systems of operational equations

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1n partial tulf1llaent of tle requlnaentbull tor tbe

ctecrbull ot

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lubull 1955

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asmiddotbull lllRODUO TIOB bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull 1

LDJIAR SYstEMS YLOllS AID MUiUCIS bullbullbullbullbull )

IXAKPlil8 bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull ~

DSlaquoRf SPAOI S bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull 9

0~ bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbull u SY8fJRS cr~middot OPDASIONAL ~UAfiOJrS bull bull bull bull bull bull bull bull 14

bull bull bull bull bull bull bull bull bull bull bull bullmiddot bull bull bull bull 36

Moat proilbullbull ui~ ln applied aathematlca lnYolva tolvtag au

equa~lon Ol WTitea ot eqQAtimiddotoDa lquatloaa _ be o~ J1J1LDV ttpaa bat

the tbaPlbullat u4 bulloat bQgtOnant la the linear equation If a aacl

b pP1teaeat lalom qaatit1bullbull aDd x Se the uabown thell u equshy

t1on vh1oh leducae to the foft u bull b 11 called a 11~ ~tlon

b1 one UDlmon AJIT equat)1on in oue UJllmo1m whiCh 1bull not llductlla

to th1a foa ia aid to bbull ~n-Uaeu lon-lhea~ equattona have amp

great D11at7 of folla and ~ genelallT lii10h ao cUtf1GUlt to aolbullbull

thaA llaeampl tnea Some laquoJAJ~Plee of non-Uneu- f1Cl1ampt1ona ~ c

(l) ~ - )r bull 0 bull

(2) ain c + bull -4a bull 0 bull

(3) Ji + 5 bull 0 bull

lqutloa (1) baa thl aoluUona (2) baa tnfbdteq JllllV aolushy

tlona aDd (3) bullbull no Jelut1ona 1t ve ar aeaklng a loot 41110IIg the

real uuaibull~bull~ The existence of a aolmtioa for a given eqwation 4ePtDda

to a large extent on the razage ot faluta which we allov the UllknoVD to

baTe lo uuapl$ U x io tequl~ted t o be an lntegd-1 theh the eqUAshy

tion x bull 5 at no aolutiQJlbull lt howne we penait fltaotloaal Tamplaaa

tor z then th1t tqUamption haa the ~ot x bull J bull Et l8 w14ot that

lf a aDd b are ~tlobal uaabaabull then ax bull ampawavbull baa a lolution

2

which is a rational number provided that a is not zero If a =0

no solution uiets unless b =0 and then every value of x eat1eshy

f1es the equation In thia ease we 8aJ that x is arbiiraa Thia

is not the aame si tuation as in emmple (2) above Equation (2) has

1nf1niteq many eolutions1 but these are a aeries of epeeif1c mlllbele

(2) 1a at1 eatisfied by all values of s bull

The baste number eystem which will be employed throaghout thie

paper is the complex number field A CoJIIPlex llUlllber is arrt ntlllbel of

the ttPe a + bl where a ampnd b ale r-eal numbers axd 1 bull J=i A Untar equation ax =b where a and b are complex llWilbere

bbaa the unique solution x abull unleaa a bull o As before if a bull o

then b mat be zero in older that a solution exist If this CoDd1bull

t1on is aat1ef1ed then x is arb1traJ7 Note that the solution

x o ~ ts generally a complex UllDiber

The cpmplg coGwmto of a+ bi te the number a - bi bull

bull bull bull bull bull bull bull bull bull bull bull bull bull

3

Only the simplest mathematical problems lead to a single e~tion

in one unknown More complex problema Will require the colution of a

set of equations in sever-al unknowns If all the equat ions are linear

in eaeh of the unknowns this set is called a linEar system of equashy

tions Sach a qstem mtq be wr1 tten in the following foltl

(21)

TJIie 1e a S7atem of m e~tions in n unknowns ~e unknowne are

bull bullbullbull bull ~ and the known quantitiea are the a1j and the 71bull he

systeJil (21) can be expreaud in a muoh shorter folm b1 using the

~tion notation

n t a1SXJ bull 71 (1 =1 bullbullbull m) bull

1-1

This ezpression meane precisely the same thing as (21) A

ao~utioA of (22) is 8Zf3 set of values tor bull b Btlch that all

m eqUationbull are satisfied simultaneouall Ve shall suppose that a1J

and are complex nWnbefs~ and that the eolut1on must be a set ofy-1

complex llWilbers

Another Ya7 of dealing with linear qatems is to use the concepts

of vectors and matrices We shall tieat these topics by maJdJlg the

following definitions (2 ppl~)

Defipitioq 2rl An n=dimepaiop complg yeq3ot is M ordered aet

ot n complex numbers w~itten (~bullbullbullbull bull ~) ~ x bull fuamp totality- of

such Yectora tor- a given n is called an n-dimeneional eomplexmiddot

vector apace

The nwnber is called the 1-th component of vectol x bullz1

The zero vector 0 211 (O bullbullbull 0) is the vector all ot whose comshy

ponentt are eeroe Note that the symbol 0 is used to represent the

EerG vector as well as the complex ll1llllbe1 eero

~fgampUon 22 The mam of tWfi vectors x =(1 bull ~) and

7 w (71 bullbullbull 711) is the veetot x + 1 bull (_ + r1 bull bullbullbull bull 2n + Tn) bull

The vqjysraquo ot a vector x 07 a eomplex nwilbel a is the vector

aJt bull xa bull (~bull bullbull bull bull axn)

Two vectors x and y are tmli if and onlyen if x 7 = o

A vector apace with the above p~perties ts called a (ineer IPMbullbull

veetora x and y is the complex nuiber

-where 7s 1e the complex conJugate ot 11 bull

The equations (22) are ea14 to define a lineMgt homogeneous tlanJJ

formation of the n-dimensional vector x into the m-dimensional

veeter 1

Defi~tjon 2J fhe Wiamp of the transfolmation defined by (22)

is the rectaJ~gU~ eJrta3

bull

If m bull n A is called a sgwe etrix of order n bull

We can now Write the aystem (22) 1n the foim

~s qmbolte notation meatus that A is thought of as something

which traneforms x into 3bull he sim1lar1tr betWe$Il 2 bull 3) and the

equation ax bull b is evident

ltttnttga ~middot5 fhe Jfill of two matrices A =(a1j) and J3 =(blJ)

ie the matr1 A+ B = (amp11 + b13) which t~anaforms tnetf x into

Ax+Bx

The mregU$ of a mtrbt by a complex nwnber a is the matrix

aA = (cra J) which transforms eve17 x into Go(Ax) bull1

In the rest of this chapter ve shall as~e that m =n bull In the

linear ayatelll (22) bull we can alWaJI suppoae m bull n by definiag a11 -= 0

and r1 bull 0 tor 1 gt m bull Timbull we ampball be concerned only with

n~tmensional bullectors and n-th order square matrices

6

for iJ =l bullbullbull n bull The te~o mat~1x transforms every vector x

into the tero vector

The 3mU majri is I = (amp1J) vhere l and 8ij = 0amp11 bull

for i r J bull fhe unit matrix carries x into 1tseli Ix =x for

every x bull

Dfipamption 2] fhb profluet omiddotf a matrix A and a vector x 1a

a vector In one case we have

(i bull l bullbullbull n) bullbull

where the quantity in parentheses is the 1-th component of the vector

Ax bull This defines pteciselr what is meant bf the tranoformetion

We may also have a product with the vector on the lett

xA bull ~ ~ x1a1J (J bull l bullbullbull n) bullt=l J Genelamplq xA I Ax bull

Detfn1t1Qn 2~sect The DtQdugt of matrix A = (a1j) by matrix

B bull (b13) is the matrix

u ~~1 bulln~J which transforms every x into A(k) bull

Dt(lA1~1oQ 2e9 If a unique matrix B exists such that

1

bull

-lAll bull BA bull I bull B is called the invttl of A and is written A bull

Detini~iQP ~lQ The matrir A 0 (aJ1) is called the tranampPi

of A (a1Jh A (aij) is the COJl1ggatt of A and r =i ie the

Hermitiy gopJyate of A bull

We are now prepared to diseues in some detail the solution of the

equation A bull 7bull Here there are three eases dependi~ on the

character of matrix A bull First 1 A has an inverse A-lbull then we

have

or

z bull A-ly bull

This gives the unique solution x which Mti$fiea the equation

Second if A bull o there is no tolution unless y = o and then evef7

vector x is a solution ~hese cases are like those for the simple

equation ax bull b bull But tor Ax bull 7 bull there is a third case since a

matrix 1JJBY not be zero and yet haTe no inverse atoh a matrix is said

to be s1Dgalar If A is a singular matrix a solution will exist

onlY if y aatisfies certain restrictions If these ~e satisfied

some of the are arbitrary The tollorillg theorem 82plains thi ilx1

completely (2 pp6-7)

mopiM 2~ The syetem Ax bull y has a solution if and only it

(z y) = 0 tor all vectors s ~ttch that sA J 0 bull Aey z satisshy

fying the equation zA =o can be witten as a linear combination ot

8

l dcertain linearly independent z a tor emmple B bullbullbull bull I bull

d ~ n bull Iiov 7 mst sat1ef7 the d linear homogeneous equations

(zk 7) bull o k = l bullbullbull d bull Exaotly d of the components ot x

can be chosen arb1traril)- and the remaining n - d will then be

detelmined fhe integer d 1a oaJled the yfeet of system Ax bull 1

and n d 1 s the ~ bull

fhe proof o-f this theorea wlll not be g1ven since it follows

veey eloeeq the proof of theorea 51 without the restriction

involving inverses

9

In definition 21 the idea of an ~imensional vector space

waa introduced Now we shall extend t he concept of bullapace First

we may consider veetQra with a denumerable infinity of components

the totality of such vectors is an 1nt1n1te-dimensional vector

space A further extension might be to vectors w1th a non-denumershy

able 1Dfin1ty of components Ve can treat such veetors as tnnct1ona

of a cont~ous variable a he totality of ~otions defined on a

c~rtain interval a ~ a ~ b 1bull called a gunction snacebull

In general an abstract linear space S consists of a set of

elements which have t he following properties

(l) It x and y aze elaenta of s the SWIl x + 1 ia

defined and is in S bull ih1a operation is associative

and commatative A zero element 0 exists in S 8lch

that x + 0 a x tor all x in S bull

(2) If x is in S and a is 811yen complex IIWnbe1t the

product ca bull ~ is defined and u is in S bull 1hia

operation has the property that lta= 0 if alld onl7 it

a bull 0 or x bull 0 bull or both bull

(3) If x and y belong to S then x = y if aJJd onlf

if X - yen a 0 bull

Another operat ion is often def ined tor abstract spaces hie

operation is called the inner product (x y) which is a complex

10

llWilbel aasoeiated with x and 7 bull ~he inner product must oatist)

the relation

(3~) (x y) ~ (y x)

where (y x) denotes the complex conjugate of (y z) bull

he ieer product for vector epaees was given in definition

23 bull For function epaCeJs the illller product is

b

(32) (x y) =Jr x(a) y(s) ds bull

a

Here it iB assumed tbat x(s) and y(a) are complex tunctions ot

a aiagle real variable o and all elements x and y belonging

to S are integrable on the interTal a ~ e ~ b bull

tn abstract spaces one utualq doee not define a product in

the oidinaq aenampe Tbat is we do not consider nml~iplieation in

which a product -q is aa element of the speC)ft In the next aecshy

tion we ahell howev-er consider transformations in abstract

spaces

u

A transformation in an abstract spaoe S ~elates to each

element x 1n S anothel element y in s Suoh a transformation

may be written tn the following notation

(41) (x y in S) bull

rhe aymbol At called an o-erato~ 1a used to reptteeent the

transfolmation We shall suppoee that A is a single-valued operatott

that 1a tor eaeh x h ta a unique element of S bull

Jltf1Nt1on lJ fhe am QJletttar O 1 s the operator which catrles

eveey- x in S into the zero element of s 0 =0 for every x bull

Phe iiMtiy QRmtgr I caniea eveey x into i tselt Ix bull z

tor all x in S bull

PefiJ1U1oa ~2 An operator L is said to middotne a4d1t1tt 1f

L(x + y) ~ Lx + ~ tor ill r and 1 1n S bull An addi t ive operator

which is continuotu is a ADetu Plttato The notion ot continuity

involves topological eoneepte which do not eoneern us here lfmiddote shal l

hereafter use the telfm lineat operator although we bave not adeshy

quatel7 defined it

Dtfiaition 41 ~he euro of two ope~tore A and B ia the

operator A + B WhiCh transforms z into Ax + llx bull be proQllQt

of two operators A and B ia the operator AB whieh can1ee x

12

into A(lb) bull

It can readily be shown that addition of oper-ators is aasooiat1ve

and commt1tative~ that multipl1catlon is associative and tbat multibull

plication is distributi~e over addition

Def1njlon 44 If a unique operator B exists ~~ that

AB bull BA bull 1 bull it is ealled the invelie of A and is written B = Abulll

Jor a linear operator L the invnse lt it exists will be

denoted by M bull lt is asSWDed that the 1nTersee of all Un-r operashy

tors considered here ue s1Dgle bull valued and linear

De11nit12A 45 The Rumttiap gon1mate of a linear operator L

is denoted by V and is defined by the lelation (Lx 1) bull (x Iq) bull

which mat be eati sfied for all x and 1 in S

In a Tector apace linear Qperatore are matrices For function

epacbulla operators are represented by integrals

b

(42) f I(a t) x(t) dt r(a) (a~ 1 ~b)

a

he Hermitian conjUgate of an integral operator ia

b 11

J x J I (t a) x(t) dt (amp ~ bull ~ Igt) bull

a

The operation xL may be defined

13

b ( ltiLbull f x(t) L(t e) dt (a o a c b) bull

In abstract epace we mq consider Lx bull 3 to be a linear

equation in Vh1ch L and - are known and x is to be found If

L bas an inverse M the equation haa the unique solution x bull M7 bull

li middot L bull 0 bull the equation has nomiddot solution unless y =0 and then

eveey x in S ts a solution However bull 1 might not be seo and

stUl heTe DO inverse It seems ltkeq that heolem 21 would a~

here as 1t does for vector epacea Aa atated for abstract apacebullbullmiddot it

is written in the followtng form

poundliiQljlf 21bull The equation Lx bull 7 baa a solution 1f and oal7 it

(s 7) bull 0 tor all s suoh thet 1L = o bull

A theor81l aimilar to this baa been proved for iltegral equations

of Fredholm type ami secolld kind (1 ppl01-1oq) We shall conaicler

onlyen those llneampl opemto~a vh1oh aaUampf7 this theolem bull

SYSmtS OF OPERATIOIAL E(tT1MIONS

Now we shall be concerned with problems which involve more than

one apace 111rst suppose we haTe two spaces s and s ~ Then the1 2

equation L =~ (_ in ~ bull 22 1n s2) indicates a transforshy

mation which carries each element of into an elampment of bull Wbulls1 s2

say that operator L ~ $pace s ~ apace s2 bull1

he detinitiona of section 4 appq in an obViOls way to the

present situation and will not be restated- Note that the identitshy

operatoll X elw~e maps a epace into itself It L bas an inverse

M 1t is an operator which mapa spa()e into bull In this cases2 s1

the eolut1on of L1]_ =~ ie bull M~ bull

Suppoee we have two aets of linear spaceamp ~-bull lb and

t 1bull bull bull bull Ibull bull m ~ n bull Oona1der elements in Xs and YJ 1n rJ bull x1

Let LJi be a lin~ operator which maps apace into space TJ bullX1

If tJi haD an inverse_ t t is written MJl and is a ltneat opettator

which maps YJ into x bull lle contider the linear 81Btem of operashy1

tional equations

~hibull qstem is aa1d to have a solution for given Ljl and 7J it amp

set of elements liJbull bullbull ~ exists wch that all m equationbull

are satiatield s1mltaneousq By defining = 0 and 7J bull 0 tormiddotL31 J gt bull ve TlliJyen suppose m bull n bull It 1s asawned that th1s has been done

Jor convenience we shall det1ne vectors and matrices fotJ

15

operational e~tiona

Definition 51 The vectgr x bull x1 = 1)_9 bull ~J is an

n-dbaenaional vector whose 1-th component is an element of apace

Det1n1tiqn 52 the uzg ttctQpound ie 0 = 0 o 0 bull

whose 1-th component is the selo element of the 1-th space

The inner product (x 7) will not be defined because product 1

of the tTPe are undefinedx171

Dtf1nit1on 513 The matriJ L e (Lji) is a matrix whose composhy

nents are linear operators Hereafter the ~bol L without anbshy

aor1pte will be used to denote this matrix

Def1n1 t ion 5)t rhe ptoduct (Jf matrix L and veetott x is the

following vector

Th1e is a veeto~ whose jbullth component 1a an element of space YJ bull

lilv1dentl3 qstem (51) 1s equ1nlent to the equation Lz bull 1 bull

Dtf1ait1on 5~ Themiddot yector OJorator zP 1a

zP = zP bull zP1 1 bull bull z where ~ 1s an operator which mampps

16

Dfapitlcm 5bull6 The prgduct of vector operator zP ond matrix

L 1a the vector

Zp L bull ~ zP xmiddot j jiJ-1

whose 1-th component is an operator which maps spaee x into YP bull1

Dttinitipn 57 he inner Rroduct of vector operator zP with

v ector y is theelement of apace YP defined by

( zP bull Tgt = zP ~

where 7j is the oomp1ez eon~ate of yJ bull

It seema as if Theorem 21t

lhould be valid for wretem (51) bull

Unfortunate]$ the attempt to prove this has not been succeasft11 A

more specialised theorem will be proved in this section vhlle secshy

tiou 6 is deToted to a diacaaaion of the general ease

he statement of the special theorem is rather involved Jor

this reason we shall prove it before stating it fhe problea 1s to

determine aome oondit1ons on y for the ex1 etence of a solution to

In the firet place if all the L31 bull o bull (1 1 bull 1 bullbullbullbull n)

then obviously there is no solution unleae y =0 bull Another way ot

saying this is that y mst eatisfy the n linear homogeneOus

equations 71 bull o (j aa 1 bullbullbull n) If this condition is tulfillecl

17

then Xi ~ are arbitra17 We Btq that the system Lx bull 7

has detect d bull n in this case

It my happen that non$ of tbe LJi bae an inverse and yet they

are not all cero This is the case for which the theorem baa not been

proved In the proof Which followa this case will be ignoJed We

shall assume at eaoh etep that either all the operators under conshy

a1delampt1on a~e sero or that at 1eaet one ol them bns en illTerae

If one of the Ljl has an invuaa renumbet so that 1t is ~l bull

The tlrat ~tion in eystem (51) ta

From this we get

n

2) bull Mu T1 - i~ Mu_ t_i _

where Mu is the inverse ot Lu bull If this is 81lbat1tutampd into the other n-1 equations the re8llt

is

hia reduceamp to

St ) bull L L K L (1 J =2 bullbullbull bull n) 51 31 fl ~ Jl ~1

(J =2 bullbullbull n) bull

Now (53) becomes

(54) (J ~ 2 bullbullbull n) bull

sratem (54) 1amp very aimilat to the original qstem (51) bull 1

If all tji = 0 (1 J bull 2 bullbullbull n) the n-1 linear homobull

geneous equations

mast be aatiet1ed in order that qatem (51) bsve a solution Retbull

bull bull bull xn ean be chosen arbitrari]yen and the defect c1 bull n - 1 bullx2 1 1

If one of t he LJi SAY has an inverse thenL22

(J =J bullbullbull n) bull

Let

Y~ ~ ~~ bull T2 ~ 112 ( ~ a 3 a)t~ bull J4fl ~ tJ

Sttem (57) becomebull

1

19

(J =3 bullbullbull bull n) bull

Not-e that ~ is a 11llear homogeneua function of y~ ud

bull which in tmn are lineal homogeneoue functions of the originaly2

Ipound one of the Ilji bas on inverse contime this process for

llow it all L~1 bull 0 the n-2 equations y~ c o must be

eamptisfied in oder that (51) have a solution Rere = bullbullbull x11

ue arbitrer- and d = n-a bull 2 we

n stepa or until a ayatem is obtained where eveq operatoi is sero

Iiecall tbat we are asswninc at each step that all operators ample eero or

that one of them has an inverse

After k steps we have

(59)

and

(510) (J =k + l bullbullbull n)

where

()U)

k k-1 Bote that 7j 1s a llJ1eampl homogeneous function of the 7J bull

bull bull bull bull bull bull

kl-2which in turn are linear homogeneous tmctions cpound the yJ etc

ao that ~ te auch a tuncUon of the original YJ bull k

Ae befor if all LJi bull 0 we have the n-k equations

~ bull 0 (J bull lt + 1 bullbullbull n) whose aat1efaot1on is neeessarr and

sufficient 1n orde~ that system (51) have a solution In this case

~1 bull bull bull xJl are arb1tlampr7 and the defect d =n - k bull

It we an able te continue for nbulll ateps we have

bull

n-l n-1 If Lnn =o y must sat1sfy the one eltllation ~n -= O a

linear homogeneous equstion in the y j bull Also llh nuv be chosen

atb1trar1ly and we have 4 =n - l bull D-1

It = has an inverse then

bull

(513) bull bull bull bull bull bull

Jl

~ bull ~l 71 1~ Mu Lt bull

21

~s case 1s equivalent to d = 0 since ther-e are no equations

which y must satisfy and none of the ate arbitraq 1

We have shown that y mat satisfy a certain system of d

linear homogeneous eqUIltiona in order that Lx =y have a solution

Now we want to prove that these equations have the form (zPbull f) bull o

(p bull n - 4 + 1 bullbullbull n) 1 whe~e the zP are solutions of the homoshy

geneous system zPL =0 bull

lfeUnitign 58 fhe gsectltf QPttator is defined aa follows

11 bull Ipw where IP is the identity operator of space YP bull

If J 1 p 11e the so~o ope ator whteh maps space YJ 1nto the

sergto element of rp bull Consider the system zPL bull 0 bull his may be Written as

n (514) ~ z~ LJl =o (1 bull 1 bull bull bull bull n ) bull

Jl

a ayatem of equations cons1at1Damp eolel)r of operators A solution ot

(514) 1s a aet of op~tora ~~ (J bull 1bullbullbull bull bull n) Which make each of

the n left members e~ to the sero operator

If eve17 LJi =0 bull zP ia arb1traey Choose

zP bull A~ bull ( p bull 1 bull bull n) bull

Nov n n t z~ 7J bull t A~ 7J yP (p =1 bull bull bull a) bull

J-1 J-1

his is the case where d bull n and the n equations which 7

mnat sstiaf1 are

(516) (zP 7gt bull T bull o (p ~ 1 bullbullbull n) p

as vas ahown before

If Lu_ has en inverse Mu we have

(518)

System (518) can be writtmiddotea

(519) (1 bull 2 bullbullbull n)

1vlth ~Jt aa before l p J)

If all Ljtbull 0 z2 bull bull bull ~ are arbltrary bull

Choose

zP r tzi ~ ~ A 2(p r n) bull

Then

his is ~he caae a = n bull l and we tind that the nl equations

23

(zP i) bull 7P Lpl 1 r 1 bull 0 (p bull 2 bullbull bull bull n) are the same

amp8 (55)

If one o the ljt aq 1

L22 baa an inverse then

bull bullbull bull n) bull

_~f eTeq Lit =- 0 bull then Z~ bull bull bull Z a1amp arbitraey

Cho~bullbull

80 tbat

and

llow

Jtter k steps we have

oP - zP (-yr-Jrl -r-nl) ( 1 )~ - ~ 4 -~ ~ r = bull bullbullbull bull k bull j=tl ~

2lf

and n

(526) t zP Lr J-r+l j 11 o (1 = r + l bullbull bull bull n) bull

1bull1 __1Let Hi bull Lji Mit bull fh1e 1s an operator whieh mapa space

Yi into YJ bull With this notation we can write (525) in the form

(527) zP = Jl E zP xJ (r = 1 bullbullbull k) bullr r1-11 -

If all L~t bull 0 then z1+1 bull bull bull z 1181 be chosen amprbitiamprU-

Define

(528) zP = tzi z~ bull bull ~middot t~( J bull k + 1 bullbullbull n)

( p =k + 1 bullbullbull bull n) bull

We llIBt evaluate zibull bullbullbull bull ~~ uabg (527) bull

n n (529) ~- t zPntbull t AP~ = llll

1k+l J 1-k+l J -1t 11

n (50) ~l =k z~ ~l bull zt a_l + ~1 = ~ ~1 + ~1 bull

ln order to eontime with this process it is necessaey to use

Jfa much more condensed notation Let ue define the symbol 1

the following recursion formula

) ok -Dk uJ+l -Dk J+2 _nk j+a(531 ~ bull trJ+l ~j + ~+2 HJ + bullbullbull + Bj+a HJ + bullbullbull

middotlf~middot~


bull bull bull bullbullbull

25

Tlma ~k is defined 1n terms of H~ (e =1 bullbullbullbull k-j) and

Rj (a bull J+l bullbullbull k p) Note that nf is an operator which maps

space yJ into yp bull ~0 give a bette~ idea as to jllst what the ak

ar-e the first threamp of them are written out ill full below

532)

With thia notat ion wamp have

Jl

z~ bull ~ z~ ttJ 8- J-k-a+l Iii ~

We shall use indnetion to prove that zt- = tor8 ~8

(a bull o 1 bullbullbull k-1) Aswme that it is trne tor s bull 0 1 bullbullbullbull t-1

where t lt k bull hen

26

1 zP bull t zP uJ lt-t Jak-t+l J ~t

JJPk vK-t+l pk ~t+2 rP= oJbull lc-t+l ~t + 1t-t+2 ~ + bullbullbull + J ~t + bullbullbull

middotrf~t+~t

iJkJe-t bull

~1e oompletbulls the induotion We ha-e proved thst

(533) ~ =~~ (a= o 1 bullbullbull bull ~1) bull

Renee

3t) (zP -) tPk -aPk utgtk (5middot bull bull J ~1 + 2 72 + bullbullbull + ~ k + 1p bull 0 bull

wh ere (p =k+l bullbullbull bull n) bull We want to show t bat equat t ons (534) are

the same as ~ =o which we had beto~e The r~eion relation for

t he tar 8 a-1 r 1 ( bull )(535) Y~ bull 1r + t1 I bull =12 bullbullbull k 1 =1 bullbull a bull8 1

How

It we aPPl7 the recutta1on relation aaan we get

bull k-3 + uPk ~3 + ~k 3 + uPk ~3 kl-7p ~ k -JIo-1 bullJt-l ~ 7k-2 bull

Jaeuae lor 1n4uct1on that

Then by (535) we have

+ uk-e+l bullz- 1 ) raquo-bull 7amp-a

+ lt( + ~ ~ + bullbullbull + ~1 u + middot~~middot +

+ ~1~+1gt ~=-1

The induction is complete Nov for bull bull k in (536) we have

~ bull Tp + ~k 7k + ~l Yk-1 + bull bull bull + ~J Ytt-j + bull bull bull + ~k 11 = O

(p bull k + 1 bullbullbull n) which la uactl7 the same system of equations

as (zP bull 7) bull o This ie the cas where d = n bull k bull

fbaa we have proved that Lx bull 7 baa a solution it and only it

(zP ~ = o (p =k + 1 bullbullbullbull n) where the zP are as defined

above

Let Z be rmy solution of ZL bull o bull Suppose (z 7gt bull 0 for all

such z hen certainly (zP f) bull o (p bull k + 1 bullbull n) bull which

implies that Lx bull 3 has a aolumiddotUon Converseq if Lx 7 baa a

solution x (z y) == (Z1 t i) bull (ZL i) bull o tor all z such that

Bde completes the proof of the fo1loW1ng theorem

WOBJiM 51 fhe system Lx bull 7 has a solution if and only 1t

(z f) bull 0 for aJl Z SllCh that ZL bull o provided that at each step

1n the elimlnation process either all the operators are zero or at

least one of them has an inverse

The tlteorem is probablr atill true without the restriction on

the existence of inverses e lhall consider in the next section

some examples of the 11 inbullbetween bull ca e

We lhall aolve two taaplea 1a wbloh x ad y atbull twOUa-

~ (o o) ~ (

Q 0)l-1 0 00

(1a) (~) (2 1 ) (xi) (ti)00 ~ + QO 4 bull ~

(61)

( oo (~) + (o o) (~) bull ( ri)-1 o) ~ o o ~ r~

fhia 1a a quem wheremiddot none ot the IJl hampve an invers~ iUl4 a111 tber

ue not aU hro fheoteua 5middot1 uat not include this bullbullmiddot fU qatea

(61) racsa to

JO

his is equivalent to

0

0 bull ~

xi bull ~~ liov xi ls un1qoely determined b7 the foutgtth ot these equations

Then either xi or ~ may be chosen arbitrarily and the othe~ will

be detelm1ned Also we must haTe 7~ c 7i = 0 if a solution is

to exist

Thus the neces~ and autficient concl1t1on for the enstenoe of

a eolution to (6 1) is

(64)

If this is eattsfied the solution is

l

~ bull ~

1 tlampl~ bull (65)

ubUfat7~ bull

~ l 2 2 ~ bull i + 7e bull 2xi bull

Jet ua conatdw i1Pt the eretea IL bull o bull Thia 1a equSyalat o

(66)

Ol t

(67) middot + 12 L 2l bull oz1 ~1

11~2 + 12lt22 0

~ ~)(68) Ia bull ( bull z~ z~ middot

SJnem 67 1bull tma

z11 ~ 2 (00) bull0( sh zta)( ) +

~ bull10z2z zl 21 22 21 22

(6 9)

c2(~ ~)( ) l ( )0~

fhlbull teducu to

bullO

610)

23middot11 ) 0 0)+ bull 0 1a

( 0 0

zi1bullzlnbull 0 bull

zia bull ~ bull o

~~ bull zl-22 bull zfl t ~~ bullbull eJ~b1i18l7 bull

Z bull ( ~ G) bull2 z21 o

3)

(middot 7) bull z1 71 + z2 72

-( h) ( 1) middot(~ ) (i)~~ ~ -z2 21

(r ~) (~ r)bull X~ ~ +

~

l7a f bull+middot(~) (~)

1t (a f) ta to equal bullero tor- tll z such that ZL 111 o bull

tha nidtDtyen we mllampt baTe ~ bull f bull o bull Convet-aeJ7 it

~ bull ~ bull 0 then IL bull 0 tor all Z nch thU ZL bull 0 bull

hibull 1a thlla tabull aame ooUt1on aa (64) bull

hnobull oample 61 aat1bullt1ebull theorbull 1 It ampppeua YflrT lstcel7

that the theorem would bOld tof all qeteu where the are t1n1te~1 ma111eea and the ~ and 7J ate ttnUbull d1aena1onal ~ecto~a Whbullther

it would be ttue tor othai ayatbullbull ia an interesting qlleamption

JiltHfH 62 tb1a xaaple 11 oae in vh1oh all the LJi Ah

ei1Dgllampl ret the fbulltem has a U1que aol11tJon Let

~ ( ) Lla (~ ~)

~ -(~ ) L22 ( )

Z aDd 1 bullbull aa 1n eturple 61 bull

fhe qate Lx bull 7 tedncea to

ext+xi+~+~- ti ~~+~+~+~bull T~

(614)

bull

2 1 1 1

2 1 1 bull 4 bull l 0 0 0

-2 0 3 -1

lroa thla we knov that qaea (614) baa a unique solution for

arb1~laq y bull llhfa 1amp ttlae efemiddotn though none ot the L1~ baa an

35

It ve oomp11to ZL aa 1~ thbull preoediDg emmple we find that

tot thia oaae Z 11101t be bullbullto in older that U bull 0 ~ fhen

eertainlyen (Z i) bull o tot arbl ttaq y aDd v find that theorebull

51 appUta to exaaple 62 bull

2hia wcgeata the tollovlnc corollat7 to theorem 51

M2HatRX 6 eJ 1ho qnem Lx bull 7 haa a aolatloa tffl amptbitlalT

7 1t aDd only ~ ZL tr 9 1Q11bulla that Z ~ 0 bull

Suppose thet Xlaquo bull r haa soluUon toI ubitrar 1 bull Let

Z be aq notor BUch that U bull 0 bull then z y) bull o by taeorbull

51 tor arbitramplT 7 bull lbt thia 1apl1es I bull 0 bull

Now npoae that ZL bull o illpUea Z bull o hen (Ibull f) bull 0 tor

arblttU7 7 and ~ z bullt1atria6 ZL bull o Then b1 theora 51

b bull 7 baa a aolution for ar-bltrampl7 7 bull

bia cotollary 111 of eourbullbullbull tuOJeot to the aame restriction

aa theorbull 51 bull

BDlLIOGRAPBY

l Courant R and D Hilbert Methoden der MathemaUechen Physik 2d ed Vol l Berlin Julius Springer

1931 Q69P

2 opicte in Hilbert spaee theoey Notes for graduate seminar Corvallis Oregon State College l95Q-51 (Mimeographed)

TPIampOID


rTfrx sf httmtlr I3 0bntlt of lbJor

Redacted for PrivacyErrO u( ernou-ampmt oil lhtbmttcr


Srfmt o $campool 0mampEtr CfiElttr


ru tf Smihrtl Schpal

htr tlmlr tr trd tr thCnrt Sahillft



























2
















3






(21)




~tion notation


1-1





complex llWilbers







vector apace















veeter 1



bull








Ax+Bx







6





every x bull












1

bull








have

or

z bull A-ly bull













8









involving inverses

9




















if X - yen a 0 bull



10


the relation

(3~) (x y) ~ (y x)




b


a








spaces

u











tor all x in S bull






quatel7 defined it




12

into A(lb) bull














b

(42) f I(a t) x(t) dt r(a) (a~ 1 ~b)

a


b 11


a


13































tional equations





15

operational e~tiona











following vector





16


L 1a the vector



















17










From this we get

n



is

hia reduceamp to



Now (53) becomes




geneous equations





Let



1

19












(59)

and


where

()U)










bull





bull


Jl


21













Jl







J-1 J-1


mnat sstiaf1 are


as vas ahown before


(518)





Choose


Then


23


amp8 (55)





Cho~bullbull

80 tbat

and

llow



2lf

and n






Define





1k+l J 1-k+l J -1t 11






middotlf~middot~



25





532)


Jl





26



middotrf~t+~t

iJkJe-t bull



Renee





How







+ ~1~+1gt ~=-1







above














~ (o o) ~ (

Q 0)l-1 0 00

(1a) (~) (2 1 ) (xi) (ti)00 ~ + QO 4 bull ~

(61)

( oo (~) + (o o) (~) bull ( ri)-1 o) ~ o o ~ r~



(61) racsa to

JO


0

0 bull ~




to exist



(64)


l

~ bull ~

1 tlampl~ bull (65)

ubUfat7~ bull



(66)

Ol t


11~2 + 12lt22 0


SJnem 67 1bull tma

z11 ~ 2 (00) bull0( sh zta)( ) +

~ bull10z2z zl 21 22 21 22

(6 9)

c2(~ ~)( ) l ( )0~

fhlbull teducu to

bullO

610)

23middot11 ) 0 0)+ bull 0 1a

( 0 0


zia bull ~ bull o



3)


-( h) ( 1) middot(~ ) (i)~~ ~ -z2 21

(r ~) (~ r)bull X~ ~ +

~











~ ( ) Lla (~ ~)

~ -(~ ) L22 ( )




(614)

bull

2 1 1 1

2 1 1 bull 4 bull l 0 0 0

-2 0 3 -1



35
















BDlLIOGRAPBY


1931 Q69P




























2
















3






(21)




~tion notation


1-1





complex llWilbers







vector apace















veeter 1



bull








Ax+Bx







6





every x bull












1

bull








have

or

z bull A-ly bull













8









involving inverses

9




















if X - yen a 0 bull



10


the relation

(3~) (x y) ~ (y x)




b


a








spaces

u











tor all x in S bull






quatel7 defined it




12

into A(lb) bull














b

(42) f I(a t) x(t) dt r(a) (a~ 1 ~b)

a


b 11


a


13































tional equations





15

operational e~tiona











following vector





16


L 1a the vector



















17










From this we get

n



is

hia reduceamp to



Now (53) becomes




geneous equations





Let



1

19












(59)

and


where

()U)










bull





bull


Jl


21













Jl







J-1 J-1


mnat sstiaf1 are


as vas ahown before


(518)





Choose


Then


23


amp8 (55)





Cho~bullbull

80 tbat

and

llow



2lf

and n






Define





1k+l J 1-k+l J -1t 11






middotlf~middot~



25





532)


Jl





26



middotrf~t+~t

iJkJe-t bull



Renee





How







+ ~1~+1gt ~=-1







above














~ (o o) ~ (

Q 0)l-1 0 00

(1a) (~) (2 1 ) (xi) (ti)00 ~ + QO 4 bull ~

(61)

( oo (~) + (o o) (~) bull ( ri)-1 o) ~ o o ~ r~



(61) racsa to

JO


0

0 bull ~




to exist



(64)


l

~ bull ~

1 tlampl~ bull (65)

ubUfat7~ bull



(66)

Ol t


11~2 + 12lt22 0


SJnem 67 1bull tma

z11 ~ 2 (00) bull0( sh zta)( ) +

~ bull10z2z zl 21 22 21 22

(6 9)

c2(~ ~)( ) l ( )0~

fhlbull teducu to

bullO

610)

23middot11 ) 0 0)+ bull 0 1a

( 0 0


zia bull ~ bull o



3)


-( h) ( 1) middot(~ ) (i)~~ ~ -z2 21

(r ~) (~ r)bull X~ ~ +

~











~ ( ) Lla (~ ~)

~ -(~ ) L22 ( )




(614)

bull

2 1 1 1

2 1 1 bull 4 bull l 0 0 0

-2 0 3 -1



35
















BDlLIOGRAPBY


1931 Q69P



3






(21)




~tion notation


1-1





complex llWilbers







vector apace















veeter 1



bull








Ax+Bx







6





every x bull












1

bull








have

or

z bull A-ly bull













8









involving inverses

9




















if X - yen a 0 bull



10


the relation

(3~) (x y) ~ (y x)




b


a








spaces

u











tor all x in S bull






quatel7 defined it




12

into A(lb) bull














b

(42) f I(a t) x(t) dt r(a) (a~ 1 ~b)

a


b 11


a


13































tional equations





15

operational e~tiona











following vector





16


L 1a the vector



















17










From this we get

n



is

hia reduceamp to



Now (53) becomes




geneous equations





Let



1

19












(59)

and


where

()U)










bull





bull


Jl


21













Jl







J-1 J-1


mnat sstiaf1 are


as vas ahown before


(518)





Choose


Then


23


amp8 (55)





Cho~bullbull

80 tbat

and

llow



2lf

and n






Define





1k+l J 1-k+l J -1t 11






middotlf~middot~



25





532)


Jl





26



middotrf~t+~t

iJkJe-t bull



Renee





How







+ ~1~+1gt ~=-1







above














~ (o o) ~ (

Q 0)l-1 0 00

(1a) (~) (2 1 ) (xi) (ti)00 ~ + QO 4 bull ~

(61)

( oo (~) + (o o) (~) bull ( ri)-1 o) ~ o o ~ r~



(61) racsa to

JO


0

0 bull ~




to exist



(64)


l

~ bull ~

1 tlampl~ bull (65)

ubUfat7~ bull



(66)

Ol t


11~2 + 12lt22 0


SJnem 67 1bull tma

z11 ~ 2 (00) bull0( sh zta)( ) +

~ bull10z2z zl 21 22 21 22

(6 9)

c2(~ ~)( ) l ( )0~

fhlbull teducu to

bullO

610)

23middot11 ) 0 0)+ bull 0 1a

( 0 0


zia bull ~ bull o



3)


-( h) ( 1) middot(~ ) (i)~~ ~ -z2 21

(r ~) (~ r)bull X~ ~ +

~











~ ( ) Lla (~ ~)

~ -(~ ) L22 ( )




(614)

bull

2 1 1 1

2 1 1 bull 4 bull l 0 0 0

-2 0 3 -1



35
















BDlLIOGRAPBY


1931 Q69P







vector apace















veeter 1



bull








Ax+Bx







6





every x bull












1

bull








have

or

z bull A-ly bull













8









involving inverses

9




















if X - yen a 0 bull



10


the relation

(3~) (x y) ~ (y x)




b


a








spaces

u











tor all x in S bull






quatel7 defined it




12

into A(lb) bull














b

(42) f I(a t) x(t) dt r(a) (a~ 1 ~b)

a


b 11


a


13































tional equations





15

operational e~tiona











following vector





16


L 1a the vector



















17










From this we get

n



is

hia reduceamp to



Now (53) becomes




geneous equations





Let



1

19












(59)

and


where

()U)










bull





bull


Jl


21













Jl







J-1 J-1


mnat sstiaf1 are


as vas ahown before


(518)





Choose


Then


23


amp8 (55)





Cho~bullbull

80 tbat

and

llow



2lf

and n






Define





1k+l J 1-k+l J -1t 11






middotlf~middot~



25





532)


Jl





26



middotrf~t+~t

iJkJe-t bull



Renee





How







+ ~1~+1gt ~=-1







above














~ (o o) ~ (

Q 0)l-1 0 00

(1a) (~) (2 1 ) (xi) (ti)00 ~ + QO 4 bull ~

(61)

( oo (~) + (o o) (~) bull ( ri)-1 o) ~ o o ~ r~



(61) racsa to

JO


0

0 bull ~




to exist



(64)


l

~ bull ~

1 tlampl~ bull (65)

ubUfat7~ bull



(66)

Ol t


11~2 + 12lt22 0


SJnem 67 1bull tma

z11 ~ 2 (00) bull0( sh zta)( ) +

~ bull10z2z zl 21 22 21 22

(6 9)

c2(~ ~)( ) l ( )0~

fhlbull teducu to

bullO

610)

23middot11 ) 0 0)+ bull 0 1a

( 0 0


zia bull ~ bull o



3)


-( h) ( 1) middot(~ ) (i)~~ ~ -z2 21

(r ~) (~ r)bull X~ ~ +

~











~ ( ) Lla (~ ~)

~ -(~ ) L22 ( )




(614)

bull

2 1 1 1

2 1 1 bull 4 bull l 0 0 0

-2 0 3 -1



35
















BDlLIOGRAPBY


1931 Q69P


veeter 1



bull








Ax+Bx







6





every x bull












1

bull








have

or

z bull A-ly bull













8









involving inverses

9




















if X - yen a 0 bull



10


the relation

(3~) (x y) ~ (y x)




b


a








spaces

u











tor all x in S bull






quatel7 defined it




12

into A(lb) bull














b

(42) f I(a t) x(t) dt r(a) (a~ 1 ~b)

a


b 11


a


13































tional equations





15

operational e~tiona











following vector





16


L 1a the vector



















17










From this we get

n



is

hia reduceamp to



Now (53) becomes




geneous equations





Let



1

19












(59)

and


where

()U)










bull





bull


Jl


21













Jl







J-1 J-1


mnat sstiaf1 are


as vas ahown before


(518)





Choose


Then


23


amp8 (55)





Cho~bullbull

80 tbat

and

llow



2lf

and n






Define





1k+l J 1-k+l J -1t 11






middotlf~middot~



25





532)


Jl





26



middotrf~t+~t

iJkJe-t bull



Renee





How







+ ~1~+1gt ~=-1







above














~ (o o) ~ (

Q 0)l-1 0 00

(1a) (~) (2 1 ) (xi) (ti)00 ~ + QO 4 bull ~

(61)

( oo (~) + (o o) (~) bull ( ri)-1 o) ~ o o ~ r~



(61) racsa to

JO


0

0 bull ~




to exist



(64)


l

~ bull ~

1 tlampl~ bull (65)

ubUfat7~ bull



(66)

Ol t


11~2 + 12lt22 0


SJnem 67 1bull tma

z11 ~ 2 (00) bull0( sh zta)( ) +

~ bull10z2z zl 21 22 21 22

(6 9)

c2(~ ~)( ) l ( )0~

fhlbull teducu to

bullO

610)

23middot11 ) 0 0)+ bull 0 1a

( 0 0


zia bull ~ bull o



3)


-( h) ( 1) middot(~ ) (i)~~ ~ -z2 21

(r ~) (~ r)bull X~ ~ +

~











~ ( ) Lla (~ ~)

~ -(~ ) L22 ( )




(614)

bull

2 1 1 1

2 1 1 bull 4 bull l 0 0 0

-2 0 3 -1



35
















BDlLIOGRAPBY


1931 Q69P


6





every x bull












1

bull








have

or

z bull A-ly bull













8









involving inverses

9




















if X - yen a 0 bull



10


the relation

(3~) (x y) ~ (y x)




b


a








spaces

u











tor all x in S bull






quatel7 defined it




12

into A(lb) bull














b

(42) f I(a t) x(t) dt r(a) (a~ 1 ~b)

a


b 11


a


13































tional equations





15

operational e~tiona











following vector





16


L 1a the vector



















17










From this we get

n



is

hia reduceamp to



Now (53) becomes




geneous equations





Let



1

19












(59)

and


where

()U)










bull





bull


Jl


21













Jl







J-1 J-1


mnat sstiaf1 are


as vas ahown before


(518)





Choose


Then


23


amp8 (55)





Cho~bullbull

80 tbat

and

llow



2lf

and n






Define





1k+l J 1-k+l J -1t 11






middotlf~middot~



25





532)


Jl





26



middotrf~t+~t

iJkJe-t bull



Renee





How







+ ~1~+1gt ~=-1







above














~ (o o) ~ (

Q 0)l-1 0 00

(1a) (~) (2 1 ) (xi) (ti)00 ~ + QO 4 bull ~

(61)

( oo (~) + (o o) (~) bull ( ri)-1 o) ~ o o ~ r~



(61) racsa to

JO


0

0 bull ~




to exist



(64)


l

~ bull ~

1 tlampl~ bull (65)

ubUfat7~ bull



(66)

Ol t


11~2 + 12lt22 0


SJnem 67 1bull tma

z11 ~ 2 (00) bull0( sh zta)( ) +

~ bull10z2z zl 21 22 21 22

(6 9)

c2(~ ~)( ) l ( )0~

fhlbull teducu to

bullO

610)

23middot11 ) 0 0)+ bull 0 1a

( 0 0


zia bull ~ bull o



3)


-( h) ( 1) middot(~ ) (i)~~ ~ -z2 21

(r ~) (~ r)bull X~ ~ +

~











~ ( ) Lla (~ ~)

~ -(~ ) L22 ( )




(614)

bull

2 1 1 1

2 1 1 bull 4 bull l 0 0 0

-2 0 3 -1



35
















BDlLIOGRAPBY


1931 Q69P


1

bull








have

or

z bull A-ly bull













8









involving inverses

9




















if X - yen a 0 bull



10


the relation

(3~) (x y) ~ (y x)




b


a








spaces

u











tor all x in S bull






quatel7 defined it




12

into A(lb) bull














b

(42) f I(a t) x(t) dt r(a) (a~ 1 ~b)

a


b 11


a


13































tional equations





15

operational e~tiona











following vector





16


L 1a the vector



















17










From this we get

n



is

hia reduceamp to



Now (53) becomes




geneous equations





Let



1

19












(59)

and


where

()U)










bull





bull


Jl


21













Jl







J-1 J-1


mnat sstiaf1 are


as vas ahown before


(518)





Choose


Then


23


amp8 (55)





Cho~bullbull

80 tbat

and

llow



2lf

and n






Define





1k+l J 1-k+l J -1t 11






middotlf~middot~



25





532)


Jl





26



middotrf~t+~t

iJkJe-t bull



Renee





How







+ ~1~+1gt ~=-1







above














~ (o o) ~ (

Q 0)l-1 0 00

(1a) (~) (2 1 ) (xi) (ti)00 ~ + QO 4 bull ~

(61)

( oo (~) + (o o) (~) bull ( ri)-1 o) ~ o o ~ r~



(61) racsa to

JO


0

0 bull ~




to exist



(64)


l

~ bull ~

1 tlampl~ bull (65)

ubUfat7~ bull



(66)

Ol t


11~2 + 12lt22 0


SJnem 67 1bull tma

z11 ~ 2 (00) bull0( sh zta)( ) +

~ bull10z2z zl 21 22 21 22

(6 9)

c2(~ ~)( ) l ( )0~

fhlbull teducu to

bullO

610)

23middot11 ) 0 0)+ bull 0 1a

( 0 0


zia bull ~ bull o



3)


-( h) ( 1) middot(~ ) (i)~~ ~ -z2 21

(r ~) (~ r)bull X~ ~ +

~











~ ( ) Lla (~ ~)

~ -(~ ) L22 ( )




(614)

bull

2 1 1 1

2 1 1 bull 4 bull l 0 0 0

-2 0 3 -1



35
















BDlLIOGRAPBY


1931 Q69P


8









involving inverses

9




















if X - yen a 0 bull



10


the relation

(3~) (x y) ~ (y x)




b


a








spaces

u











tor all x in S bull






quatel7 defined it




12

into A(lb) bull














b

(42) f I(a t) x(t) dt r(a) (a~ 1 ~b)

a


b 11


a


13































tional equations





15

operational e~tiona











following vector





16


L 1a the vector



















17










From this we get

n



is

hia reduceamp to



Now (53) becomes




geneous equations





Let



1

19












(59)

and


where

()U)










bull





bull


Jl


21













Jl







J-1 J-1


mnat sstiaf1 are


as vas ahown before


(518)





Choose


Then


23


amp8 (55)





Cho~bullbull

80 tbat

and

llow



2lf

and n






Define





1k+l J 1-k+l J -1t 11






middotlf~middot~



25





532)


Jl





26



middotrf~t+~t

iJkJe-t bull



Renee





How







+ ~1~+1gt ~=-1







above














~ (o o) ~ (

Q 0)l-1 0 00

(1a) (~) (2 1 ) (xi) (ti)00 ~ + QO 4 bull ~

(61)

( oo (~) + (o o) (~) bull ( ri)-1 o) ~ o o ~ r~



(61) racsa to

JO


0

0 bull ~




to exist



(64)


l

~ bull ~

1 tlampl~ bull (65)

ubUfat7~ bull



(66)

Ol t


11~2 + 12lt22 0


SJnem 67 1bull tma

z11 ~ 2 (00) bull0( sh zta)( ) +

~ bull10z2z zl 21 22 21 22

(6 9)

c2(~ ~)( ) l ( )0~

fhlbull teducu to

bullO

610)

23middot11 ) 0 0)+ bull 0 1a

( 0 0


zia bull ~ bull o



3)


-( h) ( 1) middot(~ ) (i)~~ ~ -z2 21

(r ~) (~ r)bull X~ ~ +

~











~ ( ) Lla (~ ~)

~ -(~ ) L22 ( )




(614)

bull

2 1 1 1

2 1 1 bull 4 bull l 0 0 0

-2 0 3 -1



35
















BDlLIOGRAPBY


1931 Q69P


9




















if X - yen a 0 bull



10


the relation

(3~) (x y) ~ (y x)




b


a








spaces

u











tor all x in S bull






quatel7 defined it




12

into A(lb) bull














b

(42) f I(a t) x(t) dt r(a) (a~ 1 ~b)

a


b 11


a


13































tional equations





15

operational e~tiona











following vector





16


L 1a the vector



















17










From this we get

n



is

hia reduceamp to



Now (53) becomes




geneous equations





Let



1

19












(59)

and


where

()U)










bull





bull


Jl


21













Jl







J-1 J-1


mnat sstiaf1 are


as vas ahown before


(518)





Choose


Then


23


amp8 (55)





Cho~bullbull

80 tbat

and

llow



2lf

and n






Define





1k+l J 1-k+l J -1t 11






middotlf~middot~



25





532)


Jl





26



middotrf~t+~t

iJkJe-t bull



Renee





How







+ ~1~+1gt ~=-1







above














~ (o o) ~ (

Q 0)l-1 0 00

(1a) (~) (2 1 ) (xi) (ti)00 ~ + QO 4 bull ~

(61)

( oo (~) + (o o) (~) bull ( ri)-1 o) ~ o o ~ r~



(61) racsa to

JO


0

0 bull ~




to exist



(64)


l

~ bull ~

1 tlampl~ bull (65)

ubUfat7~ bull



(66)

Ol t


11~2 + 12lt22 0


SJnem 67 1bull tma

z11 ~ 2 (00) bull0( sh zta)( ) +

~ bull10z2z zl 21 22 21 22

(6 9)

c2(~ ~)( ) l ( )0~

fhlbull teducu to

bullO

610)

23middot11 ) 0 0)+ bull 0 1a

( 0 0


zia bull ~ bull o



3)


-( h) ( 1) middot(~ ) (i)~~ ~ -z2 21

(r ~) (~ r)bull X~ ~ +

~











~ ( ) Lla (~ ~)

~ -(~ ) L22 ( )




(614)

bull

2 1 1 1

2 1 1 bull 4 bull l 0 0 0

-2 0 3 -1



35
















BDlLIOGRAPBY


1931 Q69P


10


the relation

(3~) (x y) ~ (y x)




b


a








spaces

u











tor all x in S bull






quatel7 defined it




12

into A(lb) bull














b

(42) f I(a t) x(t) dt r(a) (a~ 1 ~b)

a


b 11


a


13































tional equations





15

operational e~tiona











following vector





16


L 1a the vector



















17










From this we get

n



is

hia reduceamp to



Now (53) becomes




geneous equations





Let



1

19












(59)

and


where

()U)










bull





bull


Jl


21













Jl







J-1 J-1


mnat sstiaf1 are


as vas ahown before


(518)





Choose


Then


23


amp8 (55)





Cho~bullbull

80 tbat

and

llow



2lf

and n






Define





1k+l J 1-k+l J -1t 11






middotlf~middot~



25





532)


Jl





26



middotrf~t+~t

iJkJe-t bull



Renee





How







+ ~1~+1gt ~=-1







above














~ (o o) ~ (

Q 0)l-1 0 00

(1a) (~) (2 1 ) (xi) (ti)00 ~ + QO 4 bull ~

(61)

( oo (~) + (o o) (~) bull ( ri)-1 o) ~ o o ~ r~



(61) racsa to

JO


0

0 bull ~




to exist



(64)


l

~ bull ~

1 tlampl~ bull (65)

ubUfat7~ bull



(66)

Ol t


11~2 + 12lt22 0


SJnem 67 1bull tma

z11 ~ 2 (00) bull0( sh zta)( ) +

~ bull10z2z zl 21 22 21 22

(6 9)

c2(~ ~)( ) l ( )0~

fhlbull teducu to

bullO

610)

23middot11 ) 0 0)+ bull 0 1a

( 0 0


zia bull ~ bull o



3)


-( h) ( 1) middot(~ ) (i)~~ ~ -z2 21

(r ~) (~ r)bull X~ ~ +

~











~ ( ) Lla (~ ~)

~ -(~ ) L22 ( )




(614)

bull

2 1 1 1

2 1 1 bull 4 bull l 0 0 0

-2 0 3 -1



35
















BDlLIOGRAPBY


1931 Q69P


u











tor all x in S bull






quatel7 defined it




12

into A(lb) bull














b

(42) f I(a t) x(t) dt r(a) (a~ 1 ~b)

a


b 11


a


13































tional equations





15

operational e~tiona











following vector





16


L 1a the vector



















17










From this we get

n



is

hia reduceamp to



Now (53) becomes




geneous equations





Let



1

19












(59)

and


where

()U)










bull





bull


Jl


21













Jl







J-1 J-1


mnat sstiaf1 are


as vas ahown before


(518)





Choose


Then


23


amp8 (55)





Cho~bullbull

80 tbat

and

llow



2lf

and n






Define





1k+l J 1-k+l J -1t 11






middotlf~middot~



25





532)


Jl





26



middotrf~t+~t

iJkJe-t bull



Renee





How







+ ~1~+1gt ~=-1







above














~ (o o) ~ (

Q 0)l-1 0 00

(1a) (~) (2 1 ) (xi) (ti)00 ~ + QO 4 bull ~

(61)

( oo (~) + (o o) (~) bull ( ri)-1 o) ~ o o ~ r~



(61) racsa to

JO


0

0 bull ~




to exist



(64)


l

~ bull ~

1 tlampl~ bull (65)

ubUfat7~ bull



(66)

Ol t


11~2 + 12lt22 0


SJnem 67 1bull tma

z11 ~ 2 (00) bull0( sh zta)( ) +

~ bull10z2z zl 21 22 21 22

(6 9)

c2(~ ~)( ) l ( )0~

fhlbull teducu to

bullO

610)

23middot11 ) 0 0)+ bull 0 1a

( 0 0


zia bull ~ bull o



3)


-( h) ( 1) middot(~ ) (i)~~ ~ -z2 21

(r ~) (~ r)bull X~ ~ +

~











~ ( ) Lla (~ ~)

~ -(~ ) L22 ( )




(614)

bull

2 1 1 1

2 1 1 bull 4 bull l 0 0 0

-2 0 3 -1



35
















BDlLIOGRAPBY


1931 Q69P


12

into A(lb) bull














b

(42) f I(a t) x(t) dt r(a) (a~ 1 ~b)

a


b 11


a


13































tional equations





15

operational e~tiona











following vector





16


L 1a the vector



















17










From this we get

n



is

hia reduceamp to



Now (53) becomes




geneous equations





Let



1

19












(59)

and


where

()U)










bull





bull


Jl


21













Jl







J-1 J-1


mnat sstiaf1 are


as vas ahown before


(518)





Choose


Then


23


amp8 (55)





Cho~bullbull

80 tbat

and

llow



2lf

and n






Define





1k+l J 1-k+l J -1t 11






middotlf~middot~



25





532)


Jl





26



middotrf~t+~t

iJkJe-t bull



Renee





How







+ ~1~+1gt ~=-1







above














~ (o o) ~ (

Q 0)l-1 0 00

(1a) (~) (2 1 ) (xi) (ti)00 ~ + QO 4 bull ~

(61)

( oo (~) + (o o) (~) bull ( ri)-1 o) ~ o o ~ r~



(61) racsa to

JO


0

0 bull ~




to exist



(64)


l

~ bull ~

1 tlampl~ bull (65)

ubUfat7~ bull



(66)

Ol t


11~2 + 12lt22 0


SJnem 67 1bull tma

z11 ~ 2 (00) bull0( sh zta)( ) +

~ bull10z2z zl 21 22 21 22

(6 9)

c2(~ ~)( ) l ( )0~

fhlbull teducu to

bullO

610)

23middot11 ) 0 0)+ bull 0 1a

( 0 0


zia bull ~ bull o



3)


-( h) ( 1) middot(~ ) (i)~~ ~ -z2 21

(r ~) (~ r)bull X~ ~ +

~











~ ( ) Lla (~ ~)

~ -(~ ) L22 ( )




(614)

bull

2 1 1 1

2 1 1 bull 4 bull l 0 0 0

-2 0 3 -1



35
















BDlLIOGRAPBY


1931 Q69P


13































tional equations





15

operational e~tiona











following vector





16


L 1a the vector



















17










From this we get

n



is

hia reduceamp to



Now (53) becomes




geneous equations





Let



1

19












(59)

and


where

()U)










bull





bull


Jl


21













Jl







J-1 J-1


mnat sstiaf1 are


as vas ahown before


(518)





Choose


Then


23


amp8 (55)





Cho~bullbull

80 tbat

and

llow



2lf

and n






Define





1k+l J 1-k+l J -1t 11






middotlf~middot~



25





532)


Jl





26



middotrf~t+~t

iJkJe-t bull



Renee





How







+ ~1~+1gt ~=-1







above














~ (o o) ~ (

Q 0)l-1 0 00

(1a) (~) (2 1 ) (xi) (ti)00 ~ + QO 4 bull ~

(61)

( oo (~) + (o o) (~) bull ( ri)-1 o) ~ o o ~ r~



(61) racsa to

JO


0

0 bull ~




to exist



(64)


l

~ bull ~

1 tlampl~ bull (65)

ubUfat7~ bull



(66)

Ol t


11~2 + 12lt22 0


SJnem 67 1bull tma

z11 ~ 2 (00) bull0( sh zta)( ) +

~ bull10z2z zl 21 22 21 22

(6 9)

c2(~ ~)( ) l ( )0~

fhlbull teducu to

bullO

610)

23middot11 ) 0 0)+ bull 0 1a

( 0 0


zia bull ~ bull o



3)


-( h) ( 1) middot(~ ) (i)~~ ~ -z2 21

(r ~) (~ r)bull X~ ~ +

~











~ ( ) Lla (~ ~)

~ -(~ ) L22 ( )




(614)

bull

2 1 1 1

2 1 1 bull 4 bull l 0 0 0

-2 0 3 -1



35
















BDlLIOGRAPBY


1931 Q69P


















tional equations





15

operational e~tiona











following vector





16


L 1a the vector



















17










From this we get

n



is

hia reduceamp to



Now (53) becomes




geneous equations





Let



1

19












(59)

and


where

()U)










bull





bull


Jl


21













Jl







J-1 J-1


mnat sstiaf1 are


as vas ahown before


(518)





Choose


Then


23


amp8 (55)





Cho~bullbull

80 tbat

and

llow



2lf

and n






Define





1k+l J 1-k+l J -1t 11






middotlf~middot~



25





532)


Jl





26



middotrf~t+~t

iJkJe-t bull



Renee





How







+ ~1~+1gt ~=-1







above














~ (o o) ~ (

Q 0)l-1 0 00

(1a) (~) (2 1 ) (xi) (ti)00 ~ + QO 4 bull ~

(61)

( oo (~) + (o o) (~) bull ( ri)-1 o) ~ o o ~ r~



(61) racsa to

JO


0

0 bull ~




to exist



(64)


l

~ bull ~

1 tlampl~ bull (65)

ubUfat7~ bull



(66)

Ol t


11~2 + 12lt22 0


SJnem 67 1bull tma

z11 ~ 2 (00) bull0( sh zta)( ) +

~ bull10z2z zl 21 22 21 22

(6 9)

c2(~ ~)( ) l ( )0~

fhlbull teducu to

bullO

610)

23middot11 ) 0 0)+ bull 0 1a

( 0 0


zia bull ~ bull o



3)


-( h) ( 1) middot(~ ) (i)~~ ~ -z2 21

(r ~) (~ r)bull X~ ~ +

~











~ ( ) Lla (~ ~)

~ -(~ ) L22 ( )




(614)

bull

2 1 1 1

2 1 1 bull 4 bull l 0 0 0

-2 0 3 -1



35
















BDlLIOGRAPBY


1931 Q69P


15

operational e~tiona











following vector





16


L 1a the vector



















17










From this we get

n



is

hia reduceamp to



Now (53) becomes




geneous equations





Let



1

19












(59)

and


where

()U)










bull





bull


Jl


21













Jl







J-1 J-1


mnat sstiaf1 are


as vas ahown before


(518)





Choose


Then


23


amp8 (55)





Cho~bullbull

80 tbat

and

llow



2lf

and n






Define





1k+l J 1-k+l J -1t 11






middotlf~middot~



25





532)


Jl





26



middotrf~t+~t

iJkJe-t bull



Renee





How







+ ~1~+1gt ~=-1







above














~ (o o) ~ (

Q 0)l-1 0 00

(1a) (~) (2 1 ) (xi) (ti)00 ~ + QO 4 bull ~

(61)

( oo (~) + (o o) (~) bull ( ri)-1 o) ~ o o ~ r~



(61) racsa to

JO


0

0 bull ~




to exist



(64)


l

~ bull ~

1 tlampl~ bull (65)

ubUfat7~ bull



(66)

Ol t


11~2 + 12lt22 0


SJnem 67 1bull tma

z11 ~ 2 (00) bull0( sh zta)( ) +

~ bull10z2z zl 21 22 21 22

(6 9)

c2(~ ~)( ) l ( )0~

fhlbull teducu to

bullO

610)

23middot11 ) 0 0)+ bull 0 1a

( 0 0


zia bull ~ bull o



3)


-( h) ( 1) middot(~ ) (i)~~ ~ -z2 21

(r ~) (~ r)bull X~ ~ +

~











~ ( ) Lla (~ ~)

~ -(~ ) L22 ( )




(614)

bull

2 1 1 1

2 1 1 bull 4 bull l 0 0 0

-2 0 3 -1



35
















BDlLIOGRAPBY


1931 Q69P


16


L 1a the vector



















17










From this we get

n



is

hia reduceamp to



Now (53) becomes




geneous equations





Let



1

19












(59)

and


where

()U)










bull





bull


Jl


21













Jl







J-1 J-1


mnat sstiaf1 are


as vas ahown before


(518)





Choose


Then


23


amp8 (55)





Cho~bullbull

80 tbat

and

llow



2lf

and n






Define





1k+l J 1-k+l J -1t 11






middotlf~middot~



25





532)


Jl





26



middotrf~t+~t

iJkJe-t bull



Renee





How







+ ~1~+1gt ~=-1







above














~ (o o) ~ (

Q 0)l-1 0 00

(1a) (~) (2 1 ) (xi) (ti)00 ~ + QO 4 bull ~

(61)

( oo (~) + (o o) (~) bull ( ri)-1 o) ~ o o ~ r~



(61) racsa to

JO


0

0 bull ~




to exist



(64)


l

~ bull ~

1 tlampl~ bull (65)

ubUfat7~ bull



(66)

Ol t


11~2 + 12lt22 0


SJnem 67 1bull tma

z11 ~ 2 (00) bull0( sh zta)( ) +

~ bull10z2z zl 21 22 21 22

(6 9)

c2(~ ~)( ) l ( )0~

fhlbull teducu to

bullO

610)

23middot11 ) 0 0)+ bull 0 1a

( 0 0


zia bull ~ bull o



3)


-( h) ( 1) middot(~ ) (i)~~ ~ -z2 21

(r ~) (~ r)bull X~ ~ +

~











~ ( ) Lla (~ ~)

~ -(~ ) L22 ( )




(614)

bull

2 1 1 1

2 1 1 bull 4 bull l 0 0 0

-2 0 3 -1



35
















BDlLIOGRAPBY


1931 Q69P


17










From this we get

n



is

hia reduceamp to



Now (53) becomes




geneous equations





Let



1

19












(59)

and


where

()U)










bull





bull


Jl


21













Jl







J-1 J-1


mnat sstiaf1 are


as vas ahown before


(518)





Choose


Then


23


amp8 (55)





Cho~bullbull

80 tbat

and

llow



2lf

and n






Define





1k+l J 1-k+l J -1t 11






middotlf~middot~



25





532)


Jl





26



middotrf~t+~t

iJkJe-t bull



Renee





How







+ ~1~+1gt ~=-1







above














~ (o o) ~ (

Q 0)l-1 0 00

(1a) (~) (2 1 ) (xi) (ti)00 ~ + QO 4 bull ~

(61)

( oo (~) + (o o) (~) bull ( ri)-1 o) ~ o o ~ r~



(61) racsa to

JO


0

0 bull ~




to exist



(64)


l

~ bull ~

1 tlampl~ bull (65)

ubUfat7~ bull



(66)

Ol t


11~2 + 12lt22 0


SJnem 67 1bull tma

z11 ~ 2 (00) bull0( sh zta)( ) +

~ bull10z2z zl 21 22 21 22

(6 9)

c2(~ ~)( ) l ( )0~

fhlbull teducu to

bullO

610)

23middot11 ) 0 0)+ bull 0 1a

( 0 0


zia bull ~ bull o



3)


-( h) ( 1) middot(~ ) (i)~~ ~ -z2 21

(r ~) (~ r)bull X~ ~ +

~











~ ( ) Lla (~ ~)

~ -(~ ) L22 ( )




(614)

bull

2 1 1 1

2 1 1 bull 4 bull l 0 0 0

-2 0 3 -1



35
















BDlLIOGRAPBY


1931 Q69P



Now (53) becomes




geneous equations





Let



1

19












(59)

and


where

()U)










bull





bull


Jl


21













Jl







J-1 J-1


mnat sstiaf1 are


as vas ahown before


(518)





Choose


Then


23


amp8 (55)





Cho~bullbull

80 tbat

and

llow



2lf

and n






Define





1k+l J 1-k+l J -1t 11






middotlf~middot~



25





532)


Jl





26



middotrf~t+~t

iJkJe-t bull



Renee





How







+ ~1~+1gt ~=-1







above














~ (o o) ~ (

Q 0)l-1 0 00

(1a) (~) (2 1 ) (xi) (ti)00 ~ + QO 4 bull ~

(61)

( oo (~) + (o o) (~) bull ( ri)-1 o) ~ o o ~ r~



(61) racsa to

JO


0

0 bull ~




to exist



(64)


l

~ bull ~

1 tlampl~ bull (65)

ubUfat7~ bull



(66)

Ol t


11~2 + 12lt22 0


SJnem 67 1bull tma

z11 ~ 2 (00) bull0( sh zta)( ) +

~ bull10z2z zl 21 22 21 22

(6 9)

c2(~ ~)( ) l ( )0~

fhlbull teducu to

bullO

610)

23middot11 ) 0 0)+ bull 0 1a

( 0 0


zia bull ~ bull o



3)


-( h) ( 1) middot(~ ) (i)~~ ~ -z2 21

(r ~) (~ r)bull X~ ~ +

~











~ ( ) Lla (~ ~)

~ -(~ ) L22 ( )




(614)

bull

2 1 1 1

2 1 1 bull 4 bull l 0 0 0

-2 0 3 -1



35
















BDlLIOGRAPBY


1931 Q69P


1

19












(59)

and


where

()U)










bull





bull


Jl


21













Jl







J-1 J-1


mnat sstiaf1 are


as vas ahown before


(518)





Choose


Then


23


amp8 (55)





Cho~bullbull

80 tbat

and

llow



2lf

and n






Define





1k+l J 1-k+l J -1t 11






middotlf~middot~



25





532)


Jl





26



middotrf~t+~t

iJkJe-t bull



Renee





How







+ ~1~+1gt ~=-1







above














~ (o o) ~ (

Q 0)l-1 0 00

(1a) (~) (2 1 ) (xi) (ti)00 ~ + QO 4 bull ~

(61)

( oo (~) + (o o) (~) bull ( ri)-1 o) ~ o o ~ r~



(61) racsa to

JO


0

0 bull ~




to exist



(64)


l

~ bull ~

1 tlampl~ bull (65)

ubUfat7~ bull



(66)

Ol t


11~2 + 12lt22 0


SJnem 67 1bull tma

z11 ~ 2 (00) bull0( sh zta)( ) +

~ bull10z2z zl 21 22 21 22

(6 9)

c2(~ ~)( ) l ( )0~

fhlbull teducu to

bullO

610)

23middot11 ) 0 0)+ bull 0 1a

( 0 0


zia bull ~ bull o



3)


-( h) ( 1) middot(~ ) (i)~~ ~ -z2 21

(r ~) (~ r)bull X~ ~ +

~











~ ( ) Lla (~ ~)

~ -(~ ) L22 ( )




(614)

bull

2 1 1 1

2 1 1 bull 4 bull l 0 0 0

-2 0 3 -1



35
















BDlLIOGRAPBY


1931 Q69P










bull





bull


Jl


21













Jl







J-1 J-1


mnat sstiaf1 are


as vas ahown before


(518)





Choose


Then


23


amp8 (55)





Cho~bullbull

80 tbat

and

llow



2lf

and n






Define





1k+l J 1-k+l J -1t 11






middotlf~middot~



25





532)


Jl





26



middotrf~t+~t

iJkJe-t bull



Renee





How







+ ~1~+1gt ~=-1







above














~ (o o) ~ (

Q 0)l-1 0 00

(1a) (~) (2 1 ) (xi) (ti)00 ~ + QO 4 bull ~

(61)

( oo (~) + (o o) (~) bull ( ri)-1 o) ~ o o ~ r~



(61) racsa to

JO


0

0 bull ~




to exist



(64)


l

~ bull ~

1 tlampl~ bull (65)

ubUfat7~ bull



(66)

Ol t


11~2 + 12lt22 0


SJnem 67 1bull tma

z11 ~ 2 (00) bull0( sh zta)( ) +

~ bull10z2z zl 21 22 21 22

(6 9)

c2(~ ~)( ) l ( )0~

fhlbull teducu to

bullO

610)

23middot11 ) 0 0)+ bull 0 1a

( 0 0


zia bull ~ bull o



3)


-( h) ( 1) middot(~ ) (i)~~ ~ -z2 21

(r ~) (~ r)bull X~ ~ +

~











~ ( ) Lla (~ ~)

~ -(~ ) L22 ( )




(614)

bull

2 1 1 1

2 1 1 bull 4 bull l 0 0 0

-2 0 3 -1



35
















BDlLIOGRAPBY


1931 Q69P


21













Jl







J-1 J-1


mnat sstiaf1 are


as vas ahown before


(518)





Choose


Then


23


amp8 (55)





Cho~bullbull

80 tbat

and

llow



2lf

and n






Define





1k+l J 1-k+l J -1t 11






middotlf~middot~



25





532)


Jl





26



middotrf~t+~t

iJkJe-t bull



Renee





How







+ ~1~+1gt ~=-1







above














~ (o o) ~ (

Q 0)l-1 0 00

(1a) (~) (2 1 ) (xi) (ti)00 ~ + QO 4 bull ~

(61)

( oo (~) + (o o) (~) bull ( ri)-1 o) ~ o o ~ r~



(61) racsa to

JO


0

0 bull ~




to exist



(64)


l

~ bull ~

1 tlampl~ bull (65)

ubUfat7~ bull



(66)

Ol t


11~2 + 12lt22 0


SJnem 67 1bull tma

z11 ~ 2 (00) bull0( sh zta)( ) +

~ bull10z2z zl 21 22 21 22

(6 9)

c2(~ ~)( ) l ( )0~

fhlbull teducu to

bullO

610)

23middot11 ) 0 0)+ bull 0 1a

( 0 0


zia bull ~ bull o



3)


-( h) ( 1) middot(~ ) (i)~~ ~ -z2 21

(r ~) (~ r)bull X~ ~ +

~











~ ( ) Lla (~ ~)

~ -(~ ) L22 ( )




(614)

bull

2 1 1 1

2 1 1 bull 4 bull l 0 0 0

-2 0 3 -1



35
















BDlLIOGRAPBY


1931 Q69P


mnat sstiaf1 are


as vas ahown before


(518)





Choose


Then


23


amp8 (55)





Cho~bullbull

80 tbat

and

llow



2lf

and n






Define





1k+l J 1-k+l J -1t 11






middotlf~middot~



25





532)


Jl





26



middotrf~t+~t

iJkJe-t bull



Renee





How







+ ~1~+1gt ~=-1







above














~ (o o) ~ (

Q 0)l-1 0 00

(1a) (~) (2 1 ) (xi) (ti)00 ~ + QO 4 bull ~

(61)

( oo (~) + (o o) (~) bull ( ri)-1 o) ~ o o ~ r~



(61) racsa to

JO


0

0 bull ~




to exist



(64)


l

~ bull ~

1 tlampl~ bull (65)

ubUfat7~ bull



(66)

Ol t


11~2 + 12lt22 0


SJnem 67 1bull tma

z11 ~ 2 (00) bull0( sh zta)( ) +

~ bull10z2z zl 21 22 21 22

(6 9)

c2(~ ~)( ) l ( )0~

fhlbull teducu to

bullO

610)

23middot11 ) 0 0)+ bull 0 1a

( 0 0


zia bull ~ bull o



3)


-( h) ( 1) middot(~ ) (i)~~ ~ -z2 21

(r ~) (~ r)bull X~ ~ +

~











~ ( ) Lla (~ ~)

~ -(~ ) L22 ( )




(614)

bull

2 1 1 1

2 1 1 bull 4 bull l 0 0 0

-2 0 3 -1



35
















BDlLIOGRAPBY


1931 Q69P


23


amp8 (55)





Cho~bullbull

80 tbat

and

llow



2lf

and n






Define





1k+l J 1-k+l J -1t 11






middotlf~middot~



25





532)


Jl





26



middotrf~t+~t

iJkJe-t bull



Renee





How







+ ~1~+1gt ~=-1







above














~ (o o) ~ (

Q 0)l-1 0 00

(1a) (~) (2 1 ) (xi) (ti)00 ~ + QO 4 bull ~

(61)

( oo (~) + (o o) (~) bull ( ri)-1 o) ~ o o ~ r~



(61) racsa to

JO


0

0 bull ~




to exist



(64)


l

~ bull ~

1 tlampl~ bull (65)

ubUfat7~ bull



(66)

Ol t


11~2 + 12lt22 0


SJnem 67 1bull tma

z11 ~ 2 (00) bull0( sh zta)( ) +

~ bull10z2z zl 21 22 21 22

(6 9)

c2(~ ~)( ) l ( )0~

fhlbull teducu to

bullO

610)

23middot11 ) 0 0)+ bull 0 1a

( 0 0


zia bull ~ bull o



3)


-( h) ( 1) middot(~ ) (i)~~ ~ -z2 21

(r ~) (~ r)bull X~ ~ +

~











~ ( ) Lla (~ ~)

~ -(~ ) L22 ( )




(614)

bull

2 1 1 1

2 1 1 bull 4 bull l 0 0 0

-2 0 3 -1



35
















BDlLIOGRAPBY


1931 Q69P


2lf

and n






Define





1k+l J 1-k+l J -1t 11






middotlf~middot~



25





532)


Jl





26



middotrf~t+~t

iJkJe-t bull



Renee





How







+ ~1~+1gt ~=-1







above














~ (o o) ~ (

Q 0)l-1 0 00

(1a) (~) (2 1 ) (xi) (ti)00 ~ + QO 4 bull ~

(61)

( oo (~) + (o o) (~) bull ( ri)-1 o) ~ o o ~ r~



(61) racsa to

JO


0

0 bull ~




to exist



(64)


l

~ bull ~

1 tlampl~ bull (65)

ubUfat7~ bull



(66)

Ol t


11~2 + 12lt22 0


SJnem 67 1bull tma

z11 ~ 2 (00) bull0( sh zta)( ) +

~ bull10z2z zl 21 22 21 22

(6 9)

c2(~ ~)( ) l ( )0~

fhlbull teducu to

bullO

610)

23middot11 ) 0 0)+ bull 0 1a

( 0 0


zia bull ~ bull o



3)


-( h) ( 1) middot(~ ) (i)~~ ~ -z2 21

(r ~) (~ r)bull X~ ~ +

~











~ ( ) Lla (~ ~)

~ -(~ ) L22 ( )




(614)

bull

2 1 1 1

2 1 1 bull 4 bull l 0 0 0

-2 0 3 -1



35
















BDlLIOGRAPBY


1931 Q69P




25





532)


Jl





26



middotrf~t+~t

iJkJe-t bull



Renee





How







+ ~1~+1gt ~=-1







above














~ (o o) ~ (

Q 0)l-1 0 00

(1a) (~) (2 1 ) (xi) (ti)00 ~ + QO 4 bull ~

(61)

( oo (~) + (o o) (~) bull ( ri)-1 o) ~ o o ~ r~



(61) racsa to

JO


0

0 bull ~




to exist



(64)


l

~ bull ~

1 tlampl~ bull (65)

ubUfat7~ bull



(66)

Ol t


11~2 + 12lt22 0


SJnem 67 1bull tma

z11 ~ 2 (00) bull0( sh zta)( ) +

~ bull10z2z zl 21 22 21 22

(6 9)

c2(~ ~)( ) l ( )0~

fhlbull teducu to

bullO

610)

23middot11 ) 0 0)+ bull 0 1a

( 0 0


zia bull ~ bull o



3)


-( h) ( 1) middot(~ ) (i)~~ ~ -z2 21

(r ~) (~ r)bull X~ ~ +

~











~ ( ) Lla (~ ~)

~ -(~ ) L22 ( )




(614)

bull

2 1 1 1

2 1 1 bull 4 bull l 0 0 0

-2 0 3 -1



35
















BDlLIOGRAPBY


1931 Q69P


26



middotrf~t+~t

iJkJe-t bull



Renee





How







+ ~1~+1gt ~=-1







above














~ (o o) ~ (

Q 0)l-1 0 00

(1a) (~) (2 1 ) (xi) (ti)00 ~ + QO 4 bull ~

(61)

( oo (~) + (o o) (~) bull ( ri)-1 o) ~ o o ~ r~



(61) racsa to

JO


0

0 bull ~




to exist



(64)


l

~ bull ~

1 tlampl~ bull (65)

ubUfat7~ bull



(66)

Ol t


11~2 + 12lt22 0


SJnem 67 1bull tma

z11 ~ 2 (00) bull0( sh zta)( ) +

~ bull10z2z zl 21 22 21 22

(6 9)

c2(~ ~)( ) l ( )0~

fhlbull teducu to

bullO

610)

23middot11 ) 0 0)+ bull 0 1a

( 0 0


zia bull ~ bull o



3)


-( h) ( 1) middot(~ ) (i)~~ ~ -z2 21

(r ~) (~ r)bull X~ ~ +

~











~ ( ) Lla (~ ~)

~ -(~ ) L22 ( )




(614)

bull

2 1 1 1

2 1 1 bull 4 bull l 0 0 0

-2 0 3 -1



35
















BDlLIOGRAPBY


1931 Q69P








+ ~1~+1gt ~=-1







above














~ (o o) ~ (

Q 0)l-1 0 00

(1a) (~) (2 1 ) (xi) (ti)00 ~ + QO 4 bull ~

(61)

( oo (~) + (o o) (~) bull ( ri)-1 o) ~ o o ~ r~



(61) racsa to

JO


0

0 bull ~




to exist



(64)


l

~ bull ~

1 tlampl~ bull (65)

ubUfat7~ bull



(66)

Ol t


11~2 + 12lt22 0


SJnem 67 1bull tma

z11 ~ 2 (00) bull0( sh zta)( ) +

~ bull10z2z zl 21 22 21 22

(6 9)

c2(~ ~)( ) l ( )0~

fhlbull teducu to

bullO

610)

23middot11 ) 0 0)+ bull 0 1a

( 0 0


zia bull ~ bull o



3)


-( h) ( 1) middot(~ ) (i)~~ ~ -z2 21

(r ~) (~ r)bull X~ ~ +

~











~ ( ) Lla (~ ~)

~ -(~ ) L22 ( )




(614)

bull

2 1 1 1

2 1 1 bull 4 bull l 0 0 0

-2 0 3 -1



35
















BDlLIOGRAPBY


1931 Q69P








above














~ (o o) ~ (

Q 0)l-1 0 00

(1a) (~) (2 1 ) (xi) (ti)00 ~ + QO 4 bull ~

(61)

( oo (~) + (o o) (~) bull ( ri)-1 o) ~ o o ~ r~



(61) racsa to

JO


0

0 bull ~




to exist



(64)


l

~ bull ~

1 tlampl~ bull (65)

ubUfat7~ bull



(66)

Ol t


11~2 + 12lt22 0


SJnem 67 1bull tma

z11 ~ 2 (00) bull0( sh zta)( ) +

~ bull10z2z zl 21 22 21 22

(6 9)

c2(~ ~)( ) l ( )0~

fhlbull teducu to

bullO

610)

23middot11 ) 0 0)+ bull 0 1a

( 0 0


zia bull ~ bull o



3)


-( h) ( 1) middot(~ ) (i)~~ ~ -z2 21

(r ~) (~ r)bull X~ ~ +

~











~ ( ) Lla (~ ~)

~ -(~ ) L22 ( )




(614)

bull

2 1 1 1

2 1 1 bull 4 bull l 0 0 0

-2 0 3 -1



35
















BDlLIOGRAPBY


1931 Q69P



~ (o o) ~ (

Q 0)l-1 0 00

(1a) (~) (2 1 ) (xi) (ti)00 ~ + QO 4 bull ~

(61)

( oo (~) + (o o) (~) bull ( ri)-1 o) ~ o o ~ r~



(61) racsa to

JO


0

0 bull ~




to exist



(64)


l

~ bull ~

1 tlampl~ bull (65)

ubUfat7~ bull



(66)

Ol t


11~2 + 12lt22 0


SJnem 67 1bull tma

z11 ~ 2 (00) bull0( sh zta)( ) +

~ bull10z2z zl 21 22 21 22

(6 9)

c2(~ ~)( ) l ( )0~

fhlbull teducu to

bullO

610)

23middot11 ) 0 0)+ bull 0 1a

( 0 0


zia bull ~ bull o



3)


-( h) ( 1) middot(~ ) (i)~~ ~ -z2 21

(r ~) (~ r)bull X~ ~ +

~











~ ( ) Lla (~ ~)

~ -(~ ) L22 ( )




(614)

bull

2 1 1 1

2 1 1 bull 4 bull l 0 0 0

-2 0 3 -1



35
















BDlLIOGRAPBY


1931 Q69P


JO


0

0 bull ~




to exist



(64)


l

~ bull ~

1 tlampl~ bull (65)

ubUfat7~ bull



(66)

Ol t


11~2 + 12lt22 0


SJnem 67 1bull tma

z11 ~ 2 (00) bull0( sh zta)( ) +

~ bull10z2z zl 21 22 21 22

(6 9)

c2(~ ~)( ) l ( )0~

fhlbull teducu to

bullO

610)

23middot11 ) 0 0)+ bull 0 1a

( 0 0


zia bull ~ bull o



3)


-( h) ( 1) middot(~ ) (i)~~ ~ -z2 21

(r ~) (~ r)bull X~ ~ +

~











~ ( ) Lla (~ ~)

~ -(~ ) L22 ( )




(614)

bull

2 1 1 1

2 1 1 bull 4 bull l 0 0 0

-2 0 3 -1



35
















BDlLIOGRAPBY


1931 Q69P


l

~ bull ~

1 tlampl~ bull (65)

ubUfat7~ bull



(66)

Ol t


11~2 + 12lt22 0


SJnem 67 1bull tma

z11 ~ 2 (00) bull0( sh zta)( ) +

~ bull10z2z zl 21 22 21 22

(6 9)

c2(~ ~)( ) l ( )0~

fhlbull teducu to

bullO

610)

23middot11 ) 0 0)+ bull 0 1a

( 0 0


zia bull ~ bull o



3)


-( h) ( 1) middot(~ ) (i)~~ ~ -z2 21

(r ~) (~ r)bull X~ ~ +

~











~ ( ) Lla (~ ~)

~ -(~ ) L22 ( )




(614)

bull

2 1 1 1

2 1 1 bull 4 bull l 0 0 0

-2 0 3 -1



35
















BDlLIOGRAPBY


1931 Q69P


fhlbull teducu to

bullO

610)

23middot11 ) 0 0)+ bull 0 1a

( 0 0


zia bull ~ bull o



3)


-( h) ( 1) middot(~ ) (i)~~ ~ -z2 21

(r ~) (~ r)bull X~ ~ +

~











~ ( ) Lla (~ ~)

~ -(~ ) L22 ( )




(614)

bull

2 1 1 1

2 1 1 bull 4 bull l 0 0 0

-2 0 3 -1



35
















BDlLIOGRAPBY


1931 Q69P


3)


-( h) ( 1) middot(~ ) (i)~~ ~ -z2 21

(r ~) (~ r)bull X~ ~ +

~











~ ( ) Lla (~ ~)

~ -(~ ) L22 ( )




(614)

bull

2 1 1 1

2 1 1 bull 4 bull l 0 0 0

-2 0 3 -1



35
















BDlLIOGRAPBY


1931 Q69P





(614)

bull

2 1 1 1

2 1 1 bull 4 bull l 0 0 0

-2 0 3 -1



35
















BDlLIOGRAPBY


1931 Q69P


35
















BDlLIOGRAPBY


1931 Q69P


BDlLIOGRAPBY


1931 Q69P


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