geometria situs

7/21/2019 geometria situs

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cjnhjtrgd sgtus

(2<<7-4<) Mdhjs _dukfjrs



cjnhjtrgd sgtus wds onhhgssgnkjf by _ùfwjstrukfauke anr t`j Fnkdujso`gkcjr

Husgetdcj 2<4<

Agrst pjranrhjf by _yivdgk Odhbrjigkc dkf t`j _SR _gkankgjnro`jstjr Bdfjk-

Bdfjk, Fnkdujsogkcjk, 43 Notnbjr 2<4<.



prncrdhhj kntj

Cjnhjtrgd sgtus gs t`j Idtgk tjrh dfnptjf anr t`j stufy na t`j cjnhjtry na pidoj, d agjif na hdt`jhdtgos w`go` `ds bjonhj eknwk ds tnpnincy.

Vnpnincy fjdis wgt` t`j spdtgdi prnpjrtgjs na dk nbmjot w`go` rjhdgk onkstdkt w`jk ukfjrcngkc fjanrhdtgnks suo` ds onhprjssgnk nr

strjto`gkc, but knt by outtgkc nr ciugkc. _nhj nbmjots hdy bj fjanrhjf gktn jdo` nt`jr5 t`j oidssgo jxdhpij gs d onaajj oup dkf d fnkut, bnt`

na w`go` ajdturj d sgkcij `nij. Vnpnincy gs disn rjajrrjf tn ds aidt s`jjt cjnhjtry.

Gk cjnhjtrgd sgtus, dii pidyjrs npjrdtj gkfjpjkfjktiy dkf `dvj d sjrgjs na pdcjs, jdo` na w`go` ajdturjs d sustdgkjf snukf w`go` ukfjrcnjs dsjrgjs na crdfudi o`dkcjs na tghbrj dkf fykdhgo. Jdo` o`dkcj gs oujf by t`j onkfuotnr w`n hdres t`jh dt grrjcuidriy spdojf tghj pngkts.

V`j hdtjrgdi gs t`jrjanrj strjto`jf dkf onhprjssjf fjpjkfgkc nk gts pidojhjkt gk t`j pgjoj, suo` t`dt t`j sdhj hdtjrgdi hgc`t bj fgaajrjktiy

struoturjf nk jdo` `jdrgkc.

cjnhjtrgd sgtus wds onhhgssgnkjf by _ùfwjstrukfauke anr t`j Fnkdujso`gkcjr Husgetdcj 2<4<. Gt wds agrst pjranrhjf by _yivdgk Odhbrjigkc

dkf t`j _SR _gkankgjnro`jstjr Bdfjk-Bdfjk, Fnkdujsogkcjk, 43 Notnbjr 2<4<.

gkstruhjktdtgnk

Gkstruhjktdtgnk gs aijxgbij, but onuif onhprgsj=

? (? f.pgo), ?. (2 f.onr), ? (?. f. boi).?/6.?.?.2/tghp.?pjro/46.42.4<.0.1

pjroussgnk rjqugrjf= bdss fruh, tdh tdh, vgbrdp`nkj, orntdijs

Dii pidyjrs wgii kjjf dt ijdst nk duxgigdry gkstruhjkt ds fjtjrhgkjf by t`jgr sjijotjf pdcjs. _jj bjinw anr gkanrhdtgnk.

kuhbjr na pdcjs

onkfuotnr :

stdrtgkc ouj 4<

aiutj 24

nbnj 2<

oidrgkjt 41bdss oidrgkjt 4<

bdssnnk 40

`nrk 26

truhpjt 40

trnhbnkj 26

tubd 42

pjroussgnk 26

tghpdkg 0

binwk tubj 1

bnwjf pidstgo oup :

bnwjf pniystyrjkj :bnwjf wnnf 0

onaajj oup nk suradoj 1

vgnigk :3

vgnid :3

ojiin 1<

fnubij bdss 61

Furdtgnk= up tn 2: hgkutjs.



prjpdrdtgnk

cjnhjtrgd sgtus onhprgsjs d sjt na pdcjs anr nro`jstrdi pidyjrs dkf tghj o`drts anr t`j onkfuotnr. V`j onkfuotnr cgvjs oujs dt jdo` na t`j tghj

pngkts, w`go` trgccjr dotgnks dkf o`dkcjs gk snukf by t`j pidyjrs.

\dcjs s`nuif bj fgstrgbutjf dhnkcst t`j nro`jstrdi pidyjrs. S`jrj huitgpijs na d sgkcij gkstruhjkt drj prjsjkt (j.c. vgnids), spigt t`j pdcjs

jvjkiy dhnkcst t`jh suo` t`dt dii pdcjs drj usjf bjanrj fupigodtjs drj gkoiufjf. S`jrj t`jrj drj gksuaagogjkt pdcjs tn prnvgfj jknuc`

hdtjrgdi, pidyjrs hdy bj cgvjk onpgjs na pdcjs dirjdfy fgstrgbutjf, suo` t`dt t`jsj drj disn sprjdf jvjkiy bjanrj aurt`jr fupigodtjs drj usjf.

Jdo` pidyjr s`nuif bj prnvgfjf wgt` gkstruhjktdi pdrts anr t`jgr spjogago gkstruhjkt. Jdo` pidyjr s`nuif disn bj cgvjk dt ijdst nkj duxgigdry pdrt

ds aniinws=

wnnfwgkf binwk tubj5 oup nk suradoj

brdss binwk tubj5 oup nk suradoj

pjroussgnk bnwjf pniystyrjkj5 bnwjf wnnf5 bnwjf pidstgo oup5 oup nk suradoj

strgkcs bnwjf pniystyrjkj5 bnwjf wnnf5 bnwjf pidstgo oup

\idyjrs s`nuif snuroj duxgigdry gkstruhjkts t`jhsjivjs, suo` t`dt d vdrgjty na nbmjots drj usjf gk d pjranrhdkoj. S`jrj pnssgbij, dvngf

fupigodtgkc nbmjots usjf by nt`jr pidyjrs.

Jdo` pidyjr gs disn cgvjk d pdcj w`go` gkfgodtjs t`jgr stdrtgkc pngkt. V`jsj pdcjs s`nuif bj sprjdf jvjkiy dhnkcst pidyjrs, suo` t`dt t`jrj gs

dk jqudi sprjdf na jdo` stdrtgkc tghjpngkt, dkf t`dt t`jsj drj disn s`drjf jvjkiy dhnkcst t`j fgaajrjkt gkstruhjkt typjs.

V`j onkfuotnr sjijots nkj na t`j tghj o`drts. Anr s`nrtjr pjranrhdkojs, idtjr tghj pngkts hdy bj nhgttjf. V`j onkfuotnr kjjfs d stnpwdto`.



pjranrhdkoj

V`j onkfuotnr cgvjs oujs dt jdo` na t`j tghj pngkts nk t`j sjijotjf tghjo`drt. V`jsj s`nuif bj cgvjk wgt` d oijdr fnwkbjdt. Gk bjtwjjk jdo`

fnwkbjdt, t`j dpprnxghdtj tghj tn t`j kjxt tghjpngkt s`nuif bj gkfgodtjf sn t`dt pidyjrs odk hjdsurj t`j trdksgtgnk na t`jgr snukfs. V`gs hgc`t

bj do`gjvjf by crdfudiiy rdgsgkc t`j nt`jr `dkf prnpnrtgnkdiiy nvjr t`j rjqugrjf furdtgnk sn t`dt gt rjdo`js t`j tnp na t`j fnwkbjdt dt t`j

onrrjot tghj.

V`j pidyjrs dii dot gkfgvgfudiiy. V`jgr sjijotjf pdcjs s`nuif bj nrfjrjf gk dfvdkoj na t`j pjranrhdkoj wgt`nut rjajrjkoj tn o`ngojs hdfj bynt`jr pidyjrs. V`j nrfjrgkc s`nuif bj o`dkcjf anr jdo` pjranrhdkoj (dkf rj`jdrsdi ds dpprnprgdtj).

Jdo` pdcj na t`j gkstruhjktdi pdrts onhprgsjs d sgkcij `jif snukf sjpdrdtjf gktn fgstgkot p`dsjs by vjrtgodi ouj pngkts. Jdo` snukf o`dkcjs

nkj pdrdhjtjr (j.c. fykdhgo, tghbrj, pidygkc tjo`kgquj) bjtwjjk nr dt d ouj pngkt. S`jrj d snukf o`dkcjs bjtwjjk ouj pngkts, t`gs s`nuif

bj dk jvjk dkf crdfudi trdksgtgnk. S`jrj t`jrj gs d suffjk o`dkcj (j.c. hnvgkc arnh knrhdi tn trjhnin bnwgkc), w`jrj pnssgbij t`j o`dkcj

s`nuif bj crdfudiiy gktrnfuojf gk dfvdkoj na nr aniinwgkc t`j ouj (j.c. sn t`j trjhnin hgc`t crdfudiiy doojijrdtj tn auii spjjf by t`j ouj).

Vrdksgtgnks drnukf ouj pngkts s`nuif gk cjkjrdi bj shnnt`.

\idyjrs gkfgvgfudiiy sjijot d ouj cgvjk by t`j onkfuotnr dt w`go` tn bjcgk t`jgr kjxt pdcj. V`j agrst ouj gs prnvgfjf by t`j pdcj suppigjf gk t`j

pdrt (g.j. +1 wnuif hjdk tn stdrt nk t`j sgxt` ouj, dkf +4 wnuif hjdk tn stdrt nk t`j agrst ouj jto.).

Datjr onhpijtgkc t`jgr agrst pdcj, pidyjrs fjtjrhgkj arjjiy w`jk t`jy bjcgk t`jgr kjxt pdcj. V`jrj hust bj dt ijdst nkj gkdotgvj pjrgnf anr jdo`

pidyjr bjtwjjk pdcjs `nwjvjr (g.j. t`j agkdi ouj na t`j prjojfgkc pdcj hdy knt disn bj t`j agrst ouj na t`j kjxt). Ga dii pdcjs drj usjf bjanrj

t`j pjranrhdkoj gs onhpijtj, t`jy hdy bj rjusjf.

V`j agrst na t`j sjijotjf onkfuotnr oujs bjonhjs ouj 4 nk t`j ourrjkt pdcj. \idyjrs onhpijtj t`j pdcj ds fjtjrhgkjf by t`j furdtgnk na t`j

oujs cgvjk by t`j onkfuotnr. V`j o`ngoj na stdrtgkc pngkt wgii t`jrjanrj daajot t`j furdtgnk na jdo` pdcj, dkf t`j o`dkcjs wgt`gk gt.

S`jk t`jrj drj 4< oujs rjhdgkgkc, t`j onkfuotnr wgii sgckdi t`gs by onuktgkc fnwk nk t`jgr agkcjrs w`jk cgvgkc oujs. Dt t`gs pngkt, pidyjrs hdy

nkiy bjcgk d kjw pdcj ga t`jrj drj suaagogjkt oujs rjhdgkgkc anr gt tn bj onhpijtjf (g.j. ga t`j onkfuotnr `ds sgckdiijf ‐6‗ rjhdgkgkc oujs, pdcjs

wgt` agvj nr hnrj ouj pngkts hdy knt bj stdrtjf, dkf t`j pidyjr rjhdgks sgijkt uktgi t`j jkf).

Gk cjkjrdi, t`jrj s`nuif bj d rjidtgvjiy ukganrh snukf dkf bdidkoj bjtwjjk dii gkstruhjkts. V`j inw fykdhgo rdkcj s`nuif bj dfmustjf

dpprnprgdtjiy tn diinw anr t`j onrrjot snukf prnfuotgnk nk t`j jfcj na sgijkoj dt t`j inw jkf, t`rnuc` tn dk dufgbij rjsuit wgt` jknuc`

prjsjkoj anr t`j pjranrhdkoj spdoj dt t`j `gc`jr jkf.

Anr snukfs w`go` kjjf tn bj gktjrruptjf t`rnuc` brjdt`s nr bnw o`dkcjs, try tn orjdtj ds shnnt` d rjsuit ds pnssgbij, dkf dvngf ouj pngkts anr

o`dkcjs.



pjranrhdkoj tjo`kgqujs

trdkspnsgkc gkstruhjkts \drts hdy bj pidyjf nk dky rjidtjf gkstruhjkt ukijss spjogagjf (j.c. aiutj pdrts odk bj pidyjf nk pgoonin,

ditn aiutj jto.). Ga d trdkspnsgkc gkstruhjkt gs usjf, t`j snukfgkc rjsuit wgii vdry fjpjkfgkc nk t`j

trdkspnsgtgnk.

cjkjrdi gkstruotgnks

n-pppp(-----) V`j snukf s`nuif bj nk t`j jfcj na sgijkoj, dkf stnp dkf stdrt grrjcuidriy, nr `dvj dk gkonksgstjkt qudigty

fuj tn dky dssnogdtjf pidygkc tjo`kgqujs (natjk usjf nk onhbgkdtgnk wgt` vdrgnus trjhnin nr n-pppp

hdregkcs).

ppp(9;) V`j snukf s`nuif ojktrj nk t`j gkfgodtjf fykdhgo, but diinw dky hgorn-vdrgdtgnks tn jhjrcj kdturdiiy

(fn knt try tn pidy t`jh t`nuc`)

wgkf dkf brdss

Ut`rndtX D vjry rdpgf sjrgjs na sgkcij drtgouidtgnks na t`j dgr strjdh gk t`j t`rndt (knt d crnwi nr aiuttjr-tnkcuj). V`j

pdoj s`nuif bj ds adst ds pnssgbij, tn t`j pngkt w`jrj gt gs `drf tn onktrni t`j rjcuidrgty na t`j dttdoe.

Udgr/pgto`X Brjdt` snukf wgt` d oijdr pgto` onhpnkjkt. S`jrj t`gs gs ukspjogagjf (sgkcij-igkj stdvj), dky pgto` hdy bjsjijotjf. S`jrj twn-igkj stdvjs drj usjf, d inwjr- dkf `gc`jr-pgto`jf brjdt` snukf s`nuif bj sjijotjf.

S`jrj trgiis drj hdrejf nk brjdt` snukfs, agkf d agkcjrgkc w`go` diinws t`j drtgouidtgnk tn bj `jdrf oijdriy.

Udgr/kngsjX Brjdt` snukf wgt` igttij pgto` onhpnkjkt. Kngsj tnkjs s`nuif bj jhp`dsgsjf.

Huitgp`nkgo. Dii huitgp`nkgos s`nuif bj rjidtgvjiy purj, wgt` d tjkfjkoy tnwdrfs onksnkdkt pgto` onktjkt.

S`jrj hnrj t`dk nkj huitgp`nkgo gs gkfgodtjf nk d pdcj, jdo` s`nuif bj fgaajrjkt.

Dk ditjrkdtj, tghbrj agkcjrgkc (shdii hgotrntnkdi fjtukgkcs drj doojptdbij). V`j kuhbjrs drj bdsjf nk

\jtjr Qjdij‗s V`j Vjo`kgqujs na Nbnj \idygkc (Edssji, 4770) dkf rdkcj arnh < (knrhdi agkcjrgkc) tn :. V`j

`gc`jr t`j kuhbjr, t`j hnrj fgaausj t`j rjsuit.

UbijjfX _igc`tiy fjprjss nr rdgsj d vdivj arnh t`j knrhdi agkcjrgkc tn fjstdbgigsj t`j snukf. S`jrj d trgii igkj gs disn

gkfgodtjf, t`j pnsgtgnk na t`j vdivj s`nuif bj sinwiy vdrgjf.

hutjs Hutjs drj gkfgodtjf ds bnxjf tjxt. _jj duxgigdry gkstruhjkt gkstruotgnks anr rjidtjf hdregkcs

aiutj

Uw`gstij tnkjsX Onkvjktgnkdi w`gstij tnkjs. Dgh anr d rjidtgvjiy stdbij pgto`, but doojpt dky noodsgnkdi nsogiidtgnks.

UbinoejfX \idoj hnut` onhpijtjiy nvjr hnut`pgjoj dkf binw gktn gkstruhjkt.

fgdhnkf kntj`jdfs D `niinw pgto`jf tnkj, purj.

nbnj/bdssnnk

Urniigkc tnkjX V`j rniigkc tnkj gs nbtdgkjf by d strnkcjr igp prjssurj nk t`j rjjf.

D fnubij trgii usgkc d sjpdrdtj trgii ejy anr jdo` `dkf rjsuitgkc gk d adstjr nsogiidtgnk.

H

6



`nrk

`dkf stnppgkc S`jrj crdfudi stnppgkc dkf cigssdkfn drj gkfgodtjf, fn knt onhpjksdtj anr pgto` o`dkcj (diinw cigss tn

noour ds d rjsuit na stnppgkc). S`jrj crdfudi stnppgkc nkiy gs gkfgodtjf, onhpjksdtj anr pgto` o`dkcj gk

nrfjr tn hdgktdgk pgto` (tghbrdi o`dkcj nkiy).

duxgigdry gkstruhjkts

hutjs Hutjs drj gkfgodtjf gk bnxjf tjxt anr brdss gkstruhjkts. S`jrj hutjs drj gkfgodtjf anr duxgigdry gkstruhjkts

(j.b. binwk tubjs), jgt`jr t`j `dkf nr dknt`jr dpprnprgdtj nbmjot hdy bj usjf.

+ oinsjf

n npjk

binoejf ‟ ds oinsjf, but wgt` jxtrd prjssurj tn sjdi t`j tubj ds huo` ds pnssgbij.

D hutj trjhnin gkvnivgkc d shdii ukjvjk s`degkc hnvjhjkt na t`j hutj ojktrjf nk t`j pnsgtgnk(s)

gkfgodtjf.

bnwjf wnnf _jijot dky pgjoj na wnnf (tghbjr, anukf, nbmjots, gkstruhjkts), dkf bnw ds gkfgodtjf wgt` d strgkc

gkstruhjkt bnw. _trgkc pidyjrs hdy usj t`j wnnf na t`jgr gkstruhjkt ga prjajrrjf. Kngsj tnkjs drj prjajrrjf.

bnwjf pniystyrjkj _jijot dky pgjoj na pniystyrjkj dkf bnw dk jfcj ds gkfgodtjf. S`jk snurogkc pniystyrjkj, jxpjrghjktwgt` sgzj dkf fjksgty tn agkf d rgo` snukf.

Uigc`t prjssurjX Bnw t`j pniystyrjkj vjry igc`tiy tn prnfuoj d oijdrjr dkf hnrj stdbij tnkj

Uvjry sinw bnwX Qjry sinw bnw hnvjhjkt, prnfuogkc d sjrgjs na ukprjfgotdbij oigoes/kngsj.

bnwjf pidstgo oup D `drf, t`gk pidstgo fgspnsdbij frgkegkc cidss, prjajrdbij wgt` d wjii fjagkjf (s`drp) rgh. Dky sgzj hdy bj

usjf. Bnw nkj sgfj na t`j rgh dt dk dkcij na dt ijdst ?<¼, ukijss gkfgodtjf. Zp dkf fnwk bnw wgii prnfuoj

d fgaajrjkt snukf.

Uadst, igc`t prjssurjX D adst dkf igc`t bnw hnvjhjkt prnfuogkc d snukf wgt` d oijdrjr pgto` onktjkt. Bnw hdregkcs drj knt

cgvjk wgt` t`gs gkfgodtgnk cgvjk t`j kjjf tn usj d int na bnw.

Ubnw aidtX Bnw aidt dornss oup suo` t`dt t`j bnw tnuo`js t`j rgh nk nppnsgkc jfcjs.

binwk tubj _jijot dky typj na strdgc`t, npjk tubj, hdfj arnh dky hdtjrgdi (j.c. odrfbndrf tubj, frgkegkc strdw,onppjr pgpj, wdstj pgpj, `nsj jto.). Dky ijkct` odpdbij na prnfuogkc dk dufgbij snukf hdy bj usjf.

S`jk binwgkc, dk dgr snukf s`nuif rjsuit, pnssgbiy wgt` d sigc`t pgto` onktjkt. V`gs hdy bj dgfjf by

snukfgkc d ‐a‗ p`nkjtgo gktn t`j tubj. Fn knt binw gk t`j hdkkjr na d aiutj nr brdss gkstruhjkt. _jj dbnvj

anr hutj gkanrhdtgnk.

onaajj oup nk suradoj D stdkfdrf odrf tdejdwdy onaajj oup. Dky aidt suradoj hdy bj sjijotjf tn bj sdhpijf by t`j oup (j.c.

d hjtdi s`jjt, pniystyrjkj binoe, d brgoe, d wnnfjk trdy, ajit jto.) usgkc t`j tjo`kgqujs gkfgodtjf

usj bdsj na oup nk suradoj

usj rgh na oup nk suradoj (upsgfj fnwk)



UfrdcX \uii t`j oup dornss t`j o`nsjk suradoj. Dpprnprgdtj prjssurj anr t`j fjsgckdtjf vniuhj wgii bj

fjtjrhgkjf by t`j suradoj typj.

Usordto`X D rdpgf bdoe dkf anrt` argotgnk dotgnk

UogroijX D argotgnk dotgnk wgt` d rjcuidr ogrouidr hntgnk

pjroussgnk

_jj duxgigdry gkstruhjkts anr nt`jr rjidtjf gkstruotgnks

Upus`X D onkstdkt inkc argotgnk dotgnk gk nkj fgrjotgnk wgt` t`j `dkf pus`gkc t`j bjdtjr/nbmjot dornss t`j

suradoj. V`j pus`gkc dotgnk s`nuif orjdtj crjdtjr argotgnk rjsuitgkc ukstdbij mgttjrgkc, ukjvjkkjss jto.

UrniiX D vjry rdpgf rnii wgt` nkj stgoe/`dkf nkiy (knt d onkvjktgnkdi twn stgoe/`dkf) V`j pdoj s`nuif bj ds adst

ds pnssgbij, tn t`j pngkt w`jrj gt gs `drf tn onktrni t`j rjcuidrgty na t`j dttdoe. Dii rniis s`nuif bj na t`gs

typj, w`jt`jr hdrejf spjogagodiiy gk onktrdst tn nt`jr tjo`kgqujs, nr w`jrj kn gkfgodtgnk gs hdfj.

UfdhpX Igc`tiy fdhp orntdijs wgt` agkcjr tgps tn rjfuoj uppjr pdrtgdis.

UhuaaijX Igc`tiy fdhp tghpdkg wgt` agkcjr tgps

wds`gkc-up brus` D pidstgo-`dgrjf wds`gkc-up brus` wgt` rjidtgvjiy stgaa brgstijs.

strgkcs

fgdhnkf kntj`jdfs Fgdhnkf kntj`jdfs gkfgodtj d `drhnkgo prjssurj ijat `dkf agkcjrgkc. V`j gkfgodtjf pgto` dkf rnhdk

kuhjrdi fjkntj t`j pnsgtgnk na t`j agkcjr dkf t`j strgkc tn bj usjf. V`gs wgii prnfuoj d rdkcj na rjsuits

arnh rjidtgvjiy oijdr `drhnkgos t`rnuc` tn (oninurjf) kngsj.

UfdhpX Fdhp dii strgkcs igc`tiy wgt` t`j agkcjrs na t`j ijat `dkf. D oninurjf pgto`/kngsj snukf wgii rjsuit.

Ubnw tdgipgjojX Bnw t`j tdgipgjoj usgkc t`j gkfgodtjf tjo`kgqujs= t`j snukf wgii vdry fjpjkfgkc nk t`jgr spjogagodtgnk

Ubnw spgejX \idoj ojiin nk idp dkf bnw kjdr t`j jkf na t`j spgej, prnfuogkc d inw-pgto`jf trjhnin

prjpdrdtgnks _jijotgnk dkf pidojhjkt na t`j jxdot nbmjots tn bj usjf w`jk prjpdrgkc gkstruhjkts gs dt t`j fgsorjtgnk

na t`j pidyjrs. Dgh anr hdxghuh vdrgjty bjtwjjk pidyjrs dkf dvngf fupigodtgkc prjpdrdtgnks w`jrj

pnssgbij. Nbmjots s`nuif bj gksjrtjf dkf rjhnvjf arnh t`j gkstruhjkt ds qugjtiy ds pnssgbij.

hjtdi prjpdrdtgnk D shdii, igc`t pgjoj na hjtdi (j.c. d idrcj pdpjr oigp, outijry, d sejwjr, hjtdi ruijr, shdii bnit jto.) s`nuif

bj gksjrtjf dbnvj G dkf GQ dkf bjinw GG dkf GGG. V`j bjst pnsgtgnk wgii bj fjtjrhgkjf by t`j onhbgkdtgnk na

gkstruhjkt dkf nbmjot.

wnnf prjpdrdtgnk D shdii igc`t pgjoj na wnnf (j.c. d iniiy stgoe, pjkogi, o`npstgoe, tnnt`pgoe, onaajj stgrrjr jto.) s`nuif bj

gksjrtjf dbnvj G dkf GQ dkf bjinw GG dkf GGG. V`j bjst pnsgtgnk wgii bj fjtjrhgkjf by t`j onhbgkdtgnk na

gkstruhjkt dkf nbmjot. V`j wnnf s`nuif anroj t`j knk-bnwjf strgkcs tn vgbrdtj syhpdt`jtgodiiy.

odrf prjpdrdtgnk D shdii pgjoj na pdpjr nr odrf s`nuif bj wnvjk bjtwjjk t`j strgkcs nvjr t`j agkcjrbndrf. V`j hdregkc

‐odrf‗ gkfgodtjs t`dt stgkcs/odrf s`nuif bj bnwjf nvjr t`j odrf gtsjia (hnitn sui tdstn).

oint` prjpdrdtgnk D shdii pgjoj na oint` (dky typj na adbrgo hdy bj usjf) s`nuif bj wnvjk bjtwjjk t`j strgkcs nvjr t`j

agkcjrbndrf. Bnw gk t`j onkvjktgnkdi hdkkjr.



cjnhjtrgd sgtusonkfuotnr (4)

Mdhjs _dukfjrs

<=<<

<=41

<=:0

4=<7

4=?42=<?

2=44

2=2?

?=<2

?=?6

?=6?

?=60

?=:1

6 27

7=2?

7=?1

7=64

4<=44

4<=214<=?3

44=?2

42=<:

4?=<1

4?=2:

4?=?4

4?=64

4?=:4

4? :?

43=?:

43=6?

43=:?

40=<<

40=<:40=64

40=66

40=60

47=2?

47=?0

47=:?

47=:7

2<=::

24 <7




Mdhjs _dukfjrs

<=<<

<=23

<=67

<=:3

4=474=2?

4=?4

4=??

2=<7

2=?02=6?

2=6:

2=:2

2=:?

1=:<

1=:1

3=<2

3=42

3=473=20

3=??

0=64

0=6:

4<=404<=22

4<=2?

4<=?<

4<=67

46=2<

4:=67

41=<6

41=42

41=2641=21

41=?1

41=67

41=:6

43=?040=43

40=27

47=<6

47=?:



cjnhjtrgd sgtusonkfuotnr (?)

Mdhjs _dukfjrs

<=<<

<=:2

<=::

4=46

4=64

4=:<4=:6

4=:0

2=27

2=?6

2=:3?=4:

?=:3

6=<3

6=43

6 ?:

0=40

0=20

7=<?

7=41

7=2:

7=237=?3

7=:3

7=:7

4<=2<

4<=?14<=::

44=?2

44=?1

44=67

42 <2

4:=40

4:=63

41=2:

41=?6

41=?7

41=6043=<2

43=64

40=<<

40=<0

40=?640=66

47=63

2<=4<

2<=26

2< 23




Mdhjs _dukfjrs

<=<<

<=4?

<=47

<=26

<=?:<=61

<=:4

4=4?

4=41

2=2<2=2:

?=<4

?=4:

6=<<

7=<2

7=?7

7=:?

4<=<3

4<=6244=43

44=?<

44=62

42=<:

42=4<42=:6

4?=46

4?=4:

4?=41

40=6<

47=<?

47=41

47=?7

2<=<62<=<:

2<=42

2<=24

2<=64

24=6324=:<

24=:<

22=66

22=:7















































































































































































































































































































































































































































































































































































































































































































































































































































































































































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