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Formally, if is any non-zero polynomial, it must be writable as.

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## Some Related Sentences

Formally and if

__Formally__

**,**oxidation state

**is**the hypothetical charge that an atom would have

__if__all bonds to atoms of different elements were 100 % ionic

**.**

__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__

**,**

__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__

**,**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

**.**

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 hypothesis

**is**compared against its opposite or null hypothesis ("

__if__I release this ball

**,**

**it**will not fall to the floor ").

__Formally__

**,**the sequence of partial sums of some infinite summation converges

__if__for every fixed power of X the coefficient stabilizes: there

**is**a point beyond which all further partial sums have the same coefficient

**.**

__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__

**,**

__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__

**,**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

**.**

Informally

**,**G has the above presentation__if__**it****is**the " freest group " generated by S subject only to the relations R**.**__Formally__**,**the group G**is**said to have the above presentation__if__**it****is**isomorphic to the quotient of a free group on S by the normal subgroup generated by the relations R**.**__Formally__

**,**

__if__we write F < sub > Δ </ sub >( x ) to mean the f-polynomial of Δ

**,**then the h-polynomial of Δ

**is**

__Formally__

**,**a cardinal κ

**is**defined to

**be**weakly compact

__if__

**it**

**is**uncountable and for every function f: < sup > 2 </ sup > →

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__

**,**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__

**,**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 any

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__speaking

**,**a collation method typically defines a total order on a set of possible identifiers

**,**called sort keys

**,**which consequently produces a total preorder on the set of items of information ( items with the same identifier are not placed in

__any__defined order ).

__Formally__

**,**we start with a category C with finite products ( i

**.**e

**.**C has a terminal object 1 and

__any__two objects of C have a product ).

__Formally__

**,**however

**,**a face

**is**

__any__of the lower dimensional boundaries of the polytope

**,**more specifically called an n-face

**.**

__Formally__

**,**dependence refers to

__any__situation in which random variables do not satisfy a mathematical condition of probabilistic independence

**.**

__Formally__

**,**the word

**is**applied to persons who are publicly accepted in a recognised capacity

**,**such

**as**professional employment

**,**graduation from a course of study

**,**etc., to give critical commentaries in one or

__any__of a number of specific fields of public or private achievement or endeavour

**.**

__Formally__

**,**an absolute coequalizer of a pair in a category C

**is**a coequalizer

**as**defined above but with the added property that given

__any__functor F ( Q ) together with F ( q )

**is**the coequalizer of F ( f ) and F ( g ) in the category D

**.**Split coequalizers are examples of absolute coequalizers

**.**

__Formally__titled ' An Act to protect all Persons in the United States in their Civil Rights

**,**and furnish the Means of their vindication

**,**the Act declared that people born in the United States and not subject to

__any__foreign power are entitled to

**be**citizens

**,**without regard to race

**,**color

**,**or previous condition of slavery or involuntary servitude

**.**

__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 deterministic algorithm computes a mathematical function ; a function has a unique value for

__any__given input

**,**and the algorithm

**is**a process that produces this particular value

**as**output

**.**

__Formally__

**,**a partially ordered set ( P

**,**≤)

**is**bounded complete

**if**the following holds for

__any__subset S of P:

__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__

**,**

**if**f

**is**a harmonic function

**,**then f cannot exhibit a true local maximum within the domain of definition of f

**.**In other words

**,**either f

**is**a constant function

**,**or

**,**for

__any__point inside the domain of f

**,**there exist other points arbitrarily close to at which f takes larger values

**.**

__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__

**,**a redistribution of all agents ’ initial endowments of goods among all agents in the system

**is**a core allocation

**if**no group of agents can redistribute their own initial endowments among themselves so

**as**to improve the satisfaction of someone in the group without impairing that of

__any__other in the group

**.**

__Formally__however they define

**it**

**as**

__any__variable that does not directly affect the fundamentals of the economy

**.**

__Formally__

**,**a market

**is**complete with respect to a trading strategy

**,**

**,**

**if**there exists a self-financing trading strategy

**,**such that at

__any__time

**,**the returns of the two strategies

**,**and are equal

**.**

__Formally__

**,**this meant that the whole peloton finished out of time limits

**,**but the referees understandably used a rule saying that they could give clemency to

__any__group of more than 20 % of the stage's starting riders

**,**officially citing the weather conditions

**as**their reason to do so

**.**

__Formally__

**,**let X

**be**

__any__scheme and S

**be**a sheaf of graded-algebras ( the definition of which

**is**similar to the definition of-modules on a locally ringed space ): that

**is**

**,**a sheaf with a direct sum decomposition

0.194 seconds.