Знакомство с эволюционной генетикой (Гаевский)

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h= z11 + 0.5(z 12+ z21) + 0.25 z22;

i= z 13 + 0.5(z 12 + z23) + 0.25 z22;

j= z13 + 0.5(z21 + z32) + 0.25 z22;

k= z33 + 0.5(z32 + z23) + 0.25 z22.

>Ze__ ih mjZ\g_gbx hij_^_eyxl \_ebqbgm d oZjZdl_jbamxsmx g_jZ\gh\_k- ghklv ihimeypbb

IhklhyggZy gZ[ex^Z_fZy gZ ijhly`_gbb fgh]bo ihdhe_gbc g_jZ\gh\_kghklv ih kp_ie_gbx fh`_l [ulv j_amevlZlhf _kl_kl\_ggh]h hl[hjZ _keb h^gb dhf[bgZpbb Ze- e_e_c h[_ki_qb\Zxl [he__ \ukhdmx ijbkihkh[e_gghklv q_f ^jm]b_ Ijb[eb`_gb_ d jZ\gh\_kbx qZklhl ihebnZdlhjbZevguo ]_ghlbih\ aZ\bkbl hl wnn_dlb\ghklb j_dhf[b- gZpbb ihwlhfm q_f f_gvr_ kp_ie_gu ^\Z ehdmkZ l_f [he__ bgl_gkb\guf ^he`_g [ulv hl[hj g_h[oh^bfuc ^ey ih^^_j`Zgby ihimeypbb \ g_jZ\gh\_kghf khklhygbb

Ke_^m_l aZf_lblv qlh kZfZ ih k_[_ \_ebqbgZ d g_ iha\hey_l hij_^_eblv [ebahklv ^Zggh]h ihdhe_gby d khklhygbx jZ\gh\_kby LZd _keb [u \ jZkkfhlj_gghf \ur_ ijb- f_j_ ]_ghlbiu ::<< b ZZEE\ i_j\hf ihdhe_gbb [ueb ij_^klZ\e_gu qZklhlZfb b

\_ebqbgZ d0 qlh aZf_lgh f_gvr_ __ fZdkbfZevgh]h agZq_gby JZkq_l wlh]h ihdZaZl_ey ijhba\h^yl ^ey hp_gdb³^\b`_gby´ihimeypbb \ klhjhgm jZ\gh\_kby

JZkkfhlj_ggZy fh^_ev \ dhlhjhc ]_ghlbi hij_^_eyeb ^\Z ehdmkZ ih ^\Z Zee_ey \ dZ`^hf y\ey_lky kZfhc ijhklhc Ijbf_g_gb_ ^ey ]jmiibjh\db ]Zf_l lZdbo hij_^_e_- gbc dZd ]Zf_lu \ khklhygbb hllZedb\Zgby Ab, D%beb \ khklhygbb ijbly`_gby :<

DEg_h[oh^bfh ebrv ^ey lh]h qlh[u ijhba\_^_gb_ qZklhl ]Zf_l \ dZ`^hc iZj_ \dex- qZeh qZklhlu dZ`^hc Zee_eb

Ijb ZgZeba_ qZklhl ihebnZdlhjbZevguo ]_ghlbih\ ihkljh_gguo gZ ihebZee_ev- guo ehdmkZo \hagbdZxl ljm^ghklb ]jmiibjh\db ]Zf_l \ iZju ^ey ihemq_gby ihegh]h ijhba\_^_gby qZklhl Zee_e_c < kemqZ_ dh]^Z ]_ghlbi hij_^_eyxl ljb ehdmkZ ih ^\Z Zee_ey dZ`^uc \ khklhygbb³ijbly`_gby´gZoh^blky lhevdh h^gZ iZjZ ]Zf_l $%& b

DEF\ hklZevguo lj_o iZjZo ]Zf_lu³ijbly]b\Zxlky´ABc-abC, AbC-aBc, Abc-aBC).

Ihkdhevdm ijhba\_^_gb_ qZklhl ]Zf_l \ dZ`^hc iZj_ ^Z_l ihegh_ ijhba\_^_gb_ Zee_- e_c khklhygb_ jZ\gh\_kby oZjZdl_jbam_lky ke_^mxsbf \ujZ`_gb_f

[(ABC) •(abc)]3=(Abc) •(abC) •(AbC) •(aBc) •(Abc) • (aBC).

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FZdkbfZevgZy \_ebqbgZ d0 ^ey ihebnZdlhjbZevguo ]_ghlbih\ dZd b ^ey [bnZd- lhjbZevgh]h ]_ghlbiZ jZ\gZ

Ihemqblv ZgZeh]bqgh_ \ujZ`_gb_ oZjZdl_jbamxs__ jZ\gh\_kb_ ihimeypbb ^ey kemqZy ijb dhlhjhf qbkeh Zee_e_c \ i_j\hf b \h \lhjhf ehdmk_ jZaebqgh gZijbf_j : b : \ i_j\hf ehdmk_ b < < <±\h \lhjhf g_\hafh`gh ?keb qbkeh Zee_e_c \ i_j\hf b \lhjhf ehdmkZo h^bgZdh\h \ujZ`_gb_ fh`_l [ulv ihemq_gh gh ijb wlhf l_jyxl kfuke lZdb_ ijbagZdb ]Zf_l dZd khklhygb_ hllZedb\Zgby b khklhygb_ ijbly`_- gby JZkkfhljbf \ZjbZgl k ^\mfy ehdmkZfb ih ljb Zee_ey \ dZ`^hf >_\ylv \ZjbZglh\ ]Zf_lD E D E D E D E D E D E D E D E D Efh`gh k]jmiibjh\Zlv \ ljb ijhba\_^_gby dhlhju_ ^he`gu [ulv jZ\gu ijb jZ\gh\_kbb ihimeypbb

(a1b1) •(a2b2) •(a3b3)=(a1b2) •(a2b3) • (a3b1)=(a1b3) •(a2b1) • (a3b2).

Ih \b^bfhfm i_j\mx ]jmiim khfgh`bl_e_c \dexqZxsmx ]Zf_lu k h^bgZdh\u- fb bg^_dkZfb Zee_e_c \u^_eblv e_]q_ Lh]^Z ^ey khklhygby jZ\gh\_kby fh`gh aZib- kZlv

[(a1b1)•(a2b2)•(a3b3)]2=(a1b2)•(a2b3)•(a3b1)•(a1b3)•(a2b1)•(a3b2).

>ey ijhba\hevgh]h dhebq_kl\Z Zee_e_c \ ehdmkZo [bnZdlhjbZevgh]h ]_ghlbiZ wlh \ujZ`_gb_ ijbgbfZ_l \b^

FZdkbfZevgh_ agZq_gb_ d0=(1/n)n.

Ke_^m_l aZf_lblv qlh \ wlbo \ujZ`_gbyo g_l ihijZ\db gZ dhwnnbpb_gl j_dhf[b- gZpbb h dhlhjhc mihfbgZehkv \ur_ <Z`gh lh qlh \ [hevrbgkl\_ kemqZ_\ kms_kl\m_l dhebq_kl\_gguc djbl_jbc jZ\gh\_kgh]h khklhygby ihimeypbb

< aZ\_jr_gb_ wlh]h jZa^_eZ fh`gh _s_ jZa ih^q_jdgmlv qlh g_aZ\bkbfh hl jZkiheh`_gby ehdmkZ \ iheh\uo ojhfhkhfZo beb ZmlhkhfZo Z lZd`_ hl gZebqby ]jmii kp_ie_gby qZklhlu ]_ghlbih\ m ihimeypbb bf_xs_c g_h]jZgbq_ggmx qbke_gghklv b k\h[h^m ^ey kdj_sb\Zgbc ^hklb]Zxl jZ\gh\_kby b hklZxlky ihklhyggufb gZ ijh- ly`_gbb fgh]bo ihdhe_gbc JZ\gh\_kgu_ qZklhlu ]_ghlbih\ fh]ml [ulv \ujZ`_gu q_j_a jZ\gh\_kgu_ qZklhlu Zee_e_c <b^ wlh]h khhlghr_gby fh`_l [ulv ihemq_g gZ hkgh\_ aZdhgZ OZj^b <Zcg[_j]Z

22

>BG:FBD: =?G?LBQ?KDHC KLJMDLMJU IHIMEYPBB

=_g_lbdh Z\lhfZlbq_kdb_ ijhp_kku beb ^j_cn ]_gh\

L_hj_lbq_kdbc ZgZeba jheb kemqZcguo ijhp_kkh\ \ ihimeypbb k\yaZgguo k dhe_-

[Zgbyfb qbke_gghklb khklZ\eyxsbo __ hkh[_c b qZklhlu \klj_qZ_fhklb hl^_evguo ]_gh\ \i_j\u_ [ue k^_eZg > > JhfZrh\uf > > JhfZrh\ b : K K_j_[jh\kdbc ij_^eh`beb gZau\Zlv wlb ijhp_kku ©Z\lhfZlbq_kdbfbª G I >m[bgbg jZa\be wlm b^_x b \\_e l_jfbg©]_g_lbdh Z\lhfZlbq_kdb_ªijhp_kku GZ aZiZ^_ \ wlh `_ \j_fy d ZgZeh]bqguf \u\h^Zf ijbr_e K JZcl dhlhjuc \\_e klZ\rbc \ gZklhys__ \j_fy jZkijhkljZg_gguf l_jfbg©^j_cn ]_gh\ªIha^g__ ihgylb_ h ^j_cn_ ]_gh\ e_]eh \ hkgh\m l_hjbb g_cljZevghklb knhjfmebjh\Zgghc \ ] F Dbfmjhc [4].

>j_cn ]_gh\ hlghkblky d hkh[hfm deZkkm y\e_gbc k\yaZgguo k³hrb[dZfb \u[hj- db´Kmlv _]h aZdexqZ_lky \ lhf qlh ]_ggu_ qZklhlu m h]jZgbq_gghc \u[hjdb hkh[_c hlebqZxlky hl lZdh\uo m bkoh^ghc ihimeypbb GZijbf_j iheh`b\ \ l_fguc f_rhd ih

[_euo b q_jguo rZjbdh\ b \ulZsb\ gZm]Z^ lhevdh ^\Z ba gbo fh`gh h`b^Zlv ihy\- e_gby \ \u[hjd_ rZjbdh\ [_eh]h b q_jgh]h p\_lZ lhevdh k \_jhylghklvx < ^jm]hc iheh\bg_ kemqZ_\ \ \u[hjd_ fh]ml hdZaZlvky lhevdh [_eu_ beb lhevdh q_jgu_ rZjbdb

KemqZcgu_ hldehg_gby bkoh^guo ]_gguo qZklhl ijhbkoh^yl dZd \ [hevrmx lZd b \ f_gvrmx klhjhgm Ijb wlhf \_ebqbgZ hldehg_gby gZoh^blky \ h[jZlghc aZ\bkbfhklb hl \_ebqbgu \u[hjdb ?keb \u[hjdZ khklhbl ba hkh[_c mqZkl\mxsbo \ \hkijhba\h^kl-

\_ ke_^mxs_]h ihdhe_gby lh q_f hgZ f_gvr_ l_f [hevr_ Zfieblm^Z hldehg_gbc qZk- lhl Zee_e_c m ihlhfkl\Z ih kjZ\g_gbx k jh^bl_evkdhc ihimeypb_c

Kms_kl\m_l ihgylb_ wnn_dlb\gZy qbke_gghklv ihimeypbb dhlhjZy khhl\_lkl\m-

_l qbkem1hkh[_c beb qbkem jh^bl_e_c ^Zxsbo gZqZeh ke_^mxs_fm ihdhe_gbx Wlh ihgylb_ klZeh dexq_\uf ijb jZkkfhlj_gbb wnn_dlZ ^j_cnZ ]_gh\ < j_Zevguo ihimey- pbyo wnn_dlb\gZy qbke_gghklv ihimeypbb aZ\bkbl hl kljmdlmju kdj_sb\Zgby [4] GZ- ijbf_j _keb ihimeypby khklhbl ba Nm kZfph\ b Nf kZfhd mqZkl\mxsbo \ jZafgh`_- gbb wnn_dlb\guc jZaf_j ihimeypbb jZ\_g

?keb khhlghr_gb_ kZfph\ b kZfhd \ ihimeypbb wnn_dlb\gZy qbke_gghklv jZ\gZ m^\h_gghfm agZq_gbx dhebq_kl\Z kZfph\ kZfhd l _ \ky ihimeypby mqZkl\m_l \ jZa- fgh`_gbb ?keb khhlghr_gb_ iheh\ g_jZ\ghp_ggh lh wnn_dlb\gZy qbke_gghklv hij_-

23

^_ey_lky dhebq_kl\hf hkh[_c f_g__ fgh]hqbke_ggh]h iheZ Ijb jZkq_l_ wnn_dlb\ghc qbke_gghklb lZd`_ ke_^m_l mqblu\Zlv \hafh`ghklv i_j_^Zqb jZagh]h qbkeZ ]Zf_l i_- jbh^bq_kdh_ baf_g_gb_ qbkeZ hkh[_c i_j_dju\Zgb_ ihdhe_gbc \h \j_f_gb ihy\e_gb_ b bkq_agh\_gb_ bahebjh\Zgguo km[ihimeypbc b ^jm]b_ mkeh\by

F_jhc baf_gqb\hklb ]_gguo qZklhl \hagbdZxs_c \ ihimeypbb aZ kq_l ^j_cnZ ]_gh\ kem`bl \_ebqbgZ klZg^Zjlgh]h hldehg_gby

]^_ p b q qZklhlu Zee_e_c ZgZebabjm_fh]h ehdmkZ Ne wnn_dlb\guc jZaf_j < kemqZ_ dh]^Z dhebq_kl\h kZfph\ b kZfhd h^bgZdh\h N \ujZ`_gb_ bf__l \b^

FZdkbfZevgZy \_ebqbgZ σ ijb p=q b N=1.

HdhgqZl_evguf j_amevlZlhf ^j_cnZ ]_gh\ y\ey_lky aZdj_ie_gb_ \ ihimeypbb h^-

gh]h ba Zee_e_c <_jhylghklv wlh]h kh[ulby jZ\gZ \_jhylghklb qZklhl_ jZkijhkljZg_- gby Zee_ey \ ihimeypbb \ gZqZe_ ^_ckl\by ^j_cnZ ]_gh\ GZijbf_j _keb qZklhlZ Zee_- ey p lh jZ\gh\_jhyl_g dZ`^uc ba bkoh^h\ dZd aZdj_ie_gb_ Zee_ey p \ ihim- eypbb lZd b _]h ihl_jy p ?keb p lh k \_jhylghklvx wlhl Zee_ev aZdj_- iblky \ ihimeypbb b k \_jhylghklvx hg [m^_l ihl_jyg

Baf_g_gb_ \_ebqbgu klZg^Zjlgh]h hldehg_gby aZ\bkbl hl qbkeZ ihdhe_gbc gZ

]^_Wqbkeh ihdhe_gbc Ijb [hevrhfWihimeypby klZgh\blky ]hfhZee_evghc Z klZg-

^Zjlgh_ hldehg_gb_ ijbgbfZ_l fZdkbfZevgh_ agZq_gb_ σ 2 = p0q0 Kj_^g__ qbkeh ih- dhe_gbc t aZ dhlhjh_ ijhbkoh^bl nbdkZpby beb ihl_jy Zee_ey ijbf_jgh \ q_luj_ jZaZ [hevr_ qbkeZ jh^bl_e_c \ dZ`^hf ihdhe_gbb : t = 4N.

Ijhy\e_gb_ ^j_cnZ ]_gh\ fh`gh gZ[ex^Zlv dZd gZ h^ghc ihimeypbb lZd b gZ bo kh\hdmighklb Ijb wlhf h^gb ihimeypbb klZgh\ylky ]hfhZee_evgufb ih Zee_ex :,

^jm]b_ ih Zee_ex Z \ khhl\_lkl\bb k bkoh^guf khhlghr_gb_f bo qZklhl GZijbf_j ijb bkoh^guo qZklhlZo p=0.4, q khhlghr_gb_ ihimeypbc \ dhlhjuo aZdj_ibeky Zee_ev : b ihimeypbc \ dhlhjuo aZdj_ibeky Zee_ev Z [m^_l

24

>j_cn ]_gh\ jZkkfZljb\Zxl \ dZq_kl\_ ]eZ\gh]h nZdlhjZ \ hij_^_e_gbb ]_g_lbq_- kdh]h khklZ\Z ihimeypbb [_jms_c gZqZeh hl g_[hevrh]h qbkeZ hkh[_c wnn_dl©hkgh-

\Zl_eyªb wnn_dl [mluehqgh]h ]hjeurdZ Z lZd`_ ijb hp_gd_ \_jhylghklb nbdkZpbb k_e_dlb\gh g_cljZevghc fmlZpbb

Bg[jb^bg]

H^gh ba ]eZ\guo ^hims_gbc fh^_eb OZj^b <Zcg[_j]Z kemqZcghklv kdj_sb\Z- gby \ j_Zevguo ihimeypbyo j_Zebam_lky hq_gv j_^dh H]jZgbq_gb_ \hafh`ghkl_c \ \u[hj_ iZjlg_jZ \ ehdZevghc km[ihimeypbb m\_ebqb\Z_l jh^kl\h hkh[_c mqZkl\mxsbo \ kdj_sb\Zgbb <hagbdZxsb_ [ebadhjh^kl\_ggu_ kdj_sb\Zgby ihemqbeb gZa\Zgb_ bg[jb^bg]Z =_g_lbq_kdh_ agZq_gb_ bg[jb^bg]Z aZdexqZ_lky \ ihklhygghf m\_ebq_gbb \ ihimeypbb ^heb ]hfhab]hlguo ]_ghlbih\ \ dhlhjuo \klj_qZxlky Zee_eb b^_glbq- gu_ g_ lhevdh ih nmgdpbb gh b ih ijhbkoh`^_gbx l _ [_jmsb_ gZqZeh hl h[s_]h ij_^dZ J_^db_ j_p_kkb\gu_ ]_gu hdZaZ\rbkv \ ]hfhab]hlghf khklhygbb fh]ml ijh- y\blvky n_ghlbibq_kdb AZdj_ie_gb_ \ ehdmk_ dZdh]h eb[h h^gh]h lbiZ Zee_ey ijb\h-

^bl d ihy\e_gbx qbkluo ebgbc <Z`ghc hkh[_gghklvx bg[jb^bg]Z y\ey_lky ihklhyg- kl\h qZklhl Zee_e_c \h \k_o bg[j_^guo ihdhe_gbyo gZ[ex^Z_fh_ gZ nhg_ mf_gvr_gby qZklhlu ]_l_jhab]hlguo ]_ghlbih\

Wlh fh`gh ijh^_fhgkljbjh\Zlv gZ ijbf_j_ ihimeypbb k h[eb]Zlguf kZfhhieh-

^hl\hj_gb_f gZijbf_j m jZkl_gbc kZfhhiuebl_e_c J_amevlZlu bg[jb^bg]Z \ ihim- eypbb kZfhhiuebl_e_c ij_^klZ\e_gu \ lZ[e < bkoh^ghf ihdhe_gbb khklhys_f ba h^gbo ]_l_jhab]hlguo hkh[_c ihke_ kZfhhieh^hl\hj_gby ihy\eyxlky ]hfhab]hlu :: ZZ M ]hfhab]hl h[Z Zee_ey b^_glbqgu ih ijhbkoh`^_gbx >hey wlbo hkh[_c \ ihim- eypbb khhl\_lkl\m_l dhwnnbpb_glm bg[jb^bg]Z ) b khklZ\ey_l ihke_ kZfhhieh^hl\h- j_gby ]_l_jhab]hl < ke_^mxs_f ihdhe_gbb \ j_amevlZl_ bg[jb^bg]Z ^hey ]_l_jh- ab]hl \gh\v mf_gvrblky \ ^\Z jZaZ b khklZ\bl ¼ Z ^hey bg[j_^guo hkh[_c dhwnnbpb-

_gl )\hajZkl_l ^h ¾. Q_j_a Qihdhe_gbc ^hey bg[j_^guo hkh[_c khklZ\bl n Z ijb [_kdhg_qghf qbke_ ihdhe_gbc ^hklb]g_l _^bgbpu

Dhwnnbpb_gl bg[jb^bg]Z ) hljZ`Z_l g_ lhevdh ihl_jx ]_l_jhab]hlghklb gh b

\_jhylghklv \klj_qb b^_glbqguo ih ijhbkoh`^_gbx Zee_e_c \ ]_ghlbi_ h^ghc hkh[b < wlhf kemqZ_ \_ebqbgZ ) aZ\bkbl hl kl_i_gb ]_g_lbq_kdh]h jh^kl\Z __ jh^bl_e_c b fh`_l [ulv hij_^_e_gZ iml_f ZgZebaZ __ jh^hkeh\ghc dhlhjmx m^h[gh bah[jZ`Zlv \ \b^_ ko_fu Hij_^_ebf dhwnnbpb_gl bg[jb^bg]Z gZ ijbf_j_ ihlhfkl\Z kb[kh\ jh^- guo [jZlv_\ b k_kl_j >ey wlh]h kemqZy jh^hkeh\gZy bf__l ke_^mxsbc \b^

A B

C D

E

Jh^bl_eb : b < ih mkeh\bx \ jh^kl\_ g_ khklhyl H[hagZqbf Zee_eb ]_gZ : m hkh[b : Z1Z2 Z m hkh[b < Z3Z4 JZaebqgu_ bg^_dku ihdZau\Zxl qlh Zee_eb g_ b^_glbqgu ih ijhbkoh`^_gbx <_jhylghklv ihy\e_gby m ? dhf[bgZpbb Z1Z1 jZ\gZ

Ihkdhevdm k lZdhc `_ \_jhylghklvx fh]ml ihy\blvky b ljb ^jm]b_ dhf[bgZpbb

Z2Z2 Z3Z3 Z4Z4 h[sZy \_jhylghklv ihy\e_gby m ³?´dhf[bgZpbb b^_glbqguo Zee_e_c jZ\gZ

Kms_kl\m_l ijhklhc kihkh[ gZau\Z_fuc iml_\uf ZgZebahf ^ey hij_^_e_gby dhwnnbpb_glZ bg[jb^bg]Z ih jh^hkeh\ghc Hg hkgh\Zg gZ ih^kq_l_ qbkeZ ³wlZih\´ h[jZamxsbo aZfdgmlu_ pbdeu dhlhju_ \dexqZxl ZgZebabjm_fmx hkh[v b \k_o ij_^- dh\ h[sbo ^ey h[hbo jh^bl_e_c < ^Zgghf kemqZ_ wlh ^\Z imlb ? K :'? b ? K <'? ih q_luj_ wlZiZ Ihkdhevdm ? ihy\ey_lky ^\Z`^u qbkeh wlZih\ khdjZsZxl gZ _^bgbpm

\ gZr_f ijbf_j_ k q_luj_o ^h lj_o Dhwnnbpb_gl bg[jb^bg]Z hij_^_eyxl dZd kmf- fm) 3 + (1/2)3 = ¼.

< h[s_f \b^_ _keb ba\_klgh qbkeh wlZih\Qdhwnnbpb_gl bg[jb^bg]Z jZ\_g

MklZgh\e_gguc dhwnnbpb_gl bg[jb^bg]Z ^ey h^ghc hkh[b kijZ\_^eb\ b ^ey h^- gh]h ihdhe_gby GZijbf_j ihimeypby \ dhlhjhc kdj_sb\Zgb_ ijhbkoh^bl f_`^m kb[- kZfb) \ dZ`^hf ihdhe_gbb m\_ebqb\Z_l ^hex ]hfhab]hl aZ kq_l i_j_oh^Z h^ghc

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q_l\_jlb ]_ghlbih\ k g_b^_glbqguf ih ijhbkoh`^_gbx gZ[hjhf Zee_e_c \ ]hfhab]hl- gh_ khklhygb_

Hij_^_eblv ^hex ]hfhab]hlguo ]_ghlbih\ 1JJ \ ihimeypbb gZoh^ys_cky ih^

]^_ pAa qZklhlZ ]_l_jhab]hl \ bkoh^ghf ihdhe_gbb ^h gZqZeZ bg[jb^bg]Z F dhwnnb- pb_gl bg[jb^bg]Z gZ h^gh ihdhe_gb_ t qbkeh ihdhe_gbc bg[jb^bg]Z

GZ jbk ihdZaZgh baf_g_gb_ ^heb ]hfhab]hlguo ]_ghlbih\ ijb jZaebqguo dh-

wnnbpb_glZo bg[jb^bg]Z \ jy^m ihdhe_gbc

JZcl \dexqbe dhwnnbpb_gl bg[jb^bg]Z \ mjZ\g_gb_ OZj^b <Zcg[_j]Z ^ey hij_-

Ijb) qZklhlZ ]_l_jhab]hlSTmf_gvrZ_lky gZ \_ebqbgmSTl _ \ jZaZ b khhl\_lkl\_ggh m\_ebqb\Zxlky qZklhlu ]_l_jhab]hl gZ hlST< iZgfbdlbq_kdhc ihimeypbb dhwnnbpb_gl ) jZ\_g gmex b \ujZ`_gb_ k\h^blky d nhjfme_ OZj^b <Zcg[_j]Z

Ihdhe_gby bg[jb^bg]Z

Jbk >hey ]hfhab]hlguo ihlhfdh\ ijb kbkl_fZlbq_kdbo kdj_sb\Zgbyo k jZaghc kl_i_gvx bg[jb^bg]Z

Dhgp_ipbx b^_glbqghklb Zee_e_c fh`gh bkihevah\Zlv ^ey emqr_]h ihgbfZgby kfukeZ \ujZ`_gby JZclZ ij_^klZ\b\ qZklhlu ]_ghlbih\ dZd kmffm qZklhl ehdmkh\ k b^_glbqgufb b g_aZ\bkbfufb ih ijhbkoh`^_gbx Zee_eyfb

27

FmlZpbb

Kms_kl\h\Zgb_ fmlZglguo Zee_e_c \ ihimeypbb k\yaZgh dZd k ijhp_kkhf bo ihy\-

e_gby lZd b k \hafh`ghklyfb bo jZkijhkljZg_gby b aZdj_ie_gby

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dhlhjuc ij_^eh`be \ dZq_kl\_ fh^_eb ihimeypbx ^biehb^guo jZafgh`Zxsboky ih- eh\uf iml_f hj]Zgbafh\ Ih mkeh\byf fh^_eb \k_ hkh[b aZ bkdexq_gb_f h^ghc ]hfh- ab]hlgu ih ZgZebabjm_fhfm ehdmkm :: Z _^bgkl\_ggZy ]_l_jhab]hlgZy hkh[v :Z h^- gh\j_f_ggh y\ey_lky ghkbl_e_f fmlZglgh]h Zee_ey Z ]_ghlbi :Z g_ h[eZ^Z_l ij_- bfms_kl\hf i_j_^ bkoh^guf ]_ghlbihf :: h[s__ dhebq_kl\h hkh[_c \ ihimeypbb

hklZ_lky ihklhygguf gZ ijhly`_gbb jy^Z ihdhe_gbc

Ihkdhevdm iZjlg_jhf m ]_l_jhab]hlu :Z ijb i_j\hf kdj_sb\Zgbb \k_]^Z [m^_l ]hfhab]hlZ :: g_ljm^gh ihdZaZlv dZdbf h[jZahf \_jhylghklv ihl_jb fmlZglgh]h Ze-

Ijb g_baf_gghc \_ebqbg_ ihimeypbb kj_^g__ qbkeh ihlhfdh\ m h^ghc jh^bl_ev-

kdhc iZju jZ\gh ^\mf Z \hafh`gh_ qbkeh ihlhfdh\ baf_gy_lky hl gmey ^hQ<_jhyl- ghklv ihy\e_gby lh]h beb bgh]h dhebq_kl\Z ihlhfdh\ fh`gh hibkZlv jZkij_^_e_gb_f ImZkkhgZ ijbgy\ λ =2:

28

Kmffbjmy ijhba\_^_gby \_jhylghklb ihl_jb fmlZpbb b \_jhylghklb jh`^_gby hij_^_e_ggh]h dhebq_kl\Z ihlhfdh\ ihemqZxl \_jhylghklv Ploss ihl_jb Zee_ey Z aZ h^gh ihdhe_gb_

JZkq_lu ihdZau\Zxl qlh wlZ \_ebqbgZ jZ\gZH-1 beb <_jhylghklv khojZg_-

gby Zee_ey Z \h \lhjhf ihdhe_gbb \ur_ q_f \_jhylghklv _]h ih- l_jb Ih^h[gu_ jZkq_lu ^ey m]ZkZxs_c ihimeypbb kj_^g__ qbkeh ihlhfdh\ gZ h^gm iZjm λ ihdZau\Zxl qlh \_jhylghklv ihl_jb \hajZklZ_l ^h GZh[hjhl \ kemqZ_³ijhp\_lZgby´dh]^Z kj_^g__ qbkeh ihlhfdh\ khklZ\ey_l b \_jhylghklv ih- l_jb kgb`Z_lky ^h b khhl\_lkl\_ggh < ex[hf kemqZ_ \ jy^m ihke_^mxsbo ihdhe_gbc qbkeh j_^dbo g_cljZevguo fmlZpbc \ ihimeypbb [m^_l mf_gvrZlvky ihdZ g_ bkq_ag_l ihke_^gbc fmlZglguc ]_g

< [hevrbgkl\_ kemqZ_\ fmlZpbb \ h^ghf b lhf `_ ehdmk_ \hagbdZxl k hij_^_e_g- ghc qZklhlhc beb \_jhylghklvx JZkkfhljbf kblmZpbx ijb dhlhjhc ijhp_kk fmlbjh-

\Zgby y\ey_lky _^bgkl\_gguf nZdlhjhf baf_g_gby qZklhl Zee_e_c \ ihimeypbb Hij_-

^_e_gby qZklhlu \_jhylghklb ihy\e_gby fmlZpbb \ ihimeypbb ihgylgh ba ke_^mxs_-

]h ijbf_jZ ?keb aZ h^gh ihdhe_gb_ fmlbjm_l h^bg Zee_ev ba lukyq lZdbo Zee_e_c lh qZklhlZ jZ\gZ •10-5 < _kl_kl\_gguo mkeh\byo h[uqgh gZ[ex^Zxlky \_eb- qbgu -6 - 10-5 Gbadbc l_fi fmlbjh\Zgby 4•10-9 h[gZjm`_g m Escherichia soli ih ]_gm hij_^_eyxs_fm mklhcqb\hklv d klj_ilhfbpbgm gZb[he__ \ukhdbc m q_eh\_dZ ijb

ihy\e_gbb ]_gZ g_cjhnb[jhfZlhaZ

Ijb ZgZeba_ fmlZpbhggh]h ijhp_kkZ fh`gh bkoh^blv ba ^\mo mkeh\bc ijhp_kk b^_l \ h^ghf gZijZ\e_gbb gZijbf_j ]_g : ljZgknhjfbjm_lky \ ]_g Z ijhp_kk b^_l

\ ijyfhf b h[jZlghf gZijZ\e_gbyo

Imklv jh qZklhlZ Zee_ey : \ bkoh^ghf ihdhe_gbb u qZklhlZ beb kdhjhklv fmlb- jh\Zgby Zee_ey : → Z Lh]^Z qZklhlZ Zee_ey : aZ h^gh ihdhe_gb_ mf_gvrblky gZ \_eb-

IjbWklj_fys_fky d [_kdhg_qghklb qZklhlZ Zee_ey : pt klj_fblky d gmex beb ^jm]bfb keh\Zfb Zee_ev Z iheghklvx aZf_sZ_l Zee_ev :.

Ihkdhevdm \_ebqbgZXfgh]h f_gvr_ _^bgbpu \ujZ`_gb_ u)t \ \ujZ`_gbb

QZklh hij_^_eyxl qbkeh ihdhe_gbc aZ dhlhju_ qZklhlZ Zee_ey : mf_gvrblky \

jZaZ Ih^klZ\b\ \ \ujZ`_gb_ agZq_gb_ pt = 0.5po, ihemqbf t = ln2/Xbeb t = 0.69/u.

Ijb kj_^g_f agZq_gbb u=10-5, t ihdhe_gbc

>ey hibkZgby ijhp_kkZ fmlbjh\Zgby b^ms_]h \ ijyfhf b h[jZlghf gZijZ\e_gbb \\_^_f ^hihegbl_evgmx i_j_f_ggmx v dhlhjZy oZjZdl_jbam_l kdhjhklv h[jZlghc fm- lZpbb a → : < i_j\hf ihdhe_gbb ihke_ gZqZeZ fmlZpbc qZklhlZ Zee_ey : khklZ\bl

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\Zgby

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Fb]jZpby ihlhd ]_gh\)

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`Zsbo jZaguf ihimeypbyf Wlhfm ij_^r_kl\m_l i_j_f_s_gb_ hkh[_c \ ijhkljZgkl\_

30

beb fb]jZpby < ijhp_kk_ h[f_gZ ]_gZfb fh`gh \u^_eblv ihimeypbx ba dhlhjhc ijh- bkoh^yl fb]jZglu b ihimeypbx Z[hjb]_gu \ dhlhjhc wlb fb]jZglu gZoh^yl iZjlg_- jZ ^ey kdj_sb\Zgby ?keb ijhp_kk h[f_gZ ijh^he`bl_e_g ]h\hjyl h ihlhd_ ]_gh\ Wn- n_dlb\ghklv h[f_gZ ]_gZfb aZ\bkbl hl kl_i_gb beb qZklhlu fb]jZpbb F b hl kl_i_gb ]_g_lbq_kdbo jZaebqbc ihimeypbc

W\hexpbhggZy jhev fb]jZpbb g_ h^bgZdh\Z Ijb j_ijh^mdlb\ghf dhglZdl_ koh^- guo ih ]_g_lbq_kdhfm khklZ\m ihimeypbc fh`_l ijhbkoh^blv dZd klbjZgb_ kms_kl-

\h\Z\rbo f_`^m gbfb jZaebqbc lZd b khojZg_gb_ ihebfhjnbafZ < [hevrbgkl\_ kem- qZ_\ ]b[jb^baZpby f_`^m [ebadbfb \b^Zfb g_ \_^_l d h[f_gm ]_gZfb ba aZ keZ[hc ijb- kihkh[e_gghklb ihlhfdh\ H^gZdh bgh]^Z keZ[h ijbkihkh[e_ggu_ ihlhfdb ^hklZlhqgh ieh^h\blu qlh[u ^Zlv \h \lhjhf ihdhe_gbb ihlhfkl\h ijb \ha\jZlghf kdj_sb\Zgbb J_dhf[bgZglgu_ ]_ghlbiu hl fgh]hdjZlguo \ha\jZlguo kdj_sb\Zgbc [m^ml \k_ [he__ b [he__ ijb[eb`Zlvky d jh^bl_evkdhfm lbim b fh]ml \eblvky \ ]_ghnhg^ jh^bl_evkdhc ebgbb < j_amevlZl_ wlh]h g_dhlhjh_ dhebq_kl\h hl^_evguo ]_gh\ beb ]_gguo [ehdh\ i_j_c^_l hl h^gh]h \b^Z d ^jm]hfm

JZkkfhljbf \ebygb_ fb]jZpbb gZ ]_g_lbq_kdbc khklZ\ ihimeypbb Z[hjb]_gh\ Ij_^iheh`bf qlh fb]jZpby ijhbkoh^bl \ h^ghf gZijZ\e_gbb ba [hevrhc fZl_jbdh-

\hc ihimeypbb \ fZe_gvdmx hkljh\gmx \k_ ^hims_gby fh^_eb OZj^b <Zcg[_j]Z hklZ- xlky \ kbe_ ihlhd ]_gh\ ijhbkoh^bl k ihklhygghc kdhjhklvx F jZkkqblZgghc dZd ^h- ey fb]jZglh\ \ ihimeypbb Z[hjb]_gh\ gZ h^gh ihdhe_gb_ qZklhlu Zee_ey Z \ fZl_jb- dh\hc b hkljh\ghc ihimeypbyo g_ jZ\gu b khklZ\eyxl4bTkhhl\_lkl\_ggh aZ \j_fy fb]jZpbb qZklhlZ Zee_ey Z \ fZl_jbdh\hc ihimeypbb hklZ_lky ihklhygghc lZd dZd qbk- eh fb]jZglh\ fZeh ih kjZ\g_gbx k qbke_gghklvx fZl_jbdh\hc ihimeypbb

JZkkfhljbf qbke_gguc ijbf_j Imklv qZklhlZ Zee_ey Z \ fZl_jbdh\hc ihimeypbb jZ\gZ4 \ hkgh\ghc ^h gZqZeZ fb]jZpbb qh aZ h^gh ihdhe_gb_ \ hkljh\gmx ihimeypbx \eb\Z_lky Ihke_ kebygby hkljh\gZy ihimeypby [m^_l ij_^klZ\e_gZ fb]jZglZfb0 b Z[hjb]_gZfb F QZklhlZ Zee_ey Z ijb wlhf khklZ\bl q1 = 0.2•0.7+0.8• < h[s_f \b^_ agZq_gb_ qZklhlu Zee_ey Z \ i_j\hf ihdhe_- gbb fh`gh hij_^_eblv ke_^mxsbf h[jZahf

<ujZ`_gb_> 0•(qo4 @hljZ`Z_l baf_g_gb_ qZklhlu Zee_ey Z aZ h^gh ihdhe_- gb_ Hgh h[jZsZ_lky \ ghev dh]^Z ij_djZsZ_lky ijhp_kk fb]jZpbb F eb[h dh]^Z qZklhlu Zee_ey Z \ h[_bo ihimeypbyo klZgh\ylky h^bgZdh\ufb q = Q).

31

Ihe_agh ijhke_^blv aZ baf_g_gb_f jZaebqbc \ qZklhlZo Zee_ey Z f_`^m hkljh\- ghc b fZl_jbdh\hc ihimeypbyfb \ g_kdhevdbo ihdhe_gbyo ijb ihklhygghc kdhjhklb fb]jZpbb < i_j\hf ihdhe_gbb jZaebqby khklZ\yl

q1 - Q = qo - M•(qo - Q) - Q = qo - Mqo + MQ - Q = (1 - M)•qo - (1 - M) •Q, (38)

WlZ nhjfmeZ ^Z_l \hafh`ghklv jZkkqblZlv qZklhlm Zee_ey \ ehdZevghc ihimeypbb

ih ijhr_kl\bbWihdhe_gbc k gZqZeZ fb]jZpbb _keb ba\_klgu bkoh^gu_ qZklhlu Zee_-

eb[h kdhjhklv fb]jZpbb W b ^hex ]_gh\ ij_^dh\ F t \ ihimeypbb Z[hjb]_gh\ ih ijhr_kl\bbWihdhe_gbc

<hkihevam_fky ijbf_jhf ba [1]. < KR: ihlhfkl\h hl kf_rZgguo [jZdh\ f_`^m

[_eufb b g_]jZfb ijbgylh hlghkblv d g_]jblygkdhfm gZk_e_gbx Ihwlhfm kf_rZggu_ [jZdb fh`gh jZkkfZljb\Zlv dZd ihlhd ]_gh\ ba [_ehc ihimeypbb \ l_fghdh`mx Ij_^- db kh\j_f_gguo g_]jh\ \ KR: [ueb i_j_k_e_gu ba :njbdb hdheh e_l gZaZ^ qlh khhl\_lkl\m_l ihdhe_gbyfW Bkoh^gu_ agZq_gby qZklhl Zee_e_c ]_gZT0 dh^b- jmxs_]h [_ehd djh\b fh`gh ihemqblv gZ hkgh\_ ZgZebah\ djh\b m kh\j_f_gguo q_j- ghdh`bo `b\msbo \ :njbd_ >ey hp_gdb kh\j_f_gguo qZklhl pt ih^hc^ml q_jghdh`b_ rlZlh\ >`hj^`by b DZebnhjgby \_ebqbgm J ^Z^ml [_eu_ KR:

IjhZgZebabjm_f \hafh`gu_ \_ebqbgu ihlhdZ ]_gh\ F hl [_eh]h gZk_e_gby d l_fghdh`_fm b ^hex ]_gh\ g_]jblygkdbo ij_^dh\ m kh\j_f_gguo g_]jh\ gZ hkgh\_ qZk- lhl Zee_e_c hij_^_eyxsbo k\hckl\Z djh\b KjZ\gbf ihimeypbb q_jghdh`bo ^\mo mdZ- aZgguo rlZlh\ Z lZd`_ h^ghagZqghklv hp_ghd ijb bkihevah\Zgbb \ dZq_kl\_ fZjd_jh\ jZaebqguo Zee_e_c Hp_gbf jZa[jhk ^Zgguo b hij_^_ebf ^hklh\_jghklv jZaebqbc

32

kj_^gbo ih ^Zgghfm iZjZf_ljm m ^\mo ihimeypbc l_fghdh`bo ex^_c Bkoh^gu_ ^Zg- gu_ ij_^klZ\e_gu \ lZ[e

LZ[ebpZ

JZkq_lu ihdZau\Zxl qlh \ rlZl_ >`hj^`by ihlhd ]_gh\ hl [_eh]h gZk_e_gby d l_fghdh`_fm aZ h^gh ihdhe_gb_ khklZ\ey_l \ rlZl_ DZebnhjgby JZaebqby aZf_lgu gh g_^hklh\_jgu tst >hey ]_gh\ ij_^dh\ m g_]jblygkdh]h gZk_e_gby KR: ihke_ ihdhe_gbc \ rlZlZo >`hj^`by b DZebnhjgby khhl\_lkl\_ggh jZ\gu

b

?kl_kl\_gguc hl[hj

F_`^m hkh[yfb khklZ\eyxsbfb \b^ kms_kl\m_l fgh`_kl\h jZaebqbc Hkh[b \ ihimeypbb dhlhju_ h[eZ^Zxl ijbagZdZfb emqr_]h \u`b\Zgby hklZ\eyxl [hevr_ ih- lhfkl\Z b kqblZxlky [he__ ijbkihkh[e_ggufb =eZ\gZy hkh[_gghklv ijbkihkh[bl_ev- gh]h ijbagZdZ khklhbl \ lhf qlh hg l_f beb bguf kihkh[hf ih\urZ_l rZgku hkh[b mqZkl\h\Zlv \ kha^Zgbb ihimeypbb ke_^mxs_]h ihdhe_gby J_amevlZlhf _kl_kl\_ggh]h hl[hjZ fh`_l [ulv kha^Zgb_ khojZg_gb_ beb ihl_jy baf_gqb\hklb \ ihimeypbb Hl[hj ]_gh\ beb ]_gguo dhfie_dkh\ hkms_kl\ey_lky gZ mjh\g_ bo n_ghlbibq_kdbo ijhy\e_- gbc :ee_eb kihkh[kl\mxsb_ m\_ebq_gbx ijbkihkh[e_gghklb n_ghlbiZ lZd`_ ih^-

\_j`_gu hl[hjm gZ bo khojZg_gb_ Ihwlhfm kbem hl[hjZ fh`gh \ujZablv \ l_jfbgZo ij_\hkoh^kl\Z Zee_e_c b baf_g_gbyo qZklhl ]_gh\ ohly \ ^_ckl\bl_evghklb dhgdmjb- jmxl g_ ]_gu Z hkh[b b bo ]jmiiu

33

Ijb hibkZgbb hl[hjZ bkihevamxl lZdb_ ihdZaZl_eb dZd Z^ZilZpbhggZy p_gghklv ]_ghlbiZ beb ijbkihkh[e_gghklv:b dhwnnbpb_gl hl[hjZ VL_hj_lbq_kdb Z^ZilZpb- hggZy p_gghklv ^he`gZ \ujZ`Zlv dZd g_ihkj_^kl\_ggmx ijbkihkh[e_gghklv ]_ghlbiZ dhlhjmx fh`gh hp_gblv q_j_a qbkeh j_Zevguo ihlhfdh\ lZd b ^he]hkjhqgmx \deZ^ wlbo ]_ghlbih\ \ ihke_^mxsb_ ]_ghnhg^u < j_Zevghc kblmZpbb ^he]hkjhqgmx ijb-

kihkh[e_gghklv fh`gh hp_gblv gZ hkgh\_ nhjfZebah\Zgghc fZl_fZlbq_kdhc fh^_eb

:^ZilZpbhggZy p_gghklv ijbkihkh[e_gghklv dZ`^h]h ]_ghlbiZ aZ\bkbl hl \aZb- fh^_ckl\by \k_o ]_gh\ bf_xsboky m ^Zgghc hkh[b ?keb k_e_dlb\gZy p_gghklv \k_o ]_ghlbih\ kh^_j`Zsbo Zee_ev Z1 [hevr_ k_e_dlb\ghc p_gghklb \k_o ]_ghlbih\ k Ze- e_e_f Z2 hl[hj [m^_l gZijZ\e_g gZ khojZg_gb_ Zee_ey Z1 qZklhlZ dhlhjh]h ba ihdhe_- gby \ ihdhe_gb_ [m^_l jZklb

Fh^_ev hl[hjZ ]_ghlbih\ gZ mjh\g_ h^gh]h Zmlhkhfgh]h ehdmkZ F_klhf ^_ckl\by hl[hjZ \u[_j_f iZgfbdlbq_kdmx ihimeypbx Ij_^ihkuedhc hl[hjZ \uklmiZxl jZaebqby \ Z^ZilZpbhgghc p_gghklb ]_ghlbih\ n_ghlbih\ dhlhju_ \ha- gbdZxl \ ijhf_`mlhd \j_f_gb k fhf_glZ i_j\h]h ^_e_gby ab]hlu b ^h h[jZah\Zgby ab]hlu ke_^mxs_]h ihdhe_gby ?keb g_dbc ehdmk jZks_iey_lky ih Zee_eyf : b Z ih- imeypbx fh`gh jZa^_eblv ih ]_ghlbiZf gZ ljb ]jmiiu < dZ`^hc ]jmii_ ]_ghlbi bf_-

_l k\hx Z^ZilZpbhggmx p_gghklv :: Wo; :Z:1, ZZ:2 Iheh`bf qlh ijb h[jZah-

\Zgbb ihdhe_gby ab]hl ijbgylh]h aZ lhqdm hlkq_lZ qZklhlu ]_ghlbih\ m kZfph\ b kZ- fhd [ueb h^bgZdh\u b khklZ\eyeb ^ey ::± [^ey :Z± \b ^ey ZZ± ]KmffZ qZklhl

jZ\gZ _^bgbp_ x+y+z=1.

Dhwnnbpb_gl mqZklby dZ`^h]h ]_ghlbiZ \ kha^Zgbb ke_^mxs_]h ihdhe_gby ab]hl hij_^_ey_lky ijhba\_^_gb_f qZklhlu ]_ghlbiZ gZ ihdZaZl_ev _]h Z^ZilZpbhgghc p_g-

Ihkdhevdm ihdZaZl_eb Z^ZilZpbhgghc p_gghklb ijbgylh ghjfbjh\Zlv ih fZdkb- fZevghfm agZq_gbx h^gh]h ba dhwnnbpb_glh\ Wo , W1 , W2 ihemqZ_fZy kmffZ f_gvr_ _^bgbpu

QZklhlu ]_ghlbih\ mqZkl\mxsbo \ h[jZah\Zgbb ab]hlu ke_^mxs_]h ihdhe_gby hij_^_eyxlky ke_^mxsbfb \ujZ`_gbyfb

34

]^_ Ws = xW0 + yW1 + zW2 ij_^klZ\ey_l kh[hc kj_^gxx ijbkihkh[e_gghklv ihimeypbb ih ZgZebabjm_fhfm ehdmkm

Ijb mkeh\bb qlh oZjZdl_jbklbdb km[ihimeypbc kZfph\ b kZfhd kh\iZ^Zxl ^ey

hij_^_e_gby qZklhl ]_ghlbih\ h[jZah\Z\rboky ab]hl ke_^mxs_]h ihdhe_gby fh`gh \hkihevah\Zlvky \ujZ`_gb_f OZj^b <Zcg[_j]Z

Ij_^eh`_gguc ijbgpbi iha\hey_l ijh^he`blv hij_^_e_gb_ qZklhl Zee_e_c b ]_-

ghlbih\ \ ihke_^mxsbo ihdhe_gbyo ab]hl b ijhke_^blv bo ^bgZfbdm \ mkeh\byo ^_c- kl\by hl[hjZ

D mjZ\g_gbyf fh`gh ijbclb b ^jm]bf iml_f khhlghky qZklhlu ]_ghlbih\ ihlhfdh\ k qZklhlZfb jZaebqguo \ZjbZglh\ kdj_sb\Zgbc Ijb mkeh\bb qlh oZjZdl_jb- klbdb km[ihimeypbc kZfph\ b kZfhd kh\iZ^Zxl ke_^m_l h`b^Zlv qlh jZaebqgu_ \Zjb-

Zglu kdj_sb\Zgby \ khhl\_lkl\bb k j_r_ldhc I_gg_lZ [m^ml \hagbdZlv ijhihjpbh-

gZevgh qZklhl_ b ijbkihkh[e_gghklb ]_ghlbih\

< jZfdZo h[km`^_gby j_amevlZlh\ hl[hjZ g_h[oh^bfh jZkkfhlj_lv baf_g_gb_ kj_^g_c ijbkihkh[e_gghklb ihimeypbb aZ h^bg wlZi hl[hjZ >bki_jkby ijbkihkh[e_g-

³nmg^Zf_glZevgmx l_hj_fm _kl_kl\_ggh]h hl[hjZ´ Baf_g_gb_ kj_^g_c ijbkihkh[e_g- ghklb ijyfh ijhihjpbhgZevgh __ ^bki_jkbb \ gZqZevguc fhf_gl \j_f_gb b h[jZlgh ijhihjpbhgZevgh __ \_ebqbg_

J_amevlZl ^_ckl\by hl[hjZ aZ h^gh ihdhe_gb_ hp_gb\Zxl ih baf_g_gbx qZklhl

>ey kemqZy dh]^Z \ bkoh^ghf ihdhe_gbb qZklhlu ]_ghlbih\ khhl\_lkl\h\Zeb mjZ\g_gbx OZj^b <Zcg[_j]Z g_h[oh^bfh \ \ujZ`_gbb ijhba\_klb ke_^mxsb_ ih^klZgh\db x = p2 , y = 2pq, z = q2 Ihke_ jy^Z g_keh`guo fZl_fZlbq_kdbo ij_h[jZ- ah\Zgbc \ujZ`_gb_ ijbh[j_lZ_l \b^

AgZf_gZl_ev dZd m`_ hlf_qZehkv ij_^klZ\ey_l kh[hc kj_^gxx ijbkihkh[e_g- ghklv ihimeypbb ijb ^_ckl\bb hl[hjZ b h[hagZqZ_lky Ws.

Ihkdhevdm hlghkbl_evgu_ \_ebqbgu dhwnnbpb_glh\ ijbkihkh[e_gghklb g_ \u- oh^yl aZ ij_^_eu bgl_j\ZeZ ≤ W≤ \ \ujZ`_gbb fh`gh aZf_gblv dhwnnbpb_glu Z^ZilZpbhgghc p_gghklb gZ khhl\_lkl\mxsb_ agZq_gby dhwnnbpb_glh\ hl[hjZ si ijb-

Ke_^m_l aZf_lblv qlh hp_gdZ baf_g_gby qZklhlu Zee_ey kijZ\_^eb\Z lhevdh ih hlghr_gbx d bkoh^ghc ihimeypbb b i_j\hfm ihdhe_gbx ihlhfdh\ \hagbdr_fm \ mk-

36

eh\byo ^_ckl\by hl[hjZ Hgh g_ hljZ`Z_l baf_g_gby qZklhlu Zee_ey \ p_ehf ihkdhev- dm g_ hij_^_e_gu lZdb_ iZjZf_lju dZd \hajZklghc khklZ\ ihimeypbb ^ebl_evghklv j_ijh^mdlb\gh]h i_jbh^Z ^hey ihlhfdh\ b l i

< jZfdZo ihemq_gghc fh^_eb baf_g_gb_ qZklhlu Zee_ey Z \ oh^_ hl[hjZ ijy- fh ijhihjpbhgZevgh ^he_ ]_l_jhab]hl pq b h[jZlgh ijhihjpbhgZevgh kj_^g_c ijbkih- kh[e_gghklb ihimeypbb Ws QZklv \ujZ`_gby>q(W2 -W1) + p(W1 - Wo@gZb[he__ \Z`gZ ijb ZgZeba_ ihke_^kl\bc hl[hjZ HgZ khklhbl ba ^\mo keZ]Z_fuo \ dhlhjuo kh- ihklZ\eyxlky ijbkihkh[e_gghklb ]_ghlbih\ ijb ihke_^h\Zl_evghf \ul_kg_gbb h^gh]h Zee_ey ^jm]bf ZZ !:Z !:: qlh \ dhg_qghf blh]_ hij_^_ey_l gZijZ\e_gb_ agZd

^_ckl\by hl[hjZ

Fh^_ev hl[hjZ iha\hey_l hij_^_eblv mkeh\by ijb dhlhjuo g_ ijhbkoh^bl baf_-

g_gb_ qZklhl Zee_e_c [5] Kh]eZkgh ijb wlhf T1- q0 Djhf_³djZ_\uo mkeh\bc´ khhl\_lkl\mxsbo aZ\_jr_gbx hl[hjZ ihke_ ihegh]h \ul_kg_gby h^gh]h Zee_ey ^jm]bf

(p=1, T beb p=0, T kms_kl\m_l lj_lv_ jZ\gh\_kgh_ khklhygb_ ihimeypbb ijb dh- lhjhf q(W2-W1) + p(W1-Wo J_rZy wlh mjZ\g_gb_ hlghkbl_evgh p, ihemqbf

Mkeh\b_ dh]^ZSfh`_l \uihegylvky \ ^\mo kemqZyo

1)W2 -W1>0 , W2-2W1+W0>0, W2-2W1+W0>W2 -W1;

2)W2 -W1<0 , W2-2W1+W0<0, W2-2W1+W0<W2 -W1.

< i_j\hf j_r_gb_ g_jZ\_gkl\ ^Z_l:2>W1 , W0>W1 l _ ijbkihkh[e_gghklv ]_- l_jhab]hl f_gvr_ ijbkihkh[e_gghklb ]hfhab]hl hl[hj ijhlb\ ]_l_jhab]hl \h \lhjhf

- W2<W1 , W0<W1 ijbkihkh[e_gghklv ]_l_jhab]hl \ur_ q_f ijbkihkh[e_gghklv ]h- fhab]hl hl[hj \ ihevam ]_l_jhab]hl Gb`_ fu hij_^_ebf \ dZdhf ba wlbo ^\mo kemqZ_\ jZ\gh\_kb_ mklhcqb\h_ Z \ dZdhf±g_l

Ijb ZgZeba_ jZaebqguo kblmZpbc k\yaZgguo k hl[hjhf ]_ghlbih\ g_h[oh^bfh mqblu\Zlv \hafh`ghklv gme_\h]h qZklbqgh]h b ihegh]h ^hfbgbjh\Zgby h^gh]h Zee_ey gZ^ ^jm]bf b kl_i_gv hl[hjZ hl qZklbqgh]h ^h ihegh]h dh]^Z ]_ghlbiu e_lZevgu beb kl_jbevgu Fh`gh jZkkfhlj_lv iylv hkgh\guo \ZjbZglh\ hl[hjZ lZ[e hij_^_eb\ ^ey dZ`^h]h dhwnnbpb_glu hlghkbl_evghc ijbkihkh[e_gghklb Wi = 1- si b mkeh\by ^hfbgbjh\Zgby

LZ[ebpZ

Ijbk\h_gb_ dhwnnbpb_glm hl[hjZ khhl\_lkl\mxs_]h bg^_dkZ hijZ\^Zgh \ kemqZ_ dh-

]^Z ijbkihkh[e_gghklv ih djZcg_c f_j_ ^\mo ]_ghlbih\ jZaebqgZ

< g_dhlhjuo kemqZyo fh`gh h[hclbkv h^gbf dhwnnbpb_glhf hl[hjZ _keb ba\_klgh khhlghr_gb_ f_`^mV0 bV1 gZijbf_jV0=2s1.

Bgl_gkb\ghklv hl[hjZ b kj_^gyy ijbkihkh[e_gghklv ihimeypbb aZ\bkyl g_ lhevdh hl dhwnnbpb_glh\ hl[hjZ gh b hl qZklhl hl[bjZ_fuo Zee_e_c IhdZaZl_evgufb \ wlhf hlghr_gbb y\eyxlky ]jZnbdb aZ\bkbfhklb ∆ q = f(q)b Ws=f(q) k ihfhsvx dhlhjuo fh`gh hij_^_eblv wdklj_fZevgu_ b jZ\gh\_kgu_ mkeh\by jbk

DZd [ueh ihdZaZgh \ur_ ijb k\_jo^hfbgbjh\Zgbb b hl[hj_ ijhlb\ ]_l_jhab]hl

\hafh`gu khhl\_lkl\mxsb_ g_ljb\bZevgu_ j_r_gby

[s0 − q(s0 + s2 )] = 0 b (q − p) = 0 .

Z ijb hl[hj_ ijhlb\ ]_l_jhab]hl dh]^Z q = p = 0.5.

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AgZd \_ebqbgu ∆ Tihemq_gguc ijb agZq_gbyoT\[ebab lhqdb jZ\gh\_kby ZgZeb- lbq_kdb beb gZ hkgh\_ ]jZnbdZ iha\hey_l hij_^_eblv y\ey_lky eb jZ\gh\_kb_ mklhc- qb\uf beb g_mklhcqb\uf

GZijbf_j ijb k\_jo^hfbgbjh\Zgbb jbk jZ\gh\_kb_ mklhcqb\h_ lZd dZd hl[hj m\_ebqb\Z_l qZklhlm Zee_ey Z ijb agZq_gbyo qZklhl f_gvr_ jZ\gh\_kguo b gZh[hjhl mf_gvrZ_l ijb agZq_gbyo [hevr_ jZ\gh\_kguo DZd \b^gh ba jbk q_j_a hij_^_e_g- gh_ qbkeh ihdhe_gbc \ ihimeypbb mklZgZ\eb\Zxlky ihklhyggu_ agZq_gby qZklhl ]_gh- lbih\ b Zee_e_c J_amevlZl ^_ckl\by hl[hjZ fh`gh h[gZjm`blv lhevdh ijb mkeh\bb _keb ihimeypbx \u\_klb ba khklhygby jZ\gh\_kby Hl[hj \ ihevam ]_l_jhab]hl \hkklZ- gh\bl jZ\gh\_kgu_ qZklhlu WlZ jZ\gh\_kgZy ihimeypby \ mkeh\byo hl[hjZ hlebqZ_lky hl jZ\gh\_kghc ihimeypbb OZj^b <Zcg[_j]Z \ dhlhjhc kemqZcgh_ hldehg_gb_ hl jZ\- gh\_kby ijb\h^bl d ^j_cnm ]_gh\

< \ZjbZgl_ hl[hjZ ijhlb\ ]_l_jhab]hl jZ\gh\_kb_ g_mklhcqb\h_ jbk ?keb bk- oh^gu_ qZklhlu Zee_e_c g_ khhl\_lkl\mxl jZ\gh\_kguf p=T hl[hj aZ\_jrZ_lky eb[h iheghc ihl_j_c Zee_ey Z eb[h _]h khojZg_gb_f \ ]hfhab]hlghf \ZjbZgl_ ZZ)

jbk

Bgl_j_kgh gZjy^m k baf_g_gb_f qZklhl Zee_e_c ijhke_^blv aZ ih\_^_gb_f iZjZ- f_ljZ:VoZjZdl_jbamxs_]h kj_^gxx hlghkbl_evgmx ijbkihkh[e_gghklv ihimeypbb Ijbkihkh[e_gghklv klj_fblky d _^bgbp_ ih f_j_ m^Ze_gby hl[hjhf g_[eZ]hijbylgh]h Zee_ey b g_kmsbo wlhl Zee_ev ]_ghlbih\ \ i_j\uo lj_o jZkkfhlj_gguo \ lZ[ebp_ \Zjb-

ZglZo Ijb k\_jo^hfbgbjh\Zgbb ijbkihkh[e_gghklv fZdkbfZevgZ gh g_ jZ\gZ _^bgbp_ \ lhqd_ mklhcqb\h]h jZ\gh\_kby jbk < kemqZ_ hl[hjZ ijhlb\ ]_l_jhab]hl jbk Ws - fbgbfZevgZ \ lhqd_ g_mklhcqb\h]h jZ\gh\_kby

QZklhlZ Zee_ey Z T

39