~obkW Se+ ~~ (J1l) ~heV --~~ +4 St~ t (~ t 1R2 )lt~ x~+ ~~=~ ~ Stt1oo+h cvfves (I
Le~ F(x1 = )( x)~~-) thttt I2Vc st-- S flvtYl b1 F(~) =0 T~ b~ ThOle) s) 10ra
we~~- lH~ to ckck +~fI-[DF() s oh ~( 011 p~Y1ls lY1 Ov~se+
1~ vYtth~ III()~l~ Vlo-l- 19t ~V11 o~l~ ~ ~ x )) 0 J1=0 blAtshyov set 30 +e ~veVst-+ ls deed () S)Y~o+h cvIe-
N~ ~hM sII(]I-t)~)) i~ x-+~t K1l ~~ SOWlt te 7IC~ kO (-of e)(l1phd
-tt X = -l lOOK S()~t+i~ lt-e i
(111 CUI 1~ CM+ kll r ~ve 0 amp((H laquoj t~ It yt4-)
AYI1W t ~-LAV- ~~IAf s 1~ft It 011 h~ve vt 0- itJt s-$(l(t~ O tQ(h vel I~ o~ pound )~rt+ ~ 1rIY 14 X -OI(tCttofl Thw te tAU MoYlf+t(d wI1 hIflt2 1middotrIImiddotk~ JISj ei f1-t- S~u is
(lJ]l~) Svw --4- tor ~ oJ b +~ St-is- 4 cIIIJ Ylo o~ ett1A(t1ampN
r)+~) ~~ ~l ~t~-t~ ~hmiddot-h~l~ I o~ S~~ sI-rks
IV ~)+h eqlAcrh~S I ~ l6Y segtve ~ e fA) 0 VYlc1OVl o~ X oVit 1 vetvvltlf So X1 a~ y ~~ SMO-~ S~ttf
(b)( ho-t WAAt~ ~ fA ofI6 10 dDes T-oDtt~ 30 ltjvlHh~e t-t1t tk ~ef~i-IY1 )(gfo
ts 0 $~t- cu~vt1 I-crt ~eOwre~(C t1 ~-OYl --ttt t+tCvt +~ S~(e5 Qc) Yb -0( 4 ci-ter JovU cA ~ CVJ b~
-t~ t=~ R~ _gtlRl b~ ~Vtv o~ (xl tl~ 1G -0
Xtltl - b )
~ ~I1- Xcf1cgt - +t seA- Or W~tV t= N tetll Tt
PF(S)] ~ L~ )I~
ni M~X I~ Orl-D ~tI~S~ )1 clvvWIYI 0 (12 l]) +~u+ IS I 1AY)lt~~ )( )~ f 1 tYt] Now ~ ~v-- rtt~ 0 ~v-e 0+ vvhtcl- Vove( o- 0 d b (Allow -He$- to ht i x 0 Y SO
~ j (3)Jl -+ l =Ct
gts t ()y) r~ =-lc
S~~L---VI~ -ie e~h~l) ~ ~t~ L 1 - I
- y~ - + - -= - ~ 3~ -)
() ( - Y4 plusmn L - ~- b )-3
t~ t()(~~Jr too l-~o~Wtt1 Slrtgtrt+ 0+ ik ouble (I~) 1)1 wh Xtgt)Ycoil tIo+ 0 s ~ooi ~ (vi It
t
3
(1 1 oJ Let ~()~ () ~ -k ampjvtrlio V J 0- elHvl XerR ~ 0(1 s~PJt +hat- pt (~TI4 0 o(Q ( ~x
(~)FitlJ ()I elflAeA~) ~r tt co~vtt ex o-tt co-e eX Gfi over XI vJere ex lsikJr1~P1 o-lc 1 I-t +~ -Ik ~ ~ (~) - h () 6 X
ui- F(~~)() b~ -1~ e91(1i~VI for +4 JC)IUit -~+ ~ ore ~campkylJffr~ (xIf )tigt(
+~ t(l(I1)C)1o ~()(1~11)=OThUS t4 sVt~ 0+ CXo-t ~f~h+1 jlArl-ltob (I~ tk Cu(ve X S~lt k ( u~ bj 11- ~
i
WVIJ Iro~ -es ~~ ee4cko~ +l-n( tt(l~ in 14 fI(i~I()lsi+ f(xjll)=)
+e t=(xt- 1-r ~) --D OJ 1AAlyen Tf~S is bUlit4Je (l~1 WlIAI-lrple et~ (1(111 1) 1le( CrI t4 II~ -ftoV (-lt111) 0 itt Of~~IsO 4 es Y1 ex So VIe V)ef~ ~V eqVlo-iOrl so +i-a-lshy
vve Mtll~l) QII 0- X~r by ( oysfal1-t I weP s411 ~e+ o UJQ 1= (1lt ~ I =l) ~ +(-~ i-i)
(b)Tt X-holr-tlt etJu--iclo ~~1 hd4 IS- +k~vetil~ o~ exgt ~()(I)-~-X~=c) so F(x)-)~(~~)= ~-~~oI
J ~ -tl
Te etlvtlho~ ir F()(I1~) l~ - x~ r-t YA~ ~ppl) -tkW~ cd ~(AVldic) +ieoreWl 1 1M t)+e ~-t r r l ( I ~f)d G~ Tolt)~ vJ
[P~(~I~CJ - L-3~) ~J )~~1 liS (ol)d -4gt~ 0 ~ ~t~O 011 ~ X=-zO~O lJlt-lo~ is () SWl~~ J1J~-tpoundlJe ~xCfrt ~~~lb)()Y Q ~-ogtlt)
Tk ~f H-t gtlt-lJO llt Cef~I(l)3 clo~tJ bu-l I~ CtgtIfTqIIl) ~te ~iI X ijeuf (RI11
-th5 ~(HdIOyl CoY1-h 1V1( -he x OlJ ~ ()xef ~ile tleurore (he V~ X +k X-a)(i5 IS
-1e VoI+ ~~ ~S Iv X as -i-) eyenoi +~ -C-~j 5 tte hVViI-f OS -t-)Do T~r fe etO e4 Ii-+ YZ -~d ~s CClI1orlJ i ~ dSIA~ o~ )( I (IrJ 15 c))rcJ( Io X J
eqvol 10 +k lew set o~ X -~
(c) S~w+~C(J- x- 10$ 15 b s~OO+~ Sllrtc
We ()~gt 1 +k f11~ll(lt ~v u-t lOY) tLeofeVI ~VI) f(Jk tt)+ tLe TtAcob e 11 -1w r t~ X=i -~f~ )) eqval 40
[pt(Yj)]= [ -=t xJ ~ hd 4l ex d~eJ tel 41JO I ~ iJ~5 (Ils to be ~rlo CYl)l txt -k o~~VI_VVhlt~
c 5 i) 61 i 4 ~rbLl1 lllctb tLeortJrn X - (01 IS Wto~ h s IMoce-A
[cl)S~ow thrl -h~ ~Llt (amp) ~(r(~amp) S~~~+Le ~HMeh~+lo o~ X d-t09)
rvI +hlS otVII ~v sovlJ ~ve ~ +f~~e ~v~ fA V()w -kgt +e SVM(ue X
0 5~V -h~ 5 ()Votie( ~(Jf6~h~-H(9VlI ~VS+ ~~~ d 1(1 to t k btl~t-ilOv xt -1) ~O SO
r(1-sfGh-(I-se) - floco)~8 f)(l-SV~e -(CJ5~t)
-=0
~ -46 s- 0 f(ttoYYf-h-C(~DYl () X
C~t() C-)~-ueq)
10 -gtJtt OV vvl~ -th~ S~e )5 T S~ltfS1 ~X1I9 f ftY J l -Hlfl1 (J VMj Tk +~l~ w~U ~ 1ke trC1f5 brl)nhl~ te )( ClVlJ ~ -Cl(gtJ S tH r vMlrr v--tL ~e-t q JlAbk co~ b)$J oA i4gt o(~~it Oy~ 11ed YO +h~+ fs (I S Iles ~tle Xcfalt( )tJ)I~
0 l4 ~lt l ( ~1 A _
fJY ~~ Af fOItllvt(-Itk ( t eyen 0 d-0 Ohii IS i It JO re eflAA-t iOY i ho-f IA r clfJfrJfe )(
scd-Sies
() LO- 1(t) ~ (r) I bullbullu t+ h 0 ~ ~ to ~1 Ie ~ (IU bullbull~ ~Pl~middotll~
40 71(t) W~Qt Ir i ~ UjIJ4+OYl O~ Pt~
(c) Si)W ~o+ ~ 1 -4 t~ I 4e ~Cl wS Pt laquoVlel ~h tllwA1s 1Y1-ttfJeCt- () 1e ~~ (f~ te 9~vri-J~~ J-e ht- ~()t
Ntk+I lo-lt +0+ ~ ~ r etv(A-+ amp-) ~)(~- -5(~1) -0 ~gt X ~J1T1~-b =0
OT~ Pt -Iu e~4V )( +)t~ -3- -t -Jl~ -3tS =0
Je ~It 1Vrrl~i
~ U 7
x )t =-1 ~ 5-e
O I II) Cots~r amp ~te it o~ ~Igt hpfl~ eA C toa ot -eVl~ l i( iRl wte O~ eVlcJflir+ i r
coVS~~VleU -6 be 0 ~-t )( -X~ I ~id ~ otltr ~ r- tOAs1t()ed -to be 01 +~ tAVt SQ-Hshy
Ut-~R~ vAmiddot -ok ot5l
(0) (wt ~~~middothOv6 ~( JJ oS 0 ~vstt Q~ l~q welt -t~ coo~middot()t~ ~4 e +~
)(-ttgtIrtAlotA-+(
J -Q ~~ J 4 laquoJ 0 +4- x-~)c~ (trll rr t)~ 4~ o11e( +h~ cootMtlkl Gl~ 4 he oI-~ Q~~ r ~ t~~
- (~t p ~ J1+
We- crJ~~~e lbt=( t )(1~1~ )1 0 ~~ L)t-J~ -3xmiddott
1 VJt ~v~ j 1gt ~ ~Q 1shy
l)~ (lt I 00)1 =- l[0Jpound ~jl ~ ~J
nh 5 orl-o ~()r iu SO Xl 0 M()l~tOamp VI () tI~lho9t~IJDJ at f
C~ ( K)1I T )J I 0 th
~ [~ -~~
(3 Ub cohl~d)(J)Sk+~at t i=l +1pound4t olte p~s to- X APo V-hlll XI 1 ~t-I Ct Wleurolrtto1t
n t (4-e t~s~t ~Oii ~ ~e StC1- (0 0 ~ ~ SO thd+ Of) e e~ I +I-t ftJv IJ~ I -he O(~~V11 ColJ +e 01~ twY Or ik 1l~ 1it~ OVl irt (t It 11l11~ f~ +le s~l-t~ IN-h -0 Thn IPF(OrDI ~~)] = rOO J1 de]
L0 Ol~ ~e-i ~ yenIt- 01 50 X ~ VIti- () wlVI-f6J NW ihett yci~
10
(~)amptt~~ ~l osyen~a 101) -to s~ ~ )(+x~~ rAlti-tS 0 SWItJf (wr~
(o)W-M- s q ~~hv ~r tt -tw1~~ l2+O~ tvtllt t) n) tbe ~~-+ ampv-- rew(~t r()fj) X+j(~1~-)1 so ~ttf (Ivf cuf~ S Fo t-heuroy
lDf(~)l U~ lx ~ ~J
OJ ~y- ~ottw ~J~ t-~ -o~~tl-t 4fotItJ- 1vC)IA( ~~ (~) J Kef [Ol=(~)]Cvflt
[3 OJ [~l ~ 0
T-al CAO pound)~~ b I$ Mi4fIk~ 50 we (11 ckscltibe +hl I b~
(x-) =0 I Illf e~IA~twHj1 ~=t
(0) V~t+ 15 +-e ewdl~V) ~ the ~Vie~ 3 e 10 th CIA-Ne 0+ -te SO ~ ptY1+
T4 -w~eVt s~tt is- f-t[OFU)J= i(t) oo1 th+iStN j-IJiJ)VlJt WltfQrr(d~)~ 0 vtc$w- tOSt-~ 0+ --k Otgwl ~Y~ ~I +k ~-Q~lj
II
Let F(Jltl1I~)=A1-l+-~-i s-Ho+ LPP(Yfljft)] =-[Ax Jg~ -]
+~v1 04 Ovf -t~rtt ~mfL)P(ooO)] = ( 0 o IJ [OF(fofA~fl ~lA~ o -IJ b)~ (0 bgt~~)1 =[ 0 ~Sb -I]
OU -YIf~ ~vfJ off -1ll-()o or -eD
JA(X-A) - (c - Ao) 0 Jto( ~- - (c -1S b)) 0
tl ~r+ or SiI -1~ I SD
I)
t~IAVtl Ifvr-1~r +k 6l~ etl+ s~tate i-s ~e~erokd b~ 0 fD () middot0
I l~ 0
(3Jq) Pof +k U~ Gx lt~ ~(ef(~cent sLlD w~-l tie elflaI~ of +4 -b~em plare 10 CJ(
0+ OYlj plIr-t ~ C Gx -o~
N~lt -~I)- ~f Cx (vlO~ GX )+le oVlj fWrl wd- ~- (ClQalI-o-tt 0 5 tlp 6-11911 So ~ ~ 1
~S5tA~ Y (~I4) ~CAS -)OYlefro ~-co()~lVltJk TkYl - +4 eqLl(A lc of eX 5fl4If
~-tnb~ ~(~-Oye~ t-(JfI1)-O h ie eq(J((~iol oJ1kcU(ve XVIR W~t~ ~( seJ to JeiN +le CO~
)4
N~ ~hM sII(]I-t)~)) i~ x-+~t K1l ~~ SOWlt te 7IC~ kO (-of e)(l1phd
-tt X = -l lOOK S()~t+i~ lt-e i
(111 CUI 1~ CM+ kll r ~ve 0 amp((H laquoj t~ It yt4-)
AYI1W t ~-LAV- ~~IAf s 1~ft It 011 h~ve vt 0- itJt s-$(l(t~ O tQ(h vel I~ o~ pound )~rt+ ~ 1rIY 14 X -OI(tCttofl Thw te tAU MoYlf+t(d wI1 hIflt2 1middotrIImiddotk~ JISj ei f1-t- S~u is
(lJ]l~) Svw --4- tor ~ oJ b +~ St-is- 4 cIIIJ Ylo o~ ett1A(t1ampN
r)+~) ~~ ~l ~t~-t~ ~hmiddot-h~l~ I o~ S~~ sI-rks
IV ~)+h eqlAcrh~S I ~ l6Y segtve ~ e fA) 0 VYlc1OVl o~ X oVit 1 vetvvltlf So X1 a~ y ~~ SMO-~ S~ttf
(b)( ho-t WAAt~ ~ fA ofI6 10 dDes T-oDtt~ 30 ltjvlHh~e t-t1t tk ~ef~i-IY1 )(gfo
ts 0 $~t- cu~vt1 I-crt ~eOwre~(C t1 ~-OYl --ttt t+tCvt +~ S~(e5 Qc) Yb -0( 4 ci-ter JovU cA ~ CVJ b~
-t~ t=~ R~ _gtlRl b~ ~Vtv o~ (xl tl~ 1G -0
Xtltl - b )
~ ~I1- Xcf1cgt - +t seA- Or W~tV t= N tetll Tt
PF(S)] ~ L~ )I~
ni M~X I~ Orl-D ~tI~S~ )1 clvvWIYI 0 (12 l]) +~u+ IS I 1AY)lt~~ )( )~ f 1 tYt] Now ~ ~v-- rtt~ 0 ~v-e 0+ vvhtcl- Vove( o- 0 d b (Allow -He$- to ht i x 0 Y SO
~ j (3)Jl -+ l =Ct
gts t ()y) r~ =-lc
S~~L---VI~ -ie e~h~l) ~ ~t~ L 1 - I
- y~ - + - -= - ~ 3~ -)
() ( - Y4 plusmn L - ~- b )-3
t~ t()(~~Jr too l-~o~Wtt1 Slrtgtrt+ 0+ ik ouble (I~) 1)1 wh Xtgt)Ycoil tIo+ 0 s ~ooi ~ (vi It
t
3
(1 1 oJ Let ~()~ () ~ -k ampjvtrlio V J 0- elHvl XerR ~ 0(1 s~PJt +hat- pt (~TI4 0 o(Q ( ~x
(~)FitlJ ()I elflAeA~) ~r tt co~vtt ex o-tt co-e eX Gfi over XI vJere ex lsikJr1~P1 o-lc 1 I-t +~ -Ik ~ ~ (~) - h () 6 X
ui- F(~~)() b~ -1~ e91(1i~VI for +4 JC)IUit -~+ ~ ore ~campkylJffr~ (xIf )tigt(
+~ t(l(I1)C)1o ~()(1~11)=OThUS t4 sVt~ 0+ CXo-t ~f~h+1 jlArl-ltob (I~ tk Cu(ve X S~lt k ( u~ bj 11- ~
i
WVIJ Iro~ -es ~~ ee4cko~ +l-n( tt(l~ in 14 fI(i~I()lsi+ f(xjll)=)
+e t=(xt- 1-r ~) --D OJ 1AAlyen Tf~S is bUlit4Je (l~1 WlIAI-lrple et~ (1(111 1) 1le( CrI t4 II~ -ftoV (-lt111) 0 itt Of~~IsO 4 es Y1 ex So VIe V)ef~ ~V eqVlo-iOrl so +i-a-lshy
vve Mtll~l) QII 0- X~r by ( oysfal1-t I weP s411 ~e+ o UJQ 1= (1lt ~ I =l) ~ +(-~ i-i)
(b)Tt X-holr-tlt etJu--iclo ~~1 hd4 IS- +k~vetil~ o~ exgt ~()(I)-~-X~=c) so F(x)-)~(~~)= ~-~~oI
J ~ -tl
Te etlvtlho~ ir F()(I1~) l~ - x~ r-t YA~ ~ppl) -tkW~ cd ~(AVldic) +ieoreWl 1 1M t)+e ~-t r r l ( I ~f)d G~ Tolt)~ vJ
[P~(~I~CJ - L-3~) ~J )~~1 liS (ol)d -4gt~ 0 ~ ~t~O 011 ~ X=-zO~O lJlt-lo~ is () SWl~~ J1J~-tpoundlJe ~xCfrt ~~~lb)()Y Q ~-ogtlt)
Tk ~f H-t gtlt-lJO llt Cef~I(l)3 clo~tJ bu-l I~ CtgtIfTqIIl) ~te ~iI X ijeuf (RI11
-th5 ~(HdIOyl CoY1-h 1V1( -he x OlJ ~ ()xef ~ile tleurore (he V~ X +k X-a)(i5 IS
-1e VoI+ ~~ ~S Iv X as -i-) eyenoi +~ -C-~j 5 tte hVViI-f OS -t-)Do T~r fe etO e4 Ii-+ YZ -~d ~s CClI1orlJ i ~ dSIA~ o~ )( I (IrJ 15 c))rcJ( Io X J
eqvol 10 +k lew set o~ X -~
(c) S~w+~C(J- x- 10$ 15 b s~OO+~ Sllrtc
We ()~gt 1 +k f11~ll(lt ~v u-t lOY) tLeofeVI ~VI) f(Jk tt)+ tLe TtAcob e 11 -1w r t~ X=i -~f~ )) eqval 40
[pt(Yj)]= [ -=t xJ ~ hd 4l ex d~eJ tel 41JO I ~ iJ~5 (Ils to be ~rlo CYl)l txt -k o~~VI_VVhlt~
c 5 i) 61 i 4 ~rbLl1 lllctb tLeortJrn X - (01 IS Wto~ h s IMoce-A
[cl)S~ow thrl -h~ ~Llt (amp) ~(r(~amp) S~~~+Le ~HMeh~+lo o~ X d-t09)
rvI +hlS otVII ~v sovlJ ~ve ~ +f~~e ~v~ fA V()w -kgt +e SVM(ue X
0 5~V -h~ 5 ()Votie( ~(Jf6~h~-H(9VlI ~VS+ ~~~ d 1(1 to t k btl~t-ilOv xt -1) ~O SO
r(1-sfGh-(I-se) - floco)~8 f)(l-SV~e -(CJ5~t)
-=0
~ -46 s- 0 f(ttoYYf-h-C(~DYl () X
C~t() C-)~-ueq)
10 -gtJtt OV vvl~ -th~ S~e )5 T S~ltfS1 ~X1I9 f ftY J l -Hlfl1 (J VMj Tk +~l~ w~U ~ 1ke trC1f5 brl)nhl~ te )( ClVlJ ~ -Cl(gtJ S tH r vMlrr v--tL ~e-t q JlAbk co~ b)$J oA i4gt o(~~it Oy~ 11ed YO +h~+ fs (I S Iles ~tle Xcfalt( )tJ)I~
0 l4 ~lt l ( ~1 A _
fJY ~~ Af fOItllvt(-Itk ( t eyen 0 d-0 Ohii IS i It JO re eflAA-t iOY i ho-f IA r clfJfrJfe )(
scd-Sies
() LO- 1(t) ~ (r) I bullbullu t+ h 0 ~ ~ to ~1 Ie ~ (IU bullbull~ ~Pl~middotll~
40 71(t) W~Qt Ir i ~ UjIJ4+OYl O~ Pt~
(c) Si)W ~o+ ~ 1 -4 t~ I 4e ~Cl wS Pt laquoVlel ~h tllwA1s 1Y1-ttfJeCt- () 1e ~~ (f~ te 9~vri-J~~ J-e ht- ~()t
Ntk+I lo-lt +0+ ~ ~ r etv(A-+ amp-) ~)(~- -5(~1) -0 ~gt X ~J1T1~-b =0
OT~ Pt -Iu e~4V )( +)t~ -3- -t -Jl~ -3tS =0
Je ~It 1Vrrl~i
~ U 7
x )t =-1 ~ 5-e
O I II) Cots~r amp ~te it o~ ~Igt hpfl~ eA C toa ot -eVl~ l i( iRl wte O~ eVlcJflir+ i r
coVS~~VleU -6 be 0 ~-t )( -X~ I ~id ~ otltr ~ r- tOAs1t()ed -to be 01 +~ tAVt SQ-Hshy
Ut-~R~ vAmiddot -ok ot5l
(0) (wt ~~~middothOv6 ~( JJ oS 0 ~vstt Q~ l~q welt -t~ coo~middot()t~ ~4 e +~
)(-ttgtIrtAlotA-+(
J -Q ~~ J 4 laquoJ 0 +4- x-~)c~ (trll rr t)~ 4~ o11e( +h~ cootMtlkl Gl~ 4 he oI-~ Q~~ r ~ t~~
- (~t p ~ J1+
We- crJ~~~e lbt=( t )(1~1~ )1 0 ~~ L)t-J~ -3xmiddott
1 VJt ~v~ j 1gt ~ ~Q 1shy
l)~ (lt I 00)1 =- l[0Jpound ~jl ~ ~J
nh 5 orl-o ~()r iu SO Xl 0 M()l~tOamp VI () tI~lho9t~IJDJ at f
C~ ( K)1I T )J I 0 th
~ [~ -~~
(3 Ub cohl~d)(J)Sk+~at t i=l +1pound4t olte p~s to- X APo V-hlll XI 1 ~t-I Ct Wleurolrtto1t
n t (4-e t~s~t ~Oii ~ ~e StC1- (0 0 ~ ~ SO thd+ Of) e e~ I +I-t ftJv IJ~ I -he O(~~V11 ColJ +e 01~ twY Or ik 1l~ 1it~ OVl irt (t It 11l11~ f~ +le s~l-t~ IN-h -0 Thn IPF(OrDI ~~)] = rOO J1 de]
L0 Ol~ ~e-i ~ yenIt- 01 50 X ~ VIti- () wlVI-f6J NW ihett yci~
10
(~)amptt~~ ~l osyen~a 101) -to s~ ~ )(+x~~ rAlti-tS 0 SWItJf (wr~
(o)W-M- s q ~~hv ~r tt -tw1~~ l2+O~ tvtllt t) n) tbe ~~-+ ampv-- rew(~t r()fj) X+j(~1~-)1 so ~ttf (Ivf cuf~ S Fo t-heuroy
lDf(~)l U~ lx ~ ~J
OJ ~y- ~ottw ~J~ t-~ -o~~tl-t 4fotItJ- 1vC)IA( ~~ (~) J Kef [Ol=(~)]Cvflt
[3 OJ [~l ~ 0
T-al CAO pound)~~ b I$ Mi4fIk~ 50 we (11 ckscltibe +hl I b~
(x-) =0 I Illf e~IA~twHj1 ~=t
(0) V~t+ 15 +-e ewdl~V) ~ the ~Vie~ 3 e 10 th CIA-Ne 0+ -te SO ~ ptY1+
T4 -w~eVt s~tt is- f-t[OFU)J= i(t) oo1 th+iStN j-IJiJ)VlJt WltfQrr(d~)~ 0 vtc$w- tOSt-~ 0+ --k Otgwl ~Y~ ~I +k ~-Q~lj
II
Let F(Jltl1I~)=A1-l+-~-i s-Ho+ LPP(Yfljft)] =-[Ax Jg~ -]
+~v1 04 Ovf -t~rtt ~mfL)P(ooO)] = ( 0 o IJ [OF(fofA~fl ~lA~ o -IJ b)~ (0 bgt~~)1 =[ 0 ~Sb -I]
OU -YIf~ ~vfJ off -1ll-()o or -eD
JA(X-A) - (c - Ao) 0 Jto( ~- - (c -1S b)) 0
tl ~r+ or SiI -1~ I SD
I)
t~IAVtl Ifvr-1~r +k 6l~ etl+ s~tate i-s ~e~erokd b~ 0 fD () middot0
I l~ 0
(3Jq) Pof +k U~ Gx lt~ ~(ef(~cent sLlD w~-l tie elflaI~ of +4 -b~em plare 10 CJ(
0+ OYlj plIr-t ~ C Gx -o~
N~lt -~I)- ~f Cx (vlO~ GX )+le oVlj fWrl wd- ~- (ClQalI-o-tt 0 5 tlp 6-11911 So ~ ~ 1
~S5tA~ Y (~I4) ~CAS -)OYlefro ~-co()~lVltJk TkYl - +4 eqLl(A lc of eX 5fl4If
~-tnb~ ~(~-Oye~ t-(JfI1)-O h ie eq(J((~iol oJ1kcU(ve XVIR W~t~ ~( seJ to JeiN +le CO~
)4
(lJ]l~) Svw --4- tor ~ oJ b +~ St-is- 4 cIIIJ Ylo o~ ett1A(t1ampN
r)+~) ~~ ~l ~t~-t~ ~hmiddot-h~l~ I o~ S~~ sI-rks
IV ~)+h eqlAcrh~S I ~ l6Y segtve ~ e fA) 0 VYlc1OVl o~ X oVit 1 vetvvltlf So X1 a~ y ~~ SMO-~ S~ttf
(b)( ho-t WAAt~ ~ fA ofI6 10 dDes T-oDtt~ 30 ltjvlHh~e t-t1t tk ~ef~i-IY1 )(gfo
ts 0 $~t- cu~vt1 I-crt ~eOwre~(C t1 ~-OYl --ttt t+tCvt +~ S~(e5 Qc) Yb -0( 4 ci-ter JovU cA ~ CVJ b~
-t~ t=~ R~ _gtlRl b~ ~Vtv o~ (xl tl~ 1G -0
Xtltl - b )
~ ~I1- Xcf1cgt - +t seA- Or W~tV t= N tetll Tt
PF(S)] ~ L~ )I~
ni M~X I~ Orl-D ~tI~S~ )1 clvvWIYI 0 (12 l]) +~u+ IS I 1AY)lt~~ )( )~ f 1 tYt] Now ~ ~v-- rtt~ 0 ~v-e 0+ vvhtcl- Vove( o- 0 d b (Allow -He$- to ht i x 0 Y SO
~ j (3)Jl -+ l =Ct
gts t ()y) r~ =-lc
S~~L---VI~ -ie e~h~l) ~ ~t~ L 1 - I
- y~ - + - -= - ~ 3~ -)
() ( - Y4 plusmn L - ~- b )-3
t~ t()(~~Jr too l-~o~Wtt1 Slrtgtrt+ 0+ ik ouble (I~) 1)1 wh Xtgt)Ycoil tIo+ 0 s ~ooi ~ (vi It
t
3
(1 1 oJ Let ~()~ () ~ -k ampjvtrlio V J 0- elHvl XerR ~ 0(1 s~PJt +hat- pt (~TI4 0 o(Q ( ~x
(~)FitlJ ()I elflAeA~) ~r tt co~vtt ex o-tt co-e eX Gfi over XI vJere ex lsikJr1~P1 o-lc 1 I-t +~ -Ik ~ ~ (~) - h () 6 X
ui- F(~~)() b~ -1~ e91(1i~VI for +4 JC)IUit -~+ ~ ore ~campkylJffr~ (xIf )tigt(
+~ t(l(I1)C)1o ~()(1~11)=OThUS t4 sVt~ 0+ CXo-t ~f~h+1 jlArl-ltob (I~ tk Cu(ve X S~lt k ( u~ bj 11- ~
i
WVIJ Iro~ -es ~~ ee4cko~ +l-n( tt(l~ in 14 fI(i~I()lsi+ f(xjll)=)
+e t=(xt- 1-r ~) --D OJ 1AAlyen Tf~S is bUlit4Je (l~1 WlIAI-lrple et~ (1(111 1) 1le( CrI t4 II~ -ftoV (-lt111) 0 itt Of~~IsO 4 es Y1 ex So VIe V)ef~ ~V eqVlo-iOrl so +i-a-lshy
vve Mtll~l) QII 0- X~r by ( oysfal1-t I weP s411 ~e+ o UJQ 1= (1lt ~ I =l) ~ +(-~ i-i)
(b)Tt X-holr-tlt etJu--iclo ~~1 hd4 IS- +k~vetil~ o~ exgt ~()(I)-~-X~=c) so F(x)-)~(~~)= ~-~~oI
J ~ -tl
Te etlvtlho~ ir F()(I1~) l~ - x~ r-t YA~ ~ppl) -tkW~ cd ~(AVldic) +ieoreWl 1 1M t)+e ~-t r r l ( I ~f)d G~ Tolt)~ vJ
[P~(~I~CJ - L-3~) ~J )~~1 liS (ol)d -4gt~ 0 ~ ~t~O 011 ~ X=-zO~O lJlt-lo~ is () SWl~~ J1J~-tpoundlJe ~xCfrt ~~~lb)()Y Q ~-ogtlt)
Tk ~f H-t gtlt-lJO llt Cef~I(l)3 clo~tJ bu-l I~ CtgtIfTqIIl) ~te ~iI X ijeuf (RI11
-th5 ~(HdIOyl CoY1-h 1V1( -he x OlJ ~ ()xef ~ile tleurore (he V~ X +k X-a)(i5 IS
-1e VoI+ ~~ ~S Iv X as -i-) eyenoi +~ -C-~j 5 tte hVViI-f OS -t-)Do T~r fe etO e4 Ii-+ YZ -~d ~s CClI1orlJ i ~ dSIA~ o~ )( I (IrJ 15 c))rcJ( Io X J
eqvol 10 +k lew set o~ X -~
(c) S~w+~C(J- x- 10$ 15 b s~OO+~ Sllrtc
We ()~gt 1 +k f11~ll(lt ~v u-t lOY) tLeofeVI ~VI) f(Jk tt)+ tLe TtAcob e 11 -1w r t~ X=i -~f~ )) eqval 40
[pt(Yj)]= [ -=t xJ ~ hd 4l ex d~eJ tel 41JO I ~ iJ~5 (Ils to be ~rlo CYl)l txt -k o~~VI_VVhlt~
c 5 i) 61 i 4 ~rbLl1 lllctb tLeortJrn X - (01 IS Wto~ h s IMoce-A
[cl)S~ow thrl -h~ ~Llt (amp) ~(r(~amp) S~~~+Le ~HMeh~+lo o~ X d-t09)
rvI +hlS otVII ~v sovlJ ~ve ~ +f~~e ~v~ fA V()w -kgt +e SVM(ue X
0 5~V -h~ 5 ()Votie( ~(Jf6~h~-H(9VlI ~VS+ ~~~ d 1(1 to t k btl~t-ilOv xt -1) ~O SO
r(1-sfGh-(I-se) - floco)~8 f)(l-SV~e -(CJ5~t)
-=0
~ -46 s- 0 f(ttoYYf-h-C(~DYl () X
C~t() C-)~-ueq)
10 -gtJtt OV vvl~ -th~ S~e )5 T S~ltfS1 ~X1I9 f ftY J l -Hlfl1 (J VMj Tk +~l~ w~U ~ 1ke trC1f5 brl)nhl~ te )( ClVlJ ~ -Cl(gtJ S tH r vMlrr v--tL ~e-t q JlAbk co~ b)$J oA i4gt o(~~it Oy~ 11ed YO +h~+ fs (I S Iles ~tle Xcfalt( )tJ)I~
0 l4 ~lt l ( ~1 A _
fJY ~~ Af fOItllvt(-Itk ( t eyen 0 d-0 Ohii IS i It JO re eflAA-t iOY i ho-f IA r clfJfrJfe )(
scd-Sies
() LO- 1(t) ~ (r) I bullbullu t+ h 0 ~ ~ to ~1 Ie ~ (IU bullbull~ ~Pl~middotll~
40 71(t) W~Qt Ir i ~ UjIJ4+OYl O~ Pt~
(c) Si)W ~o+ ~ 1 -4 t~ I 4e ~Cl wS Pt laquoVlel ~h tllwA1s 1Y1-ttfJeCt- () 1e ~~ (f~ te 9~vri-J~~ J-e ht- ~()t
Ntk+I lo-lt +0+ ~ ~ r etv(A-+ amp-) ~)(~- -5(~1) -0 ~gt X ~J1T1~-b =0
OT~ Pt -Iu e~4V )( +)t~ -3- -t -Jl~ -3tS =0
Je ~It 1Vrrl~i
~ U 7
x )t =-1 ~ 5-e
O I II) Cots~r amp ~te it o~ ~Igt hpfl~ eA C toa ot -eVl~ l i( iRl wte O~ eVlcJflir+ i r
coVS~~VleU -6 be 0 ~-t )( -X~ I ~id ~ otltr ~ r- tOAs1t()ed -to be 01 +~ tAVt SQ-Hshy
Ut-~R~ vAmiddot -ok ot5l
(0) (wt ~~~middothOv6 ~( JJ oS 0 ~vstt Q~ l~q welt -t~ coo~middot()t~ ~4 e +~
)(-ttgtIrtAlotA-+(
J -Q ~~ J 4 laquoJ 0 +4- x-~)c~ (trll rr t)~ 4~ o11e( +h~ cootMtlkl Gl~ 4 he oI-~ Q~~ r ~ t~~
- (~t p ~ J1+
We- crJ~~~e lbt=( t )(1~1~ )1 0 ~~ L)t-J~ -3xmiddott
1 VJt ~v~ j 1gt ~ ~Q 1shy
l)~ (lt I 00)1 =- l[0Jpound ~jl ~ ~J
nh 5 orl-o ~()r iu SO Xl 0 M()l~tOamp VI () tI~lho9t~IJDJ at f
C~ ( K)1I T )J I 0 th
~ [~ -~~
(3 Ub cohl~d)(J)Sk+~at t i=l +1pound4t olte p~s to- X APo V-hlll XI 1 ~t-I Ct Wleurolrtto1t
n t (4-e t~s~t ~Oii ~ ~e StC1- (0 0 ~ ~ SO thd+ Of) e e~ I +I-t ftJv IJ~ I -he O(~~V11 ColJ +e 01~ twY Or ik 1l~ 1it~ OVl irt (t It 11l11~ f~ +le s~l-t~ IN-h -0 Thn IPF(OrDI ~~)] = rOO J1 de]
L0 Ol~ ~e-i ~ yenIt- 01 50 X ~ VIti- () wlVI-f6J NW ihett yci~
10
(~)amptt~~ ~l osyen~a 101) -to s~ ~ )(+x~~ rAlti-tS 0 SWItJf (wr~
(o)W-M- s q ~~hv ~r tt -tw1~~ l2+O~ tvtllt t) n) tbe ~~-+ ampv-- rew(~t r()fj) X+j(~1~-)1 so ~ttf (Ivf cuf~ S Fo t-heuroy
lDf(~)l U~ lx ~ ~J
OJ ~y- ~ottw ~J~ t-~ -o~~tl-t 4fotItJ- 1vC)IA( ~~ (~) J Kef [Ol=(~)]Cvflt
[3 OJ [~l ~ 0
T-al CAO pound)~~ b I$ Mi4fIk~ 50 we (11 ckscltibe +hl I b~
(x-) =0 I Illf e~IA~twHj1 ~=t
(0) V~t+ 15 +-e ewdl~V) ~ the ~Vie~ 3 e 10 th CIA-Ne 0+ -te SO ~ ptY1+
T4 -w~eVt s~tt is- f-t[OFU)J= i(t) oo1 th+iStN j-IJiJ)VlJt WltfQrr(d~)~ 0 vtc$w- tOSt-~ 0+ --k Otgwl ~Y~ ~I +k ~-Q~lj
II
Let F(Jltl1I~)=A1-l+-~-i s-Ho+ LPP(Yfljft)] =-[Ax Jg~ -]
+~v1 04 Ovf -t~rtt ~mfL)P(ooO)] = ( 0 o IJ [OF(fofA~fl ~lA~ o -IJ b)~ (0 bgt~~)1 =[ 0 ~Sb -I]
OU -YIf~ ~vfJ off -1ll-()o or -eD
JA(X-A) - (c - Ao) 0 Jto( ~- - (c -1S b)) 0
tl ~r+ or SiI -1~ I SD
I)
t~IAVtl Ifvr-1~r +k 6l~ etl+ s~tate i-s ~e~erokd b~ 0 fD () middot0
I l~ 0
(3Jq) Pof +k U~ Gx lt~ ~(ef(~cent sLlD w~-l tie elflaI~ of +4 -b~em plare 10 CJ(
0+ OYlj plIr-t ~ C Gx -o~
N~lt -~I)- ~f Cx (vlO~ GX )+le oVlj fWrl wd- ~- (ClQalI-o-tt 0 5 tlp 6-11911 So ~ ~ 1
~S5tA~ Y (~I4) ~CAS -)OYlefro ~-co()~lVltJk TkYl - +4 eqLl(A lc of eX 5fl4If
~-tnb~ ~(~-Oye~ t-(JfI1)-O h ie eq(J((~iol oJ1kcU(ve XVIR W~t~ ~( seJ to JeiN +le CO~
)4
(1 1 oJ Let ~()~ () ~ -k ampjvtrlio V J 0- elHvl XerR ~ 0(1 s~PJt +hat- pt (~TI4 0 o(Q ( ~x
(~)FitlJ ()I elflAeA~) ~r tt co~vtt ex o-tt co-e eX Gfi over XI vJere ex lsikJr1~P1 o-lc 1 I-t +~ -Ik ~ ~ (~) - h () 6 X
ui- F(~~)() b~ -1~ e91(1i~VI for +4 JC)IUit -~+ ~ ore ~campkylJffr~ (xIf )tigt(
+~ t(l(I1)C)1o ~()(1~11)=OThUS t4 sVt~ 0+ CXo-t ~f~h+1 jlArl-ltob (I~ tk Cu(ve X S~lt k ( u~ bj 11- ~
i
WVIJ Iro~ -es ~~ ee4cko~ +l-n( tt(l~ in 14 fI(i~I()lsi+ f(xjll)=)
+e t=(xt- 1-r ~) --D OJ 1AAlyen Tf~S is bUlit4Je (l~1 WlIAI-lrple et~ (1(111 1) 1le( CrI t4 II~ -ftoV (-lt111) 0 itt Of~~IsO 4 es Y1 ex So VIe V)ef~ ~V eqVlo-iOrl so +i-a-lshy
vve Mtll~l) QII 0- X~r by ( oysfal1-t I weP s411 ~e+ o UJQ 1= (1lt ~ I =l) ~ +(-~ i-i)
(b)Tt X-holr-tlt etJu--iclo ~~1 hd4 IS- +k~vetil~ o~ exgt ~()(I)-~-X~=c) so F(x)-)~(~~)= ~-~~oI
J ~ -tl
Te etlvtlho~ ir F()(I1~) l~ - x~ r-t YA~ ~ppl) -tkW~ cd ~(AVldic) +ieoreWl 1 1M t)+e ~-t r r l ( I ~f)d G~ Tolt)~ vJ
[P~(~I~CJ - L-3~) ~J )~~1 liS (ol)d -4gt~ 0 ~ ~t~O 011 ~ X=-zO~O lJlt-lo~ is () SWl~~ J1J~-tpoundlJe ~xCfrt ~~~lb)()Y Q ~-ogtlt)
Tk ~f H-t gtlt-lJO llt Cef~I(l)3 clo~tJ bu-l I~ CtgtIfTqIIl) ~te ~iI X ijeuf (RI11
-th5 ~(HdIOyl CoY1-h 1V1( -he x OlJ ~ ()xef ~ile tleurore (he V~ X +k X-a)(i5 IS
-1e VoI+ ~~ ~S Iv X as -i-) eyenoi +~ -C-~j 5 tte hVViI-f OS -t-)Do T~r fe etO e4 Ii-+ YZ -~d ~s CClI1orlJ i ~ dSIA~ o~ )( I (IrJ 15 c))rcJ( Io X J
eqvol 10 +k lew set o~ X -~
(c) S~w+~C(J- x- 10$ 15 b s~OO+~ Sllrtc
We ()~gt 1 +k f11~ll(lt ~v u-t lOY) tLeofeVI ~VI) f(Jk tt)+ tLe TtAcob e 11 -1w r t~ X=i -~f~ )) eqval 40
[pt(Yj)]= [ -=t xJ ~ hd 4l ex d~eJ tel 41JO I ~ iJ~5 (Ils to be ~rlo CYl)l txt -k o~~VI_VVhlt~
c 5 i) 61 i 4 ~rbLl1 lllctb tLeortJrn X - (01 IS Wto~ h s IMoce-A
[cl)S~ow thrl -h~ ~Llt (amp) ~(r(~amp) S~~~+Le ~HMeh~+lo o~ X d-t09)
rvI +hlS otVII ~v sovlJ ~ve ~ +f~~e ~v~ fA V()w -kgt +e SVM(ue X
0 5~V -h~ 5 ()Votie( ~(Jf6~h~-H(9VlI ~VS+ ~~~ d 1(1 to t k btl~t-ilOv xt -1) ~O SO
r(1-sfGh-(I-se) - floco)~8 f)(l-SV~e -(CJ5~t)
-=0
~ -46 s- 0 f(ttoYYf-h-C(~DYl () X
C~t() C-)~-ueq)
10 -gtJtt OV vvl~ -th~ S~e )5 T S~ltfS1 ~X1I9 f ftY J l -Hlfl1 (J VMj Tk +~l~ w~U ~ 1ke trC1f5 brl)nhl~ te )( ClVlJ ~ -Cl(gtJ S tH r vMlrr v--tL ~e-t q JlAbk co~ b)$J oA i4gt o(~~it Oy~ 11ed YO +h~+ fs (I S Iles ~tle Xcfalt( )tJ)I~
0 l4 ~lt l ( ~1 A _
fJY ~~ Af fOItllvt(-Itk ( t eyen 0 d-0 Ohii IS i It JO re eflAA-t iOY i ho-f IA r clfJfrJfe )(
scd-Sies
() LO- 1(t) ~ (r) I bullbullu t+ h 0 ~ ~ to ~1 Ie ~ (IU bullbull~ ~Pl~middotll~
40 71(t) W~Qt Ir i ~ UjIJ4+OYl O~ Pt~
(c) Si)W ~o+ ~ 1 -4 t~ I 4e ~Cl wS Pt laquoVlel ~h tllwA1s 1Y1-ttfJeCt- () 1e ~~ (f~ te 9~vri-J~~ J-e ht- ~()t
Ntk+I lo-lt +0+ ~ ~ r etv(A-+ amp-) ~)(~- -5(~1) -0 ~gt X ~J1T1~-b =0
OT~ Pt -Iu e~4V )( +)t~ -3- -t -Jl~ -3tS =0
Je ~It 1Vrrl~i
~ U 7
x )t =-1 ~ 5-e
O I II) Cots~r amp ~te it o~ ~Igt hpfl~ eA C toa ot -eVl~ l i( iRl wte O~ eVlcJflir+ i r
coVS~~VleU -6 be 0 ~-t )( -X~ I ~id ~ otltr ~ r- tOAs1t()ed -to be 01 +~ tAVt SQ-Hshy
Ut-~R~ vAmiddot -ok ot5l
(0) (wt ~~~middothOv6 ~( JJ oS 0 ~vstt Q~ l~q welt -t~ coo~middot()t~ ~4 e +~
)(-ttgtIrtAlotA-+(
J -Q ~~ J 4 laquoJ 0 +4- x-~)c~ (trll rr t)~ 4~ o11e( +h~ cootMtlkl Gl~ 4 he oI-~ Q~~ r ~ t~~
- (~t p ~ J1+
We- crJ~~~e lbt=( t )(1~1~ )1 0 ~~ L)t-J~ -3xmiddott
1 VJt ~v~ j 1gt ~ ~Q 1shy
l)~ (lt I 00)1 =- l[0Jpound ~jl ~ ~J
nh 5 orl-o ~()r iu SO Xl 0 M()l~tOamp VI () tI~lho9t~IJDJ at f
C~ ( K)1I T )J I 0 th
~ [~ -~~
(3 Ub cohl~d)(J)Sk+~at t i=l +1pound4t olte p~s to- X APo V-hlll XI 1 ~t-I Ct Wleurolrtto1t
n t (4-e t~s~t ~Oii ~ ~e StC1- (0 0 ~ ~ SO thd+ Of) e e~ I +I-t ftJv IJ~ I -he O(~~V11 ColJ +e 01~ twY Or ik 1l~ 1it~ OVl irt (t It 11l11~ f~ +le s~l-t~ IN-h -0 Thn IPF(OrDI ~~)] = rOO J1 de]
L0 Ol~ ~e-i ~ yenIt- 01 50 X ~ VIti- () wlVI-f6J NW ihett yci~
10
(~)amptt~~ ~l osyen~a 101) -to s~ ~ )(+x~~ rAlti-tS 0 SWItJf (wr~
(o)W-M- s q ~~hv ~r tt -tw1~~ l2+O~ tvtllt t) n) tbe ~~-+ ampv-- rew(~t r()fj) X+j(~1~-)1 so ~ttf (Ivf cuf~ S Fo t-heuroy
lDf(~)l U~ lx ~ ~J
OJ ~y- ~ottw ~J~ t-~ -o~~tl-t 4fotItJ- 1vC)IA( ~~ (~) J Kef [Ol=(~)]Cvflt
[3 OJ [~l ~ 0
T-al CAO pound)~~ b I$ Mi4fIk~ 50 we (11 ckscltibe +hl I b~
(x-) =0 I Illf e~IA~twHj1 ~=t
(0) V~t+ 15 +-e ewdl~V) ~ the ~Vie~ 3 e 10 th CIA-Ne 0+ -te SO ~ ptY1+
T4 -w~eVt s~tt is- f-t[OFU)J= i(t) oo1 th+iStN j-IJiJ)VlJt WltfQrr(d~)~ 0 vtc$w- tOSt-~ 0+ --k Otgwl ~Y~ ~I +k ~-Q~lj
II
Let F(Jltl1I~)=A1-l+-~-i s-Ho+ LPP(Yfljft)] =-[Ax Jg~ -]
+~v1 04 Ovf -t~rtt ~mfL)P(ooO)] = ( 0 o IJ [OF(fofA~fl ~lA~ o -IJ b)~ (0 bgt~~)1 =[ 0 ~Sb -I]
OU -YIf~ ~vfJ off -1ll-()o or -eD
JA(X-A) - (c - Ao) 0 Jto( ~- - (c -1S b)) 0
tl ~r+ or SiI -1~ I SD
I)
t~IAVtl Ifvr-1~r +k 6l~ etl+ s~tate i-s ~e~erokd b~ 0 fD () middot0
I l~ 0
(3Jq) Pof +k U~ Gx lt~ ~(ef(~cent sLlD w~-l tie elflaI~ of +4 -b~em plare 10 CJ(
0+ OYlj plIr-t ~ C Gx -o~
N~lt -~I)- ~f Cx (vlO~ GX )+le oVlj fWrl wd- ~- (ClQalI-o-tt 0 5 tlp 6-11911 So ~ ~ 1
~S5tA~ Y (~I4) ~CAS -)OYlefro ~-co()~lVltJk TkYl - +4 eqLl(A lc of eX 5fl4If
~-tnb~ ~(~-Oye~ t-(JfI1)-O h ie eq(J((~iol oJ1kcU(ve XVIR W~t~ ~( seJ to JeiN +le CO~
)4
Tk ~f H-t gtlt-lJO llt Cef~I(l)3 clo~tJ bu-l I~ CtgtIfTqIIl) ~te ~iI X ijeuf (RI11
-th5 ~(HdIOyl CoY1-h 1V1( -he x OlJ ~ ()xef ~ile tleurore (he V~ X +k X-a)(i5 IS
-1e VoI+ ~~ ~S Iv X as -i-) eyenoi +~ -C-~j 5 tte hVViI-f OS -t-)Do T~r fe etO e4 Ii-+ YZ -~d ~s CClI1orlJ i ~ dSIA~ o~ )( I (IrJ 15 c))rcJ( Io X J
eqvol 10 +k lew set o~ X -~
(c) S~w+~C(J- x- 10$ 15 b s~OO+~ Sllrtc
We ()~gt 1 +k f11~ll(lt ~v u-t lOY) tLeofeVI ~VI) f(Jk tt)+ tLe TtAcob e 11 -1w r t~ X=i -~f~ )) eqval 40
[pt(Yj)]= [ -=t xJ ~ hd 4l ex d~eJ tel 41JO I ~ iJ~5 (Ils to be ~rlo CYl)l txt -k o~~VI_VVhlt~
c 5 i) 61 i 4 ~rbLl1 lllctb tLeortJrn X - (01 IS Wto~ h s IMoce-A
[cl)S~ow thrl -h~ ~Llt (amp) ~(r(~amp) S~~~+Le ~HMeh~+lo o~ X d-t09)
rvI +hlS otVII ~v sovlJ ~ve ~ +f~~e ~v~ fA V()w -kgt +e SVM(ue X
0 5~V -h~ 5 ()Votie( ~(Jf6~h~-H(9VlI ~VS+ ~~~ d 1(1 to t k btl~t-ilOv xt -1) ~O SO
r(1-sfGh-(I-se) - floco)~8 f)(l-SV~e -(CJ5~t)
-=0
~ -46 s- 0 f(ttoYYf-h-C(~DYl () X
C~t() C-)~-ueq)
10 -gtJtt OV vvl~ -th~ S~e )5 T S~ltfS1 ~X1I9 f ftY J l -Hlfl1 (J VMj Tk +~l~ w~U ~ 1ke trC1f5 brl)nhl~ te )( ClVlJ ~ -Cl(gtJ S tH r vMlrr v--tL ~e-t q JlAbk co~ b)$J oA i4gt o(~~it Oy~ 11ed YO +h~+ fs (I S Iles ~tle Xcfalt( )tJ)I~
0 l4 ~lt l ( ~1 A _
fJY ~~ Af fOItllvt(-Itk ( t eyen 0 d-0 Ohii IS i It JO re eflAA-t iOY i ho-f IA r clfJfrJfe )(
scd-Sies
() LO- 1(t) ~ (r) I bullbullu t+ h 0 ~ ~ to ~1 Ie ~ (IU bullbull~ ~Pl~middotll~
40 71(t) W~Qt Ir i ~ UjIJ4+OYl O~ Pt~
(c) Si)W ~o+ ~ 1 -4 t~ I 4e ~Cl wS Pt laquoVlel ~h tllwA1s 1Y1-ttfJeCt- () 1e ~~ (f~ te 9~vri-J~~ J-e ht- ~()t
Ntk+I lo-lt +0+ ~ ~ r etv(A-+ amp-) ~)(~- -5(~1) -0 ~gt X ~J1T1~-b =0
OT~ Pt -Iu e~4V )( +)t~ -3- -t -Jl~ -3tS =0
Je ~It 1Vrrl~i
~ U 7
x )t =-1 ~ 5-e
O I II) Cots~r amp ~te it o~ ~Igt hpfl~ eA C toa ot -eVl~ l i( iRl wte O~ eVlcJflir+ i r
coVS~~VleU -6 be 0 ~-t )( -X~ I ~id ~ otltr ~ r- tOAs1t()ed -to be 01 +~ tAVt SQ-Hshy
Ut-~R~ vAmiddot -ok ot5l
(0) (wt ~~~middothOv6 ~( JJ oS 0 ~vstt Q~ l~q welt -t~ coo~middot()t~ ~4 e +~
)(-ttgtIrtAlotA-+(
J -Q ~~ J 4 laquoJ 0 +4- x-~)c~ (trll rr t)~ 4~ o11e( +h~ cootMtlkl Gl~ 4 he oI-~ Q~~ r ~ t~~
- (~t p ~ J1+
We- crJ~~~e lbt=( t )(1~1~ )1 0 ~~ L)t-J~ -3xmiddott
1 VJt ~v~ j 1gt ~ ~Q 1shy
l)~ (lt I 00)1 =- l[0Jpound ~jl ~ ~J
nh 5 orl-o ~()r iu SO Xl 0 M()l~tOamp VI () tI~lho9t~IJDJ at f
C~ ( K)1I T )J I 0 th
~ [~ -~~
(3 Ub cohl~d)(J)Sk+~at t i=l +1pound4t olte p~s to- X APo V-hlll XI 1 ~t-I Ct Wleurolrtto1t
n t (4-e t~s~t ~Oii ~ ~e StC1- (0 0 ~ ~ SO thd+ Of) e e~ I +I-t ftJv IJ~ I -he O(~~V11 ColJ +e 01~ twY Or ik 1l~ 1it~ OVl irt (t It 11l11~ f~ +le s~l-t~ IN-h -0 Thn IPF(OrDI ~~)] = rOO J1 de]
L0 Ol~ ~e-i ~ yenIt- 01 50 X ~ VIti- () wlVI-f6J NW ihett yci~
10
(~)amptt~~ ~l osyen~a 101) -to s~ ~ )(+x~~ rAlti-tS 0 SWItJf (wr~
(o)W-M- s q ~~hv ~r tt -tw1~~ l2+O~ tvtllt t) n) tbe ~~-+ ampv-- rew(~t r()fj) X+j(~1~-)1 so ~ttf (Ivf cuf~ S Fo t-heuroy
lDf(~)l U~ lx ~ ~J
OJ ~y- ~ottw ~J~ t-~ -o~~tl-t 4fotItJ- 1vC)IA( ~~ (~) J Kef [Ol=(~)]Cvflt
[3 OJ [~l ~ 0
T-al CAO pound)~~ b I$ Mi4fIk~ 50 we (11 ckscltibe +hl I b~
(x-) =0 I Illf e~IA~twHj1 ~=t
(0) V~t+ 15 +-e ewdl~V) ~ the ~Vie~ 3 e 10 th CIA-Ne 0+ -te SO ~ ptY1+
T4 -w~eVt s~tt is- f-t[OFU)J= i(t) oo1 th+iStN j-IJiJ)VlJt WltfQrr(d~)~ 0 vtc$w- tOSt-~ 0+ --k Otgwl ~Y~ ~I +k ~-Q~lj
II
Let F(Jltl1I~)=A1-l+-~-i s-Ho+ LPP(Yfljft)] =-[Ax Jg~ -]
+~v1 04 Ovf -t~rtt ~mfL)P(ooO)] = ( 0 o IJ [OF(fofA~fl ~lA~ o -IJ b)~ (0 bgt~~)1 =[ 0 ~Sb -I]
OU -YIf~ ~vfJ off -1ll-()o or -eD
JA(X-A) - (c - Ao) 0 Jto( ~- - (c -1S b)) 0
tl ~r+ or SiI -1~ I SD
I)
t~IAVtl Ifvr-1~r +k 6l~ etl+ s~tate i-s ~e~erokd b~ 0 fD () middot0
I l~ 0
(3Jq) Pof +k U~ Gx lt~ ~(ef(~cent sLlD w~-l tie elflaI~ of +4 -b~em plare 10 CJ(
0+ OYlj plIr-t ~ C Gx -o~
N~lt -~I)- ~f Cx (vlO~ GX )+le oVlj fWrl wd- ~- (ClQalI-o-tt 0 5 tlp 6-11911 So ~ ~ 1
~S5tA~ Y (~I4) ~CAS -)OYlefro ~-co()~lVltJk TkYl - +4 eqLl(A lc of eX 5fl4If
~-tnb~ ~(~-Oye~ t-(JfI1)-O h ie eq(J((~iol oJ1kcU(ve XVIR W~t~ ~( seJ to JeiN +le CO~
)4
C~t() C-)~-ueq)
10 -gtJtt OV vvl~ -th~ S~e )5 T S~ltfS1 ~X1I9 f ftY J l -Hlfl1 (J VMj Tk +~l~ w~U ~ 1ke trC1f5 brl)nhl~ te )( ClVlJ ~ -Cl(gtJ S tH r vMlrr v--tL ~e-t q JlAbk co~ b)$J oA i4gt o(~~it Oy~ 11ed YO +h~+ fs (I S Iles ~tle Xcfalt( )tJ)I~
0 l4 ~lt l ( ~1 A _
fJY ~~ Af fOItllvt(-Itk ( t eyen 0 d-0 Ohii IS i It JO re eflAA-t iOY i ho-f IA r clfJfrJfe )(
scd-Sies
() LO- 1(t) ~ (r) I bullbullu t+ h 0 ~ ~ to ~1 Ie ~ (IU bullbull~ ~Pl~middotll~
40 71(t) W~Qt Ir i ~ UjIJ4+OYl O~ Pt~
(c) Si)W ~o+ ~ 1 -4 t~ I 4e ~Cl wS Pt laquoVlel ~h tllwA1s 1Y1-ttfJeCt- () 1e ~~ (f~ te 9~vri-J~~ J-e ht- ~()t
Ntk+I lo-lt +0+ ~ ~ r etv(A-+ amp-) ~)(~- -5(~1) -0 ~gt X ~J1T1~-b =0
OT~ Pt -Iu e~4V )( +)t~ -3- -t -Jl~ -3tS =0
Je ~It 1Vrrl~i
~ U 7
x )t =-1 ~ 5-e
O I II) Cots~r amp ~te it o~ ~Igt hpfl~ eA C toa ot -eVl~ l i( iRl wte O~ eVlcJflir+ i r
coVS~~VleU -6 be 0 ~-t )( -X~ I ~id ~ otltr ~ r- tOAs1t()ed -to be 01 +~ tAVt SQ-Hshy
Ut-~R~ vAmiddot -ok ot5l
(0) (wt ~~~middothOv6 ~( JJ oS 0 ~vstt Q~ l~q welt -t~ coo~middot()t~ ~4 e +~
)(-ttgtIrtAlotA-+(
J -Q ~~ J 4 laquoJ 0 +4- x-~)c~ (trll rr t)~ 4~ o11e( +h~ cootMtlkl Gl~ 4 he oI-~ Q~~ r ~ t~~
- (~t p ~ J1+
We- crJ~~~e lbt=( t )(1~1~ )1 0 ~~ L)t-J~ -3xmiddott
1 VJt ~v~ j 1gt ~ ~Q 1shy
l)~ (lt I 00)1 =- l[0Jpound ~jl ~ ~J
nh 5 orl-o ~()r iu SO Xl 0 M()l~tOamp VI () tI~lho9t~IJDJ at f
C~ ( K)1I T )J I 0 th
~ [~ -~~
(3 Ub cohl~d)(J)Sk+~at t i=l +1pound4t olte p~s to- X APo V-hlll XI 1 ~t-I Ct Wleurolrtto1t
n t (4-e t~s~t ~Oii ~ ~e StC1- (0 0 ~ ~ SO thd+ Of) e e~ I +I-t ftJv IJ~ I -he O(~~V11 ColJ +e 01~ twY Or ik 1l~ 1it~ OVl irt (t It 11l11~ f~ +le s~l-t~ IN-h -0 Thn IPF(OrDI ~~)] = rOO J1 de]
L0 Ol~ ~e-i ~ yenIt- 01 50 X ~ VIti- () wlVI-f6J NW ihett yci~
10
(~)amptt~~ ~l osyen~a 101) -to s~ ~ )(+x~~ rAlti-tS 0 SWItJf (wr~
(o)W-M- s q ~~hv ~r tt -tw1~~ l2+O~ tvtllt t) n) tbe ~~-+ ampv-- rew(~t r()fj) X+j(~1~-)1 so ~ttf (Ivf cuf~ S Fo t-heuroy
lDf(~)l U~ lx ~ ~J
OJ ~y- ~ottw ~J~ t-~ -o~~tl-t 4fotItJ- 1vC)IA( ~~ (~) J Kef [Ol=(~)]Cvflt
[3 OJ [~l ~ 0
T-al CAO pound)~~ b I$ Mi4fIk~ 50 we (11 ckscltibe +hl I b~
(x-) =0 I Illf e~IA~twHj1 ~=t
(0) V~t+ 15 +-e ewdl~V) ~ the ~Vie~ 3 e 10 th CIA-Ne 0+ -te SO ~ ptY1+
T4 -w~eVt s~tt is- f-t[OFU)J= i(t) oo1 th+iStN j-IJiJ)VlJt WltfQrr(d~)~ 0 vtc$w- tOSt-~ 0+ --k Otgwl ~Y~ ~I +k ~-Q~lj
II
Let F(Jltl1I~)=A1-l+-~-i s-Ho+ LPP(Yfljft)] =-[Ax Jg~ -]
+~v1 04 Ovf -t~rtt ~mfL)P(ooO)] = ( 0 o IJ [OF(fofA~fl ~lA~ o -IJ b)~ (0 bgt~~)1 =[ 0 ~Sb -I]
OU -YIf~ ~vfJ off -1ll-()o or -eD
JA(X-A) - (c - Ao) 0 Jto( ~- - (c -1S b)) 0
tl ~r+ or SiI -1~ I SD
I)
t~IAVtl Ifvr-1~r +k 6l~ etl+ s~tate i-s ~e~erokd b~ 0 fD () middot0
I l~ 0
(3Jq) Pof +k U~ Gx lt~ ~(ef(~cent sLlD w~-l tie elflaI~ of +4 -b~em plare 10 CJ(
0+ OYlj plIr-t ~ C Gx -o~
N~lt -~I)- ~f Cx (vlO~ GX )+le oVlj fWrl wd- ~- (ClQalI-o-tt 0 5 tlp 6-11911 So ~ ~ 1
~S5tA~ Y (~I4) ~CAS -)OYlefro ~-co()~lVltJk TkYl - +4 eqLl(A lc of eX 5fl4If
~-tnb~ ~(~-Oye~ t-(JfI1)-O h ie eq(J((~iol oJ1kcU(ve XVIR W~t~ ~( seJ to JeiN +le CO~
)4
() LO- 1(t) ~ (r) I bullbullu t+ h 0 ~ ~ to ~1 Ie ~ (IU bullbull~ ~Pl~middotll~
40 71(t) W~Qt Ir i ~ UjIJ4+OYl O~ Pt~
(c) Si)W ~o+ ~ 1 -4 t~ I 4e ~Cl wS Pt laquoVlel ~h tllwA1s 1Y1-ttfJeCt- () 1e ~~ (f~ te 9~vri-J~~ J-e ht- ~()t
Ntk+I lo-lt +0+ ~ ~ r etv(A-+ amp-) ~)(~- -5(~1) -0 ~gt X ~J1T1~-b =0
OT~ Pt -Iu e~4V )( +)t~ -3- -t -Jl~ -3tS =0
Je ~It 1Vrrl~i
~ U 7
x )t =-1 ~ 5-e
O I II) Cots~r amp ~te it o~ ~Igt hpfl~ eA C toa ot -eVl~ l i( iRl wte O~ eVlcJflir+ i r
coVS~~VleU -6 be 0 ~-t )( -X~ I ~id ~ otltr ~ r- tOAs1t()ed -to be 01 +~ tAVt SQ-Hshy
Ut-~R~ vAmiddot -ok ot5l
(0) (wt ~~~middothOv6 ~( JJ oS 0 ~vstt Q~ l~q welt -t~ coo~middot()t~ ~4 e +~
)(-ttgtIrtAlotA-+(
J -Q ~~ J 4 laquoJ 0 +4- x-~)c~ (trll rr t)~ 4~ o11e( +h~ cootMtlkl Gl~ 4 he oI-~ Q~~ r ~ t~~
- (~t p ~ J1+
We- crJ~~~e lbt=( t )(1~1~ )1 0 ~~ L)t-J~ -3xmiddott
1 VJt ~v~ j 1gt ~ ~Q 1shy
l)~ (lt I 00)1 =- l[0Jpound ~jl ~ ~J
nh 5 orl-o ~()r iu SO Xl 0 M()l~tOamp VI () tI~lho9t~IJDJ at f
C~ ( K)1I T )J I 0 th
~ [~ -~~
(3 Ub cohl~d)(J)Sk+~at t i=l +1pound4t olte p~s to- X APo V-hlll XI 1 ~t-I Ct Wleurolrtto1t
n t (4-e t~s~t ~Oii ~ ~e StC1- (0 0 ~ ~ SO thd+ Of) e e~ I +I-t ftJv IJ~ I -he O(~~V11 ColJ +e 01~ twY Or ik 1l~ 1it~ OVl irt (t It 11l11~ f~ +le s~l-t~ IN-h -0 Thn IPF(OrDI ~~)] = rOO J1 de]
L0 Ol~ ~e-i ~ yenIt- 01 50 X ~ VIti- () wlVI-f6J NW ihett yci~
10
(~)amptt~~ ~l osyen~a 101) -to s~ ~ )(+x~~ rAlti-tS 0 SWItJf (wr~
(o)W-M- s q ~~hv ~r tt -tw1~~ l2+O~ tvtllt t) n) tbe ~~-+ ampv-- rew(~t r()fj) X+j(~1~-)1 so ~ttf (Ivf cuf~ S Fo t-heuroy
lDf(~)l U~ lx ~ ~J
OJ ~y- ~ottw ~J~ t-~ -o~~tl-t 4fotItJ- 1vC)IA( ~~ (~) J Kef [Ol=(~)]Cvflt
[3 OJ [~l ~ 0
T-al CAO pound)~~ b I$ Mi4fIk~ 50 we (11 ckscltibe +hl I b~
(x-) =0 I Illf e~IA~twHj1 ~=t
(0) V~t+ 15 +-e ewdl~V) ~ the ~Vie~ 3 e 10 th CIA-Ne 0+ -te SO ~ ptY1+
T4 -w~eVt s~tt is- f-t[OFU)J= i(t) oo1 th+iStN j-IJiJ)VlJt WltfQrr(d~)~ 0 vtc$w- tOSt-~ 0+ --k Otgwl ~Y~ ~I +k ~-Q~lj
II
Let F(Jltl1I~)=A1-l+-~-i s-Ho+ LPP(Yfljft)] =-[Ax Jg~ -]
+~v1 04 Ovf -t~rtt ~mfL)P(ooO)] = ( 0 o IJ [OF(fofA~fl ~lA~ o -IJ b)~ (0 bgt~~)1 =[ 0 ~Sb -I]
OU -YIf~ ~vfJ off -1ll-()o or -eD
JA(X-A) - (c - Ao) 0 Jto( ~- - (c -1S b)) 0
tl ~r+ or SiI -1~ I SD
I)
t~IAVtl Ifvr-1~r +k 6l~ etl+ s~tate i-s ~e~erokd b~ 0 fD () middot0
I l~ 0
(3Jq) Pof +k U~ Gx lt~ ~(ef(~cent sLlD w~-l tie elflaI~ of +4 -b~em plare 10 CJ(
0+ OYlj plIr-t ~ C Gx -o~
N~lt -~I)- ~f Cx (vlO~ GX )+le oVlj fWrl wd- ~- (ClQalI-o-tt 0 5 tlp 6-11911 So ~ ~ 1
~S5tA~ Y (~I4) ~CAS -)OYlefro ~-co()~lVltJk TkYl - +4 eqLl(A lc of eX 5fl4If
~-tnb~ ~(~-Oye~ t-(JfI1)-O h ie eq(J((~iol oJ1kcU(ve XVIR W~t~ ~( seJ to JeiN +le CO~
)4
x )t =-1 ~ 5-e
O I II) Cots~r amp ~te it o~ ~Igt hpfl~ eA C toa ot -eVl~ l i( iRl wte O~ eVlcJflir+ i r
coVS~~VleU -6 be 0 ~-t )( -X~ I ~id ~ otltr ~ r- tOAs1t()ed -to be 01 +~ tAVt SQ-Hshy
Ut-~R~ vAmiddot -ok ot5l
(0) (wt ~~~middothOv6 ~( JJ oS 0 ~vstt Q~ l~q welt -t~ coo~middot()t~ ~4 e +~
)(-ttgtIrtAlotA-+(
J -Q ~~ J 4 laquoJ 0 +4- x-~)c~ (trll rr t)~ 4~ o11e( +h~ cootMtlkl Gl~ 4 he oI-~ Q~~ r ~ t~~
- (~t p ~ J1+
We- crJ~~~e lbt=( t )(1~1~ )1 0 ~~ L)t-J~ -3xmiddott
1 VJt ~v~ j 1gt ~ ~Q 1shy
l)~ (lt I 00)1 =- l[0Jpound ~jl ~ ~J
nh 5 orl-o ~()r iu SO Xl 0 M()l~tOamp VI () tI~lho9t~IJDJ at f
C~ ( K)1I T )J I 0 th
~ [~ -~~
(3 Ub cohl~d)(J)Sk+~at t i=l +1pound4t olte p~s to- X APo V-hlll XI 1 ~t-I Ct Wleurolrtto1t
n t (4-e t~s~t ~Oii ~ ~e StC1- (0 0 ~ ~ SO thd+ Of) e e~ I +I-t ftJv IJ~ I -he O(~~V11 ColJ +e 01~ twY Or ik 1l~ 1it~ OVl irt (t It 11l11~ f~ +le s~l-t~ IN-h -0 Thn IPF(OrDI ~~)] = rOO J1 de]
L0 Ol~ ~e-i ~ yenIt- 01 50 X ~ VIti- () wlVI-f6J NW ihett yci~
10
(~)amptt~~ ~l osyen~a 101) -to s~ ~ )(+x~~ rAlti-tS 0 SWItJf (wr~
(o)W-M- s q ~~hv ~r tt -tw1~~ l2+O~ tvtllt t) n) tbe ~~-+ ampv-- rew(~t r()fj) X+j(~1~-)1 so ~ttf (Ivf cuf~ S Fo t-heuroy
lDf(~)l U~ lx ~ ~J
OJ ~y- ~ottw ~J~ t-~ -o~~tl-t 4fotItJ- 1vC)IA( ~~ (~) J Kef [Ol=(~)]Cvflt
[3 OJ [~l ~ 0
T-al CAO pound)~~ b I$ Mi4fIk~ 50 we (11 ckscltibe +hl I b~
(x-) =0 I Illf e~IA~twHj1 ~=t
(0) V~t+ 15 +-e ewdl~V) ~ the ~Vie~ 3 e 10 th CIA-Ne 0+ -te SO ~ ptY1+
T4 -w~eVt s~tt is- f-t[OFU)J= i(t) oo1 th+iStN j-IJiJ)VlJt WltfQrr(d~)~ 0 vtc$w- tOSt-~ 0+ --k Otgwl ~Y~ ~I +k ~-Q~lj
II
Let F(Jltl1I~)=A1-l+-~-i s-Ho+ LPP(Yfljft)] =-[Ax Jg~ -]
+~v1 04 Ovf -t~rtt ~mfL)P(ooO)] = ( 0 o IJ [OF(fofA~fl ~lA~ o -IJ b)~ (0 bgt~~)1 =[ 0 ~Sb -I]
OU -YIf~ ~vfJ off -1ll-()o or -eD
JA(X-A) - (c - Ao) 0 Jto( ~- - (c -1S b)) 0
tl ~r+ or SiI -1~ I SD
I)
t~IAVtl Ifvr-1~r +k 6l~ etl+ s~tate i-s ~e~erokd b~ 0 fD () middot0
I l~ 0
(3Jq) Pof +k U~ Gx lt~ ~(ef(~cent sLlD w~-l tie elflaI~ of +4 -b~em plare 10 CJ(
0+ OYlj plIr-t ~ C Gx -o~
N~lt -~I)- ~f Cx (vlO~ GX )+le oVlj fWrl wd- ~- (ClQalI-o-tt 0 5 tlp 6-11911 So ~ ~ 1
~S5tA~ Y (~I4) ~CAS -)OYlefro ~-co()~lVltJk TkYl - +4 eqLl(A lc of eX 5fl4If
~-tnb~ ~(~-Oye~ t-(JfI1)-O h ie eq(J((~iol oJ1kcU(ve XVIR W~t~ ~( seJ to JeiN +le CO~
)4
O I II) Cots~r amp ~te it o~ ~Igt hpfl~ eA C toa ot -eVl~ l i( iRl wte O~ eVlcJflir+ i r
coVS~~VleU -6 be 0 ~-t )( -X~ I ~id ~ otltr ~ r- tOAs1t()ed -to be 01 +~ tAVt SQ-Hshy
Ut-~R~ vAmiddot -ok ot5l
(0) (wt ~~~middothOv6 ~( JJ oS 0 ~vstt Q~ l~q welt -t~ coo~middot()t~ ~4 e +~
)(-ttgtIrtAlotA-+(
J -Q ~~ J 4 laquoJ 0 +4- x-~)c~ (trll rr t)~ 4~ o11e( +h~ cootMtlkl Gl~ 4 he oI-~ Q~~ r ~ t~~
- (~t p ~ J1+
We- crJ~~~e lbt=( t )(1~1~ )1 0 ~~ L)t-J~ -3xmiddott
1 VJt ~v~ j 1gt ~ ~Q 1shy
l)~ (lt I 00)1 =- l[0Jpound ~jl ~ ~J
nh 5 orl-o ~()r iu SO Xl 0 M()l~tOamp VI () tI~lho9t~IJDJ at f
C~ ( K)1I T )J I 0 th
~ [~ -~~
(3 Ub cohl~d)(J)Sk+~at t i=l +1pound4t olte p~s to- X APo V-hlll XI 1 ~t-I Ct Wleurolrtto1t
n t (4-e t~s~t ~Oii ~ ~e StC1- (0 0 ~ ~ SO thd+ Of) e e~ I +I-t ftJv IJ~ I -he O(~~V11 ColJ +e 01~ twY Or ik 1l~ 1it~ OVl irt (t It 11l11~ f~ +le s~l-t~ IN-h -0 Thn IPF(OrDI ~~)] = rOO J1 de]
L0 Ol~ ~e-i ~ yenIt- 01 50 X ~ VIti- () wlVI-f6J NW ihett yci~
10
(~)amptt~~ ~l osyen~a 101) -to s~ ~ )(+x~~ rAlti-tS 0 SWItJf (wr~
(o)W-M- s q ~~hv ~r tt -tw1~~ l2+O~ tvtllt t) n) tbe ~~-+ ampv-- rew(~t r()fj) X+j(~1~-)1 so ~ttf (Ivf cuf~ S Fo t-heuroy
lDf(~)l U~ lx ~ ~J
OJ ~y- ~ottw ~J~ t-~ -o~~tl-t 4fotItJ- 1vC)IA( ~~ (~) J Kef [Ol=(~)]Cvflt
[3 OJ [~l ~ 0
T-al CAO pound)~~ b I$ Mi4fIk~ 50 we (11 ckscltibe +hl I b~
(x-) =0 I Illf e~IA~twHj1 ~=t
(0) V~t+ 15 +-e ewdl~V) ~ the ~Vie~ 3 e 10 th CIA-Ne 0+ -te SO ~ ptY1+
T4 -w~eVt s~tt is- f-t[OFU)J= i(t) oo1 th+iStN j-IJiJ)VlJt WltfQrr(d~)~ 0 vtc$w- tOSt-~ 0+ --k Otgwl ~Y~ ~I +k ~-Q~lj
II
Let F(Jltl1I~)=A1-l+-~-i s-Ho+ LPP(Yfljft)] =-[Ax Jg~ -]
+~v1 04 Ovf -t~rtt ~mfL)P(ooO)] = ( 0 o IJ [OF(fofA~fl ~lA~ o -IJ b)~ (0 bgt~~)1 =[ 0 ~Sb -I]
OU -YIf~ ~vfJ off -1ll-()o or -eD
JA(X-A) - (c - Ao) 0 Jto( ~- - (c -1S b)) 0
tl ~r+ or SiI -1~ I SD
I)
t~IAVtl Ifvr-1~r +k 6l~ etl+ s~tate i-s ~e~erokd b~ 0 fD () middot0
I l~ 0
(3Jq) Pof +k U~ Gx lt~ ~(ef(~cent sLlD w~-l tie elflaI~ of +4 -b~em plare 10 CJ(
0+ OYlj plIr-t ~ C Gx -o~
N~lt -~I)- ~f Cx (vlO~ GX )+le oVlj fWrl wd- ~- (ClQalI-o-tt 0 5 tlp 6-11911 So ~ ~ 1
~S5tA~ Y (~I4) ~CAS -)OYlefro ~-co()~lVltJk TkYl - +4 eqLl(A lc of eX 5fl4If
~-tnb~ ~(~-Oye~ t-(JfI1)-O h ie eq(J((~iol oJ1kcU(ve XVIR W~t~ ~( seJ to JeiN +le CO~
)4
(3 Ub cohl~d)(J)Sk+~at t i=l +1pound4t olte p~s to- X APo V-hlll XI 1 ~t-I Ct Wleurolrtto1t
n t (4-e t~s~t ~Oii ~ ~e StC1- (0 0 ~ ~ SO thd+ Of) e e~ I +I-t ftJv IJ~ I -he O(~~V11 ColJ +e 01~ twY Or ik 1l~ 1it~ OVl irt (t It 11l11~ f~ +le s~l-t~ IN-h -0 Thn IPF(OrDI ~~)] = rOO J1 de]
L0 Ol~ ~e-i ~ yenIt- 01 50 X ~ VIti- () wlVI-f6J NW ihett yci~
10
(~)amptt~~ ~l osyen~a 101) -to s~ ~ )(+x~~ rAlti-tS 0 SWItJf (wr~
(o)W-M- s q ~~hv ~r tt -tw1~~ l2+O~ tvtllt t) n) tbe ~~-+ ampv-- rew(~t r()fj) X+j(~1~-)1 so ~ttf (Ivf cuf~ S Fo t-heuroy
lDf(~)l U~ lx ~ ~J
OJ ~y- ~ottw ~J~ t-~ -o~~tl-t 4fotItJ- 1vC)IA( ~~ (~) J Kef [Ol=(~)]Cvflt
[3 OJ [~l ~ 0
T-al CAO pound)~~ b I$ Mi4fIk~ 50 we (11 ckscltibe +hl I b~
(x-) =0 I Illf e~IA~twHj1 ~=t
(0) V~t+ 15 +-e ewdl~V) ~ the ~Vie~ 3 e 10 th CIA-Ne 0+ -te SO ~ ptY1+
T4 -w~eVt s~tt is- f-t[OFU)J= i(t) oo1 th+iStN j-IJiJ)VlJt WltfQrr(d~)~ 0 vtc$w- tOSt-~ 0+ --k Otgwl ~Y~ ~I +k ~-Q~lj
II
Let F(Jltl1I~)=A1-l+-~-i s-Ho+ LPP(Yfljft)] =-[Ax Jg~ -]
+~v1 04 Ovf -t~rtt ~mfL)P(ooO)] = ( 0 o IJ [OF(fofA~fl ~lA~ o -IJ b)~ (0 bgt~~)1 =[ 0 ~Sb -I]
OU -YIf~ ~vfJ off -1ll-()o or -eD
JA(X-A) - (c - Ao) 0 Jto( ~- - (c -1S b)) 0
tl ~r+ or SiI -1~ I SD
I)
t~IAVtl Ifvr-1~r +k 6l~ etl+ s~tate i-s ~e~erokd b~ 0 fD () middot0
I l~ 0
(3Jq) Pof +k U~ Gx lt~ ~(ef(~cent sLlD w~-l tie elflaI~ of +4 -b~em plare 10 CJ(
0+ OYlj plIr-t ~ C Gx -o~
N~lt -~I)- ~f Cx (vlO~ GX )+le oVlj fWrl wd- ~- (ClQalI-o-tt 0 5 tlp 6-11911 So ~ ~ 1
~S5tA~ Y (~I4) ~CAS -)OYlefro ~-co()~lVltJk TkYl - +4 eqLl(A lc of eX 5fl4If
~-tnb~ ~(~-Oye~ t-(JfI1)-O h ie eq(J((~iol oJ1kcU(ve XVIR W~t~ ~( seJ to JeiN +le CO~
)4
(~)amptt~~ ~l osyen~a 101) -to s~ ~ )(+x~~ rAlti-tS 0 SWItJf (wr~
(o)W-M- s q ~~hv ~r tt -tw1~~ l2+O~ tvtllt t) n) tbe ~~-+ ampv-- rew(~t r()fj) X+j(~1~-)1 so ~ttf (Ivf cuf~ S Fo t-heuroy
lDf(~)l U~ lx ~ ~J
OJ ~y- ~ottw ~J~ t-~ -o~~tl-t 4fotItJ- 1vC)IA( ~~ (~) J Kef [Ol=(~)]Cvflt
[3 OJ [~l ~ 0
T-al CAO pound)~~ b I$ Mi4fIk~ 50 we (11 ckscltibe +hl I b~
(x-) =0 I Illf e~IA~twHj1 ~=t
(0) V~t+ 15 +-e ewdl~V) ~ the ~Vie~ 3 e 10 th CIA-Ne 0+ -te SO ~ ptY1+
T4 -w~eVt s~tt is- f-t[OFU)J= i(t) oo1 th+iStN j-IJiJ)VlJt WltfQrr(d~)~ 0 vtc$w- tOSt-~ 0+ --k Otgwl ~Y~ ~I +k ~-Q~lj
II
Let F(Jltl1I~)=A1-l+-~-i s-Ho+ LPP(Yfljft)] =-[Ax Jg~ -]
+~v1 04 Ovf -t~rtt ~mfL)P(ooO)] = ( 0 o IJ [OF(fofA~fl ~lA~ o -IJ b)~ (0 bgt~~)1 =[ 0 ~Sb -I]
OU -YIf~ ~vfJ off -1ll-()o or -eD
JA(X-A) - (c - Ao) 0 Jto( ~- - (c -1S b)) 0
tl ~r+ or SiI -1~ I SD
I)
t~IAVtl Ifvr-1~r +k 6l~ etl+ s~tate i-s ~e~erokd b~ 0 fD () middot0
I l~ 0
(3Jq) Pof +k U~ Gx lt~ ~(ef(~cent sLlD w~-l tie elflaI~ of +4 -b~em plare 10 CJ(
0+ OYlj plIr-t ~ C Gx -o~
N~lt -~I)- ~f Cx (vlO~ GX )+le oVlj fWrl wd- ~- (ClQalI-o-tt 0 5 tlp 6-11911 So ~ ~ 1
~S5tA~ Y (~I4) ~CAS -)OYlefro ~-co()~lVltJk TkYl - +4 eqLl(A lc of eX 5fl4If
~-tnb~ ~(~-Oye~ t-(JfI1)-O h ie eq(J((~iol oJ1kcU(ve XVIR W~t~ ~( seJ to JeiN +le CO~
)4
Let F(Jltl1I~)=A1-l+-~-i s-Ho+ LPP(Yfljft)] =-[Ax Jg~ -]
+~v1 04 Ovf -t~rtt ~mfL)P(ooO)] = ( 0 o IJ [OF(fofA~fl ~lA~ o -IJ b)~ (0 bgt~~)1 =[ 0 ~Sb -I]
OU -YIf~ ~vfJ off -1ll-()o or -eD
JA(X-A) - (c - Ao) 0 Jto( ~- - (c -1S b)) 0
tl ~r+ or SiI -1~ I SD
I)
t~IAVtl Ifvr-1~r +k 6l~ etl+ s~tate i-s ~e~erokd b~ 0 fD () middot0
I l~ 0
(3Jq) Pof +k U~ Gx lt~ ~(ef(~cent sLlD w~-l tie elflaI~ of +4 -b~em plare 10 CJ(
0+ OYlj plIr-t ~ C Gx -o~
N~lt -~I)- ~f Cx (vlO~ GX )+le oVlj fWrl wd- ~- (ClQalI-o-tt 0 5 tlp 6-11911 So ~ ~ 1
~S5tA~ Y (~I4) ~CAS -)OYlefro ~-co()~lVltJk TkYl - +4 eqLl(A lc of eX 5fl4If
~-tnb~ ~(~-Oye~ t-(JfI1)-O h ie eq(J((~iol oJ1kcU(ve XVIR W~t~ ~( seJ to JeiN +le CO~
)4
t~IAVtl Ifvr-1~r +k 6l~ etl+ s~tate i-s ~e~erokd b~ 0 fD () middot0
I l~ 0
(3Jq) Pof +k U~ Gx lt~ ~(ef(~cent sLlD w~-l tie elflaI~ of +4 -b~em plare 10 CJ(
0+ OYlj plIr-t ~ C Gx -o~
N~lt -~I)- ~f Cx (vlO~ GX )+le oVlj fWrl wd- ~- (ClQalI-o-tt 0 5 tlp 6-11911 So ~ ~ 1
~S5tA~ Y (~I4) ~CAS -)OYlefro ~-co()~lVltJk TkYl - +4 eqLl(A lc of eX 5fl4If
~-tnb~ ~(~-Oye~ t-(JfI1)-O h ie eq(J((~iol oJ1kcU(ve XVIR W~t~ ~( seJ to JeiN +le CO~
)4
(3Jq) Pof +k U~ Gx lt~ ~(ef(~cent sLlD w~-l tie elflaI~ of +4 -b~em plare 10 CJ(
0+ OYlj plIr-t ~ C Gx -o~
N~lt -~I)- ~f Cx (vlO~ GX )+le oVlj fWrl wd- ~- (ClQalI-o-tt 0 5 tlp 6-11911 So ~ ~ 1
~S5tA~ Y (~I4) ~CAS -)OYlefro ~-co()~lVltJk TkYl - +4 eqLl(A lc of eX 5fl4If
~-tnb~ ~(~-Oye~ t-(JfI1)-O h ie eq(J((~iol oJ1kcU(ve XVIR W~t~ ~( seJ to JeiN +le CO~
)4
![INTRODUCTION TO MATHEMATICAL MODELING. A …math.ucsd.edu/~dmeyer/teaching/111fall14/probpop141013.pdf · Using this result in (5) we obtain Var[N t] = 1− rt 1− r βrt−1N 0.](https://static.documents.pub/doc/80x56/5f5a445f5f64351cd516dec2/introduction-to-mathematical-modeling-a-mathucsdedudmeyerteaching111fall14.jpg)
