' Ler'
ln/r":h""ru l'1.rh,"aJqJ1ia1:r
Cku , [',{n.15..:t"o) Frzru.j**r1,p1..!d ?,1 Shour l-hnl i:udqsl- tolo-cecln€.,t o,'.,J huro,og*',e,1-9 oL kCPlf crre un relaled conrJtfrorr: Snrny hci
i'r |hg serr14 |h-i- nerl'he. ,rrurplre: hre. 6l'he' Ii'h,b*ocje,ae r]Y tlE Y (P'"fi u
()r) = u (x (?:U)) *
u e x {&P, *Y))
(p4) a^d {l'p, l-y) are Yl-< 5a#v1e s l' p X ( Y r' Cr':yJ .,nd x (lp,l-y) ore fe* u,tl* $"h'-.,: >rCpr ifi* x {,Fp r't'yJ 'o[ cley"eg lh6- dre;t"r*n.rc! lo- evsrg g*aei >
L'
bg fheo rs*-) Z"E U"T3-*'rerfl {3o"lget [3o}o*"e*ln'*:e Ar'tu 5ym're}-ry 4 lr-rply bat)ge'Y bolattCetlm€11 do€r *oY-nete)Jcr*-tlY1 l4ornogene&g ry$6n,.18 t.;hr*h ALOr{le g}$es noF w:yt\3 hc*ro3ul,e'r}Y
<)v. fr<;.r pa1c. BI
bu.l3e.1-
tolo..".dr.ess
b9 buclge! batorc< ulnes> CLp,by)= ky
te"t
a'-,
ct
lry ]"1"e s t * w1 ! .tT
e
v*(?e,b)
=9
Z.o, ?t'Lq'3) ]; r ?.?i
* '-xi (P''9) .."'J
Lr;
i=l
x':qFf 'oq)-* u,rfh DiLD=, "l-ho,trboh ho prov.- PiCr) = S{?rtl} r: >yn-rrneh't C L$e rreed cinoFr<.r q,)5,,v*rppq.,.i
cl
2.V Suppols l-i".1- x tp,g) € lR? s"ri-rr}r.e.r budgel bal*^e.ecih€>t ..rv-,cl hor.',c,g6.ms,L-7 on"' lR i? Shou., l'h.*l- Fo" qii (p,g") € iRiS , s tp,y) . p =o *shere" s(pry) cl*rtol-ey i-he Siulskg r"ncrLri> dibo 6iralud r^rrl-h 'X (pr) t+
K*..rll Fr"r- lhe pr@D ol Theo'e",- L.t1
li*e {*"Jt *l-
t.r!.,e", burtgel- b.alsareJnc>s hcrldU u-t€ v'nctg *fu[f"srur, fieo]c
r"r,l-h rclFe*F f o p.rcs5 syrct rncerne }c, obFo'tr f,u* 'i *
f
,
.* n fi
-= *xjtp,yJ (n
h
moycwrsll'y " lwrl-l.i f,**.1
g , xi (lP, {rg) : X,. [pry) V t hne,^ rF xi [fF, Fy) = f; (l) = con5f61r'rl ar Pi&)=a
|olof
L+ hrokq. N. .L
p
cier-tv<,*!rr"e
6f
f
"(r)= f- er:ltr*blo, j;r
Vp;
$h..,: r..'(r).
f_
pi
f')
s Xi Lrf ,Fg)
Xt (t,p,Fy) ; )_::'**
+
#3
, t /, shce tg= bp.x1lp, l-g) ,a^J
g:. j't2
r(
lh"l
qre.: u5<-
r.n 1"lre>
r
(_4
by clr vr chr.S
cr
{:y [ > c:
ki {,lp,ty;
l-#,*O
o, *.
>a!j:,rr)
O = e p.i ia tj-fP,ig) n ,=r'+ apj lf
"
S ( p, u)
o = Zpi. sfpj.U] 4): s(p.g)"F
xi
(,t_r*,
ly).
1
P'J
)>,r (lp.Fy) ,. _-all'"----" XJ ( tp, l.r1 |
" ?11bpii*?-b) -
Jef
"
-J
ruHY)
u:Gp,ty)
€{.*uftq.,a
eCPt r) e l(*, n.:l' pscetl't''t1"1 qn eXY)en qlr'lur' g , sdf-r; fy [u-ol-rcr-, au.,d x (Pr VJ e lR: , hol- necetsa"'l', $ $ev"o"nJ h"rrl-ron l^ /--2 *cvtor, It'r g.l.rrorl'r.c,6 > gtven q,L p.-[-rol l-h<- sgrl'e.rn drPD
2.E Sropo*
l-L€
5l,r6c\,
L{,11cl sr'rrnry:
sal-r,ll.,ru tudg.u lr.rl--.colnc>b, Vh"^ e,Cpr') muJ- be ho^ro9ens"': oF delr<<, <>r,€ In p, tb) ID e [Pru) r,.lnor"og ev-te,otb cL A"gree c'ne rh p q'.,J Lov^ eq"h P , .i h da5u!.ae) everJ nOn ncgFal-rw volue c4) U Vorrter, l-Lr.-, >< Cpry) rrar.rs| lrc hornnogeneou5 op cl
(") tp xCprj)
4
L'
a""o'.|,^? t-o
(P,
u:
) ?j.9l!l zpi
L lr= YLe,y) ss.l-rrPre: l'l"e L" d,r.l 11
z PiXi (.prlj) * !
X cI
p.,
e
{r,u))
$c," .{,2 I "-" n
(e
lou,Lonce.lr-,e:: cc)nJ&ro'-l
For anY
Y
1-.L
conar<]
"ub'hF.uk
)= e
(pru) .rod urq, geL
z Fr Xc Cp, e Cpru)) = e- (p,u) z p. = e(p,u) = t.e,Cp,u) . l-!P-)j L ap.i
(tJ
I tl L$ by l\eor**.
I
h.2.1
f
ZX1 L l-
t). l*
E
,nl
irt*" ,s The *-** J
+!9 }Xi' I =
k -c.,J
€
tcx) !)
ho*urt*neorool
Jtgv*e k ,i *
Lovnc:geneoks o p deSree oF€ In p lhe". CZ) 'rnplte> [h-L e [p,u) '> ft e Cp.,u) rt hornay€vrsc,u) oF drSree on€ iu^? c,v.,.[ p* eqcLt
' e (" &frt-r) *
L- OrPlore.'F,.*he
{p,r) kr" **y t }o '' tEtl-i6a*.J s,td*:i [lH,t] r . :,
*
l.c:
zr
\ae /. tFp,,u/ v ,?P' l",
l,rle ,*'7h, ha;'d stcJ<. I Rl-+5) iplagce,g)l
Dt P FeF*
,r..f,tg}1*g
'Thso,oc"",
,
alg
?p
fJ
fi'e.
-b
|
f
-%
lt
2_9coi) = ?p t+ lse *P.$ S c,nJ brtng LH S **d q $-t* l'CIgeF.ei-
ar":
l=j|p^,)'. & = 1s[gf) e
€
urh,ch
[prg) yt
rr,,--.'
1clr
X
Clp,g)
=
X
x(t-p"y).\= X(r,r,\
XCp,y) rs hoh+ole.vre.$u5 of clego"*e Zgrc
h,n6
2.6 A c*,rr,r- ar ho,e gxpenoliYur< puncl1on e(Pt, P:, u) = U Pt P" ftpt t Pz) F,nJ a d,.ecl sl-rlrly Frn.'cl-roy. u(XtiX") i-hs,,l- rtril-tonql'@e) i-l'; perlot'l cJernond b<,hstr.,l og',
' eCp, , po, u)= u P,'Pz " <* @ !r',)r zt Jr' v L'r w e &>ta' Vv (ptrpz,g))= {et>?zt fFa+pt)
e> v(?rrpr,g)
=
-
(%u vz)
H.#)-
L.! crx.,v'.g th.l. g =.1, t Jhere l?t 'Xr t ?1YZ t1 a\* 1P1+?) € v(p.^ 1)H2r&./
l*
U
(X, )xz-)
z Yntr, V (pt, pz,1) s.l Ftxt ftrfn Pt+P? Pt?z lP)
LLo3u.oFJe' r
Lr F:oC
.rD
4d-
?PI
+ P2x2z
L
k, Vn - !V,lPzNt * v$z=o
Lz)
I
(P' Pr)' (tt,Pr)2
3Fz
+* z7
Pr
Pr>Cr
(a1 *t
g
L subrf-,|'vle (t7
>.!
yv It-'p,x,-Pz^z) *= ryP ft?z U,"PA : tq r vr] tu -)Xr =o
-.YJ..-
a= Y2
4 Pzx2=1
tpl
=
'
- -prn
Cs)
1-FrX1 -P2x.=g
o.n"l Cz)
Pz
-
* Cp,tfi) tt
Vrfz-Pv(z*P22. PyP,
- Py?z - Pt1
-
D2 tz 1) Y,-€ #b-ef Xz Pf 'Ft
P"2
* Pt'
Xt
Ky
cP;p4z
xrbclrlvle
V<,
Ct)
t-P,X1 -PzK> =O 45
1r' %xt
1 vt ,{, x, € t'
4
tr.re
hqVc Pt = xr* /xr,
lt uCx,,*t)= v(Pt tPz,t)
U C).r
)
xt
=
Xr +Xy
=
a-d
Pzz+Xz + '{ Xtx2
(P, + pr)
t
?r"Pz
* Z-* Xr:r'z
1
?r \ ",
t
J. ---:xr * -lX,.xr Xz{-tIxr{-r
(x, + fi[r)(xr* rf*,",
* -,lx,x, ) -r
(x2+
[x,
J;;xt
+ [xr
+-&p
*TxE]€r<=<#iil)
Cx,+-dfE)6&c+ {TFz)