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Formally, the theorem can be stated as follows:

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## Some Related Sentences

Formally and theorem

__Formally__

**,**

**the**

__theorem__is

**stated**

**as**

**follows**

**:**There exist unique integers q and r such that a = qd + r and 0 ≤ r < | d |, where | d | denotes

**the**absolute value of d.

Gödel's first incompleteness

__theorem__first appeared**as**" Theorem VI " in Gödel's 1931 paper On__Formally__Undecidable Propositions in Principia Mathematica and Related Systems I.__Formally__

**,**this

**follows**from

**the**convolution

__theorem__in mathematics

**,**which relates

**the**Fourier transform of

**the**power spectrum (

**the**intensity of each frequency ) to its autocorrelation.

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.

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