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If a polynomial time algorithm calls as a subroutine polynomially many polynomial time algorithms, the resulting algorithm is still polynomial time.

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## Some Related Sentences

If and polynomial

__If__( remember this

**is**an assumption )

**the**minimal

__polynomial__for T decomposes Af where Af are distinct elements of F

**,**then we shall show that

**the**space V

**is**

**the**direct sum of

**the**null spaces of Af

**.**

__If__Af are

**the**projections associated with

**the**primary decomposition of T

**,**then each Af

**is**

**a**

__polynomial__in T

**,**and accordingly if

**a**linear operator U commutes with T then U commutes with each of

**the**Af

**,**i.e.

**,**each subspace Af

**is**invariant under U

**.**

__If__T

**is**

**a**linear operator on an arbitrary vector space and if there

**is**

**a**monic

__polynomial__P such that Af

**,**then parts ( A ) and ( B ) of Theorem 12 are valid for T with

**the**proof which we gave

**.**

__If__D denotes

**the**differentiation operator and P

**is**

**the**

__polynomial__Af then V

**is**

**the**null space of

**the**operator p (

**,**)

**,**because Af simply says Af

**.**

__If__F

**is**algebraically closed and p ( x )

**is**an irreducible

__polynomial__of F

**,**then it has some root

**a**and therefore p ( x )

**is**

**a**multiple of x −

**a**

**.**

__If__

**the**initial value of

**a**

__polynomial__( and of its finite differences )

**is**calculated by some means for some value of X

**,**

**the**difference engine can calculate any number of nearby values

**,**using

**the**method generally known

**as**

**the**method of finite differences

**.**

*

__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__

**is**

**a**left-( respectively right -) Noetherian ring

**,**then

**the**

__polynomial__ring

**is**also

**a**left-( respectively right -) Noetherian ring

**.**

__If__

**a**large

**,**b-bit number

**is**

**the**product of two primes that are roughly

**the**same size

**,**then no

**algorithm**has been published that can factor in

__polynomial__

**time**

**,**i

**.**e., that can factor it in

**time**O ( b < sup > k </ sup >) for some constant k

**.**There are published

**algorithms**that are faster than O (( 1 + ε )< sup > b </ sup >) for all positive ε

**,**i

**.**e., sub-exponential

**.**

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__

**,**however

**,**

**the**set of allowed candidates

**is**expanded to

**the**complex numbers

**,**every non-constant

__polynomial__has at least one root ; this

**is**

**the**fundamental theorem of algebra

**.**

__If__R

**is**commutative

**,**then one can associate to every

__polynomial__P in R

**,**

**a**

__polynomial__function f with domain and range equal to R ( more generally one can take domain and range to be

**the**same unital associative 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 and time

__If__

**the**bluff failed and they ran into trouble

**,**Brannon had told

**the**others

**,**they would withdraw -- and he would come after his son another

__time__

**.**

__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__

**the**would-be joiner asks these questions he

**is**not likely to be duped by extremists who are seeking to capitalize on

**the**confusions and

**the**patriotic apprehensions of Americans in

**a**troubled

__time__

**.**

__If__

**the**world comes to this

**,**wouldn't it be

**the**very

__time__when courage and American know-how would be needed to help survivors rebuild??

__If__there

**is**

__time__after

**the**warning

**,**

**the**basement shielding could be improved substantially by blocking windows with bricks

**,**dirt

**,**books

**,**magazines

**,**or other heavy material

**.**

__If__

**,**in

**the**trustee's judgment

**,**`` reasonable market conditions '' did not prevail during any given year

**,**he was to be allowed to petition

**the**court for an extension of

__time__within

**the**ten-year period

**.**

__If__

**the**reserve components are to serve effectively in

__time__of war

**,**their basic organization and objectives must conform to

**the**changing character and missions of

**the**active forces

**.**

__If__you elect to use

**the**Standard Deduction or

**the**Tax Table

**,**and later find you should have itemized your deductions

**,**you may do so by filing an amended return within

**the**

__time__prescribed for filing

**a**claim for refund

**.**

__If__you mail A return or tax payment

**,**you must place it in

**the**mails in ample

__time__to reach

**the**district director on or before

**the**due date

**.**

__If__

**the**lever

**is**pulled to clear

**a**signal or move

**a**switch

**,**

**the**tappet moves

**a**short distance lengthwise at

**the**same

__time__

**.**

__If__

**the**background of design

**is**too smooth

**,**or you wish to create

**a**wood-grained effect

**,**it may be added at this

__time__with

**a**dull tool such

**as**

**the**handle of

**a**fine paintbrush

**.**

__If__companies will take

**the**

__time__to give objective consideration to their major problems and to

**the**questions they provoke

**,**then

**a**long constructive step will have been taken toward more effective marketing in next decade

**.**

__If__you've travelled in Europe

**a**

__time__or two

**,**it

**is**quite certain that you've had that wanting-to-be-alone feeling or that you will get it on your next visit across

**the**Atlantic

**.**

``

__If__you are trying to get us out of**the**brothel**,****the**dustbin**,****the**kitchen sink**,**and**the**tawdry living-room**,**you are probably wasting your__time__''**,**Moreland told me**.**__If__

**,**at any

__time__during

**the**assignment pass

**,**

**the**compiler finds that there are no more index words available for assignment

**,**

**the**warning message `` No More Index Words Available '' will be placed in

**the**object program listing

**,**

**the**table will be altered to show that index words 1 through 96 are available

**,**and

**the**assignment will continue

**as**before

**.**

__If__Depew had told any academic psychologist that he had

**a**weird feeling of having lived through that identical convention session at some

__time__in

**the**past

**,**he would have been informed that he was

**a**victim of deja vue

**.**

__If__baby teeth are retained too long

**,**

**the**incoming second teeth may be prevented from emerging at

**the**normal

__time__or may have to erupt in

**the**wrong place

**.**

__If__one lives near

**a**subway or an express parkway

**,**

**the**solution

**is**to have one's wines stored with

**a**dealer and brought home

**a**few at

**a**

__time__

**.**

__If__this woman had delayed until after 11:20 to start her shopping

**,**she would have had little

__time__in which to prepare

**the**substantial meal that was eaten at dinner in those days

**.**

__If__you have taken this stroll in

**the**morning

**,**and you have

**the**

__time__and inclination

**,**walk to

**the**right along

**the**crowded Corso for half

**a**dozen blocks to visit

**the**fine private collection of paintings -- mainly of

**the**sixteenth and seventeenth centuries -- in

**the**Palazzo Doria ( open Sunday

**,**Tuesday

**,**and Thursday

**,**10:00 to 1:00 )

**.**

If and algorithm

__If__they don't then for

**the**

__algorithm__to be effective it must provide

**a**set of rules for extracting

**a**square root

**.**

__If__

**a**" weak " character

**is**followed by another " weak " character

**,**

**the**

__algorithm__will look at

**the**first neighbouring " strong " character

**.**

__If__

**the**

__algorithm__

**is**symmetric

**,**

**the**key must be known to

**the**recipient and sender and to no one else

**.**

__If__

**the**

__algorithm__

**is**an asymmetric one

**,**

**the**enciphering key

**is**different from

**,**but closely related to

**,**

**the**deciphering key

**.**

__If__one key cannot be deduced from

**the**other

**,**

**the**asymmetric key

__algorithm__has

**the**public / private key property and one of

**the**keys may be made public without loss of confidentiality

**.**

__If__

**a**suitably sized quantum computer capable of running Grover's

__algorithm__reliably becomes available

**,**it would reduce

**a**128-bit key down to 64-bit security

**,**roughly

**a**DES equivalent

**.**

__If__

**a**naive rendering

__algorithm__

**is**used without any filtering

**,**high frequencies in

**the**image function will cause ugly aliasing to be present in

**the**final image

**.**

__If__implemented using remainders of Euclidean division rather than subtractions

**,**Euclid's

__algorithm__computes

**the**GCD of large numbers efficiently: it never requires more division steps than five times

**the**number of digits ( base 10 ) of

**the**smaller integer

**.**

__If__

**the**arguments are both greater than zero then

**the**

__algorithm__can be written in more elementary terms

**as**follows:

__If__R

**is**

**a**Euclidean domain in which euclidean division

**is**given algorithmically (

**as**

**is**

**the**case for instance when R = F where F

**is**

**a**field

**,**or when R

**is**

**the**ring of Gaussian integers ), then greatest common divisors can be computed using

**a**form of

**the**Euclidean

__algorithm__based on

**the**division procedure

**.**

__If__

**the**true function

**is**simple

**,**then an " inflexible " learning

__algorithm__with high bias and low variance will be able to learn it from

**a**small amount of data

**.**

__If__

**the**desired output values are often incorrect ( because of human error or sensor errors ), then

**the**learning

__algorithm__should not attempt to find

**a**function that exactly matches

**the**training examples

**.**

* Robustness:

__If__**the**model**,**cost function and learning__algorithm__are selected appropriately**the****resulting**ANN can be extremely robust**.**__If__using efficient sorting such

**as**mergesort or heapsort this

**is**an O ( n log n ) average and worst-case

__algorithm__

**.**

The

__algorithm__for deciding this**is**conceptually simple: it constructs (**the**description of )**a**new program t taking an argument n which ( 1 ) first executes program**a**on input i ( both**a**and i being hard-coded into**the**definition of t ), and ( 2 ) then returns**the**square of n**.**__If__**a**( i ) runs forever**,**then t will never get to step ( 2 ), regardless of n**.**Then clearly**,**t**is****a**function for computing squares if and only if step ( 1 ) terminates**.**__If__we apply

**the**run-length encoding ( RLE ) data compression

__algorithm__to

**the**above hypothetical scan line

**,**we get

**the**following:

__If__

**the**user wanted to see

**a**font at another size

**,**

**the**Font Manager would find

**the**closest match and apply

**a**basic scaling

__algorithm__

**.**

__If__two processes attempt to enter

**a**critical section at

**the**same

**time**

**,**

**the**

__algorithm__will allow only one process in

**,**based on whose turn it

**is**

**.**

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