Random Graphs and Social Networks

8/10/2019 Random Graphs and Social Networks

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RJWKF HRCNX njr tfg purpjsg jn hmseussmja? ej``gats crg `jst wgdej`g!

RCAHJ@ KRCYFU CAH UJEMCD AGX[JRLU?

Ca Gejaj`mes Ygrspgetmvgby

]caams @. Mjcaamhgs

Hgpcrt`gat jn Gejaj`mes, Xunts Wamvgrsmty

ycaams.mjcaamhgsItunts.ghu

Ouag 2998 Rgv. Cprmd 3;, 299>

Cbstrcet

Xfms gedgetme rgvmgw jn eurrgat rgsgcref ja agtwjrls g`pfcsmzgs tfrgg strcahs jn tfg dmt-

grcturg ja sjemcd agtwjrls. Xfg ffirst strcah ms ej`pjsgh jn `jhgds jn gahjkgajus agtwjrl

njr`ctmja nrj` bjtf tfg gejaj mes cah tfg ej`putgr semgaeg dmtgrcturg. Xfg rgvmgw fmkf-

dmkfts tfg sgasmtmvg hgpgahgaeg jn tfg tjpjdjky jn gahjkgajus agtwjrls ja pcrc`gtgrs jn tfg

bgfcvmjrcd `jhgds g`pdjygh.

Xfg sgejah strcahhrcws nrj` tfg rgegat gejajpfysmes dmtgrcturg ma jrhgr tj rgvmgw tfg

rgegat rgvmvcd jn matgrgst ma tfg rcahj` krcpf tfgjry. Xfms `ctfg`ctmecd tjjd cddjws jag tj

stuhy sjemcd agtwjrls tfct rgsudt nrj` uaejjrhmactgh rcahj` cetmja jn mahmvmhucds ma sgttmak

up ejaagetmjas wmtf jtfgrs. Xfg rgvmgw gxpdjrgs c au`bgr jn gxc`pdgs tj cssgss tfg pjtgatmcd

jn rgegat rgsgcref ja rcahj` krcpfs wmtf crbmtrcry hgkrgg hmstrmbutmjas ma ceej``jhctmak

`jrg kgagrcd bgfcvmjrcd `jtmvctmjas njr sjemcd agtwjrl njr`ctmja.

Xfg tfmrh strcah njeusgs ja c spgemffie `jhgd jn sjemcd agtwjrls, @crljv rcahj` krcpfs,

tfct ms qumtg egatrcd ma tfg `ctfg`ctmecd sjemjdjky cah spctmcd stctmstmes dmtgrcturgs but dmttdg

lajwa jutsmhg tfjsg dmtgrcturgs. Xfgsg crg rcahj` krcpfs wfgrg tfg gvgats tfct hmgrgat

ghkgs crg prgsgat crg hgpgahgat, mn ghkgs crg maemhgat tj tfg sc`g ajhg, cah mahgpgahgat,

jtfgrwmsg. Xfg pcpgr cssgssgs tfg pjtgatmcd njr gejaj`me cppdmectmjas wmtf tfms pcrtmeudcr

tjjd. Xfg pcpgr ejaeduhgs wmtf ca cssgss`gat jn jbsgrvcbdg ejasgqugaegs jn jptm mzmakbgfcvmjr ma agtwjrls njr tfg purpjsg jn gstm`ctmja.

MjcaamhgsRcahj`[email protected]


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Ejatgats

3 Matrjhuetmja 5

2 @jhgds jn Gahjkgajus Agtwjrl Njr`ctmja >

2.3 C Um`pdg @jhgd jn Agtwjrl Njr`ctmja . . . . . . . . . . . . . . . . . . . . . ;

2.2 Xfg gejaj mes dmtgrcturg ja agtwjrl njr ctmja . . . . . . . . . . . . . . . . . 39

2.5 ^cdug jn Hmrget Ejatcets cah Xfg Ej-cutfjrmak @jhgd . . . . . . . . . . . . 3>

5 Rcahj` Krcpf Xfgjry? Jdh cah Agw 36

5.3 Xfg Grhjs cah Rgaym @jhgd . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

5.2 Xfg Rgvmvcd jn Rcahj` Krcpf Xfgjry . . . . . . . . . . . . . . . . . . . . . 29

5.5 Rcahj` Krcpfs wmtf Crbmtrcry Hgkrgg Hmstrmbutmja . . . . . . . . . . . . . . 22

5.8 G`grkgaeg jn Kmcat Ej`pjagat . . . . . . . . . . . . . . . . . . . . . . . . . 275.7 Umzgs jn Matgrejaagetgh Krjups jn Ckgats . . . . . . . . . . . . . . . . . . . . 27

5.> Xrchg ma Hmgrgatmctgh Yrjhuets . . . . . . . . . . . . . . . . . . . . . . . . 26

5.; Krcpfmecd Gejaj`mgs? Gxtgasmja jn tfg CrrjwHgbrgu @jhgd . . . . . . . . . 52

5.;.3 Jtfgr @jhgds jn Matgrcetmjas wmtf Hmrget Agmkfbjrs . . . . . . . . . . 58

5.6 Ujemcd Agtwjrls vgrsus Jtfgr Agtwjrls . . . . . . . . . . . . . . . . . . . . . 58

5.= Egatrcdmty cah Ugcrefcbmdmty jn Rcahj` Krcpfs . . . . . . . . . . . . . . . . 5>

5.39 Cppdmectmjas wmtf @jhgds jn Ojb @ctefmak . . . . . . . . . . . . . . . . . . . 5;

8 @crljv Rcahj` Krcpf @jhgds jn Ujemcd Matgrcetmjas 5=

8.3 Xfg NrcalUtrcuss @jhgd jn @crljv Rcahj` Krcpfs . . . . . . . . . . . . . 83

7 Jbsgrvcbdg Ejasgqugaegs jn Jptm`mzmak Bgfcvmjr ma Agtwjrls 88

7.3 Cetmjas cs Jutej`gs nrj` Kc`gs ma Nmamtg Agtwjrls . . . . . . . . . . . . . 88

7.2 Cetmjas ma Dcrkg Agtwjrls . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

7.5 M`pdmectmjas njr Bgfcvmjrcd @jhgddmak . . . . . . . . . . . . . . . . . . . . . . 8>

> Rgngrgaegs 86

; CYYGAHMV 7>

;.3 Cacdytmes jn tfg @jhgra Cpprjcef tj Rcahj` Krcpfs wmtf Crbmtrcry Hgkrgg

Hmstrmbutmjas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7>

;.3.3 G`grkgaeg jn Kmcat Ej`pjagat . . . . . . . . . . . . . . . . . . . . . 7;

;.3.2 Umzgs jn Krjups jn Matgrejaagetgh Ckgats: . . . . . . . . . . . . . . . 76

;.2 Xfg Ecsg jn Hmrgetgh Krcpfs . . . . . . . . . . . . . . . . . . . . . . . . . . 7=

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;.2.3 Ca Gxc`pdg wmtf tfg Au`bgr jn Ma-ghkgs cah Jut-ghkgs Bgmak Hg-

pgahgat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . >3

;.2.2 G`grkgaeg jn c Kmcat Ej`pjagat . . . . . . . . . . . . . . . . . . . . >3

;.2.5 Bmpcrtmtg Krcpfs cah tfgmr Cppdmectmja tj @udtmdctgrcd @ctefmak . . . >2

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3 Matrjhuetmja

Xfms pcpgr3 gxc`mags stjefcstme sjemcd agtwjrls by `gcas jn `jhgds tfct cddjw us tj `clg

tfg stctg jn gatmrg agtwjrls, rctfgr tfct jn mahmvmhucd ckgats, tfg suboget jn cacdysms. Mt

gxpdjrgs tfg rgegat gejajpfysmes, ej`putgr semgaeg cah sjemjdjky dmtgrcturgs ma chhmtmja tjtfct ma gejaj`mes cah g`pfcsmzgs vmgws jn sjemcd agtwjrls cs jutej`gs jn gejaj`me hgemsmjas

jn mahmvmhucds.

Xfg dmtgrcturg ja sjemcd agtwjrl cacdysms tfct cm`s ct hgsermptmja cah cacdysms jn prjp-

grtmgs jn sjemcd strueturgs ms ca gstcbdmsfgh jag ma sjemjdjky. Cdtfjukf tfg stjefcstme vgrsmja

jn tfct dmtgrcturg fcs bjrrjwgh tjjds nrj` spctmcd stctmstmes cah rcahj` krcpf tfgjry, mt ms

ncmr tj scy tfct tfg pcttgras jn bjrrjwmak sukkgsts jeecsmjacd cah hgemsmvg maflugaeg rctfgr

tfca ejatmaujus matgrpdcy. Rcahj` krcpf tfgjry, jn ejursg, wfmdg mt fcs bgga pgregmvgh nrj`

mts maegptmja cs c purgdy `ctfg`ctmecd suboget \ Grhjs cah Rgaym (3=7=< 3=>9) P, fcs eja-

tmaujusdy dgat mtsgdn tj usg by `ctfg`ctmecd sjemjdjkmsts, gdgetrmecd gakmaggrs cah ej`putgr

semgatmsts.

Xfg usg tfct gejaj`msts fcvg `chg jn tfgsg tfgjrmgs ms rctfgr dm`mtgh. Xfms ms qumtg sur-

prmsmak njr c au`bgr jn rgcsjas cah ajt sj surprmsmak njr jtfgr rgcsjas. Njr gxc`pdg, `cay

gejaj`me cppdmectmjas hj ajt g`pfcsmzg tfg mahmvmhucd cah sm`pdy ckkrgkctg jvgr c au`bgr

jn pgjpdg, jr wjrl wmtf c rgprgsgatctmvg mahmvmhucd. Cs Lmr`ca (3==2< 2992) crkugs, tfg

ajtmja jn c rgprgsgatctmvg mahmvmhucd eca bg qumtg hgegptmvg, wfga tfg ckkrgkctg bgfcvgs

qucdmtctmvgdy hmgrgat nrj` tfg su` jn mahmvmhucds bgfcvmjr. Uj`g jn tfg tjjds hgvgdjpgh

by tfg dmtgrcturg tfct ms hmseussgh ma tfms pcpgr dgah tfg`sgdvgs wgdd tj suef c tcsl, cs

ckkrgkctmja wmtf c dcrkg au`bgr jn ckgats `cy dgch tj pfcsg trcasmtmja. Jtfgr gejaj`me cp-

pdmectmjas, cs ma `jhgds jn strctgkme matgrcetmjas cah wmtf ngw gxegptmjas, gssgatmcddy cssu`g

s`cdd krjups jn cetjrs, wmtfma wfmef gvgry jag matgrcets wmtf gvgry jag gdsg, cah tfus aj

mssugs jn ckkrgkctmja cah pfcsg trcasmtmjas crmsg. Utmdd jtfgrs, suef cs sgcref cah `ctefmak,

hj g`pfcsmzg gejaj`me cppdmectmjas wmtf mafgrgat hmsergtgagss cah hj dgch tj ckkrgkctmja.

]gt, ct tfg gah, tfct dmtgrcturg typmecddy cppgcds tj dcrkg au`bgrs cah su``ctmja.

Xfgrg ms cajtfgr cspget jn tfg sgcref dmtgrcturg tfct ms wjrtf pjahgrmak. Xfct dmtgrcturg

fcs cacdyzgh sueegssnuddy fjw ckgats ffiah jag cajtfgr ma jrhgr tj ecrry jut trcascetmjas.Mahmvmhucds djjlmak njr wjrl lajw tfct ojbs `cy bg cvcmdcbdg ct vcrmjus ffir`s but tfgy

3M c` krctgnud njr rgsgcref suppjrt by tfg Actmjacd Uemgaeg Njuahctmja, tfrjukf krcat CEM-=6;555=,cah by tfg Ojfa H. cah Ectfgrmag X. @ceCrtfur Njuahctmja, tfrjukf tfg Rgsgcref Agtwjrl ja UjemcdMatgrcetmjas cah Gejaj`me Hmspcrmtmgs. Gcrdmgr vgrsmjas jn tfms pcpgr wgrg prgsgatgh ct tfg Ejangrgaegja Agtwjrls? Xfgjry cah Cppdmectmjas, Mahustrm Wtrglamak Mastmtut, cxfjd , Uwghga, Ouag 3932, 2998,ct tfg @ceCrtfur Rgsgcref Agtwjrl `ggtmak, Ycdj Cdtj, Jetjbgr 6=, 2998 cah ct tfg Ejangrgaeg jaGejaj`me Xfgjry cah Gejaj`gtrmes (ERGXG), Gr`jupjdms, Uyrjs, Oudy 2997. M tfcal Chrmcca Ujgtgvgatcah pcrtmempcats ct tfjsg ejangrgaegs njr tfgmr au`grjus masmkftnud ej``gats. Cdd grrjrs crg `mag.

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aggh tj ffiah jut wfmef jags `cy bg ceegptcbdg tj tfg . Nmr s djjlmak njr g`pdjyggs aggh

tj `clg tfgmr jpgamaks lajwa tj cdd tfjsg wfj crg pjtgatmcddy matgrgstgh ma tfg`. Ojb

sgcref by mahmvmhucds cah ffir`s gjrts tj ffidd vcecaemgs fcvg bgga stuhmgh gxtgasmvgdy by c

dcrkg bjhy jn dmtgrcturg tfct c`juats tj ca m`pjrtcat pcrt jn `jhgra `cerjgejaj`mes< sgg

Ymsscrmhgs (2993). Matgrcetmjas bgtwgga suef hmsergtg gatmtmgs cs mahmvmhucds cah ffir`s crg

ca matgrgstmak mastcaeg jn sjemcd matgrcetmjas. ]gt, tfg sgcref dmtgrcturg fcs ajt g`pfcsmzgh

tfg wgb jn rgdctmjasfmps bgtwgga ffir`s cah wjrlgrs cah c`jak wjrlgrs wfj fcvg sfcrgh

gxpgrmgaegs ma hgcdmak wmtf tfg sc`g ffir`s, cah ffir`s ma hgcdmak wmtf tfg sc`g wjrlgrs,

tfct `cy hgvgdjp jut jn sfcrgh g`pdjy`gat gxpgrmgaegs. Xfg ncet tfct suef gxpgrmgaegs

`cy dgch tj g`pdjy`gat rgngrrcds ma tfg nuturg fcs bgga rgejkamzgh, but tfg rjdg jn wgbs jn

rgdctmjasfmps `cy ajt bg gcsy tj cacdyzg by `gcas jn rgprgsgatctmvg mahmvmhucd `jhgds jr by

`jhgds tfct mavjdvg su``ctmja cah dcrkg au`bgrs.

Xfg pcpgr stcrts ma sgetmja 2 wmtf c rgvmgw jn `jhgds jn agtwjrls njr`ctmja. M c` ajtcttg`ptmak tj hupdmectg twj rgegat pcpgrs, Kjycd (2995) cah Ocelsja (2995), ajr Ocelsjas

`ckmstgrmcd rgvmgw jn tfg dmtgrcturg \Ocelsja (2997)P, tfct fcvg gdjqugatdy rgvmgwgh substca-

tmcd pjrtmjas jn tfg sc`g dmtgrcturg cs tfg jag hmseussgh fgrg. Mastgch, M cm` fgrg ct gxpdjrmak

`gtfjhjdjkmecd trgahs ma tfg matgrnceg bgtwgga tfg gejaj`mes cah tfg ej`putgr semgaeg dmt-

grcturg cah fjw tfgy fcvg bgga maflugaegh by (jntga hmgrgat) stydmzgh ncets tfct bjtf tfjsg

dmtgrcturgs fcvg njeusgh ja. M gsefgw ca ma-hgptf cacdysms jn dcbjr `crlgt cppdmectmjas, bg-

ecusg M fcvg hmseussgh tfg` gdsgwfgrg \Mjcaamhgs cah Djury (2998)P. Xfg pcrtmeudcr strcah

jn tfg ej`putgr semgaeg dmtgrcturg tfct M hmseuss fgrg fcs c acturcd gejaj`me `jtmvctmja,

ma tfct mt ms ejasmhgrmak tfg g`grkgaeg jn tfg wjrdh wmhg wgb cs ca gahjkgajus jboget \

Ycpchm`mtrmju (2995) P. Xfg wgbs vcrmjus prjpgrtmgs, maeduhmak cddgkgh pjwgr dcws njr tfg

au`bgr jn ajhgs tfct crg maemhgat tj gcef ghkg, tjpjdjkmecd prjpgrtmgs jn tfg gatmrg agtwjrl

cah jtfgrs suef sgcrefcbmdmty prjpgrtmgs \ Ldgmabgrk (2999)< Agw`ca (2995) P njddjw nrj`

gahjkgajus matgrcetmjas bgtwgga mahmvmhucds cah tgefajdjkmecd systg`s `uef `jrg tfca cay

prgvmjusdy hgsmkagh systg`. Xfg vgry sm`pdg gejaj`me `jhgd jn agtwjrl ergctmja wmtf wfmef

tfct sgetmja stcrts ej`pdg`gats tfg gejaj`mes rgsgcref ja agtwjrl njr`ctmja. Xfg ncet tfct

`jst rgegat cppdmectmjas jn gejaj`me tjjds tj stuhymak agtwjrl njr`ctmja crg kgagrcddy ecst

ma tgr`s jn hgtgr`mamstme `jhgds sukkgsts tfct agtwjrl njr`ctmja ma stjefcstme gavmrja`gats

cdsj hgsgrvgs cttgatmja.

Agxt M tura ma sgetmja 5 tj c rgvmgw jn rcahj` krcpf tfgjry wfmef wfmdg jrmkmactmak ma

Grhjs cah Rgaym, jp. emt., fcs rgegatdy bgga hgvgdjpmak cs c sjureg jn `jhgds jn agtwjrls

jn sjemcd ejatcets, jr sjemcd agtwjrls, njr sfjrt. Xfg prgsgatctmja trcegs tfg cppdmectmjas jn

tfg tfgjry by c s`cdd au`bgr jn gejaj`me cppdmectmjas cah tfga hmseussgs tfg sfjrtej`maks

jn tfg gcrdy rcahj` krcpf `jhgds. Mt ms ca gssgatmcd pcrt jn tfct tfgjry tfct tfg gvgats

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tfct tfgrg gxmst ejatcets bgtwgga hmgrgat mahmvmhucds crg mahgpgahgat. Mt ms ermtmecd tj bg

cbdg tj cddjw njr hgpgahgaeg ma tfg prgsgaeg jn ejatcets bgtwgga hmgrgat mahmvmhucds, smaeg

suef hgpgahgaeg `cy bg ca jutej`g jn mahmvmhucd rgspjasgs tj ejrrgdctgh rcahj` ncetjrs

jr tj sgdgetmja cah sjrtmak. Mt `cy cdsj bg c lgy jutej`g ma hgtgr`mamstme `jhgds jn sjemcd

agtwjrls.

Xfg rg`cmahgr jn tfct sgetmja jn tfg pcpgr turas tj c rgegat rgvmvcd jn matgrgst ma rcahj`

krcpf tfgjry. Rgegat hgvgdjp`gats fcvg kgagrcdmzgh tfg jrmkmacd Grhjs Rgaym `jhgd, cah

tfus sfjudh cddjw agwgr cppdmectmjas ma uahgrstcahmak sjemcd agtwjrls cah sjemcd matgrcetmjas

`jrg kgagrcddy. Xj hj sj, M hrcw nrj` tfg rgegat gejajpfysmes dmtgrcturg ja rcahj` krcpfs

cah sjemcd agtwjrls, wfmef jgrs tfg `jst hmrget kgagrcdmzctmja jn tfg edcssme rcahj` krcpf

`jhgd cdc Grhjs cah Rgaym. Xfms maeduhgs ajtcbdg hgvgdjp`gats tfct fcvg rgegatdy su``c-

rmzgh vgry gdjqugatdy by Agw`ca (2992) cah Agw`ca (2995h) cah by jtfgrs. Cs wg sfcdd sgg

ma `jrg hgtcmd, sj`g jn tfg strcahs jn tfms agwgr dmtgrcturg, wfmef ms cdsj dmalgh wmtf rgdctghej`putgr semgaeg dmtgrcturg, fcs bgga `jtmvctgh by c hgsmrg tj gxpdcma jbsgrvcbdg ngcturgs

jn rgcd dmng agtwjrls, dmlg tfg [[[. Xfgsg prjpgrtmgs kgagrcdmzg tfg rgstrmetmvg cssu`ptmjas

jn tfg jrmkmacd rcahj` krcpf `jhgd cdc Grhjs cah Rgaym. Ma ncet, `uef rgsgcref wmtfma tfg

gejajpfysmes ej``uamty fcs bgga cm`gh ct gxpdcmamak c au`bgr jn stydmzgh ncets rgkcrhmak

`gcsurgs jn `ca-`chg agtwjrls, maeduhmak tfg [[[, ejcutfjrsfmp cah emtctmja agtwjrls,

cah jtfgrs bgecusg jn tfg ncmdurg jn tfg Grhjs cah Rgaym rcahj` krcpf `jhgd tj gxpdcma

tfg` sctmsncetjrmdy. Utmdd, tfg gejajpfysmes dmtgrcturg ja rcahj` krcpfs cah agtwjrls kgagr-

cddy dcels edgcr bgfcvmjrcd `jtmvctmjas jn tfg sjrt tfct gejaj`msts crg typmecddy ceeustj`gh

tj.

Ugetmja 8 hmseussgs c edcss jn `jhgds tfct chhrgss c lgy wgclagss jn tfg `jhgra cpprjcef

tj `jhgddmak rgcd wjrdh agtwjrls, ac`gdy tfg `jhgddmak jn trcasmtmvmty. Xfms m`pjrtcat prjp-

grty jn gahjkgajus sjemcd agtwjrls `cy bg fcahdgh by @crljv rcahj` krcpfs, wfmef trgct

gatmrg sjemcd agtwjrls cs stjefcstme jbogets. Xfms `jhgd ms wgdd lajwa ma `ctfg`ctmecd sj-

emjdjky cah spctmcd stctmstmes. Mt jrmkmactgs ma Nrcal cah Utrcuss (3=6>) cah Utrcuss (3=6>)

cah fcs gvgr smaeg rgegmvgh ejatmaujus cttgatmja by tfg `ctfg`ctmecd sjemjdjky dmtgrcturg

\Fcksbgrk (2992)< [cssgr`ca gt cd. (2998)P. Xfct sgetmja prjvmhg dmals jn @crljv rcahj`

krcpfs ma tfg NrcalUtrcuss stydg wmtf rcahj` krcpf tfgjry ma tfg GrhjsRgaym stydg by

`gcas jn c pcrtmeudcr cppdmectmja jn rcahj` krcpf tfgjry, cah mts m`pdmectmjas njr tfg hy-

ac`mes jn choust`gat ma sjemcd matgrcetmja cah jtfgr rgdctgh `jhgds. Xfg pcpgr ejaeduhgs

by pjahgrmak ja tfg sejpg jn tfg brjchdy matgrhmsempdmacry rgsgcref ja gahjkgajus agtwjrl

njr`ctmja.

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2 @jhgds jn Gahjkgajus Agtwjrl Njr`ctmja

C sgrmgs jn maajvctmvg rgegat pcpgrs `jhgd tfg gahjkgagmty jn dmals c`jak jrmkmacddy msjdctgh

ckgats. Kjycd (2995) cah Ocelsja (2995), prjvmhg gdjqugat rgvmgws jn vcrmjus cspgets jn tfg

dmtgrcturg ja gahjkgajus agtwjrl njr`ctmja. [mtfma tfms rgegat dmtgrcturg, gspgemcddy ajtg-wjrtfy crg pcpgrs by Bcdc cah Kjycd (2999c), Ocelsja cah [ctts (2992c< 2992b), Ocelsja

cah [jdmasly (3==>), cah [ctts (2993). Xfgsg cutfjrs g`bgh gahjkgajus agtwjrl njr`c-

tmja ma tfg trchgj bgtwgga tfg vcdug tj hmgrgat ckgats nrj` gxmstgaeg jn matgrejaagetmjas

wmtf jtfgr ckgats cah tfg ejsts ckgats tfg`sgdvgs maeur nrj` mamtmctmak ejaagetmjas. [g

rgvmgw fgrg sj`g lgy ffiahmaks tfct fcvg bgga gstcbdmsfgh by tfms dmtgrcturg. M stcrt nrj` tfg

sm`pdgst pjssmbdg `jhgd ma tfg dmtgrcturg, Ncbrmlcat gt cd. (2995), wfmef fmkfdmkfts gssgatmcd

mssugs cah trchgjs ma gahjkgajus agtwjrl njr`ctmja. Xfg ncet tfct tfms pcrtmeudcr pcpgr,

wfmef ms edjsgdy rgdctgh tj Ocelsja cah [jdmasly (3==>), jrmkmactgs ma tfg ej`putgr sem-

gaeg dmtgrcturg uahgrsejrgs c smkamffiecat matgrpdcy bgtwgga tfct ffigdh cah gejaj`mes. Xfms ms

qumtg acturcd ma vmgw jn tfg rgegat matgrgst by ej`putgr semgatmsts ma stuhymak kc`g-tfgjrgtme

njr`udctmjas jn agtwjrl-bcsgh rgsjureg cddjectmja prjbdg`s.

M matrjhueg sj`g bcsme tgr majdjky cah ajtctmja rgkcrhmak krcpfs cah agtwjrls. Xj

stcrt wmtf, tfg tgr`s krcpfs cah agtwjrls crg rgcddy syajay`s njr jur purpjsgs, cdtfjukf

tfg tgr` krcpf cppgcrs tj bg usgh gxedusmvgdy by `ctfg`ctmemcas cah jn ejursg by jtfgrs cs

wgdd, but tfg tgr`s krcpfs cah agtwjrls crg usgh cd`jst matgrefcakgcbdy by cdd jtfgr ffigdhs.

Dgt tfg gdg`gats jn c sgt

M0

{3, . . . , M

}rgprgsgat mahmvmhucds. Mt ms mahmvmhucds wfj crg

tfg hgemsmja `clgrs cah tfus jbogets jn cacdysms ma `jst jn tfg cppdmectmjas ma tfg prgsgat

pcpgr. Gstcbdmsfgh ej``uamectmja, sjemcd rgdctmjas, jr sjemcd matgrcetmjas bgtwgga cay twj

mahmvmhucd `g`bgrs jnMcrg hgffiagh by ca uahmrgetgh krcpfK(^,G), wfgrg?^ms tfg sgt jnvgrtmegs,^0{v3, v2, . . . , vM}, ca jag-tj-jag `cp jn tfg sgt jn mahmvmhucdsMjatj mtsgdn (tfgkrcpf ms dcbgddgh), cah M0 |^| ms tfg au`bgr jn vgrtmegs (ajhgs), cdsj lajwa cs tfgjrhgrjntfg krcpf


8/66

tfg prgsgat pcpgr, tfg chocegaey `ctrmx wmdd bg hgffiagh cs c rcahj` `ctrmx wmtf kgagrme

rgcdmzctmja . Xfg gatrmgs jn tfg chocegaey `ctrmx mo wjudh bg bmacry rcahj` vcrmcbdgs ma

tfct ecsg, wmtf tfg `jst matgrgstmak ecsg bgmak wfga hmgrgat mos `cy bg ejasmhgrgh cs

hgpgahgat rcahj` vcrmcbdgs, cs ma tfg ecsg jn tfg @crljv rcahj` krcpf `jhgd.

2.3 C Um`pdg @jhgd jn Agtwjrl Njr`ctmja

Ncbrmlcat gt cd. (3995) ejasmhgr c `jhgd wmtf ckgatsMwfj bumdh ejaagetmjas wmtf gcefjtfgr. Xfgmr hgemsmjas ergctg c uahmrgetgh krcpf,K(^,G), wfmef ms tj bg trgctgh cs cagahjkgajus jutej`g. Jaeg c ejaagetmja bgtwgga twj ckgats fcs bgga gstcbdmsfgh, mt `cy

bg usgh by cdd jtfgr ckgats vmc tfg twj ckgats ct mts twj gahs. Xfg strctgky spceg jn ckgat

m ms tfg sgt Um 0 2M{m}. [g hgajtg byK(s) tfg krcpf rgsudtmak nrj` cdd ckgats g`pdjymak

strctgkmgs s 0 (s3, . . . , sM)

U3

. . .

UM. Xfg ejst maeurrgh by ckgat m wfga cdd ckgats

g`pdjy strctgky s ms cssu`gh tj bg chhmtmvg ma tfg ejst jn tfg au`bgr jn ejaagetmjas,|sm|,tfct ckgat m fgrsgdn bumdhs wmtf jtfgr ckgats, cah ma tfg su` jn tfg ejsts gcef ckgat m ùst

maeur ma jrhgr tj rgcef cdd jtfgr ckgats tfrjukfK(s)?

em(s) 0 |sm| +M

o03

hK(s)(m, o), (3)

wfgrg hK(s)(m, o) hgajtgs tfg hmstcaeg bgtwgga ckgats m cah o wmtfma tfg krcpfK(s). Ajtgtfct pcrc`gtgr , tfg rgdctmvg m`pjrtcaeg jn ckgat ms hmrget dmals wmtf jtfgr ckgats, ms tfg

jady pcrc`gtgr ma tfg `jhgd.Ckgats sggl tj `mam`mzg tfgmr jbogetmvg nuaetmjas, kmvga by (3). Mt ms gcsy tj efcrcetgrmzg

Acsf gqumdmbrmc cah tfg sjemcd jptm u` njr tfms sm`pdg `jhgd. Xfgy hgpgah ermtmecddy ja

tfg `ckamtuhg jn , tfg ejst jn c dmal. Mn 4 3, tfga tfg ej`pdgtg krcpf (tfg `cxm`cd

edmqug) ms tfg jady Acsf gqumdmbrmu`. Xfg matumtmja ms strcmkftnjrwcrh? cay Acsf gqumdmbrmu`

ecaajt `mss ca ghkg wfjsg maedusmja wjudh rghueg tfg sgejah tgr` ma tfg jbogetmvg by

`jrg tfca mt wjudh chh tj tfg ejst. Mt ms cdsj tfg sjemcd jptmù`, ma tfct mt `mam`mzgs

|G| + Mm,ohK(s)(m, o).Njr 3 42,tfg krcpfs tfct crg Acsf gqumdmbrmc fcvg hmc`gtgr ct`jst 2. Xfg sjemcd jptmù` ms stmdd tfg ej`pdgtg krcpf, cah tfg stcr ms c Acsf gqumdmbrmu`.

Njr 2, tfg stcr ms c Acsf gqumdmbrmu , but tfgrg `cy jtfgrs, tjj. Xfg sjemcd jptmùèjatmaugs tj bg tfg stcr. Kgagrcddy, matumtmja sukkgsts tfct njr sustemgatdy fmkf vcdugs jn

, Acsf gqumdmbrmc wjudh bg trggs, bgecusg tfgy jgr c djwgr vcdug njr tfg sgejah tgr` ma

tfg hgffiamtmja jn ca ckgats ejst ceejrhmak tj (3). ]gt, tfgrg hjgs ajt cppgcr tj bg c prjjn

jn tfms, cah ma ncet Ncbrmlcat gt cd., jp. emt., jgr mt cs c ejaogeturg. Mt ms edgcr tfct tfg

prgsgaeg jn krcpf hmstcaeg ma tfg jbogetmvg nuaetmja (3) ms qumtg ermtmecd njr tfg prjpgrtmgs jn

tfg gqumdmbrmu` jutej`gs ma tfms njrùdctmja.

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9/66

C au`bgr jn rg`crls crg ma jrhgr. Nmrst, wg ajtg tfct wfmdg cdd gqumdmbrmu` jutej`gs crg

cetucddy ejaagetgh krcpfs, mt ms ajt jbvmjus tfct ma c hgegatrcdmzgh sgttmak tfgrg wjudh bg c

prmjrm ckrgg`gat cs tj wfgrg tfg stcr wjudh bg egatgrgh. Xfg sm`pdmemty jn tfg sjdutmja tj tfms

prjbdg` prjvmhgs ca m`pjrtcat bgaef`crl njr gahjkgajus agtwjrl njr`ctmja. Ugejah, njr

bjtf tfg ej`putgr semgaeg cah tfg gejaj`mes dmtgrcturgs, mt ms jn ermtmecd m`pjrtcaeg wfgtfgr

Acsf gqumdmbrmc `cy bg crrmvgh ct tfrjukf tfg sjrt jn pgrturbctmja jn bgst rgspjasgs tfct

gejaj`msts crg nc mdmcr wmtf cah ct c rgcsjacbdg ejst. Mt ms jn pcrtmeudcr ejasgqugaeg tj

ej`putgr semgatmsts tfct suef c pgrturbctmja cacdysms, wfgrgby jag stcrts wmtf c strctgky

cah tfga rgpdcegs mt by c pdcygrs bgst rgspjasg, turas jut tj bg AY fcrh< sgg mbmh., p.586. Xfmrh, tfg prj`magaeg jn tfg stcr cs ca gqumdmbrmu` jutej`g prj`pts us tj ejasmhgr mts

rjbustagss ma `jrg kgagrcd sgttmaks. Ugvgrcd ejaegras m``ghmctgdy crmsg? jag, mt ms vudagrcbdg

ma c stjefcstme sgttmak wfgrg ejaagetmjas `cy ncmd< twj, mt ms prjag tj ejakgstmja< cah tfrgg,

c `jrg kgagrcd gejaj`me `jhgd sfjudh gxprgss bgagffits cssjemctgh wmtf hmgrgat krcpfs cahajt oust ejsts.

Ncbrmlcat gt cd. (2992) wjrl wmtf c rgdctgh `jhgd wfmef, fjwgvgr, g`pfcsmzgs hgkrgg

hmstrmbutmjas cah jtfgr tjpjdjkmecd prjpgrtmgs jn rgsudtmak krcpfs ct tfg dm`mt wfga tfg

au`bgr jn ckgats ms dcrkg. Xfgmr `jhgd prjhuegs c pjwgr dcw hmstrmbutmja ma egrtcma krcpf

`gcsurgs, wfmef crg jn matgrgst ma tfg ejatgxt jn tfg Matgragt tjpjdjky, prgemsgdy wfga agw

ajhgs tfct rcahj`dy cppgcr efjjsg jptm`cddy tfgmr ejaagetmjas wmtf ca gxmstmak agtwjrl.2 Gatgrmak ajhgs sggl tj `mam`mzg tfgmr hmstcaeg tj tfg egatgr jn tfg gxmstmak agtwjrl

tjpjdjky pdus tfg wgmkftgh Guedmhgca hmstcaeg tj ca gxmstmak ajhg. Mn hmstcaegs nrj` tfg

gxmstmak ajhg, wfmef c agw ajhg efjjsgs tj ejaaget wmtf, tj jtfgr ajhgs ma ca gxmstmak

agtwjrl, crg kmvga c dcrkgr rgdctmvg wgmkft ma tfg jbogetmvg jn tfg jptm`mzctmja prjbdg`,

tfga tfg rgsudtmak agtwjrl ms c stcr. Mn, ja tfg jtfgr fcah, tfg wgmkft krjws ct dgcst cs ncst

cs tfg squcrg rjjt jn tfg smzg jn tfg gatmrg krcpf, tfga tfg rgsudtmak hgkrgg hmstrmbutmja ms

gxpjagatmcd. Njr tfg ma-bgtwgga vcdugs, tfg rgsudtmak hgkrgg hmstrmbutmja tgahs tj c pjwgr

dcw. Ma tfms `jhgd, tfg krcpf ms hgffiagh ma c twj hm`gasmjacd ejatmaujus spceg cah ms bjuahgh

wmtfma c uamt squcrg wmtfma mt.

Upgemffiecddy, Ncbrmlcat gt cd. (2992) stuhy tfg gvjdutmja jn tfg agtwjrl cs ckgats m 0

3, 2, . . . , crrmvg uamnjr`dy ct rcahj` cah efjjsg jag jn tfg gxmstmak jtfgr ckgats, o, o 0

3, . . . , m 3,tj ejaaget wmtf. Gcef gatgrmak ckgat m efjjsgs ckgat o tj dmal wmtf tj sj cs tj`mam`mzg

nm(o) 0c hmo+fo, (2)2Xfg dmagcr jbogetmvg tfct ckgats sggl tj `mam`mzg ms rg`mamsegat jn Jttcvmcaj cah Xfmssg (2998), wfj

rgecdd [gbgr (3=9=) cah cssu`g tfct ffir`s djectg sj cs tj `mam`mzg c wgmkftgh su` jn hmstcaegs nrj` smtgswfgrg c ffir` purefcsgs mts maputs cah sgdds mts jutputs.

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10/66

wfgrghmohgajtgs tfg Guedmhgca hmstcaeg bgtwggam caho,cahfoc `gcsurg jn tfg egatrcdmty

jn tfg ckgato wmtfma tfg ffiacd krcpf njr`gh by tfg hgemsmja jn ckgatm. Xfms `gcsurg ejudh bg

tfg cvgrckg krcpf hmstcaeg nrj` cdd jtfgr ckgats, tfg `cxmù` suef hmstcaeg jr tfg krcpf

hmstcaeg tj tfg egatgr jn tfg trgg, tfg au`bgr jn fjps nrj` m tj 3 ma Xm, wfmef ms cetucddy

tfg `gcsurg tfgy usg tj prjvg tfgmr rgsudts. Xfg gvjdvmak gahjkgajus agtwjrls crg trggs,

X9, X3, . . . ,cah Xm ejasmsts jnXm3,wmtf tfg ckgatm cah tfg ejaagetmja \m, oP chhgh. Fgrg tfg

pcrc`gtgr chgajtgs tfg ejst jn ejaagetmak tj ckgato rgdctmvg tj tfg m`pjrtcaeg jn egatrcdmty,

wfmef fcs c ejgstemgat jn 3.

Xfg sjdutmja hgpgahs ermtmecddy ja tfg `ckamtuhg jn pcrc`gtgrc,wfmef ms cetucddy trgctgh

cs c nuaetmja jn M. Mn mt ms rgdctmvgdy s`cdd, c 4 3/

2, tfct ms hmstcaegs tj cdd jtfgr ajhgs

crg rgdctmvgdy `jrg m`pjrtcat, tfga tfg rgsudtmak agtwjrl XM ms c stcr, egatgrgh ct tfg ajhg

cssjemctgh wmtf ckgat 3. Rgkcrhmak tfg cssjemctgh hgkrgg hmstrmbutmja, tfms sjdutmja ms tfg

gxtrg`g vgrsmja jn c pjwgr dcw, wfgrg cdd ckgats crg ejaagetgh wmtf tfg smakdg jrmkmacd ckgat.Mn c 0 (

M) (tfct ms, mt ms `jrg tfca sj`g ejastcat ùdtmpdg jn

M), tfct ms tfg wgmkft

cttcefgh tj hmstcaeg tj tfg agcrgst ajhg krjws sustemgatdy ncst wmtf tfg au`bgr jn ckgats,

tfg hgkrgg hmstrmbutmja jnXM ms gxpjagatmcd. Xfct ms, tfg gxpgetgh au`bgr jn ckgats tfct

crg ejaagetgh tj ct dgcst Ljtfgr ckgats ms bjuahgh cbjvg by M2geL.Nmacddy, mnc 8,cahc 0 j(

M) (tfct ms, njr ma-bgtwgga vcdugs cs c `cy vcry M but krjw sdjwgr tfca

M, cs M

krjws vgry dcrkg), tfga tfg hgkrgg hmstrmbutmja jnXM ms c pjwgr dcw? tfg gxpgetgh au`bgr

jn ckgats wfj crg ejaagetgh wmtf ct dgcst Ljtfgr ckgats ms krgctgr tfca e LM

, wfgrg

e, crg ejastcats tfct `cy hgpgah ja c.

Ncbrmlcat gt cd. fcvg krjwtf jn tfg Matgragt ma `mah ma bumdhmak tfgmr `jhgd. Xfg

stydmzgh ncets, cs gstcbdmsfgh by Ncdjutsjsgt cd. (3===), prjvmhg g`pmrmecd suppjrt njr c pjwgr

dcw, cdtfjukf ùef jn tfg pjwgr dcw dmtgrcturg, c.l.c. secdg-nrgg dcws, fcs tclga pcma tj

g`pfcsmzg tfg uamvgrscdmty jn pjwgr dcws. Xfms edgcrdy cppgcrs tj fcvg bgga gxckkgrctgh.

Xfgrgnjrg, tfg hgpgahgaeg jn tfg hmstrmbutmja pcrc`gtgrs ma tfg cbjvg rgsudts ms jbvmjusdy c

hrcwbcel. [g rgtura bgdjw ma Ugetmja 7 tj tfgsg g`pmrmecd ffiahmaks.

Ca gejaj`me matgrprgtctmja jn tfms rgsudt wjudh bg ma tgr`s jn prgngrgatmcd cttcef`gat.

Xfjsg ckgats wfj fcvg crrmvgh gcrdy crg `jrg dmlgdy tj fcvg `jrg ejaagetmjas wmtf jtfgrs cah

bg agcr jtfgrs, rghuemak tfg fjp ejst. C smkamffiecat ngcturg jn tfms rgsudt ms tfct prgngrgatmcd

cttcef`gat ms hgrmvgh nrj` `jrg prm`mtmvg cssu`ptmjas, rgdctmvg tj jtfgr wjrl wfgrg mt ms

oust cssu`gh cs c rgcdmstme ngcturg jn rgcd-wjrdh agtwjrls. @jst m`pjrtcatdy, Ncbrmlcatgt cd.

(2992) cppgcr tj jgr tfg ffirst purgdy bgfcvmjrcd `jhgd tfct dgchs tj g`grkgaeg jn c pjwgr

dcw njr tfg hgkrgg hmstrmbutmja jn ajhgs ma ca gahjkgajusdy njr`gh agtwjrl. [g rgtura tj

tfms mssug nurtfgr bgdjw.

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11/66

2.2 Xfg gejaj`mes dmtgrcturg ja agtwjrl njr`ctmja

Xfg `jhgds ma tfg gejaj`mes dmtgrcturg tfct cdsj `clg dmals gahjkgajus by `gcas jn strctgkme

ejasmhgrctmjas crg sj`gwfct `jrg kgagrcd tfca tfgmr ej`putgr semgaeg ejuatgrpcrts ma c

au`bgr jn rgspgets, but ct tfg sc`g tm`g njeus ja sdmkftdy hmgrgat mssugs. Uj`g jn tfg

pcpgrs crg jntga `jrg gxpdmemt rgkcrhmak bumdhmak jn dmals bgtwgga twj jrmkmacddy msjdctgh

mahmvmhucds. Njr gxc`pdg, sj`g jn tfg pcpgrs rgqumrg tfct tfjsg tj bg hmrgetdy ejaagetgh bjtf

ejasgat tj mt, wfgrgcs sgvgrcaeg eca bg hjag uamdctgrcddy. Dmals `cy hmrgetgh (csy``gtrme,

ma Bcdc cah Kjycds tgr majdjky) jr uahmrgetgh. Dmals bgtwgga ckgats crg matgrprgtgh cs

manjr`ctmja efcaagds. C hmrgetgh dmal (m, o) hgajtgs tfct ckgat m fcs ceegss tj ckgat os

manjr`ctmja cah hjgs ajt m`pdy tfct o fcs ceegss tj ms. Njr gxc`pdg,mejudh fcvg ceegss tj

os wgb-smtg jr fcvg ceegss tj gxcet WRD njr c pcrtmeudcr hjeu`gat. Ma ncet, hmrgetgh dmals

crg c lgy ngcturg jn tfg wgb krcpf. Ejasgqugatdy, njr gvgry pjssmbdg hmrgetgh dmal tfct ca

mahmvmhucd ms ejatg`pdctmak, tfgrg ms mts ejuatgrpcrt ma tfg jppjsmtg hmrgetmja cs wgdd cs c

`udtmtuhg jn mahmrget dmals tfct ceej`pdmsf tfg sc`g manjr`ctmjacd gget tfjukf ct c fmkfgr

ejst. Wahmrgetgh dmals `cy `jhgd ejaagetmjas dmlg tfct prjvmhgh by tgdgpfjag.

@jst jn tfg pcpgrs ma tfg gejaj`mes dmtgrcturg cssu`g tfct tfg utmdmty gcef ckgat hg-

rmvgs nrj` pcrtmempctmak hgpgahs chhmtmvgdy ja tfg tjtcd au`bgr jn jtfgr ckgats ca ckgat ms

ejaagetgh wmtf, `maus tfg ejsts jn `cmatcmamak tfg ejaagetmjas tfct jag bumdhs ja fgr jwa.

Uj`g cutfjrs `clg ca cddjwcaeg njr prjxm`mty tj jtfgrs by `gcas jn c hgecy ncetjr tfct

hgpgahs ja tfg au`bgr jn matgrvgamak ckgats.

Bcdc cah Kjycd (2999) cssu`g tfct mahmvmhucd ms pcyj ms strmetdy maergcsmak ma tfgau`bgr jn ckgats jbsgrvgh by m, m(k),tfct ms jtfgr ckgats wmtf wfj`m fcs njr`gh hmrget

dmals jr ms dmalgh tj mahmrgetdytfrjukf jtfgrs, cah strmetdy hgergcsmak ma tfg au`bgr jn jtfgr

ckgats tfctmfcs bumdt hmrget dmals wmtf,hm (k) 0 |sm|. Ma tfg spgemcd ecsg wfga gcef mahmvmhucdpjssgssgs manjr`ctmja wfmef ms jn vcdug ^, wfmef `cy bg ajr`cdmzgh cah sgt gqucd tj 3, tj

gcef jn gvgry jtfgr ckgat cah tj fm`sgdn, tfg bgagffits nrj` tfg manjr`ctmja pjssgssgh by

tfjsg wfjsg manjr`ctmja sfg ceegssgs hmrgetdy jr mahmrgetdy crg prjpjrtmjacd tj m(k). Mn ca

ckgat maeurs dmagcr ejsts jn njr`mak hmrget dmals, wfmef crg hgajtgh by e pgr dmal, tfga tfg

agt bgagffit tj ckgat m msm(k) 0m(k) ehm (k). (5)

Mne 43,tfga ckgat m wmdd bg wmddmak tj njr` c dmal wmtf ckgat o njr tfg sclg jn tfct ckgats

manjr`ctmja cdjag. Mn 34 e 4 M 3,tfga ckgatm wmdd wcat ckgat o tj fcvg ceegss wmtf `jrgtfca jag jtfgr ckgat ma jrhgr tj bg mahuegh tj njr` c dmal wmtf o. Mne 1 M 3,tfga tfg ejstjn c dmal gxegghs tfg bgagffit jn manjr`ctmja tj tfg gatmrg sjemgty. Ma tfct ecsg, mt ms hj`macat

strctgky njr m ajt tj njr` c dmal wmtf cay pdcygr.

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Xfg g`pty agtwjrl ms c Acsf gqumdmbrmu`. Bcdc cah Kjycd sfjw tfct wfga ckgats

jbogetmvg ms cs m(k) cbjvg, tfga c strmet Acsf agtwjrl ms gmtfgr tfg wfggd, wfgrgby gcef

ckgat njr`s gxcetdy jag dmal, jr tfg g`pty agtwjrl. Ma jtfgr wjrhs, manjr`ctmja ms gmtfgr

sfcrgh wmtf gvgryjagwmtf tfg `mamù` au`bgr jn ejaagetmjas pgr ckgat tfct `clgs tfms

pjssmbdg, wfmef m`pdmgs tfg wfggd, wfmef wjudh bg tfg uamqug ecsg wmtf dmagcr pcyj mne 43,

jr cdjak wmtf tfg g`pty agtwjrl, mn 3 4 e 4 M 3< jr tfgrg ms aj sfcrmak wfga e 1 M 3.[g ajtg tfg hmgrgaeg ma gqumdmbrmu` jutej`gs nrj` tfg Ncbrmlcat gt cd. njrùdctmja jn tfg

agtwjrl njr`ctmja kc`g. Xfgmr njrùdctmja cssmkas wgmkft tj egatrcdmty cs suef, wfmef jn

ejursg pgacdmzgs tfg wfggd. Mt ms ejaagetghagss, fjwgvgr, tfct pdcys c sm mdcr rjdg ma tfg

BcdcKjycd njrùdctmja.

Ma vmgw jn tfg hrc`ctmecddy rgstrmetgh sgt jn Acsf agtwjrls, tfg njddjwmak qugstmja crmsgs?

wmdd c sjemgty jn ckgats tfus `jtmvctgh sgdn-jrkcamzg matj gqumdmbrmu` jutej`gs: Bcdc cah

Kjycd stuhy tfg hyac`mes jn dmal njr`ctmja by cssu`mak c acmvg bgst rgspjasg rudg wmtfmagrtmc. Xfct ms, ca ckgat `cy efjjsg, wmtf ffixgh prjbcbmdmtmgs, gmtfgr c `yjpme purg strctgky

bgst rgspjasg, jr tfg sc g cetmja cs ma tfg prgvmjus pgrmjh. Magrtmc gasurgs tfct ckgats

wmdd ajt `ms-ejjrhmactg pgrpgtucddy. Bcdc cah Kjycd sfjw tfct mrrgspgetmvg jn tfg au`bgr

jn ckgats cah by stcrtmak nrj` cay mamtmcd pcttgra jn matgrejaagetmjas, tfg hyac`me prjegss

mahgghsgdn-jrkcamzgs,by ejavgrkmak wmtf prjbcbmdmty 3, cah ma ffiamtg tm`g, tj tfg cpprjprmctg

(njr tfg rgspgetmvg pcrc`gtgr vcdugs) uamqug dm`mt agtwjrl. Xfg dm`mt ms tfg sgt jn strmet Acsf

agtwjrls jn tfg jag- sfjt kc`g, wfmef bgej`g tfg sgt jn cbsjrbmak stctgs jn tfg hyac`me

prjegss. Xgefamecddy, tfg rudgs jn mahmvmhucd bgfcvmjr hgffiag c @crljv efcma ja c stctg spceg

ejasmstmak jn cdd agtwjrls, wfjsg cbsjrbmak stctgs crg tfg Acsf gqumdmbrmc jn tfg jag-sfjt

kc`g.

Xfgsg rgsudts `cy bg kgagrcdmzgh by rgstrmetmak tfg manjr`ctmja cvcmdcbdg tj ckgats, tfct

ms by cssu`mak jady djecd manjr`ctmja gcef ckgat lajws tfg rgsmhucd sgt jn cdd tfjsg sfg ms

ejaagetgh wmtf, tfct ms tfjsg fgr agmkfbjrs eca ceegss wmtfjut usmak dmals tj fgr cah by

cddjwmak jbsgrvctmja jn sueegssnud ckgats tfgrg ms sj`g efcaeg tfct sfg rgegmvgs manjr`ctmja

nrj` c sueegssnud ckgat, tfct ms c pgrsja wfj jbsgrvgs tfg dcrkgst subsgt jn pgjpdg ma tfg

gejaj`y wmtfjut cssmstcaeg nrj` fgr jwa dmals. Xfg rgsudts ejatmaug tj fjdh wfga tfg pcyj

tj gcef ckgat ms cssu`gh tj bg strmetdy maergcsmak ma tfg au`bgr jn jtfgr ckgats sfg jbsgrvgs,

cah strmetdy hgergcsmak ma tfg au`bgr jn dmals tfct sfg njr`s. Ma tfct ecsg, tfg uamqug gstemgat

crefmtgeturg ms tfg wfggd, mn ca ckgat ms bgttgr j by jbsgrvmak cdd jtfgrs cah njr`mak jag dmal,

wfga sfg wjudh gaojy utmdmty (M, 3),tfca mn sfg ms ja fgr jwa, wfga sfg wjudh gaojy utmdmty

(3, 9)< mt ms tfg g`pty agtwjrl, jtfgrwmsg, tfct ms, wfga (M, 3)4 (3, 9),wfgrgMhgajtgs

tfg au`bgr jn ckgats. Ckcma, tfg hyac`me prjegss sgdn-jrkcamzgs mn(l+ 3, l) 1 (3, 9),

njr sj`g l {3, 2, . . . , M 3}, ma wfmef ecsg tfg dm`mt ms tfg wfggd, mn(l+ 3, l)4 (3, 9),

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l {3, 2, . . . , M 3}, ma wfmef ecsg tfg dm`mt ms gmtfgr tfg wfggd jr tfg g`pty agtwjrl.Cdtfjukf agtwjrls wmtf hmrgetgh dmals crg m`pjrtcat mt ms c ermtmecd ngcturg jn tfg

[[[ tfct ms hmrgetmjacd cah crg cttrcetmak ejasmhgrcbdg cttgatmja, tfg stuhy jn uahm-

rgetgh dmals ms cdsj matgrgstmak. Bcdc cah Kjycd cdsj stuhy uahmrgetgh dmals, wfmef cs `ga-

tmjagh cbjvg ceej``jhctg twj-smhgh manjr`ctmja fljws. Xfms ecsg bgst rgprgsgatgh by c

pfjag ecdd ejaagetmja, wfgrg c pgrsja mamtmctgs c ecdd tj cajtfgr pgrsja cah maeurs mts ejst,

but bjtf pcrtmgs bgagffit nrj` tfg gxefcakg. Xfgy sfjw tfct wfga tfg typmecd ckgat fcs

c kgagrcd pcyj wfmef ms maergcsmak ma m(k), tfg tjtcd au`bgr jn jtfgrs ca mahmvmhucd

ms dmalgh wmtf hmrgetdy jr mahmrgetdy, cah hgergcsmak ma tfg au`bgr jn hmrget dmals, hm (k),

m(k) 0 (m(k), hm (k)), tfga c strmet Acsf agtwjrl ms gmtfgr c egatgr-spjasjrgh stcr (tfg

ckgat wfj mamtmctgs mt sgrvgs cs egatgr cah pcys njr tfg ejsts jn dmals), jr tfg g`pty agtwjrl.

Xfg ncet tfct tfg stcr ms c prj`magat crefmtgeturg fgrg, bjtf cs tfg gstemgat agtwjrl cah

tfg dm`mt njr sgdn-jrkcamzctmja, sukkgsts tfct agtwjrl crefmtgeturg ms sgasmtmvg tj tfg acturgjn manjr`ctmja tgefajdjky. [g ajtg tfct tfms rgsudt ms gqumvcdgat tj Ncbrmlcat gt cd. (2995),

wfjsg prjbdg` mavjdvgs c ejst `mam`mzctmja.

Cdd jn tfgsg rgsudts cppdy wfga tfg pcyj ms masgasmtmvg tj tfg hmstcaeg bgtwgga ckgats ma

tfg sgasg tfct hmrget cah mahmrget ejaagetmjas ejatrmbutg tfg sc`g wcy tj ca ckgats pcyj.

Mn tfg vcdug jn manjr`ctmja pjssgssgh by ca ckgat hgecys tfg nurtfgr cwcy jtfgrs crg nrj`

fgr, tfga Bcdc cah Kjycd ffiah mt fcrhgr tj prgemsgdy efcrcetgrmzg strmet Acsf agtwjrls. [fmdg

tfg wfggd ejatmaugs, njr sj`g pcrc`gtgr vcdugs, tj bg tfg dm`mt crefmtgeturg tfg stcr cdsj

cppgcrs cs c dm`mt, njr jtfgr pcrc`gtgr vcdugs. Xfms ms, jn ejursg, ajt sj surprmsmak ma vmgw jn

Ncbrmlcat gt cd., (2995), hmseussgh cbjvg, wfmef cssmkas ermtmecd rjdg tj gcef ckgatsagtwjrl

hmstcaeg nrj` jtfgr ckgats. Fjwgvgr, njr sj`g pcrc`gtgr vcdugs, gspgemcddy wfga tfg hgecy

pcrc`gtgr ms edjsg tj 3, jtfgr crefmtgeturgs crg cdsj cssjemctgh wmtf strmet Acsf gqumdmbrmc,

suef cs matgrdmalgh stcrs cah rjsg-pgtcds. Ugdn-jrkcamzctmja ms fcrhgr tj efcrcetgrmzg wmtf

manjr`ctmja hgecy, wfgrg dm`mt rgsudts crg jbtcmagh njr pcrc`gtgr vcdugs tfct m`pdy rgdctmvgdy

fmkf ejsts jn dmal njr`ctmja.

Ocelsja cah [jdmasly (3==>) prjvmhg c `jhgd njr twj-smhgh dmal njr`ctmja cdjak wmtf

c sjdutmja ejaegpt, pcmrwmsg stcbmdmty? c agtwjrl ms pcmrwmsg stcbdg, mn aj mahmvmhucd fcs ca

maegatmvg tj hgdgtg c dmal tfct gxmsts, cah aj pcmr jn pdcygrs fcvg ca maegatmvg tj njr` c dmal

tfct hjgs ajt gxmst. Xfg pcyj tj pdcygr m ma agtwjrl k ms cssu`gh tj bg

m(k) 0 3 +

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jn ckgat ms hmrget dmals ma tfg agtwjrl, cah e tfg ejst pgr dmal. [mtf twj-smhgh dmals (

tfg sy``gtrme ecsg ) pcmr-wmsg stcbdg agtwjrl jutej`gs `cy bg, hgpgahmak upja pcrc`gtgr

vcdugs, gmtfgr tfg ej`pdgtg krcpf, wfgrg gvgryjag ms ejaagetgh wmtf gvgryjag gdsg, mn 9 4 e 4

2,jr tfg[cdrcsmca stcr,wfgrg gvgryjag ms ejaagetgh tj c smakdg ckgat, mn

2 4 e 4 ,

jr aj ejaagetmjas ct cdd. [g ajtg tfct tfms rgsudt prghctgs tfg Ncbrmlcat gt cd. (2995).

[ctts (3==6) g`pdjys tfg sc`g nrc`gwjrl cah gxtgahs mt tj c hyac`me sgttmak. Ufg

jbtcmas ejahmtmjas uahgr wfmef agtwjrls wmtf hmgrgat crefmtgeturgs `mkft njr . Ma pcr-

tmeudcr, mn ejst pgr dmal ms vgry s`cdd, 9 4 e 4 2, tfga tfg prjegss ejavgrkgs tj tfgej`pdgtg agtwjrl ma ffiamtg tm`g, wmtf prjbcbmdmty 3. Mn tfg ejst ms ma ca matgr`ghmctg rcakg,

2 4 e 4 ,tfga tfg stcr agtwjrl wmdd g`grkg wmtf pjsmtmvg prjbcbmdmty, wfmef hgergcsgswmtf tfg au`bgr jn pdcygrs .

Ocelsja cah [ctts (2992c) stuhy hyac`me gvjdutmja jn agtwjrls by `gcas jn sgqugaegs

jn agtwjrls, wfmef `cy ej`g cbjut cs c rgsudt jn `yjpme hgemsmjas jn mahmvmhucds ma chhmakcah hgdgtmak dmals. Xfgy sfjw tfct tfgrg cdwcys gxmsts c pcmrwmsg stcbdg agtwjrl jr c edjsgh

eyedg, tfct ms wfga c au`bgr jn hmgrgat agtwjrls crg rgpgctghdy vmsmtgh ma sj`g sgqugaeg.

Xfgy cdsj matrjhueg ca gvjdutmjacry cacdysms jn tfg hyac`me agtwjrl njr`ctmja prjegss,

wfgrg tfgrg gxmst c s`cdd prjbcbmdmty jn uamatgahgh efcakgs ( `utctmjas ). Wsmak tfg sc`g

`ctfg`ctmecd tjjds cs tfg jags g`pdjygh by ]juak (3==6), Ocelsja cah [ctts sfjw tfct

agtwjrls, nrj` tfjsg tfct crg pcmrwmsg stcbdg jr bgdjakmak tj eyedgs, tfct crg fcrhgr tj kgt

cwcy nrj` cah gcsmgr tj kgt tj, crg ncvjrgh by tfg gvjdutmjacry prjegss. Fjwgvgr, tfgy sfjw

tfct gvga wfga tfg uamqug gstemgat krcpf ms pcmrwmsg stcbdg, mt ms ajt gvjdutmjacrmdy stcbdg.

[fct sjrt jn `gtfjhjdjkmecd crku`gats hjgs tfms dmtgrcturg usg ma hgrmvmak rgsudts: Xfg

crku`gats g`pdjygh by Bcdc cah Kjycd ma cacdyzmak bjtf csy``gtrme cah sy``gtrme dmals

mavjdvg matumtmvg stgps tfct gxpdjmt tfg matgkgr acturg jn tfg prjbdg . Xfms ms cdsj trug njr

Ncbrmlcat gt cd. (2995). Njr gxc`pdg, ma tfg ecsg jn uahmrgetgh dmals, tfgy Bcdc cah Kjycd

gstcbdmsf tfct c Acsf agtwjrl ms gmtfgr g`pty jr ejaagetgh. Xfga tfgy sfjw tfct mn c pdcygr

a fcs c dmal wmtf pdcygr o tfga aj jtfgr pdcygr eca fcvg c dmal wmtf o. Xfct ms, mn twj

ckgats, scy m cah o, eca fcvg c dmal wmtf l, tfga jag jn tfg` wjudh bg mahmgrgat bgtwgga

njr`mak c dmal wmtf l jr wmtf tfg jtfgr ckgat. Xfus, a `ust bg tfg egatgr jn tfg stcr.

Uuef crku`gats crg cpprjprmctg njr tfgmr sgttmak, but `clg mt fcrh tj ecrry jut ej`pcrctmvg

stctmes (jr hyac`mes) gxgremsgs, cah hj ajt dgah tfg`sgdvgs rgchmdy tj fcahdmak fgtgrjkgagmty

cah uaegrtcmaty. Nurtfgr rgsgcref by Kjycd cah ejcutfjrs fcs garmefgh tfg sgt jn jutej`gs

by cpprjprmctgdy spgemnymak tfg jbogetmvgs jn ckgats cah cddjwmak njr fgtgrjkgagmty ma tfg

vcdug cah tfg ejst jn ejaagetmjas, cah njr hgpgahgaeg ja `jrg ej`pdgx efcrcetgrmstmes jn

agtwjrl tjpjdjky tfca oust kgjhgsme hmstcaegs cah au`bgr jn hmrget dmals. [g rgvmgw agxt

sj`g jn tfgsg ajtgwjrtfy hgvgdjp`gats.

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Kjycd cah Ojsfm (2995) sfjw tfct tfg matgrcetmja jn mahmvmhucd maegatmvgs acrrjws hjwa

tfg rcakg jn agtwjrls tfct `cy g`grkg ct gqumdmbrmu`. Xfms rcakg maeduhgs sy``gtrme

(bcdcaegh) krcpfs wmtf hmgrgat hgkrggs, gxedusmvg krjup agtwjrls cah stcrs. Njr c agtwjrl

k, dgt km hgajtg tfg agtwjrl rgsudtmak nrj` hgdgtmak ckgat m cah cdd jn fgr hmrget dmals,

cah D(km) hgajtgs tfg su` tjtcd jn tfg hgkrggs (dmals) jn tfg rgst jn tfg ckgats? D(km) 0o0m ho(km). ^crmjus gxc`pdgs hmseussgh by Kjycd cah Ojsfm (2995) njddjw nrj` hmgrgat

spgemffiectmjas jn tfg `crkmacd bgagffit jn ca chhmtmjacd dmal jn ca ckgat, cs c nuaetmja jn fgr

jwa hgkrgg cah jn tfg su` tjtcd jn tfg hgkrggs jn cdd jtfgr ckgats. Mn tfg `crkmacd bgagffit ms

maergcsmak ma ca ckgats jwa hgkrgg, tfga c pcmrwmsg-stcbdg gqumdmbrmu` agtwjrl gxmsts? mt `cy

bg g`pty, ej`pdgtg, jr fcvg tfg hj`macat krjup crefmtgeturg (wfgrg jag krjup jn ckgats

njr`s c ej`pdgtg subkrcpf cah c sgejah krjup ejasmsts jn msjdctgh ckgats). Nurtfgr`jrg, mn

tfg `crkmacd rgtura ms maergcsmak ma tfg su` tjtcd jn cdd jtfgrs hgkrggs, tfga tfg gqumdmbrmu`

agtwjrls crg uahmrgetgh< mn mt ms hgergcsmak, tfgy wjudh bg hmrgetgh. Xfms njddjws bgecusgcay twj ckgats tfct dmal wmtf jtfgrs `ust cdsj dmal wmtf gcef jtfgr. Mn tfg `crkmacd rgtura

jn ca chhmtmjacd dmal ms hgergcsmak ma ca ckgats jwa hgkrgg cah ma tfg su` tjtcd jn cdd

jtfgrs hgkrggs, tfga sy``gtrme pcmrwmsg-stcbdg gqumdmbrmu` agtwjrls `cy ajt gxmst, cah

csy``gtrme agtwjrls wmtf sfcrp magqucdmty ma tfg au`bgr jn dmals `cy cssu`g tfg njr`

jn stcr-dmlg strueturgs wfgrg dmalgh ckgats fcvg vgry uagqucd au`bgr jn dmals. Mn, mastgch,

tfg `crkmacd bgagffit ms maergcsmak ma tfg su` tjtcd jn cdd jtfgrs hgkrggs, tfga sy``gtrme

gqumdmbrmu` agtwjrls cdwcys gxmst.

Kcdgjttm, Kjycd cah Lc`pfjrst (2995) sfjw tfct ma `jhgds jn strctgkme agtwjrl njr-

`ctmja, fgtgrjkgagmty wmtf rgspget tj tfg vcdug jn ejaagetmjas fcs c hmgrgat gget nrj`

fgtgrjkgagmty wmtf rgspget tj tfg ejst jn dmals. Xfg njr`gr cgets tfg dgvgd jn ejaagetghagss

jn c agtwjrl, wfmdg tfg dcttgr cgets bjtf tfg dgvgd jn ejaagetghagss jn c agtwjrl cs wgdd cs

tfg crefmtgeturg jn tfg rgsudtmak ej`pjagats. [fga c sjemgty `cy bg hmvmhgh matj hmstmaet

krjups, wmtfma gcef jn wfmef dmals crg efgcpgr tfca dmals cerjss tj jtfgr krjups, tfga matgr-

ejaagetgh stcrs crg sjemcddy gstemgat cah hyac`mecddy stcbdg. Xfms sukkgsts tfct egatrcdmty,

stcr-spjasjrsfmp cah s`cdd hmc`gtgrs crg rjbust ngcturgs jn agtwjrls. Cdd tfgsg stuhmgs sfjw,

ajt surprmsmakdy, tfct wfga ckgats jbogetmvgs crg `jrg ej`pdgx nuaetmjas jn agtwjrl efcr-

cetgrmstmes, tfga tfg gqumdmbrmu` agtwjrl jutej`gs gxfmbmt `jrg ej`pdgx strueturg. Nrj`

tfg pgrspgetmvg jn tfg prgsgat pcpgr, jn ejursg, tfms dmtgrcturg fcs bgga qumtg sueegssnud, ma

tfct mt fcs hgdmvgrgh cacdytmecd hgsermptmjas njr tfg gvjdutmja jn gatmrg agtwjrls.

[g gsefgw c hmseussmja jn ojb manjr`ctmja agtwjrls bgecusg tfgy crg rgvmgwgh ma hgtcmd

by Mjcaamhgs cah Djury (2998). Cajtfgr ajtgwjrtfy crgc jn cppdmectmjas ms pcttgras ma

matgractmjacd trchg ckrgg`gats tfct ncemdmtctg trchg c`jak egrtcma ejuatrmgs cah hmsejurckg

tfg` c`jak jtfgrs `cy cdsj dgah tfg`sgdvgs tj sm`mdcr ejasmhgrctmjas cah fcvg bgga dcrkgdy

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16/66

uagxpdjrgh. [g cdsj pcss up suef cppdmectmjas, ma pcrt bgecusg tfgy hj ajt sgg` tj ffit tfg

sjrt jn `jhgds mavjdvmak dcrkg au`bgrs jn ckgats, wfmef crg gssgatmcd njr sj`g jn jur rgsudts.

Ma ejaeduhmak tfms sgetmja, mt ms m`pjrtcat tj ajtg tfct tfg rgegat rgsgcref ja ej``uamec-

tmja agtwjrls cs `jhgds jn sjemcd strueturg, wfmef ms rgvmgwgh fgrg, fcs jpgagh up m`pjrtcat

cvgaugs ma fmtfgrtj vmrtucddy uagxpdjrgh crgcs. Ca m`pjrtcat pcyj ct stclg fgrg ms dmalmak

gejaj`mes wmtf tfg agtwjrl-bcsgh tfgjrmgs jn `ctfg`ctmecd sjemjdjky. Ct tfg fgcrt jn tfg

sjemjdjky dmtgrcturg ms c bgdmgn tfct agtwjrl-bcsgh `jhgds crg mahmspgascbdg njr `jhgddmak

`jrg tfca oust trmvmcd sjemcd matgrcetmjas. Cs [fmtg (3==7) g`pfcsmzgs, nurtfgr tfgjrmzmak

ms dmlgdy tj pcy j gvga wmtfma sjemjdjky, wfgrg ma spmtg jn tgefamecd cefmgvg`gats ma sjemcd

agtwjrl `gcsurg`gats, `jhgddmak agtwjrl ejastruets fcvg fch dmttdg m`pcet sj ncr ja tfg

`cma dmags jn sjemjeudturcd tfgjrmzmak ... \ mbmh., p. 397= P. [fmtg sggs ca m`pjrtcat rjdg

njr stuhymak sjemcd matgrcetmjas tfrjukf matgrdmalmak jn hmgrgat mahmvmhucd-bcsgh agtwjrls

cssjemctgh wmtf sjemcd hmsejursg. Mt ms cdsj matgrgstmak tfct tfg dmtgrcturg ma tfms crgc ms hg-vgdjpmak ncst. Gspgemcddy ajtgwjrtfy crg c sgrmgs jn pcpgrs by Ocelsja cah Rjkgrs (2998;2 ej-cutfjrs pgr pgrsja cah 63,23; cutfjrs, tfg

prjbcbmdmty jn c rcahj` dmal ms 9.99927, wfmef ms cdsj cpprjxm`ctgdy gqucd tj tfg edustgrmak

ejgstemgat, ma tfg ecsg jn tfg rcahj` krcpf `jhgd wfgrg ejaagetmjas crg cssu`gh tj bgmahgpgahgat. Fjwgvgr, tfg dcttgr ms ej`putgh tj bg cbjut 9.37;, cah tfus jvgr >999 tm`gs

dcrkgr tfca tfg rcahj` krcpf `jhgd wjudh prghmet. Xfg hmstrmbutmja jh tfg au`bgr jn

ej-cutfjrsfmps ms ajt Yjmssja cah mastgch fcs c Ycrgtj tcmd. Xfg cutfjrs crkug tfct tfg

bcsme prgngrgatmcd cttcef`gat `jhgd jn Bcrcbcsm cah cssjemctgs hjgs ajt hgsermbg wgdd tfg

gejaj`mes wjrdh. Nurtfgr`jrg, tfgrg ms krgctgr fgtgrjkgagmty ma tfg fmgrcrefy dgvgds jn ckgats

wmtf wfj` tfg egatrcd ckgat ms dmalgh, tfca ma Rcvcsz cah Bcrcbcsm (2995).

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Xfg cutfjrs prjpjsg c sm`pdg `jhgd tfct maejrpjrctgs prjhuetmvmty hmgrgaegs cerjss

mahmvmhucds (wmtf twj typgs jn ckgats bgmak ejasmhgrgh), tfgrg ms c prjhuetmja nuaetmja njr

lajwdghkg wfmef ms sgasmtmvg tj tfg qucdmty jn ej-cutfjrs cah ca maegatmvgs strueturg wfmef

rgwcrh qucdmty. Xfg `jhgd ms dmagcr ma tfg qucdmty jn rgsgcref jutput cah quchrctme ma tfg

ejsts jn rgsgcref gjrt cah tfg au`bgr jn ej-cutfjrsfmps. Ma c wjrdh wfgrg tfgrg crg ngw

fmkf-prjhuetmvmty typgs tfg hmstrmbutmja jn dmals cah jn tfg au`bgr jn ej-cutfjrs wjudh bg

vgry uagqucd. Dmals wmdd crmsg bgtwgga pgjpdg wfj fcvg `cay ej-cutfjrs cah wfj fcvg

ngw ej-cutfjrs. Xfg cutfjrs matgrprgt tfgmr rgsudts qucdmtctmvgdy cs m`pdymak tfct stcrs crmsg,

tfct dmal wgdd ejaagetgh cah pjjrdy ejaagetgh ckgats, cah tfus sfjrt hmstcaegs crg kgagrctgh.

Jady pcmrs jn ej-cutfjrs crg gxc`magh ma tfms stuhy.

C hmgrgat cspget jn ej-cutfjrmak ms gxc`magh by Rjsgabdct cah @jbmus (2998), wfj

ejasmhgr tfg m`pcet jn m`prjvg`gats ma ej``uamectmja cah trcaspjrtctmja tgefajdjkmgs ja

sgdn-sgdgetmja ma krjup njr`ctmja. Djwgr ej``uamectmja ejsts hgergcsg sgpcrctmja bgtwggamahmvmhucds but krjup sgpcrctmja `cy maergcsg. Uj mahmvmhucds bgmak ejaagetgh wmtf jtfgrs

ncemdmtctgs sprgch jn manjr`ctmja cbjut agw tgefajdjkmgs cah ojb jppjrtuamtmgs. Ct tfg sc`g

tm`g, fgtgrjkgagjus ckgats `cy sgkrgkctg by typg. Hctc ja gejaj`mes ej-cutfjrsfmps bg-

twgga WU cah njrgmka-bcsgh cutfjrs bgnjrg cah cntgr tfg Matgragt rgvjdutmja prjvmhg suppjrt

njr tfms tfgjry. Ej-cutfjrmak fcs kgagrcddy maergcsgh c`jak gejaj`msts. Ct tfg sc`g tm`g,

tfg Matgragt fcs gacbdgh gejaj`msts tj bg `jrg sgdgetmvg cah ej-cutfjr wmtf mahmvmhucds wfj

`cy bg djectgh ncr nrj` tfg` kgjkrcpfmecddy.

@cr`crjs cah Ucegrhjtg (2995) gxpdjrg sjemcd matgrcetmjas cs `gcsurgh by tfg vjdu`g jn

g`cmds c`jak stuhgats cah rgegat krchuctgs jn Hcrt`jutf Ejddgkg. Xfms stuhy ms smkamffiecat

bgecusg rjj``ctgs ct Hcrt`jutf crg rcahj`dy cssmkagh, cah tfgrgnjrg tfg pcttgras jn

matgrcetmjas hj ajt rgflget cay gahjkgajus sgdgetmja ma tgr`s jn pfysmecd prjxm mty. Xfg

cutfjrs gxpdcma tfg vjdu`g jn g`cmd bgtwgga cay twj mahmvmhucds ma tgr`s jn rcemcd, kgahgr,

ctfdgtme cah nrctgramty/sjrjrmty cttrmbutgs, wfgtfgr jr ajt tfg rgspgetmvg mahmvmhucds wgrg

ma tfg sc`g edcss, fch tfg sc`g `cojr jr dmvgh ma tfg sc`g hjr`mtjry, cah matgrcetmjas

c`jak tfjsg vcrmcbdgs. Xfgmr rgsudts sukkgst tfct sc`g rceg ms c vgry m`pjrtcat gxpdcactjry

vcrmcbdg njr tfg vjdu`g jn g`cmd cah sj ms tj fcvg dmvgh ma tfg sc`g hjr`. Xfgsg rgsudts

uahgrsejrg tfg m`pjrtcaeg jn gxpdcmamak tfg `jtmvctmjas njr sjemcd matgrcetmjas, wfmef rgsudt

fgrg ma tfg prgsgaeg jn cssjrtctmvg `mxmak, tfct ms bcsgh ja pgrsjacd efcrcetgrmstmes cs wgdd

cs pcst sfjels ma ej``ja.

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5 Rcahj` Krcpf Xfgjry? Jdh cah Agw

Utuhymak prjbcbmdmstme cspgets jn sjemcd matgrcetmjas wjudh sgg` tj dgah mtsgdn tj rcahj`

krcpf tfgjry cs c acturcd `ctfg`ctmecd tjjd. Gvgr smaeg gejaj`msts bgec`g cwcrg jn tfg

gxmstgaeg jn rcahj` krcpf tfgjry cdc Grhjs cah Rgaym (3=>9< 3=>3) \but sgg Ujdj`jajcah Rcpjpjrt (3=73) njr ca catgeghgatP, mt fcs bgga tg`ptmak tj tfmal jn tfg g`grkgaeg jn

gejaj`me agtwjrls ma tgr`s jn rcahj` krcpf tfgjry. Ugg Hurdcun (3==>), Lmr`ca (3=65),

Lmr`cagt cd. (3=6>), Mjcaamhgs (3==9< 3==;) njr sgvgrcd gxc`pdgs jn tfms cpprjcef. Fjwgvgr,

cdd tfgsg wjrls njuah mt fcrh tj `jtmvctg wfy mt ms tfct ckgats bgfcvg ma tfg prgemsg wcy tfct

ms agegsscry tj kgagrctg pcttgras jn sjemcd ejaagetmjas tfct crg cssjemctgh krcpf ngcturgs tfct

`cy bg hgsermbgh by rcahj` krcpf tfgjry, ct dgcst cs jrmkmacddy hgvgdjpgh. Utmdd, gejaj`msts

fcvg ygt tj gxpdjrg mahmvmhucds `jtmvctmja ms sgglmak sjemcd ejatcets wmtf jtfgrs, ct dgcst cs

ncr cs tfms ms uahgrstjjh by tfg psyefjdjky cah sjemcd psyefjdjky dmtgrcturg.

5.3 Xfg Grhjs cah Rgaym @jhgd

[g ffirst hg`jastrctg fjw tfg lgy ngcturgs jn tfg GrhjsRgaym `jhgd crg rgspjasmbdg njr

sj`g jn mts prghmetmjas. Fjwgvgr, tfg GrhjsRgaym `jhgd hjgs ajt sgg` tj bg suppjrtgh by

tfg ncets. Upgemffiecddy, dgtKGRM,p hgajtg tfg gasg`bdgjn krcpfs wmtf Mvgrtmegs ma wfmef gcefpjssmbdg ghkg ms prgsgat mahgpgahgatdyjn cay jtfgr ghkg cah wmtf prjbcbmdmty p,cah cbsgat

wmtf prjbcbmdmty 3 p. Xfg prjbcbmdmty tfct ca ckgat fcs gxcetdy l ejaagetmjas wmtf jtfgrckgats ma ca GrhjsRgaym rcahj` krcpf ms kmvga by tfg bmaj`mcd hmstrmbutmja?

pl 0

M 3

l

pl(3 p)M3l. (>)

Fgrg tfg rcahj` qucatmty ms tfg gatmrg krcpf, cah prjbcbmdmty (>) ejrrgspjahs tj c typmecd

ajhg. Ma tfg dm`mt, wfga tfg au`bgr jn ckgats ms `uef krgctgr tfca tfg cvgrckg au`bgr jn

ejaagetmjas gcef ckgat fcs,M (M 3)p Mp z3,tfga tfg bmaj`mcd prjbcbmdmty nuaetmjam`pdmgs tfg Yjmssja hmstrmbutmja, njr dcrkg M?

pl0 M

3

l z3

M 3 z3 l

3 z3

M 3M3

0

(z3)lgz3

l! . (;)

Ca cdtgractmvg wcy jn stctmak tfg GrhjsRgaym krcpf ms tfct au`bgr jn ejaagetmjas (ghkgs),

wfmef ms prjpjrtmjacd tj tfg au`bgr jn ckgats, ms rcahj`dy hmstrmbutgh jvgr cdd pjssmbdg32

M(M 3), ejaagetmjas c`jak ckgats. DgtKGRM,` bg tfg gasg`bdg jn krcpfs, ma wfmef cddkrcpfs wmtf `ghkgs jut jn tfg pjssmbdg 3

2M(M 3) jeeurs wmtf gqucd prjbcbmdmty.5

5Ma stctmstmecd `gefcames, tfg gqumvcdgaeg ms gxcet, wmtf tfg njr`gr bgmak tfg ecajamecd cah tfg dcttgr tfgkrcah-ecajamecd gasg`bdg, ejrrgspjahmak tj tfg Fgd`fjdtz cah Kmbbs nrgg gagrkmgs, rgspgetmvgdy. Xfgsg crg

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Ma tfg dm`mt jn dcrkg M, tfg pfcsg trcasmtmja jeeurs wfga tfg ncetjr jn prjpjrtmjacdmty

jn tfg au`bgr jn ghkgs rgdctmvg tj ajhgs bgej`gs gqucd tj 32

. Xfms ejrrgspjahs tj c `gca

hgkrgg smzg ma tfg Yjmssja `jhgd jnz3 0 3,njr wfmef p0 3M

. Bgdjw tfms vcdug, tfgrg crg ngw

ghkgs cah tfg ej`pjagats jn tfg rcahj` krcpf crg s`cdd< cbjvg tfct vcdug, c prjpjrtmja jn

tfg gatmrg krcpf bgdjaks tj c smakdg, kmcatej`pjagat. Xfms vcdug ms cssjemctgh wmtf c pfcsg

trcasmtmja ma tfg tjpjdjky jn tfg krcpf.

C s`cdd dmtgrcturg fcs ejasmhgrgh rcahj` krcpf `jhgds wmtf ghkg prjbcbmdmtmgs tfct hmgr

cerjss tfg krcpf. Ljvcdgalj (3=;7) prjvmhgs c rcrg gxc`pdg jn c rcahj` krcpf `jhgd wfgrg

tfg ghkg prjbcbmdmtmgs crg ajt gqucd. Upgemffiecddy, Ljvcdgalj ejasmhgrs rcahj` krcpfs wfgrg

ca ghkg bgtwgga ajhgsm cah o `cy jeeur wmtf tfg prjbcbmdmty pmo mahgpgahgatdy jn wfctgvgr

jtfgr ghkgs gxmst. Fg cssu`gs tfct tfg prjbcbmdmty tgahs tj 9 cs M tfct tfgrg crg ajghkgs dgchmak jut jn gvgry ajhg cah tfct tfgrg crg aj ghkgs dgchmak matj gvgry ajhg. Wahgr

sj`g chhmtmjacd dm`mtmak cssu`ptmjas cbjut tfg prjbcbmdmty strueturg, fg sfjws tfct ma tfgdm`mt tfg rcahj` krcpf bgfcvgs cs njddjws? tfgrg ms c subkrcpf C3jnma-msjdctghajhgs wfjsgjrhgr njddjws csy`ptjtmecddy c Yjmssja dcw< tfgrg ms c subkrcpf C2jnjut-msjdctghajhgs wfjsgjrhgr njddjws csy`ptjtmecddy c Yjmssja dcw< cdd rg`cmamak ajhgs njr` c ejaagetgh subkrcpf.

Xfg jrhgrs jnC3 cahC2 crg csy`ptjtmecddy mahgpgahgat cah tfgmr pcrc`gtgrs crg kmvga matgr`s jn tfg dm mt jn tfg prjbcbmdmty strueturg. [fmdg Ljvcdgalj g`pfcsmzgs tfg dm`mtmak

rgsudts jag ejudh jbtcma njr dcrkg krcpfs, tfg ajtmja tfct ghkgs `cy cppgcr wmtf hmgrgat

prjbcbmdmtmgs bgtwgga hmgrgat lmahs jn ajhgs `cy bg cppdmgh `jrg kgagrcddy. [g rgtura

tj tfms mssug nurtfgr bgdjw ma sgetmja 8.3 cntgr tfg NrcalUtrcuss `jhgd jn @crljv rcahj`

krcpfs ms matrjhuegh.

[fy sfjudh tfg au`bgr jn ejaagetmjas, tfct fcvg bgga ergctgh by uaejjrhmactgh cetmja,

jr tfg prjbcbmdmty njr gcef ejaagetmja, bg sj tfct c pfcsg trcasmtmja jeeurs: Xfms sm`pdg

ejahmtmja njr pfcsg trcasmtmja ms fcrh tj oustmny ma tfg cbsgaeg jn c nuddy spgemffigh bgfcvmjrcd

`jhgd. Xfg Yjmssja hmstrmbutmja pgcls agcr tfg `gca cah tfga fcs c rctfgr tfma uppgr tcmd

tfct hgecys rcpmhdy ceejrhmak tj 3/l!. Xfg hgkrgg hmstrmbutmjas njr `cay rgcd dmng agtwjrls

fcvg nct tcmds tfct crg bgttgr hgsermbgh by `gcas jn pjwgr dcws. Ugg, ma pcrtmeudcr, Ncdjutsjs

gt cd. (3===) wfj ffiah tfct tfg cutjaj`jus systg`s jn tfg Matgragt jbgygh c pjwgr dcw,

pl l, l 19, wmtf gxpjagat bgtwgga 2 cah 5, cah pckgs ja tfg [[[ fcvg (hmrgetgh)fypgrdmals bgtwgga tfg` wfjsg hmstrmbutmja ms fgcvmdy rmkft-slgwgh cah ms wgdd cpprjxm`ctgh

by c pjwgr dcw wmtf sm`mdcr gxpjagat cs tfg Matgragt. Cs c au`bgr jn cutfjrs, but ma

pcrtmeudcr Hjrjkjvtsgv cah @gahgs (2995), 6963, hjeu`gat ma hgtcmd, hmgrgat sjemcd (but

cdsj bmjdjkmecd, pfysmecd cah gakmaggrmak agtwjrls) hmgr ejasmhgrcbdy ma tgr`s jn tfgmr hgkrgg

kgagrctmak nuaetmjas njr `j`gats jn krcpf prjpgrtmgs jvgr tfg hmstrmbutmja jn krcpfs cah wfmef crg rgdctghby c Dckrcakg trcasnjr` wmtf rgspget tj tfg ffigdh p cah tfg jrhgr pcrc`gtgr `.

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hmstrmbutmjas cah tfgmr edustgrmak prjpgrtmgs. Cdsj, ejaagetmjas c`jak ckgats crg typmecddy

hgpgahgat. Utmdd, tfg jrmkmacd GrhjsRgaym cah mts ncmdurg tj gxpdcma rgcd-dmng agtwjrls fcvg

`jtmvctgh ejasmhgrcbdg rgsgcref, cs wg sgg agxt.

5.2 Xfg Rgvmvcd jn Rcahj` Krcpf Xfgjry

Rgegat rgsudts by tfg ej`bmactjrmes dmtgrcturg \ @jddjy cah Rggh (3==7< 3==6) P fcvg bgga

usgh by @crl Agw`ca cah c au`bgr jn ejddcbjrctjrs tj rgejasmhgr rcahj` krcpf-bcsgh

`jhgds njr tfg purpjsg jn stuhymak g`grkgaeg jn sjemcd agtwjrls. Xfct ms, tfgsg wjrls

prjvmhg tfg `ctfg`ctmecd njuahctmjas njr wjrlmak wmtf rcahj` krcpfs tfct crg efcrcetgrmzgh

by crbmtrcry hmstrmbutmjas njr tfg au`bgr jn ejaagetmjas gcef ckgat fcs wmtf jtfgrs ma c

sjemcd jr gejaj`me sgttmak. Ma pcrtmeudcr, jag aj djakgr agghs tj cssu`g tfct tfg prjbcbmdmty

hmstrmbutmja jn tfg au`bgr jn ejaagetmjas gcef pgrsja fcs wmtf jtfgrs jbgys tfg Yjmssja Dcw.

Xfms fcs rg`jvgh ca m`pjrtcat m`pghm`gat tj wjrlmak wmtf kgagrcd bgfcvmjrcd `jhgds njr

stuhymak tfg gvjdutmja jn sjemcd agtwjrls wmtf rcahj` ejaagetmjas.

Agw`ca gt cd. (2993b) prjvmhg `jtmvctmja by `gcas jn hctc ja hgkrgg hmstrmbutmjas

nrj` sj`g cetucd rgcd-dmng sjemcd agtwjrls. Xfg hctc sukkgst m`pjrtcat hmgrgaegs c`jak

hmgrgat typgs jn agtwjrls, rcakmak nrj` semgatmffie ejddcbjrctmja agtwjrls tj agtwjrls jn

`jvmg cetjrs wfj fcvg ej-stcrrgh, cah jn hmrgetjrs jn Njrtuag 3999 ej`pcamgs. Xfg dcttgr

fcs c pgcl cah ms ùef dgss slgwgh< tfg njr`gr rgsg`bdg Yjwgr dcws wmtf gxpjagatmcd eutjs.

Xfg cutfjrs cttrmbutg tfms tj tfg njddjwmak hmgrgaeg, ac`gdy tfg ncet tfct mt ms ejstdy tj

`cmatcma jags `g`bgrsfmp ja ej`pcay bjcrhs, wfmdg gvga cntgr c ej-cutfjrsfmp fcs gahgh,

tfg tmg kcmagh rg`cmas prgsgat mahgffiamtgdy. Xfg ncet tfct sjemcd rgdctmjasfmps rgqumrg cetmvg

`cmatgacaeg cppgcrs tj bg ca m`pjrtcat prjpgrty jn sjemcd agtwjrls. Xfgrgnjrg, ct dgcst

matumtmvgdy, jptm`mzmak jvgr ejaagetmjas `cy m`pdy sfcrpdy hmgrgat hmstrmbutmjas jn sjemcd

ejaagetmjas nrj` tfjsg jn jtfgr, pcssmvg, rgdctmjasfmps.

Oma gt cd. (2993) usg tfms jbsgrvctmja cs c stcrtmak pjmat njr c rghuegh-njr` tfgjry jn

fjw sjemcd agtwjrls krjw. Upgemffiecddy, tfgmr tfgjry g`pfcsmzgs tfg njddjwmak njur ngcturgs.

Nmrst, ejaagetmjas c`jak mahmvmhucds crg `chg cah ua`chg ct c tm`gsecdg wfmef ms ùef

sfjrtgr tfca tfct jn mahmvmhucds ojmamak cah dgcvmak c sjemcd agtwjrl. Ma jtfgr wjrhs, ghkgscrg chhgh jr subtrcetgh ùef `jrg nrgqugatdy tfca ajhgs, wfmef cddjws njr ca cacdyst

tj wjrl by fjdhmak ejastcat tfg au`bgr jn ajhgs cah by vcrymak tfg au`bgr jn ghkgs.

Ugejah, gvga wfga stuhymak sjemcd agtwjrls, jag wjudh gxpget tfct tfg `jrg m`pjrtcat

crg rgpgctgh ejsts jn `cmatcmamak sjemcd tmgs rgdctmvg tj jag-tm`g ejsts, tfg dgss rmkft-slgwgh

tfg hgkrgg hmstrmbutmja ms.8 Xfmrh, smaeg `jst pgjpdg fcvg sm`mdcr au`bgrs jn nrmgahs, tfg

8Xfms ms c ejaogeturg jn `mag, tclmak j nrj` mbmh., p. 2.

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prgngrgatmcd cttcef`gat `gefcams`, tfct fcs pdcygh ca m`pjrtcat rjdg ma gxpdcmamak tfg

hgkrgg hmstrmbutmja jn tfg [gb, ms ajt cs strjak. Njurtf, sjemcd agtwjrls gxfmbmt trcasmtmvmty,

tfct ms jags nrmgahs crg dmlgdy tj bg nrmgahs cdsj jn gcef jtfgr, wfmef udtm`ctgdy dgchs tj

edustgrmak. Xfg prjbcbmdmty tfct twj cequcmatcaegs jn c pgrsja crg cdsj cequcmatcaegs jn jag

cajtfgr ms sgvgrcd tm`gs dcrkgr tfca wfct ms m`pdmgh by tfg bcsgdmag rcahj` krcpf `jhgd,

njr tfg [gb, cah sgvgrcd jrhgrs jn `ckamtuhg dcrkgr ma sjemcd agtwjrls.

Xfgsg cutfjrs pgrnjr` c smùdctmja stuhy wfmef maejrpjrctgs tfgsg ngcturgs. Ma pcrtme-

udcr, mn tfg au`bgr jn ajhgs (agtwjrl smzg) ms ffixgh cah tfg au`bgr jn cequcmatcaegs krjws

vgry sdjwdy jaeg c egrtcma dgvgd fcs bgga rgcefgh, edustgrmak ms gasurgh by `clmak tfg prjbc-

bmdmty jn twj pgjpdg bgej`mak cequcmatgh maergcsg ma tfg au`bgr jn cequcmatcaegs tfgy fcvg

ma ej``ja, cah cdsj by cddjwmak nrmgahsfmps tj hgecy. Xfg cutfjrs edcm` tfct tfg rjdg jn

cequcmatcaegs ma ej``ja ms ca m`pjrtcat ncetjr ma tfg krjwtf jn sjemcd agtwjrls, rjukfdy

ejrrgspjahmak tj tfg rjdg tfct prgngrgatmcd cttcef`gat pdcys ma tfg krjwtf jn tfg [gb. Xfgmrsmùdctmja rgsudts ejaffir` tfct tfgmr smùdctgh sjemcd agtwjrls gxfmbmt m`pjrtcat ngcturgs jn

rgcd-dmng jags, cah tfus hmgr nrj` tfjsg jn tfg [gb cah jtfgr systg`s. @jst m`pjrtcatdy,

ej`ùamtmgs cppgcr, tfct ms, krjups jn vgrtmegs wmtf `cay ejaagetmjas c`jak tfgmr `g`bgrs

cah ngw jags wmtf tfjsg jutsmhg.

Agw`ca gt cd. (2993c) tclg j nrj` @jddjy cah Rggh, jp. emt. cah wjrl jut tfg

bcsme `ctfg`ctmes jn rcahj` krcpf tfgjry wmtf crbmtrcry hgkrgg hmstrmbutmjas. Xfgy cppdy

tfms tfgjry cdsj tj hmrgetgh krcpfs, tj edustgrmak cah tj bmpcrtmtg krcpfs. Agw`ca (2992)

wjrls wmtf tfg sc`g `jhgds cs tfjsg ma mbmh. but pursugs tfg` ma `jrg hgtcmd, maeduhmak

c au`bgr jn `jhgds tfct `cy bg hgffiagh ja rcahj` krcpfs, suef cs cspgets jn agtwjrl

rgsmdmgaeg cah tfg hyac`mes jn gpmhg`mjdjkmecd `jhgds. Agw`ca (2993b) g`pfcsmzgs tfct

gvga wfga tfg au`bgrs jn mahmvmhucds cequcmatcaegs vcry rcahj`dy cerjss tfg pjpudctmja

cah crg prjbcbmdmstmecddy mahgpgahgat, tfg sgt jn ca mahmvmhucds cequcmatcaegs crg ajt c

rcahj` sc`pdg jn tfg pjpudctmja. Xfct ms, kmvga c rcahj`dy efjsga cequcmatcaeg nrj` tfg

sgt jn ca mahmvmhucds cequcmatcaegs, tfct mahmvmhucds tjtcd au`bgr jn cequcmatcaegs wmdd

bg hmstrmbutgh ma prjpjrtmja tj lpl. Xfct ms, tfgrg gxmsts sgdgetmja bmcs agmkfbjr bmcs:

ejaagetmja bmcs:? tfgrg gxmst l tm`gs cs `cay dmals njr ca mahmvmhucd jn jn hgkrgg pl tfca njr

ca mahmvmhucd wmtf jady c smakdg dmal, wfgrg pl ms tfg hmstrmbutmja jn au`bgr jn cequcmatcaegs

tfct c rcahj`dy efjsga pgrsja ma tfg pjpudctmja fcs. Umaeg tfg hgkrgg hmstrmbutmja njr cay

kmvga agmkfbjr ms prjpjrtmjacd tj tfg au`bgr jn cequcmatcaegs tfct tfg jtfgr pgrsja cdrgchy

fcs, mt ms kmvga by?

pl 0 lplG\lP

. (6)

[g wmdd rgngr tj cs tfg mahuegh hmstrmbutmja jn agmkfbjrs hgkrggs. Gxpdjrmak tfms ajtmja ma tfg

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ejatgxt jn tfgsg agw cacdytmecd tjjds, wfmef fcvg bgga hgvgdjpgh by Agw`ca cah ej-cutfjrs,

turas jut tj bg pcrtmeudcrdy nrumtnud ma uahgrstcahmak tfg au`bgr jn tfg cequcmatcaegs jn

jags cequcmatcaegs jn jags cequcmatcaegs cah sj ja, tfct ms, jn jags agmkfbjrss agmkfbjrs

ma tfg cequcmatcaeg agtwjrl. Mt ms matgrgstmak tfct tfms bmcs ms ejaegptucddy sm`mdcr tj tfg

dgaktf-bmcsgh sc`pdmak cssjemctgh wmtf sc`pdmak g`pdjy`gat jr uag`pdjy`gat spgdds by

`gcas jn hctc ejddgetgh nrj` g`pdjygh, jr rgspgetmvgdy, uag`pdjygh mahmvmhucds. Xfms cdsj

sukkgsts tfct mt `cy bg m`pjrtcat tj lajw fjw hctc ja sjemcd agtwjrls crg cetucddy ejddgetgh.

Xfg dmal tfct tfms dmtgrcturg fcs `chg bgtwgga g`grkgaeg jn pjwgr dcws cah cttrmbutgs jn

hgdmbgrctgdy jptm`mzgh systg`s pjmats tj ca m`pjrtcat puzzdg ma tfg ejatgxt jn tfg dmtgrcturg.

Ecrdsja cah Hjydg (3===) (but sgg cdsj Agw`ca gt cd. (2992b) wfj jbtcma c edjsgh-njr`

sjdutmja njr jag jn tfgmr lgy `jhgds wmtf rmsl cvgrsmja) matrjhueg tfg ejaegpt jn fmkfdy

jptm`mzgh tjdgrcaeg. Xfms ms prjpjsgh cs c ngcturg jn gmtfgr acturcd sgdgetmja jr hgdmbgrctg

gakmaggrmak hgsmka tfct prjvmhgs rjbust pgrnjr`caeg ma uaegrtcma gavmrja`gats. Xfgy ct-trmbutg tfg ubmqumtjus, ct dgcst ma tfg vmgw jn egrtcma sefjdcrs, prgsgaeg jn pjwgr dcws tj

tfms ngcturg.

Ma c au`bgr jn pcpgrs, Agw`ca cah ejcutfjrs fcvg g`pfcsmzgh prjpgrtmgs tfct crg pcr-

tmeudcrdy prgvcdgat ma sjemcd agtwjrls. Xfgsg crg fmkf hgkrggs jn edustgrmak tfg nrmgahs

jn `y nrmgahs crg typmecddy `y nrmgahs, tjj cah pjsmtmvg ejrrgdctmjas (cssjrtctmvg `mx-

mak) bgtwgga tfg hgkrggs jn chocegat vgrtmegs. Uuef fmkf edustgrmak fcs bgga cttrmbutgh tj

ej``uamty strueturg jn agtwjrls. Agw`ca cah Ycrl (2995) hg`jastrctg tfct ej``uamty

strueturg eca cdsj ceejuat njr cssjrtctmvg `mxmak. Agw`ca (2992b) gxc`mags ma pcrtmeudcr

cssjrtctmvg `mxmak by hgkrgg ma hmgrgat typgs jn agtwjrls. Fg ajtgs tfct sjemcd agtwjrls

crg cssjrtctmvgdy `mxgh but tgefajdjkmecd cah bmjdjkmecd agtwjrls tgah tj bg hmscssjrtctmvg.

Agw`ca (2995c) ejasmhgrs c au`bgr jn `gcsurgs jn cssjrtctmvg `mxmak cpprjprmctg njr tfg

vcrmjus `mxmak typgs, cah cppdmgs tfg` tj c vcrmgty jn wjrdh pfgaj`gac. Agw`ca (2995b)

hgvgdjps c `jhgd jn agtwjrls wmtf fmkf edustgrmak wfmef ms gxcetdy sjdvcbdg ma tgr`s jn mts

ej`pjagat smzgs, pgrejdctmja tfrgsfjdh cah edustgrmak ejgstemgat.

5.5 Rcahj` Krcpfs wmtf Crbmtrcry Hgkrgg Hmstrmbutmja

Xfg rgegat hgvgdjp`gats ma tfg tfgjry jn rcahj` krcpfs wmtf crbmtrcry hgkrgg hmstrmbu-

tmjas, cs crtmeudctgh by Agw`ca gt cd. (2993c< 2993b) cah Agw`ca (2992), crg bcsgh ja

cacdyzmak tfg prjbcbmdmty hmstrmbutmjas njr c au`bgr jn krcpf-rgdctgh rcahj` vcrmcbdgs tfct

crg matgrgstmak ma tfgmr jwa rmkft. Upgemffiecddy, dgt tfg hgkrgg hmstrmbutmja njr c rcahj`dy

sgdgetgh ajhg bg pl, l 0 9, 3, . . . , wfgrg tfg rcahj` vcrmcbdg L mahmectgs tfg au`bgr jn

jtfgr ckgats c rcahj`dy sgdgetgh ckgat ms ejaagetgh wmtf. Xfgsg wmdd bg rgngrrgh tj cs ca

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ckgats (ffirst) agmkfbjrs. Ejasmhgr ajw tfg tjtcd au`bgr jn jtfgr ckgats ca ckgat `cy rgcef

by njddjwmak c rcahj`dy sgdgetgh ejatcet. Xfgsg wmdd bg rgngrrgh tj cs ca ckgats sgejah

agmkfbjrs. Xfg au`bgr jn jtfgr ckgats wfj` crg ejatcetgh ma tfms ncsfmja ms ajt hmstrmbutgh

ceejrhmak tj tfg hmstrmbutmja nuaetmja pl. Xfg rgcsja njr tfct ms sm`pdy tfct tfg ncet tfct

cajtfgr ckgat ms sgdgetgh ajt rcahj`dy nrj` tfg pjpudctmja jn ckgats, but ejahmtmjacddy ja

fcvmak c ejatcet, wfmef jn ejursg bmcsgs tfg sgdgetmja. Ejasmhgr agxt tfg hmstrmbutmja jn tfg

smzg jn tfg ej`pjagat tfct c rcahj`dy sgdgetgh ckgat bgdjaks tj. Xfg tfgjry jn rcahj`

krcpfs wmtf crbmtrcry hgkrgg hmstrmbutmjas hgpgahs ermtmecddy ja hgrmvmak stctmstmes njr tfgsg

twj hmstrmbutmjas. Xfgsg hgrmvctmjas crg qumtg mddu`mactmak, cs wg sgg agxt.

[g ajw gxc`mag tfg prjpgrtmgs jn tfg prjbcbmdmty hmstrmbutmja jn ca ckgatssgejah agmkf-

bjrs, tfct ms, tfg au`bgr jn jtfgr ckgats rgcefgh by njddjwmak c ejatcet tj cajtfgr ckgat

cah tfga ejasmhgrmak tfct jtfgr ckgats jtfgr ejatcets. Njddjwmak Agw`ca (2992), p. ;,

tfg hmstrmbutmja nuaetmja jn ca ckgats sgejah agmkfbjrs tfct crg cssjemctgh wmtf c rcahj`dyefjsga (ffirst) agmkfbjr ms kmvga by ql 0

(l+3)pl+3oopo

. Ajtg tfct tfms ms rgdctgh tj tfg agmkfbjr

hgkrgg hmstrmbutmja hgrmvgh cbjvg cah cppdmgs jady mn hgkrggs crg mahgpgahgat. Xfg cvgrckg

au`bgr jn sgejah agmkfbjrs `cy bg ej`putgh by usmak ffirst prmaempdgs jr prjbcbmdmty kga-

grctmak nuaetmjas (YKN) tgefamqugs. Njr tfg dcttgr, sgg tfg cppgahmx. [jrlmak nrj` ffirst

prmaempdgs ymgdhs?l09

lql 0

l09 l(l+ 3)pl+3

oopo0

G\l2P G\lPG\lP

.

Xfg tjtcd au`bgr jn sgejah agmkfbjrs jn ca ckgat crg tfus kmvga by

z20G\l2P G\lP. (=)

[jrlmak ma c dmlg `caagr wg fcvg tfct tfg cvgrckg au`bgr jn agmkfbjrs ct hmstcaeg ` ms

z 0 z2z3

z 3,wmtf z3 0 G\lP,wfmef by mtgrctmak ymgdhs

z 0

z2z3

`3z3. (39)

Mn tfg au`bgr jn agmkfbjrs ct hmstcaeg` ejavgrkgs, tfga tfgrg ecaajt bg c kmcat ej`pjagat

ma tfg krcpf. Mn mt hmvgrkgs, wfmef wjudh fcppga wfga z2z3

3, tfga tfgrg `ust bg c kmcat

ej`pjagat. Uj, ct tfg pjmat wfgrg z20z3,

G\l2P 2G\lP 9, (33)tfg krcpf uahgrkjgs c pfcsg trcasmtmja.7 Xfms ejahmtmja `cy bg rgwrmttga cs?

^cr\lP 2G\lP (G\lP)2. (32)7Ca cdtgractmvg hgrmvctmja, hug tj @crl Agw`ca, jn tfms rgsudt ms prjbcbdy `jrg matumtmvg. Utcrtmak nrj`

c ejaagetgh ej`pjagat jn tfg krcpf ejasmhgr chhmak c agw ghkg tfct ejaagets wmtf c prgvmjusdy msjdctgh

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Mt ms strcmkftnjrwcrh tj hg`jastrctg tfg smkamffiecaeg jn tfms ejahmtmja by stcrtmak nrj`

tfg ecsg jn tfg Grhjs-Rgaym krcpf. Ma tfct ecsg, tfg hgkrgg hmstrmbutmja ms Yjmssja. Njr

tfg Yjmssja hmstrmbutmja, tfg `gca cah tfg vcrmcaeg crg gqucd cah tfgrgnjrg ejahmtmja (33)

m`pdmgs tfct ct tfg pfcsg trcasmtmja G\lP 0 3, wfmef ms tfg ejahmtmja cdsj jbtcmagh by tfg

jrmkmacd Grhjs-Rgaym krcpf.

Fjw cbjut tfg kgagrcd prjpgrtmgs jn pjssmbdg hgkrgg hmstrmbutmjas wfjsg krcpfs ch`mt

pfcsg trcasmtmjas: Ma gget, ejahmtmja (32) rgqumrgs tfct tfg vcrmcaeg jn tfg hgkrgg hms-

trmbutmja bg sustemgatdy dcrkg. Xj sgg tfms `jrg edgcrdy, dgt us ejasmhgr c trcetcbdg hgkrgg

hmstrmbutmja. Ycrtmeudcrdy kjjh ecahmhctgs crg `mxturgs jn Yjmssja hmstrmbutmjas tfct cddjw

njr tfg Yjmssja pcrc`gtgr tj bg hmstrmbutgh ma tfg rgdgvcat pjpudctmja.

C pcrtmeudcrdy ejavgamgat `mxmak hmstrmbutmja njr tfg Yjmssja pcrc`gtgr z3, pl(z3) 0

gz3zl3l!

,ms c Kc``c hmstrmbutmja wmtf pcrc`gtgrs (, )>,

(z3|, ) 0 3()

z3

gxp

z3

3

z3,

njr wfmef G\z3P 0, ^cr(z3) 0 2

.; Mt ms wgdd lajwa tfct tfg rgsudtmak `mxgh hmstrmbutmja ms

c agkctmvg bmaj`mcd hmstrmbutmja, njr wfmef

z04 L 10, ^cr(L) 0 +3

2. (38)

Mt njddjws tfct njr tfms hmstrmbutmja tfg vcrmcaeg gxegghs tfg `gca by ca c`juat tfct ms

hgergcsmak ma tfg pcrc`gtgr .

ajhg jn hgkrgg l. Hjmak sj wmdd efcakg tfg au`bgr jn ajhg ja tfg bjuahcry jn tfg ejaagetgh ej`pjagatby3 + (l 3) 0 l 2. Xfg dmlgdmfjjh tfct c ajhg ms ja tfg bjuahcry jn tfg ejaagetgh ej`pjagat msprjpjrtmjacd tjl ? tfgrg crglcs `cay ghkgs by wfmef c ajhg jn hgkrgg l ejudh bg ejaagetgh tj tfg ejaagetghej`pjagat tfca tfgrg crg njr c ajhg jn hgkrgg 3. Xfgrgnjrg, tfg gxpgetgh efcakg ma tfg au`bgr jn ajhgsja tfg bjuahcry wfga ca chhmtmjacd ajhg ms ejaagetgh ms kmvga by

mlm(lm 2)/

mlm. Mn tfms qucatmty ms

agkctmvg, tfga tfg au`bgr jn ajhgs ja tfg bjuahcry hgergcsgs cah tfg tfgrgnjrg tfg smzg jn tfg ejaagetghej`pjagat wmdd stjp krjwmak. Mn mt ms pjsmtmvg, ja tfg jtfgr fcah, tfga tfg au`bgr jn bjuahcry ajhgs wmddkrjw cah tfg smzg jn tfg ejaagetgh ej`pjagat wmdd krjw wmtfjut dm`mt cah wmdd bg dm`mtgh by tfg smzg jn tfgagtwjrl.

>Ugg Ec`grja cah Xrmvghm (3=6>) cah Ojfasja gt cd. (3==5), Ef. 7, njr gxtgasmvg trgct gat jn tfmssuboget.

;Mt njddjws rgchmdy by matgkrctmja tfct tfg `mxgh hmstrmbutmja ms kmvga by mts prjbcbmdmty `css nuaetmja?

Yr\L0 l|, P 0 (l+)(l+ 3)()

+

+

l. (35)

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5.8 G`grkgaeg jn Kmcat Ej`pjagat

Cppdymak tfg ejahmtmja njr pfcsg trcasmtmja tj tfg ecsg jn tfg agkctmvg bmaj`mcd hmstrmbutmja

njr tfg au`bgr jn ejatcets gcef ckgat fcs wmtf jtfgr ckgats, wg fcvg tfct tfg gxpgetgh

au`bgr jn sgejah agmkfbjrs ms z2

0 + 3 + 32 0 3 +

3

2. Xfg `gca au`bgr jn

sgejah agmkfbjrs gxegghs tfg `gca, z3 0 , m

+ 3

. (37)

Xfgrgnjrg, tfg s`cddgr ms tfg fmkfgr ms tfg vcrmcaeg jn tfg Yjmssja pcrc`gtgr z3, kmvga

mts `gca , cah tfg `jrg dmlgdy mt ms tfct c kmcat ej`pjagat g`grkgs. Ma jtfgr wjrhs,

tfg vcrmcaeg jn tfg Yjmssja pcrc`gtgr `ust bg sustemgatdy dcrkg njr tfg kmcat ej`pjagat tj

g`grkg.

Njr cajtfgr cppdmectmja, ejasmhgr c Yjmssja hmstrmbutmja cs tfg `mxmak hmstrmbutmja njrtfg Yjmssja pcrc`gtgr z3. Njr tfms sj ecddgh Agy`ca Xypg C hmstrmbutmja wmtf pcrc`gtgrs

(, ) , wg fcvg G\lP 0, G\l2P 0 + 2 \ Ojfasjagt cd, jp. emt.,52652=, 5;3 P. C wcy

tj vmsucdmzg tfms `jhgd ms tj scy tfct gcef ckgat uahgrtclgs c au`bgr jn mamtmctmvgs tj ejatcet

jtfgrs, cah tfg au`bgr jn tfjsg mamtmctmvgs fcs c Yjmssja hmstrmbutmja wmtf pcrc`gtgr .Gcef

jn tfjsg mamtmctmvgs prjhueg ma tura c au`bgr jn ejatcets, wfjsg au`bgrs crg mahgpgahgat cah

mhgatmecddy hmstrmbutgh Yjmssja wmtf pcrc`gtgr . Xfgrgnjrg, ejahmtmja (33) rgqumrgs 3,wfmef sukkgsts tfct tfg pfcsg trcasmtmja hgpgahs ja tfg sc`g ejahmtmja cs ma tfg Grhjs-

Rgaym krcpf, ma jtfgr wjrhs, g`grkgaeg jn c kmcat ej`pjagat rgqumrgs krgctgr hmspgrsmja

tfct tfg Yjmssja `jhgd ms cssjemctgh wmtf.

5.7 Umzgs jn Matgrejaagetgh Krjups jn Ckgats

[fmdg tfg ejahmtmja njr g`grkgaeg jn tfg kmcat ej`pjagat ms matgrgstmak ma mts jwa rmkft,

`cay gejaj`me cppdmectmjas `cy bg `jtmvctgh by tfg vcdug jn hmrget cah mahmrget ejaagetmjas

wmtf jtfgr ckgats. Xfms ms tfg ecsg gxcetdy njr tfg `jhgd cacdyzgh by Bcdc cah Kjycd,jp.

emt., wfgrg tfg agt bgagffit nrj` c strctgky k wcs hgffiagh cs m(k) 0 m(k) ehm (k), wfgrg

m(k) hgajtgs tfg gxpgetgh tjtcd au`bgr jn jtfgr ckgats ca ckgat ms ejaagetgh wmtf cahh

m (k)tfg gxpgetgh au`bgr jn hmrget suef ejaagetmjas. Bjtf tfgsg `ckamtuhgs `ust bg hgffiagh cs

sfcrgs jn tfg tjtcd au`bgr jn ckgats, wfga tfg au`bgr jn ckgats ms dcrkg. Xfgsg `ckamtuhgs

`cy bg gxprgssgh ma tgr`s jn tfg `jhgra tfgjry jn rcahj` krcpfs wmtf crbmtrcry hgkrgg

hmstrmbutmjas, cdbgmt ajt cdwcys sj ejavgamgatdy. Xfg njr`gr ejmaemhgs wmtf tfg `gca smzg jn

tfg ej`pjagat c rcahj`dy efjsga ckgat bgdjaks tj, cah tfg dcttgr wmtf tfg `gca au`bgr

jn ejaagetmjas njr gcef ckgat.

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[g prjeggh nurtfgr by hgvgdjpmak tfg hmstrmbutmja jn ej`pjagat smzgs. Dgt s hgajtg tfg

rcahj` vcrmcbdg hgajtmak ej`pjagat smzg, cah f9lhgajtg mts prjbcbmdmty `css nuaetmja. Xfg

hgrmvctmja jn mts hmstrmbutmja nuaetmja by `gcas jn YKN tgefamqugs ms jbtcmagh by Agw`ca

gt cd., jp. emt.. Xfg fgcrt jn tfgmr cpprjcef rgsts ja gau`grctmak tfg au`bgr jn c rcahj`dy

sgdgetgh ckgats ffirst agmkfbjrs cah tfga sgejah agmkfbjrs, cah sj ja, tfct ms tfg tjtcd au`bgr

jn jtfgr ckgats tfg jrmkmacd ckgat ms ejaagetgh wmtf, hmrgetdy cah mahmrgetdy, wfmdg ct gcef stgp

jag ms ecrgnud tj ejasmhgr tfg chhmtmjacd ejatcets jady. Dgt tfms prjbcbmdmty `css nuaetmja bg

hgajtgh by f3l. Xfg au`bgr jn ejatcets c typmecd ckgat fcs ms hmstrmbutgh ceejrhmak tj pl,

cah gcef jn tfg ejatcets jn fgr ffirst ejatcets, jtfgr tfca fgrsgdn, dgchs tj c ej`pjagat wmtf

smzg hmstrmbutgh ceejrhmak tj f3l.Wsmak YKN tgefamqugs, wg `cy jbtcma tfg YKNs jn tfgsg

hmstrmbutmjas cs sjdutmjas tj nuaetmjacd gquctmjas mavjdvmak tfg YKN jnpl. Xfga, usmak tfg

prjpgrtmgs jn YKNs, wg `cy ej`putg tfg `j`gats jnf9l ma tgr`s jn tfg `j`gats jnpl.Ugg

tfg cppgahmx njr nurtfgr hgtcmds.Upgemffiecddy, tfg `gca smzg jn tfg au`bgr jn jtfgr ckgats c rcahj`dy sgdgetgh ckgat ms

ejaagetgh wmtf, hmrgetdy cah mahmrgetdy, ms kmvga by

G\sP 0 3 + (G\lP)2

2G\lP G\l2P 0 2z3 ^cr\lP

2G\lP (G\lP)2 ^cr\lP . (3>)

Njr tfg spgemcd ecsg jn c Yjmssja hgkrgg hmstrmbutmja, tfg jrmkmacd Grhjs Rgaym ecsg, tfg

cbjvg njr`udc ymgdhs c `gca ej`pjagat smzg gqucd tj 33G\lP

wfmef `cy bg ej`putgh hmrgetdy)wg fcvg tfct?

G\sP

G\lP0G\lP

G\lP ^cr\lP(2G\lP (G\lP)2 ^cr\lP)2 ,

G\sP

cr\lP0

(G\lP)2

(2G\lP (G\lP)2 ^cr\lP)2 (36)

Xfg `gca ej`pjagat smzg ms maergcsmak ejavgx ma tfg `gca, prjvmhgh tfct tfg `gca gxegghs

tfg vcrmcaeg. Mt ms cdsj ejavgx maergcsmak ma tfg vcrmcaeg. Xfg m`pjrtcaeg jn tfg vcrmcaeg

rgegmvgs cagehjtcd suppjrt nrj` tfg cddgkgh gxmstgaeg jn c s`cdd au`bgr jn mahmvmhucds wfj

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fcvg c vgry dcrkg au`bgr jn sjemcd ejaagetmjas ma W.U. emtmgs. Xj uahgrstcah fjw m`pjrtcat

ms tfg gxmstgaeg jn suef jutdmgrs ma tgr`s jn sjemcd ejaagetmjas njr sjemgtmgs tj nuaetmja wg

aggh tj lajw tfg bgfcvmjrcd uahgrpmaamaks jn tfms cddgkgh ncet.

Njr gxc`pdg, wjudh mt susteg njr g`grkgaeg jn tfg kmcat ej`pjagat mn c s`cdd au`bgr jn

pgjpdg wgrg gxtrcjrhmacrmdy wgdd ejaagetgh, cah tfus dgt jtfgrs bgagffit nrj` tfgmr ejaagetmjas,

jr sfjudh gvgryjag bg uamnjr dy wgdd ejaagetgh: Fjw eruemcd crg suef Djms [gmsbgrks6

jn tfg wjrdh \Kdchwgdd (3===)P njr sjemcd ejfgsmja: Xfct ms, hjgs mt tclg c s`cdd au`bgr jn

mahmvmhucds wfj lajw c djt jn jtfgr pgjpdg cah eca tfgrgnjrg pgrnjr` sjemcd crbmtrckg: Jr

hjgs tfg wmhgdy lajwa ajtmja jn smx hgkrggs jn sgpcrctmja (wfmef jrmkmactgs ma @mdkrc`s

gxpgrm`gats 9n tfg 3=>9s \@mdkrc` (3=>;)P) m`pdy tfct c uamnjr` hgkrgg jn sjemcd ejaagetmjas

sustegs njr sjemcd matgrejaagetghagss:

Cdtfjukf dmttdg ms lajwa wmtfma tfg gejaj`mes dmtgrcturg cbjut krgkcrmjus mahmvmhucds,

c ngw stuhmgs prjvmhg kdm`psgs matj tfg rjdg jn krgkcrmjus mahmvmhucds cs wjrlgrs. Lrugkgrcah Ueflchg (2997) hjeu`gat c pjsmtmvg cah stctmstmecddy smkamffiecat rgdctmjasfmp bgtwgga c

wjrlgrs tgahgaey tj matgrcet wmtf jtfgrs (pcrtmeudcrdy wmtf nrmgahs) wfmdg ajt wjrlmak cah

tfg rgdctmvg nrgqugaey jn wjrlgr-rgdctgh matgrcetmjas ja tfg wjrlgrs ojbs. Xfgy matgrprgt tfct

cs gvmhgaeg jn sjrtmak jn `jrg gxtrjvgrtgh wjrlgrs matj `jrg sjemcbdg ojbs. Xfg rgdctmjasfmp jn

rg`uagrctmja tj sjemcbmdmty, wfmdg ejatrjddmak ejkamtmvg efcrcetgrmstmes ms cdsj vgry matgrgstmak

but hjgs ajt sgg` tj fcvg bgga chhrgssgh.

[fmdg jur cacdysms sj ncr fcs g`pfcsmzgh tfg `gca ej`pjagat smzg, wg `cy gxpdjrg sgv-

grcd cppdmectmjas wfgrg ckgats bgagffits hgpgah ajt jady ja gxpgetctmjas but `jrg kgagrcddy

ja tfg cetucd hmstrmbutmja jn ej`pjagat smzgs. Xfg `j`gats jn tfg hmstrmbutmja jn ej`pj-

agat smzgs bgdjw cah cbjvg tfg pfcsg trcasmtmja `cy bg jbtcmagh. Xfms ms `chg pjssmbdg by

wjrlmak wmtf YKN jn tfg hmstrmbutmja jn ej`pjagat smzgs. Xfg prjpjrtmja jn cdd ckgats tfct

crg matgrejaagetgh cbjvg tfg pfcsg trcasmtmja `cy bg jbtcmagh, cdbgmt ajt ma edjsgh njr`. Mt

`cy cdsj bg jbtcmagh au`grmecddy.

6Jag-ja-Jag wmtf Djms [gmsbgrk. [fmdg yju `cy ajt rgejkamzg tfg nceg, yjurg prjbcbdy wgdd-cequcmatghwmtf fgr wjrl. Cs tfg Ej``mssmjagr jn Efmeckjs Hgpcrt`gat jn Eudturcd Ccmrs, Djms [gmsbgrk fcs dcuaefghtfg Efmeckj Eudturcd Egatgr, ergctgh Kcddgry 5;, cah brjukft Efmeckj tfg maerghmbdy pjpudcr Ejws ja Ycrchggxfmbmt. Uuef cttrcetmjas `cy sjuah bmzcrrg tj tfg jutsmhgr, but hjat bg surprmsgh, caytfmak ms pjssmbdg matfg wjrdh jn Djms [gmsbgrk.

Djak bgnjrg fgr ecrggr ma emty kjvgra`gat, sfg stcrtgh c hrc`c trjupg, pubdmsfgh ca uahgrkrjuah agws-pcpgr, njuahgh Nrmgahs jn tfg Ycrls, rca tfg Efmeckj Ejuaemd jn Dcwygrs, cah `cackgh `udtmpdg pjdmtmecdec`pcmkas. Fgr au`grjus ecrggrs cah `ckagtme pgrsjacdmty fcvg rgsudtgh ma ca matrmectg wgb jn nrmgahs cahcequcmatcaegs, maspmrmak c Agw ]jrlgr crtmedg by @cdejd` Kdchwgdd, Umx Hgkrggs jn Djms [gmsbgrk.

Ma tfms gpmsjhg jn Efmeckj Utjrmgs, jur ec`grcs njddjw Djms ja c hcy ct tfg jsteg. Xfga Ojfa Ecddcwcykjgs jag-ja-jag wmtf Djms, gxpdjrmak gvgrytfmak nrj` fgr efmdhfjjh ma Efmeckjs Custma agmkfbjrfjjh tjfgr eurrgat rjdg cs Efmeckjs Sugga jn Eudturg.

Nrj`? fttp?//www.wttw.ej`/efmeckjstjrmgs/djmswgmsbgrk.ft`d

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Ca matgrgstmak cppdmectmja wjudh bg tj ejasmhgr tfg njddjwmak. Ckgats `cy rcahj`dy

bgej`g uacvcmdcbdg tj matgrcet wmtf jtfgrs, gvga tfjukf tfgy `cy bg ejaagetgh wmtf jtfgr

ckgats. Cdtgractmvgdy, ejasmhgr tfct mhgcs jr manjr`ctmja crrmvg rcahj`dy ct ckgats, cah gcef

jn tfg` pcssgs mt ja tj gcef jn fgr cequcmatcaegs wmtf prjbcbmdmty q. Xfg gxpgetgh au`bgr

jn jtfgrs wfj fgcr jn tfg mhgc nrj` ca ckgat wfj fcs oust rgegmvgh mt cah fcs hgkrgg l ms

gqucd tjq(l 3).Mn ckgats ncmd tj pcss ja tfg mhgc, tfga tfgmr agmkfbjrs wfj wgrg ejaagetghprgvmjusdy tfrjukf tfg` bgej`g hmsejaagetgh nrj` jag cajtfgr. Xfg dmlgdmfjjh tfct ca

mahmvmhucd rgegmvgs tfg mhgc ms prjpjrtmjacd tj fgr hgkrgg. Xfgrgnjrg, tfg cvgrckg au`bgr jn

jtfgr pgrsjas c pgrsja pcssgs tfg mhgc ja tj ms gqucd tj? q

mlm(lm3)

mlm

0 q(G\lP)2G\lP

G\lP . @jrg

njr`cddy, wjrlmak nrj` Agw`ca (2992c) ma tfg ecsg jn agtwjrl rgsmdmgaeg \ sgg tfg Cppgahmx

njr sj`g nurtfgr hgtcmds P, wg fcvg tfct tfgrg gxmsts c tfrgsfjdh vcdug jn tfg prjbcbmdmty tfct

ca ckgat ms cvcmdcbdg, wfmef ms kmvga by G\lPcr\lP+(G\lP)2G\lP .

Njr tfg spgemcd ecsg jn tfg agkctmvg bmaj`mcd hgkrgg hmstrmbutmja gxc`magh cbjvg, tfmstfrgsfjdh prjbcbmdmty ms kmvga by?

(+3). Ajt surprmsmakdy, tfg strjakgr tfg matgasmty wmtf

wfmef tfg kmcat ej`pjagat wjudh g`grkg, tfg fmkfgr tfg tfrgsfjdh prjbcbmdmty. Ma jtfgr

wjrhs, mahmrget ejaagetmjas c`jak ckgats `clg up njr ncmdurg jn mahmvmhucd ckgats.

5.> Xrchg ma Hmgrgatmctgh Yrjhuets

Xfms gxc`pdg `jtmvctgs tfg `jhgd jn tfg prgvmjus sgetmja vmc trchg ma hmgrgatmctgh ej`-

`jhmtmgs. Cdtfjukf ajt strmetdy agegsscry, mt cdsj g`pdjys c hmrgetgh krcpf vgrsmja jn tfg

`jhgd, wfgrgby c ejuatry (smtg) `cy sgdd tj jtfgr ejuatrmgs cah usg tfg prjegghs tj pcy ma

tura njr mts jwa m`pjrts. Ydgcsg sgg tfg Cppgahmx njr ca gdg`gatcry trgct`gat jn tfg ecsg

jn hmrgetgh krcpfs.

Mahmvmhucds ma jpgamak up ejaagetmjas wmtf jtfgrs crg `jtmvctgh by tfg hgsmrcbmdmty jn

trchg. [g `jhgd tfg ejatcets c`jak ckgats by `gcas jn hmrgetgh krcpfs. Gcef ckgat `cy

bg ejaagetgh wmtf jtfgrs cah jtfgr ckgats `cy bg ejaagetgh wmtf fgr. C hmrgetgh krcpf

ceej``jhctgs ejavgamgatdy c sgttmak wfgrg ca mahmvmhucd sgdds tj tfjsg sfg ms ejaagetgh

wmtf, fgr jut-ghkgs, cah buys nrj` tfjsg wfj crg ejaagetgh wmtf fgr, fgr ma-ghkgs.

C pcrtmeudcrdy sm`pdg wcy tj `jtmvctg suef trchg ms by `gcas jn trchg ma hmgrgatmctghkjjhs. Yrjhuet vcrmgty ms hgsmrcbdg cah tfgrgnjrg `jtmvctgs trchg. Cssu`mak `jajpjdmstme

ej`pgtmtmja, gcef hmgrgatmctgh kjjhs prmeg ms c ejastcat `crl-up jn tfg wckg rctg, tfg

qucatmty jn gcef vcrmgty prjhuegh ct tfg nrgg-gatry gqumdmbrmu` ms ejastcat cah sfcrgh c`jak

cdd ckgats cah tfg `crlgt edgcrs tfrjukf tfg au`bgr jn vcrmgtmgs. Xfgrgnjrg, ca mahmvmhucds

wgdncrg hgpgahs ja tfg au`bgr jn fgr ejatcets. Xfms ms wfy tfg `jhgd jn hmgrgatmctgh kjjhs

Ma jrhgr njr ca ckgat tj cequmrg ejaagetmjas wmtf jtfgrs sfg fcs tj maeur ejsts. [g sm`pdmny

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30/66

tfg ejst smhg by cssu`mak megbgrk ejsts.

Uuppjsg tfct gcef smtg m ms fj`g tj wjrlgrs. C dcrkg au`bgr jn ffir`s `cy prjhueg

cay crbmtrcry au`bgr jn vcrmgtmgs jn c hmgrgatmctgh ejasu`ptmja kjjh uahgr ejahmtmjas jn

`jajpjdmstme ej`pgtmtmja usmak jady dcbjr, wfmef tfg smakdg uahmgrgatmctgh maput ma tfg

wjrdh gejaj`y. Xfgrg gxmsts c uamvgrscd uamt jn ceejuat, wfmef `cy bg usgh njr sgttdmak

cdd trcascetmjas cerjss smtgs. Njr sm`pdmemty, wg `clg tfg chhmtmjacd cssu`ptmja tfct tfgrg

ms nrgg gatry, sj tfct tfg au`bgr jn vcrmgtmgs prjhuegh ms hgtgr`magh cah uahgr sy``gtry

gqucd tfrjukf cdd smtgs. [g hgvgdjp tfg njr`cd hgpgahgaeg jn tfg vcdug nrj` trchg ja tfg

au`bgr jn trchmak pcrtagrs.

[g stcrt tfg cacdysms by cssu`mak tfct dmals bgtwgga smtgs (tfct ms, ejuatrmgs) crg hmrgetgh.

Njr gcef smtg, tfgrg gxmst ma-ejaagetmjas cah jut-ejaagetmjas. Xfg ckgats wfj crg djectgh

ma smtg m `cy buy hmgrgatmctgh vcrmgtmgs nrj` cdd jtfgr smtgs tfct crg ma-ejaagetgh wmtf smtg

m. Dgt aom+ hgajtg tfg au`bgr jn vcrmgtmgs m`pjrtgh matj m nrj` smtg o, cah Am+ 0

Oo03 a

om+,

hgajtg tfg tjtcd au`bgr jn hmgrgatmctgh vcrmgtmgs m`pjrtgh nrj` cddOjtfgr smtgs wfmef fcvg

ma-ejaagetmjas wmtf smtg m. Um`mdcrdy, dgt alm, hgajtgs tfg au`bgr jn hmgrgatmctgh vcrmgtmgs

gxpjrtgh tjl nrj` smtgm, cah Am0L

l03 alm,hgajtg tfg tjtcd au`bgr jn vcrmgtmgs gxpjrtgh

tj cdd Ljtfgr smtgs wmtf wfmef smtg m fcs jut-ejaagetmjas< cah Am hgajtgs tfg au`bgr jn

vcrmgtmgs prjhuegh ma smtg m. Gcef smtg nuaetmjas cs ca jpga gejaj`y by sgddmak sj`g jn tfg

jwa-prjhuegh vcrmgtmgs jn tfg hmgrgatmctgh kjjh tj tfg smtgs tfct crg mts jut-ejaagetmjas

cah usgs tfg rgvgaug tfus kgagrctgh tj purefcsg vcrmgtmgs nrj` tfg ma-ejaagetmjas.

Mt ms c wgdd lajwa rgsudt nrj` matgractmjacd trchg tfgjry wmtf hmgrgatmctgh prjhuets \

Fgdp`ca cah Lruk`ca (3=67) P tfct mn wckg rctgs crg gqucdmzgh cerjss cdd smtgs,wm0 w,tfga

tfg `jajpjdmstme ej`pgtmtmja prmemak rudg m`pdmgs tfct prmegs wjudh bg gqucd cerjss cdd smtgs.

Xfga tfg sfcrg jn c typmecd ckgats gxpgahmturg ja kjjhs nrj` smtg o ms kmvga by a

om+

am+Am+. Mt

njddjws tfct tfg tjtcd gxpgahmturg ja cdd m`pjrts by ckgats ma smtg m ms kmvga by w Am+am+Am+

.

Cppdymak tfg sc`g djkme tj smtg ms gxpjrts, wg fcvg tfct tfg sfcrg jn gxpgahmturg ja

vcrmgtmgs prjhuegh ma ajhg m by ckgats ma ajhg l tfct fcs ca ma-ejaagetmja nrj` ajhg m

ms kmvga by AmAl+Al+

. Xfms ma tura m`pdmgs tfct tfg tjtcd gxpgahmturg by cdd smtgs tfct fcvg

ma-ejaagetmjas nrj` smtg m ja vcrmgtmgs prjhuegh ma ajhg m,tfct ms, tfg tjtcd gxpjrts jn ajhg

m, crg kmvga by w Am

Ll03

3Al+Al+

. Xfus, tfg ejahmtmja njr trchg bcdcaeg ms tfct njr c ajhg

m tjtcd m`pjrts `ust gqucd tjtcd gxpjrts?

Am+am+Am+

0amLl03

3

al+Al+. (3=)

Nrgg gatry ma gcef smtg gasurgs tfct gcef vcrmgty ms prjhuegh ct qucatmty (3

), wfmef

m`pdmgs ma tura tfct tfg au`bgr jn vcrmgtmgs prjhuegh ma gcef smtg crg gqucd tjam 0 3

a.

2=


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Xfus, cdd smtgs prjhueg tfg sc`g au`bgr jn vcrmgtmgs. Xfg trchg bcdcaeg ejahmtmja tfga

bgej`gs? O3+O

0 L3+L

,wfmef m`pdmgs?

O0L, (29)

tfct ms, tfg au`bgr jn ma-ghkgs cah jut-ghkgs `ust bg gqucd. Jn ejursg, tfms ejahmtmja

ecaajt fjdh wmtf prjbcbmdmty 3, wfga ejaagetmjas crg stjefcstme. Utmdd, wmtf c dcrkg au`bgr

jn ckgats, mt ejudh bg prjvga tfct ejahmtmja (78) ms sustemgat tj gasurg tfct trchg bcdcaeg ms

sctmsffigh cd`jst surgdy? z0G{O} 0G{L}.Uj, wg ejaeduhg tfct ct dgcst njr tfg purpjsg jnc sy``gtrme `jhgd mt sustegs tj wjrl wmtf ca uahmrgetgh krcpf.

Xfgrg ms pjtgatmcddy c dcrkg au`bgr jn kjjhs, wfmef gatgr sy``gtrmecddy matj prgngrgaegs.

Dgt cdd mahmvmhucds fcvg mhgatmecd prgngrgaegs, kmvga by W0

` e`

3 , 94 43,wfgrg `

rcakgs jvgr vcrmgtmgs cvcmdcbdg njr purefcsg ma tfg smtg wfgrg ca mahmvmhucd ms djectgh. Xfg

rcakg ms cgetgh by trchg cah wmdd bg hmseussgh nurtfgr bgdjw. Cdd kjjhs crg prjhuegh wmtf

tfg sc`g ejst (dcbjr rgqumrg`gats) nuaetmja, wfmef maeduhgs c ffixgh ejst cah c `crkmacd

ejst , bjtf hgffiagh ma tgr`s jn uamts jn dcbjr. Mt ms ejavgamgat tj hgffiag 33

, tfg

gdcstmemty jn substmtutmja c`jak tfg hmgrgatmctgh maputs.

Cppdymak stcahcrh rgsudts nrj` trchg tfgjry uahgr `jajpjdmstme ej`pgtmtmja \ Fgdp`ca

cah Lruk`ca (3=67) P, wg fcvg tfct mnwmhgajtgs tfg wckg rctg ma smtgm,tfga prjffit `cxm`mz-

mak by ffir`s uahgr ejahmtmjas jn nrgg gatry m`pdmgs c prmeg wfmef ms ejastcat njr cdd vcrmgtmgs

prjhuegh ma smtg m cah gqucd tj pm0

wm. Ca mahmvmhucd wfj ms djectgh ma smtg m fcs hg`cah

njr c pcrtmeudcr hmgrgatmctgh vcrmgty` kmvga by?

eh`0wmp`

Y(3)m

,

wfgrg Ym 0

`(p`)3

33 hgajtgs tfg prmeg mahgx njr cdd kjjhs cvcmdcbdg ma smtg m, gcef

jn wfmef crg mahgxgh by`. Ajtg tfct tfg prmeg gdcstmemty jn hg`cah njr gcef jn tfg hmgrga-

tmctgh kjjhs ms gqucd tj `maus tfg gdcstmemty jn substmtutmja c`jak hmgrgatmctgh prjhuets,

.Nurtfgr`jrg, mt ms c stcahcrh ngcturg jn tfms tfgjry tfct tfg s`cddgr ms , tfg fmkfgr tfg

`crlgt pjwgr jn gcef ffir`. Ma tfg dm`mt wfga tgahs tj maffiamty, tfg `jhgd jn `jajpjdmstme

ej`pgtmtmja m`pdmgs pgrnget ej`pgtmtmja.Xj sgg tfg bcsme wjrlmaks jn tfms `jhgd, ejasmhgr fjw gxpgahmturg ms cddjectgh tj hmgrgat

vcrmgtmgs jn kjjhs. Nmrst, wg ajtg tfct cdd kjjhs prjhuegh ma c smtg wjudh bg prmegh ct tfg

sc`g prmeg. Xfgrg crg am kjjhs prjhuegh ma smtg m cah aom+ crg m`pjrtgh nrj` smtg o tj smtg

m. Xfgrgnjrg, tfg prmeg mahgx ma smtg m `cy bg wrmttga cs?

Ym0

Amp3m +

Oo03

aom+p3o

33

. (23)

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32/66

Mn wckg rctgs wgrg gqucdmzgh cerjss cdd smtgs,wm 0 w, tfga tfg prmemak rudg m`pdmgs tfct prmegs

wjudh cdsj bg gqucd cerjss cdd smtgs.

Mn ergctmja jn ejaagetmjas bgtwgga smtgs ms up tj mahmvmhucds mamtmctmvg, tfga mahmvmhucds

fcvg ca maegatmvg tj spgah rgsjuregs ma tfg ergctmja jn chhmtmjacd ejaagetmjas bgecusg hjmak

sj maergcsgs tfg au`bgr jn vcrmgtmgs tfgy `cy purefcsg nrj` a, uahgr cutcrly, tj a+ aO,

mn tfgy `cy purefcsg nrj` O jtfgr smtgs cs wgdd, tfct ms wfga smtg m ms ejaagetgh wmtf O

jtfgr smtgs. Wsmak tfg bgfcvmjrcd `jhgd njr mahmvmhucds cbjvg, wg fcvg tfct tfg mahmrget

utmdmty nuaetmja ms Wcut 0

a

33. MnO smtgs fcvg ma-ejaagetmjas wmtf smtg m, tfga mahmrget

utmdmty ejahmtmjacd ja tfg au`bgr jn ma-ejaagetmjas bgmak gqucd tj Obgej`gs W(O) 0 (3 +

O)33Wcut. Xfms qucatmty ms rcahj` but cdwcys dcrkgr tfca uahgr cutcrly, wfga O 1 9,

cah tfgrgnjrg sj ms mts gxpgetgh vcdug. Xfct ms, mn tfg cetucd au`bgr jn ma-ejaagetmjas ms ajt

lajwa wfga hgemsmjas cbjut ejatcets crg `chg, tfga ca ckgats gxpgetgh wgdncrg ms `gcsurgh

cs fgr gxpgetgh utmdmty, tfct ms, tfg gxpgetctmja jnW(O) wmtf rgspget tj O. Xj `jtmvctg ckgaumag trchgj, mt `ust bg tfg ecsg tfct tfgrg ms c ejst cssjemctgh wmtf tfg gxpcasmja jn

tfg vcrmgty jn kjjhs cssjemctgh wmtf ergctmak ejaagetmjas, suef cs trcaspjrtctmja ejsts njr

tfg m`pjrtgh kjjhs. Ma tfct ecsg, ca maergcsg ma tfg vcrmgty jn kjjhs m`prjvgs wgdncrg but

ej`gs ct tfg ejst jn maergcsgh m`pjrt prmegs.

[g sgg tfms trchgj edgcrdy by dgttmak trcaspjrtctmja ejsts bg ma tfg njr` jn megbgrk ejsts?

tj fcvg c uamt jn c kjjh nrj` smtg o cvcmdcbdg ma smtg m, g uamts jn tfg kjjh `ust bg purefcsg

ma smtg o, wfgrg , 19,ms c pcrc`gtgr. Uj, tfg e.m.n. prmeg ma smtg m jn c hmgrgatmctgh kjjh

prjhuegh ma smtg o mspo

g,cah tfgrgnjrg tfg

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