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Transcript of 0 JAfl 'tt^ tt-,- ,^rr;; *;%;,tf - Fmcetfmcet.in/CSE/MA2265_uw.pdf · p a 1D P,/e tpva)r7P 7 T F r...
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.42 C-tA 3-Lr\42-
lt)
,144 sCo) : 6 C- a)n.T
D>o 1'"t+b scol:
.uln*;'-9;^el
h ) o.
ltQfrun. $cn) . dGo )o
34) 6 d.lr.rz
'.ICn -'t) ' 6OD )'- t
4Lq)= (-6)
6co - rl = stn)-5
3cn) --'a scn-t)
Sca) = -6 ' 3 cn-t)
3(P) > 43Ck't) + ro I (k -t) -- o
art o i a,t'd- c'stlo9t2'\oLt) dqdaaLjL'u
- ruCK 'O + to'rCE-2 )=o
Cl,araAr,)*Z aTaat'b-'t)
12 -1Y 1to'o
y2 _ Lf -it+to = O ,
'YC'r -2) -EcT' 2) = o '
tr'<' 0'z)'(T 'a ) "o
a,at o tt''a*zd
ot-zl
UL,ScK>
JL,6
l.L., VJz 4e-L/^7'4>h'
qb*-o9e'qottS e?aafuil-,:,<rbcil . q,@u *4, cr)r
8n) 9;^^
,9 '/'itl,bl ,
Qtvz
I
uat Arr-df lu)(hi^ le'L 'rij' /x'u'u'&
,XLLllrt !.2,3'4, " '
ol-^D
- a. Lc-r) ? < CDt t)T-7
lr>o
> E2u+ 3x') "'--z I
=(r-x)-=;;,
lA,unrg^"uA+) =la,l +tnrl+tA'l+ le4l - I
! a,n n")-ln,naal -!At n a") -[A" n
+ I a,no, n %l +lA'on '-o aal + lA 'o
+ lh2oAa oA4l - lAtoA'oA"hAa)
9r,>
Lttu /".fu l* lA'oe' uAguA+
ul,lt a',a ocPoA"il cn a"tl%o^Lou't
e.n=DQr-,-|Qn-r+4h
Xtn a,$oua,&d JaocnXzat*t't
Qo= DQr-t -f 6Qn', = o'
Jtz daracA,.iat'iY2-nt"t6=o
I
n,AAz) -
nol - lAi
n son4 )
nn+ I
&eL/ltun^ f-a't-3+) ild
s:l
&*u aAz r'2'3o -
(h)
t .'. $alah,it*' & ( n
(,ct
._ d ,(L)n * d, ca)o
!8
,l
I* l,'br^, r'ryl4hftErnhl1cA L
Suhl,a,-=a-
MA THEI.4AflCJ
E- r
-i.-
Fd +\L4rlqlt6n',
I
*T laa "at/- Wt
l, ',Marqr).I (r,t -1
20t3)
+.vwv, t
t +i,* ,
4;,,
+";"^
A, &
M- d
T - A hnu4
C*t +wnz ,s-l+;w.
o
,F-1.*v"h'e
,^? *t tlv- .Utru "r"", L" l.*Tt. )
val-,.,-r -& --' r*
- \ na =' -f :".+1----.f}a
= )o L=,,-.11, + i 1"- - til-'\)Q 3aE < = '>-o -<-2-
\-z-r 2-t > \*'-( = q
-r : "v +.>- A '>- ':-o
6.rive6
tu(ni'€ aoc.{
tr -..." @
lt .l
aldral r!r!' I rl t I
,-oL.*1c-oLa,. '>-
LA'u F{lD-'r/tL')2'o'
-(!-@
E ,..-ot' Pa<e;b?tE1n (D
@
P..o rz
Jol:
&pt )e".
.;^1- er;* H)+22+ ' - "+ 2h1'' 9n = 2n*'-l'( A'tr' Al9 >--az- N I 12'>'ov)
?xrhc,i,,lt 4 mall-e-rna.tfctl or*atot o* t!* P
be
,fL
a ,P1^o/o4
t.L.r Pcnlo
Lt)
n ,.tt N toff*"
fepottttz.ei) Pc t)
1ii,l Pco+r)
Xl-t P ,i
',1 ;,* /r/"^ad @r\- Va*fie J,.'tAt''
l"ku fo.t aaa
-? clt,oz;tt/, Jr-Lt ,
P .hat 'llt'
.f, ar- '
"& *rrl, utlt)alz,
,hu/, X* *^7
ttit
"aL
141/,
fecnl ,'k
f*n*'n
,hrL.
L,t l2fl?)
Ptn):Jo.r 2'+)L+'' +2h =rl*'',
Pcrl ' 9o72t ' 22 -t
llI =22-r3 ,*''
3 r3-o
"' PCt) uu t 4'''tl2 '
A +tLlt4-a. Ptn> 3 dLu-t- .
Jnh*t6?"+&t& +.. .12 =2 't
olo ,*a yz h(blt) r, .lr,Lt,
teo+&'+JQ+ . .. 14h-t2o+l
.[an*! ,1 u ii'r'D+t
' 2 f et1 ,J''
= 2n'')-t -- 2n*2- t
-'.PCn+O h ,*.rra . l1oni p(h) ) dtup-.
0,
6 al,ug
Sol:
d)
t,StK) -/ke1t-t)1*sck-a) =3t +2, 6co). r,sct) >/
Cn.ul alH :-oo:)J, lr"d $/-t' uldilAf e^Lout adltl tuh"
ttu- ctaoacn*i'sa| Q?^"hf- '1 nLe^ 'l'7
x2 -4d+ 4.o
-'.t =A.A z.L
-l o,n-o|eatoa -s ola l,tA^' = C A + Bk )Y/-L
t1)) Jo fu -ttz Vaxlta,Ll,at i*et3'<'al
dr* P1, -c+dx
-rfu-a' t,bzliL'um scl ) 'cld<lCK- t) ' CtdCr<-t)
aAd 8Ck'1) = c+dCx-z)
l&L ,ScK) - AsCk-D+ksCk'D =Jk a "'d
. 44,t'r ;t; rb^'b
o'e.t n* ttx-iJ asfc+dtr-zrJ = t*fa-+d r] 'h fc'+c
(t'oo)*e)k:3kbr4 * ?4 . "ut n*
o d =3'C:r2' :. P1,. : tL+38 " .
(ii:,) No
{nt*^^ cIe yn t-c't'&i'+ ' i"fir"at f '*'
?I, ' ak2 2r(a'ru A& a- Cha,4a or),1 4 tlc
e-oo* q.r"dr. r1
6a*aga* 8(k) - ek, c*tt'
ECK-t) =qCr-r)2 2r''fufr. ,tcK) -&sc(-t) ++sck:2) =2& a-^-d lLu,
411*4litu*Lt'')o 7*'q- rL zk - 4f+ 1x -, )'] zk
-' + 4 fo c* + tJ -tk
-'''o
,il* *a*" iau Jn tL, cDe tt-catc2- 4a.(r<2 'zr+D {') * t fucr2-ttt..+4) I <+l
. -2 k 2 k-l'.PI,=!Crt t =K 2
YJZ q 4,,./.Lal -Sdlu 'LLit ' eL'/- ,<-l
<Nk) 4A+Bk)2k + (rztat<) t R'2
Qtnen dk* 9co) =t => t = A'1 t2
.fCr) . t =>t =(A +B) 24 tL
.'. A:-r, , E = AZ
." scr) = (-tr + ] o) z* n (o+ 3r) +kt eE-'
6ck) .(kz +tx'tt-)zR) 1712aBk)'
&a "r-.4 = -:-
,-4^d.sa?.uth ce-
9" &* q(F,ocr42 3u,*
Lt'(e- |er$ah;Y /*^ . ^ # -o,'y'-L b o rUZC.a
(n.u a1.q >ooz)
z) ,h ,ltz Vzuaal-|'',1 6--c*i'" 4ti lo-nacc-! 4 e?u"-/t cL
/ot Qcr,il =! FCn)zn.
P dh'p/,/ .A,4^tt- .*dq ,? ,'. aon t t<nmahb
J.a.a- o, 2 1o n =eo
., I f rrr, - Fco-t) -rco-:lf xn=2
2 f(n)z^- f Fb-r) zn -2 rcr-nd n=A n-L
f*Li or,, zD^ F (o,t -rc,tz-l - .fi-,rrn-,
- ,'fg Fro-z) '"-'l --oL:,, J
f+tr,rt -,-zJ ,19- Fcn-') 7n't 't,:,
t1
a
)
,rrrl
-l
=F(o) + FCDz + €0) {a-q" lr,.r"hb f,-*+
.qt-la.crir-.
.' Fc^) - FCn-t) - PCn-z)-o.
@
le
/5e7ua^ @
,"t LtrLt- Vil.o-ae;
-_o.
) zn -0.
-z ftt
cr,z s -l - t [o,cr.,]'1, -r -Z'*f ..oJ.
il t-zlt lt-z-t$(r,z) =
10,,,
Q (c'rz)
I
I - r-2"
"45 ,u.- .r"g"La 7.^ury l,*cta*
4)I
.hlt, .,(-(- ct+L-A2./ur -ta-l4hdL + *4 q)/nLod -l - I, l)n' pn-t*tr-r, h)lco.ta-Ui+< . 1,., . lr, ,.
..lol: CA,t), m t) :gra r.r Ip racr.-r".e
2,tz .sarLlQrae q Nil,tno"ei ,x*tn*ent ,"hrQt-dXr, 4.e- e.u,a4irtt (. "r"/"Ir;*
l, = l,-, *tn-, -oar'd aaaT-<i-t -t<- n';b-aj /,: q f .=
doL?e
lSe4.Lac/ttz-
dtuhq,!
(t)
d4*
ayal,,ix
@
y"i&onocd
oJ,ot*i
=>/ -trn ln-t
Lv" -t .. q u, 2-^.,o
F'1 :o -rAU n -a
a aotl-,bia .r/ Lilr-
(D 5> rn -Yn''- f''" =o
.r, l- , rl =or L-T'71.'. XLt crla*act ^)t,i j"a*"" -P-e ,2-7-'"o
T: tj 'f C;FJZ:;
4)
T = ll 'lt+tr
*e* r, .
." rb.1UJ0n
-f , =,* f,,x,(tr),"(,*)",
(9I
( '*6)a, n (t'76)oL= Q ------>@
=, *il,=r,(ff)1"o (*)a,fuGf ._ (prif+ o(, t t-'> '
-th
*, (t-,1?)' + ot, (r-G)e=4 -->@
(r-G) ->
( r+ rts)(t-{B) + (t-ti)t d, -- a (t-G) -->@
&t, @) 1t .) dr {r+'G)1 + *' (r'ti)L=4 --> @
(O -tc)'>ot, 0+ fil fr' G .l' t;J = 2' 2G')+
d,
(
(r+r)f'trrJ = -2G'2(*{,) GaG) ' UD(t+ti)
*,G =tIO( =-,ti
U) =l (t+ n) J- + ( r-G)o(t'4G
-tIt,
4vq,
! +r +(t-{7)qt,G
Q-G) d, ,-P
Q-,|-a) x,
:t =rJ,=k''JtkYry\.?[f)'
%z
aL
Eol:
"V ?na.dtmalfcal
Va+hie J\rrlrrl
,@-itra.c, cttin , -trlora ,-ta-ab t*.
Ln+t n+) ^n, 3 +2 ,ri aa;lLtz o,V t.LA',.-) ql11 zoa?_ rtl>u<)
s9 ,'{t P(n) = B2'*' + ,n'" f, d.it;i,silL 3y r
fo tpanve Pr,) -Z ,tu.u'2.t t t+2 3 3
i"' + t'-'.8"+2" =ll-t8,36 .(6)tt)
urthl ,2, ao,nilL OT t-'. Pc D ,;4 ,-bj''-t- .
63!_:' j a ur,v utua-t' Pu) ,i" dt-u2' )K+t k+tt.e.t { t L , (4)ct4) fo< z a^.
{ At ! , Jo p<ovz pC k t D tt .l-i<r2-tt. _rek+t)+t l.+t)t2 LrtJ k+J|(.e.,, 3 1r*"' -, a + z
, e'k'' cg)' + "o*o c')
. k+2.
- {t'cil + z*" (')
i"lae-t
2E+ t
&*t*/au-r- .' Ba
ll,l-nd,uc,$br,
r LK+t ' '"" 1 +(it s)- tls '1 2- J'"'t-
Lk +l
= ) (t)(m) 't + G)o o 4r;dbt- b.,t I
J4t4tch u u o
.PC r+r) i,t a^a+ "/'<'l'^2'v.3^ P(k) ..''''.r^r- ?;""?o q *"-'+*'*"*YtPcd "2" v{,u,L loo a* V*rZd "
-tlfuXot n-.
6) dLrivhd
uarTttn ,fanunttd cer"i^tt 4 a- &e,Au. ,r1
t&X, elflla,lat '/*ttoaza I'11'' W'""fl
{aL Jr+h
le fa-tnel.
U) J.t-e. -b.ta/- gtu-dl?, "af VaawAt JAa* ca-^-
q'tata a-"oad,t ,h ,ilufuU v'orCn.o . r-711 o-oo1, N/D a'aq)
3ol:
d)r, ct .a,u.t-4- La4z "L6 "/fM .^a t" "dnie
a,^-d -"- "A;U ee-.^- bz dlV ealtn, "'V L"u' foda&
tul-k- Lr/Le- q./"t ^a^b2,1 fr 3- cna.cqc*oL "pa*s u-'-o,sot 'fi
pg.16.to.d6ooco
6.<-tu/ axh Llrt- 91-,t Lru-bel $ 4 Cbazaca*ou
faNuoxd "tt Ae . to. to. to .a6ooo
1t "-n
cL dtt- ,Xnra, ,X,t on^bet oatr ?u4 t/o/d. Ll,
d6oo+a6oo6 = a861o
Ol aL"r* L.t^r. dr;;b 4'u' ?t'at 'L'|f z4r2d ' "a"-* 4* a+haJzt &.^" lL ,.Ab-tu f-tn "'
^;^rir c drirrh e-nd ao ot.y*t?YX44 Lbl]- ct*a,bza 'o$ s' chq'4achl' /b'81-;'
-& d6'tos =dJ4o.
G) X^t- '|hil-hlzt- *tdr_4L1,1 claza.lt
llt
-f,-
^1-ea-/h
d;wilAr)'l
|amfiO,<d uL
,th,
16. to .?.e ' t77a,
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,,tt, |en^ati'szd
"r,^,1 . f,La .ut-e-
-L a- cler$ ,h
*kr.n.An t ,r;^"h/'IU
otiiwrln o^///'*2r'Di
lo *tzt- "tla* a'fus*
,-tl'r- 4o!v!- Q1/a *h
C A.u , 14- l:) D'a?r l"i la't-'oo1)
A rr*r"
"ttz ttuJ
o
Ar r-1"1 ^' Cu-c- b @ -lt "*-'1)
qwo *L
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tAA
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*fu- 4a)r4 +1-LDA.tL
,Lat/- U)*,n
o
-r- k -tlr-
rJ
'\.t//t4.bu 4 ,4-cha-+acb'l
qAltlLa $ P'ut'n"a '1J,54o+tgXto zdto6o '
sol: o ..-. I o* t. o
|a.ooa.un' 9-14- 2^t""i2"*'b- f e'x--l''-- 'n*
t^j- f+e -fnk*u--o--z^t ,,)to ^r^icfi,a a" n iqz,*L"tfu p qrt rp- a'4-2' "-tf* '* ''"n.
A.bn- "rue- ,f dlz' 'fflta,buA.'* L#'l -, -ph,^''
flat be
fi7*nlnl* ' ''Ltr'tuU*rt-bor $.
a
*"ndw"rias eoen
n=tl, Na-i, ,re
@
l?n utL^*
Q. 3. , a-/\_d
d-n ;1,L
{u*
&;it;ttL
letd;n/W
4 94//rila 4
4 at<n+z.z 4 ;rq-,
4 ArrwkL 4 'tttu-2'o'zt
Inoc1
leoe
vt .f-4= as
+) ff,J.ss=121 =""l_zx sJ
, =les:f _ ,"Ler gJ
* P.t -*
bl:LL "E Lr'ta'
qrttx)h -A a fu*da-h .
U
Ltie. O1.t t-/t\abz.L
a)-e- "UrA;ll-7.
3:1,
/Lt A be J^L -t(*1),i
^f L/r- bq "2 .
ll
X,* e '/'L Ltr-'' 4t'tLx t'
C JrL dJ-L [*
/'v b'
8 lz ,a/:- 4*
r'toty. tat =[jaJ=ttn
t"r =fLll - *Iea'd ffi'+t
Uoo) ffi: tv.t
agfiil-.l qu,.3;,ru,.,-,o
g ioqro ,**: t and
LV "n 4 Ltt-z- dra.tz-l-
CA.rl . sr lD roo? z.oog.)
q ato,rxbz.a. ,4 4./&4rut
lboo)
lgocoo)
Z"l_
LLla* a44
-', rll*uf*./Auguartl
.tu.urubot *td.;-,";/"lL
fin oocl
Da;l' L-ii-1 "" 'I
lcot) =lzoilI l= 7L9" 9J
ltE-o I= t--_ / "yt,r,1l = "
f Aueucuol
= 2 . laoeoc"r, = f-:."-1.,I -2'l' t 'il
,-a.^A+, aztV +4s Q
-Ldcout /-dso
L1" /n u au cuD)
?LA+/L q Xn ur-k^ -ErcL-,-t io*
= tA) +le) + lct +lot - /aoBl - lAoc) -(fr n Dl -lLoc t - (o o ol -k o t / -t (AoBoc/+ l1o eoa,
+ lA n(oD) + lBocna )- le o eo cn.p)
. fu Y, AS, ro + 3q) - ( 4 H 2 s + t7 + t64 tt+ 7 )(8+nta,t)-t
-tti +ts -l
-&SCn-r) - 3s(n-t) =6 .62
I
=a%= t93.
1) 3 Cn) 2 t,t<-tL 6Ca)=3
lt ao1=E*,ll =t 'lAncD'D)
6dbea-nd 3Ct)=l b7 ,,-f"1 2z,a;a'a-fy
l:' (n'' u8!r^ gcn) - gscn-D -J,scn'2) -
,. e. Qn - &.Qn-, - !'olr-" c-o
wIo.,aJ
Hettt
Qivt* :
UL)+ Ca) =>
(L) =>
-"1,t =2 o.,
,r) dalue.otpt L
9o l,
uH/. )fL C,t t L )tzt1C2- .,t/-/'h-tv.Y,!o A)\d Ir. t'ct' .
. LA'U
vn*" - bY"r/lt Yn* tbl'
- F-t l_p >-oo8)
Qiur, ' , Yo* u'6u*,
+ 't.y4..t-
LYn=o
XA Uana*ut:frc e?u-*in
4 oe+l A)t- | 2 a, 'r . 3
'f2 -lt-a . o
(r -s)b-tt-l =o
Qn -- t, Gt) n +xr-($' '->6
6Co) -ggCD .'
4or3.
1.Q.,
a., --t )
1. e., a6 ;3
t.2., qr.lo
4" : *, ('')' +r(. C Z) '3d, -r 4, =3 ----> @
Q, = 4, (-fi' t- t<2 (3)t '
-o/, t 34z =' -)
@
Jrap = 4
42 =ld,+t =3qt = 2
.. 2C-t)n.t.Uf .
oro, dnr, -tQrrr+ tlfli-t -A4, 'o
1a
1-^ t 4 .<oet
o
o' + J oti
42 =2 1Q,
Vr *44t
G)-tal => d:+ Ja, =-J ----->@
-t- 3xs =o -__> @>ot, Ct)z + d, ct)L I oL Cz)2 = -J+9 da : - c- ----) @
Olaaaetor?,.s.tz\ g/^er.r;L -Zf - tr" -+n, -6 =o
o.-L.t
I ll -6I
y, -5t +6 =o(Y -s) (.r -2) =6t.2,J
Hu.u LrlL "to a.lt A.12 t.2,JHLn t2- en = 4, (t)n +or, O)n* 4u (3)D _>O
9tven: Qo =4, e. =o.ez=-f,ao =2 + 4a = xttr)o 1*, (z)il t" (a)6= 2,
et.e./ ai +otz + a\ -- i __-_> @
Q,.o + Q, -- vr, Cr)'+of, (z)' aolr(2)t=o
lt -L-qe
Qt-G) ? aie +cort - -a > @Ct) x 2 -2 JoL + Ads . _4 __-_> @
(b) xr * ed, +6,<n = -a ----+ @Ct) -C+t 4 eo*.2
l*;:,1
t'5yt-
(6) + Dr+A > -J,
@(D =Y d, 'h+ I -d
cl , -2 =)
"zlz ,tQ- clt-11hep-
n22 a^d qo>t,q,
()n: I ( a^-,- ar-r)
* Q, - oZ Q.n -, + &, Qo-,
'-. lJL o4a^'4c// bfuc
)L ,-aA+& 'o'.,1 -2rG-;;)
o^ = 5( t)^ + c -t'L'' t(94
6aA.)e
Ue4Laz
9ol :
,rz,tt*iw a^. a (an-la'-r)
( A.u M l= .>-oz)
--o -) Ao2t,-atfu'- ,2 Vl"e- 6At
2! 2.1
- l+t
:. A = t+i,r-i ( r-oo7-, a^, aitdr* ,eaay*x)
-'. 6ol,*,k^, ,,tt d-n =fr (r+i)o +o(r-i)"' d/4r^r- 4 orn d B 0'4'e' az't-tl*-'a-1 . W4't aa*^^-U - C J6 "fi4e'1' @ta- 'LLq' '*t'- taille)J)
a,t-a -<-e-al ,*i:*'", co'AXa^*)
rla"" 11>o -l>ats e'
{ ' r+?Y. ,r[c[, o +i oon eJ
-lrer-ott t, ['ifi,O = 1"" (9)
a-r.d fi umo?vst't t*l'-o^'t'tu'
@4e4-"4."- rt
(Cote -ri 6b,o) = c^lne ai sia t>6 , /oT n >,o
;. nt-lTJ
b'" P'i'"'
D-Ll-t) -/l-- ) Itosnr- -ur/ l-- _ _t
LA
n (of l,'7.,,- TJ)r[Ll-
,l
u:e .r>a,.te , (,+i)^ =
b ,r^f *tr, *)l
A/\d
an =
te* C, , 416 , eL =;c+'s)4maL.
(2) +(2) +
.Qo =t . q, =2
Q6 =l rQrr?rq. -- (G) fC, co" 6 te, sio
. (G)" k*f -t ,; :t )]
->@
.C,
a, = (a)' fc.L'
, a lc.)L12
,C,tc2
) a * =L'/-')
cotToprsh
, ,riJ
"
[*q)+tc'^el-'. ( t)
u^ia- L'ttt' tbtt
9'-, ,2 -t . 25'l ,
,,oa)il2 ""^axa't
, h )a
7a, tn € Z r
a-,\d m odd , g,t'ove cltaL cltu)-!- z)Lt-'/'
* -p oi-h;tZ i-lr{,^ n ;+*ch 'lit* - d)'i dt^ pn-'
(A'''sal:l >eo3J
lA+ I,
B1 '1r' ?qfa*"1* ,PaZ*otPt' a-^/ crr''-
'dr.,.,in;r' fion-caru' "1* *': s, t e X'+ Lo'bL
rlstttmt -t'llt''- )t - t and g€ - t u\g^ie L'l-i'"
4 a/r-;- '1t DL;"k^ 't'f o*
'b'tt?' t'e^'
,l(& -t )
^t ,s* 'a
argl 1mc-a- ft)
Itert u ttl I 2- -rhzt-s.t
Xt eaz-z Js't = X,m +t al'dqt , -'a m+r # 1,,1,€N a-AtDL -t - lt- ' oz yLrll
r./ a-^d
"'r;4e- la ^
-(ts-r) =(7,t*') - (1,
=. (?, - 7,) m', h t^' j.b't'- ;s6nL-"-r)
-t, odol u)o Jttu)L tut (t''-)' II and cbr' 'tzs
t'1t 'f o Ltouts L'2t +b
/3) Urn/ \rtaltua6fi'*1D c2
-ZcXnoQ-enba,
n(h+t) C2h+t)
I
?*vu -Lt*
Sol :
6
,2)2: t 12 13 1-
cA.u At
J.+ D'
orlA
t k+t)
d* P (o)
(r)
I nt
.tu4/^/-
0D c-Latr:
ilt,o,
H ea c9-
8*suno.PCt) : 12 =
l.tsLLnn-L
P(rc) , r2 + zL+
,olLu
r(t+r) (l.r+r)
6t/<rtr. .
(2k+ t)yt U-t4
t--->(u
- (p+r) fzscz **!-Y-,--.-.,/l
- 0<+D fcCtcrnr)+,)
F(r+D h ":2,r: ncn+t)(2r.-t
tt?<r.^+ H :-
*rrr-a-u'h
,)'/ o-afe
"^/ u.tVot
P Cx +,) ou 12r,2
PCr+D : tL1z2 '1 ' ' 1t<e+ Cl+')'
^ k(x*r) (2Rtt) , -L - o- --' +lk+t) /turb9 CD)4'u
l) >o t1-)-6 12-
nco+t)(2ryt:
l:l b
lh) l'..o, e
"It4<rV
bl:
oOLa+
tud.lad
\7rE
h- sn .ri a- .)t*r.z,t* 7,az
iad-z< ctz-da
F l r,,o
Xo* Ptn) t
Jo
u4tLdA
Hzn@ , A
g '''- 3'" = ,-u..
-/-rr.r- ,
t-,1 a
s3/-:,
,rurr*r7,t .-e{ ->
-'. Pct) -, u6^r- .
A +t tr-tu-e ,-*)rn pct) & Al,<,,.-z
Bk-r* =G)(rh) /* "y"- i*9a- *
Jo .bAi ue Pckat),i- *-r,,--.ok., k+t R. k6 -3 ,Ba-s-s.
=(r,n *,tt) I - u*"
-,(ar'n +ECal)-tsCtk)
a-
oru)*7tz ,4 s-.utr*u*.
"1,<,<,-4,t!op.b
{e* *t dz-.q*-tz- L-tat- cA'u' Y1:-n/p '>-eoL)
/4<ca.n rhi ,r* ;**:;#,ff*f* 'v'
qai,hiea..2 dr:r;:C'a
--ai c
V4+ z-.x-u-z& .
,4, a-
-'. P ( r+r)
-3 u
= E fem+ sJ
q,\aa-tV *t tt.
ts). ,7u^tI ooo
Sol:
F4zuQ PC r)
/A. d ".1
/D
a- fnu,la"7l- -sS8n- aD
-.1)
o-eo;+ )
')rg+
{'* 0
oLat .*-:*
&12 4* ,4
a,* a))ibt*
,0ttt.al =uzYt ^.
J,ll
zx czt d-an
/, laLte) '
J,2
/669 '/'a*
ht = ft:9o'lL1)@,W -ltosJ =to
I n n el -- 119991 .fr t' 't) = r z
l?v tt --t
4,ur-ila *t v'*)e i"t'lza h'st
,*, U* -aLt
&/;;)'4'- ";/a'^ + o)t- tt
1 trL
J,U
wit7-
al r'"i'*IPa)"
et 9 nclrutu' - ExcL'naio-
tAl+lBl -lnnylruLtqo-t2t)o.a&o + vi .i,rl-2o.+ %9t z%'L"db
e' hA x @'L tt'
6 -dqx-
J4J,LQ-
a;',.s9tl-t4^4-
a
Y;^d uUr' 4 8-V t/2./1c!- -bel-aro A "'-'tla) l'/-cA;*
"i/ 6 - aq L ,r,rn )"q qPa*abt '/aaeB^ '
zoz'-iii v ' ,K u. V tj -oo\)6 -Aqx
' go*t'n z +l
C F4 x) C r -6"<)-->a
\t/rlal-" ,
A-ac,Q'rr0+t)
n-+-1-57 t_6k
-A(t-Lr> +B(t'5\)
,e -ff = a(,-f)p*6 -as
*- =]
l.L't L "9 =26A
Fu* 9<' L- 41
t -AItA=-t tg
(t -st)
=A(_l_q/ ,- 9r)
t.9.t
4"ilr"*; #Y0cr)
t7) daLve ;/J2' tt e ut'<'tttttt' vttla't';'$" a' -'lQn '+lo$t '=o
4ox n >.t gt"te* d'14't Qo'to' Q ' ' ht u'snm6 A"'- "fi|Lnxrj,t.' (A'\r ' N(9 >-oa)U;, t"- yfu^ ..1*ct4^,LP-nct- ,4ilar-b* ;'=.-i u
J-an at-V*i^-d ae-7'-nc' -2' ce'?l4f i^* '4
7" L acr)rD , -.n8cn) =-C+6) ra(6)
=\ f6;:l.tl
o '^1 ae 9zt. 1 utt'- L" "
Q -et)
-,[,'r{ '' * aY+xJ-l
-, f ,* Et- 1( 5*1'+ " '-)+L-__1afr+ot +(br)"+-" )
-,9 tooln **) (e)nri -b
I
a;!3"-Jto4=z_. o -
# o"'n -, Eon-,
zq,uAh.;h, -r,y { atd-r./*I
x'+toZan-a
aL oo
alnyg
,lD4
rttQ n lD -'1 2 < ao'-,r-D-'l. rox, ;! ao- ,
7n .o
l) -2-2 :on-L
lo t2-x7+)
to -29tGtr),(J-tiLr-ot)
G(t) = A +9-l-a| l-D,
, A( t - t z)'' -t rZ (,..nrr') -'
-- o [ * L.^+ c2 x)2+ " ] *'f' t -t+xtt+''
Q(*>= a 2 r" n'* 4;f lnn"n>2-
A^: b$4/-cli^r % xn i Gt>)
4n= At\ +8 6D.t n.Zz.
q, -- tb2 P?t n>o i'n (a) , ae--yaaU
Q., ACt)o + gcs)o
@ = A.rg
A +$ =tt
mr&rpbt**r"r""y -fan
axt+z
-fA,A
Qtl<n
=> ferr*>ro.-o.{ -ztfGot-<"] +ro*g(\) =o
=2 Gczt'ro-4t^ -azfba)- toJ + to xt(a]11 --.
=> G(D (, - lrn,or,2 + 2qz -,o =oQlzT ' Io'zq*
te ., Cl lveo a , z/1r , pub h >) "- (a> , * o
. .Q, > 5cz>' + gcr).1
4r:LA+r5Bl'.2., Ln 1D g ''ll
5otvm6 C t> qCD , @e_ Au fi: g, B-1
1k4. e n,J.2a +a.5b
Q.+ b >o
a-=--q
'. Y)-Q. Aalu-tziY' J.Aj, dAsOAat d
e^arun tot r-o u-s
\
@9
1d,
datyz dck)+ 6gck -o =9.5r,).5.(a'rr' s 11 >-oa'l )
Qiru* 9e-crr.uu)ca- adt^-h-eb- k
Utrr) +a6Cr-r)=-9
J ht- a+zo ei of-& -A*angz*t-out ail"bb,- 3^
SCK) + asCr<-r) =o
Xtz' ctt"n acbrtl,sn". n? &i* -:,
(Cx) +6 8Lt't)='o
4L,l)
$or m tc )
H zr.-to *U 1Lr* eTuaf,rbl^-J
ick) +EtCk-0 =q L<ctn-s
C+Fc -96L =9
\'-,
,.tolrhoh.-'. J^L f7.; cua*f?)
-,ok -;
J.l.t- Au.L4 a)(h)
$(x)' = Sr +
aabb&-- tj)I
Llt
= q,(z)k '+3 4)
fn C"> tDz- 3x*
=o(t (-a,o* + -4
i.u., q, ** --4
qt. [': .3, --t@Su-b+*.fu,cay (Bt -Z^ tc> c,:z g&
&(x) = 3, c-r r* **
dctve'. Dct<) -1.9(l-r) 6 p(*'l) 'o ueib''' Dco)r&'
DCD=8 ot Dct>)22 C A. \, . \-t tl ,Nl P >€' d)o
' o 2a uAia'aa' 4J.ttp- Ct>a;nZ,ttr., a lte k-7" -
y t- 77+6 -o '-fiY" I ; a '<oe*
t o -l L
o t I -g
y u* K'-o
': Co )
tq)
-+ - I -<L
rlr-(=oy '.t 3r - 2"r _L: o
Ic.a+l) -&Cr+l)>-o
Ly+t)C,r-:-)=o:,.--2.-3
Jlt- aolph,ot* utD&) -' d , (, )K + x, c>')t< +d s t' tK -->@
Put R=o , DCo) = X, +- "(, +<L : Q
put k = t, gcl = e(, r- dr'L * oU-+
pu* k=r,. Dcz) =4,+4t.4+ Xr.1 ..22
d. +.(. +o( = IJ
d, +d4z - 343
d | + !+4a +rt d3 =
4oLomg Cs) ,Q\) ,(5)U
41'= fr t o(, =2
----) @
- s ---->@a> ------) @
L,ta TAhU
dA-t-rL Vd'/.<'.'J ;4 12)
=n+) tD4 + r('3)x
b" A"L4 t :-l{-i .httt njU
DcR)
,tut'in^ ,k
&k+l t o1
= rr+2 +t'-5/ol
.tlt- rt-?uxzq zal--t'h^ '
@
S4^0,,y ttak 6l% arrd-,t*r- 1p" SrbrA ?U^upf
*^rrlVrrw,odivd" i^nl&-" ) ' l6e1a l\o'y o" ?oziliw-
ara;r+',s) lt *n I ;6 '2;+f"14 4 ?xiwru owtttr4"u o-l
L Lortifi'a. -q4 ?aodr,,* 1 lai'n'* n+w"Lan'
t'i f; 'en\bJ
$
YY"i-F lAr'L
BuL}'", ?fdA'ft ; ,\ot",n#vwvrs
l.t-r,,Jr,f,;* i"Wd'o's;' Pk)
ptr) Pt e) P(s)"'fC()t,
wAv;"t'tbr+* 6
0L
.AAl VLUI
l-Lb, ru7ln-+'-
)-a,\s@tdw
tknotrt-w ,,q^ lrr<-. r ^!.
/L^OJ.u^-Ldr'
o-l1
6', -
@, A*r-* le.F.r lie. firl*,-* O fu y
Te* tu&ttiv ?rra7u
Po,rf& "tr 667 "^"n*-'a*A
t\llPx '* ?r'l'ou at Pad""r)-+ eOl: t't7& 'il '
fa;nrvs ,
F@)is M.L a . a Z ,o PaiY'ne .
ll+nu, Pfe) \ ffi. .
S zn n 4.\.
T, fD t?ft+t) -R k,q
fu 1?, 3^,h* u+r ,\ trnrio ^ ?ri,wu,W,
A+ak-".f v 1il, % k+r ,ri -nt: a Par',* ,tr,rnk+l f,rm h, WhHt^,,u Pq,, f. ,*nys a*-psk
-A/rd\. A<q/1k.
Atu-ry1;4 5 tr,- rurh-- l1foNfw;^.pq0u.t
?,t;,^r) -ot ta ?aodu,r $ k;nn, . alt" q/ L3;'lr,<-1 "a,
xt'ttut ,,a
Y4, i,n
41.r4nn,b<l
{il A,b,c',a^udn
Paimu,a Yad.* "6 ?<;,mn. tunq,,llfP*;; 4 &,*,* l'
PA$r//^L,te +fv /yu,wyk< _tl lea^|,a, i,t,kl,;.;;*:;:\;xff"'ll* - ,-o.,, 'f
&'g'uf
^"*dr 3, 5 ,ail'
&a,wr -dr^D+^ it<-Lrfr,,- t- &ow fou* -au-
d:"rqLZ '7.
@
lll . I dooo -lL,J' farr.rJ, arr
h.>4
n .o anl at"l ,
---
[ananc nD).LH*-; = Eff:]= ['r]-1.
u^rX t"^onr^to-t i" '
" 0n-5An*r
SoLb;,Qturn '
f,nn'*g '" o
"Oad
.lDl -lFn: 38r-9.
: &'76,
anc n oj
, AorP
fi", a,,4
lff***, ,&oL- #s. *c'tt*"u-
vn"u,
v&7=l ,
7- an- 5an _, I f,an-p.-o 2 e, qu
q,,ib,^tX._Sv*L=6,
[rr_a; (r_s;
eAil,td)^ sokh^
A. grd.
qnr.4t.Joo {, dD
?ut n=, .rL (*)
@
--o
----, @
46 : a(s l,( p..
0 -_<, +4-e +
; [*)
=41 8+D(a,L
2 84r +d 4a
aa --sp. *7*r'-
Pu-l. n=,
tul
t
(o.)
t- - 84a
l= ^4a.S&, 4,,nm), 44 ,,tn
tdaa-4a "*t
(*)
t> n* ?"*- .
?.r} -g
A o-r*tr \,) qrve.esr-r y a"n?r*- o. -J Qu=.oo".r..D 15
S-l^j=.gcr)
gC\. )
inl+l-.t-an
- +SC\4-l)
n n'-lr+.}-+t+ . tz-' +;' -- )- -\(A,u.t{o'r/orc-z-oro) ( r' '\a I rG)
( ^'u AEaiMA'l- )-'".a,
-+ !+ s are-1) = u\. +J' , sLd)"| '(A.iJ qse /u+r-sg) e'*o rta)
Fi^a,{F{-L q,rrul,.H^J ^ *.^ ,1 f,-Ve-r'.tr-ci seavLnJ-Q'.tujlcwY _
(A'U ae e- t na'r 1 -za) c-e'...o" >d
., e\,oti,.o "* L'hq- Fi k'nJ'z d
+"= {--' t {o- > ' 'r)) -st Lh-Sa-L!a t+a-3eauL4rrc.a
.-6 4c-tL1a 4i:L.A
-P -,t.<LLarA
-inatr-I c.naiH6r^ , {.=1, S>r\ -rr e.^ra:+.-G.a' *tt 2da, NID '>o'<"$dl)''
^ r x-a-k *€r --!'L{ ^Au-cE
ian " 1*!o r r , -, 1 .r',-ri ut-. o'r *
3 +L ^2,
( ,s r; n f1 ,ra1 - .l-\ I D >oe1) 'L?'sl'
"2r
.Ja"gpoe"ullre
t r-o.t:t^-r.,--t:1'.^-[
|it3t(e ^,
A ea"":p"iar-*^
e- €, zL-e't.Ea-'t
rjd^a '
=g[,-*Cr a\n.a],eJi
Fr^A El^.o-
r> -f r^-,0- t t'I
"i) *\,-.&
^--i-. 'EtLa- 3et o-'.a.li...- 2.o\ pl6 atttx-la- '"-X'^ieA't-.:5r-.cl:.-
w-eo-.' *\r) . 1.6d tsha '-&"3*"-n v"r'rrn4p-L +.& sru,-u!
.!.- .,l-o'e':' -!-., U'a -auaa *h-al c.S{ra-sl z --} +\a'
') --"t
i-n -v'a 34rn-e- r<rocsb
La.u nt t: .>-ea, Ar,z+q)L€.No:16;
e .[fr"fl^{ - -o- -.n-a\-...r.*"-ot ! ,rt r-'aaa-ug.tJ LLat eao u' '-
;;;;r* :'., 'stui'b *". *:.*::.(n.,; orrl.f, >aol, F\)q)'a1) LrtF\:;a!
Pa$*urlzsY4c.-r&..tb
5,a[L6qt'd 'aJ
a.
q.
?aEea.11i11!- *h.e i..il5er-a \-a!u:ta!'! 'a turd
\aJ -'q, e+ '<1^q 'uirqta-L
sla)a \
.ol.pr a
Far .o e {r -.J .n
h"=i+i"o *r-rt-o3e-a--
oo=\, ar:a
(A' U r'^ ll lgaa)
urau e tho* =\iat. '2-5J/J*' ^a. .1-.,
',<*-J1 t-r^-t: r' 'JJ!'*- t a't-l q t11 14a3, (?.N,;.':
., *3- 6 Y,'*)-r. \ r Y'.,n,- 6Y', =o
r\f -o >aaa) L? '\q"-q'
a. -- a L'r'-' - ar,-a )
.>orl,z-oo- ) 1" +"'' 34)
( A.\)
,.A^r-*:,.." ts<6\,e iJ'4-t
n Co-rt) a2-rt+l)
6( p, 'u vr(J
e ::1:4j,aEr\9
r,.r-...4yae., Z?
>-so +ka-e a^L )Lvirihlz
2-2 z. g ^^d +
.3alrre
.,.i^trb
U.e^ q
-@t Pa4\!e'e
3c<'t)'-a1 cr) --l
+1.@ .{e c-\L-'). {4<t<a
ta-1j tt-u
!t\q 1_e-cL '4'4rl'-ar\ >-'l- -'d
rr\4thrjr.afiJ2 ,''2- .1- '-
(A-rJ' rla/oer-'aa i e"'a)
'-' > 2 -''{Eh:Lgcn-t) -es(r)-1)
1o '
t=, -
.-.a* 3e"'^-o.}r.5 -{LL^'Li61 '
(A'u Fro' loE( - t'"? )
?--..e f=r..-'l e:- f i- 4- ""'Jtid- a& u
" G%
4 i-^,tru c LiaD
tA.\r. nlov IDE' -D@+2
\+06
-rte-r+.rpd
^',foAu-,ga o'(
(e'
L?-F\o:31
rq.
A,f ;'31-A2- !=11
u MlJ,NIo roa6)t e"ro: g<
A<f - A
d; we'tli.o'- .1__
9fut a2t E--.^71-Ln u M"r/
% d.,'1- E"Sr-
5*^e zct{ B
c,1Ar-C-n iAP, tie'
A-E-c-D-E-B*A<Sin;.-e- , 'rL il'cltA"'- d "Lo'r\r-e , line oLeue ?dr-
j5 a-
Aace- t[c. Xa* conL^ia:-
:,t, a- f.).- T'fL'
t+ee e)1e,= o4,
x-^rL
Eder;--t-Et*Let;o,n-
21'A-.t4cle , rE
B-p,r. a- -teluJa'r X"f Co-rL
b-o o- oq.,ti X-'fL I L n .l
a- c-o,^yA.eLe X^4-
Tfie O^r-!
*"-'7 we'rtex CPb
-:-i,^7le ,-,..f' '*?'-'{'}-. -iD r.-rlleJ ou
o-
the
Sel"l'o.' ,
aeqd-oa
d.-q,.-
/ex
nl^*,a'("-r)
T"FL ka,.
J-q",.-- , th
TfFve,tr71
ri. , tkc^
, F:.1Fv e-:.;{
Vert€r>< a-rl- a- 4 e'q.l-a)1.tq5aPh -r.:E cq"tl ed k --l I
CFr'.l)\eLe q5tePh -rl-I -r iCe'.'p\e\e q".o.eh witE rlnt
tl ot I De c 20\64
.tre,qu-to-,, X--f*
:r?"J-.-r
Ji-^.te, lfe
W'LeL q = [", r) Le-
u/;th '.' ei-q='-. -fh e-n-
h.-J-,5Lt^ 1 *)',eoo-,,. - ^I fr \)
tan u,A;"tJ
...1
NID eolQ' i
c'^F
,|uuX(-) = e.i'"., 1fre sr,uvL T d-e4-"o'-
ff, d,L r.,,,,d,;.-.}--)" X-^fL a-=.
4 *.-- B the- ?-,-ft -,.*
wiolE;a^?
sr,'.fte ?-"ft" q ;-7 tcl*? W'f nu Nle aorrl
Sirny)z- Ga oe\- q 'L'> *^;) L' Ln
.:J o'g w[cT) ca^ te-
{-,,e &jer *t n-erL ''^fLt, '5er"-kf.,'ttl- I ;(;f^i^ttt>-'.-J" ' ;,^lavl 4J v,
\r u '\,, = t Lo.) , aw-*-
tr.a -t J Ve>rbe>< .^i^ w'
thaL e-e''1
^^) arlgttse'r ,-^A
i. vr. f se tFr q.L n-o eJ-q e: ; q csn ne<'L ..**
Fe-r F<a--fle , Cqr-i Ae4 tae. 1-'1* 9
'B*[e..ee
the werhLe-s
tEc r./u^te-a
FCc) hn5
vevtLe>.
TF,e- G\ :'D o-
a-y,^. ;.,^fe- X-^fL ^ :"'"s,L*fl f A u r19 zorr]
- ; -- .T*L ' a *-= -]o'te-Lrr--* e ve,u{ Pa-irr "t YethLe>> '
q-+ ;'- .')tu| c-e'^pi'ete X-ofl'
a- f*.**t'-
c^!\. var.-tqs,"- t5urlv e-
,*^-*
.-!q!1!i-,
P*n-Aa x-+ f^u FiM *,f^^A
?-',JLe.L
-ADorA i--; wi";'J"- ,"-l-,
c,;--Ued o- Po'd*X"+ 'A F^f'".AA* oze c\"
&\,^rlie.'
Ce" yl'e
f c u Alq 9o,l]tba- L+e 2^^F\- % O.,- o,.J'r...
1 K",a T*
s{-.le*q&
'lSi,tu\l;.t^e d;4".'o{i1 J5| .\ I
51.*-e l[,e Nece'*o:u1 ""f, S\^-WctehL
is,1 ll<. e>;-oLe''c'e. X a'ra Eui"ez;'^n-
!1..^"o* -T'fL io, N,D '2oroJ.Jeun;, .
-o c'n"r-*'eA r^r'k fuaa *
,
t.rL n,sL a)1 Ful-e'r C)'',LLL a::fr.o- z= atL)4 t*e 'v ettvc e>> "B
Gd-d'
'Conruzc.EeA.
f nu N1D
A
Co^*e*eA ,
f.fl, tefh
t* DtLe-
14* 7xItro neAe-aaotfre''r
db
tEe
?dE "i- cr-'
EulP- PoF;]
bee.
e nlrr.t;Le4".
X-"FL .
A,o roJ
+o;L +!n-oJ.e,-
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8.6
o-vu1 tro elar".,entJ ob o- ?'to.{p ( o, ,*).o-.X\, e & .. By IJ^-a- Sx-utof,ca. oi
o ^cr en''' , i' , b-' , 1o x rrj ' tr^k rh G .
-t- .-l rl- b *o-
o-rj sv'. 6' ,it e G --> f,'x".-'e G.
(q-*b) = b-r * (d, *a.) r. u
L B7 a,rao aoti",' l*1]
b lF e>Fb
tw:^1 c,pb e W-? o[ orcdor n riLo .t'\e,
? xoLrp ( zn r +^ )fAu A/.{ 2-5I)trl
Let th-e c!,li. ? r*r,f" tc^ ,x ) oN o>td-er n X"nnrq-a.r1 .gr.,yant a. € G .
.lo .' tr.rq.L ' . th4. - dlrnenl. .( g arte ^,a-Lc_,..J =.9:eL;9 , 9, 2.. --) G -{trc}L t}L^ t ltrl = o.
Note rr..ot LrI i^ b,\a- g e.n orratox o! L z-^ +.Jlrlr = aj Jo, j = o,r..., n-r and t,qr.ca *t ia
U o rno:tphizrn -
e-e) Tt t L - l"omo..o >tp ki:* -! o- gxo"p crr crlo trqI ,a",rp &' thar.. Prrove tr.La..L ! xoup rL L"or.-orn-rrx4hirn?to-teyvu L"b,.,litllr . Leg_tg"_j]Solr.tlo" :.: Let o- a crr., or"A t ,] 6 l^.o"".^o..-o-pr,.,-a'- {rrar'.
Cn .;lo c" ' 1,,11.e1''_ f, t-: e cr/.
+ tLo) * e' = {to; = d ta r.e1 = t(a) * t(eJ=) e' = tr c.) t uy tnlt cck,.6 r, oLbr;n L^,.rJ
'4) Shou., .er,.o-t
1,.1o rYto ,,i.Phr-6n-\
Sotr^lzt-:
o[ o. m-o"roid- uJt LicJ-'. r-r .,ot
Xyot?.
.r,-ovroid. t -rl,uiCX
o-ea, * 1^1.
L noh a- frcup
L1r 9+; - rt-r;X ar.d Xwe ar,. Ereo^Ph.
=eb:-9-tdn of4e-rov.o-ir- xryh^. ( S, + , .) ; cc.rro/. o- rrrn7,
+ r*\.e b&svry opero.Lrirn.t -1 onA . gh S -s^L;zt,flru ".$o.t:I.o.o&g. 1*"- .prtopenhor: !
(. (S , tJ u: o.. alrqt^o-n qrlo"+.
3 . The o pol o-tion 'u &b<-i6rli.e 617e, + ; .o*-oL i,Jorr o-, b, c g S.
i + tb+.J = o-,b +s-.c a-r.d. Lb+cJ-a ab.a-+c-aI
l"r; Stob r o ?xor"X.l .bueo1-crn frr ixotfz 14 &1e-
I c-."o",ru 1.,,^a? fAO Nl) zotz\ try_yg!I Sot ^ht" t
I tt G u a [i*]" 7'<ou+4 ar\d' H t.E a lula
I q r-"+, ob G tlro-n r*tc oxd-u o$ H r'r a diwi'rortI
I ok orrds:r of, cx '
| -
- 'It^-e- clhverLse o[ Lo?.''?e's th'alru Yt ;" J'l''"'
l,I
l '"J p".b;* aat ,Llxoup a-nd .t ortoi4 . crive an € xar"ndeLI "b -- -y.r'.;XrtD.+ rrrt Lir.r-L ir not a mcv"oid.I
I SoU^ Uo" : Det. Serniqrrouat:'I
cet s ba a- none^."fu Jeb q.^d o lo< a. bi^atud
I "P"r.oh;" sya S. Tte- 6d-?el' -qic .aVtl.,., L-g,") ra cc.u.e.{
I a; 4eri1rtrru+ 4 I,}.. oporo't.io o u a.rroddti\.e.
I
I In o.tra-o:r rr.ro-rtd- ( s, oJ i,a o. ,aer lTxo"p ^i j"r .tr^.
I "'V x,9,2 € 8 ' (cr-.,g) 6 z -- t<- o lloz)I
Honoid:
4 aerniq xoup ( H, oJ trr ibn on iduhQ gLer.,1e,ot qJ. rr.\
lo,lpect to frra- opo:ro Lon o .: CQI'"d\ a, r<-onoi.d. ?nO f:..'-e, tr)oxru , g^ a.l{€-LnoJ.c .ag,, hr.," ( x , o) Ls Cqlr. .{o-
'-o,ro ta * t"" ,,i,ry e H..
(rc; Yo.) o'ze-nd, t '.el_q ,rcr4tl
:. e H. ?ox
= xo L!52-) '
oJ 9(crnent e- € H z,-.<-l.r ll,.ot /or.
I
t tu] ,1r;1 = { tt 3, t,* = Lrlr u
t tsl, lsiJ = ltst,yzlJ = [2J + H
= l3J+ H . H
=[q]+H, [2J+H:[sJ+H
3vO"*.nod..,, t.,'r ?r,.oy ( z6 , +b1 .
-foLr^hir":--Lc-b .t =1 [o] , f ,1, [21 , b1 ,lti,ls).b] J r,z a-
?t-{l ona H = { 1"7 ,\il) bz o. 4til. nr.4uf ob zt"crr\.,ter +b ( ooldilin mod 6J.
Tha {qtt c-oaet s ol H are_
[o]tH =tt"l ,LilJ =n[,J +n =,[tr)., 1rr3][2J rfi ={yr1 ,lsl}[r] +
tr+l +
a'r^du
[s] +H =
[o]+r[,]+a
ai{- I11o drlhr.,"F t_"it "r.4'r,lo" ol H o" z 6
I
-.). tJhan
Jotr^trbn
l-{ l^o.,
t -r\ll a- 4ek ob e\er.ae.ruta {os<rn 61 r.ro r.oid. 2
f6rl ar.r rqql]:
4nOtL
alqefarto.ic
iole,ltity4V6$(H,.) & mcwro'rd 4 aa azl
..-lr-1r.. ,l-o-rpec[ to I*-o oPejlot;.'
3t) Det; .:"r t crive o-r. 9x o".pta o$ 1 ru;gL\a-! z at\o d.iJ, zoxl - -'lfrv A l*.-z-oS I
-So [-,,^hir.-r:.-
.,J t "icl^
"b
An afq e5,.q; c 4W6,n LS, +,tl^r- ta1nq'jy o pcrrotio 6r l and
.) t)
. qr..d
q,- r r^-l
onSo. ,u-rl?
4.\-ts;Ll.l
(- (S., +) & o-n o.betiqn Wu?.J. ts, .J i o- zennL ?e"oy.3 ' Tlra- o pqr.Ll-. itr . i5 dr.rtNi tuut ra. ovex + j tnc.t d, Jon
4rb, c eS.o-. (b+c,l = q-b+b.C or..d
[b+ c,) - a. = b. o,. 1 c- a-
txor'^a1o.-
+t.e- jo[o,- &g propert_r]* .
'lhe- :l-,i.g (zrodo'-or^ -(Gce- g x ro L
, +to, xto.)
=o, Yet IU hoL ofi
+o, 2 +o
u"bj ".o
t
"i. Zr o.
y) t\ 5 ¬ar l"l- .<oL of, poai.l.i.,ra ,.l..t"gn * 4 too ,,"" ec,9 eS, dr&;ls axY -'nL{r,Jt.v"-;h
uJt re.U^crl ( S, x ) .a ^ rnonoicl arr,3t tra1i;1l ,17"-11.t * b
o-yrociqAi.,<r. f At-:-@J
.9oLtb" : /oo ia r*,- i&^Uh s'te''€,"L ;' (s,x):
J ti"co x t< too = -.-. { x, too} = :' -rGQ.
x alroo Jof * r€S.Thotejoxo ( 8, ,c) u q -*roi"l. .
s) Cai re a-n 5.ro,^r+.e ofi x.,-to Ze.^i7xot p. f6,., nt ts2a!-SoL^U;,., ,
FoY tl'e Ze.".i gxouj C "l ,tzr€rn - na{a-tl-ir'Q, t"t -9or4
LE, +) o$ [ rvr +J,
bJ S1^o.r, thqi -l efY1i ?xoqp l,uo,-_o_uapt uia n- p tn-te:.'.\tet LnzPe.roPenq tao-.eoluc,7. f R u n, rp *!
Let t =(",*) -)
4), b\a aeL e
r) oc ,aq-h -ter.n r
L t1 , A ) 6e- 4e,-i17t<.ot 7
dtr.; a f, cr_;
o[ o-u
? "tD"1,
SoL.hbn .
ho".l"o"'.b:ghi,rm.
I v\a./r
.
x Lnd.ernpo kent €ft u.
r- xJ( = r_, t[x *rj
t.*j: tq*;atc*.lt'.,**:t? pxn*xve{
1,. ,. Ii
H.
i ) t4 H is o .acabqr{Dup o[ +a.e ?* r+, ,,'r r arno,rT {rLe_
x{ht cn+e*^ of H tn Q, Prtove t}ro-L -t"er.o ." o{rfgr^r. -4Ldoq r{!up vij . . l.l . ft ,-, u lp z_o-Ll
.foL^trbn:: Lok Flo- * bL 94;?.,1 co,t* ol H ord. c^ q)ha-:rc
a-9 G. ++ Ha &- a- -a,. r^nrr-o"{, .ob U, U4er\ € € lt o-
"uhqia e ; fhri ia-or.ti{t1 et-a,-n eJnt u,,. G .
H o,- [ o.rt ggcnvofmcq Claryi cDn torirtir-7 o. - r..t
ort 21Lr;-r,rctle.6o 9teto.F,).r1. -!o e e Ho. =) Ho. :He .
B,rt He=H Ho,:H
31) FdrcL or zul. 9 rlouf o& o>tdor L,,o oJ .,u 7,.o,rpo r L. L":^':-=::{ (zs.+s')
S i t.l ,1,.:) ; tnr onlrtr 4\rb 1r<,,.,f ofr o:rot-erg.
38J Qirn on e](enrga o[ aat',- ?e.oul,. b...L ,.ot o. mo}.,oid.C n" srs ",- "ilJoLhi,;: Let D= t.... -\r -.t-,6,t,\.J t
'rharL t D, xJ x o. aerli g rtorp b"rt hot a monoid-
3, P>t-ore "^-t
nt oup a b ,r"- "r1
e.Lqrne^t. C ao e rr.t -.-:J
fd-cr.^fo tlrrt €(ernent , p(^ ater,,ent a € G r-l ,1c.id, +D
bs d+e,*pohnt t^: itL rrorpec[ to x + o( )E a. = cr.
acfor*ib r.teqcnt : An C(errq,,k e ob c{ rA ,<cid LDbe idlttlr\ ebr"e.nt, .s.T. e*a : q*e = o_, {on eve:lla-e(n.
Trt a ?r{oy lru. Ld"rrtj-q elor-ent e orrk -irrf;; f,; e,s
th4 Ld4hpobant cond,bLbn e.*e=e.Llencq tlLq pr.o%.
4"J L,ho! d.,o yst! c,olr to o' hor..o,,.o np hir,,u ob a 2",...i -?r-"f irto it-rerb I ff" o,, ".,-g
Sot-h.in..-
A h^o-"-o*,,or,aolia. o$ a ,lernllrroup J*.*""tb n co.u!.I- a- 4ery\.r-?,.-oy e..do'.,"or.pt^",)- .
Eva,.,j Cacltc- 9'"*F 1.o o-rt ,t-\,/-aa4 <ae"F
Cge9i a g-tts(F Gi go ge,rata-E r
\---co&,
+-at Cq,+) Ba
- \-dY D(. ) e 41
Vr.!, :a" .l : a. Ser< ivrtaaa-t6 r, t .,)
=t.4 g = o-o *^t = ^**t = ott *=
ot* ^o =91.
.rc*\, . 3+1Ca,.x) -'-'' a'r' "^!rP\i-t 6'i':ar'
{42^tits &l,nt'.Ir.- 4 o- eu-io3'r:a'"'P 'l-
!l^a,{; ^T .r\c 3--*r'
f n' v r-rarloec "-atc- l
-sa,t".Etor':l-e-l
G,r----P q'
e-t-o-.tA-.-l:-=
H \.a rr.c- ='-'1-'3-o'-p 4. =:,,.-a d P'e + l^'-o ;'&*L'"]
--\d e- a,.s .z_.aigJy.
at q ^'"*i L1 'te's \
* az\^.=^':..^t::^:;,
e; f \-g 'i-U *L; \
r-lzY"-o
V>e-" ^ , \-r8:
"+I
a,2 = a- e-
--A-!-r^a"t
-/ A g.a-io ae t-.1
RL LAl.t:Ly
I
)trl^o.l a'e' +).,--, e-z zJ ^i
1t -^ -g.5+jc h goza a 9 o-:- d '
L& f : q-- d l>e a- fl'arwevnoYri*-^ T-;ti,
D ke-, tte,!- K ' -rFren- P..e" e ttr& t<
r.rs-,^-l "Sf-*f % q 'il e"to -:ire''.e..pfu. {.- tr. \'"tf "t B
I ll5se\utrbr' f Au /5 'o\u ., tu I
X"'f:
'-,{
we, kJlBuJ tts a_{- ,k* tt) = l(.- eo Igc-; - .')J
sj^ce I te) = er -i,= a-\rra-qt {.,ue. , J\e--.t
e e [<e'(-ft)Tr. B\r'en\rro:A:- )<", ( fi) ; neL eY-f\1 ; q
LeC" &e -l-'.r,s ele'. e-nt--. o- ; V L k- C E, ) ,lFeae{os,,e. , tt"-)= d o--J !Lu)=r'G*', [[o*b-') = ft')x B(E')
- f,t-) r fttul' f6 ,. t"**-*.p";{= et ,* d
tLo* b-') = e,
t<en ( [)a-,Le k*Cg) :) a>3'rrex.,-,6gl:. n- ( E) :.D ^ ""t-l--T % Gl
tet q .^A ct be-iJe,rbdt l ele-.enL e-
-^X 1.,,e X"-f w.itE
-.1 et
=-f t : Cn -+ ort be ct'6b\; it) ..- a r\o'''*'.\
A e' :.e-y..tc.el"hernerno-r pi[iorvr , t\i er.*1---r
Qi.ren e -; a.n -id e^U\ ,Ln q ^J JarL -iAe.,tjtq .Ln c,t'
Let !c = ke.rt€) .1 zeG \[t-) = e]*-: - , lce'. L6) _,-, o. S*tfl*F s[ o [au.e"d;1
r{e,.,, , ,,.1s }.o5 g +s pao.-s tt-, .r.,,' ke. ' R..,
"a Itr.ce"..^ o-l-
I uteY€ lh c1-L t" i g) i-,=' r
TD P-ro".,e, o_\C _ l<, q_Foor , lr-L >. 5 6., orrd h g r.,* [-* Lx >.*r) = {t*) x E(a)r gt."_,)
= t('.)*.,r11__,; [,t-= Er,) * ! r*._.) f. l'I]*T"-!= tL*r :--r) ^"*=x*f,)= [(o
!
C|,aj-r.l- '.
="il^-t1 +
et I r<, 4 c"t
: $'.9\r-+q
+ (K+ t)dT^"d
q
(t<* a-) = BCa) j €ear o-\ ?eq
IrI1
@
--> {(-) = 6ru)-' $ (*+ o-) =
* tu N'tti- orp, - onra_ :
-ro p.,,e.,- S txr_o) = f [r_tb):) h-r a- =V.t.x.r, t t"r a-) = Q [r* U)
-> t(^) = trb):> f;t-) + {[u_,) = ttq la €(b_,)= { tu* r')
- tt-)bL^r+{tu-,) =ef, Ca,+ u*,) = ero.* b-l e lak+ q- = K* t
.i- eYra - ghs
iAe'."hl.g j. Cr, ):- hsme.,e.Fli.d
x d(b)
k+t
=*
0 ; e,.Ls ,
Cl-co',.:f;snLtsLeL Heq'
SirtC.q
'ltsd-t'B .i= ent_o
f,t-)=,+ (k* o)
"- C,^!s .
, ttse1e ex-,=E a-e q s,t -I-1
= !C^)
-t .i,.- o He,'.e".sll\"r-on .
Ns.,c t (** ox k+ L) = 0 [** o*r1! C'+ u;
= fc*) r Bcu)
H eh ca-'+ -t
A{c.-.},e orl-J
€.\*f"f ,
Leu (c^, *) t=
Porgve. L*q_o,._, .f eu ,.,rrr *,".;l;1'^ -rfiao.:'e,^',
nu rvie ""i"')a"''rir';iijtA 26,
oLc-) = r-
a- X-t*f w|.e.e e-Ae. .i-. _r-
reL (H,k).L,o rn
L= o., Sr)'a"-f T
H
Co >eY. H * a
O[H) = '^LeE h, , hr, \,o..., \-gl-s.. e,.uta
L- tF. ..r,,, J;g6o-e-.,b
}"- Richta1.* '' otl-1 * o- = f h,+ a., l,i* a,-.., L-*.1 , a€9
.S,c'rco ffieoe j-E, or-..e1.^-,e... tn- -r=-,-TL- Q ,*:;=Fil"'-T;.e\e "' =n1.. -E H+ 4 *= o a-au'.-u '
He^ae e.acL .:ccr}didrnsr ;;;-;f"r
.eset eg r+ ;'ru q h,- tv-
!'I ' r< . 'r- -dfl |Lt ces et- T H J.rr q a-zr e eiih!,rd,r'j.r\t J d:".L;---" 1
t=J:*L=, iA,:., Tu'* fi;;,"DJ-iE_o- u'ie. y the:e \< d;.Jrn LH j,.! q * t.al* _t__6 G1
co=era 1.
Le-L \Ee,-:e \+ .l-: ..[,..,--g .j..!qtr-t cese.F-= teH* qr? l-.{ t aa, Ht 4a, -'.' . u*ooaEq.' q= (*t o,)u 1ux ar)ur....;... o(_<,) = oL** o.,)i- (r+* ac)+ ._. -r-
'u(Flx 4*)) (rrx o-t)yL = r-'r -\_ yn{_ -.. t }a [X U"^ e,=)
tr = k,tw
rrr , -;- = \< t.., oj9 = JC
o (".)
etH-i/Lq-rD
l-^r\-rg=e: €j^J.LJl'
lsjruce \<, -no a'L l4Leqen ( {i.ne) . rE .ji> d-t-J;..,.iuosr sp rL
6- t.:) o L r_r\ .ib d- di.ri,-rsu e,p
--r o I* ) orr*= . (G^) b
TRi.'D p-,re.w e,: ltr e I oq- o..,-t.,o -I-Fc eE ver^. -t -l
t'l Lr) .&5.-- E,ej::ll=__J1gp! :aL- - ^ ' _n
i. .|]a'h;\]'r dt* clssu" P-y,dtJLr) .{^rrooial:cwe
c"rr*<l el .z
t Cq, b)
Alo^,
(o, t)
* j-,x)) * 6.,4y/ Co*, or1 *L) x C(,J)l
f o.rc: qld ,r ( o,] * g)l-''-)
L ac:<, ed,q * o1l *t] =-(,)
* fC*,y) r (.,d)]
o Cc")
LeL S-exq, tre tFe seE q a-le. er.A_.J
".::,T ;Ti H;-;' ?'Lwe'.I-H
i1 c!'elr<, (s, +) :,- o z5 errLi-q.4 6.,-P " r_,.., iEca-,-r,ruuL aiIye 2
ir'; /\l^. 4t^S rx- -,J,e.LbiIr1 el".,errt -g s'=\5, LAU wro z-orz)
= fo.,u) *_r'
Lo.-x-L 1
: I o.-'*' c- ,- L alx-.,
f ,.c, rA 1 Xl.- f:.A + 1)+ blo--J
J".-l?cn^ (tI a*d @I *- l_cv-
f C-.,u1 + C,(,ti 4 (c, d)= Co,D+ [L>.,t1 )*
* j> 4r>scfaub:ve_
ts,*) j,? .s€*,,ie^-"f,
t.,d)l
+ o.-r1 -i. Bl
*1il * t] - cr).
(a-:c, trc+\)
t*-,1) + Co,t)Cornrntr-Eali>e
L3) C"..r^*t*l-"= f-ep*r,t,..{
.(^,r) * 1*,[=[]l*.1(t'U) * Lq,t) = C.,ca, r-t+ y)
F.''ot^ (B) *A G)(o,t)k Ir.tr\ -l d/ ti' (s, x) jD rust
) =Je'.t;{L{ P-r g,oe,-rt Eu
Let f ., , .*) te- lr=. +Jerrh.{g g[er_ er,!' { Cs,+;*T 1"" ^"11 f a., b) e s ,(a,t)*(.,:ea)=Cq,t)(o., , de2 *t) = ta,t;
er:Iarde2=L+=o,(a*_)rh. id.nL:Lr1 el1:.enL _ Cet , e.z)
. (tro)
F
P:reve t\a-t the n ece+dtul d-A s u-UtJ:cL e,LL- t-l"--L-oh-dJ-w 6yL -y O, a. nOn_ ernDLd B\rL.+ eL ll eE P?--^F { o , "+" 3 +.6 te <)- )tJ,qr^t1 u a,L e H
<9oLrrt-:s,. .
- Le! L{5 . q'*\.ryie_ thd-E H- -.L_. ia_-. sul,4-r o,_,_c,
T q s,;h.La- H .Lto.U .ii, o- qjl c,. 't I
{* a-,LeH -> q-- "1
- ,lr-.LT' t^.,e Lo'vs
6inco bel ) t-,e r,-.. Fo-:r ct,b efi _| a_r
'-t eU
%,Let a+b-Lr_r , I* o,be,lMs, L\e hor- f- p:r or,s th a-L H%cn
t't =d en}-tu
rr:I
ele-'.en! -t-O, J-b
i-,- a Su11.no.?
Le-t a- eH:) o._l e H
=) o- * o.-, e !.1:) ee uHeh{.o r-r-tie -tderublg
Lii) Tnwer-,e
ele.ne.rt te'a ,,
o:e O Ha- + a.-l eH
enq?l
i
=
f(ro-'€H\a"'
% H La..- il5, ir,.r.ro- inH
1ir\ ) clexrrrte :
Let b e tt :> E-le rrFBq arteH -) art-'e u
-> a* [b-'.)-, e u -> a-,rte u
H'.H
:- c!,+>eJ
-i:D a- sutq--*7 % q
('-..,= lEdL ik. 6eL =+ = Ifol ,f ,-1 ,Iat,[.1]j.., o- Cst"nrrur-Ldi5/ e er;,L1 L,rith o-r +peoL
IEe b-tn-.or4 fer a-tccrn aiAihb^- rtoL{r'Xe
o,^A uuliy.ai*,* r''aedrrl& -r4 o-,.'J x*
sstr-Lhb'1- ' f fi\) NID 2a \2-)
-111e c-r-.re,--iLu o-' iaL\e- qonUr:d-u-l.o ',* ^-A. f.a u,,*..p^.cJ=",-d'1e X* -i^ +4L1e
+-
aAA)*;e,rLNoAl^Lo \
x. - --)-\'-
f"l f,l [.] t"]
[al
{
f.l ['] ["] f=lrd
r'l
f.l
[a]
ooo olI3l
osZO6<j
,47
-*r41 "",qd( b+a c)
( bxa c)
Q+\o--oB*q\\o-€Z\a-x\ I =a- {"- o)\ o_e z*
iil (o*u t) xa c __ o*4Fev ;:.>c-.,..o\ -\'-
Corr-:1dq,. o-=\1 t =a,, C=Bt'\s',r-
(r +'* r)-ta a -- 3++ a=e,At",' I+qta++B) = r -r-+ \ =aa^d ( txu a)x\B : ax43 = Q,
Al^- \ x\ Le x\ 3) = \ *\ r, = a,TErrJ> , a>r."o cr:atr3,g
^-4*, i,2_ ,s.lifl;eJ gor-t\ -^A *u tH zr+
Ll*) rlR,J , a\'.cgo r-q o =
I x,. o- ='t
\.r:e h-q'.,e- s .1r:
ar.d t -i:D ttr._
A^L th" erulaAe-,- -i"/t te-iEr ttie+e z\ . l)a\ce ) z+ io cte.=A
a-,V,c ez\, it ca-n terl (a-r4 t) -F4 C = o_ -r4
-TF e e.ntrli e-r- -.,- th e {&.st -:'rotr: <\rteo,, hh s-=- T .H= f!*+ col.r-,. n j*.tFe-se .E '*- f.:i"U ce .u. n ;-l+tl e-- "1..1=r,',o-
=? o .c ey'v. rn ut.oJ,:w e-7re:Feot +e beth +4 a_J x4
-tg.t \e- te\"r.,uJ.l ++ oJ ,.*
tts e -sar'!e-tetr q--teti,, 1ts"
t litts
-, tfh djc
tF.- Jd;t6,-- -t le,n
\-{tr\U:pt!cah:ve- -#T=r
'E =+
L5) Fge.- the
inrzerr:oe %5ebfiic'e\
+ob\e , we k a-ve oA);*;u ao, 1r 2'3 atl e o'=2-2r1
o-r-. J r< r.rLti p,l,icativs --f'nv ez>e-I
e.l-..rerrL t, a,V a,te 112,4
,++ ? = 6-{- h i.a >."
C grn rnu-! qilw e
a- A--*f her*e,*erpL;*1(,,' , A) tke.n Psrs.ve- -trid-t
L--') = f 6c.>1,fe,r * 19z"rz-)
Y)-ora-;aene
sre-' pe-et-irelrl .
F:n crr- ltr e *,a|,1 q , * c<'+1 te e::4;,t'4ve+g;eA tF'dt
^ X\ (t +a c) = (o x* t) + 6o>.ao'.4 f b** c-) xu o_ = tb** a) +-fc^,t
lrex Ercor. p\e ,
< *t* L 3 r-,., r) = ax* o _o( a*. >\\ -'-\ -) r\ (c x* r) =.G, d:oL',itubwe Bve.r
) L<+ , r-+ I X+) i-D o-r^-{ifh ;d" e.n-L.{-u.l
co")
''+ftenco
.q.irr-o.t
-> q' ir=
,x\ re to eq , E
ftntlr,'
^^a
=1
F"-.-Ps'rB
SeL'.,"tie.r
Clrvert , f, :
Vo, t
t t'i' u)
Iq,x)€q,= EL')
io a- l-\omsmo"3|i s,n-> (qt, .a)tre- \-a-v:
^ tcol _ L1)
Nols , I to.) A.t'-
:) ffo-). c-:) ff a:r
xcd--') = E
EL=) (et
a f,t;') = e_
) = (g to)-'
(o.+ o:r) (u,'i*.g r)
'- aX o--'= O,'
t -r,= ke '-e.ne'ry!.i,=t'){te) = e.
r-e-L ( S, {) be a- 6''t?-''Y -T}i-orl fue'we
1.,.& tEexe ex -i>B o- 1s'r-'g66npLr:-r^
2'. S --+, ss r 1a.fuo:-r e Cts , o) :', o- ,5.-.^r-q-*f
% tu,-.1r6!-,: E-* s +-o 3 r'-nA e-r t!ic.
.p"-"^tr-"- T LyO) co'n|e>'Lon
se\utien . Uu NID zor\)
feL o- e. €
8., : s__> _'"; ;3; = JJ.d.-*6(') = ["-xt)+c
- ar( tu*c)= f"- Cs+ c) f"'*' ;
axro c;_alt_-1= ut f,oL.)) - t €* o Eu) c ") :.r
.-P to*u = 6*o tuNe.r;
^t*, o_ |nof x: e -*s3 u3 x(.) = !--ggt o,l e e
r6-ah {('-*. b1 = T-* u
-- t*otu= ttO o XLu)
-.X!^,ts; = iL.)o2Lu)' .t'' G.q- lte.ns tn s,ryl,-.n -gie.;. 'S i..{s s,s
P=.or- tH aL ev en:t {+n,,Ee ?- --t, T Bat Aea
rL iD - 'terrus-r;, t";c, *.e c!- p"" ,,-"-go-tio""-
ff'r % s'r J-e'r rL cAu Nlg 20 / A( ,,q1r."t31
<Se\uLgru'.
S:te1 '- t
v.{e- E;h-a"\i Erove 1k!- tkee.=,e.'. ;* = Sb7,'
v^re aho"\\ tJcvrt 1-i^l a- seE "' %
Pe-,' rr"wE a-blevt
.S-reP rl
sLel .- -1 \^J e Pleve qr j: -- ?--tS.lLeT: a F-!,;kt o.J.l, =--:s.^"o-rp}.-c:;; g , q_> r.J
Lel q Ee fi-^;.!-= _q="..,p ef e,,'J.-, .1-b, '\ | bIel a-ehDel.r-r.'e t^, Q + q tl {,-C-) = a-:.
' 6i'tqo t*aD -- E-.
t'{) :> a-).- = -y :> a= y-r J5 \ -r6a-
6in-"c., a X..n, tEe., lo(;,X)=d,.tr=1t"- ia ev'"Le
-rF,-tt>- , ftz- -.,- o- L:-f,e'J:'-eru
6fin ce q I'a.-- 'r,' =]*=rtL , f,^ ;,: 2ij-UPe"-,"n glaJi sn Eh ' n' -tgmleL-.
!€; c'.= t f,_ l-ec,l!.Srep.g
Le-L
+ba-
R6e
Q' ;.- a'
€-'€ob f,, C"-) =
P o P.6G bb
= q \-r{>
?"-"F
{. [[ut,)) = focu,") =attE"U H erL(q c"' j,- cJ€tef,
iAe."ti\
= P C t"\boL '
*fF-o- rEtrv e>'r>, e .B L.-Cn'+ * .1--f
i- q, o E;,
.STEP i 3
v{e P-ro*e q o-,'A qt a.,r -- ,>e.-enP!,-ir-
-?n- + : e_>4 b? 4L-\= E"-{ca-) = + tb) => f,-: !r. =} f,*C.") = tsc-)
:) a-:.- = Bz- :> a-.t.Hence_ q i.> \ _tEi.uco-
t- o e rd-s 1 .,b BrLoar'"' tLo-b) = !-u= f,-.Bu= pCo-) o $tu)r. + ; Itet, e,"j\.i"=m
-' G,\ '1 qtHer.rl- 'p.ror\ .
a-!'1 =
NOu>
-n-R
(n,
-r-d
?=row e tHe tkeo.:r e" . \et t(a, *) l=e- a'
fir,jl-e Cgcbc .151..,? qe-''-q-' --LJ bq an'
.-t*=^u j' e q ' =1 ca .io % e-'J'- '' ' ttsa}
j.z I G'l'l =n, lts en 6\-n=e, SB rho'!
q -'t - , @1 , o-', ... .I = . I Fr-r:rttre-l r'! e..lq hj;l ci &r--t peti b:r,e =ntetr e-' f€l rp)-.r& "a-n =e f 6u ".il ,orf
't 5
So\r,},-'en ..
=t ffL .i,.- tFe g:rJ =, T a- , tE e.ru
EPE
*i€a'
o, -t=o.l-5
-* u -t 2 z-9,ta
t] F-trLd
iil trrr.d
(i I w\ocl-rlM-o!
e-TFerL F L i;
n -- !c"."
a-\ = o,Y,',,
_- [.'9*=ek
=e
eY'- t:F.e ,SeLtd ,<-rr y=
e.xL,t=
b-ir''--rr1 -7:":Nr.r-mteruo
"T*
*) -i- a- Ser'.i,q.:tot7.
;ud-.^trd| eler.' s''g tt
'rhvQrrEe-e\e.'r,e-n\; h,-ale !H21 3
fRu nla zoll /Au rrlszorz]
i
So\uUs'. -,
(\) e, x) -;Li) c L-s:rr,le P=repe"rLlr
"t: x' ! e e' :> a-t-x e ,
zr-t+ a z-ri e R,
:k s-b-6g.re>
6errLiqs!
2 xrr C D
>
('r*\)t z =
Con;ytJ.el
-t (1+ =)F.-tgo.,
tx t,
t (-*,
clerll,rq E.f*l:l
= :c--1_ t _i ar<_\ _t' z -r . C_*: * , *y ) =: )c.F \ -r 2 + I r_r1 r_ azr \-r-z.t .-,uJ +. "- *1=-J*"1-;-\
"x(Ir>) = -+ (-\ -F >-r p!-\' l-)2-ry-r -=* a\= ra_ L1 *=* a,.., =.,-+y-r=* r\= * r-y * ar.r-i
)
tr"r
= -:** z-t a>cir- oy=* ,=- _ n*, j 11qtLd (.41 r \=-,,
tlr=)' -iD
r) .:.-
- (=r\\r" --ctrrbecio_fy
-q- 6e-rYlt:ljrro?
iI
I
!
l+- ea_
er= oeR=J:*^-+-'+q elernsr.t elx j:__t:
LeE a-r te He .{-,r\/er\ri g
a-ela TE!J1 4x o--r =
;' ( r+ aa-) -- o-a-d- = -tt
=o f --'ttaa+o]
elet', e.rL-
| 1- 2a-a-*
f .=-lai' q1-i: l:
*:T*- .rB
| -r e. ca,
oe{r."e rr-.e sq"J-qr X"-T-G?lL.- -iB C.or.fs>jt,:crL .La-t\e
TRc .-de-^tiXX eJer',ent a,-.d
e-a-c}' e}o,-er-rt
4nJ
,k) odHe4ag p&l
6*i-^/eiDq
G2 \ lro fl^l i J e,.ti\ el-e,^enL r
-
> .OJ
ca- - a,
(3)
=t ;/-.k;q e\e..er.L e:c.,>L , .lsL it t-. e' r-\o'.r, o-e R.
&>E e - ^e -{. e_ + e a_e-
e ( 1*a"-) -e= o
fAu * tA Lot)
-q
Se\l\igq ;
e 3Ll\ \ J]
.3 ul,a.,1 *_la 3I+l3 a,-_]A 3 lF -fa I \l
[=
f-. r
/'
i :l ',.= [ ,]
==:] .*=f =
-1a =
'+ -
2 3 ,_l, a =_l
e 3 ll3 \ rl
L;l-rL"
1
sl<-Tl L-qr = J-rIr \
rra n5 ng
vt3
_rl h
55
5.
f,
q8
ql
5g
5S
Jl3
.5.ll
.,\3
q3
53
5\
.x-T
}lB
5-1
:,\5
.\3
5-I
stg
5rB .55 qr+
-t11 st 5
ns 5rg
r\g jr
5, -nl
5tb
5lE
q-I
tls
5lt
sl\ \3
$r srL
-'ta .na
q^
ri+
H€1e ttse
-e1+ = [- r
LrHeaq I - \-\
L "'' J
C o+)-t' = .n +
Lqr J = .r\ --1
:A-rtbt-.J e\et,eoL -i:a z r+f, 3 t+]
, \.\,qB r L-a) = ora
, (ts)-' = -'= , L- u)-i, (:d s\ -\'- 5s
LJrSJ = !r
- -rE ,
She.r:' tk dis 4- qn o'rPb'tr?'-f ( H.,
69\Ar!On',
ke"''e r,.e,1p\-;o1-^
a-net}r e.:'l
TG'
tFe !<e:r n-er\ eP a-K
t,,, *) i,.t-- " a.,.
A) j- cr- tCX--ff6o ara =or1)
w\{.T
- k",'(t) = {-..n lgt,.>=-'].Sin c- |
t t.) = -' i,, o),-,.'ay,, {r*. , a-ll eo--Le e ken (B)
Tvt s\t-.e,,u, e o-.J,= \.. t8) :-- neL -,-ptH *t^ gLet rF.e +,.r-e .\.r...8 arj e ke' 191 .*t=',= , j to-) = ei o,-.d f,Lu) - er 'E) 2
Ne.,r, tL-*t-') = f C^) r Etb_,)= t (-) + [f r":J-t
= e l+ el{totb-r) =e'-) o-; 6-t e *e,r ( g;
i.', o-,be !.-"-(tr):) ax6-r a k-({).'. l<en tg) ; a-
"*q:".rt,o T q
{ [c^, *)Co* t) "
SoLuh.ort ..
A:r5t -.r, e
a-t E *-
r\e.r, a' 4 t' ' =
o-be-X;,c-n
E2 fn,i, o-rt
q
= a-tX.=t.-f , S!"sr-r- tt-t
N l D 2.lo /, A rf rl I: zo f3]
tts a-t q .ie a-be] rq'b+ o- , o.,t € c,l _-_e)
(-a rs o-) + (ba t)
. f".*blrfa+ulf^t bl_
. (a+t)"= dr+ b,cot vq'.,--e-\ , ory5u.ne 1..l
(o*u)a =. o-2,6 gc-
o-t f ax (u* t[ f r. a^.ooc^F;-)4+ I-(a*u)x tl
^8 f tur o.) t tl By L,)
f I-1L co',cefrot tr-tar_,-]
[u* *)t* o.
C{ .i-
*b = (ox__a,Er tcUel,h-r-
t){< tREqha
<t Ca.vLc4ltd-L-o,".r /a./-.r]
.S[gtrr tka-L (2, e-, ><)
r,,r..lh-o:t s > -i-r-
fa
.-i,> arL -f;d.-ea.z-L
rhe seL e] a'Lt u4*u *19 =.ro)
Jo,,,or,-
Selrrb'e,. .
=LI
FelO.r..J
Fe.''
(2, -r) ltreoe7-
Fes orr-1 o-rte> , rue Lo., -ar-t = tr- o-. V qrE
-" t=, -F) -iD a..n- ol el;o-r,'- ?-r"PMo,r- , c-or".:;io\ o' L=, t)
rt -&- oB"w-i-o.r:o LH cut t x ' ,5 <Lbi=4jeJ
Az*eciaEte' P",re7ez!1d.JFo.:r
\LZ
--nA
Ze>to Jiv-t-o-.-=,
--tnYeata-l Ss.t ait -
where z- .:E i[Fe .3e-t
o{E ''b tEa .Seb
tt-.qb tFle {r-to
t r , -") cul.o- -i-s.^s',*.-a.
4*socra-bve P.-rf .,j.--d--'U.lq e-t-;e;t i- o
ele-. ertL c,- Gz , & -&1ve-r-'-->a
-a-ez
\:-\e:^ssva lA , >.) . -l,oJ--^bhttut -= €)ve,r t
aAo- fz, r- , x) har>
L=, +. x) j'
=t L=,+) --i tE,+)-E dl\ -i,rJLe€e'b o^A
everu in!,e-qeio , El'o'r:
6."-,11ore-1a f:,-t) oJ
rLB
a-n
e[-,.r.eraL i:rr're n eeJ
C-ar-ty...|-La-h)'we. artl x
closu,,e-
C eo Nl9 zo to)
l
r!.
j:I
ii
LeI t. C=, +) --+ (r, +) ba .le#--t t:1
{ L',-) : a-
.. E_a-.-X) = Bcx.)+ TLy)s._. L ''€ Hern o rns,ty!u:,no'. t'. (>,+) -> CF,. i> ... u1..Ue .! H s mems,, p1.^;,,,B .,: he.-e".ery!,.i,-"n
G: -t) o..4 f E , -r) ane =e,.,e,rplcc -Lo eo.h en
"
6oLrtien-.'. :
jII!f
ii:tI
crrieci-q}I-\e .s'- g
Fqe?
€
req "^4 cotwiltra-u
nt q-LL .tJ-''-fEe..tStrb*sr.siJ
cer.r"ula-k-e *o'"er,J, (*, *)el=" =-rL,
(au
, tFe Set
€€E rn -D a-
lp
r-1
2ott)'P-ssF
-r,t*
C. I,I) F= o.- .A-L\.<'L . rne"siJ -^lrJe'^pete-'rt e].".erB
.ir= o- r5uLtenetd
-[-",.nLo
-Jenh''.-{4 6;1e-'' errL ! e' -s.tzog,'.-.-
% s a,e -l-a-fter-t fl-.- *,
tec-cr.^ e'- rJ er,tPeLe a t' .c z: C-
ub [a,r:n. - 4 : a.-r16 e 9:] a-+Les
A*+,,,,',-e. a,t e s-'. e,Fn=? q^-d b*b=g --_- Lr)
l{ow, Lo+ l')* C-r u) _- a.x (ua (a,+r,)) f .r_u,_= a-r(.->st6t)= o. 1.Ca*b)
@*..)*g= 4_+ t
@*U) * (arb) _- a-r".t*=Lr"--
sAe-"pEentCo+ b) j6
b eg
.S l-e,t-ltF'. ae.t
cl-a;^ '.
(s, *).S.i-n-c-e, tF e
lrJe }.ot. S
Suldot m -r :
<S.r)r'-c-e- tHe
e+e =e, e
o-+.
- *ec-"f..{L . X *-- C-em-,-,Lob"e.-l
tr=it-!o t,\d-'( + -.- a,.le.r:ot;.,e)u.oin1 Cr)
Pqeve 'k iDe-O>',\
-. S -i-- Srt*-'sneiA
". -T6,e 6e-L
ot- SlrrL'^et"eiA
i< ib
\+. 'ffi* c e*
SsLutisru
Q-1- j> tHex{urnte>,:--
(-l )
La)
% d-ll "rAe..petent e\t"tnt€--
a.g"t-.A, * ) -ir-..o-t1 aLeLta4- q-n W u..rhe,,e.
bu a-x l.- = aj- I -
-a -----=,tb,lalt a.,b e. grL>( p.) M tJ 2Dl a_]
He>re
.r{ o.fisrLdlaet { o-rt 1-b>-.rEwe
d-* B = at. eq-
o.bc
.- c4)
F-re-
Lox
(A)
b)x c
+[B) r."3e-
= ax (b+ ")
Let'e, t-
.- (s;qe-L'l
L3 ) =de',bc*.,------------'---l
-lEal ,
T6n o.U o,L, c €gffie- iJehtd{! Je,,,s,rL
d-+ e -- a-
?=4e =a
Ar'aecl..*,lve B.egs.t.4'.
Cor u)1 c = eb_+c2
= aLclt
a- x Cb* c) ba a-bc
=Je,..tcLq ele.'.ent ; s:a€e,
4LL!+ ) JnwQ^r>e ._
-Ttreft ,
L€-t be- 1tr- ri,nve:u>e, of, cu
(=Je..t:!1)
a
Leo'' IInv er>,e BE o- :r- cli.t -I> eq-a-
L5) C€rr.^u,batir,e :
No.!...l
B1
a&t
' Iq*
= sL_b
-- aL"';-
. t) -r-,
-Eo,r a-tl
o-tL aleLiar.
art
f".
eE*
\s. Shou, th a..L
ur)lLere * icD
6o:lr a-U o., t
S o{-snbyt :.-
(-t-t'\,+)a-+ t=
CR- l'\ )
dbdiqnd-+b r-al>
geL
d-edl-.J +tX-'Y'
ER
Hene k - {- ,1 rileo-ru.5 -tH et lur"a te,.r-, exc-lL _l T o.-t
() cle>t-- P-..?=.Lg ,
"l*\ o.xb = a-+E+ o.t e f- t-'
P=repe, !q .
-.-.T---..4L-?)
Ca+.B)x c = (a--ptr1ob)+ c= o--f br of +- ct fa-+ b-+ e!) a= a+b+ a-tr- c--r a-c- f bc._r a-b c_
o*fux c) = a*. fb+ c+Bc) -6-r
= q--{-b-F C..ttc 1 o-f b+ ct_B)
tr *" (A) -^da+ fbxc) =
a- r-t+, C+ t.:_+_ aJ. +. az_ I q5--
('lLB) *- q=t
1a+t) + c ) Esl otr q,t,e e (R_
.ideyrtit-r1 eler.ent .
1'\)13) =Je-bi\ .. Le-L ,e., Le, tl{e
-,aEe>a-
--> 4-i e {- a_e. = q-
e Lr+ a-) = oe=
Hence .o, -ro
L\) Tlvetrbe. : LeL tfretTE-erl , aA dl= o
a- -l a-'a a-a-- | = 6
_o--, (, r- o-) = _ a-
a-' - - o-
tf . ,il enb{q eler," e.rrL --^J oe e_1q
l*.ver.>e d- be a'"B
)
t'\)-O-l+ d-
C.r/er.tr+H
E CR-
Cornr''r.^lalr:ve
a-+ E = or-b+ ot: b+ o_+ E, o_
:b* a-'I o r. t- - 1
:.- -^ o ,E* ?] a.,b e[n- 1,11 + arl o,LA-Lq^L-
C B- i'\)?"':f
-]Ee- Uni-on
n-J h-pE
?nool ,
---t) l-eL ur? Paeve- -TqLt-
we knc'.r: r6-o-L L= , * ) .s
t.J=, c. JA; tc er.
o^A Ha =t-[r-=3n,".ezJ= { ., *+, .rU,. .\
e\--,r\ !* , o..A H 3 a:.e. A.rt1,roul+
Hr U l-t. = t:. l.r<- e H, s'r r e Ha-\
Here 2
". Hr uH.z-'. FrruHiHerrto \-{ r
Tc
-t L,,o 5"bf-*T- % '- ?"'f',u- 3''-.t?--f
the.eo.e-rn t-l .-- Ex.or"y\e .
o.- X".-.f T *t a=-,-
= 1o,ta,i5,r\,=,t,-..\€ Hr anA B e!{2 t,rL ,r_3 =Eg!H,uH&.io rueL c-\eaed ur,A..r o_dAib:o aJr-. ngb a- q-.*F
ur{1 j> neL q_ *"loq*-t T q
-tqe- D^i-aYL % U" a..tq-,--f % - ?---tr ." +a- .ar-rtq-r-*7 one ib cr,ruLo:"-.4 .)-1L LF.e
otFie,' .
foelo^l ' .t<
C4 'lTLe-r-L
H c!9 si
sutrycr',p-,- %.:.j:D ^ 3"!?-r1f,: ,&
ole l-,r,e "dfl.-,1.- g e
- . l-.1 u t<. = K e'n HU}<" = H
Henc.g Lr u rc -i.o a- gulqja ouf .
Co-riwe:r-.e.ll , ;]u.pps+e- Huk -io o- SuIq \^re 'j-r"^ tL--" H c- k ., o .T-.T %
Ejgeae th a! H -iD rust Conta-irrc_J _; k- o.^JK .Eo yLBL corl"ta"UruqJ .irt H .
-TFar! irh e:re e><-l-L g[e'-r'enEo a , B +u4h bh o-E
a- ett a-^J o, {H -L')h'erc o.J b+H -:-(a)cAo.-a)4 a., t e. H utc . Sirrc-a.- HuK ..;.b c
dL cn, a-t g H uk, . K- Dt.'C'o- l-tuK jio a- s .'bq.,ttsr?
l-l er.c.o ob g I-l c'-r a-B e k-ca>e -t:. Le! a.b €,H .Eti^ cE- qe1 , dt eHl-l en co- o:'6a-b) = b eH , urhich ; e\ :>4= .t- (.)
C-a.->e - a : [gf oL € ]c..Sirtcs bgk, lc-, e u"
P-r o.1 ,
LeJc H
?-1rej-lH e'r
Ia eruca-
-l<o 11 1
-' . o\1,
!re- L-u,-e
!l ukK4H
Att-r-ne- t-r a-^f, |cl-.{ c t( o,1 kr u
b-r 1o.-U) =d-eK, t uhiz,l.' -;o a- co,t6 aAirt;e,a-
ar*u.r^plC.on .jD u, an1 :. H9 k e{ tC!; H
''I-ka
T{
ar\ta.z-=et:xi,s-t -* ^ta -=.,,o ..at--{ =*S3 *p. -* ^ go"..
-- no".^--l -sL^lo€1r6trf .
(a-)u -^4 t- -r-r+ .t-a',r.^J -sulo3r'6ue< -{ ^ g -*ts q,!n+oHn)rro - rr-oxrvr<I :3Gbg"5srf
fAd M-y 13cae l-orRJ
k oz-q ,-r-ar.naf <u-Egaa.,ps ,a>tL € \^-t q-i-6r{!>< ...[-, 4'a- -<y.Joglrduf a+ q.
{,.4lQ4veq H -'.a+ H*4 r-1 \-{o k Xz,
t'lotO uJQ r^rauoe G p,-r-av e Hv-]. Aaw fo q-ar,,-!.
Let -C e q a^4 V\ e Hfl\4 ,r
{ cq. -.,\d h€H ar\.J \.\aE
')t +h*:lr eHd +h rs :il 1\La'"4
E .r H o-..d fe a.lz- ^arrno-l
,ai^})qtelP.J l
F.,-"--t O2@UrzgeE
-r+ h * "C e .Ho t' .
* une <2- -.rrJ 3v-t-q'r'6'{P a'f 4 '
II
i!I
gnno" t!;ve's'9 W W,*, 4a
i;- w<2<t--u
l-of ]'. u -+ u' Le o Ao-,o' otPicr,l + 9n*^uJil6 ketnai K, k2o p*w t-r'dk K b q n'.'^o0
^ i cr\ k t-d* " U".- oryLic [; tt'1,'W*) % \rt ot'c
%^r % +. f nu "1r uot,-., Av {ili*,,-?ir,' o]'ve 4^<z lall' r q e lS Jte o'ern 'akl{ ,rd ' t lty or-,", ol,),.1-2'r.rr,,' Fp,,
l)*no n
Q)
(D
p'ote rr'a* nn,., naz.a-7u-b4T'*';*:-;.!*T,''7t(; *a ft.rv-\ EnpV A-)'ae-*
,.t _": ,:,: _ 76,,tla.o )-ot7 -v k 6- ,sJ,s'*." A o'bcP "> *a-'ft\1"',itiiil
6.& .r.u-+.u ,: L:*{*.X"'#";rqti,^t E 4t',^9 r*" fPo ',tD 'otLJ
i,n,r. 8,,1-' lTw w'' at )e v?-) €"i6 g@*? E *a"- n i^
Pt"'e fior ,, , ^ulo) r1* a.".p 'n erzLe'v n
f-,h,-aq)";" : .^ ^,1:' t',ot , @,t i4ti e-ot'\)I-co Nlp *n,,"fo*r.,o: ea)
a4 ^, ,, i ?,-"?-^,"ff*!nt'-*;:-,f-,:ttJ %J Nu-r'7be^{a d'c
ab * ii " ot* iu ' ^' 'nlr bt4 Pa2e !""'!
S [o-..>
tt, a-
ia o-
n'a* l<ezo"'!
5"L2,-^"1"
( hr a't
-7 (t
,lr* t6 a^ ., f
^ /)ia *-t''
t*?o+ a- Qo, u1"t-7'1"'a''n
i., k; a-no hon X",--P (a'' a)
I oo P:l- tott)
f P94 1"-o: 96)
^td;n g'F.P .St;,.s hJ-
6.r, *lo c-olo t Plg >oD)
f!*" t 1 31)Ly
V*r)a
-F=.SI\D,J Yrek ( z, + ' x) Ia o.rr 9^kg.q! Ao,.;^ )
(,u'€ z la w aet "6 "lt g"lagot f e o ".Jlo z"tu
(Pqzt'rw: 9t'1
@
f\. o^A
q oJl
&>o-d-ltva
i&'"p. trn t'rrorial (Mr Q lto '9e*
e)o ^o nus % !!. N: a
Ca u N/D t'orta
t+-a-h (o+' *>J r\bd
o>; ar ia J-ab\"'d')
lO " $@9)1,tt !-oto-
)ul..,ore\J-
9*,6.te"h'c"-''N tn
i4 o,n o)*)"o-l
b7 cn*b'lL
Lf,f r': 4t)
9*"lD.J
+...J| a
.L\A, n*.-.1 *"4*-P'i q
L^"{' ;il*-P' le'': w^a) s'uo
lg^+" 16'. 411 ,
a--o19-
i
-i-J!
Par<-r - A
Z- \ PslB\e tkai a,a (o"-.t D = -,
f"a. Lo"tGc-u- (w,
a\L arb tL
th. GILB
- Lr)
a,
n L*rlo\, weho-"e
F.rcrrx 1r; o,*"J ge, > \Ne !r-..ra
a7T*7v>-Z-^ - (?)
ot";eur{ a- L-
? rfic dTurEo,^- %
a 4 & v (a-'ab)BX Cz) *")
a,v (a^D - a.
6t"ar)"alr\ we c-a,!L
a. a(a"vt) = ov
3el-rrhior,-
.-.Jt",&
J-t =F.rt "^-* "e-.l_
e,llta|erLv
ce"^ple-*e*
fq u r'4i r 2s 2lSo\u!.bw'.
.Siruce a" b \^t>
r, /\b 4 a.
i'. -, A g-J'eor -
LuE , we ha-ve.
L',) , -. ho"
P.:ngve
oo)c-a,'\- Al-q&^a- LHU N lD 2- ot 2--)
I n u H 15 zol2-J fau "ro z"t1]
ce,,.upLe,nen1eA J.ioh;t"f;.,e )-dffL<e-
Boo)t-ax- )\a&ta-
Lakfce u,rffh ' o';^ wh,,;dr eve/t(t1 det*et l+ot'
a.- Esol"eaYu
rl
a-
er fgv-v.te"I! IoLeeL.n* JP a.
te$e^,rruX cerL&fLb^,- '
Vorb.,c-€L
Lr: 6-,..o,- o-
L, '. a" /\ b = ba q.
L2 .. a- A (u n c) = [a_at)a c
eyeta.b-'o-'.o-tkc
o.\ o-, ' o-
sw \2 = tv a,
o.Y tL\c-) = (o'rQvc
o.v (@at) = a-
1 2 b*.a-,rasqr.-EteJ ti'
a,^(o".vtr)=^a- Y (b
^ c) = Co.., b),a. ( o,v c)
Da i a.^ Cbv")= (a_at)r,(a-ac)
-rF.e^-e ex.i'-t el-ewuerJSo ' o ' a.'rl t , ' 5, ,r-a[ i!'t a-t,
4 ,,\o = o , o. v o - a. t o- | = "' o'""d a-v I = I V
a. * o- eL , tkere exJ3?L ce'rt'1 e>pr'Arnq sl e-e'Lt
a,r -;,. L Swc,}" tF& a Aat -o tr"J avJ=t
t.
3-
A.
5.
b.
f
L\
b,
\n heJ'.- j: a-
>Btr$r-U^ - :
lo&rOce- AarL to be
f nu rrrlo zor{Le,t CL, , vl te a- A:--e^ l-tc ee
Lto-, tst!',, \o' eLo-,.^errjc o'--A 't' e'1r-r''n-e-n-E
;r, ar.;A +- t= tret --f,.J Xa-VC''u ' at ;t'(u,^,v, o, 1)
Fx.-,*g\- t (+(") , g) i,- o- s's'-'-.A=J \-o'tlsi<-e
L{e-,e rrc' o;-,"-...! ,i-e S t t,, e\ernen-E j-z' A
Let .4 = f a.,b,.\ .-A f Lail V' -r-B Pcn': e:r -5gg
e5ars a- l-]d.-'De- Ai-Z.-q.^cr,.4 a (eG), e)
ser.$rb,.,. f nu Nlp z-orflo. lo,L,.\
? (A\ = [ t.tr , lu], {.\ , to.u\, {o,.! ,1r,4 ,lo'r',;5,S3
l-'
{,\ ( ^LL" )
Jio-gr..--
( e6"t , q1
-Ga^ te-
t^:h-o:n e A
Jtep:'re-re-CUeA
= l".,t,e]
| ^,u, "]
.dJ cevwpl{-te_trsrg zortJ.A tA c-eqas(1,f,g .t'Sulabet. h-r- a-
A lalx,r-:c-e i.>et i-t* h-crh e',.otub t.l
ADL.
che*" \^.*te\t-.eT tFe
{t,,t,ar, ias).,>3
=*bH
TFe Hante cl!o-q-ta*'^ €- tke Ps=eL '
I C,, a,u, r\ , o] i-E
pei,€lto I C .,., b, q) , .3 o-".d
aae lottiee=r crvr n-eb .
fcu A IM ap\l]
Hs,e l.^,u 1u , eJ' . q-i,ve-n pe'=el
e ) Tfie ll ot-r-e-
Je.-r= "teE e>.ir.td.- Jld.tt^-'c'e
3* {Ct,-=2as,ras)PJ
= b vq;,, n-gt:
@
A4o
Fey I Fd;'1q<-.j-t
' q"i./e-r.- Pe'*:b
% n-sde-- te1t' Xl"u a^ d J"ul
)-*tE;c-e
DetrYte s\no - ld.!wc--e t C., N rp zo r "]Se\^$,ie^^- :
L€)c f u,nrv) be q- Lo-tE<--e>
Ag L E= o- g\rtrae! T t -rF-!-r' (S,
o- SuLLo-\tic,e "t CL,a,v)rB s.l-E1^-.A=, tstq gp,e.'r a}!er.r' . .-A V
I
\'-'" Va-rb e g :) q , b egqrrJ a-.v b €3
^^A ta-t
,t, v) ;_c-te>-A
6Lrorr: tha;' & a- Laltj cee@b : E?- c --,-l f a+r-)
6 e\.r-r-Eb.,' . f nu N t5> 2orolL\ ) G1'werr- a- Z.b 4 c>
Eir.c-e, o- 4 B =y, a, v LE 4c :.> Lv 6, =qz-_C- :) q_V c =
trcrs* 6; a_^d La)4"v E -- t -- bac' a.vb = Ea C
(ar Llrs -- Co ^u) v Lb^ a
d- L L 4 c. , t\ e-,-
f u+ c) = ( d-e)b) +(b
-i,6o
o-.
tL
bac =
, a- c-
- tr)
-G)
RHS - (a"u)a1a.vc)= bac- = b ( u,"i*Xt.r) :- (a:) .- (=;)
p-a qrn (u) a" d 1 s)(a"atr)'(u^.) = (a\B)^ (o.vc) =t
{3lteu, tts 4! ;^ a-
c-tr"^pl.er^e-'uE %r\\rDE l-e rr,\i-y-e
S€\rrtLb''\ -.
t-e]'
Cew.>\elr. en-tI
Sinc-e -a-^ z--
Ji'-.Lr',tulr-e ,ta-ttc<-e -, rLztrl ef et'.e'uL exi'?Y'5 |6en- i-L
f n u N/, zoro]
olb\rhe >c_ *A .{ z1j't =- trrog4
a- c-er^p\eqs.r-t q c^ , ,r., e }1o.rerB4Vz- = t .- O)
c-e.-*p\e-.^err-L q ^ , r^., e }r-c,.., srb
.o.w\ = I -
Lr)
f -..r.vo=z_)
EFb
Sin<-e \l.\o'^ t =
No.rc z- >
q = L{ \/o
F.:.tE,rn
7-
t^) ^" _A LB-)- -tu = 1.r."
a<-=1
) \,\Je Yra.Ve-
d
seVI)
'" v(a-.a{) [u-_^1 Cr)(x rzo-) ,\ CE .v t) (si:-$..itu_[-.,)( a' w a\ ,r (x- w t1) [ ce."",.,wr.ua,b.:-e)
l,rf-vq) (*^t L,))- tl G)
= U ", ( a- a '.; f r1,=i,-1 C.t)-- (U ., "-) n (t v -) L D:,+yit*bc-t= Co.vB) ^ C\r.x-) (ce"-".,r-t^.b:-e)= I L.t ., =-) f r:-r^X Ca)
\ = \ vo LB)
Co-r^yIe-^eruL io:- \)rl;:q.1st
laak-a. n oJaeLt o- & -r-o-b=o
A lo,\ "--\il f nu ,rra.o,!6 / r.a zorr]
(n9tv Car4 c-q,ll d.b.'oyv
Fo-ra,n LA) ^, -A (pl)a': a-+-L -b
- a-:b
1P, a-V
o-=L
,; a-
feorBo
Lebe,.r+,orr) b- oy e,ooLeo,v, AlqeLro.
IEL a',b Q > o'*A a- = L
ct oi* : a-E -+ d,b -o Lt)
Noro, aE =r at = a-.E-+ -a,. L
: o-. d _r_ E. q- L,*,.^9 c,:1= o*o {i, o [ . a.4'=o)
c.Lb _f o-b =<)6str".,'oe\ , A,.rs.r-,,. -
.gel-!b:s^4 r
=)
4L, + a,b = o
;"7 =-: =
7^,-:;:--1:i'*'"'(**a)'(cr-tr)= o f ar:t..,t*to,= r_*.,,)t 'f o-r L) = o" (o-*E = Dq.+ t - a- (a.t=a-)
( a L.:e,. i"bron
14r,, \Lol-)
( Qn-+'itu!:/e I4.,-)(b+E-D
sl,\r.n^ *ke
lfre l-l qr.be
6,e\lligtr ;
-rFe Ho.r'>e.
t
^Jhrrt-e-a\&J
?4n\laL e.:rAor.in-q =rep-'"-en[eA br4
<Ii,o!qc a^ ^ . fA "A lM 2-o r{
Af4,g."1o,.v\, fl Lx , R) GD
R- 5It
(. 'u> 1r,
ta) (" , .D (.r,=u)(e,u)(:,r
{ra,e,.7 (r.,.1)
eEcu.*pleT o- ,!,o-ltj.- r,.rhic-L .-,:o
Ju-Eitur've fA.r atrr ,.,i]
q"i-, o\ 4
r^.oJurl_ol
SeLu&.b* ,
t.rf, N o-l:
(ixo,....ge-.l
dJs- ty :L
i
M6; e).,1 Dro.r,,raewv J Lo*tt -e -i,e a,r^
\..red-ul o-,, !-hic-e- tuf rLe,r^. -La-ErLe
%uldve-
P--.tca,L es,Je.r
Bin-aln :trel"ah--ott
d- P.rt*FaL gtrJe-,.=regte-^ive ,
a-n-l!*qmm eb"ic- ,
9eJ.i"e Fa.,etbA 34 P leqatEe> uitE a- Fa'\trdj"
o:.r A s..,tr.g R b> 'L',u"A o- P*E d\ e-,d =''='l.Sc-L -- c'- ?ez€LNoEe ", ---1-t: i> ce^.\ve;\i--6'rLaJ- .go .1-- na6= o_
P,^,ni J elr d!^L13 U1 tF.- '=4' Le{' + TE;'S)\'.^t't d-e== n6; {e-e>*.t;\ rneo^ ,, ,l er,sl1, -.no, eau.al to " ,>.t- i''- rr--J- B* ,.o4L r.7,^,-te-.ro
.-r^-crt4.'1^lt Ld.$rc-e L*.)
9e&n^e -ruEo-ltr1 o-d""oA .3e-t
Le-U G .1\ Ee a-
Q,\Qru] L , q
, Fne-r" I v)
L P ,-'je
ca-\eA
e:rlh ezr 1-t tlPa-lt { e-^ d--'-J >-eL .
=t tru-ho"=
si.-f\e,t
t-' f,=-',-2 .n-
-\!n-* O:nJerrna e'rL
a- -|gEalfq u;.AQ,.Je.:1 d eLa-ri'n
P o,^A t P,9sr ,brrnflt
z-_) l: eo-tf -Jg-,Ae---lr"
3eL
tb. 9e"ge^e gYrle.n\Y\^- .-J -- fi'""* ..
u=u tp, g\ be d' Po:'rtfa-\ 'e':'Ae-. eJ 8e!.,-':. ,,eE A aP Aru{ e\e"-en-E rc e' P i5 d-
\rt,Ea,l teu^d fet_ ,A\!hsn-e t .iD '""a "Yf.1 :t'"i B€-!- 'A@t , ** q;*-v '^-^.'T!:no' *'o;'l'-t^ ' +e' A Jb o,'* de'--et* 4- e P"-=;\;'; '*- b; .- -iewe, b''''^L ^-'A
iZ: {e;'r a}L xe"'e'' t-'^-'d-' q '
*T^--A
e-Ao,J
We\t - e"'A:JP-'.tCa-i\u e'-J"xJ s'eB
; L-=1 . .-"e"' --fL'l
XL-,-+ t\rr""be-t
j,> .of,f,,JEube& %
u-: elL -rt
18, gt* !-to*>e '[ago'* o..'r P'srt;allr{ s''J"J g=;
Aii,q'r.-* -
A P^'*f,* o''J'-'rn1 ^-' ^-* h:: "*:-t'-
-repgre-enE=A tq *t'T 1_
*_ ,-.L-^r-1 e r.,l-cr J
L a- Ha.,.,=e "iTT-. 1^_ *r*f'^
aLa.a>.,otv^ .
J&-ffi;% -1"'-7="'*'t^ bq a- s''a'lL
C;J1c-tr..- €''r 4 AgL
-iFe C)r AP-cine\-e 8e.r.b
,ip clnq.rr)n-
eeveire =e.-
B- -- oP
NgP .tt-Jo.lr-ee-ru
-.1,2 &o rr ." telou- 'f1,oI O-
=- t- \ , a^,.!al a- /^-r-rLe
,..^-.-Au&Lld6d
J---- h-Bt c€v e,l a , tts
c-enn_e<EeJ Al3"ect\
c.- st^-\= Jtner l-leme-we-x , th e-,1 ane
c-..-.!Jl thr eueh ctrle e{ tY'l€ri e 'YTe-O TP. trt -ib pe,,oitie te gbtaf-'ru tFe .g:: %e.y,J-- J Pdrre ';^ , T* *w& a-
-l-Ee_ qLBEe &-,-eE-.\ \Y7e- BsuJi.e -, CrlLB {o, Ul =
Q*eoXe-Y. Lo,oe,,, BoulA ( cnua) e-lP-.,-) ( r"B)
.Sut>eL {*,Bi -c L ':'r !lL' o'*-A tF.e Le-.=Ea-6 b.
6. 16 B f .^= e'E e..r p-.edrrrt__ o"^A t )
aO t C d-'- e.:r "* T*J r)-"8-= Ala.t--,:e .5g.:*.ern
A L4lff.e,e .rL o\^^ a l.qel-' -ic -.SV+ter,r
( r-, * , O) \!ith Lu:s t''"'-'1 f :r a!;o-n5
* -/"f, @ er!^. L rr'#.t-ch-' an'- Lett' (rf
Cs*,r*'-uta,Eiwe o'.^d L'a) 4 ,,*ot-r;4!irr e- tt-'A(5 t,tBd1 lFe cr-Erc ts..rpb: czn Lavt= .
ot^- Di'1e..!
Le! Lt, * ,
Latttc-e. -The A&et- a,i"c-
J}\^ wkic!' tE e v;^ ^^xt:t1. L >< g art e S,rel-" fFd, 3a- *'1 (o , , tr)o^,.4 (c".1oj\ ;" L-9
(o,,\-,). (oo,b) -- (a, + a., tr ^ti)
(*t 'L) + fo-o,b") = Co, O aa, t, \ ba)-i'- c-a.tteJ tFre A,i^ e.k P.:rela-u-cE .l Tte)a.*r<-E'o 1u, * , O) ^^^-A ( e "^ ,'v5
Eau**qroEontt-'A one - to .' one
ele."-^jE ef d- 3eED
YnD.rA,enal)-a'n
P=.o&roY
o) o'.4
6 oJl1't--}-7o."Jo..nc-e urttFr
-ib ' eo'lleA o'tv,
J \-anr;e-e-2.
L, , n, \) be L',o,5qeL-,er" Lf- X g,, , *)
g?eir anbr"= -r-. a.*J
EeL-cr"e Lall;ce He*so^o,*gk:"-
r-e-V fr-,*,@) .-l (s,.t,w) Ee L.,"e
Ldkjtiae>> - A -'fF-^X X:L-+s .:> c-artJ o-
Lqttjce H or^o-rno{ PhiDF^ F* iH- La-tt':c-e (t,*,O)
t- Cs, ^, v) n i- o^X o'bel2 C-x Q - XL"-)^ Xtq d.,^A XL.e b) --tCo-)vXtu)
A l6r,lir9ee
3s- alt'
[?-w\tr) ^ z
(t-, 't , v)z, \, z e
fr'ntJ-".].o"
-,b caltJL 2 z- bzVXuat"-n-)
:> zv [1a>)
lFe
4 ecrruiJe, Co+ , a) .-t A
ft.o'.b!"- uY'eXe:t n, Dn
Eai!,:*e Jivri.o''o eR n-
J&-bit,U.tr ebLd;^ fk".\
Paat"a&rr-rr oct Je'' el =ebP-"# P"',b:J g."Jeur
'la6
I
.A(5:
Ae{rldev! '.
5\ =tr'bq = i. r,
D,r ,. sq =
{srr, 4) ,t>
P-e'e Ifidts Evql\la e drrla'' ttru h-Bb
6e\r"rbbn r
A Lo.!oie= r-1
r,.rh-cr.e-ves a- 1 cw Cb^ c) = fa..v E)
a 1c- tkea d-
d- w Lb^ c-)
> - L\,1> ,Z-t,q) (a,r),aa,?)>,<u.,,)Zt,:)(\,">J
(..i qr'a)
1r r'i)c'. )
Lr,a)(r,\
(Pq , e) , whs'' e Tt-Jergv.r= ttie ==t "E
dlfor-J L iF lFeHa>.se )t'1-T % *'
L = :D4 ). & rrar' dezr tFe
t,,n
,, l*ja, g\
I z- r ,r1a 'z
o\i=,LvibulfweConw e:r:=e111
La-tte ce
r'4eAlrl-orl
Lo,rrr.*Ld +o te
!.o' - tt<eiebd-
-iPtHeru c-e
\lre/\C
ve cg) z' ( av g) = (o_w r),^. a
L'^'o)
= fa-v
Na,-rt e-d,w
Latt^ic--e .i>P:ne,e tkdr
l.4sAd.altF- rrsrr- <In-Evitulfv e
3r- Dna^r= tke H q+-e>< j€ tE- set %4Lnd the .rrel_d-f!'erh
lL.,h'.71 Q,A 2
&,l"Atr--c,n :
h+'rttr,eL .ii- ,awclu
Jco-g-to,.^^Z- ) t^,ha,rq
\e A ^ 6* ar..,:J..:5)]
% (x.dL.Leo,,=tFd.E j, T .*-s
=B A = {o, =\ e.x = t o, B,t, ta, a4, 3t3,auzt lho.E ,. 4 t .-ib'lh. .:re]-akiew L Gb
JilrA-= u r.r'.Ab-q.:lre.r! e>,t ele-r', e.r.L.tSa\lhsr. r
-TfteA_e )r,
ttie .l-ea:--+ €-t4rYter...t -,. JB*'' A
Slto,v th a,b tFo .x-att i-e 'i-> n-eb Jr>txitu-tl-we
SeLul})nrbr * (b,@ E)
vcl,!\e I u, * tr) o-The LalI.:c--e .i.E hlet
fu,*t)-oJi'*+.tujrwe
EbLoi- tHe
t6a Hdli"e
Pa:r\iaIAi=q^...''i
B.:, J e-'\ no1 a-1- e-err-t-J \
C",'),tE,t),Ib,c)LE,a) tr,e)t d, t)J
gE\uJEei^ I\.dt
a = | [o,^), (u,u), Cc,.)C3,d),(o, c)(o,r) C',dL''t)
, tt,, 1) Cc,a)(c,J[4,.)
=:> tF- Lclltic'e T cli-s'.:'t: T =* a-
o^l.-D,'^" "&Jgo' ?
&ll!:*: t
No , \it- &vj:e;r:= -L ?' ib <r'- cAaZ'-
" -.A !+e-nr:-e- -iE JF rL et Dc-et. v)'c-t..'-*=AI
geline 6- Jl:ot."-,tt'!:"' La-tt'"'e-e aJ Pae'e tgoj;b - -l-.,.1"ftu$;'e I6JIr-ee .
*-':1 c-na^'vt JE c- CAv ntts zotl
Se\lrlr'ea^: A l-a]fiic-e Lt,*, €) i,- c-a.\leL a,
d:'rL"'trl}'ve )atrE-e ,L 3* t^t o-,L,c e1q-+ CbeO = t-+)-)ea@ Cb,r.) = Co_o t),+
a>tlcC= c-
fa+c\(o o.)
9vet1
P-ro'eFverrq Cltq;^ ;.b a- D-t -.{-rAt u-b.--e
4
Le}- ( r- , z1 te a- e-l-roi^LeE arb, c eL
- -lcovrri de.,, l}le E>ll
owtnq Fo^,si ble can>e>>
t.-i: q 1L Gt, o- L c- -" JLn) a->L .-J o- 2 c
\^'/e sh-a.\l :.touo -sI" o,.u |E dr tF. Jir-bi,t.rlz.,= o"-."'
ox(E6O = fa*t)o(a+c-)c--a-re [t o- 4E gr 4 4 c thea \re l'"-.,e a4. ! - 6,-
c\O o- = c\ , a.+ c - a- c^r*d
s- o. + (b6 c-) = a _-tu)a^\J Co.+D O (ar c) = o.6 ct =ct
- (o_l
(t ) r- 1a; rroe. X-ta.jF Cb@ c) = lox$ 6 Co+ c_)
En c,o,:-e [i)
^2c !6r* toe h*we a-+t -- LL@c+o.
eh"i .i'. a- &t"'lorrlr'"e {^!tZe.
JTL
l-9D
-1
^-J
so tlh aic d. r. f b€)c.) = be) c _@ c("LJ
C**b)@(a+c)-- u@c -L+)6:..,'., la; "",.J t \) \,r5 e ae}-
ar. (b@c) = g--xt)O [ar" c)
Q.o,er"rgr'- B
ffie
s={rthln e
1,,R=
e , - ,- lo\ o'*I a-
I t,.,1) l--r- 1 = r']% tfi e- :" e14'b-'o''
ote[o]::s^r'r R,
u.:halz or\s-'
R?
Gf .ir- o'v- =*o^,t p\. o- srdotr"'on rd"i-h iJ.tetf., .aXr^m<Fic- --.J onE sqt r"rteE r c- '
Se\^r-b;en :
Lel R1 Le 6r,,'-e\He'\ .:.el.qll--en errr ,,a
i'e- Ra 9 a\ A ?ve.!-L tX c.=tLr,'), L-i,\),t3,i(",,j), ta, <1\
Fe-:. Ra {e te *{rnrrtevu- e
\Heo ta, -) 5\.,'EnA te\e..1C,,O e ea o',-i f-a " | .-\:-.- te\e.2> t- R2
!.;-frr,ri=e ( a ,a) -,..J L=, -) te\e.1 {F R5 . -TRe, e.i,a h-e e\tre, Fa-i, L'., 1) uniti - o'T'd X A;yXeae+t '
Her.!c,e R2 -i-b -a-Y.r^ej6! c ' E\r-u t* O- {-5 t:
a,^t.:- ,aq",....=liie + t-,1) o^^^d tX,*-) I R.c,
lh en ac-:'-\ ' \-{ele [t,=> "^ a ta 'c) e ea t-ub e
o., A 3 a,1e- &=\,:..& e-\e'.''e-''E eB A ' Henco- R'
-n- a,*tf - agt'.wlelx; c- -fE'^-re? Re nD bettr
.5q"."'.e-hiic- ^J tuE*q"'"'etEc-'
, ,t C-,{) e e*+s Ra. l+ere
P-'epeaE'--'
D
9a + t-\4
Cs's,:=S- "B F4fi> C - , X) S'rclu
= lo o^"-A >c, , \A e g ie ., 1E con.t>B(r,o) C*,s) ta,r) L'.,,O (s,s) (L.r)a) Cl , \ . .rle p.,r o;.e-ati:e-- are a--
{oltotr.r>
.:r erl e:<ive a-> it Aee-- t sL c-entoi^a6
hhd!
\i) R, -iD
Ao.-.
trf )
-B Pa}E
tr' 5 fe,Aeu
stb
neL c€"s.;I fr,D (-r.z\ --' (to, to)nie
R*-
tEe-
R,
-i,= i:r:'.eX\e;<i'ze , €iirlc'e -B e
o = ,- '- R, Ae e: tueE c-grLbd'':'r
I"ai'r= Lr' r) la.,i-)' - lro, rO
, B.'. -t'" -d
{E
\jii) js .sHrnr,'ekEc-
RrL .1,,.-) a-\>6 'ib
eR ; (r,a) te,a)
Lv)
ti!) R io? r'eE ^'".Vl -sq"''tt., e-E"; c-3." R" Csh-ta}n,=,
e\e.^ e1\= L-' Y) "".Af-e:,r Eioco.\,v!y\ e L.'o)
R.ia rret {-'oro'ru:i\ive 5
Cq.,D ee t:c Ltrl Ao.-,
, d.l- 8..l Pd; L :. ,1t) irt
"n'\ R i-e., lrrg) -.*d 1q,i;e R, o-J 30 erL '
( u - z-\ 1r-^itts 7- F uJLq,D eR t.* l+?
.}o.r ( r,e) e *, .-ln-eE bef. ?^- to R
3a. :>etll-1-e a-
lE-' s,el,a-F!'.o' " .te^r.
denerEe, \tte s,' elotr,-e^^
z Ai..j,Je-- S Be\t\He Se! ir, a,3,b\aeB -"^d Fii.A L nDD or\e- Ee tE-
t'. o,r,.t "t\[.,et
>te),a.-Fu-.'lg'' Le,E L detusrEe
fE o.,yu s- e?-r-a.l to', o, ,,-J S
" ->iwi J e-=,' .o!l-qr,-e Lly r^edrr
L a^d 'D d:re de4r^sf, s.4L
. \rrsr"r*-e L a^J =, o.t)
. Skolro dl-- ttro-E L o.,*Jsef,t-o-x\e o.rwE.tq.,^,r,teEvic -,. d
So\rlden .
LeV a ' !rr2,3,b\= Jrr.D (r,r)(r,a)
T-( a.,u1 , (+,O La,\)
lz.rr\ a I tr>h e'' eY eti 2- i'=
tot:r)' $C,, D 1r, a) fr,3)Lr',u)
' 1+ , u) tb,b)\ si'. Q-
De- d'fxcAe-' t1
Lr,b) la,e\ fatt,b)\ sl'"*-
I'err= tfio'a'r e>'t e1'r.f
L, , "\ Cr't) (a,s)C-. tI g D tlh'o'r'e-"e:1
-TYren ,n
-rF,en LftD = {t-'', D t t,a) Ltr+)Cr'u)La'a)L''t)ts,R) (a,hD th'b)5
$\nce eL\\ B* Pd;6: f r '' r) (-a' a) LB '=) [-s 'L)be\en X 'rco i ,D , LnD ' o'\t \tn'ee X tFern
t.e., L,> o.,\e =reE)e,'i" a '
-rEene .i,2 Y\-E Pdi' L*'9) 'i' L \r.l.,:tts "+ to'"^A \lttli La , -) i^ L , E "- '*''*1oLe -,
.
( r, c) e u o^^A I + 4 b tr-t Lo ,r) Ae'-- v'-et
be\en-X Lo L F"l en c-e L "&5 a-^-tt - SqF \rneErlc '
Sitnr.\a-r\ D i':, d': a"lt -- '59rnrn et:ric
Lr D o\:A e t"o'""-'JE - z le"^ i L-- 'y) t'*d
tt .,r\ te\e.. q +€ ^*X % f6ea'e tttve-e
Ae! \k en (-'.-, z) a.!"e tr e\e .as !- tRe
,6d.^,\^e- .seg Fetr gx a^'*p\ e ( r 'c) e L t" J
La,u)e! al+o Lt,b)al
Lel, A t- d- AeX nBln'a."r'r s''eLolc o e,^/\' A
.l,j.^ F" q- b<ro\ d- c-€bf'-
DPetr.re:r D;o- Haa-t' toq
4;' .vre'\e>ttv e ? 99t 'r'eltt e ?
'DJ3,ronz.,'\i.re "
Booft-: ' 1-s1 F' ter o-
<ltr-c,H fk d-t C c( , lE) -ED
ine,f\e -r''d C-srutar;u--\- . T-rL q er'.e7al , .ib.- -tAruE Iiq nr'.^. eL'Yt c. ?
Ss\.lrlet. .
%,o -i'z -i- t.e BeLatoen *''-A *3'ce,+tt F^orre a.^^A cen-t-al't'> tt'-: ?t1- it<arL
Booh \. R\ 3> n-et ;re{\exiv 5 ' gi..c-e C o ' o)
;'; "." u-tt''1 lro e'DB., ''-X b':o\ a' ' a->
; eon"Yo,.'*o- Ln--' ?'X.' T: bc'ofr cr-
j.+'aeAJ R' t > ^;
h-oL '6!\'6r^ etYi c ' sin-ce-
.-\1*. b L o.b) e. Rr Lb' o-) Ao- !!e't b etcrru2 tcRr
f seE- r'd, ,\ oTt l*"*kj:"T"^-"oT ttqen b*\ E c-a'lv'nev
o,*d .l-i-fte-r^rr-- e , iL bctoB a" cc"tt.oi.8' L*po.q.: rJ *^,". u ' u '3q u c-auaneL I-''"" 1 te* -"::
;.,- a) R' b azwt:- '+1'*r-ehi". d'':>
u*i. Lo ,b) -,: L\" ' a) a' :i" - l'1=[: .
reletl Yo Fr trJ :1..:? i"l; j: f -,.ts-' 1- t1".:-- ; c*rLboi*)
$ien L"e\ d c-e
-P errjevt Po4e'2 lE a\'v\ E ^'*A Eee&- t Ce-s\=
3."r. ;A'' Cendo'in:, B* faae= Yr cr"w
-,.A c- -so \t dE t*t a c*'v "rr-o"re o' A
Ce..to''i^a> tn*- F a- !r o-' 3 a"a"-A '3-s
C^,.) e Rr
{o. 6Vre\}, \ts a-b 'i^,'
6o-r-D Co'* O3e\rrlfsr. :
rev (_e,)- ; . ;.o-.,A o., t, c e
L. H. g
e{':"eSe\rr.lie r. '.
_rrae
Relq.trb""r R
B-\eorr-
db r-tc*'...x
o-\qeEto-
TF,e f,ettou:rnla- Frt..tIX
r.uhe-ltsen -iE
-r/> !tre l{o+t>e- J-fr-r* .Eogtde-,'ed s.t.
".-nH 1)
.cl> a- L4l,Ecc-e
8-
Ss\ub)en :
DOe'c n-€E
a-(D b aee:, h-eE.neX-ie--ehE
e><bE
a- naJ*&-a, t..-re
o\qeEr-e-
t ) be B",o\eo-n
B
= Lo-*t) 1a.r+ ) = (a+t) or* 1o"r-t)e
= cr-cul r- bJ 1 ac- 1 be
= 6.{- qt to -r d-c- )-Lc-
= a-e-f at 51 bc = R' H 'S
Pa')r:all4 .J---J
9e}. !!!\ts, .rlts
.i- aa\teJ Bser-Lop )
Pd:&:d}. grJez
lurtti o- Pd:'}E J g"td"'ti"q R'
orn i.e-,r J set e.r o- pe=etA 3e*. P de,qetts-e--t
.,; aaltJ cr- f"br.X{
4a. =u fu+eL h oo q- A.e--a-tL e)L,,,.enL ,,t-\h5-,- Fneve .,-v :D
:--t
ss\rrFr-:er., f cu AlPl 2-o\3)
Itl- e-E t Lt-, !) ' at r Qr on-e
e\-r e.ut .rin L-T'F.en ar l:._ aa 2 ea 4 o-1
Bg o.n-\i.*,rl r- r' e,b-r.-\ a: = et ao R.-o.ot e\e'..e.n-t
tt e)<.E t= -ib
4\ 'theBou6urr-n:[
8a-1trlc-e 4..,-,-Af, (E'(b ta)*E"
SeL,lIEen.' :
br kF) b3 ge-nc--e- LE, O tri t, = t,.L, =to
l+s G',.. a^/\ F'c-o-' p\e- % Lt^'e - etre""' e"E
Ec,c,.Le-a.^ Alqel-nt-'Se\rr.tsign :
(g,x>.,s>) luhs,e B =fo,rJ
\b. glcole
kogearl r
faeyentle> $ La:e;e-e--.
4) te o- J.altjc-e
4 ,b,, c g L r^'a !"awe
a .^A a-6 o-=of -Aet^peVe't- Lt"t]
Lr,4
.f-..E.". .BLalt ice
l;rog
LeE
Fg.:r
a-+
6}.eu:rneeL
ane
hsP*rvrt4
hoFEr<r) Z '- g}rsrP \tr ajr \terr.ee-E *,,-J ls,ln erl o-
koperry \ ALae-ryt-'o^ \. ^^)
"\^^J4o LD-* b)
tk db ttreaore jd^ err
ai,rjgcfdjrve
tT':'t"* tB. o- La-ttr]ce
arl a - CB^..^rrtJrd5a,U:ve L-r,'/J' ' l'-Q ':
a+b - tre o- ,.-i oOE - 1>O o'
e.tLaOt):o-= o-.
t-
Paer - g'=-=::===:
6Yr-en: trrdE tFe *"e* f:,rodrrc! ? ^^f +-u, o
Aiob;b.r$'"- Lahr"ee> 'i:- o- &iobctulct'e Lalr':c-e
fn u W' i"'4 (" 611'4 -o'fQg\uf,ien : -
' o.nA -v d@ Atd oJoe"t stfu1; cemmrltolive.,
4r.r,sci.o.Jiwe -J obae:^ybb. !o"jt' '(' L * \"1
Let (rc, ,\,> , (*,,.t.),(-., t{a) e L " l'n
1il q:<,r\r) + (.-rr.{.) = (zrrl'cc,H)Ott;)(*.v=,[email protected],)
= 1*a 'to) r. (:-,,\r)
5i-,'lail1 , (,-,,8,) . (>.r,\.) = (-r, Ur)' (*'',tf')
o.' A - 4^ e cs-rnrr,bajtfze '
\,) (-,,1,,) *[(-,,{.)* (-a,1">J= (-,.,"1 ,) -t (-'Y''e"1 a.vtlt)
= (.,cr \ (t, w"ci, \\,t'(S"'uX)= ( t*,v,(-.) w ,.a , ['a rv.t r) tt"))= lnrvz-2, grw\r-> + {>-},Xr>= f<,., :y,).r- 1*-'1.1*(*a'U)
5t'' i\.rttr .
{-','r'': fr:;:-i:,1"i"1 :a,Y:)
"' + o.,. A - q'\ e a:6-9galh-""= '
;
A() Z-zr',Y,) r- f {,., . rtr,)
.(:. r , ttr,) r-'1u'(.*' 'r
,1">]z:r\t
{-,'v (ac1 Arcf rX., V 1.1,1.\a))^ \")
$r'^1,.\.t
1*,,. -t qnA.
H ehc-e
{- 2,, U,) ( at,'e'pb:o.
,[')-f(-',y,)-- (*,,"{.{ , (>.' r$,)
gd.h;Pisr atae"pbb'.sl.\LL><M,.,r-)rD
t aL B be- a l'cn:)ce Booleon a-tqet-ra- ''r.dq 6D a \l a},a'nn e[ B rneru
I€* A te tlie ge! % -:pasve l.'nal rte B-J; ^f€'ti ^- ,
* 'ED
-ir-..'"'.e-r phJ c +D lFre Botolea-n' d5et'l.. 'PCA)'
;; t ; a) i,- ttse Po"'eo '5et' eB '4 '
*!*:a' tAu Mls zovz-J.-
'.-t B t- .-X C.o"t ,
E,.o\to"'' d'Icetn '-\!e q-l,!\\ u€ce -i,.Arr &s"' en \ E \
\!e o.>b!^,\.,lre th ab 1t e. :ne:orrl'b i'= Lv \re-
u.:h"sn \e\ ry\
c loi", : TF e .vr e->rA,t; .,i.. l-y.r-e f o.r I el = n-
Le' B Be IEe Bqolean ^& e! i'- utr.;tF lBl =Ft
Le-t ra-' be ."{ elern-eruE ef, ta 'auuh ltsare L a- L\
goolea^ dJ4eLn* fo, a-] & Ba-
f o., o'fcJeulu..,-
)"o.r.^J
a. !a$i, c-e
62.
LeE 12 t
t- tF.
;
te Tt-'e
t,crDlea^^
B 2- Bt xB-
rPB' -> le, lzlef "r,
^aSi' r\."-r*1 , t {.^ :Y \e"\ z \et\ = rr
- €f ;r.Jrt&st' o*s u'"^Fbi en , tkere e><"i>--E' o-
{,r^aLe +eU x 'J 1 s"r'& tHqL
Br tt (et-),U, o) ^-f,Bc :r t? L\) , !, n)
N. li.T Boo.f,eal' a.I-q-laro-
t e t*ul), u, o) =, e L.),u,o)- (ptr),u,o)bn 2. >< u ).
-TE-on
( e t=),u, n), (p[*), u, n ) - [et1),u,n)crB! >< Ba 1l B
" B .r (e(=),r-:, l.r) .3o" 4 suitat\e g^;Ee- *eEz
Nors , tY-re .srno,\\ crcL A"t"'t='t % e\-"'en]= ';"
^\ ^o1o.'*o *niil .z e\-""e."B c-orLt,ai"' eYJt o& t
=t z= St.q\'Len SeJt
1?t=>,u,O :: th' P'J-^' o-\qt'o i::;e]erYrerub
''' Bq SJt"otien Hqye\h e-=,': ^'. T" -'e'-'rl't i>
tr.-- T," --'). Ps-l=an A rX"l"'-
Dsr d$r the- l-i a>r'e- cltd-q''rou ^' -T- dt^f
the p*^E:J o.'.Ae.t''X [Ca , e) '- A e e\ en
)He Po.,-r e,. .ae}; Pas) u.rhezr e s = | a-,t, c-!
F-"c'.J the rno.x.crn-a.L , tnfn'i'.al , 4st eaL*a-,) A-.-,,t>L ele,Y"enL. nE fu- 7e-*' '
+e\^&rer.. , fA I N19 uo\;JS= f o,L,.\
PCc) = t iq, 1"1, i.3, fa,b\' t.,J\,1b,"1,1",b,"\,s\
1",d5
t&
tet">-,.; ,^l-"- s,= to,L,"3l-leve
t r) !e---,1- el=..,..L
L'zl G:'e-o!er=\- e\e""ent,
(3) sao-xi.na& e\e"r1g6g
(\> \4\r;.nd\ elsr^ ent
to' e)-"'.er1b , F
=tr)e\er^enb-S
- s=t''u'.-3=+
- l',u,*
\a,c\
I', o. Buo\oo'" Alqelro'
tgv cL\ arUeBD fAU6e\.l en :
o Co+tr) = aa +ot= ar- o-t= a-1r*t)=O-
t..o'e\e-..ent
Lei. \D ants\-Lt.n e,
Cs.n p\qtt-.en! eP, .Llb
, P.",cve tk<.L a(ol-!) =o-
Ni, .",4
Jr_t it.rk j.,e La_ttic-e {,_, v, a \ "X ..nq" el a c-gwr3\ern eiL tE en_ s.t "-i:
fnu N/, zot{
=. a...A H cr,n e t.^:o
f '.' a.o- = e]f s "t-t'i ttlt;we Lur-')
f *+r --r]
Ce.^p\e'neruE et a , rrre. ho'..-IDd.V -- =l -.......- ()
c ern p\ er.Le-rLL op q- , *a |ro.r=lF.a.'., t =1. -- tjz )
f '-' "" vo = -)
1 A (->cv 1) Lr,r-t .'Dt vt G4)
$i.'.e'x. .i-q ie =
\21n.c-e _1 !r->
o ,\t =
r-\s.p , x- >
ao,
o,
io,
,e vo
)c:
\- Xvo = y\ LaA= [,2 va) a
:ev faarl) f tei,,2tr))L="'o') a (-tI) Lb.:L.'ituhr".)( o-v =.-) a 1- ".1 ) ( ce" ...rr-ta.Fr_:v e)
") L*'-,1 t.!(X"*1 [ e"t-t*,!ut""")
)
F-l-e.>c
+
= fo-vl)^ Ll1v,,-)= \ A t,1 .r"')
XV,. - LB)
["] cr-J Le) 1^, e
, =-vL\ = \v?c =
--\
f ce- rr..^-Uaiiv e)
L.]-1-X La))
\.o-., .
trd
Cer';\ernerlt
g, &''.f*t11 tE- Eoo\-o" Ex;-,e'+qsn
/ at. tt . c- + ..-' tl- c -r o'- tl- et tt-otr.cl Bc"oteon
oJ-A&aa- JAelllib-- f. C ,l ^
ll z" t {4e\^rJdo.' '
Q'r"e+ exPsrer>tla '-r:
o!.b)- L -t a-. bl .e + o.l-bt-"1
I (a.r =a)
( c+"'= r )( e-r =c)
a- 5* Pf s) i:a lFe
aJ u, n dne
PJ\e."- \Fa-E LrCs> ,
Ps...,- ttre VreArrj'oYr
(t, +),vir t"(b -r e) 4 f avb)ac
.:(e\,-,lr:. t - ..
Leb A t-
Lo(o), q)Tezabco,t
l-lereet,l.b
.=e!oc-- jeio- Xattf-e
6
[e
,b, c- e,L )--i
o Nlr 20lU
-i$.ert
rt ee-t\rru.€rYr,
o.^rl
.r:-
(")
. Pe.vebult'ir? u--tr-al
^^)taln eTeaab:o" Lt,,)
-inte;'.raecbjonHe:' e , t6 ' e-tet"et ir- +
t l' e\err'=.tL .ir.- a = f o
eat)- Fd.;n % e\e'en'b& l/^12 exi>d
( e C") , 9) :o o' la-tr-i'e
,r-" "\|.o.o belts
(- u, n , v) s:a d- La-ttrze , FFen ttt c^":a
c gL
Lc() o-v(tr,.c) z (a,t),rc
u)
vroA.u\o" \elua}i\q :
b
P-al ;
Ar+n r-" c-t- L t
o'-.v C =C
.l,;tr';tuLve
Brr.=..Lo\en
.{.'^-Ee eeic
a- Lallic-e'jt-
^6r1!.i^;1
c! Se.E g
.-J r-eet ,- A\>o ,
d ,lalrtcea-+ c- gi aV
-l:n-eaual[U'i
B,.t-t, -rrt€au.bk/ : 1tse- ho-t.
a. vCb^ c) L (o.wt") r[avc)
( u-t^1 Lr)qec -> a"V (b^4 3 (o'vb),t6 _(_a>
e7v ea>-eli1 , a>,t \),r',-e-.
q\(b,r.) a (a.ru) na-N ort , Bq dXr.otc--
T L\rB ..^ l qls , *. io-r,a' 4 a- v Lb^c) z (ar L) ^. 1c
_\-/ d4 c-
a.v (u,rc). e (a-vt).a c :) a. 1 cF.,,crm Ca) o,,A Lg) ure !.o., e
a. Lc (:)o- v(o".) 4 (avU)^.
p-teve- lF.o); Bene:rrqanr-'a Lc'-ru \refJ q'J T"', \iz
(ow\,)r= Jatt o-J fo-at)'- o''vbl
\-",L:e,. f nu M/J gpt3 t ft: * lo z-",1, L\J'\r.\,n l- f' -\ Au ,.r l_D zoro]Clai,- - J ' Iaab)r= a'.rgr
L-
-to p:.cre tEe "Joe."el:t" , rt i.o e-ne u-q F to
Pc..owe tlao-t
t. loat),4fa'rt')=o _62' [..au)y(*,vb,\=rl. (o.,rt),r (a'y5r)
t
=) (o.aJz,u) vf o.,rta1"')
=) q:vo-)o
I a,l t) o ( o.., L,) =o
a. (- .a r-) w 1a\i b')-) (e"f.'vU)) a (U., [ov u)+ Lo.\@\tDnl^rvtvu,)
(a,v,e,r;a.}:c"e. 1?"f.)
(aaJ = o)
[oa ". =s)
U)
(>:"*-ritut-'ve E'u\)
I Aa.,ocab'e nrtu-)
(_awo"'-- t )
( a.vr = r)
- La)
Clortn - A- I I o,vt)' = ,,ta tr
=t .i:, en-cr,.-Xh +-.o P;r, owe.
t- (-tU) ^ [o, at.) =. .a [-vL)vL''at') =r
I. (avu)^ (o,at')) (o,rIo'at')) v (Ur.(a,',.t'))
tli 4b
D'G$..Jbu-E-veRUL)
:) Lo' r.o',rgr1 v tbA t a d)
_\ OVO =o.: (o..,r) a (dau) =o
( a^>e.rat-"e nu.r")
. ( '.' ". "d =")( ..' e ao =o)
( >iobli Lulr-'"e' **)
( e,*oc!ah.\e R"t")
['.'ovJ=O(
=Je.^pote^E t*)
- L3)
a. (onl) v (""'1 b,)((a- v b) v o,\) a (( av E) v sr)
(av",rvt) a [cvtvt,)(rvu) a[avr)
+
?P "x/1ett".-r',1
(o'RvtJn,t) =r
Frq.., (B) c^"^J t+)( arr u)' = o-,,'t,
.-L\)
B*\=on -d fr1=t-a , P'o'- \tr 't lt{e
a\obe-*e-n!o o^e e1"f--d*^L
G) *'b - cl- Le) o'\r- l' = I
f nu rsro z"r!0-) a-. br - o
Lrr a+B=E
.5ol.,r:r-'..en
Lr)
Le}-
I h.e-n
-) rfi.l
q1E
o-aL =o- ^^J a...r t -- t - cr)
a ^
tr . (( * ,^. t) ,l tr ) (u,=t^1 t'))
= (.rat,rb\) L-. ata,.r;ah:"-
Herce
C: babr=o)
t', c^ao=o)o, z\ tt = bcr +t --y
Qrr :> 6ri;Le!- o..a, t\ = o'rafte comp\em enL sn- tsth -51da , r^re fi.re
f a- at') t = (o)'q\vb =l4^tr=^ -\
(iii) :) \-,)Let atvb =l
_: (a',rt') v [r-ar;=',
t ,'ffH:;r'
(oJ,.,tr)vo = E, (.. t,nU=q)
. !l6 =o-
a\vt =) *> L, I dC!) =) ti>
LeE L'el-tE en ota E =t'
rcr\e ce..^p\ -.,. er*- ea/r tstf SiJe-' .
( ". a U)' ' Lu)'
g! q-z- t
c-crnv erv'e L"n 1 :r'*c!-l H*n/M a-o tf
&\.ulfen :
-,
( t, ^, v) te in- g1'Lze-n &-t"i t,.rLv' l-olt"-'e
D, '. av Lb^O = (a-.lE) a 1o.vc) Y.o\J-- qJ -u)v- o.. t, c €Lu"- X o-4<, ,Iken a.vc.=g _ L.)
Lr) + o-vfbac) = la.,b)a(a.vc;-- (o.\tr)aq 1 "rr.,.X t.)
*L ^ 4 c tE e-n a"v ( bz. c) = gav b)a c
Ever. -1 Jx'"t'itutruwe L",-It-,:c-e 'ip MedllolBJ..b, c6h!er\r=e- -iD rLEE -I;n tLe
i'. , Eve,rq taeArr'\ ovr L a-tri <-e n-eef, rr-eL be-
Ji:.,$"lbun:-eF€v Ex a-.y'l e , \vs
1'1sAr-i o.7r t.,--t it
Pr.or- ftrdtFfgA.rLa-r,. =a ttieel.a^r"^ f n r',
J-t>drl tulll're J'a'H'tc-- '-ro
lot.,.",.en-A ) La-tl--,.- .Lj,. r.Eli Jio!^zitlrw*"e
D.nanr, 'lF-,- t--l q'>*e
I +, =,o, ra, a,.,]36 Lr)cr.nrJ 4, iD c! .:.e\ o-k.3-t
>"11 fnu A/11 203st,lr)"r:s"'" '
P: = 1',g".ch
,r]
D_tt-
, rc]4t "tt
gs. a)ArB.r{,t
fF ar aa
(t) -iFe Relah:on
C Z.z,lo) (e,rc>1a,aq)(e,au) (3,b><+,ra) |I (a,eLr) <R,3t>(b,\?>(r,,ar) (u,au>
[
[- <r?,4q> 1 ra. at) J
A;,o-q.>-.,o,"
R.
(.r rt) (r, ra)(a,u..,4o,t)!re) (a,L) 1=,r:)<""ttJ
TFle AbTq t* C*' R) G
Re\&'ell R i-
< r,a> < \,3> (t,\)<a,
--ite
R= 5
I
Poer- ko.j,&,"-i=e,lr=
%a- B--[e--"^Sut dlq-t''a,-
D rr o , ht e seL T q-l I hlsiti.,ea pe>it&e .i 'ute-qe-r llo, -,t
dlq -1". .- ^,^A t,-J dt .il6
fo, n/a z-orf
qSe\rrlfsn :]-:D,o = l,r.,E, \o, tr,r.,==,t,d
Jtr"c-e , D ,aa-h!{.c -'- 'setextve 2 an,t;+qn* e-h-i'
a^J -ltr *itt,=" P=nep -'.h:=--' ' D L> c'-
Palbai e.Ae-, Retoi.'en en D11o
f o"." , D) .iD cr. Po-seL
He;.e o- ,1 t = ?-a {o, t}-* to,tb \* o-,b e,D lro
\r) .i'- 4€x-bt
't' CDtto ,,1,-LE ,5 H d^r,e
Hene , _[-ru=t eler^enE [o ere.^ent)
'r!f? IGlste-.lte---l s\e'nent- fr e\o.^ent)
I-lq>lel\o
e.a.cY" ev=\ J-. enL
dio2>.a-.-
LcL\ !C-e
o'.Ac-or.g\ e-"ne_nL
F.:te.n
YF dI
]L C-e"^p\ernenteA .-LqI\!l)&-o
\He Hcrr-.5e Jtogrro_Fn ,, i-t. i& A;>.L,-,t,,rIi...e
iL iD gtvGs\r>l -ttr- -
Csrro, o) ia q- Boo\eo.. Atqst ro-.-The- Elr-t bo.ol-a^ eJq-t-:r.e,t ase
L') {", R = lrrrro}('j) f rre, s, ro,1r, aa, Fs,lroJL3) 1..,drr,,r ro\ ,r- {eS
Sts.o tko-t
Aeb r-oYte-we
lr" -
^l
=)
2_
Pa>rlr*of\ oJ.rr=d
:.oY
rLgn
,%,Ed
-we rntEe-l
d.\\ Pea-rb-'ve
tr'1 .n +- lffi- h?dtsrl 'e inteq.rn '
#-*FI -rh
r\e.o ,
.^ra Jq- flgn
Se\u!"'en .. f nu N l, 2olo]
G1-wen
-iF.e X&en
z--*-,e R5-
r.{ .i> tF-Re\okr-ien
11 - rYl S.t,
e;\
r* >c e r.l
** .L qlL +ve ir}-e?e,
rtB
-ve \nte2ea
-ve irdrel.a-|
:> t - - X) -t (-'1 *z) -iz a- DerL -we i rutoae,l
-) =--> 'j'o d- rr-eh -ve- itteXe'r
*-Rt ,- t1R= :> 2c-R>1
R "1, Lxoaeitt-e-" R &, Po,rEi qL e:rJe-r R eJqb'-crt
h6n - Va
R ib el=[lexi\,eccrn>i Je-r
- RX & %R >c,
<Sirrc.e a R1 :) >c- - r1 .6- rl.,rL _ve inleq e>r _C.r;
tRz- :> t- E i.b n€ry, -ve \n\_*qe:r
-> - 1=-- g) S- rLEr! -ve 'nt eqer
-> =--y f r'o.. e) &(e)),g, -b Aif,i,:qrn- gbyic
Ar+ss"e *-Rt 8- tR>.z. R1 -> a<--r1 .i:o Y!grr
YL€YI
- 1-zr
St" 4 La.lfrf--
t>cvt)^Sebhjen .-
C-lairn - I :
f u , a) P-e.,e- ff,qJ; "v C Y A >\ L
L-- v =)f au NID 2o\o]
LuR r'rt -.iD
iE .i: etvi€ub \ttd-
"fF:
,"., Ll n-) I [,c vI) -'r f rvz)F.rr ernr, ttre dqrr-.r,tie-*etv;]srr> tts o-U'
>c- L )cY t -
tr\}ail \n>1y42c-v.A
Fsre.n q1 o..A L.a) , :crX .i:o cal.,! \rfpq t.*.4 1{*-,ts^=3
\1er. gg acv vtr ) .t- w fXaz)Fv's..' \He. Aq.*h:en eB LuB ,
x- L '<-'v z- -
t?)o-'lJ 1u\= LZ1*-^vz
F.r el^ (a; -"J tO ) x!z- 3'>
te.,"J "L f -, .1 a z.\
e)
Hehce >c- \l= ) z--vF.:'em (41 "\^d (B) ure
ty^=) -_ G))
L.a,.r e- - -Ct1zt>) -i.-, d,J.e*.z te.,^.J "E f ="X , -.r=!
-' . x- \'/ tt" =) . L-ry) a lxwz-)llenx-a- the V--J
Ia
IE
>"i*il*\ 2- a Yat -a)
=t E\"o ,ib
PJtevet-d))),<--e
ltie act
-tJalfon
f6ae:" &'-i-e -
s! -i- o-
2(, r{
lts.e
tF.-u
cd,X .IL-t -eo.:o e! +e ^^^J
t" €y, .sq€
c-e".yI-'-e-&J
6e\-,r\f en :
55. =
6,n" _ t--\Fro \\or>e Jic'?=.,
He-t.e ,o, e\ e-...,er.L
t t ) ele..'e-.'-t
r-c-r,r I r, 4\G* lr,qa\ =
I v \1\ 4a
: t1 L
=4r, 4 4 il - {-
F)' = zr
\a,\
5- \eCe"^y\ e.nerrt 'gt ,B
i'e., qr)1
, !',
LAr Nr.u
S& -g o.r\ div.aes> % U"
,l,t,r\rlarl
1".
ar , z1\
L "u., >)
I
b4ee,=,r le->\- el-or*=,.rt i:
'gsl <.'l-o!e.o\ ele*'en-+-
e.,rt\ -b 2- zv\\ - It+
(-); I = rr,.
t ^ -1 - u & t vt = 1
Iu)'=;Btw"t er,.-\ (,r)'=t, It'n)t: + , (at)\= a o-n{ Lr,2)'=t
6ih c-e , evei'\X ele-."e-r"E e'B E42 I'oe C-e..rp\e'-e-nt
C ur.. , =l ib a- 4ernflerne",t-A Lalh,'ee- '
EYrs,.r: tEaL i,rz. o! Lotic-e beYR -i+6,ten ePaec erkq o-J Jil-"itu-ic" e in-o9\r'olt-hj€'- c,r1e' I a lno]c ;,- a- 'Ai-\-..',t.rrc-.tru.e AXrr. -6Yuew
I,ojxc-- co,*c-ett atds^,' p'4r j'- bru-'-
,qs\,gfiien . f Rt.) t"t l9 zoro]
Lr) uelo (u, ,A,Y) Ve' qL"! e\L Lo-t"'c-e ' Fe-r '-^i1'
a,t, c gL, trc- !"o-'e '
V + c =) r) o.-,.1o 4 o'','.c
a) <x"tb L o.Y c
P"rte$ :Gil.,e.r bL c:. c,rr-e!b, c-3 = b^ c- : b ---u)
Lua I b,c) -* iovc- = c' --La)c.-[ou'i.na - r ',
6,,t8 2 a-as -Ca)-r:o P",o... L ?) , iL i: e.rs1rqh lso P..---
cnLB loz,,t, o..r.\ = a- ti. e -rb p-,owe-
@"U)aLc,.a) =a^t
( a^ae ct,l.u-e- 1-'-)( ce.,-t wEa.V--.,e Lo,',,)( *.r.se cco-h:.re t-.r^-)
( =d.e.-g
eEent Lo!*)
( r,rs,tna u)
6o'a e ) = o-rrt
L a..\c
o,wb-E> ?g,o'u e L\) ,
!r.re f ovt, o.r c\
i-e -\-o ?vrowa
f ..v \.)r Lo- .) -- o-v e
LHg
C a--b) "v (o,vc)
;j a.w (avt)ve+ Q,va-) .v [brze):> o-w tU"c-)
>z @-v C
( o,wb)v to.,D = 4v t- a^t L o- Ac-
! o,w c --Llt)'rE 1- eto.tah
= o-v C
.go Pnq).,e lfi&
L A,q oci. )rYe f'a-u-)
C co"^.n r-boh:.,. I*.4
f a,,ro ocah". Lor.,.)
(_ rAe-mpoEertt*')
(.r^'r-2L$)
(a) ue\, f r- , a , y) b' o- Xi'veJ' Laltri c-e " E-.a , t, c gr- tF.e.
Be\t6ur-r.-? -"Xy^tqti) a.vlbac.) a (o"vb)a [ovc.)$l o, a [u.ra >.- (o aD v 6a,a c)P-:teef, :
cLa.>,. - t :
lua c) a
Je-6rnili-'srt
d-4 o-vE _ (J)a",r-J \> a.c 4..L 4 ouv b
Fore^-r \rt c^,^d L4) , 4lt j.- o\^^-
{'.,tael!le.r. c-r.- crvt ! .- v(_L,^.)
-u\)Fr.e. tfi- d e1i.s-bbn T_ Lus, iL6bo,- L o.-* _
"fF"" tE^^"^J T
a-'"d t^" 4- c- 1 o-.vc-
Fs.o"- L3) o.-"1 Lr+) , a v c J..-o o*n wfp.n to...d"t 1"., b ,^.cfH e-",-c.e o-v c_ > o_ v ( br.c) _ (B]l
"r-XhelJa
l-srsrt'i Ihe
(o.rb) a fa.w c-)
% LUB - rt- J-b etrisu:, tk&
-i> eb.viou>- \i cdg
Fsre."^ 1n) -,.^Aa- ,Re.o.-, torr,-d
o,w Iua")f-lence tF.
a
( B) , \A)e hav e a-v I ba c,) i5"B f ovb,4vcl
3 fa.vu) ngovc)o"%
% tr)
C \D^,'^ - 9- '.
a_ ^ (b^ c)
w.k . -ro" ;: o.ab
a.n.J Bv c- > t
Fs.e- L\) o.-"d
] ['',b vc\He-nc-e-, q- a b
l,^) .}9 .T
a > o. ^. c-
anl Lvc >:, c_\r bvc >
F-s.., LA) *J1q,bYc5
) o- Ac
a-Ac .- (t r(\), cra6 J> au/r
Uel-qa c\ ,1 c =:
q. AForern Cc) qnd LD)a^.! .^?p: te,.,^A
o^ C bv") >
) (o.a t) v (aa c-)
) a-a B
a- aE -
(")G) , a^b -rb a.^/! .!.o.rre,', berr-J
4 o.a fbvc) _Lc)
ce)
-!-e.,"e-, to.,-.J eg
( b.'/ c-) -- Lb)t re' !,o-., e o_ a f U v c) .;.o
f "'.tt, "'z..\(o-at) v6o.ac)
ct)
Heh ca fie p-:roor
Let ( t. ^, ,) u3
tv f ba a-)
twIoab)
s[ tri )
^-.: q,trc €,L fii.He\ A,>
L3) Cen --.. Ae, ,b
xt-=n
( C-ev'.,,vt.rta!.:.,e
LqJfuc-e. Fe.r
fmau-)n\
ta-)L*)
eeUB\^,1?
( ab+e-.' pLJun
=bvfoac),(_t"v.-)a fuvc)= Lo"'u) a ( uw c)= (o"")z.fuve)= (cvo)n (cvr)- cwLcat)
= cvlctac)= il, (ca""),c
t=e
( Gl.,,="t Ce.J"rtie-')
Lo, - ce*'!bo( c e"*r"u&o-\i..e L*)
C !r-'.a qileil.Ce.'Arlie"1)
C ce.-^tt r"jg.t'l*.'r=-RuL,-\( *H -, - ."fuo*)
f <a1.,..' Ce .lJ.t!e,..,)
C ce*w.,',Lai.,,lr. L"^,( ate,s.pL,is1a Lc,.,*)
Ske\, thoi, n-La,tf,,'c-e
& 9L e> o--x tr=s€)o,'@u=,+>N/J got3 / Ao N /,
b' 1ol .
2-o t{
_ C,)
C ,t,*"a .,D
"Z..,sectali:-e La.r,,)
=oo .,, . -_ .=)
a- c e-^yle."^ erd;eJ --J diog"-,!.,1;.,=
seLubbn, fn ,r1) :> d,'tLet q +LrFLc-r! o- aE -- a- cr,r",J q-v t = tNe..': a_ zr1,) = ((*,rt)a b,)
= (aata!') r-L= q_,,1, o ( -.. u at ,
=o (.l{ err-ca" a_ aL =) q,1 tl = o
di) -> Cii{)
Le-Y; a_ AD.=9
fat(e cerng\e.,r-r' er& €rL
( o,nbr)'= toJ'
tet'tt ,s'rAe=>, rr:e
f e'11 ca.a^c- eltolCevr Lo]-)( s.a+"ibut le r
".r..)C ...r^B'=o)
,side>.
!to^r e-
a.w E = r
aabr=o :) dwb =rtia) => L!)
I-eEat.rb=1
=> (.,nur) v (u,..U) = g
":aa_'vL =t :l L, e d
Li.r) -) Li)Leb rb 4= o-
-IEsn al n Ll : U-r-oSte
c.eF^d e+^ erlE o1y.' t e thI
(o,,,^,tr),= tU),=> o,w t - t
s! a- LV
lot4o'->a-LV
t-ev ( u, a")
Xxb La.t)ffien 6\.ovsCe.,'',$r,tok6l e,
f Aewyef e.,rtt
Se\.r-^r^.bn,
!e o La,\tic-e
",*A * @t -
\tso"L te\ia x, drise qLd}iw e
La.rJ-\r-r .
tA, $/r 20\3 /
€L,
Paoeenl r : L=Ae,c-esuentt
LeE (-r-, , v) Le f-Lven.\ cr,t,c e L,c)- \t o- =* oJ a4
Psroof :t
a_\ e = LuBLq,o) :o,\/ o" z q_
Nsrr:, q,l 4 = Gnue [cr, o")
d.Aa, =o-PnopsRr) a..
C
Let (LrA,re)oY o,t, c
a, vt -_
).o-^r)
L alLi-e
Cer^r.nut:qhjwe
b e- Xrr.r"CL
t v o- o,"..J
P.:LR:a-vt = Lua(a,t) = Lue:(b,O = tva_
L .,. L\rB (r,U) ;o "^gqo-.,t -- F,v a.
o.At = Gur: Ia,U)
o. AL> = b^ cL
L
, i1 .r.,e defr"e c *.Cut Lo, , b) 1er. - ,t
^^,_J o *{t, atoe-ptre-- o-J
Ai-r N/D 2o\o]
- -rFo,l , Bssr
Lue(.o) = o-
= Gr ue (o) = o,
L^ 'Jr )
L o:t-'ce, . Tt\s-'r- { o',6
cd,^b - L^ a.
64LB Cb,(l) = l> zt q
" 6te Co,u) ;- tr-^".1d
N sroc
ve o\ivlA o" a(ba) = (oaBac
hogee=1 a-. ( a,^aec,cah:--t-t I L,,\, \) !e qr_ven
*11 o,t, L e L
a, vC bv- = lo-wu)
f-arrr)
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