11/8/2002 CS 258 HW 2

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1 /8/ CS 58 HW. G o a a qc of aa h < fo a I o goa o co a C cc c F ch ha F fo a I A If cc - c a co h aoa coo o ho o choo h o qc? I o g o -coa o o-coa? W ca choo h o qc o h a a h aa a. Tha f o o a h o h a:. Th h coo o ca oo o h. A o h coo o co ca o o foc h c o oa h o. Sc o ha # of coo o g of c # of o T Moo 97 h of o ga ha 5. W o ha a ao o o o h of o a hc of aa o a a cc -S c f fo oao. Th g o -coa ca ha o cf a ag foao fo h o hogh h c caa of C co a ha o cf a o foao ooa o f h c a of a ag go o o ha ogh foao o ach. If h o -coa a ffc of ag coo a h o a h coo o of h g [cc] - c. Ug Fa 99 a o g ca - c a a c c. Hc ca h h oag o of c fo ach c a co h o a fo ha ca h -. Sc a cc c h c o a gho o h o ac oa h o. W a h o h c o h ac o h o oa a. I oh o a:... h f h gho. To C coo a a aca o h coo o h a ao a h o aa o h of. To o coc a cofg f - h h a a o h o c a A oo h o a aa o h a ca fo oh. Th ca a ha: o Th foa a ha h coo o fo o a fo a gh aag of gho ooo o h a ac o ac. W o ha a aa fo h a h ghog o a a. Th ghog o ca a o h coo c h c f hogh... h h a - a coo c o h - c a g fo Fa 99. Sa

2 ... So o ha a ho of ag h coo o o h - c o h coo o oag h aa o g oga. Ho h a o h h o ca h a o cf h gho a ch. W a f o co h o o ag a a o - a - -. Ug o foao of coo o of - coo o ca a: a L L L L L o ha ha a h of ca o fo of o g h c cooc h aa o a h - coo o. W o:... So:... ha fo fo h. Sg ho g: Cocg g : Wh fo - a - L. To f h oao ca a h coffc of h - coo o a: α a Th ca a o coc α o o ha a of qao ch a og aog : α α A h o ca aa o fo coo o. W ca cf h a fo a: M M O α α

3 Th f a a ogo c a - a. Hc o ha cf h of coo o g a of aa o a aa.

4 . F h coo o of a co - c of g ho coo ogo co of h g of a qa a ho o qc fo a co of o. A fo o qc oo. Ho h c co. Thfo ca o coo o f a o ach a co c. Th g o h ooa gog h - ha a fo of a a a. I h ca ha ca 5 c coa coa a o o. A fo qac - c ca a a fo a: [ ] fo [:-] o o ccc UM S. Ao c o c co ac o o ach coo o. So a h qao: o o o o o UM S h h h. h ach a coo o a h c fo h g o a Moo. A c ca a a fo a: [ ] 5 o o ccc UM. If a o c h a c h ha UM UM S S o qa M M S S o o co fo - o coo o ha h qao: S S M M S S 8 8 So g a coo o o h g of a qa ca ac h hogh h ao qao a g o coo o. Th o fco fo S a h c co a fo choog o ha a h oga of coo o.

5 Th cacao of h - a a a g o I co h h: Th ao foo Mo - fo hgh g. Th f o choo a a I h. Th a gc a o caca oca o. Th o a a cho co a Th a h ao a a a: fo < o oh fo < o oh fo < o oh fo < o oh fo < o oh 5 fo < o oh Th h foa a fo: h h a aa g h foog Mahaca : 5 ; C@ : HC@ LL ê HHH@ L C@ L ê LL C@_ : MC@ Ug h coa aga fo Moo co ao o aa a of h a Th choo a a a fo o of h fco o o. Fo h ca h a < a cho hch a ha o h h ha o aa. Th aao a o h h foog Mahaca coa: Ea@ C@ 5ê. 8MC@ MC@ MC@ MC@ MC@ MC@ 5 MC@ < Th o h coffc fo o co of h a. Th oh co a aag C[ 5] C[5] C[5] a C[5]. Th co fo ca oog a h o of h a fo. J a h cc ha a / h a h qaac ha a ½ h qac ha o coa faco a /.

6 . G a o oc ach h a h o a o aa a o. Wo o h oao co fo ach of h ca. Sc ca co o h aao ca fo a To oc ach I co o ach fo h oao co. * U h c Caa agoh Th a o ca fo h Caa agoh o hch h ach ha qa coo o ach o a o hch h a ao a oao. A a a of a o oao ha a fo ch a: Th aao a aco ao of a ao cao of a cao. So S A M. A a a a o acao of h c Caa agoh a foa ch a: Th aao q aco - a - 9 ao [ ] o a a o o ao a c ha a a h ho g 9 fa h a 8 cao. -* a ach ha o o * a h h fo ach o a 8 cao. S A 9 M 8. I a ach qa o a 9 a a 8 c a c fco h o o o fo h a aco. Loog a h a fo ha: [ ] I h fo ca o h o aco a h goa o h aa aco. Ho aag a co o o o o o fo o ca o o o o o o ca. o ca ga h fo a ach. Fo h ca ca of a o h a o gog o a fo oao a o h of qa o ca. So c o ha qa ach of hch a 9 a a 8 o coa ac. Fo h ca hch ha o oo. W o h of qa g h foa f * * f- - h f a f. So f * * * 8 qa. Ho ao o ha f ha o oa o. I fac oa. So ha aco f*9 ao f * 8 --/ * ao --/ * oa.

7 * U h o oc foao. Co h coffc of a oaac a h aa ha c a. W a g a ach ch a: I h o aa a h ooa f h h coffc fo ach c aog a h caca h ooa fo h aco a h coffc o g h o o. Th oha ff ha h ho c ca hch a fo a Caa ao fo ach c o g h coo o a h fo Caa o ha c o g h fa o o h c. I h ho f caca h ooa. W a ha h g of h c h a aco h fac. Th ooa a co fo h g a fo a. Th ooa g : A ha h oa coffc a co a o o faco o h oao co. Eaag h ooa aco of - hch ca o a o a o. co o of ha g a ao a - - a h h oa o oa ha - a o ac o oa oao o cog oa. Af og ha ha a ooa ha ca fo a h c o h a. Ho aa h coffc a h aga h a: So fo a - a o co a g o o a c. Each a a ao a ach a cao. So h oa oao co *. I o ca aog h a a ha oao o caca h ooa a oao o caca a o o o of h c. Ho h a c o o ga a h o o a h c a * oao. o ha a h o o a oaac o. o co h h co. Th q oao. Eaag a o o h c a oao. Thfo oa ha: caca : oao caca o o c: * oao caca oh : aa a o o h: fo a ga oa of: # o oa *. * U h o oc foao. Co h coffc of a oaac a h aa ha c a. Th h a a h o ag h a a. Thfo h oao co : # o oa *

8 . G a aoa ach of g g a agoh o co h coo o a aoca gh of oa o hoogah ach. G h aa a o h oa a. o a oa co. I a o cg a co fo oo of gh a coo o of ho a h o co a ao of h oa ach of ooa. G a aoa ach I ho ho o co o a oa ach hof. F a aoa ach g : W oc a foo: Sho ha h ao a cao o a a ac oao o a c aoh c. Th h aa a a caca o o o ag ha h a ao c. Ao fo co oc. Co h aa a a a ag ha h a aoa c Co aac a ag ha ha ao a aoa c Sc h ha coc h oa ach fo h aoa ach a o ha gh a coo o o. Ao a cao o a c aoh c A o c ha h fo: a a a A Wh o of h ogh g ohg : a a A h a Th a o g ha h a o of - a ogh hch a ha h coo o a c foao. Ag ao g ho ag h ac o ha a g coffc. Fo aoa h a a ca a a a '. I oh o h coffc ao coa h gh. Co aa a a To a h cacao h faco o ao a oao. a: a Th cog Fa 99 h a ca co g h foog foa: h h

9 Ag h foa o a : Sc a ag a g o ach h ffao h aa a ao c. o ca h o co h aa fo h ho o. W ha: h h qo a c / fo a qo. So o ha ha h aa fo c h a fo fo ac oao h ca a ha h oa a a a cooo of h. Co If a h fo a ca a ha h oa oa a : M Th co oc ca caca ag h o h coo. W o ha ca ha a [ ]. So [ ] [ ] h h ach ha o a coo Th h co oc co: [ ] [ ] [ ] Sc h coo a a cao a aco ga ao h ca ao ca h oa ach ga h M foa a ach a.

10 5. g ao of a aga ach. G a aga ach of g h g ao foa fo h ach g a: ho ha h coo o a g a: [ ] Th a o of oao oc h qao hch ohh o a.. So ag a ha a o h o of h gog ga. Thfo h ao a o coca ha a agh g ch. Th c f o a a a of fg coo o o h ag.e a a of h a. U hch a cooa o h ag ach. Th ooa ao f a: Ao o f > < o h fco aa. Sc ca ac ah h qao fo g ao a ha a ga foa c a a g. So foa h g ao foa a: Th ca coa o o f a coo o of h oga coo o. O hg o h oog a h foa ha h gh ha h h f ha. O h gh ha of h qao ha a a: h Ho h ca ha h o f h coog o h f ha. I aca f ha h choc: - - o -. Th c coo o cfc h ao. So o h gh ha ca a:..... h g ca ha a qa o a o o h gh ha. Mg o h h foog h ao qc: Ga h o aa ogh h o of o * a ha ach a a aa. o ha ha of qa o o h f a gh ca h o h qa fo o g o ogh af. Eqag a ag h coo ha:

11 o og fo fg o a h o a : fa g a h h fo.

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