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Since Af are linearly independent functions and the exponential function has no zeros, these R functions Af, form a basis for the space of solutions.

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Brown Corpus

## Some Related Sentences

Since and Af

__Since__

__Af__is ferromagnetic

**,**we felt that any results obtained from

**the**magnetically contaminated

__Af__would be suspect

**.**

__Since__

__Af__

**are**distinct prime polynomials

**,**

**the**polynomials

__Af__

**are**relatively prime ( Theorem 8

**,**Chapter 4 )

**.**

__Since__D

**and**D'

**are**both diagonalizable

**and**they commute

**,**they

**are**simultaneously diagonalizable

**,**

**and**

__Af__is diagonalizable

**.**

__Since__

**the**complex

**of**singular lines is

**of**order K

**and**since there is

**no**complex

**of**invariant lines

**,**it follows from

**the**formula

__Af__that

**the**order

**of**

**the**involution is

__Af__

**.**

A final class

**of**exceptional lines is identifiable from**the**following considerations:__Since__**no**two generators**of**__Af__can intersect**,**it follows that their image curves can have**no**free intersections**.**__Since__two curves

**of**symbol

__Af__on Q intersect in

__Af__points

**,**it follows that there

**are**

__Af__lines

**of**

__Af__which

**are**tangent to Aj

**.**

__Since__by

**the**principle

**of**optimality

**the**remaining stages use an optimal Af-stage policy with respect to

__Af__

**,**we may enter section

__Af__

**of**

**the**table at this state

__Af__

**and**read off

**the**optimal policy

**for**stage

__Af__

**and**

**the**resulting state

__Af__

**.**

Since and are

__Since__

**the**hazards

**of**poor communication

__are__so great

**,**p can be justified as

**a**habitable site only on

**the**

**basis**

**of**unusual productivity such as is made available by

**a**waterfall

**for**milling purposes

**,**

**a**mine

**,**or

**a**sugar maple camp

**.**

__Since__

**the**slogans have little application to reality

**and**

__are__sanctimonious to boot

**,**

**the**applause is faint even in areas

**of**

**the**world where we should expect to find

**the**greatest affection

**for**free government

**.**

__Since__

**the**recognition

**of**Israel as

**a**nation state

**,**claims

__are__made in many cases which identify

**the**claimant as

**a**member

**of**

**the**new body politic

**.**

__Since__accurate base maps

__are__necessary

**for**any planning program

**,**

**the**first step taken by

**the**planning division to implement

**the**long-range state plan

**has**been to prepare two series

**of**base maps -- one at

**a**scale

**of**1 inch to

**a**mile

**,**

**and**

**the**second

**a**series

**of**26 sheets at

**a**scale

**of**1 inch to 2000 feet

**,**covering

**the**entire state

**.**

__Since__conventional methods

__are__insensitive at

**the**low frequencies

**of**

**these**molecular transitions

**,**

**the**paramagnetic resonance method is being used instead

**.**

__Since__broadcast frequencies

__are__very limited in number

**,**

**these**objectives

__are__to some extent inconsistent in that not all

**of**them can be fully realized

**,**

**and**to

**the**extent that each is realized

**,**there is

**a**corresponding reduction

**of**

**the**possibilities

**for**fullest achievement

**of**

**the**others

**.**

__Since__

**the**provision

**of**skywave service requires adequate freedom from interference

**,**only class 1

**,**stations

__are__capable

**of**rendering skywave service

**.**

__Since__fly cutters

__are__one sided

**and**not balanced

**,**they should be used at

**the**slowest speed available

**,**

**and**fed very slowly to avoid binding

**.**

__Since__there

__are__almost

**no**schools in

**the**country offering graduate work in interior design this rule cannot at present be observed

**.**

__Since__charges

__are__relatively

**the**same

**,**reserving

**a**car before you leave

**for**Europe will assure you

**of**having one on tap when you want it

**.**

__Since__practical washing procedures

__are__both lengthy

**and**expensive

**,**

**a**number

**of**laboratory tests have been developed

**for**

**the**numerical evaluation

**of**detergents

**.**

__Since__

**the**circulating thyroid hormones

__are__

**the**amino acids thyroxine

**and**tri-iodothyronine ( cf. Section C )

**,**it is clear that some mechanism must exist in

**the**thyroid gland

**for**their release from proteins before secretion

**.**

__Since__emotional reactions in

**the**higher vertebrates depend on individual experience

**and**

__are__aroused in man

**,**in addition

**,**by complex symbols

**,**one would expect that

**the**hypothalamus could be excited from

**the**cortex

**.**

Since and linearly

__Since__in oral languages

**the**elements

**of**sound

**are**

**for**

**the**most part produced

__linearly__in time ( that is

**,**in

**a**word like cat

**the**

**a**sound comes after

**the**c sound

**,**

**and**

**the**t sound comes after that ), they can generally be easily written in

**a**linear ( one-dimensional ) writing system such as an alphabet

**.**

__Since__this is

**a**second-order differential equation

**,**there must be two

__linearly__

**independent**

**solutions**

**.**

__Since__

**the**amplitude at

**these**reflections grows

__linearly__with

**the**number N

**of**scatterers

**,**

**the**observed intensity

**of**

**these**spots should grow quadratically

**,**like N < sup > 2 </ sup >.

__Since__

**the**field inside

**a**parallel plate capacitor depends

__linearly__on

**the**potential

**,**

**the**index

**of**refraction depends

__linearly__on

**the**field (

**for**crystals where Pockels effect dominates ),

**and**

**the**phase depends

__linearly__on

**the**index

**of**refraction

**,**

**the**phase modulation must depend

__linearly__on

**the**potential applied to

**the**EOM

**.**

__Since__

**the**power dissipation

**,**P = I < sup > 2 </ sup >

**R**

**,**increases with

**the**square

**of**

**the**current but only increases approximately

__linearly__with

**the**number

**of**windings

**,**

**the**power lost in

**the**windings can be minimized by reducing I

**and**increasing

**the**number

**of**turns N proportionally

**.**

__Since__

**the**recording is imposed

__linearly__on

**the**medium

**,**

**and**

**the**medium is easily cut

**and**glued

**,**sounds recorded can be easily re-sequenced

**and**separated onto separate tracks

**,**allowing more control in mixing

**.**

__Since__

**,**

**for**

**a**given bullet shape

**,**frontal surface increases as

**the**square

**of**

**the**calibre

**,**

**and**mass increases as

**the**cube

**of**

**the**diameter

**,**then sectional density grows

__linearly__with bore diameter

**.**

__Since__barycentric coordinates

**are**

**a**linear transformation

**of**Cartesian coordinates

**,**it follows that they vary

__linearly__along

**the**edges

**and**over

**the**area

**of**

**the**triangle

**.**

__Since__illumination

**and**reflectance combine multiplicatively

**,**

**the**components

**are**made additive by taking

**the**logarithm

**of**

**the**image intensity

**,**so that

**these**multiplicative components

**of**

**the**image can be separated

__linearly__in

**the**frequency domain

**.**

__Since__calculations show that

**the**Kuzyk limit increases faster than

__linearly__

**,**it is best to link together as many molecules as possible

**.**

__Since__

**the**

**space**

**of**meromorphic 1-forms is

**a**1-dimensional vector

**space**over

**the**field

**of**meromorphic

**functions**

**,**any two meromorphic 1-forms yield

__linearly__equivalent divisors

**.**

__Since__

**the**total flight-time difference between " fast "

**and**" slow " lightsignals would be expected to scale

__linearly__with distance in simple emission theory

**,**

**and**

**the**study would ( statistically ) have included stars with

**a**reasonable spread

**of**distances

**and**orbital speeds

**and**orientations

**,**deSitter concluded that

**the**effect should have been seen if

**the**model was correct

**,**

**and**its absence meant that

**the**emission theory was almost certainly wrong

**.**

__Since__electric field components parallel to

**the**wires

**are**reflected

**,**

**the**transmitted wave

**has**an electric field purely in

**the**direction perpendicular to

**the**wires

**,**

**and**is thus

__linearly__polarized

**.**

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