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If r is the degree of the primitive generator polynomial, then the maximal total block length is, and the associated code is able to detect any single-bit or double-bit errors.

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## Some Related Sentences

If and r

__If__we define

__r__< sub > i </ sub > as

**the**displacement

**of**particle i from

**the**center

**of**mass

**,**

**and**v < sub > i </ sub > as

**the**velocity

**of**particle i with respect

**to**

**the**center

**of**mass

**,**

**then**we have

__If__

**the**relative priorities

**of**these substituents need

**to**be established

**,**R takes priority over S

**.**When this happens

**,**

**the**descriptor

**of**

**the**stereocenter

**is**a lowercase letter (

__r__

**or**s ) instead

**of**

**the**uppercase letter normally used

**.**

__If__u ( t )

**is**

**the**control signal sent

**to**

**the**system

**,**y ( t )

**is**

**the**measured output

**and**

__r__( t )

**is**

**the**desired output

**,**

**and**tracking error

**,**a PID controller has

**the**general form

__If__

**the**weight

**of**

**the**roadway per unit

**length**

**is**w

**and**

**the**weight

**of**

**the**cable

**and**

**the**wire supporting

**the**bridge

**is**negligible in comparison

**,**

**then**

**the**weight on

**the**cable from c

**to**

__r__

**is**wx where x

**is**

**the**horizontal distance between c

**to**

__r__

**.**Proceeding as before gives

**the**differential equation

__If__F (

__r__) represents gravity

**,**it

**is**a negative term proportional

**to**1 /

__r__< sup > 2 </ sup >, so

**the**net acceleration in

__r__in

**the**rotating frame depends on a difference

**of**reciprocal square

**and**reciprocal cube terms

**,**which are in balance in a circular orbit but otherwise typically not

**.**

__If__

**the**source

**of**Φ (

__r__)

**is**a dipole

**,**as it

**is**assumed here

**,**this term

**is**

**the**only non-vanishing term in

**the**multipole expansion

**of**Φ (

__r__).

__If__it

**is**generally by temporary uninhabitability

**,**for example a nuclear winter

**,**

**then**n < sub >

__r__</ sub > may be relatively high

**.**

*( EF1 )

__If__a**and**b are in R**and**b**is**nonzero**,****then**there are q**and**__r__in R such that**and**either__r__= 0**or****.**__If__

__r__represents an arbitrary element

**of**R

**,**f can be viewed as a right R-homomorphism so that

**,**

**or**f can also be viewed as a left R module homomorphism

**,**where

**.**

Thus

**the**set**of**all polynomials with coefficients in**the**ring R forms itself a ring**,****the**ring**of**polynomials over R**,**which**is**denoted by R**.**The map from R**to**R sending__r__**to**rX < sup > 0 </ sup >**is**an injective homomorphism**of**rings**,**by which R**is**viewed as a subring**of**R**.**__If__R**is**commutative**,****then**R**is**an algebra over R**.**__If__R

**is**an integral domain

**and**f

**and**g are polynomials in R

**,**it

**is**said that f divides g

**or**f

**is**a divisor

**of**g if there exists a

**polynomial**q in R such that f q = g

**.**One can show that every zero gives rise

**to**a linear divisor

**,**

**or**more formally

**,**if f

**is**a

**polynomial**in R

**and**

__r__

**is**an element

**of**R such that f (

__r__) = 0

**,**

**then**

**the**

**polynomial**( X −

__r__) divides f

**.**The converse

**is**also true

**.**

__If__F

**is**a field

**and**f

**and**g are polynomials in F with g ≠ 0

**,**

**then**there exist unique polynomials q

**and**

__r__in F with

__If__an object moves with angular velocity ω around a circle

**of**radius

__r__centered at

**the**origin

**of**

**the**x-y plane

**,**

**then**its motion along each coordinate

**is**simple harmonic motion with amplitude

__r__

**and**angular frequency ω

**.**

Consider now

**the**acceleration due**to****the**sphere**of**mass M experienced by a particle in**the**vicinity**of****the**body**of**mass m**.**With R as**the**distance from**the**center**of**M**to****the**center**of**m**,**let ∆__r__be**the**( relatively small ) distance**of****the**particle from**the**center**of****the**body**of**mass m**.**For simplicity**,**distances are first considered only in**the**direction pointing towards**or**away from**the**sphere**of**mass M**.**__If__**the**body**of**mass m**is**itself a sphere**of**radius ∆__r__**,****then****the**new particle considered may be located on its surface**,**at a distance ( R ± ∆__r__) from**the**centre**of****the**sphere**of**mass M**,****and**∆__r__may be taken as positive where**the**particle's distance from M**is**greater than R**.**Leaving aside whatever gravitational acceleration may be experienced by**the**particle towards m on account**of**ms own mass**,**we have**the**acceleration on**the**particle due**to**gravitational force towards M as:

If and is

__If__his dancers are sometimes made

**to**look as if they might be creatures from Mars

**,**this

__is__consistent with his intention

**of**placing them in

**the**orbit

**of**another world

**,**a world in which they are freed

**of**their pedestrian identities

**.**

__If__a work

__is__divided into several large segments

**,**a last-minute drawing

**of**random numbers may determine

**the**order

**of**

**the**segments for

**any**particular performance

**.**

__If__Wilhelm Reich

__is__

**the**Moses who has led them out

**of**

**the**Egypt

**of**sexual slavery

**,**Dylan Thomas

__is__

**the**poet who offers them

**the**Dionysian dialectic

**of**justification for their indulgence in liquor

**,**marijuana

**,**sex

**,**

**and**jazz

**.**

__If__he

__is__

**the**child

**of**nothingness

**,**if he

__is__

**the**predestined victim

**of**an age

**of**atomic wars

**,**

**then**he will consult only his own organic needs

**and**go beyond good

**and**evil

**.**

__If__he thus achieves a lyrical

**,**dreamlike

**,**drugged intensity

**,**he pays

**the**price for his indulgence by producing work -- Allen Ginsberg's `` Howl ''

__is__a striking example

**of**this tendency -- that

__is__disoriented

**,**Dionysian but without depth

**and**without Apollonian control

**.**

__If__love reflects

**the**nature

**of**man

**,**as Ortega Y Gasset believes

**,**if

**the**person in love betrays decisively what he

__is__by his behavior in love

**,**

**then**

**the**writers

**of**

**the**beat generation are creating a new literary genre

**.**

__If__

**the**existent form

__is__

**to**be retained new factors that reinforce it must be introduced into

**the**situation

**.**

__If__we remove ourselves for a moment from our time

**and**our infatuation with mental disease

**,**isn't there something absurd about a hero in a novel who

__is__defeated by his infantile neurosis??

__If__many

**of**

**the**characters in contemporary novels appear

**to**be

**the**bloodless relations

**of**characters in a case history it

__is__because

**the**novelist

__is__often forgetful today that those things that we call character manifest themselves in surface behavior

**,**that

**the**ego

__is__still

**the**executive agency

**of**personality

**,**

**and**that all we know

**of**personality must be discerned through

**the**ego

**.**

__If__our sincerity

__is__granted

**,**

**and**it

__is__granted

**,**

**the**discrepancy can only be explained by

**the**fact that we have come

**to**believe hearsay

**and**legend about ourselves in preference

**to**an understanding gained by earnest self-examination

**.**

If and degree

__If__she has not had such experiences

**,**

**the**female's normal adolescent

__degree__

**of**indecision will be compounded

**.**

__If__a

**is**algebraic over K

**,**

**then**K

**,**

**the**set

**of**all polynomials in a with coefficients in K

**,**

**is**not only a ring but a field: an algebraic extension

**of**K which has finite

__degree__over K

**.**In

**the**special case where K = Q

**is**

**the**field

**of**rational numbers

**,**Q

**is**an example

**of**an algebraic number field

**.**

__If__five judges

**then**

**the**highest

**and**lowest scores are discarded

**and**

**the**middle three are summed

**and**multiplied by

**the**

__degree__

**of**difficulty ( DD ), which

**is**determined from a combination

**of**

**the**moves undertaken

**,**in which position

**and**from what height ).

__If__a mathematical function

**is**used

**to**represent

**the**production function

**,**

**and**if that production function

**is**homogeneous

**,**returns

**to**scale are represented by

**the**

__degree__

**of**homogeneity

**of**

**the**function

**.**

__If__

**any**player interferes with an eligible receiver's ability

**to**catch

**the**ball it

**is**pass interference which draws a penalty

**of**varying

__degree__( largely depending upon

**the**particular league's rules ).

__If__v < sub > n </ sub >

**is**

**the**number

**of**vertices

**of**

__degree__n

**and**D

**is**

**the**maximum

__degree__

**of**

**any**vertex

**,**

*

__If__f**is**an irreducible**polynomial****of**prime__degree__p with rational coefficients**and**exactly two non-real roots**,****then****the**Galois group**of**f**is****the**full symmetric group S < sub > p </ sub >.__If__every formula in R

**of**

__degree__k

**is**either refutable

**or**satisfiable

**,**

**then**so

**is**every formula in R

**of**

__degree__k + 1

**.**

__If__

**the**

__degree__

**of**underrepresentation

**is**small

**,**

**the**sample can be treated as a reasonable approximation

**to**a random sample

**.**

__If__

**the**monopoly were permitted

**to**charge individualised prices ( this

**is**termed third

__degree__price discrimination ),

**the**quantity produced

**,**

**and**

**the**price charged

**to**

**the**marginal customer

**,**would be identical

**to**that

**of**a competitive company

**,**thus eliminating

**the**deadweight loss ; however

**,**all gains from trade ( social welfare ) would accrue

**to**

**the**monopolist

**and**none

**to**

**the**consumer

**.**

A number a

**is**a root**of**P if**and**only if**the****polynomial**x − a (**of**__degree__one in x ) divides P**.**It may happen that x − a divides P more than once: if ( x − a )< sup > 2 </ sup > divides P**then**a**is**called a multiple root**of**P**,****and**otherwise a**is**called a simple root**of**P**.**__If__P**is**a nonzero**polynomial****,**there**is**a highest power m such that ( x − a )< sup > m </ sup > divides P**,**which**is**called**the**multiplicity**of****the**root a in P**.**When P**is****the**zero**polynomial****,****the**corresponding**polynomial**equation**is**trivial**,****and**this case**is**usually excluded when considering roots: with**the**above definitions every number would be a root**of****the**zero**polynomial****,**with undefined (**or**infinite ) multiplicity**.**__If__

**the**areas

**of**

**the**two parallel faces are A < sub > 1 </ sub >

**and**A < sub > 3 </ sub >,

**the**cross-sectional area

**of**

**the**intersection

**of**

**the**prismatoid with a plane midway between

**the**two parallel faces

**is**A < sub > 2 </ sub >,

**and**

**the**height (

**the**distance between

**the**two parallel faces )

**is**h

**,**

**then**

**the**volume

**of**

**the**prismatoid

**is**given by ( This formula follows immediately by integrating

**the**area parallel

**to**

**the**two planes

**of**vertices by Simpson's rule

**,**since that rule

**is**exact for integration

**of**polynomials

**of**

__degree__up

**to**3

**,**

**and**in this case

**the**area

**is**at most a quadratic in

**the**height

**.**

__If__

**the**overall system

**is**pure

**,**

**the**entropy

**of**one subsystem can be used

**to**measure its

__degree__

**of**entanglement with

**the**other subsystems

**.**

__If__we use

**the**

**generator**

**polynomial**

**,**where

**is**a

**primitive**

**polynomial**

**of**

__degree__

**,**

**then**

**the**

**maximal**

**total**

**block**

**length**

**is**

**,**

**and**

**the**

**code**

**is**

**able**

**to**

**detect**single

**,**double

**,**

**and**triple

**errors**

**.**

__If__their test scores suddenly diverged by a large

__degree__

**,**this would be strong evidence that studying (

**or**going

**to**

**the**amusement park ) had a causal effect on test scores

**.**

__If__this wave interacts with a material at which

**the**horizontal component ( green sinusoid ) travels at a different speed than

**the**vertical component ( blue sinusoid ),

**the**two components will fall out

**of**

**the**90

__degree__phase difference ( required for circular polarization ) changing

**the**Kerr Ellipticity

**.**

__If__f

**is**a

**polynomial**

**of**

__degree__less than

**or**equal

**to**d

**,**

**then**

**the**Taylor

**polynomial**

**of**

__degree__d equals f

**.**

"

__If__one has a living person inhale this liquid when it**is**at its highest__degree__**of**concentration**,****the**body**of****the**patient for a few moments becomes as light as a balloon ," Robert-Houdin claimed**.**__If__

**the**public sector company

**is**making money

**and**selling their product for half

**of**

**the**price

**of**

**the**private sector companies

**,**

**the**private sector companies won't be

**able**

**to**simultaneously gouge

**to**that

__degree__

**and**keep their customers:

**the**competition keeps them in line

**.**

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