Snrny hci ' 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:

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))

bccau:<, brclge,L (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

lh6- dre;t"r*n.rc! lo- evsrg g*aei ><dLt"JY) '* h*vr'"r{,:.#€ m*au} 'o[ cley"eg

L*rer F{,-t**cl l*a *t b*d g*}" 1:*[ou"6ecl ,,"-res)

L' bg fheo rs*-) Z"E

{3o"lget [3o}o*"e*ln'*:e Ar'tu 5ym're}-ry 4 U"T3-*'rerfl

t.;hr*h ry$6n,.18 l4ornogene&g do€r *oY-nete)Jcr*-tlY1 lr-rply bat)ge'Y bolattCetlm€11

<)v. bu.l3e.1- tolo..".dr.ess ALOr{le g}$es noF w:yt\3 hc*ro3ul,e'r}Y

fr<;.r pa1c. BIb9 buclge! batorc< ulnes>

te"t CLp,by)= ky

a'-, ct lry ]"1"e s t * w1 ! .tT

boh ho prov.- PiCr) = cl u,rfh DiLD=, x':qFf 'oq)-*L$e rreed cinoFr<.r q,)5,,v*rppq.,.i

"l-ho,tr- S{?rtl} r: >yn-rrneh't C

e v*(?e,b) = 9

Z.o, ?t'Lq'3) * '-xi (P''9) .."'J]; r ?.?i

Lr;i=l

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 =ocl*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

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 ,

L p; ?:gjv) -= *xjtp,yJ (n<l Z p; "'c?Y) *tb?'t aE

*shere" s(pry)

li*e {*"Jt *l- €{.*uftq.,a

.* n fi

: X,. [pry) V t(l) = con5f61r'rl ar Pi&)=a

s Xi Lrf ,Fg)

N..L ) Xt (t,p,Fy) ; cr_::'** " (_4#3 r

clr vr chr.S {:y [ > c:

1

P'J

( tp, l.r1 |-J

- ruHY)

u:Gp,ty)

L+ h "

moycwrsll'y

lwrl-l.i f,**.1 p <r^d g , xi (lP, {rg)

hne,^ rF xi [fF, Fy) = f;L+ hrokq. |olof cier-tv<,*!rr"e 6f f r( f')

"(r)= f- er:ltr*blo, +j;r Vp;, t /,shce tg= bp.x1lp, l-g) ,a^J by

g:. 2 pi ki {,lp,ty;j't

$h..,: r..'(r). f_ l-#,*O o, *.

>a!j:,rr) " xi (,t_r*, ly).

O = e p.i i tj-fP,ig) n )>,r (lp.Fy) ,.,=r'+ a apj _-all'"----" XJ

lf qre.: u5<-

lh"l r.n 1"lre>

S ( p, u) " ?11 i*?-b)bpi

o = Zpi. sfpj.U]Jef

4): s(p.g)"F

2.E Sropo* l-h..rl- lh'e. fut'ol-tc,'.r eCPt r) e l(*, n.:l' pscetl't''t1"1

[u-ol-rcr-, au.,d x (Pr VJ e lR: , hol- necetsa"'l', $ $ev"o"nJ

l-h<- sgrl'e.rn q,L p.-[-rol drPD <r<nVta1[ g.l.rrorl'r.c,6 > gtven

5l,r6c\, l-L€ L{,11cl sr'rrnry:

(") tp xCprj) 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

da5u!.ae) everJ nOn ncgFal-rw volue c4) U Vorrter, l-Lr.-,lrc hornnogeneou5 op cl<gr<.e ZC"<, th (frrg)

4 L' a""o'.|,^? t-o (P, u: ) ?j.9l!l - X c I p., e {r,u)) $c," .{,2 Izpi

L lr= YLe,y) ss.l-rrPre: l'l"e L" d,r.l lou,Lonce.lr-,e:: cc)nJ&ro'-l

(e

conar<]<v"

PiXi (.prlj) * ! For anY Y

e (pru) .rod urq, geL

Fr Xc Cp, e Cpru)) = e- (p,u)p. l-!P-)j = e(p,u) = t.e,Cp,u). L ap.i

f E irt*" ,s The *-** J

+!9 I = k -c.,J € tcx) !) ho*urt*neorool}Xi' Jtgv*e k ,i *

lhe". CZ) 'rnplte> [h-L e [p,u) '> Lovnc:geneoks o p deSreet). l* ft e Cp.,u) rt hornay€vrsc,u) oF drSree on€ iu^? c,v.,.[ p*

' e (" &frt-r) * fi'e. {p,r) kr" **y t }o ,r..f,tg}1*g l.c:L- OrPlore.'F,.*he tEtl-i6a*.J s,td*:i [lH,t] r '' . :, ,

\ /.ae tFp,,u/ |

-%

, f

,?P' v

l"-

lt

alg zr

qn eXY)en qlr'lur' gh"rrl-ron , sdf-r; fy

l^It'r *cvtor, /--2

Lov^ eq"h P , .i

h

>< Cpry) rrar.rs|

"-" n

11

z1-.L

)=

zz"ub'hF.uk

(tJ

?p

,nl

I I tlL$ by l\eor**. h.2.1

ZX1 l-L

oF€ In peqcLt fJ'Thso,oc"",

h,n6

Dt P FeF* l,rle ,*'7h, ha;'d stcJ<. I Rl-+5)iplagce,g)l =

-b 2_9coi)* ar": ?pt+ lse *P.$ S c,nJ brtng LH S **d q $-t* l'CIgeF.ei-

l=j|p^,)'. & = 1s[gf) e € x(t-p"y).\= X(r,r,\urh,ch rr,,--.' 1clr<s X Clp,g) = X [prg) ytXCp,y) rs hoh+ole.vre.$u5 of clego"*e Zgrc

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'lcJernond b<,hstr.,l og',

' eCp, , po, u)= u P,'Pz " <* e {et>?zt V (ptrpz,g))= @ -fFa+pt)

w v L'r &>ta' v !r',)r zt Jr' (%u vz)

e> v(?rrpr,g) = H.#)-L.! crx.,v'.g th.l. g =.1, t Jhere l?t 'Xr t ?1YZ t1

€ v(p.^ a\* 1P1+?)1)H2r&./ - -prn

l* U (X, )xz-) z Yntr, V (pt, pz,1)tpl

= ftrfn Pt+P? >.!' lP) Pt?z

LLo3u.oFJe' *= ryP yvft?z

s.l Ftxt 4 Pzx2=1

Pr>Cr + P2x2z L

It-'p,x,-Pz^z)

-)Xr =o (a1*t I

+* 1-FrX1 -P2x.=gz7L subrf-,|'vle (t7 o.n"l Cz)

a= - Vrfz-Pv(z*P22.Y2 Pr Pz * Cp,tfi) tt PyP, - Py?z - Pt1

cP;p4z

Lr F:oC

xrbclrlvle V<, Ct)

t-P,X1 -PzK>

tr.re hqVc Pt =

lt uCx,,*t)= v(Pt

U,"PA : tq r vr] tu(P' Pr)'

k, Vn - !V,lPzNt(tt,Pr)2

r .rD4d-

?PI

-.YJ..- g

3Fz* v$z=o Lz)

Cs)

D2tz-ef- P"2 1) Y,-€ #b-Xz Pf 'Ft* Pt'

=O 45 1r'4

%xt 1a-d

(P, + pr)

vt ,{, x, € 1

-rt' ?r \ ", * -,lx,x, )

Pzz+-Xz + '{ Xtx2 (x2+ J;;xt + [xr +-&pxr* /xr,

tPz,t) =

tt J. ---:-xr * -lX,.xr Xz{-tIxr{-r [x, *TxE]€r<=<#iil)

?r"Pz(x, + fi[r)(xr* rf*,", Cx,+-dfE)6&c+ {TFz)

U C).r ) xt = Xr +Xy * Z-* Xr:r'z

XtKy