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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 >.

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## Some Related Sentences

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 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 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**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__

**,**

**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**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__

**,**

**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**.

Formally and just

__Formally__named " Olallie ",

**it**has usually been marketed

**as**olallieberry

**,**

__just__

**as**" Marion " is sold

**as**marionberry.

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__

**,**

**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__

**,**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 )

0.172 seconds.