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Identification of the parameters when the density of the minimum is given

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Title:
Identification of the parameters when the density of the minimum is given
Physical Description:
Book
Language:
English
Creator:
Davis, John C
Publisher:
University of South Florida
Place of Publication:
Tampa, Fla.
Publication Date:

Subjects

Subjects / Keywords:
Tri-variate normal
Parameter identification
Minimum variate
Asymptotic order
Tail probabilities
Dissertations, Academic -- Mathematics -- Doctoral -- USF   ( lcsh )
Genre:
bibliography   ( marcgt )
theses   ( marcgt )
non-fiction   ( marcgt )

Notes

Abstract:
ABSTRACT: Let (X1,X2,X3) be a tri-variate normal vector with a non-singular co-variance matrix, and where each covariance is negative. It is shown here that it is then possible to determine the three means, the three variances and the three correlation coefficients based only on the knowledge of the probability density function for the minimum variate. We will present a method for identifying the nine parameters which consists of careful determination of the asymptotic orders of various bivariate tail probabilities.
Thesis:
Dissertation (Ph.D.)--University of South Florida, 2007.
Bibliography:
Includes bibliographical references.
System Details:
System requirements: World Wide Web browser and PDF reader.
System Details:
Mode of access: World Wide Web.
Statement of Responsibility:
by John C. Davis III.
General Note:
Title from PDF of title page.
General Note:
Document formatted into pages; contains 57 pages.
General Note:
Includes vita.

Record Information

Source Institution:
University of South Florida Library
Holding Location:
University of South Florida
Rights Management:
All applicable rights reserved by the source institution and holding location.
Resource Identifier:
aleph - 001920239
oclc - 187990354
usfldc doi - E14-SFE0002113
usfldc handle - e14.2113
System ID:
SFS0026431:00001


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Identification of the parameters when the density of the minimum is given
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ABSTRACT: Let (X1,X2,X3) be a tri-variate normal vector with a non-singular co-variance matrix, and where each covariance is negative. It is shown here that it is then possible to determine the three means, the three variances and the three correlation coefficients based only on the knowledge of the probability density function for the minimum variate. We will present a method for identifying the nine parameters which consists of careful determination of the asymptotic orders of various bivariate tail probabilities.
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Dissertation (Ph.D.)--University of South Florida, 2007.
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Includes bibliographical references.
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System requirements: World Wide Web browser and PDF reader.
Mode of access: World Wide Web.
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Advisor: Arunava Mukherjea, Ph.D.
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Tri-variate normal.
Parameter identification.
Minimum variate.
Asymptotic order.
Tail probabilities.
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PAGE 1

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PAGE 6

4 4 nn 2nnnn / n n&n/n &n/2n nnrAB@C nr nn/%D< nn/n nnn n { } W = W q q% nnn 9 q : q nnn q n q E q 6 3 6 3E q q = n n/ q F 2nnn n9nn nnn n//&n/n nnnn /&n/9 nn3n9 6 !n/ nn /&n/9n n ( ) ' '4 nn' { } ' 4 = nn#n'n / &n/' 'n/nnn ' '4 n nn nnn&n/2 nnn n%

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4n ( ) ' '4 nn n nn ( ) ' '4 /nnn;nnn /n { } ' 4 = nn = G -nn ( ) ' '4 n G 2nnn nnnn nnn nnnn n<4 nnnnn/ nnnnn/ n 4 n//n nn /n nn //% 4 4 4 4 + + = + + = b a b a :4 a 4 b nn4 /4 6 ',34s!'a b nn/6 ',3-s4 a '4 b '4 s 'a 'b 's nnn / nnn 34 < 6n 4 9nn n n= n'/nn n34 < 6 nn /n n // { } 4 = : n nn'n nn 9n n n34 6 nn36GnB4Cn

PAGE 8

+ /nnn/ n4 = ( ) ()4 4 4 4 H Hs b a s b a+ + #nnnn I = n4 ''4 Is m/nIs n #nnnnn I I = n4 I I I I n m 24 a '4 b 'a 'b nn/ n% I I I I I I I I I I I I4 4 4 4n b a n b a m b a m b a = + = + = + = + )nn ( ) 4 'nn/ %D! nn 'n nnnn 9n4 F2n % { } { } 4 4' 9 = = n' = nn?n/ { } 9 = /4' = = 2n ( ) () () ()4 4 4 4 4' H H H Hs m s m s m s m:n9 n ( ) { } 4' = n ( ) 4 -' !nnn

PAGE 9

J nnn9 { } 4' 9 = n ( ) 4 -' /4 nn 'nnn/ 'n n %!n n nnn //nn346 n nnnn nnn '3-6n/n n ' 6 3 6 3 n n,n nK< ' '4 n/n; ' '4 n { } ' 94 = { } ' '4 nn n ' '4 nnn n /n; / { } ' 94 = = ' '4 ;n ' '4 3n6 !nn /n nn9n #n/ nnn/ $n n< nn £ £ 4' < { } ' 4 = 2 n/ ' '4 nn&n/ n n rnn3 nn ' '4 nnn 6 n nnn nnnnn nnn/ nn':/ n nn £ £ 4'n { } ' 4 = nn/n 0nn nn3nn 6n n nnn: nn { } ' 94 = rnn

PAGE 10

@ 3 nn ' '4 nnn 96 n n n/ nn9 ' '4 n /nn ' '4 { } ' 94 = 9/nn ( ) { } ( ) ()()()() ! ! ! 4 4 4 4 ' ' ' 9 = £ £ £ = £ £ £ = £ = £ = ?n/n ' '4 nnn' n /nn = =n nn9nn nnn :n/'nnnn nn nnn9 nn AB@C n9% > = > = £ = £ = !nn J + 4 l l l l + = + { } 4 = { } J + = n ( ) ( ) ( ) ( ) ( ) ( ) n n n n = + = + =+ + J + 4 J + 4l l l ll l l ln < £ n 2n nnnn /

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8 n;nnnn'9 nn 2nnnn nnn nnn/nn n9n nn nnnnn n2'n ( ) ( ) ( ) ' ' '4 4 nnn0n n n/ ( ) ( ) '-'4 n H = S m ?n/ { } ' 94 4 = { } ' 94 = 2nn ( ) 4'" '-'4 n = S m nB4Cnn n nnnLn'n0 nn'n0 n9&$'?&' B44C9 nnnn n nnLn &$'?&'nn/ nn' nnnnn n nn92n n nn/ %< () ()' 4'-' 4 4 4 = + = # p nnnnnn n< () () 4'-' $' 4 4 4 = + = r $ p nnnnnn n?n/ n $ $ $ # # # 4 4 = nn/ = ' 4 n nn r r r ' 4 7///nn nn nnnn nnn &nn/ nn :B5Cnnn n/n9n nnn/nn n nn nn

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. 2n/ n nn nn nnn !n ( ) ( ) ' n n%n&' 2 n/n nn 44 4 ' ' r s s m m / 4 4 s m nnnn3 /n n/ 6 s m nn n'/n/ n n /n n nnnnn / nnnn ' ' 4 'nnn n n9n n2' n' { } = = = ' 9 / % 4 nn2n &n/ nnn 2n ?B4+Cn n 0nn nn/ n 4 > s s r AnB+Cnn 0n7 &$ B.Cn0n /n, 4 > s s r n2nn0 n /n/ nnn Ln/n0n /nn nLn

PAGE 13

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PAGE 14

5 ( ) ( ) ()() ()() ()() ()() ()() ()() ()() < = = < = = + < = + < = < < n n n r n n n r n n n n n n r n n n n n n r n n n n n n n n n n n n n n rnn 6 3 6 3 n n n + ,-=9 ()()()() + + n r n -4 A<;M; n = 6 3 6 3 n n ( ) < n 6 3 6 3 = n n ( ) > - n = 6 3 6 3 n n ( ) = ( ) > r - n 6 3 6 3 = n n ( ) = ( ) < r + n = 6 3 6 3 n n ( ) = ( ) < r + n 6 3 6 3 = n n ( ) = ( ) > r ( ) = ( ) = r 6 3 6 3 n n n + ,-= n n = 6 3 6 3 n n ( ) > ' n 6 3 6 3 = n n ( ) < N

PAGE 15

4, n<6 3 6' 3 6' 3 n n + n n n 6 3 n 6 3 6 3 n + n + = #n n ( ) n n2 n nn/ % 4 6 3 n 6 3 H n 6 3 n 6 3 n + n + 6 3 6 3 n n 6 3 6 3 H n n rnn6 3 n 6 3 H n - n 6 3 6 3 n n O4- n 6 3 6 3 6 3 n n + n + - + = n 4 6 3 6 3 n n + O4- n 6 3 6 3=n n + '6 3 n 6 3 n + n - n 6 3 6 3 6 3 6 3 n n n n 6 3 6 3 6 3 6 3 6 3 6 3 n n n + n n n *n + = - 6 3 6 3 6 3 6 3 6 3 6 3 6 3 6 3 n n n + n n n n n *n n - - + = 4 4 6 3 6 3 4= + = + = - n n +nP6 3 6 3 n n 6 3 6 3 H n n #'6 3 6 3 n n 6 3 6 3 H n n - = n 4 6 3 6 3 6 3 6 3 n n n n *- = n 6 3 6 3 n n 6 3 n 6 3 H n Nr<6 3 6' 3 6' 34n n n n n 6 3 6 3 6 34n n n + + = /()()6 3 6' 3 4 n n n nn n2 n '()()6 3 6 3 H 6 3 4 n n n + + n!n s s 'n, < r 2() 4 r s s s s r s s+ = n n r n r = 4 4 4 4 -s s s s r rnn

PAGE 16

44 36 n r n r = 4 4 4 4 -s s s s r n r n r = n r 4 4 4 -s s r s s -4 4 + = + s s s s r s s r s ss s s s r s s r s s= ( ) r s s s s r s s s s= + () 4 r s s s s r s s+ = 3n6() + =-4 r s s s s r s s + >-s s s s r s s -4 4 s s s s s s+ = + > ( ) -4 s s s s+ > n r n r 4 4-s sn 4 2n() + =-4 r s s s s r s s s s s s s s r4 4 4 4-+ n r n r = / r < r n r n r = 4 4 4 4-s s s s r Nn 4 4 r s s m s r m s b r s s r s s a= = #n £ £ + 4

PAGE 17

44 + = + 4 4a s a s + = --s m b a s m b a + + = + -b s m b s m ,' + *n rnn4 = + 4 a s + 4 s = n r 4 r s s r s s ( ) () 4 4 r s s r s r s s s r s+ + -() = + = 4 r s s r s s s s ( ) () 4 4 r s s r s r s s s r s+ + +-4s = n r 4 r s s r s s -4 a s+ = -s m b a = n r n r --4 4 s m r s s m s r m s r s s r s s () = n r + --4 s m r s s m s r s r m s r s m s s r m s s ( ) () = + 4 4 r s s r s m r m s s r m s m s r m s s() = + 4 r s s s m r m s s m s r m s s ( ) () = + 4 4 r s s r s m r m s s r m s m s r m s s() = n r + --4 s m r s s m s r s r m s r s m s s r m s s

PAGE 18

4+ = n r n r --4 4 s m r s s m s r m s r s s r s s s m b a+ = +b s m +s m = n r --4 r s s m s r m s ( ) () = + + --4 4 r s s m s r m m s s r m s r s m ( ) () = + + --4 4 r s s m s r m m s s r m s r s m +s m = n r --4 r s s m s r m s b s m+ Nn !n 9 S n0 n92 S n ( ) '6 3 4 > S < S rnnAnnOn r = S 4 4 4s s rs s rs s/, < r 2() n r = S 4 4 4 4 4 4 4s s rs s rs s s s r !n ( ) 4 > S ?n/nn = nn < S n ( ) ( ) 4 4 + + n9/6 3 < S rn S

PAGE 19

4J n r S S S = S + 4 -4 444s/ 44 S 4S 4 -4 S 4 ?n/nnnnn 9 n r = S + 4 /nn 0 n n ?n S 'n'n0 4 44 4 ' S S , n'n'n0 ?n/ 4 4 + = SS '/ n r = n r + S + S S + S S + S + + 4 , 4 4 4 -4 4 -4 444 444 + + ,s s /nn/ n% ( ) () () ()J4 +, -, 44 -4 4 4 -4 4 444 444 = + S = + S = S + S = S + S + + + + s s #n3-+6/n ()@4 -4 4 *S = +s ?n//3-46 ()84 -4 44 44 , = S S S +s n//n () ()>4 .4 -4 44 44 4 4 -4 44 44 n r S S S = n r S S S = + + ,s s ?n/ , 4 n /6 43 £ £

PAGE 20

4@ ()() () ()()()5, 4 4 '4 4 4 -4 44 44 4 < S S S = S S S = + + + + ,s s ( ) < S = S / 44 'n4 + n ( ) ( ) '4 4 + + S S /6 43 £ = £ = ( ) ( ) ()() ()()4, 4 4 4 4 '4 4 4 4 4 > S S = S S S = = + + + + + ,s s 23-4,6n 4 & 0 < Sr ( ) () S < S < + + + + 4 4 4 4 '4 s s s s :4 0&n9/ ( ) , < / ,4 Ann £ £ > 4 n 7nnn-4 S 'n ( ) £ £ < S 4',4 -4'n nnn -4 44 S = +sn !4S '.* + = nnnn + n #'n3-J6/ , 4-4 -4 4< S > S = + + + -s

PAGE 21

48 n <() ( ) = < = S S 4 / ', H ' '4 n' ( ) d d d d ' 4 = nnn0n ( ) 4' '4'4 = r #' r 4 4 S n r b b b 2 n ( ) H ' 4 4 n n n d d d + > + > + > () [ ] () = S S + S + S . n n n 4 4 4 4 4 9b p d d dr r r 3-446 rnn ( ) H ' 4 4 n n n d d d + > + > + > () () + + S S =44 4 9 4d dpn n r d = () ()() [ ] + S + S = n n d d p4 4 9 4 ( ) ( ) ( ) ( ) ( ) . . d d d d d4 4 4 4 S + S + S = + S + [ ] () ()()[] S + S S S = n . n d p d d4 4 4 4 9 4 9 r r n n n = = / [ ] () () () () S + + + S + S S = 4 4 4 4 9 4 9 . n n n d p d d ( ) ( ) ( ) ( ) . .n n n n r r r 4 4 4 4 S + S + S = + S + ( ) ( ) ( ) . n n r 4 4 4 S + S = S +d d d

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4. ()()() [ ] () ()()()[] n n n . . . S + S + S S S + S + S = 4 4 4 4 4 4 4 9 4 9d p d d dr r r r < ', / > = n n ()()()[] n . S + S + S 4 4 4 4 9dr () ()() n n n . S + S + S = 4 4 4 4 4 9 4rd n ()4 4 9 4 S --. n ()4 4 9 4 S --. nd () {} = S = 4 4 9 9br () ()() = n r S + S + S . n n 4 4 4 4 4 4 4 9b dr 2n' n ( ) H ' 4 4 n n n d d d + > + > + > () [ ] () '= -S S + S + S . n n n 4 4 4 4 4 9b p d d dr r r Nr < ( ) ( ) 4 / P4'4P,', H 4 £ < r r n' 4 / ', ', d d b a > > n2 n ( ) H 4 4 d b d a + £ + £ n n n -p9 () + + rn n 4 3-4-6

PAGE 23

4> () ()() ra b rb a r= + 4 4 -r b rab a+ = ( ) 4 4 4r bd ad bd r ad + + = r 4 4 4 -r d r d d d+ = 2n ( ) 4 4 d b d a + £ + £ n n n ( ) n r + + 4 4d a d a fn n ( ) n r + + -d b d b fn n rnn! ( ) ( ) ! > > = £ £ 4 4 ' () () + = 4 4 9 4 4 r r r p < ) = = () ()) ) ) + = 4 4 9 4 4 r r r p ( ) £ £ = 4 ()()4 4 ' = nn/ ( ) ( ) ( ) ( ) 4 4 4 4 ' d b d a d b d a > > = + £ + £ n n n n ()() () () () n r > > = > > =b d b a d a d b d a4 4 4 4 ' n n n n 2 n ( ) n <-8 () () H 4 4 n r > >b d b a d an n ()() [ ] = S S + S + S 4 4 4 4 4 9 . n n nb p d d dr r r S n9n n r b a4 2n

PAGE 24

45 n r = S 4 4b ab r ab r a' -4b a r= S n r = S -44 4 4 4r b r rab r rab r a' n r = S n r 4 4 4 4r rab b r rab a b b. r 4 4 -r b rab a+ = S.r r n r =b d a d d4 4 4 4 4 -r d r d d d d d+ = S -. ( ) 4 4 4 4r bd ad bd r ad d+ + = S -.r 2 nn/% n ( ) H 4 4 d b d a + £ + £ n n () () ()() ra b rb a p r d r d d d r bd ad bd r ad r b rab a rn r + + n r + + + n r + 4 4 4 4 + 4 4 4 4 9 4 n n n #n' ( ) ( ) {}' 9 4 4 4 -b a a r ra b b r rb a r b rab a> + = + = + <-4 ( ) 4 4 d b d a + £ + £ n n n ( ) n r + + 4 4d a d a fn n ( ) n r + + -d b d b fn n n N < ( ) ( ) 4 / P4'4P,', H 4 £ < r r n' 4 / ', ', d d b a > > n2 n ()[] d p d b d a+ + > + > n n n H 4 4 4 9 () + 4dn 3-4+6

PAGE 25

-, / { } {} > < = = = = = = b a d b a d b a d d d d d b a d d b a b a 9 n 4 4 4 4 4* rnn ( ) [ ] 4 4 4 d b d a + > + > n n ( ) 4 4 d b d a + £ + £ = n n ( ) ( ) ( ) 4 4 4 4 d b d a d b d a + £ + £ + £ + + £ = n n n n A;)n' n ( ) H 4 4 d a + £ n ( ) 4 4d a d a f + + n n p d a44+ = n9 () + + -4 4 -4d ad an n ( ) H d b + £ n ( ) -d b d b f + + n n p d b4-+ = n9 () + + --4d bd bn n nn-84' ( ) 4 4 d b d a + £ + £ n n !n ( ) n r + +4 4d a d a fn n ( ) n r + +-d b d b fn n nn/<-4 ( ) 4 4 d a + £ n !n ( ) d b + £ n d d a + = + n n 4 / b a < n/ 4 d d b a > = ( ) d b + £ n !n ( ) 4 4 d a + £ n d d b + = + n n / b a > n/ 4 d d b a < = 2n<--'/4 n d d b a ()[] d p d b d a+ + + n n n 4 H 44 4 9 () + --4dn

PAGE 26

-4 '/4 d d b a = = ()[] d p d b d a+ + + n n n H 44 4 9 () + --4dn N< ( ) ( ) 4 / P4'4P,', H 4 £ < r r n' ' ' 4 d d d b a nn-8-2 n 3-4J6 ()()[ ] "n n n n + + £ + > + > -p d d b d a4 H 44 4 9 () + --4 "n / ( ) ( ) d + £ n : = = = = = = 4 4 d d d d d d a b b a n4 d d b a = = ' n ( ) ( ) [ ] d d b d a + £ + > + > n n n 4 4 4 ( ) H 4 4 d b d a + £ + £ = n n n -p9 () + + rn n 4() ()() ra b rb a r=+ -4 4 -r b rab a+ = ( ) 4 44r bd ad bd r ad + + = r 4 4 4 -r d r d d d+ = rnnnnnnn-8-'/ ( ) [ ] 4 4 4 d b d a + > + > n n ( ) 4 4 d b d a + £ + £ = n n ( ) ( ) ( ) 4 4 4 4 d b d a d b d a + £ + £ + £ + + £ = n n n n <= b a < 2 ( ) ( ) [ ] d d b d a + £ + > + > n n n 4 4 4 ( ) ( ) [ ] 4 4 4 4 4 d a d b d a + £ + > + > = n n n

PAGE 27

-( ) ( ) 4 4 d b d a d b + £ + £ + £ = n n n nn-84 ( ) 4 4 d b d a + £ + £ n n ( ) n r + +-d b d b fn n ( ) ( ) [ ] 4 4 4 4 4 d a d b d a + £ + > + > n n n ( ) 4 H d b + £ n ( ) "n + £ = "n +p4 H9 () + --4 "n ( ) ( ) d + £ n d d d b a = = = = 4 ( ) ( ) [ ] 4 4 4 4 4 d a d b d a + £ + > + > n n n ( ) ( ) ( ) [ ] 4 4 4 4 4 4 d b d a d b d a + £ + £ + > + > = n n n n ( ) H 4 4 d b d a + £ + £ = n n n -p9 () + + rn n 42/nnn-8-Nn< ( ) ' '4 n0n/ ( ) -' H s m £ £ 4 n/, < r nnnn ( ) £ < £ 4 n ( ) s s s > > > 4< ( ) ' 4 = 2n 6 3 n O ()= = + > n r n n4 44b a s m f s/ 4 4 r s s m s r m s b r s s r s s a= ='() n r =-4 9 4 n np f #n { } ' / '-'4 '

PAGE 28

-+ ( ) ( ) r P4'4P,', H '/ 4 4 r r r r r r= rnn<6 3 n nnn ( ) ' 4 = ( ) n n £ = 6 3 n r = £ == n !4' () = £ == n !4' / { } { } [ ] = = = ()= = £ = n 4 () = £ £ = n 4 () n = - = = = = P n r = 4 4 4 4 ' j'/ j n ( ) ' '4 ()() n = - = = + = = P n r = 4 4 4 4 4 4 4 4 Q ' ' j j ( ) 4 j n4 ( ) Q ' ' '4 4 4 4 + j nnn ( ) ' ' '4 4 4 + = () n = - = = = n r n r =4 4Q 4s m f s f n

PAGE 29

-J n/n '6 3 n #2n n% []6 3 6 3 n n n = ()= = = n r n r = n n n 4 4 Q 4s m f s () = = n r n r = n n 4 4 I 4 s m f s ( ) I I 4 I 4 I I 4 ' ' ' + 30460n/ ( ) () n r + I 4 H n r s s m r s m() I I 4 4 r r r r r = 2n 3*<2n B4JC6 ?n/ ( ) () I 4 n r s s m r s mn r + = n ( ) 4', H () 4 4 r r r r r = n/ 6 3 n O ()= = + > n r n n 4 4 4b a s m f s/ 4 4 r s s m s r m s b r s s r s s a= =

PAGE 30

-@ 220?n/? n n ( ) + 4 ' nn/ ( ) H s m+ 4 n £ £ /, < r nnnn ( ) + 4 n £ < £ + 4 s s s > > < { } + 4 ' = 6 3 n n //nnn nn ( ) + 4 ' 2/nn/ 6 3 n -+ 4+ 4+ 4 + 4 ' ' ' ' r r r s s s m m m A<-./n 6 3 n O ()= = + > n r + 4 + 4 4 n nb a s m f s3+46 / 4 4 r s s m s r m s b r s s r s s a= = '() n r = 4 9 4 n np f ( ) ( ) r P4'4P,', H / 4 4 r r r r r r= ( ) n ' n ( ) +'-'4 n

PAGE 31

-8 ?n/()4 '+'-'4 , + 4 = + > = > < = - n n b a a r2n n n p s s m f s4 4 H 6 3 n n = n r 9 n r + n r + --4 4 n ns m s m s3+-6 4 n 4 44> < s s'4nnn/ p s4 H 6 34n 9 n r + n r + 4 4 4 4 4 4 4s m s m sn n n r -- -6 3 n n n n r =44s3++6 2 4 s ()6 3 4 n 6 3 n 9 4 -4sn3+J6 A--' n() p s4 H 6 3 4 n 9 n r + n r + n r -4 -4 4 4 n ns m s m s s3+@6 4 -4 4s s/ O4'4n() p s4 H 6 34 4n 9 n r + n r 4 4 4 4 4s m s mn 3+86() n r - n n n6 3 4 4 4s m = 3+.6 2 4 m

PAGE 32

-. 63 4 n 6 3 n n r -4 4 44s m f sn3+>6 ()[] () ()-+ -+ -+ 4+ 4+ 4+ + + + +++444+4 +4 +4 -4 -4 -4 4 4 4 4 4 4 4b a b a s m f s b a b a s m f s b a b a s m f s+ > + > n r + + > + > n r + + > + > n r = n n n n n n n n n Ann-8-' n 4 n 63 4 n p d sH4 4 4 4+ n *9 () + + n r -4 4 4 44d s mn n p d s4 4 4 4 + = n *9 + + n r + n r + -4 4 4 4 4 4 4 4 44 4d s m s m d sn n 3+564 ' 4 -4 4 -4 4 = = = b d a /+4 -4 a a < 4 '4 +4 4 +4 4 = = = b d a /-4 +4 a a < 4 '4 -4 4 +4 -4 4 = = = = b d a a /+4 -4 +4 -4' b b a a > = 4 '4 +4 4 +4 -4 4 = = = = b d a a /-4 +4 +4 -4' b b a a > = '4 +4 -4 4 +4 -4 4 = = = = = b b d a a /+4 -4 +4 -4' b b a a = = n 63 4 n p s4 H -9 n r + n r + n r --4 4s m s m sn n + n 63 4 n p s4 H +9 n r + n r + n r -+ + + + +4 4s m s m sn n + -4 4s s< n+ n

PAGE 33

-> 4 4 +4 4 4+ + -4 4 4-4 4 4 4 4 4 4 4 + < + = + < + = + +s s 4 4+> -s s'n-n() n r - n n n 6 3 -s m = 3+4@6

PAGE 34

-5 2 m 63-n 6 3 n n r n r -4 4 4 4 4s m f s s m f sn n3+486 ()[] ()[] ()-+ -+ -+ 4+ 4+ 4+ + + + +++444+4 +4 +4 -4 -4 -4 4 4 4' 4 4 4 4 4b a b a s m f s b a b a s m f s b a b a s m f s+ > + > n r + > + > n r + > + > n r = n n n n n n n n n nn-8-' n 4 n 6 3-n p d sH4 4 4 4+ n 9 + + n r + n r + -4 4 4 4 4 4 4 4 44 4d s m s m d sn n n H 63-n p d s--+ n 9 + + n r + n r + --4 4d s m s m d sn n 3+4.6 4 '44= = = b d a /+4a a < 4 '++= = = b d a /4+a a < 4 '4+4= = = = b d a a /+4+4-' b b a a > = 4 '++4= = = = b d a a /4++4-' b b a a > = '+4+4= = = = = b b d a a /+4+4-' b b a a = = + n 63-n p s4 H+9 n r + n r + n r -+ + + + +4 4s m s m sn n A-J + = + < + = + <+-+ + + +4 4 4+ + +4 4 4 4 4 4a s a s s a s a s s3+4>6

PAGE 35

+, + + < 4 4 +4 4 4 s s sn 44-4 4 +4 4 4 = + = + s 3n4-4 4 4-4 4 ' b d b d a a = = = = 6 =)/ 44 4 -4 -4 44+ + 4+ 4 + = n r = < =s m b a s m b a s m s 63-n n r + =4H n *p d s9 [ ] + + -444-4 n + n 3+-,6 *0 %2+n/ 4+4
PAGE 36

+4 -r [ ] n r + - n n n n n -4 -6 3 n r = n 4 n 3+-J6 /n-r */ %-r = !-4-, = '-+'4r n r n r =4 44 4 4 4 s s s s4r n/4 4' ' m m s s /'/n/ -4 4-4 4-' ' b b a a 63 n &6 3-n () ()444-4 -4 -4 4 4 44 4b a s m f s b a s m f s+ £ n r + £ n r n n n n 3+-@6 ()()[ ] ()()[ ] ()-+ -+ -+ 4+ 4+ 4+ + + + 444+++444-4 -4 -4 +4 +4 +4 -4 -4 -4 4 4 4 4 4 4 4 4b a b a s m f s b a b a b a s m f s b a b a b a s m f s+ > + > n r + + £ + > + > n r + + £ + > + > n r = n n n n n n n n n n n nn-8+ ' +4 4 +4 4 -4 4 -4 4 b a b d a = = = = n ' ++44b a b d a = = = = n 4 n63 n & p b a s4 H +4 +4 4 + n 9 + + n r + n r + +4 4 4 4 4 +4 +4 +4 4 4 4b s m s m b a a sn n 3+-86

PAGE 37

+n63 n & p b a s4 H +++ n 9 + + n r + n r + ++++4 4b s m s m b a a sn n 3+-.6 + n 63 n & p s4 H + 9 n r + n r + n r + + + + + 4 4s m s m sn n + + < ++4 4 + 4 4 4a s a s s'+n63 n &n n r - 6 3 n n & n + 4s= + s n4 ()6 3 + n 6 3 n & 9 + 4sn 3+->6 n 4 n()6 3 + n p b a s4 H +4 +4 4 + n 9 + + n r + n r + +4 4 4 4 4 +4 +4 + +4 4 4 4 4b s m s m b a s a sn n 3+-56 n()6 3 + n p b a s4 H +++ n 9 + + n r + n r + ++++ +4 4 4b s m s m b a s a sn n 3++,6 + n()6 3 + n

PAGE 38

++ p s4 H + 9 n r + n r + + + + 4s m s mn + + < ++4 4 + 4 4 4a s a s s'+n()6 3 + n n() n r - + 6 3 n n n + +s m = '3++46 + m n4 *0 % r = + 4s= + s n!n r = + + -s m' + m n#nn/ + 4 + 4 ' ' s s s m m m 63 + n 6 3 n n r n r n r + + + 4 4 4 4 4 4s m f s s m f s s m f sn n n 3++-6 ()[] ()[] ()[]-+ -+ -+ 4+ 4+ 4+ + + + +++444+4 +4 +4 -4 -4 -4 4 4 4 4 4 4 4 4 4b a b a s m f s b a b a s m f s b a b a s m f s+ > + > n r + + > + > n r + + > + > n r = n n n n n n n n n

PAGE 39

+J '/nn n 63 + n Ann-8-' n 4 n 63 + n p d sH 4 4 4 4 + n 9 + + n r + n r + 4 4 4 4 4 4 4 4 4 4 4d s m s m d sn n n 6 3 + n p d sH + n 9 + + n r + n r + 4 4d s m s m d sn n + n H 6 3 + n p d s+ + + + + n 9 + + n r + n r + + + + + + + + + + 4 4d s m s m d sn n 3+++6 4 ' + 4+ + 4+ + = = = b d a / -+ 4+ a a < 4 ' + -+ + -+ + = = = b d a / 4+ -+ a a < 4 ' + 4+ + -+ 4+ + = = = = b d a a / -+ 4+ -+ 4+ b b a a > = 4 ' + -+ + -+ 4+ + = = = = b d a a / 4+ -+ -+ 4+ b b a a > = ' + -+ 4+ + -+ 4+ + = = = = = b b d a a / -+ 4+ -+ 4+ b b a a = = #n +'-'4 = '% { } 6 3 n n Q + n 1 + 4 I 4 = + ,s' = + I I 4 Q 1 s3++J6 nr 2/nn n 6 3 + n % rnn

PAGE 40

+@ A-4'n I 1 n n n ' a = b d = I 4 4 4 ,a s a s s+ = + = + = nn/ a = n'J' s m b a s m b a= / b d = 2 nn n2n 1 1 n/n 1 /n /nn n 6 3 + n NA-( ) 1 n n 63 n H 63 + + n r + + + n r + n r + 1 n n n *p d s d s m s m d s4 4 9 H 3++@6 A-4'n 1 / + 4 4 4 = + = + s s I 9 1 = -s m d s m d 63 + n n r + - 1 n *p d sH 9 n r + n r + = n n 1 I + 4 9 4 4s m d s3++86 / n r + = 4 9 d s m

PAGE 41

+8 + n r - + 6 3 n n n 6 3 n 6 3 + n 9 + 4 n + n r - n n n 6 3 3++.6 n/nn/ + 4 + 4 = + = s I + 9 1 =s m d3++>6 n// 63 + n n r + - 1 n *p d sH 9 [ ] + n n + + 4 3++56 n nn 1 % = = + + + 4 Q 1 s m d s3+J,6 n/ = = + + + 4 Q s m b a a s3+J46 3+J-6 ?n = / n = / n n nn// { } -'4', n n rnn = = + = + = + + 4 4s m b a s m b a a s a s N

PAGE 42

+. ?n/% + 4 s + + 4 s s m d+ = n r + = 4 9 d s m3+J+6 n 63 + n n ' = d d= = ', = / nn Pn' 63 + n n r + = + 4 H n -p d s9 [ ] + n n + + 4 3+JJ6 ?n/ + r []n r + - n n n n n + + + 4 9 -6 3 p3+J@6 2nn/ n/n/9 nnn n ( ) + 4 ' 2n + r + 4 ' /n nnr < 'nnn+'-'4n' ( ) + 4 ' n/nnn3+J-62 'nn/ n% 4 ?nnnnnnLn nnnn nLn'n/nnn nn2n nnnnn ', = 4 = = = + n r n' = nnnLn' nnnn /n/nnnnn n' 4 = = '= , = = + n r r n' = =

PAGE 43

+> + /nnnn/n' nnn /n , = = = + nnnn n / = = rnnn4%!n, = 2 nn !/nn n nn 2n n / #n'n+-/ { } { } = = n4 = = b d a b d a = = = = ' nn/ = + = + = + 4 4a s a s3+6 n r n r = + + + 4 4 4 4 s s s s r n'-J' n r + = 4 9 d s m n r + = 4 9 b s m n r + = 4 9 b s m = n r + = 4 9d s m n = rnnn-%<4 = = n, 2 n : n % + + 4 4 4 = = = = + = + = +s m d s m d s m d s s s !4 = { } = n { } =

PAGE 44

+5 !n { } = + { } = 4 = = /n, n { } { } ' = = { } = = n/ b d b b d b d = = = = ' a a a a = = = = ' 4 4 $& $ + = = + = + = a s a s/-+ n r n r = + + + 4 4 4 4 s s s s r n r n r = + + + 4 4 4 4 s s s s r n/n/ = = % n r + = 4 9 d s m n r + = 4 9 b s m n r + = 4 9 b s m = n r + = 4 9d s m n n r + = 4 9 d s m n r + = 4 9 b s m n r + = 4 9 b s m = n r + = 4 9d s m n n = =

PAGE 45

J, rnnn+%< = = { } = { } = = n/ b b d b b d b b d = = = = = = ' a a a a a a = = = = = = ' nn/ $& & $ + = = = 2'-+ n r n r = + + + 4 4 4 4 s s s s r n r n r = + + + 4 4 4 4 s s s s r n r n r = + + + 4 4 4 4 s s s s r nnnn-'/ = = % Nn/n n ( ) + 4 ' ( ) ( ) ( ) ( ) ( ) ( ) ( ) -'-'' -'4'4 '4'-'4 '4'4''4'4', '4','4 ,'4'4 2n #n9/nn + 4 ' 3 = / ' nnn +'-'4 6n + 4 = = ?n = n /, 4 = = = !n + 4 = = / + r An + r []n r + - n n n n n + + + 4 9 -6 3 p ?n/ + r n r + + = = 4 4 4 + + + + + 4 4 + 4 s s s s3+J86

PAGE 46

J4 / + 4 = = = = + r nn #nn + r O + r 2/ + 4 = = + r + r 2n+-L nn 2n ( ) + 4 ' nnnnn nn + r + 4 ' nn/% 4 + 4 = n = = = + 4 + r 4 4 4 s s + = n r + = 4 4 4 4 + + 4 s s ( ) ( ) ,'4'4 ' + 4 = n r n r = + + + 4 4 44 4 4 4 s s s s r + 4 = n = = = + 4 + r + + + 4 4 4 s s + = n r + = 4 4 4 4 + + + 4 s s ( ) ( ) 4','4 ' + 4 = n r n r = + + + + 4 + 4 4+ 4 4 4 4 s s s s r + 4 + = n = = = + 4 + r + + + s s + = n r + = 4 4 4 4 + + + s s ( ) ( ) 4'4', ' + 4 = n r n r = + + + + + -+ 4 4 4 4 s s s s r

PAGE 47

JJ = = = + 4 + r + + + 4 4 4 s s s + + = n r + + = 4 4 4 4 4 + + + + 4 s s s ( ) ( ) 4'4'' + 4 = n r n r = + + + 4 4 44 4 4 4 s s s s r n r n r = + + + + 4 + 4 4+ 4 4 4 4 s s s s r @ = = = + 4 + r + + + 4 4 4 s s s + + = n r + + = 4 4 4 4 4 + + + + 4 s s s ( ) ( ) 4'-'4 ' + 4 = n r n r = + + + 4 4 44 4 4 4 s s s s r n r n r = + + + + + -+ 4 4 4 4 s s s s r 8 = = = + 4 + r + + + 4 4 4 s s s + + = n r + + = 4 4 4 4 4 + + + + 4 s s s ( ) ( ) -'4'4 ' + 4 = n r n r = + + + + 4 + 4 4+ 4 4 4 4 s s s s r n r n r = + + + + + -+ 4 4 4 4 s s s s r

PAGE 48

J+ = = = + 4 + r + + + 4 4 4 s s s + + = n r + + = 4 4 4 + + + + 4 s s s ( ) ( ) -'-'' + 4 = n r n r = + + + 4 4 44 4 4 4 s s s s r' n r n r = + + + + 4 + 4 4+ 4 4 4 4 s s s s r n r n r = + + + + + -+ 4 4 4 4 s s s s r N#n/nn nnn2n nnnn '// nn/ n% 6 3 4 n () n n n d s m f s+ £ n r -= + 4 + 4 6 3 4 + n r - 4 6 3 n n n ()6 3 6 3 4 4 n n 9 4 + 4 n 4 + () n r - n n n 6 3 4 4 + 4 4 s 4 + 4 + 4 4 s s m d= n r + = 4 4 4 9 d s m

PAGE 49

JJ % ( ) ( ) ,'4'4 ' + 4 = =/ + 4 = n + 4 = = + r 4 4 4 s s + = #n2n+-/n 4r n r n r = + + + 4 4 4 4 4 4 s s s s Ann-8+/ n 4 n 63 4 n p b a s4 H +4 +4 4 + n 9 + + n r + n r + +4 4 4 4 4 +4 +4 +4 4 4 4b s m s m b a a sn n n 63 4 n p b a s4 H +++ n 9 + + n r + n r + ++++4 4b s m s m b a a sn n + n 63 4 n p d sH + + + + + n 9 + + n r + n r + + + + + + + + + + 4 4d s m s m d sn n / + 3+++6 n/nn ( ) 4 + 4 4 4 ' / = = = n/ 4 n/ n 4 n/ n 4 4 4 4 4 4 + 4 4 4

PAGE 50

J@ {} = = n/ n 4 -+ 4+ + -+ 4+ +b b d a a / { } = = n/ n 4 / + + -+ -+ + + 4+ 4+ 4 + -+ 4+ 4 +s m b a s m b a , !n'// 63 4 n n r += + 4 4 4 4 4 H n -p d s9 [ ] + n n 4 + 4 + 4 =+n 63 4 n n nn n n 2n nn% 4 =4+n n ( ) = 4 + 4 4 4 ' ( ) 4','4 4 + 4 4 = /nn4 =-+n n ( ) = 4 + 4 4 4 ' ( ) 4'4', 4 + 4 = /nn4A + n n ( ) = 4 + 4 4 4 ' ( ) -'4'4 4 + 4 4 4 = = /nn4 n/ 4 + r [ ] n r + - n n n n n 4 + 4 + 4 4 9 6 3 p = = + 4 4 4 4 -s / 4 + 4 4 4 = = / 4 + r n r + + = = 4 4 4 4 + + 4 + 4 + 4 4 + 4 4 4 + 4 s s s s/ 4 + 4 4 4 = = = = 4 + r nn0 #nn 0 4 + r O 4 + r 2/ 4 + 4 4 4 = = + r + r

PAGE 51

J8 !n2n+-/%2r/, % ( ) ( ) 4','4 ' 4 + 4 4 4 = n 4 4 + 4 4 = n 4 + 4 4 4 = = 4 + r 4 + + + 4 4 4 4 + = s s :n4+n n '4+ 4 + +4 4 4a a= = 4+ 4 + = 4+ r n r n r = 4 + 4 + + 4 + 4 + 44 4 4 4 s s s s 2n -+ r 6 3-n ()4 4 4 + 4 4 4 6 3 = + £ n r n n n d s m f s 'nn-8+/ n 4 n6 3-n () + 4 4 H n -p s9 + + n r + n r + 4 4 4 4 4 4 4 4 4 4s m s m s n r n 4 4 4 4' ' r -nn-8+ n6 3-n p b a s4 H++-+ n9 + + n r + n r + ++++4 4b s m s m b a a sn n + n6 3-n p b a s4 H-+ -+ ++ n9 + + n r + n r + -+ + + + + -+ -+ -+ + 4 4b s m s m b a a sn n

PAGE 52

J. ?n/ + n r - 6 3 n n n '()6 3 6 3 n n [ ] + 4 n + () n r - n n n 6 3 + r []n r + - n n n n n + + 4 9 -6 3 p */3 + r 2-+n% = -+ + -+ r n r n r = + + + + + -4 4 4 4 s s s s *03 + r 24n% = + 4 4 4 +s 4 -+ +4 +4 -4 4 -+ -4 4 4 4 + + =r a a a r a s +4 -4 4+ 4-+ 4 -+ 4 4r r r r r r= -+ r F 2r/+ % ( ) ( ) 4'4', ' 4 + 4 4 4 = n 4 4 4 + 4 = n 4 + 4 4 4 = = 4 + r 4 + + + 4 + = s s 4+ r -+ r n n 2r/, 2r/* % ( ) ( ) -'4'4 ' 4 + 4 4 4 = n 4 + 4 4 4 = = 4 + r 4 + + + 4 4 4 4 4 + + = s s s

PAGE 53

J> : -+ 4+ 4 +a a= = -+ 4+ 4 + , = = 4+ r n r n r = 4 + 4 + + 4 + 4 + 44 4 4 4 s s s s -+ r n r n r = 4 + 4 + + 4 + + -4 4 4 4 s s s s % ( ) ( ) 4','4 ' + 4 = % ( ) ( ) 4'4', ' + 4 = nnn % ( ) ( ) 4'4'' + 4 = =/ + 4 = = + r + + + 4 4 4 s s s + + = : 4+4 -4 a a = = -4a = + 4+ a = 4+ 4+ 4+ 44 + + + , = = = = 4r n r n r =+ + + 4 44 4 4 4 s s s s' 4+ r n r n r =+ + + + 4 + 44 4 4 4 s s s s' n 4 n 6 3 4 n () + 4 4 H n -p s9 + + n r + n r + 4 4 4 4 4 4 4 4 4 4s m s m s n r n 4 4 4 4' ' r -nn-8+

PAGE 54

J5 n 63 4 n p b a s4 H++-+ n9 + + n r + n r + ++++4 4b s m s m b a a sn n + n 63 4 n p b a s4 H-+ -+ ++ n9 + + n r + n r + -+ + + + + -+ -+ -+ + 4 4b s m s m b a a sn n n4n -+ r n 63 4 n n 6 3 n 4 n % ( ) ( ) 4'-'4 ' + 4 = % ( ) ( ) -'4'4 ' + 4 = nnn % ( ) ( ) -'-'' + 4 = 4+4 -4 a a = = -+4a a = = + -+ 4+ a a = = = = = = = = -+ 4+ 4+ -+ 4+ 4+ + + + , 4r n r n r =+ + + 4 44 4 4 4 s s s s' 4+ r n r n r =+ + + + 4 + 44 4 4 4 s s s s'

PAGE 55

@, -+ r n r n r =+ + + + + -4 4 4 4 s s s s

PAGE 56

@4 J 79 794< + 4 ' ?3,'46 r r > > > > ', ', ', + + + + 4 4 r = = = 2 ( ) ( ) ( ) [ ] ()()()()()() ()()()[]()()() ()()()[]()() r r r r r r r = = = = = = < + = + + = = 7 7 n 7 7 7 n 7 7 7 7 7 7 n + + + + + + + + + + 4 + + 4 + 4 + + + + + 4 4 + + 4 4 !n ( ) + 4 ' 0n/ nnn

PAGE 57

@79-< ( ) 'n0 0n/Ln n0 n< 4 + 4 ' ' r r r nn + 4 + 4 ' ' n ( ) { } () { } ()() ()()[]()()[]+ 4 + 4 4 + 4 + 4 + 4 4 + 4 4R *' S' n 9 0 n n 9 R *' S' n 9 0 n n r r r r r r + + + + < < + + + + + + < + + > < 3J46 ?n/ r r r + 4 + 4 + 4 = ¢ = ¢ = ¢ < ( ) ( ) ' n ', H nn/ ( ) ¢ ¢ ¢ ' n #' ¢ = ¢¢ ¢ = ¢¢ + ¢ = ¢ ¢ ( ) ¢ ¢ ¢ ¢ ¢ ¢ '0nn/n/n0 = ( ) ( ) ( ) ( ) ( ) [ ] ()()()()[ ]() ()()[]{} ' ' 9 J7+ 4 + 4 + + 4 + + 4 4 4 + + 4 4 + + 4 + + 4 4 4 + + 4 4> + + + + = + + + + + + + + < + + + + + + + + = ¢¢n

PAGE 58

@+ ( ) ( ) ( ) ( ) ( ) [ ] ()()[](){}()(), ' ' J+ 7+ 4 + 4 4 + 4 + 4 4 + + 4 + + 4 4 4 + + 4 4 < + + + + < + + + + < + + + + + + + + = ¢¢r r r r r r r r r r r r r r r 2 ( ) ( ) ¢¢ < < ¢¢ 7 7 3JJ6 ( ) ( ) ( ) [ ] () ()()()[ ]()[]()[]()()()[]()() () ()()()[]() () n n + 4 / , , 7 7 7 7 n + 4 + 4 + + 4 + + 4 4 4 + + 4 4 4 r r r r r r r r r r r r r > ¢¢ < + ¢¢ = ¢ ¢ = ¢¢ ¢¢ < ¢¢ < ¢¢ = ¢ + ¢ ¢¢ = ¢ + ¢ = ¢¢ ¢¢ £ £ < < + + + + + + + + + + + + + = ¢ + ¢ ¢ ¢ = ¢ + ¢ = ¢ ¢ ¢ ¢

PAGE 59

@J 79+< ( ) ( ) S ', H ' + 4 / n r = S+ -+ 4+ -+ 44+ 44s s s s s s s s s0&n 9/ 4 4 4s s s< < ?n/ + 4 + 4 4 4 r + + = + = = 3JJ6 / + + + = n r + = = > 6 36 3 J ', 4+ -+ 44 4+ 44 444 44s s s s s s s s s s s s s sr r r r =() () () ()() 3J.6 n ', / n n ', / ', 3J86 n 3J@6 4 n4+ 4+ 44 44 + 4 4+ + 4 4+ 44 4+ 44 + 4 4 444 4 4< + + + £ > < £ < + + + = < = n r + =s s s s s s s s s s s s s s s s s s

PAGE 60

@@ ( ) ( ) 4 6 J 6 / n / ?n 3J>6 n 44 4 44 4444 -+ 4+ 444 + > < < < > + + + =s s s s s s s s s s s s s s s s s s sr r r r r [][] ()()() (), 6 36 3 n , n / nn/ n/ J n-+ 4+ 4+ -+ 44 444 -+ 4+ 444 + -+ 4+ + -+ 4+ 4 44 4 44 4 444 444< + + + + + £ + + + = > + < < + < n r + + + = + = +s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s sr r r r r r r r r r r r r r r

PAGE 61

@8 ) B4Cn'2''!345..6 nnr nn9)nT 1n&5 2nnn2n r '++.0+JB-Cn'2''!345.>6" #nLnnrnn A?nn#n ,n&nnn2nnn "n&n '8+085 B+CA'r'n'(345.>6 nn= n"n )&n 1"n)n, 6 'J4+0 J-5BJCA'r'n'(345>,6 nn"n )&n)&nnn !'2n0 534J64@4@04@-@B@CA'r345>46nrn 2nnn n)&%!2 n' 36' 2nnn-nrn 2n 'Tn@'++@0+J.')r n'n ':n B8C''&$'3-,,46n nrn ?nTnnn9 1.&n!rrn 6 Tn r '?n4'-8.0-5> B.C7 ''&$'345556 nnrn2 ?nTnnn 1'2nnn! 6 '-+0+. B>C#')''345.-6nn 7nn& n 'J5.0@4JU B5C'':'345>,6 nn=A ?nnT 1','2nnn',' 3+.46'8@408@J

PAGE 62

@. B4,Cn ''&$'3455,6!nnn rnnnn rnn9)n T% ?n 1.&n!rrn 6 Tn '?nJ'.>+0.5, B44C&$''?&''' 345>86nnr nn9)nT% 2n02n 1"n)n, 6 r '4.>04>8 B4-C&$''!')3455,6 nnrn n9)nT%2 ?n !r'.& 5'6 '->50-58 B4+C?'345.,6=7 nn)nTn= !nn= .&n r 3J6'5-+05-J B4JC2n '*< .&"n)n-nrn ?/*n&%! 0T '455,

PAGE 63

nn n'/n n': n=&/nnn :'/A4 5>8:"n !n#n'2'#://( #n/455> :/n& n"n!n# n $nn:n/n&n: nn nn $nn2n3'-,,.6 /n0n/ &$ 1"n)n,