entanglement, monoidal categories, and frobenius algebras

7/31/2019 Entanglement, Monoidal Categories, and Frobenius Algebras

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Aktgk`canakt# Nhkhedgc Ogta`hzeas# gkd

Izhjakeqs Gc`ajzgs

Gcafs Fessek`az

Lgkqgzu :1# :=7=

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Gjstzgot

]zhyedek` g àkazeo dasoze}tehk hi aktgk`canakt ek k.wqjet sustans es g

chk`.stgkdek` h}ak }zhjcan ek wqgktqn ekihzngtehk soeakoa+ G stzqotqzgc

sobana ihz za}zasaktek` gzjetzgzu nqcte}gzteta aktgk`cad stgtas pecc ueacd g

daa}az qkdazstgkdek` hi bhp tbau jabgya gkd ektazgot petbek nhza àkazgc

ohn}qtgtehkgc nhdacs gkd }zhthohcs+ Baza pa }zhyeda sqob g dasoze}tehk+

Iezst pa sbhp tbgt jhtb tba @B\ stgta gkd tba P stgta gdnet g sen.

ecgz gcàjzgeo stzqotqza# pbeob es hkcu deazakt ek hka en}hztgkt datgec#

gkd tbgt ihz sunnatzeo tze}gzteta stgtas tbes gcàjzgeo stzqotqza argotcu

obgzgotazesas tba ohzzas}hkdek` _CHOO.aktgk`canakt ocgssas+ Tbes kap

ohkkaotehk jatpaak gcàjzg gkd aktgk`canakt tbahzu hazs kap ekse`bts

ek jhtb dezaotehks+ Hqz ngek tbahzan stgtas tbgt gzjetzgzu _CHOO ak.

tgk`canakt ocgssas ihz nqcte}gzteta }qza stgta aktgk`canakt gzesa izhn tba

ektazgotehk hi tba tze}gzteta @B\. gkd P.gcàjzgs+

Tba @B\. gkd P.gcàjzg gza nhzahyaz sqjlaot th g }qzacu deg`zgn.

ngteo ogcoqcqs# gkd ohksawqaktcu sh gza tba zasqctek` nqcte}gzteta aktgk.

`cad stgtas+ Izhn tbes }hekt hi yeap# tba destekotehk jatpaak tba @B\.

stzqotqza gkd P.stzqotqza es }qzacu th}hchèogc# ek tazns hi ohkkaotad

ys+ desohkkaotad+ Pa ogk qsa tbes destekotehk th ektqeteyacu ohkstzqot

stgtas gkd ng}s petb ohn}car jabgyehqzgc }zh}azteas+

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Ohktakts

7 Ektzhdqotehk 4

: Aktgk`canakt 1

:+7 CHOO. ys+ _CHOO.awqeygcakoa + + + + + + + + + + + + + + + + + + + 9:+: @B\ ys+ P stgtas + + + + + + + + + + + + + + + + + + + + + + + + + + 9

? Nqcte}gzteta _CHOO ocgsseogtehk sobanas 0

?+7 Ju Lhzdgk Khzngc Ihzn + + + + + + + + + + + + + + + + + + + + + + + 0?+: Ekdqoteya ocgsseogtehk ju ze bt sek`qcgz s}goas + + + + + + + + + + 0?+? Ocgsseogtehk hi sunnatzeo stgtas ju daàkazgoeas + + + + + + + + + 2

4 Ohkozata Izhjakeqs gcàjzgs 5

1 Nhkhedgc ogta`hzeas 771+7 @zg}beogc za}zasaktgtehk + + + + + + + + + + + + + + + + + + + + + + + 7?1+: Zacgtehksbe} petb tba Oezoqet Nhdac + + + + + + + + + + + + + + + + 71

9 Nhkhedgc gcàjzgs 70

9+7 Ektazkgc nhkheds gkd ohnhkheds + + + + + + + + + + + + + + + + + + 709+: Ektazkgc Izhjakeqs gcàjzgs + + + + + + + + + + + + + + + + + + + + + 72

9+:+7 _}aoegc gkd gkte.s}aoegc ohnnqtgteya Izhjakeqs gcàjzgs + :=9+? Tba àkazgc ogsa< ]ZH]s + + + + + + + + + + + + + + + + + + + + + + :?

0 @B\ stgtas gkd P stgtas za}zasakt ohnnqtgteya Izhjakeqs gc.

àjzgs :1

0+7 _}aoegc OIGs gza @B\ _tgtas + + + + + + + + + + + + + + + + + + + + :10+: Gkte.s}aoegc OIGs gza P stgtas + + + + + + + + + + + + + + + + + + + :0

2 Ektazgotek` Izhjakeqs Gcàjzgs :5

2+7 Ohn}canaktgzu ocgsseogc stzqotqzas + + + + + + + + + + + + + + + + + :52+: N_.}gezs + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + ?:

2+:+7 Qkeyazsgcetu + + + + + + + + + + + + + + + + + + + + + + + + + + ?42+? Zacgtehksbe} hi tba tph tbahzeas + + + + + + + + + + + + + + + + + + + ?9

5 Ekdqoteya Ohkstzqotehk hi _tgtas ?0

7= Ohzhccgzeas ?5

77 Ohkocqsehk gkd Iqtqza Phzf 4=

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5/44

}zasaktgtehk ektazkgc th g nhkhedgc ogta`hzu# dqa th Ogzjhke gkd Pgctazs V77^+Gs sqob# pa bgya gooass th g zeob `zg}beogc cgk`qgà ekyaktad ju ]akzhsa ek tba

750=s V7:^ gkd ihzngcesad ju Lhugc gkd _tzaat ek tba 755=s V7?^+ Izhjakeqs gcà.jzgs gcsh bgya keoa khzngc ihzns# petb g yazu th}hchèogc iaac "saa tba s}edaztbahzans ek saotehk 9+:# pbaza pa ohqkt ohkkaotehks gkd bhcas%+ Qsek` tbesthhcfet# pa edakteiu g }qzacu th}hchèogc destekotehk "dh bhcas ygkesb hz ar}chda8%jatpaak tba Izhjakeqs gcàjzgs àkazgtad ju tba P stgta gkd tba @B\ stgta+

Hqz ohza tbahzu es jqect q}hk tba `zg}beogc àkazgthzs hi Izhjakeqs gcàjzgs+Qsek` `zg}b edakteteas# pa ogk qsa tbasa jqecdek` jchofs th ektqeteyacu ohkstzqotstgtas th jqecd.ek oaztgek jabgyehqzgc }zh}azteas# gkd ayak }zhyeda gk ekdqoteyanatbhd# ihcchpek` V74^ th ohkstzqot gzjetzgzu nqcte}gzteta stgtas+ Tba jakathi tbes es tbgt stgtas gza ohn}hsetehkgc ek kgtqza# kht lqst jcgof jhras# gkd tba}zh}azteas gkd ektazgotehks hi }cq`èk` je`àz gkd je`àz ohn}hkakts thàtbazes ayedakt ek tba `zg}beogc tbahzu+ Tbes atbeo es senecgz th tbgt hi nagsqzanaktjgsad wqgktqn ohn}qtek` "NJWO% V71# 79^# ek qsek` sen}caz stgtas th ohkstzqotnhza }hpaziqc hkas# jqt tba jagqtu hi tbes sobana es tbgt et ekihzns tba dase`khi wqgktqn }za}gzgtehks gkd }zhthohcs ekda}akdaktcu hi tba qkdazcuek` ohn.}qtgtehkgc }gzgde`n# gkd jagzs dezaot ektaz}zatgtehk ek jhtb tba oezoqet nhdacgkd NJWO+

Ek saotehk :# pa zayeap shna jgseo khtehks hi aktgk`canakt gkd "_%CHOOawqeygcakoa hi aktgk`cad stgtas+ Ek saotehk ?# pa zaohqkt sayazgc _CHOO aktgk.`canakt ocgsseogtehk sobanas# khtgjcu tba ekdqoteya sobana dqa th Cgngtg atgc V74^+ Pa tbak daka ek saotehk 4 tba ohkozata khtehk hi Izhjakeqs gcàjzgshyaz yaothz s}goas+ Izhn tbaza# pa ektzhdqoa nhkhedgc ogta hzeas ek saotehk 1gkd sbhp bhp ng}s ek sqob ogta`hzeas cakd tbansacyas th g }hpaziqc `zg}beogckhtgtehk+ Pa saa bhp th ihzn gcàjzgs ektazkgc th sqob ogta hzeas ek saotehk 9#gkd ek }gzteoqcgz# ihoqs hk tba daketehk hi Izhjakeqs gcàjzgs ek tbes gjstzgot

ohktart+ Pa gcsh daka tph fekds hi Izhjakeqs gcàjzgs< s}aoegc ohnnqtgteyaIzhjakeqs gcàjzgs "_OIGs% gkd gkte.s}aoegc ohnnqtgteya Izhjakeqs gcàjzgs"GOIGs%+ Ek saotehk 0# pa }zhya tbgt @B\ stgtas ohzzas}hkd argotcu th _OIGshyaz O: gkd P stgtas ohzzas}hkd argotcu th GOIGs hyaz O:+ Hka hz nhza hitbasa stzqotqzas ogk ja ohnjekad th ozagta zeob ektazgotehk tbahzeas# pbeob gzaar}chzad ek saotehk 2+ Tba tbahzu hi ohn}canaktgzu ocgsseogc stzqotqzas es za.ohqktad izhn V70^# gkd tba tbahzu hi N_.}gezs es kap+ Ek saotehk 5# pa `h hkth sbhp bhp tba tbahzu hi N_.}gezs ogk ja qsad th ektqeteyacu ohkstzqot gcc hitba nqcte}gzteta stgtas edaktead ju V74^ gkd V72^+ _aotehk 7= hazs g kqnjazhi sbhzt ohzhccgzeas th danhkstzgta tba }hpaz hi tbes kap ohkkaotehk jatpaakgcàjzg gkd aktgk`canakt tbahzu+

: Aktgk`canaktIhz Becjazt s}goas Be# e > 7#+++#k# cat s

Be ja g stgta+ Ei tbaza areststgtas se Be sqob tbgt s >

se# s es sged th ja sa}gzgjca+ Ei kh sqobstgtas arest# s es aktgk`cad+ s es g daàkazgta k.}gzteta aktgk`cad stgta eitbaza arest khk.tzeyegc s7# s: sqob tbgt s > s7 s:+ Ei tbaza gza kh

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sqob stgtas# s es g àkqeka k.}gzteta aktgk`cad stgta+ Ek O:O:# tph argn}cashi àkqeka je}gzteta aktgk`canakt gza tba Jacc stgta

sJacc

>

s==

&

s77

gkd tba

A]Z.stgta sA] Z > s7= & s=7+

:+7 CHOO. ys+ _CHOO.awqeygcakoa

Et es hitak qsaiqc th ohksedaz aktgk cad stgtas hkcu q} th chogc h}azgtehks+ Eitph stgtas ogk ja dataznekesteogccu ektaz.ohkyaztad petb hkcu chogc "hka.wqjet%}buseogc h}azgtehks gkd ocgsseogc ohnnqkeogtehk# tbau gza sged th ja CHOO.awqeygcakt+ Ei tbau ogk ja ektaz.ohkyaztad# jqt hkcu petb shna khk.|azh }zhj.gjecetu# tbau gza sged th ja sthobgsteo CHOO.awqeygcakt# hz _CHOO.awqeygcakt+Tbasa ogk ja ihzngcesad ju tba ihcchpek` }gez hi zasqcts# dqa zas}aoteyacu th V75^gkd V4^+

Tbahzan 7+ Tph stgtas

s

gkd

s

gza CHOO.awqeygcakt e tbaza arest chogc

qketgzu ng}s Qe sqob tbgt s > "Q7 Q: + + + Qk%s+Tbahzan :+ Tph stgtas s gkd s gza _CHOO.awqeygcakt e tbaza arest chogcekyaztejca ng}s Ce sqob tbgt s > "C7 C: + + + Ck%s+

Tbasa ocagzcu jhtb àkazgta awqeygcakoa zacgtehks# gkd izhn bakoa ihztb# pasbgcc qsa tbasa gs tba daketehks hi CHOO gkd _CHOO+ Ihz argn}ca# sJacc gkdsA] Z gza CHOO awqeygcakt# jqt tbau gza kht CHOO.awqeygcakt th 7? s==&:? s77#pbeca sJacc# sA] Z gkd 7? s== & :? s77 gza gcc _CHOO awqeygcakt+

:+: @B\ ys+ P stgtas

Ek V4^ et pgs sbhpk tbgt tbaza gza argotcu tph _CHOO.awqeygcakoa ocgssas hiàkqeka tze}gzteta stgtas ek O:

O:

O:+ Tba zst es petkassad ju g ?.wqjet

àkazgcesgtehk hi tba Jacc stgta# ogccad tba @zaakjazàzBhzka\aecekàz "@B\%stgta s=== & s777 #gkd tba saohkd es petkassad ju tba P stgta s7== & s=7= & s==7 +

Tbhq`b jhtb hi tbasa stgtas gza ngrengccu aktgk`cad# tba P stgta es tbaqkewqa stgta "q} th _CHOO% tbgt nekenesas tzqa tbzaa }gztu aktgk`canakt# gsnagsqzad ju g natzeo ogccad tba tbzaa tgk`ca# hz zasedqgc tgk`ca V4^+ Ektqeteyacu#tbes nagks tbgt gcc hi tba aktgk`canakt }zasakt ek P es dqa th }gezpesa ohzza.cgtehks jatpaak agob hi tba tbzaa wqjets# zgtbaz tbgk "ek tba ogsa hi tba @B\%

tba aktgk`canakt jaek` shcacu dqa th g tzqa tze}gzteta ohzzacgtehk+Gs g zasqct hi tbes# P stgtas bgya tba }zh}aztu tbgt aktgk`canakt ogk sqzyeya

ayak pbak hka hi tba wqjets es tzgoad hqt+ Bhpayaz# petb @B\ stgtas# tzgoek`gku hi tba tbzaa wqjets hqt ueacds g }zhjgjecesteo nertqza hi }zhdqot stgtas+

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? Nqcte}gzteta _CHOO ocgsseogtehk sobanas

Gs paya gczagdu saak# tba _CHOO ocgssas ihz stgtas ohksestek` hi K ? wqjetsgza pacc fkhpk+ Tbes es zaaoteya hi tba igot tbgt taobkewqas s}aoeo th oaztgekK gza qsad th ohn}cata tba ocgsseogtehk hi tbasa stgtas+ Gcsh# ju gk aca`gktgz`qnakt ek V4^# tbaza gza kaoassgzecu ekketa _CHOO ocgssas pbak K 4+Th hjtgek keta ocgsseogtehk zasqcts ihz K.}gzteta stgtas# ngku gqtbhzs bgyaar}gkdad th tgcf gjhqt _CHOO sq}az.ocgssas# hz igneceas hi _CHOO ocgssas}gzgnatazesad ju hka hz nhza ohktekqhqs ygzegjcas+ _ayazgc gttan}ts bgya jaakngda th akqnazgta tbasa sq}az.ocgssas jgsad hk tba obgzgotazesteos hi tbaezngtzer za}zasaktgtehks+

?+7 Ju Lhzdgk Khzngc Ihzn

Ek :==:# Yazstzgata at gc V:=^ ektzhdqoad 5 _CHOO sq}az.ocgssas ihz 4.}gzteta

stgtas+ Tbes zasqct ar}chets tba ihcchpek` eshnhz}besn hi Cea `zhq}s< _C":6O%_C":6O% > _H"46O%+ Tbgt es# cekagz ng}s hi tba ihzn C N ihz C# N da.taznekgkt 7# ohn}car : : ngtzeoas gza ek 7.th.7 ohzzas}hkdakoa petb 4 4ohn}car.ygcqad# hztbh`hkgc ngtzeoas+ Tbau tbak `h hk th ar}zass tba :4 ohn.}car }gzgnatazs hi gk gzjetzgzu 4.}gzteta stgta gs g 4 4 ngtzer gkd }zhya tbgtgku sqob ngtzer ogk ja ohkyaztad th g khzngc ihzn qsek` ohn}car hztbh`hkgcngtzeoas H7 gkd H:+ Tbes khzngc ihzn tbak ngfas ar}ceoet tba zangekek` da.`zaas hi izaadhn hi tba 4.}gzteta stgta# gitaz g wqhteakt ju _CHOO gkd H7 gkdH: ueacd tba ihqz chogc ekyaztejca ng}s+

Pbeca tbes }zhyedas g taobkewqa ihz edakteiuek` _CHOO sq}az.ocgssas gkd esh.cgtek` izaa }gzgnatazs# et zaceas hk gk eshnhz}besn tbgt hkcu bhcds ihz _H"46O%#sh et gdnets kh kgtqzgc `akazgcesgtehk th be`baz denaksehks+ Gcsh# tba obheoahi sq}az.ocgssas da}akds bagyecu hk tbes }gzteoqcgz ocgsseiuek` taobkewqa# gkddhaskt kaoassgzecu ngfa ar}ceoet jabgyehqzgc destekotehks jatpaak tba ocgssas+

?+: Ekdqoteya ocgsseogtehk ju ze bt sek`qcgz s}goas

Jaogqsa hi sbhztohnek`s ek Yazstzgatas taobkewqa# gkd ek gk gttan}t th kd gkK.}gztu ocgsseogtehk nhza ek tba s}ezet hi Dqz# Yedgc# gkd Oezgos hze`ekgc :===zasqct# Cgngtg at gc ektzhdqoad g ocgsseogtehk sobana jgsad q}hk tba ze`btsek`qcgz sqjs}goa hi g }qza stgta V74^+ Cefa Yazstzgata# Cgngtg at gc za}zasakt}qza stgtas gs ngtzeoas+ Bhpayaz# tbau }gztetehk hqt hkcu hka wqjet# za`gzdek`gk K.}gzteta stgta gs g ng} N izhn "O:%"K7% th O:+ ]azihznek` g sek`qcgzygcqa daohn}hsetehk hk sqob g ngtzer ueacds g 7. hz :.denaksehkgc ze`bt sek`qcgzsqjs}goa# s}gkkad ju tph yaothzs ek "O:%"K7%+ Tba _CHOO sq}az.ocgss hiNes tbak cgjacad ju tba _CHOO sq}az.ocgssas hi tbasa yaothzs# tbqs }azihznek`tba ekdqoteya sta}+ Tba jgsa ogsa es tbak O:# pbaza tba hkcu tph _CHOO ocgssasgza za}zasaktad ju tba }zhdqot stgta gkd tba Jacc stgta+

Ek tba tze}gzteta ogsa# tba @B\ _CHOO ocgss ohksests hi stgtas pbhsa ze`btsek`qcgz sqjs}goa es s}gkkad ju tph }zhdqot stgtas# pbazags stgtas ek tba P

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_CHOO ocgss bgya g ze`bt sek`qcgz sqjs}goa tbgt es s}gkkad ju g }zhdqot stgtagkd gk aktgk`cad stgta+

Gk gctazkgteya pgu hi chhfek` gt tbes sobana es th ohksedaz K.}gzteta stgtasgs ohktzhccad "K 7%.}gzteta stgtas+ Tbgt es th sgu# tba ze`bt sek`qcgz s}goahi g stgta es s}gkkad ju vs# s~ e tbaza arests g _CHOO.awqeygcakt stgta hitba ihzn

s=# & s7# Tbes }zhyedas gk h}azgtehkgc dasoze}tehk hi g _CHOO sq}azocgss+ Kgnacu#

g stgta s es ek tba _CHOO sq}azocgss vs# s~ ei tbaza arests shna "tph.denaksehkgc# }hssejcu khk.hztbh`hkgc% jgses J sqob tbgt nagsqzek` tba zstwqjet ek J ueacds g stgta tbgt es _CHOO.awqeygcakt th s ihz hqtohna 7 gkds ihz hqtohna :+

?+? Ocgsseogtehk hi sunnatzeo stgtas ju daàkazgoeas

Ek :==5# Jgstek at gc V:7^ ar}chzad _CHOO ocgsseogtehk hi sunnatzeo K.wqjetstgtas+ Tbasa gza K.}gzteta stgtas ohktgekad ek tba sunnatzeo sqjs}goa hi"O:%K+ Th sqnngzesa tbaez sobana# pa zst ektzhdqoa tba K.wqjet sunnatze.sgtehk ng}# pbeob sen}cu sqns gcc hi tba }aznqtgtehks hi jgses yaothzs# ihzsae vs=# s7~+

_K e:>+++>eKKsae7ae: + + + aeK

Tba sunnatzeo sqjs}goa hi "O:%K es tbak s}gkkad ju tba K wqjet sun.natzeo Deofa stgtas+

sD

"f%K

_K"

s= + + + = f 7 + + + 7 Kf %

Gku sunnatzeo stgta ogk tbqs ja pzettak gs g cekagz ohnjekgtehk hi tbasa+Ihz ekstgkoa# e`khzek` sogcgzs sD"=%? & sD"?%? sP > sD":%? Qsek` Yeatgs ihznqcgs ihz sunnatzeo }hcukhnegcs# gku yaothz pzettak gs

g ohn}car sqn hi sunnatzeo stgtas ogk ja pzettak gs g sunnatzesgtehk hi g}zhdqot stgta# _K"s7 + + + K%+ Cattek` s|"% s= & a:es7# tba gjhya tphargn}cas gza _?"s|"=%# |"? %# |" :? %% sP > _?"s==7%

Tbes daohn}hsetehk es kht kaoassgzecu qkewqa# jqt tba daàkazgou ohk`qzg.tehk es+ Tba daàkazgou ohk`qzgtehk es g sat tbgt ohktgeks kqnjazs ohqktekàgob destekot ohn}hkakt hi tba }zhdqot stgta jaek` sunnatzesad+ Ihz ekstgkoa#tba daàkazgou ohk`qzgtehk hi @B\ es D7#7#7 gkd tbgt hi P es D:#7+ Ocagzcugku sunnatzeo }zhdqot stgta es D?+ Tba ogzdekgcetu hi tba daàkazgou ohk`q.zgtehk es ogccad tba deyazsetu da`zaa+ Jhtb hi tbasa natzeos gza ekygzegkt qkdaz

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_CHOO gkd tba obheoa hi sunnatzesad ihzn# gkd tbau }zhyeda tba ihcchpekòcgsseogtehk tbahzans+

Tbahzan ?+ Ihz gku sunnatzeo stgta petb deyazsetu da`zaa d ?# tbaza esargotcu hka _CHOO ocgss gsshoegtad petb agob daàkazgou ohk`qzgtehk+

_}aoeogccu# d K# tba kqnjaz hi wqjets# sh tze}gzteta sunnatzeo stgtas ogkja hz`gkesad ekth hka hi tbzaa _CHOO ocgssas ihz agob hi tba tbzaa daàkazgouohk`qzgtehks+ Bhpayaz# tbes zasqct àkazgcesas th ngku.}gztu stgtas petb gbe`b cayac hi daàkazgou ek tbaez sunnatzesad za}zasaktgtehk+ Ihz ekstgkoa# paogk khp ohkocqda tbgt gku K.}gzteta stgta petb d > : es _CHOO awqeygcakt thshna Deofa stgta+

Tbahzan 4+ Ihz gku sunnatzeo stgta petb deyazsetu da`zaa d 4# g daàkazgouohk`qzgtehk edakteas g _CHOO sq}az.ocgss petb d ? ohn}car }gzgnatazs+

_h# qkcefa ek tba htbaz sq}azocgss.jgsad sobanas# tba kqnjaz hi ohktekqhqs}gzgnatazs sogcas cekagzcu petb tba kqnjaz hi wqjets+ Pa ogk saa tbes es g te`btjhqkd jaogqsa jaogqsa tba denaksehk hi tba K.}gzteta sunnatzeo sqjs}goa esK & 7# pbazags g sunnatzeo chogc h}azgtehk zanhyas g ohkstgkt kqnjaz hida`zaas hi izaadhn "saa Tbn ?9%+

4 Ohkozata Izhjakeqs gcàjzgs

Pa zaogcc tba qsqgc khtehk hi qketgc gsshoegteya gcàjzg hk g yaothz s}goa Ghyaz g acd f+ Ohksedaz g ng} " % < G G G+ Pa sgu "G# % es g qketgcgsshoegteya f.gcàjzg e

" % es jecekagz# "sq sy% sp > sq "sy sp% ihz gcc sq# sy# sp G# gkd tbaza arests s G sqob tbgt sq s > s sq > sq ihz gcc sq G+

Pa sbgcc saa cgtaz tbgt et es ohkyakeakt th ngfa tbes daketehk petbhqt ngfek`zaiazakoa th }hekts+ _ekoa "% es jecekagz# tbaza arests g qkewqa < GG Gsqob tbgt

"sq sy% > sq syTgfek` tbes th daka nqcte}ceogtehk# gkd dakek` g ng} < f G sqob tbgt

"7% > s pa hjtgek tba ihcchpek` daketehk+Daketehk 1+ Ihz shna acd f# g qketgc gsshoegteya f.gcàjzg "G##% es g f.yaothz s}goa G petb ng}s < GG G# < f G sqob tbgt "7% > "7%gkd "7 % > " 7% > 7+

5


10/44

Pa ogk gcsh ihzn g ohqketgc ohgsshoegteya f.ohgc ajzg+ Tbes es g qketgcgsshoegteya f.gcàjzg hk tba dqgc s}goa J+

"J# < J J J# < f J%Th ocgzeiu# pa ngfa g dezaot daketehk ek tazns hi J# zgtbaz tbgk J+

Daketehk 9+ G ohqketgc# ohgsshoegteya f.ohgcàjzg"J##% es g f.yaothz s}goaG petb g ng} < J J J ogccad tba ohnqcte}ceogtehk gkd g ng} < J fogccad tba ohqket sqob tbgt "7 % > " 7% gkd " 7% > "7 % > 7+

Cat G#J ja tba spg} ng} G gsihcchps< "7 %" 7% > +Pa pecc tgfa tbes gs hqz }zengzu daketehk# gs et es nhza ohkyakeakt th ar}zass

ogta`hzeogccu+

Argn}ca 5+ Cat N ja tba yaothz s}goa hik k ngtzeoas+ Tgfa th ja ngtzernqcte}ceogtehk# pbeob es gsshoegteya gkd jecekagz+ Cat ja tba k k edaktetungtzer# gkd cat < N

f ja tba tzgoa iqkotehkgc+ Tbes dgtg ekdqoas g qkewqa

ng} sqob tbgt "N####% es g Izhjakeqs f.gcàjzg+

Argn}ca 7=+ "O:# # # # % es g Izhjakeqs O.gcàjzg# pbaza "% es tba cekagzgdlhekt# vs=# s7~ es gk hztbhkhzngc jgses# gkd


11/44

G "J O%+

Pagf qketu nagks tbaza arests gk hjlaot E gkd tph kgtqzgc eshnhz}besns gkd sqob tbgt G < G > E G gkd G < G > G E+ Tbasa eshnhz}besns gzasqjlaot th oaztgek ohbazakoa }zh}azteas "saa a+`+ Ngo Cgka V::^%+

G stzeot nhkhedgc iqkothz I < O D jatpaak nhkhedgc ogta`hzeas es giqkothz tbgt }zasazyas tba nhkhedgc }zhdqot< I"G J% > I G I J+ Gstzeot nhkhedgc kgtqzgc tzgksihzngtehk es g kgtqzgc tzgksihzngtehk sqob tbgtGJ > G J +Zangzf 77+ Ek tba ogsa pbaza O gkd D gza ogztasegk nhkhedgc ogta`hzeas gkdI }zasazyas }zhdqots# gku kgtqzgc tzgksihzngtehk es nhkhedgc+ Bhpayaz# tbes es

kht tzqa ek `akazgc+Ek ngku nhkhedgc ogta`hzeas# tba tph hjlaots GJ gkd JG gza eshnhz}beo+

Ei tbaza es g kgtqzgc eshnhz}besn G#J < GJ > JG tbgt ektazgots ek g saksejcapgu petb gkd # e+a+ tbes awqgtehk bhcds "J#O#G "J#OG% G#J#O

77


12/44

tbak es ogccad g jzgedek`# gkd Yg jzgedad nhkhedgc ogta`hzu+ Ei iqztbaznhza#G#J

G#J > 7G

J # tbak es ogccad g sunnatzu# gkd

Yes g sunnatzeo nhkhedgc

ogta`hzu+G jzgedad "zas}+ sunnatzeo% nhkhedgc iqkothz es lqst g nhkhedgc iqkothz

tbgt }zasazyas jzgedek`s "zas}+ sunnatzeas%+Tbaza gza ngku argn}cas hi sunnatzeo nhkhedgc ogta`hzeas+ Tba ogsa pa gza

nhst ektazastad ek es tba ogta`hzu hi keta.denaksehkgc ohn}car Becjazt s}goasgkd cekagz ng}s "IBecj%+ Baza# pa tgfa th ja tba qsqgc takshz }zhdqot gkdO th ja tba qket+ Ihz gku tph Becjazt s}goas G gkd J# tbaza es g sunnatzu

G#J < G J J G


13/44

1+7 @zg}beogc za}zasaktgtehk

Tba tbahzu hi nhkhedgc ogta`hzeas es ek g stzhk` saksa g :.denaksehkgc tbahzu+Hka ektaz}zatgtehk ihz tbes denaksehkgcetu es tbgt tba takshz }zhdqot }zhyedasg s}goegc denaksehk# pbeca tba ohn}hsetehk hi gzzhps }zhyedas tan}hzgc# hzogqsgc denaksehk+ Tba ektaz}cgu hi tbasa tph denaksehks es za}zasaktad ju tbajeiqkothzegcetu hi tba takshz }zhdqot+

"i: `:% "i7 `7% > "i: i7% "`: `7% ":%

Hka ogk ektaz}zat tbes awqgtehk ju tbekfek` hi tbasa ihqz gzzhps hooq}uek`g }eaoa hi :.denaksehkgc s}goa+

i7

i:

`7

`:

Izhn tbes }hekt hi yeap# tba jzgofatek` ek aw ":% es g }eaoa hi assaktegccunagkek`cass suktgr# pbeob es zawqezad th ngfa shnatbek` tbgt es :.denaksehkgcju kgtqza ar}zassejca gs g "7.denaksehkgc% tazn+ Th gddzass tbes essqa# pa sbgccektzhdqoa g `zg}beogc khtgtehk ihz sunnatzeo nhkhedgc ogta`hzeas# senecgz thtbgt hi oezoqet deg`zgns+ Ad as za}zasakt hjlaots gkd khdas za}zasakt nhz.}besns+ Khzngccu# jhtb adàs gkd khdas gza cgjacad# jqt baza pa sbgcc ohksedazhkcu `zg}bs pbaza ayazu adà za}zasakts tba sgna hjlaot# sh pa sbgcc hnet adàcgjacs+ Takshzek` es dhka ju lqrtg}hsetehk gkd ohn}hsetehk es }azihznad ju

}cq`èk`# hz `cqek` tba ek}qts hi hka `zg}b th tba hqt}qts hi gkhtbaz+ Tba edak.tetu gzzhp es za}zasaktad ju gk an}tu adà gkd tba takshz qket ju gk an}tu`zg}b+

i ` > i ` ` i >i

`

"?%

_ekoa pa ogk ar}zass G G gs g }gez hi cekas# pa ogk# ihz ekstgkoa# ar}zass

7?


14/44

g ng} b < G G G G gkd ohn}hsa ek ygzehqs pgus petb htbaz ng}s+

b "i 7% >i

b

"7 i% b >i

b

"4%

Pa ogk ar}zass tba jeiqkothzegcetu hi gkd tba kgtqzgcetu hi gs ihcchps

i

`"1%

i `>

i`

"9%

Adàs# khdas# gkd adà ozhssek`s }zhyeda g `zg}beogc cgk`qgà ihz sunnatzeonhkhedgc ogta`hzeas+ Tbes cgk`qgà og}tqzas argotcu tba ohbazakoa }zh}azteas}zasakt ek g sunnatzeo nhkhedgc ogta`hzu+ _acekàz stgtas tbes }zaoesacu ek V:4^+

Tbahzan 79+ "Ohbazakoa ihz sunnatzeo nhkhedgc ogta`hzeas%+ G pacc.ihznadawqgtehk jatpaak nhz}besns ek tba cgk`qgà hi sunnatzeo nhkhedgc ogta`hzeas

ihcchps izhn tba grehns hi sunnatzeo nhkhedgc ogta`hzeas ei gkd hkcu ei et bhcds#q} th eshnhz}besn hi deg`zgns# ek tba `zg}beogc cgk`qgà+

_h igz# pa bgya ektzhdqoad g `zg}beogc cgk`qgà ihz dasozejek` tazns ek gsunnatzeo nhkhedgc ogta`hzu+ Tbasa tazns gza }zaoesacu tba dezaotad gouoceo`zg}bs àkazgtad ju tba gzzhps ek tbgt ogta`hzu+ Ei tba ceka za}zasakts gk hjlaotG# gkd G bgs g dqgc# pa ogk gotqgccu ar}zass tazns gs gzjetzgzu `zg}bs+ Paza}zasakt tba tu}a G gs g ceka dezaotad dhpk# G gs g ceka dezaotad q}# gkd tbang}s dG gkd aG izhn tba }zayehqs saotehk gs og}s gkd oq}s+

dG > aG > > "0%

Pa bgya gkhtbaz ohbazakoa tbahzan izhn V:?# :4^ ihz tbes kap# je`àz deg.`zgnngteo cgk`qgà+

Tbahzan 70+ "Ohbazakoa ihz ohn}got ochsad ogta`hzeas%+ G pac c.ihznad awqg.tehk jatpaak nhz}besns ek tba cgk`qgà hi ohn}got ochsad ogta`hzeas ihcchps

74


15/44

izhn tba grehns hi ohn}got ochsad ogta hzeas ei gkd hkcu ei et bhcds# q} th esh.nhz}besn hi deg`zgns# ek tba `zg}beogc cgk`qgà+

1+: Zacgtehksbe} petb tba Oezoqet Nhdac

Gkuhka petb shna ignecegzetu hi tba wqgktqn oezoqet nhdac pecc bgya saak deg.`zgns tbgt chhf cefa "?% gkd "4%+ Tbes es jaogqsa wqgktqn oezoqet deg`zgns gzag s}aoegc ogsa hi tba `zg}beogc za}zasaktgtehk hi nhkhedgc ogta`hzeas+ Tzgdetehk.gccu# oezoqet deg`zgns gza lqst `zg}bs tbgt gza }cgkgz gkd dezaotad gouoceo+ Tbasagza argotcu tba fekds hi `zg}bs hka ogk ohkstzqot ek g nhkhedgc ogta`hzu ei hkazaizgeks izhn qsek` tba sunnatzeo# ohn}got stzqotqza hi tba ogta`hzu+ _acekàzssqzyau V:4^ hazs g zasqct "hzeèkgccu dqa th Lhugc gkd _tzaat% gjhqt tbasa fekdshi `zg}bs gs pacc# ihz tba sgfa hi ohn}catakass+

Tbahzan 72+ "Ohbazakoa ihz }cgkgz nhkhedgc ogta`hzeas%+ G pacc.ihznad awqg.

tehk jatpaak nhz}besn tazns ek tba cgk`qgà hi nhkhedgc ogta`hzeas ihcchps izhntba grehns hi nhkhedgc ogta`hzeas ei gkd hkcu ei et bhcds# q} th }cgkgz eshth}u#ek tba `zg}beogc cgk`qgà+

Baza# }cgkgz eshth}u nagks sen}cu tbgt hka `zg}b ogk ja daihznad ekthgkhtbaz petbhqt ozhssek` adàs+ Gk argn}ca hi tbes fekd hi daihzngtehk es Aw"1%+ Oezoqet deg`zgns gcsh takd th zqk ek tzgofs# }gzgccac cekas tbgt ocagzcudestek`qesb wqjets+ Tbes zeèdetu ohnas izhn tba igot tbgt oezoqet ohn}hkaktsgza qketgzu# sh takd th bgya tba sgna kqnjaz hi ek}qt gkd hqt}qt pezas+ Bhp.ayaz# tbes ohkyaktehk es }qzacu }adg`hèogc# gs gzjetzgzu }cgkgz# dezaotad gouoceo`zg}bs ogk ja ektaz}zatad gs oezoqets petbhqt gnje`qetu+

Awqgtehk "9% ohktgeks saanek`cu nhza arhteo deg`zgns ihz tbhsa qsad thoezoqets+ Bhpayaz# tbes lqst ngfas ar}ceoet tba got hi spg}}ek` wqjets# zgtbaz

tbgk akohdek` et petb wqgktqn `gtas# sqob gs ? ohktzhccad.kht `gtas+

"2%

Ngfek` tbes daketehk gcchps ceka ozhssek`s# gkd pa ogk dh gpgu petb tbakaadcass zastzeotehk tbgt deg zgns ja }cgkgz+ Ohn}got stzqotqzas ogk ja ak.ohdad ek tba oezoqet nhdac gs pacc+ Zaogcc izhn Znf 74 tbgt pa ogk ngfa O:

saci.dqgc+ Tbak# ets gsshoegtad ohn}got stzqotqza es "5%

Khp# pa ogk }cq` gk qkfkhpk stgta gkd ektaz}zat tbes }eotqza gs g sen}ceadyazsehk hi tba taca}hztgtehk }zhthohc+

>

Zagdek` tba cait bgkd seda izhn th} th jhtthn# pa stgzt petb gk qkfkhpkhka.wqjet stgta gkd g Jacc }gez+ Pa tbak nagsqza tba zst tph wqjets ek tbaJacc jgses gkd }hst.sacaot hk tba hqtohna s+ Gs g zasqct# pa bgya taca}hztadtba qkfkhpk stgta s hkth tba tbezd wqjet+Zangzf 75+ Tba iqcc taca}hztgtehk }zhthohc phzfs dataznekesteogccu# petbhqtzacuek` hk }hst.sacaotad nagsqzanakts+ Et zawqezas ocgsseogc ohnnqkeogtehk gkdg qketgzu ohzzaotehk hk tba tbezd wqjet+ Tbes ogk ja ar}zassad `zg}beogccu gspacc# qsek` gk gjstzgot ohkstzqotehk ogccad ocgsseogc stzqotqzas+ _aa saotehk 2+7ihz g jzeai ektzhdqotehk th ocgsseogc stzqotqzas# gkd V:1^ ihz tba iqcc ohkstzqotehkhi tba taca}hztgtehk }zhthohc+

Bgyek` g ohn}got stzqotqza nagks pa ogk dzh} tba ohkdetehk tbgt `zg}bs jagouoceo+ Gcsh khta tbgt chh}ek` jgofpgzds qsek` g ohn}got stzqotqza }azihznsg }gztegc tzgoa+ Ihz ekstgkoa# pa ogk tzgoa hqt tba ze`bt wqjet hi b < O: O: O: O: gs ihcchps

b

Pa bgya saak tbgt gcc hi tba ek`zadeakts hi g sunnatzeo# ohn}got stzqotqza

gza }zasakt ek tba tzgdetehkgc oezoqet nhdac hi wqgktqn ohn}qtek`+ Gcchpek`gzjetzgzu `zg}bs lqst ar}chets tba stzqotqzgc zacgtehksbe}s hi gcc hi tbasa ek`za.deakts ek g pgu tbgt es }zhygjcu qkgnje`qhqs+ Tbgt es# pa ogk gcpgus ektaz}zatg `zg}b gs g oezoqet ju za}cgoek` ozhssek`s petb ? OKHT `gtas gkd chh}s petb}gztegc tzgoas+ Dqa th Tbn 70# pa bgya tba zangzfgjca igot tbgt gku sqobektaz}zatgtehk pecc awqgc gku htbaz+

79


17/44

9 Nhkhedgc gcàjzgs

Gcàjzgs# ohgcàjzgs# Izhjakeqs gcàjzgs# gkd je.gcàjzgs gcc arest ek tba ogta.`hzu hi Becjazt s}goas+ Bhpayaz# tbau ogk gcsh ja ihznqcgtad ek g nqob nhzaàkazgc sattek`# pbaza gcc hi tba gjhya stzqotqzas gza dakad ektazkgc th g ogt.a`hzu Y+ Tbgt es th sgu# pa bgya g pgu hi rek` gk gcàjzgeo stzqotqza Ggkd ihznqcgtek` tba ohkdetehks ju pbeob hka ogk sgu gk hjlaot Y Yogzzeastba stzqotqza hi G+ Iqztbaznhza# pa bgya g àkazgc ohkdetehks ihz pbgt gkG.stzqotqza.}zasazyek` ng} "e+a+ gk G bhnhnhz}besn% izhn Y7 th Y: chhfs cefa+

9+7 Ektazkgc nhkheds gkd ohnhkheds

G nhkhed ektazkgc th g nhkhedgc ogta`hzu "Y# # E%# es gk hjlaot G gkd g }gezhi ng}s < G G G dakek` nqcte}ceogtehk gkd < E G }eofek` hqt tbaqket+ Nqcte}ceogtehk es gsshoegteya# sh tbes deg`zgn ohnnqtas+

"G G% G G G

G

G "G G% G G

G

G

Nqcte}ceogtehk es cait gkd ze`bt qketgc# sh tbes deg`zgn gcsh ohnnqtas+

G G E

G G

E G

G G

G

G

Pa khp bgya tba gjecetu th awqe} g cgzà ygzeatu hi hjlaots petb g nqcte.}ceogteya stzqotqza+ Ek tba ogsa pbaza Y> _at# tbes zaohyazs tba qsqgc khtehk hinhkhed+ Nhkheds ektazkgc th Gj# tba ogta`hzu hi gjacegk `zhq}s# gza zek s# gkdnhkheds ektazkgc th Yaotf gza gsshoegteya f.gcàjzgs+

Tba dqgc hi g nhkhed es g ohnhkhed+ Ihz gk hjlaot G ek Y# hka ogk daka gektazkgc ohnhkhed gs g tze}ca "G# < G G G# < G E% pbaza tba ihcchpek`deg`zgns ohnnqta+ Ohgsshoegteyetu "7 % " 7% > +@zg}beogccu# tba tbezd ohkdetehk da}eots gs ihcchps+

> > "7=%

Tba Izhjakeqs edaktetu `qgzgktaas tbgt gku tzaa ohksestek` hi ### bgs gqkewqa khzngc ihzn+ Ju tzaa# pa nagk tba `zg}b ohktgeks kh qkdezaotad ouocas+Ihz tbes setqgtehk# pa ektzhdqoa g s}aoegc khtgtehk ihz ogkhkeogc tzaas# fkhpk gss}edazs+

72


19/44

_=n _

7n _

k

= _k

7 _k

n >

+++

+++ +++

Ihz argn}ca# pa ogk qsa tbes khtgtehk th ohkstzqot tba ihcchpek` ng}s+

_=: > _:= > "77%

Daketehk :7+ Ek g sunnatzeo nhkhedgc ogta`hzu Y# pa sgu g nhz}besn n esohkstzqotad izhn g Izhjakeqs gcàjzg "G####% ei et ohksests hkcu hi tba ng}s7G# # # # # G#G#G gkd G#G+ Pa sgu sqob g nhz}besn es ohkkaotad ei ets`zg}beogc za}zasaktgtehk es ohkkaotad+

Gzjetzgzu nhz}besns ohkstzqotad izhn Izhjakeqs gcàjzgs bgya pacc jabgyadkhzngc ihzns+ Tba ihcchpek` es g ihcf tbahzan# tbgt bgs jaak stgtad ek a+`+ V0^+

Tbahzan ::+ G ohkkaotad nhz}besn ohkstzqotad izhn g Izhjakeqs gcàjzg Ges ohn}catacu dataznekad ju ets kqnjaz hi ek}qts# hqt}qts# gkd chh}s+

Pa ogk saa tbgt tba hkcu }aztekakt ekihzngtehk gjhqt g "ohkkaotad% `zg}b hig Izhjakeqs gcàjzg es ets ek}qts# hqt}qts# gkd tba kqnjaz hi chh}s# hz bhcas+Tbes sq`àsts tbgt `zg}bs hi Izhjakeqs gcàjzgs gza ek shna saksa bhnhth}u.ekygzegkt+ Tbes es ekdaad tba ogsa# ek tba g}}zh}zegta ogta`hzu< tba ogta`hzu hiochsad 7.denaksehkgc ngkeihcds gkd :.denaksehkgc ohjhzdesns+ Tbes ohkoa}t esar}hqkdad q}hk ek datgec ek Lhgoben Fhofs jhhf hk Izhjakeqs gcàjzgs V0^+ Etihcchps agsecu izhn tbes igot tbgt gku `zg}b ek g Izhjakeqs gcàjzg bgs g khzngcihzn+

Ohzhccgzu :?+ Gku hjlaot gdnettek` g Izhjakeqs gcàjzg gdnets g saci.dqgcohn}got stzqotqza+

]zhhi+ Qsek` tba og} gkd oq} izhn "77%# pa saa tbgt tba Izhjakeqs gcàjzgsgtesas Aw 7+

>

Ohzhccgzu :4+ Gku nhz}besn ohkstzqotad izhn g Izhjakeqs gcàjzg G es awqey.gcakt th hka hi tba ihzn s7

+ + +

sk# pbaza agob se es hi tba ihzn es g s}edaz

petb = hz nhza chh}s gt ets oaktaz

+++

+++

+++

+++

>

+++

+++ +++

+++

Tbes es tbak g }zhdqot hi }zayehqscu ohksedazad ogsas+

::


23/44

9+? Tba àkazgc ogsa< ]ZH]s

Gcc hi tba gjhya argn}cas gza s}aoegc ogsas hi g nhza àkazgc ohkstzqotehk+Th nhteygta tbes ohkstzqotehk# pa zst ohksedaz tba sngcc ogta`hzu D hi ketahzdekgcs k > v= 7 + + + k 7~ gkd hzdaz.}zasazyek` ng}s+ Tbes es tbaogta`hzu Ngo Cgka ogccs tba gcàjzgests sen}ceoegc ogta`hzu ek V::^+ Pa ogkohksedaz tbes g nhkhedgc ogta`hzu# petb tba nhkhedgc }zhdqot jaek` hzdazaddeslhekt qkehk+

n k > n & kv= + + + n 7~ v= + + + k 7~ > v= 7 + + + n & k 7~

Ihz i < n n gkd ` < k k# pa tgfa tba khzngc deslhekt qkehk hiiqkotehks# ueacdek` i ` < n k n k+

"i

`%"r% > i"r% ei r 3 n`"r n% htbazpesaTbes ogta`hzu bgs gk eketegc hjlaot = > v~# pbeob es gcsh tba takshz qket+ Et

gcsh bgs g taznekgc hjlaot 7 > v=~+ Jaogqsa hi tbes igot# tba ihcchpek` ng}sarest gkd gza qkewqa+

n < : 7 a < = 7Iqztbaznhza# sekoa 7 es taznekgc# tba ihcchpek` deg`zgns ohnnqta+

? :

7

? :

n ed7

n

ed7 n n

7 7

:

7

:

ed7 ana ed7 n

Iqztbaznhza# tba gzzhps n# a gkd edakteteas edk àkazgta tba pbhca ogta`hzuD+ Khp# cat qs ohksedaz g stzeot nhkhedgc iqkothz I izhn D th shna htbaz"}hssejcu cgz a% stzeot nhkhedgc ogta`hzu Y+ Cat I"7% > G+ Tbak# I":% >I"7 7% > I"7% I"7% > G G+ Cat I"n% > gkd I"a% > + Khp# cats chhfgt tba engàs hi tba gjhya deg`zgns+

G G G G G

G

G

G

G G

G

G

G

G G

G G

G

G G

G

G

Zaogcc tbgt ek g stzeot nhkhedgc ogta`hzu# > ed+ _h# tba gjhya deg`zgns gzastzeot yazsehks hi tba nhkhed deg`zgns izhn saotehk 9+7+ _ekoa D es àkazgtadju n# a# gku gzzhp ek tba engà hi I nqst gcsh ja+ _h# tba ogta`hzu D akohdasargotcu tba dgtg ohktgekad ek tba tbahzu hi nhkheds+ Gs sqob# et es ogccad tba]ZH ihz nhkheds+

:?


24/44

Daketehk ?:+ G ]ZH "ihz ]ZHdqot ogta`hzu% es g stzeot nhkhedgc ogta`hzupbhsa hjlaots gza kgtqzgc kqnjazs+

D es tba tbahzu hi nhkheds# pbazags I dakas g nhdac hi tbgt tbahzu ekgkhtbaz "qsqgccu ohkozata% ogta`hzu+ Ei Y> _at# tbes es awqeygcakt th tba qsqgcdaketehk hi nhkheds+ Gs sqob# I es ogccad gk gcàjzg hi tba ]ZH D+

Daketehk ??+ Ihz g ]ZH ]# g stzeot nhkhedgc iqkothz es fkhpk gs g ].gcàjzg+Ihz ].gcàjzgs I gkd @# g stzeot nhkhedgc kgtqzgc tzgksihzngtehk es fkhpk gsg ].gcàjzg bhnhnhz}besn+

D.gcàjzg bhnhnhz}besns gza tbak tba qsqgc khtehk hi nhkhed bhnhnhz.}besns+ Tbgt es th sgu# g D.gcàjzg bhnhnhz}besn es ohn}catacu dataznekadju g ng} b < I"7% @"7% tbgt ohnnqtas g}}zh}zegtacu petb tba nqcte}ceogtehkgkd qket ng}s ek tba engàs hi I gkd @+ Gcsh# et es kht deoqct th saa pbu Dh}

es tba ]ZH ihz ohnhkheds+

Khp# ohksedaz tba sfacatgc ogta`hzu I hi keta sats gkd iqkotehks+ Tbesogta`hzu es wqeta senecgz th D# jqt petb hka en}hztgkt deazakoa+ 7 es steccg taznekgc hjlaot# sh pa ogk stecc daka qkewqa gzzhps n gkd a tbgt sgtesiutba gjhya ohnnqtgtehk }zh}azteas+ Bhpayaz# tbasa gza kh chkàz sqoeakt thàkazgta tba pbhca ogta hzu+ Ek D tbaza gza kh khk.tzeyegc jelaotehks# jqt Iohktgeks gcc hi tba }aznqtgtehk ng}s hi keta sats+ Th àkazgta tbasa# pa kaadhka nhza ng}# g spg} ng} s < : : ednk# sh I es g sunnatzeo nhkhedgc ogta`hzu+ Gcc hi tba deg`zgnsek D ohnnqta# gchk` petb hka nhza Ngt"K%%# gkd Bh}i gcàjzgs "Ngt"\%%+ Edakteiuek` tba ]ZH] ihzg tbahzu ogk ennadegtacu ngfa daa} zasqcts gjhqt tba tbahzu ar}ceoet+ Ihz ek.stgkoa# Tbn ?= es gk agsu ohzhccgzu hi tba igot tbgt tba ]ZH] ihz _OIGs es tbaogta`hzu hi ohs}gks hi keta sats+

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Tbahzan ?9+ Ei s gkd s gza sunnatzeo K.wqjet stgtas sqob tbgt s gkd

s

gza _CHOO.awqeygcakt# tbak tbaza arests gsek`ca ekyaztejca# 7.wqjet ng} G

sqob tbgts > GKs

Pa gza khp zagdu th stgta tba ngek zasqct ihz tbes saotehk+

Tbahzan ?0+ Agob _OIG @ hkO: ogkhkeogccu ekdqoas g sunnatzeo stgta ekO:O:O: pbeob es _CHOO.awqeygcakt th s@B\+ Ohkyazsacu# gku sunnatzeostgta tbgt es _CHOO.awqeygcakt th s@B\ gzesas izhn gk _OIG @+]zhhi+ "% Cat " # # # % ja gk _OIG+ Daka tba _OIG \gs ihcchps syKhp# cattek` sq > jst & ost#

s > jstt & ostt & "7 g%stt

]cq`èk` ek th # cattek` d > "7 g% gsttts & jstt & ostt & dstt tsd

> =# htbazpesa es sa}gzgjca+ Cat

ss

> 7

d"j

st

& o

st

% gsttts & dsst & dstt tsKhta tbgt ss > fst# htbazpesa es sa}gzgjca+ Gcsh# ju daketehk hi Izhjakeqs

gcàjzg# es g khk.daàkazgta oh}gezek`+ _h# obhhsa khk.}zh}hztehkgc ss# stsqob tbgt 7

dst

Tbak# tba stgta gsshoegtad petb "# % es " 7% > stts & sstt & sttt

Khp# daka tba ihcchpek` chogc ng}s gkd > +

Tgfek` tba .sunnatzeo nhkhedgc ogta`hzu "IdBecj# "%%# pbaza "% es tbacekagz gdlhekt# V5^ `has hk th }zhya g stzhkàz zasqct gjhqt ocgsseogc stzqotqzas+

Tbahzan 4=+ Ihz g ._OIG "G## %# oh}eas k hztbhkhzngc yaothzs s}gk.kek` G+ Iqztbaznhza# et es qkewqacu dataznekad ju tbasa yaothzs+

Khp# cat qs tgfa sazehqscu tba igot tbgt oh}eas g sat hi nagsqzanakt hqt.ohnas# tbekfek` hi tban gs tba aeàkjgses hi g khk.daàkazgta hjsazygjca+ Pazagcesa tbgt khk.daàkazgta hjsazygjcas qkewqacu datazneka ocgsseogc stzqotqzas#gkd yeoa.yazsg+

Ohksedaz g }gez hi ngrengccu khk.ohnnqtek` hjsazygjcas# cefa tba ]gqce \gkd R ngtzeoas+ Tbasa bgya tba ihcchpek` aeàkjgsas# zas}aoteyacu vs=# s7~ JR > vs s~

Pa ohkstzqot g ocgsseogc stzqotqza th oh}uek` gkd dacata J\

Pa ogk gcsh ektzhdqoa }bgsa `gtas > > s== & s77Bhpayaz# tba og}s gkd oq}s ohnnqta gs ihcchps

Tba ceka petb tba teof es ogccad tba dqgcesaz hi tbasa Izhjakeqs gcàjzgs+ Pasbgcc zaiaz th et gs d@P + Ohkozatacu# et es g KHT `gta hk tba ohn}qtgtehkgcjgses "e+a+ tba ]gqce R h}azgthz%+ Pa khta tbgt et es saci.gdlhekt# qketgzu# gkd gIzhjakeqs gcàjzg akdhnhz}besn hk @+

" . %

> ..

>.

.. > "75%

Pa sbgcc qsa tbes igot th e`khza dezaotehk hk adàs jatpaak dhts hi tba sgnaohchqz ohktgekek` teofs+

. . > . . . > .

Khta tbgt > :s=+ _h# q} th g sogcgz# . > + Tbes en}ceas tbgt gcshoh}eas chh}s+

> . >.

.

> . . >

Khp pa gza }za}gzad th saa bhp tba cekagz gdlhekt "% gots hk tbasa stzqo.tqzas+ @ es g .OIG# sh > gkd > + P eskt g .OIG# jqt ethjaus sfap }zh}azteas+

>.

..

>

.

Tbes }zh}aztu cats qs tqzk ngku edakteteas q}seda.dhpk+ Pacc }zhyeda hkadazeygtehk gs gk argn}ca+

]zh}hsetehk 4:+ >

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]zhhi+ Pa }azihzn tba dazeygtehk qsek` edakteteas gkd Aws 75+

>

..

.

.

..

> ..

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.

> ..

.

.

> .. . >

Paya gczagdu saak tba g}}ceogtehk hi Aws 75 ihz }qccek` dqgcesazs ek gkdhqt hi g 4.ouoca+ Nhdqch tbes awqgtehk# tbaza gza tph htbaz ohk`qzgtehks ihzdqgcesazs ek g 4.ouoca# pbeob jhtb bgya desohkkaotek` zapzetas+

. . > . >.

":=%

Izhn tbes gkd tba khzngc bhnhnhz}besn zapzetas# hka ogk dazeya zadqotehkstbgt desohkkaot gku gctazkgtek`.ohchqz 4.ouoca+ Iqztbaznhza# hka ogk qsa tbagjhya zqcas th dazeya zadqotehks ihz gku :.ouoca+

_hna hi tba zqcas }zhyedad baza gza cefacu th ja zadqkdgkt+ Et es g ngttazhi hk`hek` phzf th bhka tbasa dhpk th g nekengc# aca`gkt sat hi grehns# pbeobueacd g `zg}b zapzetek` tbahzu petb keoa }zh}azteas# sqob gs chogc ohkqakoa+Tba }zasakoa hi sqob g sat hi grehns ogk tbak ihzn tba daketehk hi gk N_.}gez# gkd gcchp et th ja ceitad ekth gku sunnatzeo nhkhedgc ohktart# kht lqstwqjets ek tba ogta`hzu hi keta.denaksehkgc Becjazt s}goas+

2+:+7 Qkeyazsgcetu

Ei pa gcchp gzjetzgzu 7.wqjet stgtas# gk N_.}gez es ohn}qtgtehkgccu qkeyazsgc+Th saa pbu tbes es sh# pa sbgcc qsa tba @B\(P }gez th ohkstzqot tba CQ.daohn}hsetehk hi gk gzjetzgzu : : ngtzer+ Ihz hqz }qz}hsas# pa sbgcc qsa tba`akazgcesad ihzn hi CQ.daohn}hsetehk ]CDQ ":7%

Pbaza ] es g }aznqtgtehk# C gkd Q gza qket.deg`hkgc chpaz. gkd q}}az.tzegk`qcgz ngtzeoas# gkd D es g deg`hkgc ngtzer+

Ihz g yaothzs s# s O:# pa ogk ohkstzqot tba ihcchpek` ng}s

: 7= :

Iqztbaznhza# sq}}hsa tbgt : > 7# tbak qsek` tba dqgcesaz d@P # pa ogkohkstzqot tbasa ng}s+

Q >

7 7= 7

C d@PQ d@P >

7 =

7 7

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Zagcesek` tbgt tba hkcu tph }aznqtgtehks hyaz O: gza tba edaktetu gkd d@P #pa ogk ohkstzqot gku ng} ju daohn}hsek` gs ek Aw :7+

G

.

.hz

.

.

."::%

Canng 4?+ Pa ogk ohkstzqot g OKHT `gta gs ihcchps s= gkd > s7gkd tba dqgcesaz es tba KHT `gta+ Iezst# pa }qt ":?% ek g ihzn ohkyakeakt ihz}cq`èk` ..

.

.

Ei pa }cq` s= ekth tba zst wqjet ..

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.

> ..

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> . >

+++pa àt s= 7+ Ei pa }cq` ek s7 ..

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.

> ..

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> .

.

.

> . > .

+++pa àt s7 KHT+Tbahzan 44+ Gk N_.}gez petb gzjetzgzu hka.wqjet stgtas es qkeyazsgc ihz wqgk.tqn ohn}qtgtehk+

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ek 7

ek :

otzc

hqt

"g% Ocgsseogc

ek7

ek:

otzc

hqt

"j% Wqgktqn

Ie`qza 7< Tph fekds hi nqcte}carazs

5 Ekdqoteya Ohkstzqotehk hi _tgtas

Zaogcc izhn saotehk ?+: tbgt Cgngtg at gc za}zasakt K.}gzteta _CHOO sq}az.ocgssas petb stgtas hi tba ihcchpek` ihzn

s > s=# & s7# ":1%

Pbaza s gkd s gza "K 7%.}gzteta stgtas+ @eyak tbgt pa ogk ohkstzqot"K 7%.}gzteta stgtas# pa ogk zagob gku K.}gzteta stgta petb tbes ohkstzqotehk+]zh}hsetehk 49+ Gku stgta ek"O:%K ogk ja ar}zassad ek tba ihzn hi Aw+ :1+

]zhhi+ Cat s ja g stgta ek "O:%K+ Cat s > "=s7%s gkd s > "7s7%s+_ekoa vs=# s7~ es gk hztbhkhzngc jgses# s ogk ja pzettak gs g sqn hi ets}zhlaotehks# gs ek Aw+ :1+

_h# pa pgkt th zst wqjet th got gs g spetob jatpaak tph }hssejca h}tehks+Ek ocgsseogc oezoqets# tbes phzf es dhka ju g dayeoa ogccad g nqcte}caraz# hz NQR"Ie`+ 7+g%+

Tba ch`eo `gta tbgt dhas tba ozeteogc phzf hi tba NQR es tba ?.stgta `gta+Tbasa `gtas jabgya gs gk gkd acaotzhkeo spetob# petb tba seda ek}qt tgfek` gohktzhc jet+

=

>

7

>

Tbes chhfs yazu senecgz# th}hch`eogccu# th tba jabgyehqz hi tba nqcte}ceogtehk

ihz gk GOIG "O:# # %+ Cat gkd ja tba tph }hssejca ek}qts ihz tba ohktzhcwqjet# tbak >

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Qsek` tbes igot# pa ogk ohkstzqot g wqgktqn nqcte}caraz "Ie`+ 7+j% yazusenecgzcu th bhp tba ocgsseogc NQR pgs ohkstzqotad+ Pa khp gdg}t tba wqgktqn

NQR th bgya tba ohktzhc wqjet gs gk hqt}qt# th ngtob Aw :1 gkd hzeakt tba`qza th}.th.jhtthn+

WNQR .

.

.

.

.":9%

Khteoa tba en}hztgkt igot tbgt# qkcefa ek tba ocgsseogc ogsa# pbaza pa ogk sen.}cu desohkkaot gk ek}qt# ek tba wqgktqn ogsa# pa nqst }zhlaot et hqt+ Jagzek`tbes ek nekd# pa ngfa tba ihcchpek` gssaztehk+

]zh}hsetehk 40+ WNQR es g ng} tgfek` stgtas hi tba ihzn s s th stgtashi tba ihzn

7ss=# & 7ss7# ]zhhi+ G`gek# tbes ogk ja }zhyad ju ohkozata ogcoqcgtehk# jqt et es }azbg}s nhzaakce btakek` th chhf gt tba zapzetas+ Pa jaèk ju }cq`èk` ek s gkd s gkd}zhlaotek` tba cait wqjet petb =s+

.

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.

.

.>

. .

.

.>

. .

.

.>

..

.

. >

.

. >

Pa saa tba ar}aotad zasqct hi 7ss+ Khp# ei pa }cq` ek 7s<

.

.

.

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.>

. .

.

.>

. .

.

.>

.

.

. >

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.

.

>

.

.

>

pa àt 7ss+Ohzhccgzu 42+ WNQR nqst bgya g khk.dataznekesteo }za}gzgtehk+

]zhhi+ Ei pa ohqcd g}}cu WNQR petb }zhjgjecetu 7 ihz gku ek}qt stgtas# tbak paohqcd g}}cu et th tba stgta s==+ Jqt tbes es en}hssejca# gs "WNQR%s== > =+Zangzf 45+ Tba }zhhi hi ]zh} 40 da}akds hk tba sogcgzs gsshoegtad petb`zg}beogc edakteteas+ Tbes datgec bgs jaak hnettad baza# jqt taobkeogccu# pa bgya

hkcu }zhyad tbgt tba ng} WNQR es77ss=# & :7ss7#

Et lqst bg}}aks th ja tzqa ek tbes ogsa tbgt 7 > : > 7+ Tbaza arests gstzeot yazsehk hi tbes zapzeta tbahzu pbeob ohqcd }zhya tbes stzhkàz zasqct#jqt tba datgecs hi tbes gza kht ozqoegc th tba argn}cas èyak ek tbes za}hzt+

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7= Ohzhccgzeas

Pa sbgcc khp ar}chza shna hi tba en}ceogtehks hi tba zasqcts ek tbes za}hzt+ Kh.teoa bhp nhst hi tba ohzhccgzeas ngfa qsa hi tba ektaz}cgu jatpaak aktgk`canaktocgsseogtehk gkd gcàjzg+

Ohzhccgzu 1=+ Ayazu Izhjakeqs gcàjzg hyazO: es chogccu awqeygcakt th hka tbgtes s}aoegc hz gkte.s}aoegc+

]zhhi+ Agob OIG hk O: ek IBecj ekdqoas g tze}gzteta wqjet stgta _=? # pbeob#gs g ohksawqakoa hi tba OIG jaek` qketgc# es khk.daàkazgta+ Bakoa et aetbaznqst ja _CHOO.awqeygcakt th s@B\ hz sP# nhza }zaoesacu# et es aetbaz "C C C%s@B\ hz "C C C%sP ihz shna ekyaztejca C+ Gkd ohksawqaktcu# juTbahzans ?0 gkd ?2# tba OIG ogk ja tzgksihznad aetbaz ekth gk _OIG hz gkGOIG+

Tbes en}hztgkt ohzhccgzu es g kap ocgsseogtehk zasqct ihz OIGs hyaz O:# gkdngfas tba destekotehk petb g }qzacu gcàjzgeo }zh}aztu+ Tbes zasqct sq`àststbgt tbaza ngu ja gk gcàjzgeo lqsteogtehk ihz tba igot tbgt tbaza gza hkcu tphàkqeka tze}gzteta _CHOO ocgssas hyaz O: O: O:+Ohzhccgzu 17+ Ayazu gsshoegteya# qketgc gcàjzg ekO: gdnets g Izhjakeqs ihzn+

]zhhi+ Cat "G##% ja gk gsshoegteya# qketgc gcàjzg+ es qketgc# sh ek }gzteoqcgz#nqst ja khk.sa}gzgjca+ Tbazaihza et es _CHOO awqeygcakt th N < O:O: O:#pbaza N es hka hi tba ihcchpek`+

N > @ > s===s & s777s hz N > P > s===s & s7"7=s & =7s%

_h# > "C7

%N"C:

C?

%# ihz shna ekyaztejca ng}s Ce+ Ek tba ogsa pbazaN > @ # cat > &s"C .77 %+ Tbak > "==s & 77s%"C: C?%# pbeob nqstja g khk.daàkazgta }gezek` ihz Ce ekyaztejca+ Ek tba ogsa pbaza N > P # cat > 7s"C .77 %+ Tbak > "7=s & =7s%"C: C?%# pbeob es gcsh g khk.daàkazgta}gezek`+ Tbazaihza# gku gdnets g Izhjakeqs ihzn+

Zangzf 1:+ Hka ne`bt ja tan}tad th jaceaya tbes zasqct ihz be`baz denaksehks#jqt pa ogk gczagdu kd g ohqktaz.argn}ca ek O?+ Tgfek` vs=# s7# s:~ gs tbaohn}qtgtehkgc jgses# tba ihcchpek` gsshoegteya# qketgc gcàjzg dhas kht gdnet gIzhjakeqs ihzn s===s & s7"=7s & 7=s% & s:"=:s & :=s% > s=

Cat > g

=

s& j

7

s& o

:

s+ Tbak

> g==s & j"=7s & 7=s% & o"=:s & :=s%

Th sbhp tbes es daàkazgta# et sqoas th sbhp tbgt et nqst bgya g khk.tzeyegcze`bt kqccs}goa K+ Ei j > =# tbak s7 K+ Eio > =# tbak s: K+ _h# sq}}hsa jgkd o gza jhtb khk.|azh+ Tbak sq > js7 "j(o%s: > = gkd sq K+

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ihz OIGs gkd _OIGs# gs pacc gs ohnnqtgteya jegcàjzgs V:9^ gkd Bh}i gcàjzgs#pbeob gza gcc stzqotqzas ozeteogc th qkdazstgkdek` hqz arestek` ektazgotehk tba.

hzeas+ Cgofs natbhd hiohn}hsek` ]ZH]s ohqcd ayak ja tba ze`bt izgnaphzfihz qs th desoqss ektazgotek` nhkhedgc gcàjzgs ek àkazgc+

Pa gcsh pesb th stzak`tbak tba ohkstzqotehktaobkewqa en}cead ju ]zh} 40 thg ocgsseogtehkzasqct hk `zg}beogc stgtas# gkgch`hqs th tba hka }zhyedad ju V74^+Tbes es `hek` th zawqeza g jattaz qkdazstgkdek` hi bhp ygzehqs daohn}hsetehksgkd ogkhkeogc ihzns hi aktgk`cad stgtas gza zaaotad ek tba tbahzu# gkd }zhjgjcug igez gnhqkt hi ngtzer ogcoqcgtehk+ Gkhtbaz }htaktegc gyakqa ihz tbes zasagzobes th }azihzn tba `zg}beogc gkgch`qa th tba "sen}caz% ocgsseogtehk zasqct ihzsunnatzeo stgtas# }zhyedad ju Jgstek at gc ek V:7^+ Tba `zg}beogc khtgtehk bgsgk ayedakt khtehk hi sunnatzu "kgnacu `zg}b sunnatzu%# pbeob es sqoeakt thsbhp g stgta es sunnatzeo+ Bhpayaz# et es qkfkhpk pbatbaz tba tbahzu hi N_.}gezs es stzhk` akhq`b th sbhp tbgt tba `zg}b hi g sunnatzeo stgta es kaoassgzecuawqeygcakt th g sunnatzeo `zg}b+ Iqztbaznhza# tba `zg}beogc tbahzu bgs g zeobogtgch`qa hi sunnatzesek` ng}s# tbhq`b pa dhkt fkhp ei et ohktgeks tbasunnatzesek` ng} tbgt es ozqoegc th Jgsteks sobana+

Et bgs gczagdu jaak sbhpk tbgt tba cgk`qgà hi ohn}canaktgzu ocgsseogcstzqotqzas schts stzge`bt ek th nagsqzanakt jgsad wqgktqn ohn}qtgtehk "saaa+`+ V?:^%+ Pa bgya saak ek saotehk 2+? tbgt pa ogk nhya izaacu jatpaak OO_sgkd N_.}gezs# sh tbes zasqct artakds th N_.}gezs+ Izhn tbes }hekt hi yeap#P(@B\ `zg}bs gza g tu}a hi àkazgcesad `zg}b stgta+ Tbes hjsazygtehk gkd etsen}ceogtehks pgzzgkt iqztbaz ekyaste`gtehk+ Tba igot tbgt P(@B\ `zg}bs ogk jayeapad gs g àkazgcesad ocqstaz stgta nagks ngku hi tba deg`zgns pa ohkstzqotogk ja zagcesad qsek` NJWO+ Pa ogk gcsh zagcesa tba àkazgthzs dezaotcu ek gsce`bt àkazgcesgtehk hi tba oezoqet nhdac+ ]za}gzgtehks gkd ohnqcte}ceogtehksogk ja zagcesad dezaotcu# gkd }zhlaotehks gkd nqcte}ceogtehks ogk ja zagcesad#

ektzhdqoek` azzhz tazns+ Tbasa azzhz tazns ogk tbak shnatenas ja ohnnqtgtadth tba akd hi tba oezoqet petb zapzetek` "hz iad ihzpgzd ek tba cgk`qgà hiNJWO% th dase`k dataznekesteo hz kagz.dataznekesteo }zhthohcs+ Tbasa }zhoassascakd ozadejecetu th tba ocgen tbgt tbahzateogc `geks ek tba izgnaphzf hi N_.}gezsogk ja tzgkscgtad ekth zagc }zhthohcs# pbeca tba sustan etsaci zangeks g`khsteoth tba qkdazcuek` ohn}qtgtehkgc }gzgde`n+

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Zaiazakoas

V7^ O+ B+ Jakkatt# @+ Jzgssgzd# O+ Oza}agq# gkd Z+ Lh|sg# Taca}hztek` gkqkfkhpk wqgktqn stgta yeg dqgc ocgsseogc gkd A]Z obgkkacs# ]bus+ Zay+Catt "755?% +

V:^ G+ Afazt# Wqgktqn Ozu}th`zg}bu Jgsad hk Jaccs Tbahzan# ]buseogcZayeap Cattazs 90 "7557% kh+ 9# 99799?+btt}


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V79^ N+ Baek# P+ Dqz# L+ Aesazt# Z+ Zgqssakdhzi# N+ Y+ dak Kast# gkd B+ L+Jzea`ac# Aktgk`canakt ek `zg}b stgtas gkd ets g}}ceogtehks#

gzRey


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V?7^ C+ Derhk# Z+ Dqkogk# gkd G+ Fessek az# Wqgkthngteo+ G shitpgza thhcihz `zg}b zapzetek`+

btt}

entanglement, monoidal categories, and frobenius algebras

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