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Formally, let S and T be finite sets and let F =

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## Some Related Sentences

Formally and let

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__

**,**

__let__

**be**a stochastic process

**and**

__let__represent the cumulative distribution function of the joint distribution of at times.

__Formally__

**,**

__let__A

**be**a real matrix of which we want to compute the eigenvalues

**,**

**and**

__let__A < sub > 0 </ sub >:= A.

__Formally__

**,**

__let__G

**be**a Coxeter group with reduced root system R

**and**k < sub > v </ sub > a multiplicity function on R ( so k < sub > u </ sub >

**=**k < sub > v </ sub > whenever the reflections σ < sub > u </ sub >

**and**σ < sub > v </ sub > corresponding to the roots u

**and**v are conjugate in G ).

__Formally__

**,**for received words

**,**

__let__denote the Hamming distance between

**and**

**,**that is

**,**the number of positions in which

**and**differ.

__Formally__

**,**

__let__P

**be**a poset ( partially ordered set ),

**and**

__let__

**F**

**be**a filter on P ; that is

**,**

**F**is a subset of P such that:

__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

Formally and S

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__

**,**cl (

__S__) denotes the smallest subset Y of M that contains

__S__such that for each reaction ( A

**,**B )

__Formally__

**,**it is a coalition of U.

__S__.

**and**European CEOs

**,**largely from transnational corporations

**,**launched at the urging of former United States Secretary of Commerce Ron Brown in Seville ( Spain ) in late 1995 at a conference attended by CEOs from more than 100 U.

__S__.

**and**EU companies

**and**by top government representatives led by the Commissioners for Trade

**and**Industry

**and**by the U.

__S__. Secretary of Commerce.

__Formally__designated as the Manhattan Engineer District

**,**it refers specifically to the period of the project from 1941 – 1946 under the control of the U.

__S__. Army Corps of Engineers

**,**under the administration of General Leslie R. Groves.

__Formally__

**,**a Hopf algebra is a ( associative

**and**coassociative ) bialgebra H over a field K together with a K-linear map

__S__: H → H ( called the antipode ) such that the following diagram commutes:

__Formally__

**,**an unlabelled state transition system is a tuple (

__S__

**,**→) where

__S__is a set ( of states )

**and**→ ⊆

__S__×

__S__is a binary relation over

__S__( of transitions ).

__Formally__

**,**given a partially ordered set ( P

**,**≤), then an element g of a subset

__S__of P is the greatest element of

__S__if

__Formally__

**,**a partially ordered set ( P

**,**≤) is bounded complete if the following holds for any subset

__S__of P:

__Formally__

**,**given two partially ordered

**sets**(

__S__

**,**≤)

**and**(

**T**

**,**≤), a function f:

__S__→

**T**is an order-embedding if f is both order-preserving

**and**order-reflecting

**,**i. e. for all x

**and**y in

__S__

**,**one has

Formally and T

__Formally__

**,**the ith row

**,**jth column element of A < sup >

__T__</ sup > is the jth row

**,**ith column element of A:

__Formally__

**,**if

__T__is a complete sufficient statistic for θ

**and**E ( g (

__T__))

**=**τ ( θ ) then g (

__T__) is the minimum-variance unbiased estimator ( MVUE ) of τ ( θ ).

__Formally__

**,**the bounded Borel functional calculus of a self adjoint operator

__T__on Hilbert space H is a mapping defined on the space of bounded complex-valued Borel functions f on the real line

**,**

__Formally__

**,**we define a bad field as a structure of the form ( K

**,**

__T__), where K is an algebraically closed field

**and**

__T__is an infinite

**,**proper

**,**distinguished subgroup of K

**,**such that ( K

**,**

__T__) is of

**finite**Morley rank in its full language.

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 " 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__

**,**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 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 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.

__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__

**,**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 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__

**,**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.

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