Upload
others
View
10
Download
0
Embed Size (px)
Citation preview
Solutions to
the August 2008 Qualifying Examination
Any student with questions regarding the solutions is encouraged tocontact the Chair of the Qualifying Examination Committee. Arrange-ments will then be made for the student to meet with a faculty memberwho can answer their questions.
1
1-L <9) f -r LAJ OOY1 Dte ~f flooY' v4 '>(.1 i C +l-:re (, "'II'rj v€
\ Y) ~"'S' \ t. L>< J :c x <C. L'>(J+. \ ~ n) ~erloc?li(·~tJ u1 J) LKJ X I 'x
F(~) :: jS f- J~ "LxJ f ++ rf () LXJ 0 LXJ
F(~I - ~ F(\) =- (l j ~",) HI) ;- Sf L>r<J
VJ,(.J t 0>"11 0", [oJIJ cY",cl (l<'(jva!'c :A fE'Y1"oI I
-=> 1:( I f Co", [ 0/1- OJ) / /)':J 1:( lA--e \"~.+ )<
0(\.r={LKJ-X) FCI) +- S~
-r ~~ 1<:p(>V/:: ". Fn) +C Jl .. t
"F(\)+-c W L~
~ dJ ~ 0 UYl okJ. 1 \-re Co", 1; V) vi +'1 04 ¢ Iv /I OW , Wl "'"E' ell 0>1 e I~ I) I Me ¢ I"j & oIlJ+e ~(e v:I + Gl\., J; Y1 w tJ {",,,, rh VI S"
(F(\\ ~'; f"",o( th 01 edeL/lv! I rJ",d )c ~(\) )
(
c - ---
t W
S ~} =II'~ , (. H.-7+V.J
UJ
b) (£~J Xl
i
y F(\) I . __ T
\~
b
~ f vWA L) D -\i)./))
r 11 ().: -l/u) Jj +x"',, (C;J) t-'- 8
(/ '" / u 7 <J v' Y.; I
']'{:oJ. ::;-; (0)")l"
,/~ ~ (v, J)l.. ;J L; L
If+-)(~ (0,01eN\ ?J T ~(~"')
IR()<.,'1) I< cP(>--!)). h" "-1 ,4 o,;:L,1 ) '+ Wv-t,
~l -- ') (J 7
JJl-l . 1,( I ({7) I I -il . {C f4v,~ {,(t)(Z--1.,jl ~+)
J j-<'w>o V. ~ k -)1" &vi )-'1\ \7 d ~ i<:, ~ :> 'I, \
I[olcb-+» I slh'Oilll1-i 1\ If-I. It ( CE l -£J 17:-7-)\ Lt.
--:) i??) C1 v) ~ r/J ("t) ') 0 Vh,P ¢ tJ h ~ I) tl C ~ vJ 1. r? V) ~
cJt t'). --=) r [" ~ ~ pv)-e ov<k(' t<- ~of ~-
~ vi- If C V1 yr t - u+ c)vO{P( L (C rltCJ ) :t I) I) I ~et ~~ ~ {'OJ t D 'ro ,,/lei 7:" ev ho{
Ir(c- ) \ \~--+o 11,,- =, ftl-) \17- Z I I d>((1) I 11 )~ ~f- ~ -teo cV) t -./6D .
-i J 't- ~vl\
~~. :.( l1&J QI~\v/ d p;)L~ 0{ uVrhf I c)/ d
~ lVcilCl \--e J.)\ r6 J)cJv Jy . {,v
~)/f( P h Nv.,l<- vi OV~.( I 104, [! cV 7~ Co J--:- -+ \ r '" tP ,,,ft?~
~~ 0 Yo u~ t IZ~f~)=-\"l-u) t·.~, R~C~lVJ.7(2v1
(r. T b€ I vW 11ncc f M"'J N r,1 a+ (:OfJ
)
J f(,;)Ch. r~o Jf <.
~ r~HJ
~ 7.n· X eff J~ (~J~q (U, 1'\ ,,[ /-<l- 0",
{v, fLl f('~) c \t1 ~ ~ ~k ~J-t~ f Cv"-lh~v()J~ ~lhQ {,Yi1 f =
-, ).-\:' (l
f t --s, Jc R{i l l9 f (~ el~)
(L [7\::1\
J f
1111 ?<u
"" '\ t 01J(l1') X o