personally

**be**

**the**" head

**of**government "

**(**ex officio or by ad hoc cumulation such

**as**

**an**absolute monarch nominating himself

**)**but otherwise has formal precedence over

**the**Head

**of**Government and other ministers

**,**whether he is their actual political superior

**(**absolute monarch

**,**executive president

**)**or rather theoretical or ceremonial in character

**.**

__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__

**,**

**a**unique factorization domain is defined

**to**

**be**

**an**integral domain R in which every non-zero and non-unit x

**of**R

__can__

**be**written

**as**

**a**product

**(**including

**an**empty product

**)**

**of**irreducible elements p < sub > i </ sub >

**of**R and

**a**unit u

**:**

__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__

**,**

**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**poem

__can__

**be**understood

**as**

**the**combination

**of**two sonnets

**,**though

**the**spacing

**of**

**the**stanzas is irregular

**.**

__Formally__

**,**Mbenga – Mbuti Pygmy music consists

**of**at most only four parts

**,**and

__can__

**be**described

**as**

**an**

**,**" ostinato

**with**variations ," or similar

**to**

**a**passacaglia

**,**in

**that**it is cyclical

**.**

__Formally__

**,**

**the**complexity class MA is

**the**set

**of**decision problems

**that**

__can__

**be**decided in polynomial time by

**an**Arthur – Merlin protocol where Merlin's only move precedes any computation by Arthur

**.**

__Formally__

**,**

**a**system is said

**to**

**be**observable if

**,**for any

**possible**sequence

**of**state and control vectors

**,**

**the**current state

__can__

**be**determined in finite time using only

**the**outputs

**(**this definition is slanted towards

**the**state space representation ).

__Formally__

**,**

**the**definition only requires some invertibility

**,**so we

__can__substitute for Q any matrix M whose eigenvalues do not include − 1

**.**

__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__

**,**

**the**issue is

**that**interfertile " able

**to**interbreed " is not

**a**transitive relation – if A

__can__breed

**with**B

**,**and B

__can__breed

**with**C

**,**it does not follow

**that**A

__can__breed

**with**C – and thus does not define

**an**equivalence relation

**.**

Formally and be

However

**,**shortly after this positive result**,**Kurt Gödel published On__Formally__Undecidable Propositions**of**Principia Mathematica and Related Systems**(**1931 ), showing**that**in any sufficiently strong axiomatic system there are true statements which cannot__be__proved in**the**system**.**__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__

**,**

**a**frame is defined

**to**

__be__

**a**lattice L in which finite meets distribute over arbitrary joins

**,**i

**.**e

**.**every

**(**even infinite

**)**subset

__Formally__

**,**

**the**convex hull may

__be__defined

**as**

**the**intersection

**of**all convex sets containing X or

**as**

**the**set

**of**all convex combinations

**of**points in X

**.**

__Formally__

**,**

**a**detailed balanced process is

**the**one on which

**the**probability

**to**

__be__in

**the**state x and transit

**to**x ' must

__be__

**the**same

**as**

**to**

__be__in

**the**state x ' and transit

**to**x

**.**

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**government would lay out its reasons for going

**to**war

**,**

**as**well

**as**its intentions in prosecuting it and

**the**steps

**that**might

__be__taken

**to**avert it

**.**

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