Знакомство с эволюционной генетикой (Гаевский)

3

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H^gZ ba jZkijhkljZg_gguo \ kh\j_f_gghc [bheh]bb w\hexpbhggZy ]bihl_aZ ba-

\_klgZy dZd ³Kbgl_lbq_kdZy l_hjby w\hexpbb´ KLW jZa\beZ hkgh\gh_ iheh`_gb_ ^Zj\bgbafZ h l\hjq_kdhc jheb hl[hjZ b jZkijhkljZgbeZ ^_ckl\b_ w\hexpbhgguo nZdlhjh\ gZ ihimeypbhggu_ kbkl_fu W\hexpbhgbklu b ]_g_lbdb klZeb ]h\hjblv gZ h^ghf yaud_ yaud_ ihimeypbhgghc beb w\hexpbhgghc ]_g_lbdb Bf_ggh KLW bgb- pbbjh\ZeZ [mjgh_ jZa\blb_ klZlbklbq_kdbo f_lh^h\ k\yaZgguo k bf_gZfb N =ZevlhgZ D IbjkhgZ < Bh]Zgk_gZ J Nbr_jZ K K Q_j\_jbdh\Z b fgh]bo ^jm]bo

KlZlbklbq_kdbc ih^oh^ iha\hebe kgylv ijhlb\hj_qb_ f_`^m aZdhgZfb deZkkbq_- kdhc f_g^_e_\kdhc ]_g_lbdb b gZke_^h\Zgb_f dhebq_kl\_gguo ijbagZdh\ Mki_ob ih- imeypbhgghc ]_g_lbdb hkh[_ggh aZf_lgu \ k_evkdhf ohayckl\_ b f_^bpbg_ FZl_fZlb- q_kdbc ZiiZjZl ihimeypbhgghc ]_g_lbdb hdZau\Z_l g_hp_gbfmx ihfhsv \ ihgbfZgbb f_oZgbafh\ ^_ckl\by ]eZ\guo w\hexpbhgguo nZdlhjh\ fmlZpbc ^j_cnZ ]_gh\ fb-

]jZpbb ihlhdZ ]_gh\ b _kl_kl\_ggh]h hl[hjZ

< aZ^Zqm ^Zggh]h ihkh[by \oh^bl agZdhfkl\h k hkgh\gufb ihgylbyfb ihimeypb- hgghc ]_g_lbdb b ijbgpbiZfb ihkljh_gby fh^_e_c ^Zxsbo ij_^klZ\e_gb_ h ^_ckl\bb hkgh\guo w\hexpbhgguo nZdlhjh\ < ihkh[bb gZjy^m k deZkkbq_kdbfb iheh`_gbyfb ihimeypbhgghc ]_g_lbdb jZkkfhlj_gu hjb]bgZevgu_ imlb hp_gdb j_Zevghc b fZdkb- fZevghc ]_l_jhab]hlghklb ihimeypbb qZklhlgh aZ\bkbfh]h hl[hjZ b hl[hjZ ih ijbagZ- dm k\yaZgghfm k ihehf

FZl_jbZe baeh`_gguc \ ihkh[bb ij_^gZagZq_g klm^_glZf bamqZxsbf l_hjbx w\hexpbb ]_g_lbdm ihimeypbhggmx ]_g_lbdm k_e_dpbx Hk\h_gb_ fZl_jbZeZ ^Zggh]h ihkh[by ihfh`_l ijb jZ[hl_ k gZmqghc b mq_[ghc ebl_jZlmjhc ihk\ys_gghc ijh[e_- fZf w\hexpbb ]_g_lbdb b kf_`guo ^bkpbiebg

IHIMEYPBHGGH =?G?LBQ?KDB? I:J:F?LJU

QZklhlu ]_ghlbih\ ]_gh\ b bo Zee_e_c

H[uqgh qZklhlZ p dZdh]h eb[h kh[ulby hij_^_ey_lky dZd hlghr_gb_ qbkeZ m ih- y\e_gbc wlh]h kh[ulby \ ^Zgghc ihke_^h\Zl_evghklb bkiulZgbc d h[s_fm qbkem bkiu- lZgbc n – (p=m/n) < gZr_f kemqZ_ bkiulZgb_f y\ey_lky bamq_gb_ n_ghlbih\ ]_ghlb- ih\ hkh[_c beb bo ]Zf_l ^ey \uy\e_gby m gbo hij_^_e_ggh]h ijbagZdZ BkiulZgb_ kqb- lZ_lky mki_rguf dh]^Z bkdhfuc ijbagZd h[gZjm`_g IjbagZdZfb ]Zf_lu \uklmiZxl

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hij_^_e_ggu_ Zee_eb beb khq_lZgb_ Zee_e_c ijbagZdZfb ^biehb^ghc de_ldb iZjZ Zee_e_c beb dhf[bgZpby iZj ijbagZdZfb hj]ZgbafZ djhf_ \k_]h i_j_qbke_ggh]h ± n_ghlbibq_kdb_ oZjZdl_jbklbdb

=_ghlbibq_kdh_ jZaghh[jZab_ hkh[_c h^gh]h \b^Z ijhy\ey_lky \ bo n_ghlbibq_- kdhc baf_gqb\hklb AZf_lbf h^gZdh qlh bamq_gb_ n_ghlbih\ ^Ze_dh g_ \k_]^Z iha\h- ey_l j_rblv h[jZlgmx aZ^Zqm aZdexqZxsmxky \ hp_gd_ jZaghh[jZaby ]_ghlbih\ gZ hkgh\_ bamq_gby n_ghlbih\ IjbqbgZ wlh]h djh_lky \ keh`guo \aZbfhhlghr_gbyo f_-

`^m ]_ghlbihf b n_ghlbihf >ey hp_gdb ]_ghlbibq_kdh]h khklZ\Z ihimeypbb©ih^oh-

^ylª ^bkdj_lgu_ beb f_g^_e_\kdb_ ijbagZdb k q_ldh \ujZ`_ggufb dZq_kl\_ggufb jZaebqbyfb < lh `_ \j_fy \uy\e_ggu_ gZ hkgh\_ bo bamq_gby aZdhghf_jghklb ^_ckl-

\by w\hexpbhgguo nZdlhjh\ ih \b^bfhfm kijZ\_^eb\u b ^ey gZke_^m_fuo dhebq_kl-

\_gguo beb ]Zevlhgh\kdbo ijbagZdh\ .

Kj_^b f_g^_e_\kdbo ijbagZdh\ gZb[he__ m^h[gufb k lhqdb aj_gby ]_g_lbq_kdh]h ZgZebaZ hdZau\Zxlky l_ \ ihy\e_gbb dhlhjuo mqZkl\mxl Zee_evgu_ ]_gu g_ \klmiZx- sb_ \ hlghr_gby ihegh]h ^hfbgbjh\Zgby < wlhf kemqZ_ dhebq_kl\h gZ[ex^Z_fuo n_ghlbih\ khhl\_lkl\m_l dhebq_kl\m ]_ghlbih\

IjbagZdh\ k lZdbfb k\hckl\Zfb ba\_klgh ^hklZlhqgh fgh]h Ijb ZgZeba_ q_eh\_- q_kdhc ihimeypbb qZklh ijb[_]Zxl d hp_gd_ baf_gqb\hklb ih ]jmiiZf djh\b \ kbkl_- fZo $%2beb01JZkiheZ]Zy g_h[oh^bfuf gZ[hjhf ^Zgguo ihemq_gguo gZ hkgh-

\_ fZkkh\h]h ZgZebaZ ]jmii djh\b fh`gh mklZgh\blv qZklhlu beb \_jhylghklb \klj_- qZ_fhklb \ ihimeypbb khhl\_lkl\mxsbo ]_ghlbih\

JZkkfhljbf wlh gZ ijhklhf ijbf_j_ k ]jmiiZfb djh\b ijb h[ke_^h\Zgbb q_- eh\_d m [ueZ \uy\e_gZ ]jmiiZ djh\b F m01b m1Kh]eZkgh ^Zgghfm \ur_ hij_^_e_gbx qZklhlu n_ghlbih\ jZ\gu ^ey 0 01

1 Ihkdhevdm ^ey dZ`^h]h n_ghlbiZ ba\_klgu khq_lZ- gby Zee_e_c0LmLm; MN - LmLn; N - LnLn ihemq_ggu_ qZklhlu kijZ\_^eb\u b ^ey khhl\_lkl\mxsbo ]_ghlbih\

>jm]hc oZjZdl_jbklbdhc ^Zgghc ]jmiiu ex^_c fh]ml \uklmiZlv qZklhlu Zee_e_c mqZkl\mxsbo \ nhjfbjh\Zgbb jZaebqguo \ZjbZglh\ ]_ghlbih\ < gZr_f ijbf_j_ h[- s__ qbkeh Zee_e_c Lm , Ln jZ\gh m^\h_gghfm dhebq_kl\m h[ke_^h\Zgguo q_eh\_d •2

qbkeh Zee_e_c Lm jZ\gh m^\h_gghfm dhebq_kl\m q_eh\_d k ]jmiihc djh\b F iexk dhebq_kl\h q_eh\_d k ]jmiihc djh\b01• qbkeh Zee_e_c Ln,

5

khhl\_lkl\_ggh jZ\gh • QZklhlZ \klj_qZ_fhklb Zee_ey Lm - p(Lm) jZ\gZ p(Ln) - 1200/1460 =0.822.

H[uqgh jZkq_l qZklhl Zee_e_c ijhba\h^yl gZ hkgh\_ ba\_klguo qZklhl ]hfhab]hl- guo b ]_l_jhab]hlguo ]_ghlbih\ LZd qZklhlZ Zee_ey Lm hij_^_ey_lky dZd kmffZ qZklh- lu ]hfhab]hlgh]h ]_ghlbiZ LmLm b iheh\bgZ qZklhlu ]_l_jhab]hlgh]h ]_ghlbiZ LmLn:

Z qZklhlZ Zee_ey Ln dZd p(Ln) = p(LnLn) + 0.5 p(LmLn beb p(Ln) = 1-p(Lm).

Ijh^he`Zy jZ[hlm k ij_^eh`_gguf \ur_ ijbf_jhf ihemqZ_f p(Lm) = 0.030+0.296/2 = 0.178; p(Ln qlh kh\iZ^Z_l k ihemq_ggufb \ur_ agZq_gbyfb

JZkkfhlj_gguc kemqZc ^hklZlhqgh ijhkl lZd dZd ehdmk \dexqZ_l lhevdh ^\Z jZa- ebqguo Zee_ey H^gZdh bkihevah\Zggh_ \ g_f ijZ\beh jZkijhkljZgy_lky gZ \k_ iheb- fhjngu_ ehdmku BkdhfZy qZklhlZ Zee_ey jZ\gZ kmff_ qZklhl ]hfhab]hlgh]h ]_gh- lbiZ b ihem kmff_ qZklhl ]_l_jhab]hlguo ]_ghlbih\ \ khklZ\ dhlhjuo \oh^bl ^Zg- guc Zee_ev

Jbk Qbkeh \hafh`guo \ZjbZglh\ ]_ghlbih\ \ ihimeypbb \ aZ\bkbfhklb hl qbk-

eZ Zee_e_c \ h^ghf ehdmk_

Dhebq_kl\h ]_ghlbih\ Ng \hagbdZxsbo gZ hkgh\_ kemqZcgh]h khq_lZgby n Zee_- e_c \ h^ghf Zmlhkhfghf ehdmk_ hij_^_eyxl ih nhjfme_

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]^_ i_j\h_ keZ]Z_fh_ wlh qbkeh ]hfhab]hlguo Z \lhjh_ ]_l_jhab]hlguo \ZjbZglh\ G_ ljm^gh m[_^blvky qlh k jhklhf qbkeZ Zee_e_c ijh]j_kkb\gh gZjZklZ_l dZd qbkeh \hafh`guo \ZjbZglh\ ]_ghlbih\ lZd b ^hey ]_l_jhab]hl jbk

FZeh\_jhylgh qlh ijb gZebqbb fgh`_kl\_gguo Zee_e_c \k_ \ZjbZglu ]_ghlbih\ [m^ml magZ\Z_fu ih n_ghlbiZf GZijbf_j \ kemqZ_ ]jmii djh\b ih kbkl_f_ :<H dheb- q_kl\h \ZjbZglh\ ]_ghlbih\ jZ\gh lh]^Z dZd ]jmii djh\b lhevdh Fh`gh kqblZlv ijZ\behf qlh ]_g_lbq_kdZy baf_gqb\hklv \ ijbjh^guo ihimeypbyo gZfgh]h \ur_ q_f fh`gh aZdexqblv ba ijhkluo gZ[ex^_gbc gZ^ bo fhjnheh]bq_kdhc baf_gqb\hklvx >ey \uy\e_gby ]_g_lbq_kdhc baf_gqb\hklb jZajZ[hlZgu b ijbf_gyxlky jZaghh[jZagu_ f_lh^u hkgh\Zggu_ gZ ZgZeba_ j_amevlZlh\ bg[jb^bg]Z [ebadhjh^kl\_ggh_ kdj_sb-

\Zgb_ hj]Zgbafh\ we_dljhnhj_aZ bffmgheh]bb b ^j

Ihebfhjnghklv

=_g_lbq_kdh_ jZaghh[jZab_ hkh[_c \ lhc beb bghc ihimeypbb aZ\bkbl dZd hl qbk- eZ Zee_e_c \ hl^_evghf ehdmk_ lZd b hl ^heb qZklhlu ihebfhjnguo ehdmkh\ Ehdmk kqblZxl ihebfhjnguf _keb qbkeh \ZjbZglh\ \oh^ysbo \ g_]h Zee_e_c n ≥ >hey ih- ebfhjnguo ehdmkh\ y\ey_lky h^ghc ba oZjZdl_jbklbd ihimeypbb gZau\Z_fhc iheb- fhjnghklvx h[hagZqZ_lky eZlbgkdhc [md\hc J hl gZqZevghc [md\u \ keh\_

polimorphyzm).

Hij_^_e_gb_ khklZ\Z Zee_e_c ^ey dZ`^h]h ehdmkZ g_\hafh`gh ihwlhfm h iheb- fhjnghklb km^yl gZ hkgh\_ bamq_gby \u[hjhqgh]h g_ \k_]^Z q_ldh hij_^_e_ggh]h gZ-

[hjZ ehdmkh\ IhemqZ_fu_ ijb wlhf hp_ghqgu_ \_ebqbgu iZjZf_ljZ J ^hklZlhqgh \Z- jbZ[_evgu >Z`_ ijb kljh]hf kh\iZ^_gbb ZgZebabjm_fh]h gZ[hjZ ehdmkh\ \_ebqbgZ J g_ klZ[bebabjm_lky IjbqbgZ jZaebqbc \ hp_gd_ J fh`_l kdju\Zlvky \ gbadhc qZklhl_ h^gh]h beb g_kdhevdbo Zee_e_c \ ehdmk_ Ijb wlhf iZ^Z_l \_jhylghklv h[gZjm`_gby hkh[_c ghkbl_e_c j_^dh]h Zee_ey b \_ebqbgZ J hdZau\Z_lky aZgb`_gghc

>ey lh]h qlh[u mf_gvrblv \ebygb_ ©j_^dh]h Zee_eyª gZ j_amevlZl hp_gdb J \\h^yl lZd gZau\Z_fuc³djbl_jbc ihebfhjnghklb”: ehdmk fh`gh kqblZlv ihebfhjn- guf lhevdh lh]^Z dh]^Z qZklhlZ gZb[he__ jZkijhkljZg_ggh]h Zee_ey wlh]h ehdmkZ g_ ij_\urZ_l aZ^Zggh]h ij_^_eZ h[uqgh

Ke_^m_l h[jZlblv \gbfZgb_ gZ lh qlh ihebfhjnghklv ebrv hlqZklb hljZ`Z_l ]_g_lbq_kdh_ jZaghh[jZab_ hkh[_c \ ihimeypbb < jZ\ghc kl_i_gb ihebfhjnguf hdZ- au\Z_lky ehdmk k ^\mfy b iylvx Zee_eyfb B wlh \ kblmZpbb ijb dhlhjhc kh]eZkgh k^_-

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eZgghfm \ur_ \u\h^m qbkeh \ZjbZglh\ ]_ghlbih\ ijh]j_kkb\gh \hajZklZ_l k m\_ebq_-

gb_f qbkeZ Zee_e_c

=_l_jhab]hlghklv

>jm]hc f_jhc ]_g_lbq_kdhc baf_gqb\hklb kem`bl qZklhlZ \klj_qZ_fhklb beb

^hey ]_l_jhab]hl \ ihimeypbb JZkq_l \_ebqbgu ]_l_jhab]hlghklb G ihimeypbb ijh\h-

^yl \ ^\Z ijb_fZ KgZqZeZ ih dZ`^hfm ba \u[jZgguo ^ey ZgZebaZ ehdmkh\ hij_^_eyxl

qZklhlu ]_l_jhab]hlguo hkh[_c Hi Z aZl_f ihemq_ggu_ agZq_gby mkj_^gyxl ih dhebq_- kl\m1bamq_gguo ehdmkh\ \dexqZy dZd ihebfhjngu_ lZd b fhghfhjngu_

N

>hihegbl_evguc bgl_j_k fh`_l ij_^klZ\eylv kj_^gyy ]_l_jhab]hlghklv kj_^b

ihebfhjnguo ehdmkh\ Np:

H[gZjm`_gbx ]_l_jhab]hl gZ hkgh\_ ZgZebaZ n_ghlbih\ f_rZ_l kihkh[ghklv h^gh]h Zee_ey ^hfbgbjh\Zlv gZ^ ^jm]bf Kms_kl\m_l ijb_f hp_gdb h`b^Z_fhc ]_l_jh- ab]hlghklb Gh`b^ gZ hkgh\_ l_hj_lbq_kdbo ij_^ihkuehd ihimeypbhgghc ]_g_lbdb <_- ebqbgm Gh`b^ dhgdj_lgh]h ehdmkZ hij_^_eyxl ih qZklhlZf _]h Zee_e_c ?keb \ ehdmk_ fh]ml ijbkmlkl\h\Zlv iZju khklZ\e_ggu_ ba n Zee_e_c k qZklhlZfb I1, f2, f3, .. fn lh ijb \uiheg_gbb mkeh\by kemqZcgh]h kdj_sb\Zgby h`b^Z_fu_ gZ hkgh\Zgbb khhlghr_- gby OZj^b <Zcg[_j]Z qZklhlu ]hfhab]hl \ ihimeypbb jZ\gu d\Z^jZlZf qZklhl Zee_e_c (f1)2, (f2)2, (f3)2,.. (fn)2 LZd dZd kmffZ qZklhl ]hfh b ]_l_jhab]hlguo ]_ghlbih\ jZ\gZ _^bgbp_ h`b^Z_fZy ]_l_jhab]hlghklv ih wlhfm ehdmkm [m^_l jZ\gZ

Gh`b^ = 1 - [(f1)2 + (f2)2 + (f3)2 +..+ (fn)2] . (6)

Kj_^gxx h`b^Z_fmx ]_l_jhab]hlghklv ihimeypbb ih \k_f bamq_gguf ehdmkZf eb[h ih ]jmii_ ihebfhjnguo ehdmkh\ gZoh^yl ZgZeh]bqgh kj_^g_c gZ[ex^Z_fhc ]_l_jhab]hl- ghklb mjZ\g_gby b

Hij_^_ebf dZd aZ\bkbl Gh`b^ h^gh]h ehdmkZ hl qbkeZ Zee_e_c b hl khhlghr_gby bo qZklhl G_ljm^gh ihdZaZlv qlh fZdkbfZevgZy \_ebqbgZ H h`b^max [m^_l ^hklb]gmlZ ijb

jZ\_gkl\_ qZklhl Zee_e_c ?keb qbkeh Zee_e_c jZ\ghQlh]^Z qZklhlZ h^gh]h Zee_ey kh- klZ\bl Q<ujZ`_gb_ ^ey fZdkbfZevghc h`b^Z_fhc ]_l_jhab]hlghklb h^gh]h ehdmkZ kh]eZkgh \ujZ`_gbx bf__l \b^

ij_^klZ\e_gZ ^bkdj_lgufb \_ebqbgZfb \ bgl_j\Ze_ hl ^h

>ey ihebfhjnguo ehdmkh\ dh]^Z qZklhlu [hevrbgkl\Z Zee_e_c g_ ba\_klgu b lj_[m_lky hij_^_eblv \_ebqbgm Gh`b^ fh`gh h^gm ba ba\_klguo qZklhl k^_eZlv g_aZ- \bkbfhc beb³\_^ms_c´b aZ^Z\Zlv agZq_gby __ qZklhluS\ bgl_j\Ze_ hl ^h Imklv

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qZklhlu hklZevguo ©g_\_^msboª Zee_e_c [m^ml h^bgZdh\ufb Lh]^Z dZ`^uc ba gbo [m^_l bf_lv qZklhlm jZ\gmx²p)/(n—1).

< khhl\_lkl\bb k \ujZ`_gb_f \_ebqbgZ Gh`b^ fh`_l [ulv aZibkZgZ ke_^mx-

< mjZ\g_gbb \_ebqbgZ Gh`b^ baf_gy_lky hl gmey j→ ^h agZq_gbc H h`b^max ijb

\u[jZgghf agZq_gbb n.

Dh]^Z agZq_gby n b G ba\_klgu ba mjZ\g_gbc b fh`gh hij_^_eblv qZklhlm

Ihebfhjnghklv b ]_l_jhab]hlghklv m Phoronopsis viridis

Ijhbeexkljbjm_f \ur_kdZaZggh_ gZ dhgdj_lghf ijbf_j_ < dZq_kl\_ bkoh^guo

\havf_f ^Zggu_ ih hp_gd_ ihebfhjnghklb b ]_l_jhab]hlghklb ihimeypbb fhjkdh]h

q_j\y Phoronopsis viridis > @ k^_eZgghc gZ hkgh\_ we_dljhnhj_lbq_kdh]h ZgZebaZ

[_edh\ ehdmkh\ <u[hjdZ khklZ\beZ hkh[_c < lZ[e \g_k_gu lhevdh qZklhlu

Zee_e_c ihebfhjnguo ehdmkh\

:gZeba lZ[e ihdZau\Z_l qlh ihebfhjnghklv ihimeypbb [_a mq_lZ ^hihegb-

l_evgh]h djbl_jby khklZ\ey_l J Ihke_ \\_^_gby djbl_jby ihebfhjngh-

klb jZ\gh]h qbkeh ihebfhjnguo ehdmkh\ khdjZsZ_lky ^h Z ihebfhjnghklv

klZgh\blky jZ\ghc J0.95 = 11/39 = 0.282.

Kj_^gb_ \_ebqbgu gZ[ex^Z_fhc b h`b^Z_fhc ]_l_jhab]hlghklb \dexqZy fh-

ghfhjnguo ehdmkh\ jZ\gu khhl\_lkl\_ggh

GgZ[e = 2.794/39 = 0.072 ; Gh`b^ >ey qZklb ihebfhjnguo ehdmkh\

(Np=27) GgZ[e b Gh`b^ = 3.782/27 = 0.136.

9

; _ k i h a \ h g h q g u _

GZ bkihevah\Zgb_ djbl_jby ihebfhjnghklb kkuehd g_l

IZjZf_lj hij_^_e_g \ ^hiheg_gb_ d [1].

KijZ\_^eb\ \hijhk dZd bgl_jij_lbjh\Zlv ^hklb]gmluc mjh\_gv jZaghh[jZaby ]_ghlbih\ \ ihimeypbyo jZaebqguo hj]Zgbafh\ ?

Ij_`^_ \k_]h h[jZlbf \gbfZgb_ gZ \_ebqbgm iZjZf_ljZ ihebfhjnghklb WlZ \_ebqbgZ fh`_l hdZaZlvky aZgb`_gghc _keb [ue \\_^_g djbl_jbc ihebfhjnghklb gh ^Z`_ k mq_lhf wlh]h ih \b^bfhfm iheh\bgZ \k_o ehdmkh\ fhghfhjngu

<_ebqbgm ]_l_jhab]hlghklb gZihfgbf gZoh^yl dZd kj_^gxx ih \k_f bkke_^h-

\Zgguf ehdmkZf \dexqZy b l_ dhlhju_ fu kqblZ_f fhghfhjngufb Ihwlhfm kj_^gyy ]_l_jhab]hlghklv ih \k_f ehdmkZf y\gh gb`_ ]_l_jhab]hlghklb dhlhjmx g_kml iheb- fhjngu_ ehdmku

LZd dZd ]_l_jhab]hlghklv G _klv hlghr_gb_ kmffu qZklhl ]_l_jhab]hlguo ]_gh- lbih\6Jd h[s_fm qbkem bamq_gguo ehdmkh\1\dexqZy b fhghfhjngu_+ 6J 1Z ihebfhjnghklv J wlh ^hey ^b b ihebZee_evguo ehdmkh\ Ng) P = 1J 1hlghr_gb_+ 3

= Sg/1Jkhhl\_lkl\m_l kj_^g_c ]_l_jhab]hlghklb m ihebfhjnguo ehdmkh\ G_h[oh^bfh h[jZlblv \gbfZgb_ qlh \\_^_gb_ djbl_jby ihebfhjnghklb mf_gvrZ_l iZjZf_lj J b ih\urZ_l hlghr_gb_ G J < khhl\_lkl\bb k wlbf ihemq_ggu_ agZq_gby kj_^g_c ]_l_jh- ab]hlghklb ihebfhjnguo ehdmkh\ fh]ml hdZaZlvky aZ\ur_ggufb

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Bkihevamy mjZ\g_gb_ ^ey n = (2 ÷ b \_ebqbgu ]_l_jhab]hlghklb m iheb- fhjnguo ehdmkh\ G J ba lZ[e ihemqZ_f qZklhlu \_^ms_]h Zee_ey lZ[e

LZ[ebpZ

n=2 n=3 n=4 n=5 n=6

;_kiha\hghqgu_ 0.286 0.827 0.837 0.840 0.841 0.842

Iha\hghqgu_ 0.243 0.858 0.865 0.867 0.868 0.868

JZkl_gby 0.174 0.904 0.906 0.907 0.908 0.908

Q_eh\_d 0.238 0.862 0.868 0.870 0.871 0.871

K mq_lhf \hafh`gh]h aZ\ur_gby mjh\gy ]_l_jhab]hlghklb J agZq_gby j fh]ml lhevdh \hajZklb LZdbf h[jZahf ijb \uy\e_gghf mjh\g_ ihebfhjnbafZ b ]_l_jhab]hl- ghklb \ ij_^_eZo ihebfhjnguo ehdmkh\ ih qZklhl_ ^hfbgbjm_l dZdhc lh h^bg Zee_ev KemqZb dh]^Z Zee_eb ij_^klZ\e_gu \ jZ\guo ijhihjpbyo \klj_qZxlky ^h\hevgh j_^dh Wlh aZdexq_gb_ kh]eZkm_lky k ^Zggufb lZ[e

FZdkbfZevgh \hafh`gu_ mjh\gb ]_l_jhab]hlghklb ihimeypbb

Ijh^he`Zy bkke_^h\Zlv ]_g_lbq_kdh_ jZaghh[jZab_ ihimeypbb fh`gh kjZ\gblv

gZ[ex^Z_fu_ \ _kl_kl\_gguo mkeh\byo b l_hj_lbq_kdb h`b^Z_fu_ ij_^_evgu_ fZdkb- fZevgu_ agZq_gby ]_l_jhab]hlghklb ihimeypbb <ur_ [ueh ihdZaZgh qlh fZdkbfZev- gZy ]_l_jhab]hlghklv hl^_evgh]h ehdmkZ ^hklb]Z_lky ijb jZ\_gkl\_ qZklhl _]h Zee_e_c b aZ\bkbl hl qbkeZ Zee_e_c Ihkdhevdm \ ]_ghlbi_ ihimeypbb _klv fhgh ^b ljb b l ^ Zee_evgu_ ehdmku g_h[oh^bfh agZlv beb hij_^_eblv \ dZdbo khhlghr_gbyo hgb ij_^klZ\e_gu G_kfhljy gZ lh qlh \ ijb\_^_gguo \ur_ ijbf_jZo kj_^gyy ihebfhjn- ghklv qZklh f_gvr_ wlh Z\lhfZlbq_kdb ij_^iheZ]Z_l ^hfbgbjh\Zgb_ fhghZee_ev- guo ehdmkh\ p_e_khh[jZagh jZkkfhljbf b ^jm]b_ \ZjbZglu dh]^Z qbke_ggh ^hfbgb- jmxl ehdmku k ^\mfy lj_fy b ^Z`_ q_lujvfy Zee_eyfb JZa^_ebf \k_ bamq_ggu_ ehdm- ku gZ deZkku ih qbkem \oh^ysbo \ khklZ\ ehdmkZ Zee_e_c >ey hibkZgby qZklhl \klj_- qZ_fhklb jZaebqguo deZkkh\ \ ihimeypbb \hkihevam_fky jZkij_^_e_gb_f ImZkkhgZ:

]^_ iZjZf_ljNbaf_gy_lky hl ^h +∞ Z λ hl ^h +∞ < gZr_f kemqZ_N^he`_g hij_^_- eylv qbkeh Zee_e_cQ\ ehdmk_ b g_ fh`_l [ulv jZ\_g ihwlhfm \_jhylghklv deZkkZ k h^gbf Zee_e_f [m^_l khhl\_lkl\h\ZlvS1 ijbN bebQ NIZjZf_lj λ aZ^Z_l qbk- eh Zee_e_c \ lhf deZkk_ qZklhlZ dhlhjh]h \ ihimeypbb fZdkbfZevgZ < lZ[e ^Zgu

12

agZq_gby pk ^ey q_luj_o jZaguo khklhygbc ihimeypbb λ b ijb dhlhjuo gZb[he__ qZklh ij_^klZ\e_gu ehdmku k h^gbf ^\mfy lj_fy beb q_lujvfy Zee_eyfb < lZ[e lZd`_ kh^_j`Zlky g_h[oh^bfu_ ^ey jZkq_lh\ agZq_gby fZdkbfZevghc h`b^Z_fhc ]_l_jhab]hlghklb

LZ[ebpZ

QZklhlu \klj_qZ_fhklb deZkkh\ ehdmkh\ k jZaebqguf qbkehf Zee_e_c \ ]_ghlbiZo

ihimeypbc k jZaebqguf jZkij_^_e_gb_f ihebfhjnguo ehdmkh\

Hq_\b^gh ^ey lh]h qlh[u hp_gblv fZdkbfZevgmx ]_l_jhab]hlghklv ihimeypbb

+PD[ \ dZ`^hf ba q_luj_o \ZjbZglh\ g_h[oh^bfh kmffbjh\Zlv ijhba\_^_gby \_jhyl-

545

ghklb \klj_qZ_fhklb deZkkZ gZ \_ebqbgm fZdkbfZevghc h`b^Z_fhc ]_l_jhab]hlghklb

GZijbf_j ^ey i_j\h]h \ZjbZglZ

+PD[ =0.368•0+0.368•0.5+0.184•0.667+0.061•0.75+0.015•0.8++0.003•0.833+0.001•0.857

545

=0.368.

AgZq_gby + PD[ ^ey hklZevguo \ZjbZglh\ ijb\_^_gu \ lZ[e <b^gh qlh h`b^Z_-

545

fu_ fZdkbfZevgu_ agZq_gby ]_l_jhab]hlghklb ihimeypbb agZqbl_evgh ij_\hkoh^yl __ j_Zevgu_ \_ebqbgu lZ[e >ey fhjkdh]h q_j\y Phoronopsis viridis lZ[e oZjZdl_j jZkij_^_e_gby bamq_gguo ehdmkh\ ih deZkkZf ihdZaZg gZ jbk JZkq_l fZdkbfZev-

13

Qbkeh Zee_e_c \ ehdmk_

Jbk QZklhlu deZkkh\ ehdmkh\ k jZaebqguf qbkehf Zee_e_c m Phoronopsis viridis.

>ey kjZ\g_gby ihdZaZgh jZkij_^_e_gb_ ImZkkhgZ ijb λ =1

H[sbc \u\h^ ba ij_^klZ\e_gguo jZkkm`^_gbc b jZkq_lh\ k\h^blky d lhfm qlh ijbjh^gu_ ihimeypbb ^Ze_dh g_ iheghklvx j_Zebamxl k\h_ ihl_gpbZevgh_ ]_g_lbq_- kdh_ jZaghh[jZab_ ;hevrbgkl\h ehdmkh\ ih \b^bfhfm fhghfhjngu Z \ ihebfhjn- guo ehdmkZo ij_bfms_kl\hf ihevam_lky h^bg j_`_ ^\Z lbiZ Zee_e_c

Q:KLHLU :EE?E?C B Q:KLHLU =?GHLBIH< < IHDHE?GBYO JH>BL?E?C B BO IHLHFDH<

:mlhkhfgu_ ijbagZdb

JZkkfhljbf dZd gZke_^mxlky Zee_eb h^gh]h Zmlhkhfgh]h ehdmkZ \ iZgfbdlbq_- kdhc ihimeypbb ijb hlkmlkl\bb ^_ckl\by w\hexpbhgguo nZdlhjh\ fmlZpbb fb]jZpbb hl[hj Ih^ iZgfbdkb_c ihgbfZxl k\h[h^gh_ hkgh\Zggh_ gZ kemqZcghf khq_lZgbb \k_o lbih\ ]Zf_l kdj_sb\Zgb_ jZaghiheuo hkh[_c b i_j_dj_klghhieh^hl\hjy_fuo hj]Zgbafh\ \ ij_^_eZo ihimeypbb beb ^jm]hc \gmljb\b^h\hc ]jmiiu hj]Zgbafh\ IZg- fbdkby \Z`gZ ^ey fh^_ebjh\Zgby ]_g_lbq_kdbo kblmZpbc gh \ ijbjh^_ g_ gZ[ex^Z_lky

[2].

Dh]^Z kdj_sb\Zgb_ kemqZcgh qZklhlZ \_jhylghklv kdj_sb\Zgby hij_^_e_gguo jh^bl_evkdbo \ZjbZglh\ ]_ghlbih\ aZ\bkbl lhevdh hl qZklhl dZ`^h]h ]_ghlbiZ \ ihim-

14

eypbb b jZ\gZ ijhba\_^_gbx wlbo qZklhl >ey hij_^_e_gby \_jhylghklb jZaebqguo \ZjbZglh\ kdj_sb\Zgby m^h[gh ihevah\Zlvky \kihfh]Zl_evghc lZ[ebp_c

QZklhlu Zee_e_c : b Z \ jh^bl_evkdhc ihimeypbb fh`gh jZkkqblZlv gZ hkgh\_ ba-

\_klguo qZklhl ]_ghlbih\ LZd qZklhlZ Zee_ey : h[hagZq_ggZy dZd j jZ\gZ p= x + 0.5•\ qZklhlZ Zee_ey Z h[hagZq_ggZy dZd T jZ\gZ q= z + 0.5•× \ < dZ`^hf \ZjbZgl_ kdj_sb\Zgby \_jhylghklv ihy\e_gby jZaebqguo ]_ghlbih\ hij_^_ey_lky gZ hkgh\_ ijZ-

\beZ F_g^_ey h g_aZ\bkbfhf jZkij_^_e_gbb ]Zf_l J_amevlZlu kdj_sb\Zgby m^h[gh h[h[sblv \ \b^_ lZ[e

LZ[ebpZ

Ijb wlhf qZklhlu Zee_e_c m ihlhfdh\S1 , q1 dZd ihdZau\Zxl jZkq_luS1 = p2 + 0.5•2pq = p(p + q) = p; q1 = q2 + 0.5•2pq = q(p+q) = Tg_ baf_gbebkv

LZdbf h[jZahf ijb khojZg_gbb mkeh\by kemqZcgh]h k\h[h^gh]h kdj_sb\Zgby

(iZgfbdkbb \ hlkmlkl\bb nZdlhjh\ baf_gqb\hklb qZklhlu Zmlhkhfguo Zee_e_c \ jy^m ihdhe_gbc g_ baf_gyxlky b ihimeypby hdZau\Z_lky \ khklhygbb jZ\gh\_kby ijZ-

15

\beh OZj^b <Zcg[_j]Z Ijb jZ\_gkl\_ qZklhl Zee_e_c m dZ`^h]h iheZ khhl\_lkl\b_ f_-

`^m qZklhlZfb Zee_e_c b qZklhlZfb ]_ghlbih\ mklZgZ\eb\Z_lky \ i_j\hf ihdhe_gbb < kemqZ_ g_jZ\_gkl\Z qZklhl Zee_e_c m fm`kdh]h b `_gkdh]h iheh\ jZ\gh\_kgu_ qZklhlu ihy\eyxlky \h \lhjhf ihdhe_gbb lh]^Z dZd \ i_j\hf mklZgZ\eb\Z_lky jZ\_gkl\h qZklhl Zee_e_c fm`kdh]h b `_gkdh]h iheZ

JZ\gh\_kgu_ qZklhlu m ]_ghlbih\ hij_^_ey_fuo khq_lZgb_f jy^Z Zee_e_c p, q,

U «fh`gh hij_^_eblv gZ hkgh\_ h[s_]h \ujZ`_gby OZj^b <Zcg[_j]Z

Kp_ie_ggu_ k ihehf ijbagZdb

JZkkfhljbf kemqZc dh]^Z ]_g hij_^_eyxsbc ijbagZd k\yaZg k iheh\hc ojhfh-

khfhc Hij_^_e_gb_ iheZ m ^biehb^guo hj]Zgbafh\ hkgh\Zgh gZ ]_l_jh]Zfbb <uy\e_- gh g_kdhevdh lbih\ iheh\hc ]_l_jh]Zfbb >ey ^jhahnbeu q_eh\_dZ b fgh]bo ^jm]bo hj]Zgbafh\ `_gkdbc ihe oZjZdl_jbam_lky gZebqb_f ^\mo ;ojhfhkhf OO fm`kdhc

;ojhfhkhfu b<ojhfhkhfu;<DZjlbjh\Zgb_ iheh\uo;b<ojhfhkhf q_eh\_dZ \uy\beh gZebqb_ m gbo ]hfheh]bqguo kh^_j`Zl j_p_kkb\gu_ ]_gu h[mkeh\eb\Zxsb_ jZa\blb_ h[s_c p\_lh\hc ke_ihlu ib]f_glghc dk_jh^_jfu [he_agb H]mlb kiZklbq_- kdhc iZjZie_]bb wib^_jfZevgh]h [mee_aZ b g_]hfheh]bqguo mqZkldh\ <;ojhfhkhf_ wlh j_p_kkb\gu_ ^ey `_gsbg ]_gu ]_fhnbebb ^ZevlhgbafZ fur_qghc ^bkljhnbb ]bih]ZffZ]eh[mebg_fbb ^hfbgZglgh]h ]_gZ ]bihnhknZl_fbq_kdh]h jZoblZ <<ojh- fhkhf_ ]_gu hl\_lkl\_ggu_ aZ ihy\e_gb_ i_j_ihghd f_`^m iZevpZfb b \hehkZluo mr_c

IjbagZdb k\yaZggu_ k ]hfheh]bqggufb mqZkldZfb iheh\uo ojhfhkhf \_^ml k_[y \ ihimeypbb lhqgh lZd `_ dZd ijbagZdb k\yaZggu_ k ZmlhkhfZfb Ijb bamq_gbb ehdm- kh\ ehdZebah\Zgguo \ g_]hfheh]bqghf mqZkld_ O ojhfhkhfu g_h[oh^bfh mqblu\Zlv g_jZ\ghf_jgh_ jZkij_^_e_gb_ Zee_e_c f_`^m km[ihimeypb_c kZfph\ b km[ihimeypb_c kZfhd Ijb jZ\ghf khhlghr_gbb hkh[_c dZ`^h]h iheZ ^\_ lj_lb \k_o O ojhfhkhf gZ- oh^blky \ km[ihimeypbb kZfhd b lhevdh h^gZ lj_lv \ km[ihimeypbb kZfph\ < kemqZ_ dh]^Z \ fm`kdhc b `_gkdhc km[ihimeypbyo qZklhlu Zee_e_c g_ h^bgZdh\u kj_^gxx

JZkkfhljbf dZd \_^ml k_[y \ jy^m ihdhe_gbc qZklhlu Zee_e_c ehdmk dhlhjuo gZ- oh^blky \ g_]hfheh]bqghf mqZkld_ O ojhfhkhfu

16

Imklv \ i_j\hf ihdhe_gbb qZklhlu Zee_e_c \ km[ihimeypbb kZfph\ b kZfhd g_h^bgZdh-

\u b ij_^klZ\e_gu ke_^mxsbfb \_ebqbgZfb kZfpu j : TZ kZfdb j

: TZ Kj_^gyy qZklhlZ Zee_ey : \ ihimeypbb kh]eZkgh jZ\gZ jkj = 2/3•0.4+1/3•0.8=0.533.

<h \lhjhf ihdhe_gbb qZklhlu Zee_e_c \ km[ihimeypbb kZfph\ [m^ml lZdbfb `_ dZd qZklhlu Zee_e_c \ km[ihimeypbb kZfhd i_j\h]h ihdhe_gby lZd dZd hgb [m^mqb bo fZl_jyfb i_j_^Zeb bf k\hb O ojhfhkhfu j : TZ Wlbf \Z`guf k\hckl-

\hf hij_^_ey_lky baf_g_gb_ qZklhl b \ ihke_^mxsbo ihdhe_gbyo QZklhlu Zee_e_c \ km[ihimeypbb kZfhd \lhjh]h ihdhe_gby fh`gh jZkkqblZlv ba qZklhl ]_ghlbih\ kZfph\ b kZfhd i_j\h]h ihdhe_gby IjbgbfZy mkeh\b_ k\h[h^gh]h kdj_sb\Zgby ihemqZ_f j : •0.4+0.5 (0.8•0.6+0.2•0.4)=0.6; TZ

jkj. = 2/3•0.6+1/3•0.4=0.533

< lj_lv_f ihdhe_gbb qZklhlu Zee_e_c \ km[ihimeypbb kZfph\ [m^ml jZ\gu j :

0.6, TZ M kZfhd j : •0.6+0.5(0.4•0.4+0.6•0.6) = 0.5; TZ

jkj. = 2/3•0.5+1/3•0.6=0.533.

KhihklZ\e_gb_ qZklhl Zee_e_c \ lj_o ihke_^h\Zl_evguo ihdhe_gbyo iha\hey_l k^_eZlv ke_^mxsb_ aZdexq_gby kj_^gb_ agZq_gby qZklhl Zee_e_c hklZxlky ihklh- yggufb jZaebqby qZklhl h^ghbf_gguo Zee_e_c f_`^m kZfpZfb b kZfdZfb \ ihdh- e_gbb ihlhfdh\ \ ^\Z jZaZ f_gvr_ q_f \ ihdhe_gbb jh^bl_e_c qZklhlu Zee_e_c i_jbh^bq_kdb f_gyxl k\hb agZq_gby hlghkbl_evgh kj_^g_c qZklhlu jkj ihkl_i_ggh ijb[eb`Zykv d g_c

Ihdhe_gby

Jbk >bgZfbdZ qZklhlu Zee_ey j : ^ey ehdmkZ k\yaZggh]h k O ojhfhkhfhc \ km[ihimeypbyo kZfph\ b kZfhd

17

GZ jbk ^ZgZ ^bgZfbdZ qZklhl ba jZkkfhlj_ggh]h \ur_ ijbf_jZ gZ ijhly`_gbb r_klb ihdhe_gbc Ohly Z[khexlgh]h kh\iZ^_gby qZklhl Zee_e_c \ km[ihimeypbyo kZf- ph\ b kZfhd [ulv g_ fh`_l \k_]^Z kms_kl\m_l ihdhe_gb_ \ dhlhjhf jZaebqby klZgh-

\ylky klZlbklbq_kdb g_hlebqbfufb dZd ^jm] hl ^jm]Z lZd b hl kj_^g_]h agZq_gby qZk- lhlu Zee_ey < wlhf kemqZ_ jZ\gh\_kgu_ qZklhlu ]_ghlbih\ fh`gh ij_^klZ\blv ke_-

^mxsbf h[jZahf

KZfpu : jkj Z qkj (16)

KZfdb :: j2kj; :Z jkjqkj; ZZT2kj (16’)

IhebnZdlhjbZevgu_ ]_ghlbiu

:gZeba qZklhl ihebnZdlhjbZevguo ]_ghlbih\ g_h[oh^bf ijb bamq_gbb iheb]b[- jb^gh]h kdj_sb\Zgby Z lZd`_ gZke_^h\Zgby fgh]bo dhebq_kl\_gguo ijbagZdh\ gZ- ijbf_j jhklZ b fZkku l_eZ

<k_ qZklhlu Zee_e_c \ Zmlhkhfguo ehdmkZo dh]^Z bo jZkkfZljb\Zxl g_aZ\bkbfh hl qZklhl Zee_e_c \ ^jm]bo ehdmkZo ^hklb]Zxl jZ\gh\_kby \ i_j\hf ihdhe_gbb _keb hgh \hagbdeh ijb kh[ex^_gbb mkeh\bc kemqZcgh]h kdj_sb\Zgby H^gh\j_f_ggh mklZ- gZ\eb\Z_lky jZ\gh\_kb_ b ih qZklhlZf khhl\_lkl\mxsbo ]_ghlbih\ H^gZdh qZklhlu ihebnZdlhjbZevguo ]_ghlbih\ \ hij_^_e_gguo kemqZyo ^hklb]Zxl jZ\gh\_kby lhevdh ihke_ kf_gu g_kdhevdbo ihdhe_gbc

JZ\gh\_kgu_ qZklhlu ihebnZdlhjbZevguo ]_ghlbih\ fh]ml [ulv hij_^_e_gu gZ hkgh\_ ba\_klguo qZklhl Zee_e_c ih dZ`^hfm \oh^ys_fm \ _]h khklZ\ ehdmkm

]^_D1, a2...an ), (b1, b2,...bn), (k1,+k2,+...+kn qZklhlu Zee_e_c ehdmkh\ : < b D khhl\_l- kl\_ggh

JZkkfhljbf ijbf_j gZke_^h\Zgby [bnZdlhjbZevguo ]_ghlbih\ \ iZgfbdlbq_kdhc ihimeypbb >ey ijhklhlu ij_^iheh`bf qlh \ dZ`^hf ehdmk_ \klj_qZxlky ih ^\Z Zee_- ey k khhl\_lkl\mxsbfb qZklhlZfb j : TZ U< s(E QZklhlu fhghnZdlhjbZevguo ]_ghlbih\ ijb jZ\gh\_kbb \ khhl\_lkl\bb k aZdhghf OZj^b <Zcg[_j]Z jZ\gu j2(AA),

jTAa), q2(DD^ey ehdmkZ : bU2(BB), 2rs(Bb), s2 (EE^ey ehdmkZ < QZklhlu ^_\ylb [b- nZdlhjbZevguo ]_ghlbih\ kh]eZkgh jZ\gu j2r2 ::<< j22rs(::<Ej2s2(Aabb), r2($D%%jT UV$D%EjT V2(Aabb), q2 r2(aaBB), q22rs(aaBb), q2 s2(aabb).

18

Ijb ^b]b[jb^ghf kdj_sb\Zgbb h[jZam_lky q_luj_ lbiZ ]Zf_l $% Ab), (aB), (DE?keb h[hagZqblv qZklhlu ]Zf_l dZd h(AB), i(Ab), j(D%b k(DElh ijb kemqZcghf khq_lZgbb Zee_e_c ^he`gu \uihegylvky ke_^mxsb_ jZ\_gkl\Z

< wlhf kemqZ_ ijhba\_^_gby qZklhl ]Zf_l :B)•(ZEb Ab)•(D%jZ\gu ^jm] ^jm]m hk = LMlZd dZd h[Z hij_^_eyxlky \_ebqbghcSTUVMdZaZggh_ jZ\_gkl\h hljZ`Z_l kh]eZ- kh\Zggh_ ih hij_^_e_gbx X : Nbebiq_gdh dhgdhj^Zglgh_ khhlghr_gb_ qZklhl ]Z- f_l

Ijbgylh kqblZlv qlh ]Zf_lu :< b ZEgZoh^ylky \ khklhygbb ijbly`_gby Z ]Z-

JZkkfhljbf ihimeypbx kh^_j`Zsmx \ bkoh^ghf ihdhe_gbb lhevdh ]_l_jhab]hlu :Z<EHgb h[jZamxl \k_ q_luj_ lbiZ ]Zf_l qZklhlu dhlhjuo ijb hlkmlkl\bb kp_ie_- gby h^bgZdh\u b khklZ\eyxl <_ebqbgZ d < j_amevlZl_ kemqZcgh]h h[t_^bg_- gby ]Zf_l \hagbdg_l ]_ghlbih\ kh ke_^mxsbfb qZklhlZfb m ]_ghlbih\ ::<< ::bb, ZZ<< ZZEEm ::<E:Z<< ::EEZZ<Eb m :Z<E< ke_^mxs_f ihdhe_gbb ]Zf_lu [m^ml \hkijhba\_^_gu k jZ\ghc \_jhylghklvx b bo qZklhlu \gh\v khklZ\yl Z bo kemqZcgh_ h[t_^bg_gb_ ^Zkl l_ `_ qZklhlu ]_ghlbih\ Ke_^h\Zl_ev- gh \ wlhf kemqZ_ jZ\gh\_kgu_ qZklhlu ]_ghlbih\ ^hklb]Zxlky \ i_j\hf ihdhe_gbb \hagbdr_f \ j_amevlZl_ kemqZcgh]h kdj_sb\Zgby

>jm]Zy kblmZpby \hagbdZ_l dh]^Z ihimeypby ij_^klZ\e_gZ hkh[yfb k ]_ghlbiZfb

::<< b ZZEE\ khhlghr_gbb Hgb h[jZamxl lhevdh ^\Z \b^Z ]Zf_l :< b ZEk qZklhlZfb b <_ebqbgZ d bf__l fZdkbfZevgh_ agZq_gb_ Ijb kemqZcghf khq_lZgbb ]Zf_l \h \lhjhf ihdhe_gbb \hagbdg_l lhevdh ljb \ZjbZglZ ]_ghlbih\ k qZk- lhlZfb ::<< :Z<b (0.5), ZZEENhg^ ]Zf_l \ wlhf ihdhe_gbb ij_^klZ\- e_g ke_^mxsbfb qZklhlZfb :< :b, 1/8 Z< b ZE <_ebqbgZ d < lj_lv_f ihdhe_gbb ihy\eyxlky \k_ ^_\ylv \ZjbZglh\ ]_ghlbih\ h[jZamxsbo ]Zf_lu k qZklhlZfb :< :b, 3/16 Z< ZE<_ebqbgZ d LZdbf h[jZahf hl ihdhe_gby d ihdhe_gbx m\_ebqb\Zxlky qZklhlu i_j\hgZqZevgh hlkmlkl\h\Z\rbo ]Zf_l :E b Z< b ijhihjpbhgZevgh kgb`Z_lky qZklhlZ ]Zf_l :< b ZE <_ebqbgZ d \ dZ`^hf ihke_^mxs_f ihdhe_gbb mf_gvrZ_lky \ jZaZ Ihdhe_gb_ \ dhlhjhf \_ebqbgZ dn g_ [m^_l ^hklh\_jgh hlebqZlvky hl gmey fh`gh kqblZlv jZ\gh\_kguf

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Mf_gvr_gb_ \_ebqbgu d \ jZaZ \ dZ`^hf ke_^mxs_f ihdhe_gbb kijZ\_^eb\h lhevdh ^ey g_kp_ie_gguo ehdmkh\ dh]^Z qZklhlZ j_dhf[bgZpbb k fZdkbfZevgZ b jZ\gZ

>ey kp_ie_gguo ehdmkh\ qZklhlZ j_dhf[bgZpbb k b ijb[eb`_gb_ d jZ\gh\_- kbx ijhbkoh^bl f_^e_gg__

K mq_lhf kdZaZggh]h h[sbc \b^ \ujZ`_gby ^ey hij_^_e_gby agZq_gby d \ i ih- dhe_gbb bf__l \b^

Ihkdhevdm wnn_dl kp_ie_gby ebrv aZf_^ey_l gZklmie_gb_ jZ\gh\_kby ijb dhlh-

jhf ijhba\_^_gby qZklhl ]Zf_l \ nZa_ ijbly`_gby b \ nZa_ hllZedb\Zgby h^bgZdh\u \ khklhygbb jZ\gh\_kby ijZdlbq_kdb g_\hafh`gh h[gZjm`blv kp_ie_ggu_ ehdmku gZ hk- gh\_ ZgZebaZ qZklhl ihebnZdlhjbZevguo ]_ghlbih\

AZf_lbf qlh bkoh^gu_ mkeh\by ijb dhlhjuo \ ihimeypbb ijbkmlkl\h\Zeb lhevdh ]Zf_lu \ nZa_ ijbly`_gby \u[jZgu ijhba\hevgh >ey h[s_]h kemqZy dh]^Z lj_[m_lky hp_gdZ jZ\gh\_kghklb ihimeypbb m^h[gh bkihevah\Zlv Ze]hjblf hkgh\Zgguc gZ aZ-

iheg_gbb fZljbpu qZklhl [bnZdlhjbZevguo ]_ghlbih\> @

]^_ zij qZklhlu ]_ghlbih\ aZibkZgguo \ e_\hc fZljbp_

QZklhlu ]_ghlbih\ ih ehdmkm : gZoh^yl kmffbjmy qZklhlu ]_ghlbih\ ih kljh-

dZf

j :: ]11 + z 12 + z 13,

p(Aa) = z21 + z22 + z23, p(aa) = z13 + z32 + z33.

Kmffbjmy qZklhlu ]_ghlbih\ ih klhe[pZf gZoh^yl khhl\_lkl\mxsb_ qZklhlu ]_- ghlbih\ ih ehdmkm%p(BB), p(Bb), p(EEGZ hkgh\Zgbb ihemq_gguo qZklhl fhghnZd- lhjbZevguo ]_ghlbih\ ih mjZ\g_gbx fh`gh hij_^_eblv qZklhlu dZ`^h]h ba q_lu- j_o Zee_e_c - p, q, r, s < iZgfbdlbq_kdhc ihimeypbb g_ ih^\_j`_gghc ^_ckl\bx w\h- expbhgguo nZdlhjh\ qZklhlu g_ baf_gyxlky b jZ\gh\_kgu_ qZklhlu [bnZdlhjbZev- guo ]_ghlbih\ gZoh^yl ih mjZ\g_gbx

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