tuililtifffi[[uiltilil;

da B.Eng iFleclbli.s lnenreedu). tir! i\Irf$tt.crori.s rr{nu rsl

t:1':

I t, 1,,., , I ,,, , 'l

Id,T

: / i t

r ' f '1 t ' \4',r:r

Koch $nowflalsei , Jroch,{r{15tnq|r . t .daunof l . {oo do5lrr

d:s u rlclgc \or hoch r.ltti.{:,1,{i:L'lt5ttitao

. ly ! : ,6t06f1Lr. . . . . i t . I , . . J0, q. . l_ on .-

cgDtin,rous cuire irithoui tangcnl. .orstruciibte i.r

elen.ntary scomer[. rnfi offr:6aori]u6nrrir ul,l5rri'o

(Fractat) 66riudirEr.ofi !& iiri.:iarrj6Soir::Eiiarfi r6oi. ". r" : . . : i"""

-.ul:" l"ui, . . f i f i - i" " , . f . t^

L$.L. , ' - r ' " f , ' ar . . uvlEnf ia i . . :1r . . I " i rd ' "u,

i6ait.:mfi ofi e.e6ooduiou6rir

1r1 rroTlauwFrrdrur.rir

1:; rLl.:uclavdruaonrfiu : eirurrir 1fiu LLiruruq'e,L ' ra :" . . . , , :6r"6. :" i . r " : r iuf i " ; : . r . . r i rldX{1 i '1 ' tL l .vt .x" t

- . . . v1. i . 2 a U . .

a.arrvd rardruryirdfi onr-rLurrdrurvirfirs 1]vt. 1 . lllxnLr.:rd:

" ' . j !J UdLr e ' l r lq (2) r . ^ , ! " i . r . r t ; .

""ui z ntl. r , l . ,,", q , i...- o.d r"16 r." ri i

j .o. :vr . r . ' . r "u, . . , .v- . u.- i , ld ' iu, , raur',i6erardruMr t:i6ifi arfiLn6rur6Enfi r1icl.lyld rr rfil{ol

1140-ri.lut,t L

,ro!-r&lltxt?|nlj L

i .t ;irurLoiov;iue|r L7;; dfii']

oxrarurx;rr..rrr'ra! (4/3)L

Lrsloiriiradi I dru"lu:oriufir

fi a ciru [sin!6ir !1r (L/3)/3

6.!fiuol"l}]if lrrer MrfiiJ (4/3)rL

irrmuFr&iufioudrniiir n :o!l

rr'16'orra.,rrr:r:.1 1+7r;1'r d.ruiui'orauirdo

' r{r{ariud ;irdrilnuot n't6auugt 11u1,iiJ1 UOt',t'.]'I

ld1-lldl]tlllrliOV'.ltO0O 3.lO ) r3t-

ur-'rliluofi d

MY MAIHS s€pe.b.' ,oos 41

qdufi fir}iaurvierardrdufi riuirvlrfirr .q lurr:. i " , l

^ : l

- -v ' .

! r . r . Lt t iu i .14 " 1 - .vF

1 .d1 6l

r. r" r ' q in :J .. 1 . ? ^ . ". r ir" ir : . .uFulr

i1L"1 rr r )c"L1: . r 1 - . : . , r , , _.

l ix .a. t l t r tLr rN

* i o. r r .6 L . . t .v ' t , " ' : , i t iu, ' . ro. . '

o.r , i r ""u"" . r i ' l ! : J ? ' . t ,dv?' f i f iLt

ACt\ ' , / | C. '_.Lv " {LuC .1-4aluutdl '. . :

' . ' f i r_r"^ . , 12 F' uvF:qr iJ$l l . l i A v ' f ia

- - l.L i l l . . ' r . '1 J : i l . &.r4rU&[ci 6 'd [1{ t1 l^u dn

- ! ) i

.2, s. . " . . :n ' f , f iun A q. f"Judrd:1.0! i

- . t . :*

-1 .3 ja i \ .v ieL.d1 i l . . '1. \ . f . i r 'a. t 3{4r ' F ' r

- . i , i

- - : i

r .Gi : f l f r f ix , t A/q v-p l t \x. i j l .&rt _ '11 A l3A/9

+ 3(4)A/e1+ 3(4fAle3 +... +3(aI ' rAle" - s(n)' ;tlrdr,.lrfnMror S(n) $a n tlll-{ouuql lo

I r r a a, r r r ' 'af

l ls-A | | - r t . t t+. . .1

:J l ,1 9l ' ! ,1 LCI l

t r r l: . r r+l l f l l

Jl , ,*I e 'll

- - l r*115l

: !a5

LrtcdTiai" lttv&l ldLtdillTlu rdur,v lLotll^ufl" r0 L'

fi olll.rdeqrreiEin'illiu lunrvlxurlmdhiLn6o,sll?6aoq:

cl 5. t . l3 l f ld 'crGl 6l&t}vue1-:"pl iJuv).reo lu) l

, !^. ) . i 13, . t1.1?r lCLqo?1..1r161,c u60i "ev,rvoio t t r

,ra!qiu rfl rliurYur fi ruutlol otaao trar tilq tvfi ou

llit!" t.,slur" .ll1"$1-lL1,l"::. [x,qaL("Lt'.'x-. 1-|1-{ t

rn6ofir:6aorfluuviSnfinuaqfiererfllvfii fi.:era,rdrt r

i: 'r ' ,trf cJ r!irnir..nir dr.,duirfic dai;i., ". L

raru 5nfia 6a orrllodrsdirm.:'lri'jrqu{elrua:aro.r

a ri t: Lih r11r. "b3.f zoom),1 1!v, I F 6l {: 1\l I v srNu

- - - ' "o. . ro l i!v: .L.o& (vt01tc ' .ot [x.-5r-u.ndovt]r6

.jrryfi aut6atnislsurorFna.jGacl,l B,{rrurlvlJ't nlrdirn

rir.. ir. r:r,u r.:&,u",i "!;.d." nlri6ir '.c . i

o-rqqillj[a1ni (aE]'r.3uoEr4r1rluno v1.r.r6s) oa o tNlllu

q xr" "G1L nlo t l r L l ' . , l t lL lOl-Sl l&. ! lL i U1Ulq! . .J

I f lO " : -1r 1r rLA " l . "oO I I .LLh .J " ̂ l l .1 iddt l i i dd'

1'! P 1F ',tt.]?]l6]1',1'.] $',t Liluct I fi 11,{cT1-n1,ltcl r.ln

flF]L? LtvdN:.or Lr-.1lLa 1.1.1'.915. !1j"tr"161 I 1nloO'

f ic. . 1" 'n: , rv l r : d i ' r 'c \Sprp , ,e I r . ' .F) ' ' "x" .

." : ui r i6 rFt^,.n o :J 66 .""' ;- ' '*

qtri.t

d.r l Fi r ( , t , i6 : .sr i """ : r i i . i r " .arnvr

l l_.2 &, c , l lG lx" . .d 'L, . . . , ' t " : ?_11) l le

e" i r r l i o i ' i . iuo: t . i r . [ "" ' L i r" . , r \ , ] 1;vt l - '

- - : -qr.i/v'r &0t !ltn,,d)r.(t' $. | ,..lsl l,!r"tl ) ll'rral ivtllxl

:vur! avl6.i ovliru r,iiaavaq]"Lu;r1r:nno+r (n6..i) fi 2

" - : -r-s .!,r rs..LtuGlx" ."d1, q.n.c" d'-ltucd' tl J tl't Flll.l

o.)ail-ti I uct fll1 2 o

aanillufi':1rlllfl urilvin I fi 6 ;irurrorairlrflu:

- i - - . i

v l l ! 2 3J, ldo. l l : . :oa". : . lv t1&nusl ' !1111. '1"Ud 6l !dq

[ua1n]fln].lo alol't tit.l'tEl !&[vt]aunu llll'lllr..ln:..lal Lv

tt'to't1lflq9!.1! a n 6l$.l1tnl]0illlfl5.lul,lo'] El ql rtl lstl Lilu

? dl :0 ld '1 ' c J $U .dLft ' t@ 6l F [510L _!q1]al i6 t0

, : . l , . Z . \ A Zqov .r1] 0a!uld1.l1-|1.l r5lnqong !1"rt!q71o1n{ ].roflnlox^uu" i ' " ' , , r , r ; r t : r<, , r rT {L.{€i . lb l ; { i . ' } .11

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. - " - :

D

1 p . y..y1.t y. tL, C 6t!-L y.r ,qlr dr1":_[haU!]utro

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^ . . :

- q

^:f il0dr.]lrl11i3G]uLr.r!1fl.r'r0]16{ n- l\utr1 no to I a3 tl\o. ;

u x!'? 5.' s JtrL Lta 2 lJsr 6{.1.rYlt:01" Q6 lO il ii-tltllol-l

rl r.or .ardr rir, : iF d'ur. ' - 'rrru,i r,. l : i lrotds:-2. i61fl flr!OnSr1Uv1-NnO tO'l nluaqln!r?rlaf lJfl-rl4.!nouu dl

r!.\\01,!U 1,!, HI1{..1JU t, t e' " i?UYlT iO'v 1dl )r.ll1,lltUl

Ae@

rriil[ilfil|ifr\fi!1lr

42 MY MATHS sep,emb( 2oog

[ll

ITI r

(ndr.i) fiu'lsifir&r:oardarodluornro z fi616 r!'u6orrr: eirau6ar:rdar1#i"u . riatfjaofliuniailoiirrfiu r, "hia?nrd J fi6 "lulru:fidrrosluro.:rflu::lr:r .z fid L.; fio rirorar r,7. .fr'rmfiriufirdufi ,r = r,xl n'::dufirGanfiGra.rornrodaiuarfierodir nnmaarng ai-"o"inn r6n 1dufi err mdarlduir5n 1<iruru l7e, durfio

ir i6li$rorioTad (Topolosical dinensiox)

;irvSuilmr "

fi6 (fi6!6lyorolaE = .1 u.ro5lq

^.r . ' :r rrJdr tfd dL01 t1"? J tc n 8.9 Lv .nL 1]\Lr. J . .t

, [6 iu.rx"11i FmbFddi l8 oim"! , ion 2. 1rr e\ . " . ! '

Embedding dinension:3, ? y l r i Klein Bott le f i

Enbpooi lc. i i , Fn":oo = r l r r i l leetr : r r ' . . i . " .uI"

u lutn_'hxat"J *s KlFio Bolr ip 6)" I0"Lx"r , te, o

tl]]fl! r 2

n:'l_0d: 111xa1,l 1.tlu .64'o1a ildlLsll '511,1 | { rLll,l

r,lfr'r 1rr:' ."r,'. rr:r-uyf;rruJu r6ui r i6 :lu lfi

2 rc ,!\d. .r ' ' r.F .,ilx"fll.;."6 i.Fu - flr oqiuuirx-tt rrl

f l r 'u ia!q ' t 2 8ro l f iDnN.i l r ]1. . l . : " . . . .y1"." '{ : " i " .

.1,{01-rl x"fl atrlL[.uu";,J 1,"i. c h ui i. .-tii .. .-".. :.

- ) -3 'r ' ) r1^i0di . r . ' r . . ,ut" l r . ' {x_, d. " ; i r d. ) \" , t ly. . .?G

ntno 2l i l t1r ' .1 0' .1111J41.Ly&l ' ] te

tv - . alq6.rndf

- i, ta{ J?ra_r" r , , lJ! "1- l i iye.r ' toI l . r i tha:r : l1orrqol} i

renerfi 6adrqiuviardeit? qoqatl,ir'ldrata1t.: aErr.:nu

tgqnoiao [ciavaofififiMrn'! o fi] rflr-irqom Gsisiafiu

dr-{l,{o n9 tllutduo:1113.&9 [}1 1r I 0 T.iUL tAyV{Otl Ultr.t-

!fl u{adi rnq'lfi nirlrn5ofi err5oo16'rfu fi 6r6wanala 6to.rn6efirc6aorylrfi! i rwl:l'urflurfiu uoifiir:, rr6an z 1olo .l uu.;x". Foiier*_aq !t rd:-r'{L jtolr:lu nnolrFu

::u . ' loq.r l ,u. f l uof iuoi , rur"6f i q1e.vi aurfrudn6n::rordn rylr'l: uayuuuau.itfiu'lr]}itvu rr r r.r

fr fir;mn4rnroranrvil!onoruaiairufild.irrdurnSofi lcEaourqyfifiGelnnir I uqi1xi6.! : rfiuorurGofir,ir'hlei

Nsllllullll,li ll)1,lto& .&? &1-l€l {.!1.i S)7l [:.J!!1-ll tLrt![Str]

i6fierrfi fiirunialrala6fi qnh rfl didaleralyrfi ur:ri6

iotr6.,ll1l lflsi .re: ri.flr"ifl' r" a r..l5nriaf,ya ruff6 orfr

Haus. lor f f d imFtrs ion. Box-.ourt ing dimeosroo,

IDformation dimension, Couelation dinension ti!6aq

' ' r i ir.,u16o.rru-i r 1 nobi.f ir"nrn lrirq 'r

dndclr

,r run 1n ["d lLurt , c Bo\-.nurr:ng [t1{c.]du Ini

lta')nu

a:+r6r:ifiriucnr t fildrurarrfiurfin 1 urr e

ilo r, x r,'lrifi o riruorr6srfiufirrrfi anrlrqlfrlSs]lot

v : LxLxL firqnr]roridrfiou exsx€ lr1fia.:"lidixr1".,, t

' ,.ar,f i. io v }!f io F:.-qi ,"o{ri,F'.ra, "

riurvlrr'rfi 6riryouolaErfl u uasE.i e fi drfiaEr,irurudu

6rfioufi rouriloErff eirlrn urufi r:rac'lfrdudrfiaerriu

riurfin 1fir.r riudufirr5n 1 :it riarfiu Slro:Fn 19a- a A^'ll I tT'l vUO lyrJ1-{llJtlnaelndl' tflf tFt rl y&Ol tt n. t.

Lr dr ,vri! e hld .nr :-.: i ,r ai :n.iu..Fr1tu oe :.rilo:rjy:.r (ro6lnfianJri:.lirr,ro s) ufrrririruruncio.:rfia16yrfi6flr$,rac fi 6a r i nr:flcr r ir r?iufi6l'l6fio,irBox-countins rlinension (r,ria Nlinkowski-Boulieand

fir1fr N.1s1 unudurunriar#n lruro e 6o e>oir:rrqairr ui d o tr1#'r ls' fic rN n rincfi yr-

i "o-l rdr ru d lolfrf i"[ f N.(sJ ., i-sl rr. t /rr .Tnll iur d

]-lulUtral Box-.ouDtjry lOtttl?ltt S

.tvr d [6arnq) IrJd'&fiudrw.j r N,rSr -"L tf fr u6lrJfl]l

. .N.rs) i6:lT, i - - r ' r iJd k ruu6ro'v:_ln

F1450

t]ffi

,l/lY wIATHS sepkn|tr 2oos r13

l

. t t l

l :n 1, , \ - S] d l r )e):hk

l i , , r in \ , {Sl T d]nD lD € : h k

lnk- l in ln\ ,1S)d= - 'o -

li)D ln e

, , . hrk - l rN"(S)tD€

, , . I INJS)

a'rng drJn]rfi (i11 6'a.ifin?nirFir d a!60au

,1al: In , , ' r fr iuo .r 1o r .) d u d Srriu* ' trn

:^ X A' lq,.|4 Ilitr1flru.!Yl{ornLv!Mu lo n1rdtJn it{!no11-|,u0

.&c l l imlr $Jd. l1 l l . )v_lM.t V16_ d !&!O..s lo l l&OO

"- ^ i i . , : ! " , : . , qa.t{,n h.1l UU€IO1l IUHIO :q U!l1t l ['] llodeJn':1lnU l4i t{ lO

ruir odrfiaero.:niu'ir ,1 lsi,ir(fludolrflu,ir Tu!61 !t1

8.nofll{l t n dofl}Jlo aourJsl uJurylt $ |,2

l. 6!1-tv1,{110,![n60y].rtodo s

- :. fr r : t/9 ri1.r Nr,"(s) Idlrlrfilr 12. .l:

: ! ' , t l - t "o*

ln 12--T

h48ln27

4. fi € = 1/27 ! Nt ,{s) 16lYirftr 4s. . hr,r8

27

-l-17457

nr flIH.t14 r - 1/3' ivUU N, ,"(S) to,v rn:r

3x4"-r ni ia i di ;u

. . . l r r \ . {S)

"-o ln€

' ' lD(3 4" ' l

3.. . ln{3 a" r)

"-r nln3, . hr3 . . {D l l ln4

- n-- n ln3 "-r ntn3

=-\11618rtn3

c^c1 )-rnno14rrrnoolllr.{:ij11i.ir1t.t 1.26285 r..r61

fr e!h> { , .

d L1i r'-.r t4,

t .1

* lEcf - f d* " l f -

-*^? q-.r# q-r'\-; q-r^r*

. ln3-

3

1/3 l i r r Nr/3(s) '16'rz i r f i r : ,

44 MY ,{ATHS s"Bsse, 2oo3