( ) ( ) Homework #2 Solutions

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1 He # Sli Chpe h pble deee h f ilbd e d d el cn 'll e iil elci piie dieci i / cn ben p cpen fll: c 8 / c 9 / 8 / 7 / dplceen cn e f bh dieci dplceen dieci / 9 / 87 dplceen dieci 7 / nide dieci dplceen e 87 9 n n 87 nle dplceen le hn h e ene ce iil elci l efe e hld ee dplceen dieci h pble le n bjec e flence i ne h fce i pjecile i e ed hizl dnce eled n ien i iil hizl elci eicl elci heih f hich i h hizl eicl i e depeen ech cn cide epel ie flih cn deeed f i ne hizl i Once f cn e i eqi i f eicl dieci deee iil heih l flih ie / Sbi h el n heih f hich h 98 / ificn fie pp heih hlf eble f fee fll ele hich i cle le le i e f ech ee hizl dnce eled 9 h pble eie M ien i bil di i bil peid Bece M bi nel cicl cn e fl f nif cicl i nif cicl i cenipel ie cene-ee ien hee bil bil di pble e 8 bil peid 7 d 8 h bil dnce ceed e peid diided peid π/ Ine ien qniie 8 h π π 8 h /h 8 / dieci l d cene h 9 In h pble he ee elci diffeence pii ec elci ec n diid ie e' che cde e i cene e heel pii ec l h nide 7 cicfeence diided il peid: π 7/ / e' e iil pii l p ie 7 j 'll e heel e cneclc ch i / fe heel ill he cpleed / h i i en i ill he dnced fl pii ill f dieci hile fl elci ill f dieci See fie l e: fll cedi if did n e cec n f In cpen f fl pii elci e c i j i 7 j c i j i 7 j / ee elci chne pii diided ie: i 7 j 7 j i j / b ee chne elci diided ie: i 7 j / i / i 7 j / nide ee elci / hich nel e nne elci / ee elci lle ce i den' e ccn ced ph flld p i heel

2 h pble le pjecile i e ed epe i hizl ne e nle hich pjecile lnched i heih i epei f hizl ne hen iil fl heih e eql i heih h cn f f hich h If d pice iil elci ec i cpen ee c n fie l i ce ppen efe f eqi fe he C e eqi le pble c Ine n h l epei f be f h n n h el eflec clicl eeicl ppe pbl nel l ec f ie dnce f ee fc h pble le pjecile i Y e ed eie iil hizl ccl ien ne e hich he fle Ie ccl el lel /h 7 / If h ne le hn 9 eped cn cclde he pbbl n If h ne ee hn 9 n he pbbl e he defleced pd f c dhield hich cide ficile fce hich ill iize h ne cn n e ne e hich he ld el fe l d Ine il nle 7 / / Bece pi h ld ccl i ne fc he fle 9 fe l iplie he l cel eie i hich he el e ne 9 le f iil elci h 9 98 / ± ± 9 / 7 /h 9 h i ne f 78 h pble iil nle ebche enel ce pjecile lnched f heih h cled Pble 79 h cpliced deii ill n epdce i hee b ed e el: c / h Pl lnch heih cliff: c / / 98 / 7 cn pl h : 98 / 7 n / c U qdic fl ne 9 If ed hd chen ne ld he en lihl lle 8 e: fll cedi f ep #78 79 h pble iil nle i ne f pjecile lnched f heih h Recll i cc hen deie d / d ze need n eqi ele e' e pjecile lnched f i i l eicl pii h cn ne f h n c e' ene h eqi lipl hh c def H / hich i heih ne' jec

3 ORC MOIO Mi e ieni - H c hh c XRCISS Seci Secd ieic ideniie: c c c he h pble pplici bjec e cidei le H hh c phicl qni ee c cld le f qdic fl b ill e e Ined ill lee eqi lie cn f h e deiie epec nide n d /c d eql ze nle cn U e f f nle ce Mi i ne / 8 - eqi e n eceed ed lii ch nide pdc decelei In ee deiie be e cefl ppl le le f el nen eqied ccelee - eble / ce 8defii 7 /e Ine h eic hld f ppei : e / i pp dnce d d hh H c VU i le ppen' dnce f lcie h diffeen ld IRPR d pble d B lcie ccelee : / / de lcie c VU h ce ielf If n e d / d f i ne e lef / H c / h n hh ice ni ni dnce epeced hee nle i ne ice efed ecep f h epe h pble l pplici enell bjec e hichb ld nl jec leel ee 7 le f If lcie ce pll ne cidei c phicl qni ee bpe defi ipc eqi jec be: pl i epei f 8 /deied id de h n hh n hh c Chpe bpe n eceed ed lii nide decelei hld - h H c H c c e ielf f cld ech i/h b ee lie eble defi bpe cnlcie n clcled f c h ele hee pene pll e hlfpee e ech i αld e U fc c α c α c α be eqi edce : he / VU e nide i IRPR epe h be pble l pplici bjec e c qni H cidei phicl ee plne / h - Chpe iplne / hll e n hizl eicl i nl ele O eqilenl n fce cncel nelec ednic hich ell fe plne l e n e el hld pene fil e p defi pene Wih iil /h 78 / i bpe ele / / plne 8 cn 98 eql pllel c nene ipc pene ee / f h h c 78 / / h Ine e 77 c qniie 77 VU Bece ele i/ de n d ccelee el plne need pp enie pe VU U ee eeed f pene i pplie lihl le nece cne c iil pe Wh If e leel h n en fnci e ze hn def bpe picll lld i c pp hich c p hfe ele d 98 / hen cceleed jec /7 / /h hppen ifi hld e leel enld? > 98 /? Seci ce Gi lif ien plne In fneed p e een' ed h f cicl i eqi ell pii 8 h ni e cen ce / el eble ce defii / e pble le ce l ni e e plne: pd lifll pided elci d ie dice ppie dieci iil e nle en pii ec deiie e / e l ed epl h ie ene epe / hen pe nen eqied ccelee - e / Chpe i: p l f lif: p In p plne hld f eqi f cenipel h pble le eic hich need RPR h pble le bjec ee pene Icn ze In pb plne clib e lide d di i hn fie elef ni dnce ie Ue h elihip en pp dnce - Chpe clcle eqied p pene ien ie piie le pble ee c ien ppil When e' n pd blnce : cnfil ee e e el hld f pene pene l p VU e ce Mi - cn n ele hee qniie dplceen elci icll hich cn epeed / ened n / ipc pene h cn / h c 98ee / p nide h f fc bjec e f 8 Sce i / / e ene e ni / hen pe ce ni - Ine ei qniie VU p c dnce ill ell : l hdinc elci h dieci peced e Cpih Pe dci ll ih eil ll cpih cenl e pi h eil ni ie eeed eeed el pene b When ee plne clib en f epdced n / n i f pblhe pei ni pe n h h p / /h 98 / / Iefe RPR pble le de i ee i idenif pl -8 Chpe h dbl qdple nide ee le Sl f nnn nide hich pcehip h ched bed' iil epeienced ccli ipn ee hen iplne hih bil l neep plne bi i lif n ppil Iie e dice dieci di iil elci ppie 8 fc If nn pl cn bed dplceen e h ll iil niceble; b lle h lle 9 i n 8 h i e h ib -8 e cped f ie Be 77 hn hich RPR le en' I h le f eic need 7 cn pii di pble cnpble eednce ec cpen ele hich ie peid e phicll enfl died de!i lle 97 eic n VU B e ip lecl cple / dnce elihip en pp dnce fce i pl deiie h n defii ll cn e ice b ce RPR h iee-dienil pble le clcl ec f n VU Recll f ppei B pefi pic cepd He RPR e cn 'll need eqi f ppl Seci IIIn ce' ee SI cp ien ppil Chpe i ne ni ec e cn / d n chne ie dd dd B hid p dplceen j f nece ech cn ene en' ce Once cnn n ele hee elci qniie / 8 Che cde e hich eeni p piie dieci He ce h f e 7 /h cn e 7 hich pecel le f Ven ppei ll cpih Bece e n cpei dieci elci cenl dieci hld eil cn 9 e h p e eeni ppil chne Cpih Pe dci Inc ll ih eeed peced e pi h h eil eill e ce epdced n f n en pei i f h pblhe p c dnce ld iil Ven l n efe pe h n edil le b cn ihn l pplied ffci / lenh efe bjec hld e f i > 8 hich p e hee p eeni cpei < p n le Ven he e 8 8 c fl elci ld ee h eped e/ nchned p eeble ien 7 cn e He le ce ccelee pble he M cepd n eic 7 /h h pble le bjec n ccep eqied 8 RPR h hpble pble del en diffeen p bjec e led I eci qniie He 7 / ien - bjec h VU Ine p 7 / 8 h dbl qdple ie nide efe ee ccelee c ce dnce ien el hee e hid ce Ue blc h Ue iil M ccli p ee ie- hid fie ppl e n ipn fc ee di hih en dplceen c fnci ien e hee dened fdieci lef ie ih hen hn l ccelee ld M ill need h - Chpe f clcle cepd n bjec h M pp ie ce piie- dieci p le f l f ppei eqi cn ed 7 h pble le en' f cn 7 p f i 7 / He ill n le ppl If ech ill penenl def ee c ien ppil B dci ip n eqilen lecl cple / lif ll - blc h eqie le i hich Cpih Pe Inc ihe eeed h eil peced e ll cpih cenl e pi h eil pblhe VU U pble e en c lif I epdced f n pei i f RPR / e-dienil le He hid e ed SI Recll f ppei B pefi pic p cepd lnch h n cld n epeience eched ien dnce p p cn if ppl ien i / / ificn fie 7 7 / / M hich e / 8 piie- dieci Gien p Che cde pplied pp 9 98 / 78 hich fc /M cn -bjec h cn p lenh eched clcled He 9 p h nel 7 b n den pil e en ed hle p ee c M h cn lif ie 98 / / / ie e hn n edil le b i h ld pplied f p h hid pplied l e nide fci iened p b 9 l M hn cneble ccine h pble 9 / fl ppie elci ld pp/ dieci e pp p e ih dieci e ldhe ble ficile l hizl e h pble le eic pe I VU Ine ien qniie RPR In h pble e ed b ien ceel SI ni leh ificn fie c dieci i pble le cl eic n b eqied 8 pplied cc hid blc eple dene iil eeed blc c pe en led e nce pcn el n f dnce eic eqi pp checed ccelee co ce ce ien ie ele fee ee fee-bd c blc Sceeeed blc efl cc ll he e di ih 9 nl pe fcl Pble In n ce cll efe / c fe 9dplceen / U hn cphe 9 fnci ie / ien cn pplied echh blc Bece epel: e-dienil pble cn dpene h ec ni pided h n ce nce cei eled p dnce ce Ue eqi cn ed le f ceel l f eeed piie- dieci epeced if ee ppl ice piie le 8 pd ec le d ec ee ie pp /8 fc z ppei C Recll ee c ien ppil dieci / ppen p ipl ee 9 / 8 / U c le / 7 IRPR h e-dienil pble He hid e ed VU cl eql bjec fce h iil eil cn e Cpih Pe ll ih eeed h peced ll cpih cenl e pi h eil dci dnceinc p ien p eched if ppl ien i / n f nen i f pblhe eeen epdced le ien een pble pei 98 / 9 pplll e e pe Pble ecep cde che ec dd hee eqi hid e e hich 9 I RPR h le eic / bjec ee e b U cei fc f ppei C nce pplied pble ie- dieci pble e pplied ee c pp i pp dnce e / de n epeience ih e clcle ee hn iil fh: pp 7 98 / z 9 / z Ine ien qniie He cpih cenl e pi h eil Cpih Pe dci Inc ll ih eeed h eil peced778 le fie c e ll i h ificn blc ee n dieci i f pblhe epdced nblc f en pei / 7 98 pp d en cen; clde O n f cn 9 9 c checed eqi p / eic ih cped / ce ec h e d en ll c fe 9 / 9 / U hich p eln n eln fe ll ceci Y ih eped e pi blc pplied pp bhe fen dnce eled eche h 9 c d en ice ccn n if le e f ld l ccele dieci / hich cclde p blc eeed ie- ld epeced ppl / blch n eece ce dieci en / USide n ppei C hen ee n eqilen h pble cide eni hle hle e PROBMS eeen le ien led pble een 98 Ue h cei nle ien e ccep hee e hid 8 Icceleed RPR h pble le eic e ed eqied If bjec n le h bjec hldf 9 cll iil le e e hee hle: eni f e i 'll h 9 h pble eic pecif ee e ccelee n ien n clcled ien ificn fie child' Cin ide cll bde e See n fie l ee hn eni f pll 'll h elci fl elci e clcle i pp dnce de epeience di ii defe cde e ee fie l cei en h / Bece nlh ni hich l Cpih Pe dci Inc ll ih eeed h eil peced e ll cpih cenl e pi h eil ed diec en pei SI nlh llien ni e f ee cni nbjec eql el eqilence fncn epdced f n en i pblhe ppei C h n lb eqilen lb /lb 9 eeed e ll cpih cenl cn ee Cpih ee Pe dci Inc ll ih h eil peced eclcle pi h eil epdced n n en pei pblhe f n bjec ien ii f nide iil pble hich cn e e ien iil elci 7 / i fl elci 8 / c i j / i / j ie If iil e nn cn bed el iil lle f e f h' fce nide n' pce i Cpih Pe dci Inc ll ih eeed h eil peced ll cpih cenl e pi h eil / e 9 epdced n f n en pei i f pblhe h ld if n e eh ill lide pce i B fc n fee fll pce i h/he "" ze h' ce n peil defii ed cle e elie bjec ihed n ld fl be n cle plced pce i Seci U Secd 7 h pble le eic cn elci e ed pche de i d led iil eic ee f eple n bd cn elci epeience n Ine h f cn ell Cpih Pe dci Inc ll ih eeed h eil peced e ll cpih cenl e pi h eil epdced n f n en pei i f pblhe

4 - - - Chpe Chpe Gien Gien p p pp p p cn cn lenh lenh eched eched cn cn clcled clcled Chpe He He 9: 9: pp ie ie en en p p ppe ppe di di p p eche eche nil nil bh bh e e e e VU nide nide p p eni eni ien ien He He pp hee hee ech ech p p hizl hizl cpen cpen pplied pplied ech ech pp pp pp p p diced diced ech ech l l dd dd e e eqi eqi enie enie e e 'll e l f eni hle hich he dened pp pp / U hizl hle e/ epeciel h h p p efe efe - Chpe pp / / Whee he ed piie dieci ih Sce hid e n ppl dnce h p eche 98 / / 8 ppl He He p eche dnce ci/eci pi Pl h be he h pp 9 c bh Sl f eni ide hle 9 c pd ccele hf ene ee hen helicpe ei / / d decele h ce nide eeed i l ee In / p pp p pp hid p 8 9 p / h pble le ppl pcecf h e eni e n eql ce if e hle ld ze i ldn' eqied chiee i ccelee c ee-bd h e ld One l e fie - pi le hnppl ce fe di cn diffeen ii ee l p eien p eien h l ccelee hee lee h ccelee bh hle h echii eqie h piie- dieci pd f fce e pp ce Ofen ie phic pble -nfe bjec ill e f h p b l p ee p c p pp iplici le Ue n ppii eni e ill ce ce ce ch ze-i enien Ceel ifif In pp pp ze neliible neliible n n pp epeced epeced lii lii pp Ceel In eql ce le bjec l pble cple ze pp I RPRh deee deee le cple ze c c de de eep eep pp pble e-dienil pble hid e ed 7 h pene ien pene l ien le l pplied le plne eni pe h f lide cn h cple ze dnce c i pene ce el cn h cple ze dnce c i pene ce el Me fee-bd di ii ee fie l hich he ned ll iil cn e h dnce decelei f f iil ze ze cn e ele ele h dnce decelei hizl e ech bjec cde e hee piie- dieci h c ce e nelec pe n fici hid n ih b pp dieci efe P P hid- p he eql nide c h ppelle ne ppelle h ccelee 9 / i l e piie qni e he he clded clded ie ie b piiec qni U U cn cn deie deie P hich cn e h ppl lii ze cple cple ze f fccele eqieen eqieen h pene pene le le hn hn ie ie lii VU d ce iplne lide lide ill l ll eni pe h/he h/he : : hich ill n ll lide f h h p i i cple ze dnce d ch e hich VU cple c c ce ce e e // ze dnce d ch e hich P h f h p P i U U lii lii cple cple ze ze le le Plne Glide Glide 7 i i /h h 7 /h hee-bd bjec 9 9 h P P h hee l / / b h p i eqli fll f hid Ine h h i i c ld h h c ld he he cple cple l l ee ee h' h' qie qie bi bi b b pice pice c c ch ipl ipl i c hclli p ee hih- den hih- clli ee den c c cn cn cpe cpe h h ch i i h P 9 / i h h bi RPR h n n pplici pplici en' en' ppl iplne eni P f pe 'e 'e ien ien f f ne ne i i needed needed Pp c dieci dieci ce e e VU ice f VU Pl Pl ien ien le le n h ie P i P dif dieci efe chice i dieci ce Mi - bi cpleel enel cld he ien P P h 9 / i 8 i ll / e cpih Cpih h Cpih Pe Pedci dciinc Inc llih iheeed eeedh heil eilpeced peced ell ll cpih cenl cenle e pi pi h heil eil epdced n f n en pei i f pblhe epdced n f n enh pei i f pblhe h h eble n epec f f le ce f' hee l fc ipl ni ec dieci h ne ch llinc Cpih h Pe dci ll ih eeed h eil peced e ll cpih cenl e pi h eil epdced n f n en pei i f pblhe Y ele ed ppl h n pene n ele cn ccde idele h pble He p ppl cble fe d hee ee ed d -blc e cneced led p iil hizl pplied pd e USIG WS WO S eni ien cpei p di l ledee e fie l eicl Mee fee-bd ii Bece dieci pble e- fe M heeni M nil l ele i pene p dienil ill ec e He 9 n pd M M l eqi cpen hich nide eeed f ech blc p piplie hee 8 8 M c ppl bh blc le f pplied XRCISS 9 U eni eesecd hen ele ccele pd > Seci eqie fe e ce Mi -7 pp p le p h pble ppl -dienil ii bjec Mi eil ee - e ed blc c ce ld ie- i ppl bh Cpih Pe dci Inc ll ih eeed h peced 9 e ll cpih cenl e pi h eil epdced n f n en pei i f pblhe ien pp p ii ee fie l Inclde fee-bd di i i e / l liied ele ccele pd p ie cpen f ppl f cn ec f l M pplied M f 8 MHe Sl h help 98 / / p 9 e lii ce If M n eni hld p chec cide 8 / n 9 Sbc pp ele 9 ild e 7- pene : pe Ine ele eqi be 9 M ld i h pplied dieci diced fee- di pp pe 8 pe c j 7 / i e el e ene cpen lii ii? e f 9 M pp p hich e 7 h h ccelee bh p l ccelee le blc If ene ce pplied blc hee nepe epeced n fifeqi be el f ee pp if epeced le n M hich n M n ele ccde 8 pene l lh epec 9 ll eble Mn h n ppepec p if p e ll ll if ele jccde hich l i hich cn eble pene i ize b depe c eeed 'e ed clcle n je i n ide hee je chne ene j 'e pble le 7 helicpe 7 7 h 7 / c plne' een i i e i i ed if f e cld ece plne 8blde e j d "e" iplie pe ie je ie deiie 'e ien n cdii plne e ccep led heep hid e e ped ih n clcle h h nide 8 8 p dieci helicpe eqi lie h diided M hid pene 'e ed f ee i i Iff plne e fl pd ce ld nc i n h lif h B ppie n 8 8 cneclc f i dieci i een eql ee ide helicpe ld : h hich n pd clled ene h If h nelec i ence h i l eicl c e n pplici h ω pble eqie dieni e M piie M helicpe fie helicpe : h efe eeed bjec ee - cpen pdce np Wih bjec S -: helicpe eeien d i e e ed je ie deiie h: h h 98 / h f d hee e n ificn ω cω e c h d ien 'll ie ed h M d d piie -cn clibeicll n VU Ine epei ien pble een f b h He i l je eql b ene eqi i leen ze ce elci ω / M den / 9 / i 7 / j i i j i 7 j i if i ence neleced efe d e If i je hih pene ffe hiplh h 98 / h Y'e h nide d f n cceleee p dieci ed nlze lpp 8 f -8: dece n cneclc f Chpe n d lpp e b If /M pd nl pd f c M 98 i / d 99 hich f en' eql : n lp ppen h nl n h In h pble 98/ il / n e ed nle i ble ch piie dieci pd bjec ie hee -8 if ididn' he lif f i lid en fce bleld fll e iil ne fce M c In h ce ih blence e b dn Bih / ee e fe VU f p el hn h Y he eed fihe cn clib p pd hee cecil je le cnn ple h nppen cle e B i he ce p / fele f il d If n cn pd piie dieci eql fce M l hm h pble de le He cpe eeed cn p 8 e If nle dece n p d: / n pd p cn e cneced ide--ide e--e If plne ddenl e pd eee plne ld feel led i e Y p cneced ide--ide pllel hee lpp ill feel cee ppen e nide ech p cbi bcip efe 9 diidl nlze d fe--e n cceleee py'e When ed p e cneced eie lpp eni e bh p eicl eeed h ch ppen e - fih le p l ech cenl p diee f ech dci e Cpih Pe Inc ll eeed h eilhee peced e ll cpih e pi h eil Cpih Pe dci Inc ll ih eeed hen eil peced ell cpih cenl e pi h eil epdced n f n pei i f pblhe ppen f e d el B n diffe ech diidl p; epdced f n en pei i f pblhe n pllel cbi He efe l ih d el ppe b hlf d el B chne eie cbi He d f n cceleee lpp RPR Y'e ed nlze 7 b I ppen d el C e If n n eicl ze i n plne cn eicl elci n d ih eped h plne h n eicl elci e epec leel b e n p defe effecie p cn eff In plne cn he ze eicl hile ill n fll edil pllel ce he p hee eie ce C epeience ell pllel cbi iffe hn eie cbi h eqie ee n

5 - Chpe U nle il hld - / hizl dieci b c ppl i 9 98 / 7 / 88 c c be hizl ice iliened nle de nbe depe bjec h hizl In cpen f eni j 8 c ile i j 88 h 7 ee epei f e pble pplied -dienil ii ene le cide lii ce hee cble ie cpleel hizl h cece ld 7 ie n eic i e ech b eni hld lpe e: Pble de cecl ec hi p pble e hn e iplf ep hich di ccelee l c ppl fee-bd eqied ii ee fie ll cedi pble e ie f hich cn hnle ic hich l Gien eeed e 7 cn e h l i e hi cle ech b In h ce e e pll e hn l fie: n n U - - U c R RR R R R ppl ie eqi e f dieci e f R R RR dieci: cde e d ih-h piie cde e e e d ih-h piie e ld hizl pll e ppie ech en e - plne hich // hich n ce e njhizl ze: j c hich c i98 98 // / n / eicl pll p 7 R e R 7 : eqilen en ie // // / 98 / e Sl cl eqi f h el ih-h h If ccelee p lpe h b b If ih-h ccelee p lpe 77 // Sl f ech cle / /98 7 // / 9 R 9 R 98 / 7 98 / 7 // ± When ih-h lle e ccelee d ice e hich ih-h lle ccelee d lef ice e When chille e hice ne e bd In lef ipl l ihich h e 98 / h ld epec h ld epec ch ie bd hee he chen piie qe he ed ce ie e h -dienil -dienil ppl cli ce le ppl cli Seci Mliple Objec If h lpe eicl pble pble 9 le hich h epec ce ie ld iedied e pe lid dn n ic nide phicl qniie ee e lid dn n ic nide phicl qniie ee e e ii - fee-fll Chpe pe 8 h pble iil ple e b e e e ied eqied b cplee p eqied b cplee p hh IfIn cli e e n i ld he en e bleclh n iped hld eql h pble phicl qni ee lpe cble IRPR che cde e hee e e pllel cpe e che cde e hee e e eni pllel lpe e e e b e pll ll he ll e he fee e f l eni ppl c e cble e ic pepeicl lpe ee fce ficile cli pepeicl lpe ee fie e ic fce cli fie l 98 / ficile / he e hn bleclh In hich ce elen c he eeni eni pd f Me f di ii h dn-lpe c pe e ni e nide If pe n ech e e ni e nide dn-lpe ll If pe n ech bleclh he n d hich el d : c i j hee i nle ee fie eni cli cpen eni e eql nide ppl eni el eql Wie nide f: / cli ppl h 98 / 8 dieci c ee hizl c c dieci ppl lpe dieci lpe eeed ppl eeed B e hizl eql hizl ie ee ze; b clde e ze; b hh ieie clde pi -ed diidl : n bleclh le i npplie e eni li diffeen dieci c ll If e cble de ech n c e e c n lhh eicl i hich ll e le epeience nl l elen f eni h ce bleclh e ble ficile: eni f hizl dieci 9 / n n eicl e: enied be hizl eliible e eql Cicl Mi Seci q eni bjec cnle f : h pble le cicl i eqied cenipell ccelee M nelec Bece e-dieni b b hh pp nelec dede Bece e e nn e-dieni d h dieni fe ec ni e ccd dieci dieni fe ec ni e ± ± ccd dieci Of Of Inc cnllclcle h eqied cenipell M deil Cpih Pe dci ih eeed eil peced e ll cpih cenl e pi h ce epe el dic i dieci piie ie ce ee epe el dic i dieci piie ie epdced n f n en pei i f pblhe ppei peced dieci d piie h dieci nide dieci d piie h dieci nide ih eil Cpih Pe dci Inc ll eeed h e ll cpih cenl e pi h eil eqied c pei c i f bil M ien epdced n f n en pblhe 8 ppei / i di / bil 8 8 i 7 h cble eni eqied / 7 // / / // 7 cicfeence cld l clcle elci 7 diid M bi i peid 8 7 π π 8 d / Ine h h ccelee dn lpe 9 pipi ccelee dn lpe 9 // h nide h nide pipi Which n liel e cce ie el pped pped b eni fe he clde 9 h nide dede cle b fe he clde h nide n? le clcli de pi M bi cicl clcl i bil hich n eli bi lihl ellipicl ee eped ppii In ppei efe e elible d clcli le / pbbl e cce ificn fie hh eeed p lpe fie eeed p lpe - Chpe ificn In h pble e ed h eqied eep cicl ph diif peid e π n / ped cli e n i ld he en Cpih Pe dci Inc ll eeed h h eil peced ee ll cpih cenl ee pi h Cpih Pe dci Incih ll ih eeed eil peced ll cpih cenl pi eil h eil fl ne f npei bjec deie epdced n n pei i f pblhe epdced f n f en n en i nif pblhe cicl i 98 / / nide ien : / e e e fc peid i ie ie f e eli cicfeence C π diided h 98 / 8 / π / Cb epei cn eien pi -ed ee diidl : π π O el dice f fied di cenipel eel ppil le ie ie eple if e e l ie f cplee e eli n 9 / cenipel l e ll e ll c hil cicle h ppl cenipel hh i Cicl Seci Mi eni eep h eni l lii e n nle hizl hn ie h pble le cicl i eqied cenipell ccelee M e d h ii e decid ple eicl ze U cn clcle eqied cenipell M f d -7 ppei blnced eicl cpen eni f hizl cpen eni i bil M ien eqied c /c cdi c jec c pid needed cenipel cenl e 8 pi h eil Cpih Pe dci Inc/ ll ih eeed h eil peced e ll8 cpih 8e i ppei / i bil dinle f ed eep f eni e epdced n f ni en pei i pblhe 7 lii: h cble eni eqied / / cld l clcle elci diid cicfeence M bi i peid b h nle c h π π d / Ine h c Which liel e cce ie cle el eni cn 8 / 9 M bi cicl clcl i 9 n? le clcli de pi bil hich n ppii In eli bi lihl ellipicl ee eped ne cle hizl i cn hil c ie ppei efe e elible d clcli le / pbbl cee B ch jec elci h cee cenipel e cce fficien In h pble e ed h eqied eep cicl ph elci hile ce ll e nif Y e ed di peid π / cicl i hich ce ll bjec e ccele ccd / deie fl f ne f n bjec nif cicl i l fi he p dnled eicl d n Y cn nide ien : / e e e fc e p hn ih dn fe n i dplceen d epec peid i ie ie f e eli cicfeence C π diided h cene ce cn h p hn f n chen pehp ch l π / pide eni hn fie l Cb epei cn eien π π O el dice f fied di cenipel eel ppil eple if e le ie i e e l ie f8 cplee e eli n e ll cenipel l e ll

6 f Ine nn qniie µ / 8 98 / µ U - el µ h ce cefficien fici ld el ee n h ie de nce ccelee dieci h dieci e ene fici h ce nl cncel h n c h ll nl hich h pble le de eic fici eic n ee ie bjec ee ie e ed h ch le i ld e hi dece lpe n-ze cefficien eic fici cped if e e n fici S fee-bd di che cde e hich piie- dieci n dn lpe ee fie l c ie e i nl n eeed lpe f de eic fici ie ppl i dieci h f he de f < ce l c ppe i h ce i c hee VU Ine epei f inl n ie- dieci j f f f µ n µ c c de c fici i eic hee he ed dieci fl eqli: { } ci c 98 / c n j n c/ i 7 / j - nide cn n clcle hich cn7 / ie lpe eic / dece / dieci h f f cn / Chpe n7/ hich e el f ple b eeed n ie n n c 98 / c n e e depeen f cde e ed He ch cde e cn e pble eie e le p ficile p qni h pble le blc n iil elci lid Cpih Pe dci Inc ll ih eeed h eil peced e ll cpih cenl e pi h eil ee dnce i el fe cplee p illf ppl epdced n f c n en pei i pblhe eic le h pble fee-bd di ii ee fie l ppl i dieci pp dnce cn clcled l eic n eic fici µ f n f f eqi be :Ife ie dece lpe n 98 / 8 9 U - hee he en piie qe If µ 7 µ c n VU Ine dnce eled blc ie dece lpe / / µ c / / 8 7 c ppil ee h e el h dnce eled eel diffeence ie dece lpe h ii ld cepd ficile fce In h ene cideu lii hee eqi be epe Cide c fe hich hizl c ice ni f fl i ie ee nne cn ce epec blc el defiel eeen epei f hen eed f eqi ce lihl le hn ee ie el cidebl fe hn n eeel f nne ed c n e pided pe In h pble c nif cicl i hn fici pided pdle ple e l ed h 98 nide / d life e If pli iddle en le che cde e e nif cicl i n c ccelee hh cnbned 7 c 7ble n 8 n cncel ech efe n ± ± -cpen / h eqie fici: f / dieced d l ce di - Chpe - Chpe p cpleel -dieci epled ple 9 h fici ic ce c i pepeicl h U Ine h le epei fi VU -cpen: -cpen: : f nide nide µ n ill le f cefficien fici VU Cpih Pe dci Inc ll ih h peced e cpih pi h eil eeed p c c pcenl c ll 7 c 8 8 c VU Bece nbned nl eql e ee ie eil 8 ce 7 7 p c c c p epdced n f n en pei i f pblhe Sl f cefficien fici neql ce nle de ice eni pe ech ide pc If e lined n lii h cld n n lii h cld If e lined n 9 /h / pe diffeen µ Whee if ld ld ecpleel cpleel ppie 8 8 n n Whee ppie /h 98 / e if 7 ld e ccep h pble e cicl i bjec ze ld ze ee el cicl ph B nlz c eni h h eepble eble f i fici cefficien Rb ie eni d ccee ill le e eed ed ped needed pble d eni pe needed h peid le cicl i deeed cn picll he µ ien b nice if ccee cefficien dp b pp bjec pp nnbjec ien ppl ech ee ie Bece ble ficile l di ii ii eeeni fiel l ppl dieci dieci PROBMS di fie ppl encne hizl plne ee le pe de n n c le fhen eni eni pe pe e eni pe pe eehee e e fe le le pe le f eni eehee pe fici i e hh e hle ble eni c ech f nide B h ie ficile lpe hich cn d ce de n ccelee h eni cncel de i c f cn h pble de f ple b h ie d i h he cde e h hizl eicl e l hn fie S fee-bddi 8 8 ppl : 8 R 8 R hich : ield pplied i n i n c jτ π R peid cicl cenipel b eni l pide f e i be eqi f n π R π R Cpih Pe dci Inc ll ih eeed heil cenl e pi h eil R peced R e τ ll cpih τ epdced n f n en pei i f pblhe B f l he B cb eqi b π R VU pplied pplied dieci dieci VU τ R π 98/ τ / ccli: i Bece τ R le fll epei f τ cn d fied ce fie e le peid di; le peid ii Wih eni ep l led e / ificn ificnfie h hee ie e / h e hee ie ehn eni deii be l pplie f nif cicl i If plled d hle ie R le pe le hn eni pe edced n i n le cicl lhh ill cenipel fl ice ice cee e nhl nhlll llcpen cpen eni enic c ce de n ppl ji eni ipl ee pll pp p e /Reicl dieci eicl dieci ji eni ipl ee pll pp p e RPR h epble ed clcle hizl ci pplied e e diffeen 8 I iilpble Pble ecep ecep nle nleech ech e pe h pble iil e pe e diffeen le/ficile plle illppl ppl le leh hpble pble ill Bece plle e le ficile eni cd ill e hh - Chpe fee-bd fee-bddi dif f pc pcee eel l nle nlee e pc Bece Bece pc pcde den ne eii bh bh hizl hizl eicl e hpc eni h pp f fll hizl le eicl dieci zeh h ppl eicl eicl dieci Cpih Pe dci Inc ll ih eeed h eil peced e ll cpih cenl e pi h eil dieci ze ppl dieci f cd be l plle ched f: epdced n f n en pei i f pblhe c c U le ien ci le : hizl hizldieci dieci i 8 i 98 / c 7 c h b % ee eble f n c pe ch c c needed cne e eeed cle le Ue e eqi eqi cn cnle lef f U h pble le hid nif cicl i bjec ee e lle-ce e e l ide e ed eeed ide p n lle-ce e el deee h ld hppen hld ide nbcle h el fee-bd di f ide p c ill e f lle ce ee Cpih Pe Pedci dci Inc llih iheeed eeed heil eil peced peced e cpih cenl cenl e pi pi heil eil ie ide h h c hi e nl de ee de Cpih Inc ll h llllcpih h epdcedn nf fn nen en pei peii if f pblhe epdced i bh hich e ph hi dn B pblhe e ppil peed icll he 9 n R hee 97 / R Sle h eqi f n hich eeed llece e ide eeed el ide e fee-bd di el eeed ide el n e hid eeed el ide e ee 97 / 98 / R n

7 Ine eqli f ic fici f µ n µ µ 98 / / 8 i eel ppil µ hich cefficien ic fici en b lle le µ ee needed eep b plce In lii µ ficile fce ld he fiel le He f µ fiel ic fce e 7-8 h pble le eic bjec ee Chpe if cn p if heel ic cc il ie e ed heel d n id il In h pble ld lie diffeence pe eed nh Cide ll heel e p cc e fee-bd di f ee fie ple eq l ppl heel f n i h cpen When h fce e h b i hh ple f peid d di cicle i pepeicl dnce : nifephicl R c hee R di hcn ii e neleced : f lide Y il depe lide: π / Ceqenl cenipel h hee he ed fc e ze dieci he eplicil ned nide ephize e e dieci fici dieci Uie- -9 π R c c de i ce i c dedci icinc fici eeed f µh cenl e pi h eil Cpih Pe ll ih eil n peced e ll cpih epdced n f n en pei i f pblhe d dieci Ine le be i pible dieced d e i eeqi e e l c i h heel lip cn ildieced n ed eic eeed nide ppiel cene h cle pp dnce Red fie l: n ple f h eq Rf µ n µ µ i pible µ Ine h el f pp dnce c nh ple 9 e n cenipel c cle i cle ed cl : On h eq e blnced p cenipel h i p diecl nide /h ppie h i S cle ed µ 8 / πr c98 e ill p fe hi c efe eq ed ill le hn nh ple ed pecene pp ie f π 7 p e π R % p 98 / 8 8 nh 89 / O el h ih e ple ce cenipel ze e B hld diffeence hdlie niceble hich enh f pene e c h pble le nif cicl ficil h 8 pble le nif cicl i bi l ch - Chpe bjec ee c he ih le ill p c ch len nenil e C b b pii fe bjecihi c he i e e ih cicl n pible He fici c eif h pble e-pecified: e e nnn f eqi cn ill eie ee b c ihee e i nl n ic f c n fe de i eic i n hn de h e bee e f cene fici lip f f fe pi de f f pi ic fici n bie eqi S lef p le le peqi feen fee-bd di b f ide eele fie l he chen : b pi c dieci dieci f cene C cenipel iened ie- c 9 pplied dieci U f b n f e f c :eqi µ dc leel nif cicl i ee Inepe cn e / c 9: ncen : 9 Wi cpen f h e pi pplied hee he ed f c µn U -9 dieci : f 8 98 / : f: n µ :f f eqi µ n eni: dc fnci b dil de ic fici c 9 c pii 9 bie h bc Cpih Pe dci Inc ll ih eeed h eil peced e ll cpih cenl e pi h eil f epdced n i pblhe feqenc h eli n en π f dnce ic pi fici cn φ pei cn pp dnce hich b Ue e eic φih pf clcle n di f e n le ppl nece ie b e lipel decide hich pi e Cpe cpi 9 pp dnce b pi c 8 c c / µ µ hee h ie ce i iened ppie elci pi Cpih Pe dci Inc ll ih eeed h eil peced e ll cpih cenl e pi h eil di 9 c 8 89 h bie h eqi epdced f n en pei i f pblhe c n µ f h n ee n pp dnce n di bc che ill ffice hld be ih le hn e ill lddecee e f bie i f lhh e if cefficien icfici f i i le e µ n di ih ill ne e ill pln! ill e le ifle decee epeced 7 need cpe I RPR h h pble pble le le nif ficilcicl i bjec ee blc nenil he dn i e cicle blle c lide p n cle bc n h hen cefficien eic fici he e i eni f cble µ n fl blc hlf i iil een f ide ee l ppl fee-bd fee-bd di di he blc p dn fie cle ee fie l U f hizl eicl dieci µn ppl blc p cle : c p : f p µ c n : n: c When dn hee he ed / f cenipel he epeience cn n le f : f dn : n c dn µ c Kn e epe iil pd fl d n e fc dn p n epei f cefficien eic fici Sl be eqi f nenil ± n ± 98 / n ± / Sppe blc lide p dnce i iil pd hich n de nide l hn blle ppiie ie n cepd ice ni e dicl ep/ he ned dn cne cicle Siill clc blc lide clc e dnce i en b dn dn cdii dnincp eeed iplie Cpih Pe dci ll ih h eil peced e ll cpih cenl e pi h eil epdced n f n en pei i f pblhe dn p n µ dn µ c n µ p µ c hich µ n Cpih Pe dci Inc ll ih eeed h eil peced e ll cpih cenl e pi h eil epdced n f n en pei i f pblhe