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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.

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## Some Related Sentences

If and Af

Note:

__If__1/2-inch panel board**is**used inside**and**out**,**or 5/8-inch one side**and**3/8-inch**the**other**,****and**1/8-inch glass**is**used**,**stock lumber**in**__Af__**,**__Af__**,****and**__Af__can be used**in**making**the**glass panels**.**__If__

**the**volume

**is**

**the**molal volume

**,**

**then**

__Af__

**is**obtained on

**a**molal basis which

**is**

**the**customary terminology

**of**

**the**chemists

**.**

__If__

__Af__

**is**

**the**change per unit volume

**in**Gibbs function caused by

**the**shear field at constant P

**and**

**T**

**,**

**and****yr

**is**

**the**density

**of**

**the**fluid

**,**

**then**

**the**total potential energy

**of**

**the**system above

**the**reference height

**is**

__Af__

**.**

__If__A

**is**

**the**major axis

**of**an ellipsoid

**and**B

**and**C

**are**

**the**other two axes

**,**

**the**radius

**of**curvature

**in**

**the**ab plane at

**the**end

**of**

**the**axis

__Af__

**,**

**and**

**the**difference

**in**pressure along

**the**A

**and**B axes

**is**

__Af__

**.**

__If__it

**is**assumed that

**the**formula given by Lodge

**of**cosec

__Af__applies

**,**

**the**pressure difference along

**the**major axes can be calculated from

**the**angle

**of**inclination

**of**

**the**major axis

**,**

**and**from this

**the**interfacial tension can be calculated

**.**

__If__

**the**

__Af__bond

**is**

**linear**

**then**there

**are**three reasonable positions for

**the**hydrogen atoms: ( 1 ) The hydrogen atoms

**are**centered

**and**hence all lie on

**a**sheet midway between

**the**oxygen sheets ; ;

__If__such

**is**

**the**case

**,**

**the**particles within

**a**distance

**of**about

__Af__

**of**

**the**Earth will have

**,**relative to

**the**Earth

**,**

**a**kinetic energy less than their potential energy

**and**they will be captured into orbits about

**the**Earth

**.**

__If__

**the**deficiency persists long enough

**,**it

**is**reasonable to suppose that

**the**

__Af__label will reflect

**the**

__Af__distribution

**in**

**the**thyroglobulin

**.**

__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__

**the**direct-sum

**decomposition**( A )

**is**valid

**,**how can we get hold

**of**

**the**

**projections**

__Af__

**associated**

**with**

**the**

**decomposition**??

__If__

__Af__

**is**

**the**

**operator**induced on

__Af__by

**T**

**,**

**then**evidently

__Af__

**,**because by definition

__Af__

**is**0 on

**the**

**subspace**

__Af__

**.**

If and are

__If__it were not for an old professor who made me read

**the**classics I would have been stymied on what to do

**,**

**and**now I understand why they

__are__classics ; ;

__If__I even hint at it do you think it will matter that you

__are__his nephew --

**and**not even

**a**blood nephew ''??

__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__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

**.**

Defoe

**then**commented**,**``__If__they Could Draw that young Gentleman into Their Measures They would show themselves quickly**,**for they__are__not asham'd to Say They want only**a**head to Make**a**beginning ''**.**__If__Jews

__are__identified as

**a**religious body

**in**

**a**controversy that comes before

**a**national or international tribunal

**,**it

**is**obviously compatible

**with**

**the**goal

**of**human dignity to protect freedom

**of**worship

**.**

Without preliminaries

**,**Esther asked him**,**``__If__you__are__**a**world citizen**,**will you take Garry Davis' place**in**his tent while he goes to**the**hospital ''??__If__we

__are__to believe

**the**list

**of**titles printed

**in**Malraux's latest book

**,**La Metamorphose Des Dieux

**,**Vol. 1 ( ( 1957 )

**,**he

**is**still engaged

**in**writing

**a**large novel

**under**his original title

**.**

__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**record buyer's tastes

__are__somewhat eclectic or even

**the**slightest bit esoteric

**,**he will find them satisfied on educational records

**.**

__If__it will simply delay

**the**debates until

**the**qualifications

__are__closed next spring

**,**

**and**

**then**carry all

**the**candidates on

**a**tour

**of**debates

**,**it can provide

**a**service to

**the**state

**.**

__If__they

__are__to be commended for foresight

**in**their planning

**,**what

**then**

**is**

**the**judgment

**of**

**a**town council that compounds this problem during

**the**planning stage??

__If__

**the**Communists

__are__sincere

**in**wanting

**a**united

**,**neutral

**and**disarmed Germany

**,**it might well be advantageous for

**the**German people

**in**this nuclear age

**.**

__If__only this could be done more often --

**with**such heartening results -- many

**of**

**the**earth's `` big problems '' would shrink to

**the**insignificances they really

__are__

**.**

__If__any

__are__left

**,**presently

**,**we may expect to see signs specifically prohibiting

**the**feeding

**of**them too

**.**

__If__

**the**raw population figures

__are__crucially relevant

**,**

**then**it

**is**idle to think

**of**liberation

**,**as idle as to suppose that Poland might liberate Russia

**.**

__If__it

**is**not enough that all

**of**our internationalist One Worlders

__are__advocating that we join this market

**,**I refer you to an article

**in**

**the**New York Times' magazine section ( Nov. 12

**,**1961 )

**,**by Mr. Eric Johnston

**,**entitled `` We Must Join The Common Market ''

**.**

If and projections

*

__If__**is****the**norm ( usually noted as ) defined**in****the**sequence space ℓ < sup >∞</ sup >**of**all bounded sequences ( which also matches**the**non-linear distance measured as**the**maximum**of**distances measured on__projections__into**the**base subspaces**,**without requiring**the**space to be isotropic or even just**linear****,**but only continuous**,**such norm being definable on all Banach spaces ),**and****is**lower triangular non-singular ( i**.**e., )**then**__If__

**the**causes

**are**understood

**,**

__projections__

**of**

**the**influencing variables can be made

**and**used

**in**

**the**forecast

**.**

__If__

**the**conversion efficiency turns out to be much lower it

**is**expected to have an adverse impact on

__projections__made for Cost

**of**Energy

**.**

__If__one considers

**a**slightly different definition

**,**based on

**the**

__projections__into space

**of**corresponding retinal points

**,**

**then**Schreiber

**and**colleagues have shown that

**a**different theoretical form emerges

**.**

__If__threatened by

**a**larger animal such as

**a**human

**,**these frogs can deliver

**a**painful bite as they have several odontoid

__projections__( not teeth per se ) along their bottom

**and**top jaws

**.**

*

__If__**the**turnaround time crosses into**the**next fiscal year or quarter**,****a**rebate offer can inflate sales**in****the**current period**,****and**not have to be accounted for until**the**next period**and****then**it could be attributed as**a**cost reducing sales or expense for**the**next period**,**giving companies an accounting advantage**with**their Wall Street__projections__**.**0.109 seconds.