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Thus if E is sufficiently small, there can be only one intersection of C and Af near Q, for if there were more than one intersection for every E then the difference between C and Af near Q would not be a monotone function.

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Brown Corpus

## Some Related Sentences

Thus and if

__Thus__

**,**

__if__corporations are

**not**to run away with us

**,**they must become quasi-governmental institutions

**,**subject to public control

**and**needs

**.**

__Thus__

**,**

__if__public pressure sets

**the**effective limit to

**the**price that

**the**industry may charge

**,**this pressure

**is**itself

**a**

**function**

**of**

**the**wage rate

**.**

__Thus__

**,**during any given year ( A )

__if__

**the**revenues from

**the**residential service are $7,000,000

**,**( B )

__if__

**the**operating expenses imputed to this class

**of**service come to $5,000,000

**,**

**and**(

**C**)

__if__

**the**net investment in ( or value

**of**)

**the**plant

**and**equipment deemed devoted to this service amounts to $30,000,000

**,**

**the**cost analyst will report that residential service

**,**in

**the**aggregate

**,**has yielded

**a**return

**of**$2,000,000 or 6-2/3 per cent

**.**

__Thus__

__if__

**the**gyro

**and**platform-controller combination maintains

**the**platform with zero angular deviation about

**the****yf axis

**,**

**the**system

**can**

**be**rotated with an angular velocity

**Af**

__if__

**a**torque

**is**supplied to

**the**gyro output axis Aj

**.**

__Thus__

**,**

__if__we are to observe American folklore in

**the**twentieth century

**,**we will do well to establish

**the**relationships

**between**folklore

**,**nationalism

**and**imperialism at

**the**outset

**.**

__Thus__

**,**

__if__what

**is**at issue

**is**whether `` All S

**is**P ''

**,**it

**is**indifferent whether `` Some S

**is**

**not**P '' or `` No S

**is**P ''

**,**since in either case

**the**judgment in question

**is**false

**.**

For

**every**group G**there****is****a**natural group homomorphism G → Aut ( G ) whose image**is****the**group Inn ( G )**of**inner automorphisms**and**whose kernel**is****the**center**of**G**.**__Thus__**,**__if__G has trivial center it**can****be**embedded into its own automorphism group**.**__Thus__throwing

**a**rock at someone

**for**

**the**purpose

**of**hitting him

**is**

**a**battery

__if__

**the**rock in fact strikes

**the**person

**and**

**is**an assault

__if__

**the**rock misses

**.**

__Thus__

**,**

**a**court officer taking possession

**of**goods under

**a**court order may use force

__if__reasonably necessary

**.**

__Thus__

**,**

__if__P

**is**

**a**program in L < sub > 2 </ sub > which

**is**

**a**minimal description

**of**s

**,**

**then**InterpretLanguage ( P ) returns

**the**string s

**.**The length

**of**this description

**of**s

**is**

**the**sum

**of**

__Thus__

**,**

__if__

**every**historian

**were**to claim that

**there**was

**a**solar eclipse in

**the**year 1600

**,**

**then**though we might at first naively regard that as in violation

**of**natural laws

**,**we'd come to accept it as

**a**fact

**.**

Thus and E

__Thus__

**,**it might

**be**used to challenge

**the**sharp distinction made in U

**.**S

**.**v

**.**

__E__

**.**

**C**

**.**Knight ( 1895 )

**between**commerce

**,**which was subject to federal regulation

**,**

**and**manufacturing

**,**which was

**not**

**.**

__Thus__Euclidean geometry corresponds to

**the**choice

**of**

**the**group

__E__( 3 )

**of**distance-preserving transformations

**of**

**the**Euclidean space R < sup > 3 </ sup >, conformal geometry corresponds to enlarging

**the**group to

**the**conformal group

**,**whereas in projective geometry

**one**

**is**interested in

**the**properties invariant under

**the**projective group

**.**

__Thus__optical amplification

**,**

**and**laser operation

**,**

**can**take place at

**a**frequency

**of**ν < sub > 32 </ sub > (

__E__< sub > 3 </ sub >-

__E__< sub > 2 </ sub > = hν < sub > 32 </ sub >).

While this equation suggests that

**the**activation energy**is**dependent on temperature**,**in regimes in which**the**Arrhenius equation**is**valid this**is**cancelled by**the**temperature dependence**of**k**.**__Thus__**,**__E__< sub >**a**</ sub >**can****be**evaluated from**the**reaction rate coefficient at any temperature ( within**the**validity**of****the**Arrhenius equation ).__Thus__

**,**according to Paul L

**.**Maier

**,**

**the**majority

**of**Herod biographers

**and**theologians hold that

**the**Massacre

**of**

**the**Innocents

**is**" legend

**and**

**not**historical ": Geza Vermes

**and**

__E__

**.**P

**.**Sanders

**,**

**for**example

**,**regard

**the**story as part

**of**

**a**creative hagiography

**.**

__Thus__

**,**by charging

**a**higher per unit price

**and**

**a**lower lump-sum fee

**,**

**the**firm has generated area

__E__

**more**profit

**than**

**if**it had charged

**a**lower per-unit price

**and**

**a**higher lump-sum fee

**.**

__Thus__

**,**

__E__( k ) dk represents

**the**contribution to

**the**kinetic energy from all

**the**Fourier modes with k < | k | < k + dk

**,**

**and**therefore

**,**

__Thus__

**,**what we call

**the**year 3053 A

**.**D

**.**(

**the**year

**the**novel begins )

**is**in

**the**novel

**the**year 1108 A

**.**

__E__

**.**

Light's wavenumber

**is**proportional to frequency**,****and**therefore also proportional to light's quantum energy__E__**.**__Thus__**,****.**__Thus__

**,**

**if**

**a**road

**is**built across

**the**fault as in

**the**figure panel Time 1

**,**it

**is**perpendicular to

**the**fault trace at

**the**point

__E__

**,**where

**the**fault

**is**locked

**.**

Given two tensor bundles

__E__→ M**and**F → M**,****a**map A: Γ (__E__) → Γ ( F ) from**the**space**of**sections**of**__E__to sections**of**F**can****be**considered itself as**a**tensor section**of****if****and****only****if**it satisfies A ( fs ,...) = fA ( s ,...) in each argument**,**where f**is****a**smooth**function**on M**.**__Thus__**a**tensor**is****not****only****a**linear map on**the**vector space**of**sections**,**but**a****C**< sup >∞</ sup >( M )- linear map on**the**module**of**sections**.**__Thus__

**,**on July 19

**,**1990

**,**

**the**Konkan Railway Corporation Limited ( KRCL ) was incorporated as

**a**public limited company under

**the**Companies Act

**,**1956

**,**with its headquarters at CBD Belapur in Navi Mumbai

**and**

__E__

**.**Sreedharan

**,**

**a**senior railway official

**,**as its first Chairman

**and**Managing Director

**.**

__Thus__

**,**replacing

__E__by K we may assume that f < sub > n </ sub > converge to f pointwise on

__E__

**.**Next

**,**by

**the**definition

**of**

**the**Lebesgue Integral

**,**it

**is**enough to show that

**if**φ

**is**any non-negative simple

**function**less

**than**or equal to f

**,**

**then**

Thus and is

__Thus__

**,**

**there**

__is__freshness

**not**

**only**in

**the**individual movements

**of**

**the**dance but in

**the**shape

**of**their continuity as well

**.**

__Thus__

**,**it

__is__no mystical intuition

**,**but an analyzable conception to say that man

**and**his tradition

**can**`` fall out

**of**existence ''

**.**

__Thus__human perception

**and**human volition

__is__

**the**immanent cause

**of**all social change

**and**this most truly when

**the**change reaches

**the**civilizational level

**.**

__Thus__

**,**circular motion

__is__itself

**one**

**of**

**the**essential characteristics

**of**completely perfect celestial existence

**.**

__Thus__Burns's `` My love

__is__like

**a**red

**,**red rose ''

**and**Hopkins' `` The thunder-purple sea-beach

**,**plumed purple

**of**Thunder '' although clearly intelligible in content

**,**hardly present ideas

**of**

**the**sort with which we are here concerned

**.**

__Thus__science

__is__

**the**savior

**of**mankind

**,**

**and**in this respect Childhood's End

**only**blueprints in greater detail

**the**vision

**of**

**the**future which

**,**though

**not**always so directly stated

**,**has nevertheless been present in

**the**minds

**of**most science-fiction writers

**.**

__Thus__

**the**copywriter in

**the**world

**of**

**the**space merchants

__is__

**the**person who in earlier ages might have been

**a**lyric poet

**,**

**the**person `` capable

**of**putting together words that stir

**and**move

**and**sing ''

**.**

__Thus__

**,**as

**a**development program

__is__being launched

**,**commitments

**and**obligations must

**be**entered into in

**a**given year which may exceed by twofold or threefold

**the**expenditures to

**be**made in that year

**.**

__Thus__

**,**

**the**Span

**of**its ossification was shortened

**and**

**the**center's ability to `` catch up '' in ossification

__is__demonstrated

**.**

__Thus__

**,**

**the**study

**of**

**the**solutions to

**the**equation

**Af**

__is__reduced to

**the**study

**of**

**the**space

**of**solutions

**of**

**a**differential equation

**of**

**the**form

**Af**

**.**

__Thus__in

**the**three-dice example

**Af**

**,**

**Af**

**,**

**and**

**the**independence assumption imply that

**the**probability that

**the**three dice fall ace

**,**not-ace

**,**ace in that order

__is__

**Af**

**.**

__Thus__we do

**not**score

**the**number

**of**bull's-eyes

**,**

**and**

**the**random variable

__is__

**not**

**the**number

**of**successes

**.**

__Thus__

**Af**

__is__also continuous at

**Af**

**,**

**and**in

**a**neighborhood

**of**

**Af**which does

**not**contain

**a**tangent point

**.**

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