The rabbit reproduction model. Partitions. Leonardo Fibonacci. Solve in integers. Recurrences and Continued Fractions. x 1 + x x 5 = 40 x k rk;
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1 V. Amchi D. Sleto Get Theoeticl Ies I Comute Sciece CS 5-5 Sig Lectue 8 Fe, Cegie Mello Uivesit Recueces Cotiue Fctios Solve i iteges = ; = = 5 ; 9 Ptitios Solve i iteges + + z = ; ; z Fi the ume of ws to titio the itege = ++=+ =+++=++=+=+ [X ] ( -)... ( -)( -)( -)...( - ) Leoo Fiocci I, Fiocci hs ecome iteeste i its wht the o The it eouctio moel A it lives foeve The oultio stts s sigle ewo i Eve moth, ech ouctive i egets ew i which will ecome ouctive fte moths ol F = # of it is t the egiig of the th moth moth its 5 8
2 F =, F =, F =F - +F - fo WARNING!!!! Wht is close fom fomul fo F? This lectue hs elicit mthemticl cotet tht c e shocig to some stuets. Chcteistic Eutio = F =F - +F - Cosie solutios of the fom: F = fo some (uow) costt iff - ( - - ) = iff - - = =, o = -/ Chcteistic eutio 5 must stisf: = ( hi ) is the gole tio =, o = -(/ ) So fo ll these vlues of the iuctive coitio is stisfie: - - = Do of them he to stisf the se coitio s well? F =, F =, + (-/ ) stisfies the iuctive coitio Ajust to fit the se coitios. =: + = =: + (-/ ) = = / 5 = -/ 5
3 Leoh Eule (765) Fiocci Powe Seies F ( 5 ) F?? Fiocci Bmoozlemet Cssii s Ietit F + F - - F = (-) We issect F F sue ege ieces ito F + F - sue Fom the evious lectue Chcteistic Eutio,,, ( )
4 ( ) ( ) Chcteistic Eutio, ( ), Chcteistic Eutio Chcteistic Eutio Theoem. Let e oot of multilicit of the chcteistic eutio. The λ, λ, λ,..., - λ e ll solutios. Fom the evious lectue: Rogue Recuece The theoem ss tht = is solutio. This c e esil veifie: = - 5 Chcteistic eutio:, 5 8, 8,
5 Geel solutio: 5, c c 8 8c, c -, The Gole Rtio Some othe mjos hve thei mste umes, lie e. I Comute Sciece the mste ume is, the Gole Rtio. S.Ruich Gole Rtio -Divie Pootio Rtio otie whe ou ivie lie segmet ito two ueul ts such tht the tio of the whole to the lge t is the sme s the tio of the lge to the smlle. AC AC AC AC AC A B C Gole Rtio - the ivie ootio 5 = Phi is me fte the Gee sculto Phiis Aesthetics ls cetl ole i eissce t chitectue. Which is the most ttctive ectgle? Afte mesuig the imesios of ictues, cs, oos, suff oes, witig e, wiows, such, schologist Gustv Feche clime tht the efee ectgle h sies i the gole tio (87). 5
6 Which is the most ttctive ectgle? The Gole Rtio Gole Rectgle Divi Pootioe Luc Pcioli (59) Pcioli evote etie oo to the mvelous oeties of. The oo ws illustte fie of his me: Leoo D Vici Tle of cotets The fist cosiele effect The seco essetil effect The thi sigul effect The fouth ieffle effect The fifth mile effect The sith ieessile effect The seveth iestimle effect The ith most ecellet effect The twelfth icomle effect The thiteeth most istiguishe effect Tle of cotets Fo the se of slvtio, the list must e hee Luc Pcioli Divi Pootioe Luc Pcioli (59) "Nith Most Ecellet Effect" two igols of egul etgo ivie ech othe i the Divie Pootio. C B A AC 6
7 Eig Recusivel Eig Recusivel A (Simle) Cotiue Fctio Is A Eessio Of The Fom: c e f g h i j... whee,, c, e whole umes. A Cotiue Fctio c hve fiite o ifiite ume of tems. c e f j... We lso eote this fctio [,,c,,e,f, ] g h i Cotiue Fctio Reesettio Recusivel Defie Fom Fo CF 8 5 CF whole ume whole ume CF = [,,,,,,,, ] 7
8 8 Poositio: A fiite cotiue fctio evlutes to tiol. Covese: A tiol hs fiite cotiue fctio eesettio. Eucli s GCD = Cotiue Fctio Eucli(A,B) = Eucli(B, A mo B) Sto whe B=... Eucli s GCD = Cotiue Fctio... Eucli s GCD = Cotiue Fctio... A Ptte fo Let = [,,,, ] = = [,,,,, ] = / = [,,,,, ] = / = [,,,,,, ] = 5/ so o. Theoem: = F /F - whe -> Divie Pootio lim lim F F lim
9 Hes-o Mgic covesio 5 m = + + How to covet ilometes ito miles? 5 = F 9 + F 7 + F F 8 + F 6 + F = miles Qutic Eutios A Peioic CF X = X = X + X = + /X X = + /X = + /[ + /X] = X... A eio- CF Poositio: A utic solutio hs eioic cotiue fctio. Covese: A eioic cotiue fctio is the solutio of utic eutio 9
10 Eucli s GCD = Cotiue Fctio Wht out those o-eioic cotiue fctios? Wht is the cotiue fctio esio fo?... π ?? Wht is the tte? Wht is the tte? e 6... Eve itiol ume gete th is the limit of uiue ifiite cotiue fctio. Eve itiol ume gete th is the limit of uiue ifiite cotiue fctio. Suose tht > is itiol = [ ;,,...]. whee >. Thus, Simill, fo ll....
11 To clculte the cotiue fctio fo el ume, set = the Wht cool eesettio! Fiite CF: Rtiols Peioic CF: Qutic oots A some umes evel hie egulit. Let us em ow o oimtios π π π CF fo Aoimtios We s tht ositive itiol ume is oimle tiols to oe if thee eist ositive costt c ifiitel m tiols / with > such tht c We will see tht lgeic umes e ot oimle to itil high oe.
12 Positive tiol umes e oimle to oe Let = / e tiol ume, with gc(,) =. B Eucli s lgoithm we c fi, such tht =. The Positive tiol umes e oimle to oe Positive tiol umes e oimle to oe Liouville s ume Simill,! m m m Tsceetl ume.... Liouville s ume e e tsceetl ume Wht out e? Ae s costt itiol ume ζ() Riem Zet fuctio tsceetl ume -???
13 Review GCD lgoithm Recueces, Phi CF Solvig Recueces Liouville s ume Stu Bee
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