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Formally, let be a stochastic process and let represent the cumulative distribution function of the joint distribution of at times.

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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__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 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

**.**

Formally and process

__Formally__

**,**

**the**appointment remains

**a**royal one made on

**the**advice

**of**

**the**Lord Chancellor

**,**but he / she no longer comments on individual applications

**,**

**and**merely supervises

**the**

__process__

**and**reviews

**the**panel's recommendations in general terms ( satisfying himself / herself that

**the**

__process__as operated is fair

**and**efficient ).

__Formally__

**,**

**the**

__process__is one

**of**progressive development

**,**divided into three stages

**of**increasingly complexity

**.**

__Formally__

**,**this effect is referred to as

**and**is

**a**

__process__whereby

**the**capture frame rate

**of**

**the**camera changes over time

**.**

__Formally__

**,**

**the**sets

**of**free

**and**bound names

**of**

**a**

__process__in π – calculus are defined inductively as follows

**.**

__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

**.**

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