Samenvatting Toepassingen van de Meetkunde in de Informatica

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Dnljp|b}lb qàb `b }lglbwàqlvq`b glnvmlbm `bm`lb `bw $Y ) u}llmu h`bju h`g} ykb ∖∖∖∖↖ p p`*;" Kbkhnngfl}ljlb} wàqlvq`b mk} `bw $Y ) u}llmu vled}u h`g} ykb ∖∖∖∖↖ p p`*;"

Llb enbyltl ylvqkclh`bg C ù enbylt `bm`lb ∏ p;( p6 ∄ C 1 p; p6 ⊆ C

Llb Enbyltl yllhdnlj ù llb yllhdnlj wkkvfà `bw $Y ) enbylt ù"

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h`gglb"

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9 Llbyn|m`gl cll}j|bm`gl pvnfhlclb

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9"; H`gg`bg ykb }wll hàbu}|jjlb

Wl yvkglb nbu ki ni glglylb 6 hàbu}|jjlb p; p6 lb p5 p=( mlql hàbu}|jjlb lhjkkv ubàmlb ni b`l}" M`}flvljlblb wl `b }wll u}kpplb" Ml llvu}l u}kp ù llb ublhhl h}lv m`l ml cllu}l glykhhlb kh jkb flkb}wn#nvmlb" D`lv yvkglb wl nbu ki ni ml }wll ncd|hhlbml vled}dnljlb ykb ml hàbu}|jjlb llb glf`lm glcllbdlfflb cl} lhjkkv" @bm`lb b`l} mkb ù lv gllb lbjlhl cnglh ajdl`m mk} ql lhjkkv ubàmlb" M`} }lu}lb wlmnnv }l }l u}lhhlb mk}1

ckt$t $ p;) ( t $ p6)) ≬ c`b$t $ p5) ( t $ p=))∧

c`b$t $ p;) ( t $ p6)) ≭ ckt$t $ p5) ( t $ p=))∧

ckt$ $ p;) ( t $ p6)) ≬ c`b$ $ p5) ( $ p=))∧

c`b$ $ p;) ( t $ p6)) ≭ ckt$ $ p5) ( $ p=))

$=0)

Ylvmlv dnlylb wl khhllb bng }l }lu}lb ni ml }wll l`bmp|b}lb ykb hàbu}|j 6 kkb ylvued`hhlbml jkb}lb ykbhàbu}|j ; h`gglb" Lb ncgljllvm" M`} mnlb wl khu ynhg}1

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9"6 H`gg`bg ykb llb p|b} }lb npqèd}l ykb llb yllhdnlj

Wl u}lhhlb llb yllhdnlj Y 3 p; p6 " " " pB " Dl} khgnv`}cl gkk} bk ni llb p|b} p ∄ `bw $Y )" M`} j|bblb wlmnlb mnnv dl} }lhhlb ykb dl} kkb}kh ubàp|b}lb }|uulb ml yllhdnlj( lb llb dkhyvled}l ylv}vljjlbml ykb|`}

dl} p|b} m`l bkkv nc dl} lylb wlhj p|b} jkb hnplb" @bm`lb m`} kkb}kh nblylb ù( ylvwked}lb wl mk}dl} p|b} `b ml yllhdnlj h`g}( kbmlvu h`g} dl} lvf|`}lb" M`} hlylv} led}lv pvnfhlclb np `bm`lb ml ubàm`bgb`l} np llb hàbu}|j( ckkv np llb dnljp|b} cl} ml yllhdnlj h`g}" @b mk} glykh jkb clb b`l} fluh|`}lb ni clb ykb mllh $`bwlbm`g.|`}wlbm`g) ylvkbmlv}" Wl qn|mlb j|bblb np}lvlb nc llb b`l|wl dkhivled}l }lblclb" M`} dn|m} led}lv ylvh`lu ykb wlvj `b" Llb kbmlvl cnglh ajdl`m ù( llb `bb`} u`ckhl vn}k}`l kkb}l fvlbglb np ml vled}l" M`l pku glhm} ykbki dl} dnljp|b}" Mkb ubàm} ml dkhivled}l m|u dl} dnljp|b}b`l} cllv lb wnvm} ylvmlv gl¤lykh|llvm cl} ml b`l|wl dkhivled}l" Wl lykh|lvlb ubàm`bglb dl} flu} cl}llb lylb}#mv`ylb mnnvhnnphàbkhgnv`}cl"

9"5 Ubàm`bg `b llb ylvqkclh`bg hàbu}|jjlb

D`lvfà cnl}lb wl bkgkkb ni `b llb ylvqkclh`bg ykb B hàbu}|jjlb c`bu}lbu }wll hàbu}|jjlb lhjkkv

ubàmlb" @bm`lb `lmlv hàbu}|j cl} `lmlv kbmlv qn|mlb ylvglhàj`bglb dlfflb wl b$b∖;)

6

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b6

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Jkb khhllb `b }wll m`clbu`lu glfv|`j} wnvmlb

Gllb nblylb#yn|m`gl ubàp|b}lb

Dl} llvu}l jkb lylb}|llh nbglmkkb glckkj} wnvmlb mnnv llb vn}k}`l |`} }l ynlvlb" Dl} wlvj} `b }wllu}kpplb1

Unv}lvlb np flg`bblbml t#wkkvml

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llb ubàp|b}

dl} }nlynlglb ni ylvhk}lb ykb llb hàbu}|j `b ml glglylbuu}v|e}||v

Ylvmlv wl}lb wl mk} llb hàbu}|j lbjlh jkb ubàmlb cl} qàb nbc`mmlh ajl fnylb# ni nbmlvf||v" U}lh b|mk} wl wl}lb mk} glhm} fà }wll hàbu}|jjlb l lb i 1 l 2y i lb i 2w l lb y 8 w" Mkb ∅q qnmk} y 8 q 8 w"Wkkvfà nbmlv t 3 q l lb i f|vlb qàb" Khhl }|uulbhàbu}|jjlb cnl}lb m|u ynnvm`lb glu}np} qàb" Dl}ynhu}kk} m|u nc lbjlh np lylb}#p|b}lb bk }l gkkb ni f|vlb cl} lhjkkv q|hhlb ubàmlb" Wl }lu}lb m|ulbjlh wkbbllv llb hàbu}|j ke}`li wnvm} lb wkbbllv dl} pkuu`li wnvm} ml f|vlb ykb dl} }nlglynlgml ni dl}ylvwàmlvml hàbu}|j cl} lhjkkv" Mkkv lv ckt`c|c 6b ykb mlql lylb}u qàb" $@lmlv hàbu}|j ylvuedàb} lbylvmwàb} uhled}u ½l½lbckkh) lb dl} unv}lvlb `b ml u}v|e}||v $KYH# ni Vnnm#Qwkv}#fnclb) `b N $hng$b))jkb j|bblb wl fluh|`}lb mk} dl} khgnv`}cl `b N $b · hng$b)) wlvj}"

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clh`bg

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yllhdnlj" Wl j|bblb mlql ncd|hhlbml flvljlblb cl} lbjlhl cl}dnmlb" Dl}khgnv`}cl ykb Gvkdkc

niwlh ml Gvkdkc uekb( dl} khgnv`}cl ykb Akvyù niwlh ml `bpkjcl}dnml" Niwlh cl} llb ylvmllh lb

dllvu khgnv`}cl mk} wl `b cllv ml}k`h flupvljlb"

7"; Gvkdkc uekb

Dl} gvkdkc uekb khgnv`}cl gkk} khu ynhg}" Wl unv}lvlb khhl p|b}lb fldkhyl llb}al $m`l dl} vlilvlb}`lp|b}ù) np vn}k}`l ykb vled}u bkkv h`bju" M`} fl}ljlb} m|u mk} p` llvmlv glunv}llvm wnvm} mkb pa `bm`lb mlpnhk`vl dnlj ykb ∖∖↖ p? p` jhl`blv ù mkb m`l ykb ∖∖↖ p? pa" Dl} vlilvlb}`lp|b} ù llb p|b} ykb ml enbyltl lb wnvm}glynbmlb mnnv fàynnvfllhm c`b`ckhl #ennvm`bkk} $`b N $b)) m|u" Ylvynhglbu fn|wlb wl llb u}kej wkkvvlilvlb}`lp|b} pj np glp|ud} wnvm}" Ylvynhglbu p|udlb wl lhj nfale} `b ynhgnvml ykb unv}lv`bg np mlu}kplh" Ykbki dl} cnclb} mk} lv mv`l lhlclb}lb ni cllv llb lhlclb} qàb ykb ml enbylt ncd|hhlbmljàjlb wl ni mlql llb dnlj bkkv h`bju ckjlb" M`} ù }l }lu}lb mnnvmk} ∖∖↖ p6 p; × ∖∖↖ p; p? ≭ ? cnl} qàb" @bm`lb

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7"6 Cl}dnml ykb Akvyù

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7"5 ylvmllh lb dllvu khgnv`}cl

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7"5"; Nbmlv# lb fnylbfv|g

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4 Bkfàdl`mupvnfhlclb

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4"; Ynnvfllhmlb ykb Bkfàdl`mupvnfhlclb

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4"9 L|eh`mùedl C`b`ckhl Mnnvhnnpfnnc

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Kkbglq`lb dl} llb khu lb uhled}u khu ù cnl}lb wl }wll m`bglb flwàqlb1

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Jkb ml mllhylvqkclh`bglb nnj khhllb ckkv flgvlbum qàb"

6" U}lh mk} dl} Ynvbn`#glf`lm flgvlbum ù( mkb ù p` gllb dnljp|b} ykb ml enbylt ncd|hhlbml1 U}lhmk} m`} wlh qn ù" Ylvynhglbu flu}kk} lv m|u llb vkkjhàb h kkb ml enbylt ncd|hhlbml m`l mnnv p`

gkk}( lb wkkvfà khhl kbmlvl p|b}lb kkb mlqlhiml jkb} h`gglb $u}lh d`lv h`bju( vled}u ù kbkhnng)"Wl mlb`lvlb ml hnnmhàb j np h m`l mnnv p` gkk}" Khhl p|b}lb np j vled}u ykb h h`gglb( h`gglb plvmlb`}l mèd}lv fà p` mkb fà llb kbmlv p|b}" Dl} Ynvbn`#glf`lm ncyk} m|u qljlv dllh j( mkkv jbkkv nbl`bm`g hnnp} ù dl} glf`lm m|u nbflgvlbum"

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0"6 Glglylbuu}v|e}||v

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66



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― _n`b}lv bkkv }wllh`bg $Ynvbn`#dkhiqàml)

― _n`b}lv bkkv dl} Ynvbn`#glf`lm

― _n`b}lv bkkv ynnvgkbglv ykb ml dkhiqàml $Ynvbn`#dkhiqàml)

― _n`b}lv bkkv npynhglv ykb ml dkhiqàml $Ynvbn`#dkhiqàml)

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0"5 Enbu}v|e}`l ykb llb Ynvbn`#m`kgvkc

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65



Wl ckjlb glfv|`j ykb llb lt}vk l`glbuedkp1 lhj p|b} np ml Ynvbn`#qàml u}lc} np llb glglylb cnclb}nylvllb cl} llb fvlljp|b} ykb }wll gnhvnb}lb"Flwàu1 Wl u}lhhlb llb Ynvbn`#qàml }|uulb ml Ynvbn`#flf`lmlb ykb u`}lu p lb {" Lb wl ml`blvlb d`lvllb p|b} | m`l np ml Ynvbn`#qàml h`g}" Fàglynhg glhm} p| 3 |{ $mlb`}`l Ynvbn`#qàml)" Wl mlb`lvlbu khu dl} ubàp|b} }|uulb ml mnnvhnnph ab lb llb hnnmhàb np ml mnnvhnnphàb m`l mnnv | gkk}" Wl wl}lb mk}np llb glglylb cnclb} `b ml }àm $ml mnnvhnnphàb ù `cclvu ml lb`gl ylvkbmlvhàjl) p| 3 |{ 3 |u

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Ynhglbml l`glbuedkp ù ckjjlhàj |`} ml ynv gl hlccku }l flwàqlb1 Np `lmlv nglfh j nglbfh`j flyk}dl} gnhvnb} ckt`c|c 6b ∖ ; fnglb" Flwàu1 Ml lb`gl ckb`lv nc cllv mkb b#fnglb }l fljnclb ùmnnvmk} llb kbmlvl fnng llb fnng `b }wll uph`}u}" Fà dl} }nlynlglb ykb llb u`}l( jkb ml b`l|wl fnngckt`c|c ½l½lb fnng `b }wll b`l|wl fnglb uph`}ulb" Lb ylvdnng} m|u dl} kkb}kh cl} 6 $; lt}vk fnng( lbml l`glb fnng)" Ncmk} ml fnng ykb ml hkk}u}l u`}l m`l wl }nlynlglb b`l} cllv jkb} gluph`}u} wnvmlb( ùdl} ckt`ckkh kkb}kh m|u 6 $b ∖ ;) * ; 3 6b ∖ ;"

0"5"6 Dl} khgnv`}cl

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6=



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Enbevll} wlvj} dl} flwàu m|u khu ynhg}1 Wl j|bblb dl} c|ul|c npmlhlb `b mv`ldnljlb $m`} jkb fnylbm`lbnp ylvued`hhlbml ckb`lvlb)" @b mlql mv`ldnljlb glhm} mk} `bm`lb llb flwkjlv np llb glglylb p|b} u}kk}(dà ml cnglhàjdl`m dlli} nc ml }wll kbmlvl p|b}lb }l q`lb( lb fnylbm`lb lhj p|b} `b ml mv`ldnlj jkbq`lb" Ylvynhglbu pkuulb wl llb jhl|v`bg }nl np ml gl}v`kbg|hllvml gvkil" M`} jkb kh}àm cl} mv`l jhl|vlbflvl`j} wnvmlb" M`} jnc} ncmk} `b mlql mv ldnlj khhl p|b}lb cl} lhjkkv ylvfnbmlb qàb" Fàglynhgjvàg} `lmlv ykb ml p|b}lb llb ylvued`hhlbm jhl|v" Llb kkbglhlglb mv`ldnlj dlli} ckt`c|c 6 p|b}lbglcllbuedkpplhàj $kbmlvu qn| dl} mlqlhiml mv`ldnlj qàb)" Fàglynhg ù lv bng kh}àm llb yh|ed}#jhl|v nylvykb ml 5" Mnnv np m`l ckb`lv mv`ldnlj plv mv`ldnlj `b }l jhl|vlb ù ml gvkil kh}àm enbuù}lb} gljhl|vm"Fnylbm`lb wl}lb wl mk} kkb dl} l`bml ykb ml jhl|v`bg dl} ylvued`h }|uulb ml cllu} ynnvjnclbml jhl|vlb ml c`bu} ynnvjnclbml jhl|v ; ù" Fàglynhg j|bblb wl |`} dl} m|`ylb#dnjpv be`pl qlgglb mk} `lmlvljhl|v b.5 ni b.5 * ; jllv ynnvjnc}" Lv ù m|u kh}àm ml glhlglbdl`m nc dl} jhl|v ykb b.5 }l j`lqlb"

Mkkv lhj jhl|v `b `lmlvl mv`ldnlj llb jllv ynnvjnc}" Fl}ljlb} m`} mk} lv `b lhjl mv`ldnlj llb flwkjlvu}kk} $fà kkbgvlbqlbml uncu mlqlhiml flwkjlv) m`l ml mv`ldnlj jkb flwkjlb"

69



;; Vnfn}èk

Npclvj`bg1 M`} mllh ù nplbfnlj

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;;"; _kmphkbb`bg ynnv Vnfn}u

Wl fluedn|wlb llb ncgly`bg m`l flu}kk} |`} d`bmlvbùulb $m`l llb cnglhàj nbmnnvmv`bgfkkv glf`lmev`l¤lvlb)" Ynnv ml llbyn|m fluedn|wlb wl lbjlh yllhyhkjjlb" ylvynhglbu fluedn|wlb wl llb vnfn}"Mlql vnfn} dlli} nnj llb yvà }l j`lqlb ynvc" Wl qnljlb llb $jnv}u}l) pkm nc ykb llb p|b} p bkkv llbp|b} { }l gkkb" M`} ù b`l} bnnmqkjlhàj cnglhàj"

Wl uph`}ulb dl} pvnfhllc np `b ylvued`hhlbml mllhpvnfhlclb m`l wl ½l½lb ynnv ½l½lb nphnuulb( mlql qàb1

jnv}u}l pkm ynnv llb p|b}vnfn}

jnv}u}l pkm ynnv llb enbyltl yllhdnlj $gllb vn}k}`l)

yvà pkm ynnv llb hkmmlv $vn}k}`l cnglhàj)

Kbmlvl glkykbellvmlvl pvnfhlclb wnvmlb b`l} fluedn|wm"

;;";"; Ml _|b}vnfn}

Wl gkkb lv ykb|`} mk} ml vnfn} llb p|b} ù( lb m|u gllb `bdn|m.npplvyhkj}l dlli}" Wl fluedn|wlbd`bmlvbùulb khu c ylvued`hhlbml b`#yllhdnljlb" Cl} `b }n}kkh b dnljp|b}lb" Wl j|bblb m`} pvnfhllcnphnuulb mnnv llb qèd}fkkvdl`mugvkil }l fn|wlb m`l ml dnljp|b}lb ykb ml d`bmlvbùulb p` flyk} lb dl}flg`b# lb l`bmp|b} p lb {" Lv flu}kk} llb ylvf`bm`bg }|uulb }wll p|b}lb y` lb ya `bm`lb y` ya jkb q`lb"Mlql enbu}v|e}`l jkb glfl|vlb `b khgnv`}clb `b N

b5

( N

b6

lb N $b · hng $b) * l)( cl} l dl} kkb}kh|`}l`bmlhàjl lmglu" Ylvynhglbu pkuulb wl dl} jnvu}l pkm#khgnv`}cl $ni dl} khgnv`}cl ykb Màju}vk) }nlnc dl} jnvu}l pkm }l y`bmlb" Màju}vk‐u khgnv }cl wlvj} `b N

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fàglynhg ù m|u dl} }n}khl khgv`}cl

N b6

"Nc ml a|ù}dl`m ykb dl} khgnv`}cl }l }nblb u}l|blb wl np llb hlcck mk} qlg} mk} dl} jnv}u}l pkm llb

mllhpkm ykb ml qèd}fkkvdl`mugvkil ykb ml dnljp|b}lb ykb ml yllhdnljlb lb flg`b# lb l`bmp|b} ù"

;;";"6 Ml Uedàivnfn}

@bm`lb ml vnfn} llb uedài ù( ù dl} cnglh aj mk} mlql bl} nylvkh cllv mnnv glvkkj}" Fnylbm`lb qàb wlb`l} cllv gl¤Ļb}vluullvm `b dl} jnvu}l pkm( ckkv llb pkm `b dl} khglcllb" Wkbbllv wl llb uedài $ni llbkbmlvl npplvyhkj}l) mnnvdllb d`bmlvbùulb cnl}lb jvàglb j|bblb wl m`} cnmlhlvlb khu llb p|b}vnfn}wkkvfà ml d`bmlvbùulb |`}glql} qàb" @bm`lb dl} nfale} `b jwlu}`l b`l} jnb pkuulvlb fl}ljlb} m`} mk} lvmkb gllb mnnvgkbg cllv ù `b ml |`}glql}}l d`bmlvbùulb"

M`} mnlb wl cl} fld|hp ykb ml C`bjnwuj` unc" Ml C`bjnwuj` unc ù ml unc ykb }wll npplvyhkj}lb

K ⊜ F m`l khu ynhg} glmlb`llvm ù1

K ⊜ F 3 zt * xt ∄ K( ∄ Fs $<4)

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K 3 z p ∖ txt ∄ Ks $<0)

Lv flu}kk} llb pkm }|uulb u lb } `bm`lb ml }wll p|b}lb `b dl}qlhiml encpnblb} ykb ml b`l|wl d`bmlvbùh`gglb" Mnnv ml qèd}fkkvdl`mugvkil kkb }l pkuulb( lb e`vjlhfnglb }nl }l ynlglb" J|bblb wl llb khgnv`}cl

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67



;;";"5 ]vkbuhk}`l ykb llb Enbyltl yllhdnlj

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llb d`bmlvbù flwllg}" Qkh m`} vlu|h}lvlb `b ml |`}glfvl`ml d`bmlvbù khu llb hàbu}|j lylbwàm`g cl}mk} ykb ml nvg`blhl yllhdnlj" Mnnv mlql hàbu}|jjlb fà }l dn|mlb( lb wkkv bnm`g $f a dnljp|b}lb)}wll hàbu}|jjlb }l ylvf`bmlb cl} llb mlvml( lb cl} dl} ylvwàmlvlb ykb ubàp|b}lb lb }l ylvykbglbmnnv dnljp|b}lb" M`} j|bblb wl qllv lste`¤lb} mnlb mnnv dl} npu}lhhlb ykb llb u}lv#m`kgvkc( m`} ù llbglglylbuu}v|e}||v m`l ml vèd}`bglb ykb ml hàbu}|jjlb ykb qnwlh ml d`bmlvbùulb fàdn|m} $cl} m|`mlhàjnbmlvuedl`m }|uulb vnfn} lb d`bmlvbù)" @bm`lb wl fà llb d`bmlvbù llb flpkkhm dnljp|b} dlfflb mkbwl}lb wl mk} wl khhl hàbu}|jjlb cnl}lb }nlynlglb }|uulb ml }wll nylvllbjncu}`gl hàbu}|jjlb `b dl}u}lvm`kgvkc( m`l ykb ml vnfn} qàb" Ylvynhglbu ml}le}lvlb wl phkk}ulb wkkv ml b`lwl yllhdnlj b`l} cllvu`cplh ù" Mlql gnn`lb wl lv|`}"

;;";"= Encphlt`}l`} ykb }vkbuhk}`l ykb Vnfn}u

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N

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;;";"< ]vkbuhk}`l lb Vn}k}`l

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d`bmlvbùulb kidkbjlh aj ykb ml vn}k} l ykb ml hkmmlv" Ml vn}k}`l ykb ml hkmmlv fluedn|wlb wl khu mlq#encpnblb}" Wl fn|wlb cl} kbmlvl wnnvmlb 5M d`bmlvbùulb" Mlql $y`v}|lhl) 5M v|`bc}l bnlclb wlml enbg|vk}`l#v|`c}l" M`} j|bblb wl mkb nphnuulb" Ynnv led}l 5M nfale}lb m`l `b ml mv`l#m`clbu`luynnv}flwlglb lb j|bblb vn}lvlb cnl}lb wl m|u llb 9M v|`c}l fn|wlb+

Lv flu}kkb nnj flbkmlv`bgukhgnv`}clb1

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Khgnv`}cl glfkullvm np Ynvbn`#m`kgvkcclb $ni vl}vke}`l)

;;";"9 Elhhlb mlencpnu }`l

Fà llb elhhlbmlencpnu`}l uph`}ulb wl ml enbg|vk} l#v|`c}l np `b elhhlb" @b qn‐b elh ù ml pkmphkbb`bgllb llbyn|m`g pvnfhllc $nc m|u ykb lvglbu `b ml elh bkkv lvglbu kbmlvu `b ml elh }l gkkb)" Ynnvllb flpkkhml nv`lb}k}`l ykb ml hkmmlv $dnlj) qàb elhhlb cl} lhjkkv ylvfnbmlb" Wl flued`hhlb m|u nylvllb enbble}`y }l`}ugvkil wkkvfà lv llb ylvf`bm`bg flu}kk} `bm`lb elhhlb lhjkkv vkjlb" Wl j|bblb fàml fn|w ykb ml enbg|vk}`l#v|`c}l jv`}ùedl dnljlb fluedn|wlb $dnljlb ykbki wkkv lv llb mnnvgkbg}|uulb d`bmlvbùulb cnglhàj ù)" M`} qàb ml `b}vluukb}l dnljlb lb mlql qàb mùevll}" Mnnv ylvynhglbullb jnv}u}l pkm khgnv`}cl }nl }l pkuulb y`bmlb wl llb jnv}l vn|}l ykb u bkkv }"

;;";"7 Vl}vke}`l#cl}dnml

Ml vl}vke}`lcl}dnml ù glvèd} np pvlylb}`l" Wl enbu}v|lvlb llb $pul|mn) Ynvbn`#m`kgvkc" M`} Ynvbn`#m`kgvkc flyk} dl} p|b} t `bm`lb m`} fl}ljlb} mk} `bm`lb dl} vlilvlb}`lp|b} h ykb ml hkmmlv qèd qn|fly`bmlb np t( dl} l{|`mù}kb} ù cl} }lbc`bu}l }wll d`bmlvbùulb" M`} ynvc} llb jnlvufkkb vnbm ml

hkmmlv" Cl} m`} khgnv`}cl pvnfllv} clb ml hkmmlv qn ylv cnglhàj ykb `lmlvl d`bmlvbù }l dn|mlb"

64



;;";"4 Vljlbencphlt`}l`}

Ml khglclbl vlglh ù mk} llb fnylbgvlbu ynnv llb flwlg`bguphkbb`bgupvnfhllc wkkvfà ml vnfn} m yv`# adl`mugvkmlb dlli} glhàj ù kkb N

bm · hng $b)

;;"6 Flwlg`bg ykc llb vnfn}kvc

Wl w`hhlb llb vnfn} kvc cl} b u}kbglb lb m|u b*; uedkvb`lvlb flwlglb qnmk} ml kvc bkkv llb glglylbp|b} flwllg}" D`lvfà ù dl} llvu}l uedkvb`lv yku}" M`} wnvm} nnj ml uedn|mlv glbnlcm" Dl} khhlvhkk}u}lp|b} bnlclb wl ml dkbm" Wl gkkb |`} ykb llb gl¤Ļmlkhùllvm cnmlh( mk} m|u gllb flplvj`bglb `b mluedkvb`lvlb `cphèllv}" Lb fnylbm lb qàb lv gllb d`bmlvbùulb `b ml ncgly`bg" Khynvlbu dl} pvnfhllcnp }l hnuulb flupvljlb wl lbjlhl l`glbuedkpplb ykb ml vnfn}kvc"

;;"6"; Flvl`jfkkvdl`m

Wl u}lhhlb llb vnfn} cl} b u}kbglb wkkvfà ml hlbg}l ynnv `lmlvl u}kbg h` ù" Wl yvkglb nbu ki ni mlvnfn}kvc `b u}kk} ù llb p|b} p }l flvl`jlb" Lb `bm`lb m`} jkb q|hhlb wl ylvmlv flupvljlb dnl m`} cnglhàjù" @b ml `mlkhl wlvlhm d`lvfnylb gluedl}u} ù m`} llb llbyn|m`g pvnfhllc m`l c`}u pvlpvneluu`bg `b N $;)glmkkb jkb wnvmlb" Ml pvlpvneluu`bg qlhi ylvlù} N $b)" @bm`lb mlql `mlkhl wlvlhm b`l} flu}kk} ù dl}

pvnfhllc B_#dkvm"

U}lh mk} ml vnfn}kvc |`} uhled}u ½l½lb u}kbg flu}kk} ù dl} hngùed mk} dà khhllb ml p|b}lb jkb flm`lblbm`l np ml e`vjlh h`gglb cl} khu u}vkkh ml u}kbghlbg}l ykb mlql u}kbg" @bm`lb lv }wll ni cllvmlvl u}kbglbqàb" Blclb wl ml C`bjnwuj`#unc ykb ml e`vjlhu cl} khu u}vkhlb ml ylvued`hhlbml u}kbghlbg}lu" Clvj npmk} ml ynhgnvml m|u gllb flhkbg upllh} `b dl} flvl`j ykb ml kvc" M`} dn|m} m|u `b mk} ykbki b ≬ 6 m`} llbv`bg ù" Ni nbmlv uple`khl enbg|vk}`l llb uedài" @bm`lb ml kvc |`} }wll u}kbglb flu}kk} ù m`} khhllb qn`bm`lb ml hlbg}l ykb ml }wll u}kbglb ltke} glhàj ù" Khglcllb j|bblb wl u}lhhlb mk} wl llb v`bg fljnclbcl} khu f|`}lbu}vkkh ml unc ykb ml ylvued`hhlbml u}kbghlbg}lu v|`}w" 3

b` h`( lb khu f`bblbu}vkkh ml

hlbg}l ykb ml gvnn}u}l u}kbg c`b ml unc ykb ml kbmlvl u}kbglb1 v`bw" 3 6 · ckt` $h`) ∖b

` h`" @bm`lbv`bw" ≭ ? fl}ljlbm m`} mk} wl m`} khu e`vjlh j|bblb fluedn|wlb" Ml |`}wlvj`bg ykb ml |`}wlbm`gl lb`bwlbm`gl u}vkkh qàb hngùed mnnv encc|}k}`y`}l } ykb ml C`bjnwuj`#unc" Wl j|bblb m|u u}lhhlb mk}

dl} flpkhlb ykb ml flvl`jfkkvdl`m uhled}u N $b) flwlvj`bglb ylvlù} $ynnv ml unc lb dl} y`bmlb ykb dl}ckt`c|c)"

;;"6"6 Enbg|vk} l y`bmlb

Ylvynhglbu w`hhlb wl b| ml enbg|vk}`l ykb ml vnfn}kvc y`bmlb" Lv qàb led}lv cllu}kh nbl`bm`g yllhnphnuu`bglb" Ynnv ; kvc ù m`} }v`y`kkh" Ml kvc vn}llv} qèd `cclvu glwnnb bkkv dl} p|b} `bm`lb cnglhàj"Kbmlvu ù dl} nbcnglhàj" Ynnv }wll u}kbglb ù m`} pvnfhllc nnj ckjjlhàj np }l hnuulb" Wl enbu}v|lvlbllb e`vjlh ykb|`} ml uedn|mlv cl} ml u}vkkh ykb ml llvu}l u}kbg" Ylvynhglbu enbu}v|lvlb wl llb kbmlvle`vjlh ykb|`} ml dl} }l flvl`jlb p|b} cl} khu u}vkkh ml hlbg}l ykb ml }wllml u}kbg" @bm`lb ml }wll e`vjlhuc`bu}lbu ½l½lb p|b} glcllbuedkpplhàj dlfflb" J`lqlb wl ½l½lb ykb ml flued`jfkvl p|b}lb" M`} wnvm}mkb ml pnu }`l ykb ml lhhlfnng ykb ml vnfn}kvc" @bm`lb lv gllb qn‐b p|b} flu}kk} ù dl} }l flvl`jlbp|b} b`l} flvl`jfkkv" Wl qn|mlb kbkhnng cl} llb 5 lb cllvmlvl u}kbglb j|bblb mlb`lvlb" @b mk}

glykh fluedn|wlb wl ml C`bjnwuj` unc ykb khhl u}kbglb( fldkhyl ml hkk}u}l" Wl enbu}v|lvlb llb e`vjlhcl} khu u}vkkh ml hkk}u}l u}kbg vnbm dl} mnlh" Ylvynhglbu j`lqlb wl llb glcllbuedkpplhàj p|b} ykb m`le`vjlh cl} ml C`bjnwuj` unc lb m`} ù dl} p|b} mk} flvl`j} cnl} wnvmlb mnnv ml b ∖ ; u}kbglb" M`}khgnv`}cl ù m|u vle|vu`li lb ù c`}u np}`ckhùk}`lu $|`}gkkb mk} ml hkbgu}l qàml kkb ml uedn|mlv ù( wlcnglb `cclvu ml kvclb yvà }vkbuhk}lvlb) `b N $b) |`} }l ynlvlb"

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@bm`lb ml f`bblbjkb} glublmlb wnvm} ni dl} p|b} qèd `b ml C`bjnwuj`#unc fly`bm} cl} khu ml e`vjlh(

60



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