Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://www.chronos.msu.ru/old/RREPORTS/vengerov_khronoartefakty.pdf
Äàòà èçìåíåíèÿ: Sat Dec 14 13:20:44 2013
Äàòà èíäåêñèðîâàíèÿ: Fri Feb 28 12:30:37 2014
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KemqZx iZjZ^hdkZ khhl\_lkl\m_l kblmZpby dh]^Z \b^bfuo hrb[hd g_l gh j_ amevlZlg_khhl\_lkl\m_lh`b^Z_fhfm DZd \b^gh ba jZaj_r_gb_ iZjZ^hdkZ khklhbl \ h[gZjm`_gbb \ kdju luohrb[hdZ b bebab
) [ LZdh_ hrb[hqgh_ bkihevah\Zgb_ [ b ) \f_klh [ b ) ghkbl km[t_dlb\guc
oZjZdl_j b g_ y\ey_lky g_ba[_`guf _keb h[t_dlb\gh kms_kl\m_l©ijZ\bevgZy


k\yavª :^_d\Zlguf wlhc kblmZpbb l_jfbghf y\ey_lky©Zjl_nZdlª ± ijh p_kk beb j_amevlZl g_ k\hckl\_ggu_ ^Zgghfm h[t_dlm h[t_dlb\gh b \hagb dZxsb_\oh^__]hbamq_gby[16]artefactum ­bkdmkkl\_gghk^_eZggh_ijb\e_ q_ggh_ eZl Wlhl l_jfbg rbjhdh bkihevam_lky \ [bheh]bb b ]_heh]bb Zjl_ nZdlu \ iZjZ^b]f_ ]hjghc l_iehnbabdb h[gZjm`_gu Z\lhjhf [29] OjhghZj l_nZdlZfbfugZau\Z_fZjl_nZdluk\yaZggu_kh\j_f_g_f[30] < ebl_jZlmj_ m^_ey_lky [hevrh_ \gbfZgb_ iZjZ^hdkZf h^ghfm ba \b^h\ Zjl_nZdlh\ \ l_jfh^bgZfbd_ [2,6,9,11,21,31] \ ^jm]bo jZa^_eZo nbabdb [27,32­39] \ fZl_fZlbd_ [24,25,28,40,41] Wlb iZjZ^hdku h[uqgh h[tykgy xlky beb©jZaj_rZxlkyª ± mdZau\Z_lky ijbqbgZ bo± [ beb ) dhlhjufb gZ ^h aZf_gblv bagZqZevgh ij_^iheZ]Z\rb_ky [ b ) qlh[u h[hkgh\Zlv ihemq_g gh_ \ \f_klhh`b^Z\r_]hky \ GZr ih^oh^ bghc fu mkljZgy_f Zjl_nZdlu dZd j_amevlZlu hrb[hqgu_ h[mkeh\e_ggu_ijb\g_k_ggu_ bkke_^h\Zl_e_f bkihevah\Zgb_f [ \f_klh [ bebb ) \f_klh ) l_f qlh gZoh^bf©bklbggu_ª [ b ) dhlhju_ h[_ki_qb \Zxl \ \ MkljZg_gb_ Zjl_nZdlh\ ^he`gh gZqbgZlvky gZ lhf `_ mjh\g_ iZ jZ^b]fu ]^_ hg \uy\e_g b lhevdh ijb g_\hafh`ghklb lZdh]h mkljZg_gby ©jZaj_r_gª i_j_oh^ gZ [he__ \ukhdbc b_jZjobq_kdbc mjh\_gvijbgpbi HddZ fZ Ki_pbnbdZ l_jfbgZ©Zjl_nZdlª aZdexqZ_lky \ _]h Zihkl_jbhjghklb Zgh fZebyijhlb\hj_qb_ fh`_l [ulv gZa\ZgZ Zjl_nZdlhf ebrv ihke_ __ mkljZg_ gby ?keb ZghfZebx mkljZgblv g_ m^Z_lky hgZ g_ y\ey_lky Zjl_nZdlhf Fu bk ihevam_f l_jfbg©Zjl_nZdluª Zijbhjgh ©ih ih^haj_gbxª Wlh hijZ\^Zgh bo ihke_^mxsbfmkljZg_gb_f VIjbf_juZjl_nZdlh\ Ijbf_ju ZghfZevguo j_amevlZlh\ ± ij_^iheh`bl_evgh Zjl_nZdlh\ ± ijb\h^ylky \ [30] b hlghkylky d h[hehqd_ iZjZ^b]fu l_jfh^bgZfbdb [_kdh g_qgZykdhjhklvl_iehi_j_^Zqbg_h]jZgbq_gghklvgZqZevguoihlhdh\l_ieZgZ ]jZgbpZo kbkl_f kdZqdb l_fi_jZlmj gZ ]jZgbpZo dhglZdlh\ \ khklZ\guo kbkl_ fZo [_kdhg_qgh_ \j_fy aZ\_jr_gby g_klZpbhgZjgh]h ijhp_kkZ G_dhlhju_ ba wlbo wnn_dlh\ g_h^ghdjZlgh ljZdlh\Zebkv dZd©iZjZ^hdkuª lj_[mxsb_ h[t ykg_gbyJZkkfhljbfbodjZldhZ\h\lhjhc]eZ\_±[he__ih^jh[gh : ;_kdhg_qgZy kdhjhklv l_iehi_j_^Zqb J_r_gb_f aZ^Zqb Dhrb ^ey mjZ\g_gby l_iehijh\h^ghklb \ h[eZklb 6 5 ^W ° `

5 5 5 = {[ - }


7 W

D

7 7 [

7 [ W [ 5 W 5+

kgZqZevgufmkeh\b_f

7 [


M [ [ 5




y\ey_lkynhjfmeZImZkkhgZ[42­44]

7 [ W = DW

-



H[S -




([

- ) G DW



]^_ D ±dhwnnbpb_gll_fi_jZlmjhijh\h^ghklbBa ke_^m_lqlh^eyex[hc iheh`bl_evghc \ h[eZklb [ °>F G @ b jZ\ghc gmex \g_ __ gZqZevghc nmgdpbb

iehi_j_^Zqb [28,45­50] Wlhl iZjZ^hdk fZl_fZlbdb©h[tykgyebª g_lhqghklvx nbabq_kdbo ij_^ihkuehd e_`Zsbo \ hkgh\_ \u\h^Z mjZ\g_gby l_iehijh\h^ ghklb [48,49] hlkmlkl\b_f mq_lZ bg_jpbhgghklb fhe_dme beb bo \aZbfh^_ckl \by [28,45,46,50] L_iehnbabdb mkfZljb\Zeb _]h ijbqbgm \ ijb[eb`_gghklb aZdhgZNmjv_[43,51­59]

M [ 7 [ W 7 [ W ! ^ey ex[h]h kdhev m]h^gh [hevrh]h _ [ _ b kdhev m]h^gh fZeh]h W ! Wlhl j_amevlZl b ljZdlm_lky dZd [_kdhg_qgZy kdhjhklv l_

T [ W

O

A^_kv T [ W ± iehlghklv ihlhdZ l_ieZ O ± dhwnnbpb_gl l_iehijh\h^ghklb Ml\_j`^Zehkv qlh kijZ\_^eb\h ebrv ^ey f_^e_gguo ijhp_kkh\ b fZeuo ]jZ^b_glh\ l_fi_jZlmju Z ^ey bgl_gkb\guo ijhp_kkh\ b [hevrbo ]jZ^b_glh\ g_h[oh^bfhbkihevah\Zlvbgu_g_ebg_cgu_\qZklghklbkhhlghr_gby K [_kdhg_qghc kdhjhklvx l_iehi_j_^Zqb k\yau\Zxl b ^jm]hc ZghfZevguc j_amevlZl± h[jZs_gb_ \ [_kdhg_qghklv kdhjhklb -i bahl_jfbq_kdhc ih\_jo ghklb \ gZqZevguc fhf_gl \j_f_gb Dhhj^bgZlZ wlhc ih\_joghklb [G GW iheZ]Z_lky jZ\ghcih ZgZeh]bb k \uqbke_gb_f ^bki_jkbb \ l_hjbb \_jhylgh kl_c [22]

7 [ W [



G W



[



0 W 0 Q W = 0 W



-



[ Q7 [ W G [



]^_ 0 Q W Q ± fhf_glu l_fi_jZlmjgh]h ihey± j_r_gby aZ^Zqb DhrbBa ke_^m_l

G W

DW -i

G GW GW

õDü é W ç



OLP -i
W



:gZeh]bqgu_ b ih nhjf_ b ljZdlh\d_ j_amevlZlu \ l_hjbb ^bnnm abbba\_klgudZd©iZjZ^hdku[jhmgh\Z^\b`_gbyb^bnnmabbª[23,51,60­62] Ki_pbnbq_kdbc ^bnnmabhgguc iZjZ^hdk©[_kdhg_qghc kdhjhklb jZa[_]Z gbyqZklbpª\^\mojZaebqguonhjfZojZkkfZljb\Z_lky\[61,62]<[61]Z\lhju ibrml © k jhklhf \j_f_gb djZy©h[eZdZª qZklbp jZkoh^ylky kh kdhjhklvx

-

K W a K K ijb K LZdbf h[jZahf \ ij_^_e_ fh]ml gZclbkv







qZklbpu i_j_f_sZxsb_ky aZ _^bgbpm \j_f_gb dZd m]h^gh ^Ze_dhª A^_kv K ± \_ebqbgZ kdZqdZ qZklbpu \ jZkkfZljb\Z_fhc fh^_eb©[em`^Zgby qZklbp gZ ijyfhcª Z ± ijh^he`bl_evghklv kdZqdZ Wlb \_ebqbgu k\yaZgu khhlghr_ gb_f K = ' ' ± dhwnnbpb_gl ^bnnmabb < [62] ml\_j`^Z_lky qlh kdh jhklv^bnnmabbhij_^_ey_lkynhjfmehc


9' = -

'



=

[ - [ W - W



]^_U±iehlghklvqZklbp [±dhhj^bgZlZlhqdbgZijyfhc 2[ \dhlhjhcijb W W [ue hkms_kl\e_g G \b^guc©\ijukdª qZklbp NhjfmeZ bgl_jij_ lbjm_lky dZd©iZjZ^hdk [_kdhg_qghc kdhjhklb jZa[_]Zgby qZklbpª hl lhqdb [ [ ijb _ [ - [ _ b ijhba\hevghf dhg_qghf W ° W Ijb\_^_ggu_ ijbf_ju± g_ iZjZ^hdku beb Zjl_nZdlu lj_[mxsb_ h[tykg_gby beb mkljZg_ gby Z y\gu_ hrb[db qlh [m^_l ^Ze__ ihdZaZgh Fh`gh [ueh [u ijb\_klb b ^jm]b_ih^h[gu_d\ZabZjl_nZdlu
7 W

D

7 7 [

7 [ W [ ° 5 W ° 5



7 [
bf__l\b^[43]

7

FRQVW [ ° 5 7 W
7F 7 [ W 7F 7

7F

FRQVW W ° 5


[ü 7F ! 7 HUI õ é DW ç Ba ^ey]jZgbqgh]hihlhdZl_ieZ T W ke_^m_l - [ W T W


7 O õ ü é [ ç [

O 7F 7 OLP T W = SDW W




ijh\h^ghklb \ khklZ\ghc kbkl_f_ 5 5 ]^_ [ ± lhqdZ dhglZdlZ ih [ b [ ! k jZaebqgufb l_iehnbabq_kdbfb iZjZf_ljZfb b emijhkljZgkl\ gZqZevgufb nmgdpbyfb [43,57­59,71] Kbg]meyjghklv gZqZevgh]h ]jZgbqgh]h ihlhdZfZkkuoZjZdl_jgZb^eyZgZeh]bqguo^bnnmabhgguoaZ^Zq[51,56,] Ijbf_ju ZghfZebc k\yaZgguo k nhjfmebjh\dZfb djZ_\uo aZ^Zq i_j_ghkZ mjh\_gv [ZabkZ iZjZ^b]fu beb k hkgh\gufb iheh`_gbyfb g_jZ\gh\_kghc l_jfh^bgZfbdbmjh\_gvy^jZiZjZ^b]fu \ ba\_klghc gZf ebl_jZlmj_ hlkmlkl \mxlaZbkdexq_gb_fdeZkkbq_kdbol_jfh^bgZfbq_kdboiZjZ^hdkh\[2,11,21])

Wlhl ©iZjZ^hdkª nZdlbq_kdb Zjl_nZdl qZklh ijb\h^blky \ ebl_jZlmj_ \ [22,51,56,57] \ qZklghklb :gZeh]bqgh ih\_^_gb_ T b \ aZ^Zq_ l_ieh

^

`




=E:<::JL?N:DLUH;HEHQDBI:J:>B=FU
IhdZ _s_ gbdhfm g_ m^Z\Zehkv mklZgh\blv ohly[uijb[ebabl_evghqlhlZdh_:jl_nZdl DKZcfZd©AZih\_^gbd]h[ebgh\ª ;_kdhg_qghklv ± wlh kljZgZ fZl_fZlbq_ kdbo nhdmkh\ LZf pZjkl\m_l ghev±qZjh^_c lZf mijZa^g_gu ex[u_ jZfdb b[h ghev lZd beb bgZq_\k_k\h^bldh^bgZdh\hfmmjh\gx GDZjmk

VDZghgbq_kdb_fh^_eb < gZklhys_c ]eZ\_ Zjl_nZdlu h[hehqdb iZjZ^b]fu jZkkfZljb\Zxlky im l_f ZgZebaZ j_r_gbc ijhkl_crbo fgh]hdjZlgh \klj_qZxsboky aZ^Zq Dhrb b djZ_\uo aZ^Zq ^ey h^ghf_jgh]h ebg_cgh]h mjZ\g_gby l_iehijh\h^ghklb± dZ ghgbq_kdbofh^_e_c[42­49,71,72] Fh^_ev ± AZ^ZqZ Dhrb ^ey h[eZklb 6 5 5 ijb G \b^ghf gZ qZevghfjZkij_^_e_gbbl_fi_jZlmju

7 [

M [

$G [ $

'4 6F9

7 /



A^_kv 4 = 6F9 7 / - dhebq_kl\h l_ieZ ih^\_^_ggh]h d lhqd_ [ ijb W F9 ± h[t_fgZy l_ieh_fdhklv kj_^u \ h[eZklb 5 6± iehsZ^v ihi_ j_qgh]h k_q_gby h^ghf_jghc h[eZklb 7 /± l_fi_jZlmjZ b©rbjbgZª wd\b \Ze_glgh]h G bklhqgbdm dhg_qgh]h l_ieh\h]h bfimevkZ J_r_gb_ aZ^Zqb Dhrb kmkeh\b_f

7 [ W =

[ 7 / H[S - DW DW

[ ° 5 W ° 5



Fh^_ev± AZ^ZqZ Dhrb ijb gZqZevghf jZkij_^_e_gbb l_fi_jZlmju lb iZ©l_ieh\hcbfimevkª

7 [


M [

í7 [ °> / / @ î [ ° / / > @ ï




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7 [ W

7

ê õ [ / ü õ [ / ü ø ëHUI é DW ç HUI é DW ç ý [ ° 5 W ° 5 ì ù

Fh^_ev± AZ^ZqZ Dhrb ijb gZqZevghf jZkij_^_e_gbb l_fi_jZlmju lb iZ©klmi_gvdZª

7 [
J_r_gb_bf__l\b^

M [

í7 [ ! 7 ! 7 î7 [ ï



7 [ W


7 7 7 7 [ü HU I õ é DW ç



Fh^_ev DjZ_\Zy aZ^ZqZ k ihklhygguf ]jZgbqguf mkeh\b_f I-]h jh^Z ^eyh[eZklb = 5 + â 5+ ijbihklhygghcgZqZevghcl_fi_jZlmj_

7 [
J_r_gb_bf__l\b^

M [

7 7 W = 7F > 7 [ü 7F 7F 7 HUI õ é DW ç



7 [ W



Fh^_ev± DjZ_\Zy aZ^ZqZ k ]jZgbqgufb mkeh\byfb IV]h jh^Z ^ey ^\mokehcghckbkl_fu 5 5 kihklhyggufbghjZaebqgufbgZqZevgufb l_fi_jZlmjZfb\h[eZklyo 5L

^



L

`



7 [

M [

7 [ ° 5 7 [

M [

7 7 [ ° 5

J_r_gb_aZ^Zqb^ey 7 [ W ^ey 7 [ W ZgZeh]bqgh bf__l\b^

7 + . 7 7 - 7 [ 7 [ W = + HUI D W + . + .



]^_ .H H H HL O L F9L DL O L F9L L Fh^_ev±DjZ_\ZyaZ^ZqZk]jZgbqgufbmkeh\byfbI-]hjh^Z^eyh[eZk [ ° / / kehc dhg_qghc lhesbgu =jZgbq lb 6 : 5 : gu_bgZqZevgZyl_fi_jZlmjuihklhyggughjZaebqgu

^

`

7 [ 7 / W
J_r_gb_lZdh\h

M [

7 7F

FRQVW [ °: FRQVW W ° 5 7F 7



7 / W




- [ W
]^_

7 [ W 7F 7 7F PQ



Q [ü FRVõ P Q H[S PQ ) PQ é / ç Q

ã









S Q )

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/







kbkl_fZ { }±g_pbdebq_kdZy

Fh^_ev± =jZgbqgZyklZpbhgZjgZy aZ^ZqZ k pbdebq_kdbfb ]jZgbq gufb mkeh\byfbIV]h jh^Z ^ey ^\mokehcghc kbkl_fu >: : @ L_fi_jZlm ju\kehyo± 7L [L [L L = {[L ( /L )}L >eyi_j\hcdjZ_\hcaZ^Zqb

7 [ [ = = 7 [ [ = = 7 7 [ [ / 7F7 [ [ /
L_fi_jZlmju 7L [L \h[eZklyo :L m^h\e_l\hjyxlmjZ\g_gbyf

7F

L

G 7L

]^_ I L [L ±nmgdpbbiehlghkl_cbklhqgbdh\l_ieZ\ :L Fh^_ev ± G_ebg_cgZy djZ_\Zy aZ^ZqZlbiZ Kl_nZgZ JZkkfZljb\Z xlky mjZ\g_gby \b^Z ^ey ^\mo nZa©l\_j^hcª b©`b^dhcª jZa^_e_gguo ih^\b`ghc ]jZgbp_c nZah\h]h i_j_oh^Z [ I [W k aZ^Zgguf gZ g_c mkeh\b _f

+ I L [L = L G [L





oh^Z^eydhlhjhcbaj_r_gbyaZ^Zqbke_^m_l[43]

7 7 G W - = [ [ = W [ [ =W GW ]^_ O L ± l_iehijh\h^ghklv nZa J ± iehlghklv l\_j^hc nZau V ± m^_evgZy kdjulZy l_iehlZ nZah\h]h i_j_oh^Z [W ± dhhj^bgZlZ njhglZ nZah\h]h i_j_ [ W E W E FRQVW


Nhjfmeu lbiZ oZjZdl_jgu b ^ey j_r_gbc [he__ keh`guo aZ^Zq lbiZ Kl_nZgZ[55,73] VAZ^ZqbDhrb Bkke_^h\Zgb_ Fh^_e_c ± kljhbf lZd Ih j_r_gbyf ± nmgdpbyf 7L [ W L b TL [ W = - ë

±ghf_jZfh^_e_c gZoh^bf nmgdpbb 3 [ W L

7L Ih\_^_gb_nmgdpbc 7L 3 TL bamqZ_fgZ]jZgbpZoh[eZklb L [

7L W


6 Z \gmljb __± ih\_^_gb_ 7L [ W < i_j\hf kemqZ_ \uqbkey_f h^ghdjZlgu_ ij_^_eu [ [ W W Z aZl_f ± ^\hcgu_ ij_^_eu dhf[bgZpbb h^ghdjZlguo Ihke_^gb_ gZoh^bf ihhq_j_^ghgZijbf_j \gZ qZe_ OLP 7L W Z aZl_f OLP 7L [ b h^gh\j_f_ggh dh]^Z OLP 7L [ W = OLP 7L [ W W = 7L wdklj_fmfiheZ]Zy [ ±iZjZf_ljhfZ W ±g_aZ\bkbfhcdhhj^bgZlhc bgZoh^bf nhjfmeu± hp_gdb wnn_dlb\ghc ehdZebaZpbb 7L [ W H^ghdjZlgu_ b ^\hcgu_ ij_^_eu ^ey [ b [ - g_ \uqbkey_f \\b^m bg\ZjbZglghklb j_r_ gbchlghkbl_evghij_h[jZah\Zgby [ [ GZoh^bfnmgdpbb 3 [ W L
[ W W W [

7L W

[ 7 / [ 3 [ W = - H[S - DW DW DW 3 [ W õ 7 ü è û é SDW ç





ê [ VKõ / [ ü / FKõ / [ ü ø H[Sõ [ / ü è û ë é DW ç é DW ç ý DW ì ù é ç

3 [ W

õ 7 7 è é SDW

ü õ [ ü û [ H[Sè DW û é ç ç



O

7L W [ [ H[S - DW DW


ë7 / T [ W = ÿ DW T [ W

õ O7 ü VKõ / [ ü H[Sõ [ / ü è û DW é SDW ç é DW ç é ç

[ ë(7 - 7 ) T [ W = H[S - DW ÿ DW >eynmgdpbc 7L [ W bf__f 7 [ W [





7 / 7 [ W [ = 7 [ W W = 7 [ W W = SDW



/ 7 [ W [ = 7 HUI 7 [ W [ DW 7 [ W W

7 [ W W



í7 [ / ÿ7 î [ / ÿ [ ! / ï 7



7 +7 7 [ W [ = 7 [ W W = 7 [ W [ 7 [ W W [ í7 ÿ 7 7 î [ ÿ7 [! ï



>eynmgdpbc 3 [ W ba ± ke_^m_l L

3 [ W [ = -

7 / 3 [ W [ = 3 [ W W = 3 [ W W = DW
H[S - (/ ) 3 [ W = 3 [ W W = [ DW

7/ 3 [ W [ = DW



[ / 3 [ W [ = [ = /



3 [ W [ = 3 [ W [ = 3 [ W W = 3 [ W W =
>eynmgdpbc TL [ W ba ± ke_^m_l

T [ W [ = T [ W [ = T [ W W = T [ W W = [ / T [ W [ = T [ W [ = T [ W W = T [ W = W : [ = / (7 - 7 ) T [ W [ = T [ W [ = T [ W = T [ W W = DW W
< ihke_^gbo ba khhlghr_gbc b nmgdpbb





7 õ [H / ü è SDW û é ç W


:H

:H [ W

íõ O7 ü ê [ / ü ø÷ ÿ ÿ H[ S ë õ H ýú îé ÿ SDW ç ýÞ ë é DW ç ùÿW ì ï

>ey ey:H [ W iheh`bf [H / 1 DW H 1 FRQVW H! Ihkdhevdmijb [ H / 1 Zijb [ H / 1 ! bf_ _f

<

OLP
õ 7 ü OLP é 1 W H ç W 1 + = OLP [ W W [ (/ )+

H ° í ÿ 7 î 1 H ÿ H ! ï = --

>ey:H [ W g_aZ\bkbfhhlagZdZ [ - / ihemqZ_f

(- 7 :± = OLP [ W = OLP H[S - 1 W = W _ [ _(/ ) W DW

(

)

)



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ey 7 [ W bf__f L

/ = / = / = / = / = / = / = / = / = / = / =



< hlebqb_ hl ^jm]bo ij_^_eu ijb [ W g_ kh\iZ^Zxl / / Ihwlhfm / \uqbkey_lky k ihfhsvx

[H W

1 DW H H!

/

í H ° î H ï


+

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_ 1 _

/

/

/

7 /

/

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/ / / /
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/ /
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/ /

1 ! [H / 1 [H /

7 ô < ô < 7 7 ( + = ( + HUI _ 1 _) ô = ( - = ( - HUI _ 1 _) ô = 7 ô> 7 ô >



>ey 7 [ W ihemqZ_f

/

7 [ + 7 7 / = / 7 [ -

7 7

7 +7 / = / / / /

/
/

/
7 7



H ° ÿ 7 7 õ 7 7 ü í îHUI_ 1 _ H éçÿ ï H !

ij_^_eu g_ aZ\bkyl hl ihjy^dZ bo \uqbke_gby b kh\iZ^Zxl k ^\hcgufb \u qbke_ggufb ijb h^gh\j_f_gghf i_j_oh^_ [ D W E [ [ H W \h \lh jhc ]jmii_©ZghfZevghcª ^\hcgu_ ij_^_eu aZ\bkyl hl ihjy^dZ i_j_oh^h\ jZaebqguf_`^mkh[hcbg_kh\iZ^Zxlk^\hcgufbh^gh\j_f_ggufb >ey nmgdpbc 3 [ W b TL [ W ©ghjfZevgu_ª ij_^_eu himkdZ_f Z L ©ZghfZevgu_ª^Ze__ijb\h^bf

- 7 + 7 7 - 7 / = + â HUI _ 1 _ = > Ba ± ke_^m_l qlh _klv ^\_ ]jmiiu \_ebqbg /LM DE \ i_j\hc ]jmi i_©ghjfZevghcª /LM DE /LM D E /LM D E l_ ^\hcgu_ ihhq_j_^gu_

/

/

/

í H ° ÿ î H ÿ H ! ï

/ /

/ / / =

/ / = +

- >

/

/

1 7 - 7 =



/

/

/ =



- > -


1 / 7 = D / / = :± í H ° î H ï


/ /

/ /

/

/

/

A^_kv < b : khhl\_lkl\mxl b Iheh\bgZ ^\hcguo ij_^_eh\ ijb ihhq_j_^ghf \uqbke_gbb kh\iZ^Zxl h^gZdh g_ kh\ iZ^Zxlk ^\hcgufb h^gh\j_f_ggufb M ^jm]hc iheh\bgu ihhq_j_^gu_ ij_^_eu g_ kh\iZ^Zxl gb ^jm] k ^jm]hf gb k h^gh\j_f_ggufb < ihke_^gbo \u[bjZy khhl\_lkl\mxsb_ H b 1 ^ey /LM b / / fh`gh ihemqblv ex[u_ gZi_j_^ aZ^Zggu_ agZq_gby \ bgl_j\Ze_ beb ± LZdbf h[jZahf m jZkkfhlj_gguo nmgdpbc \ lhqdZo [ b [ / bf_xlkyjZaju\u JZkkfhljbf ih\_^_gb_ nmgdpbc 7L [ W \h \gmlj_ggbo lhqdZo h[eZklb 6 Bamkeh\bywdklj_fmfZnmgdpbc 7L [ W ih W [ b]jZ_ljheviZjZf_ljZ

7L W
ihemqZ_fiheZ]Zy [

3L [ W





[ [ 3 [ W = - H[S - = [ ! W ! DW DW AgZq_gb_ W W _klv lhqdZ fZdkbfmfZ nmgdpbb 7 [ W b k\yaZgh k dhhj^bgZ lhc [ [ ° ke_^mxs_cba aZ\bkbfhklvx

[ 7 / DW

W W

DW FZdkbfZevgh_\lhqd_ [ [ agZq_gb_ 7 [ W ^hklb]Z_lkyijb W W bjZ\gh õ 7 / ü PD[ 7 [ W 7P [ [ W °6 SH é [ ç Ijb W W l_fi_jZlmjZ\lhqd_ [ [ \hajZklZ_lZijb W ! W m[u\Z_l lZd qlh kh\iZ^Zxs__k hp_gdhc wnn_dlb\ghc ehdZebaZpbb l_fi_jZlmjgh]h ihey

[


^Z_l dhhj^bgZlm [ ± ehdZevgh]h fZdkbfmfZ l_fi_jZlmju >ey 7 [ W bamkeh\by 3 [ W ke_^m_l

/[ / /[ [ VKõ ü FKõ ü é DW ç é DW ç
Iheh`b\ [ / K!

[ ! W !



K DW /

) ba ihemqZ_f ^ey [ ! /


K K WKõ ü é ) ç

?keb j_rblv mjZ\g_gb_ lh ihemqbf K K K ) beb \ha\jZsZykv di_j\hgZqZevgufi_j_f_gguf [ = [ W Ij_h[jZam_f d\b^m

K )

õ ü K û OQè è û K è û é K ç )

]

/ ] [



bj_rZ_f_]hijb[eb`_gghjZaeZ]Zyeh]Zjbnf

õ ] ] ] ü ] è û é ç )



Ba ijb ] ke_^m_lmjZ\g_gb_]hijb[eb`_gby
]







Imklv D hagZqZ_lqlh D Lh]^Z kijZ\_^eb\h^ey

[ /
J_r_gb_



([ )

= DW

l_kh\iZ^Z_lk Ba b ihemqZ_f

/ W D
MjZ\g_gb_]hijb[eb`_gby



)

õ ] ü ] è û ç é






ke_^m_lba ijb\uiheg_gbbmkeh\by

]

] ]





hldm^Zke_^m_lmlhqg_gb_hp_gdb

[ /



[



õ / DW / DW # DW è é DW

>

@

ü û ç



Ihke_^g__ khhlghr_gb_ \uihegy_lky k lhqghklvx \ K lhqghklvx \ fh`ghaZibkZlv

[



# DW
/ D



ijb

W





Nmgdpby 7 [ W fZdkbfmfZg_bf__llZddZd 3 [ W \ghevgb]^_\g_ h[jZsZ_lkyH^gZdh 3 [ W bf__lwdklj_fmf\lhqd_ [ [ hij_^_ey_fuc bamkeh\by 3 [ [ = DW l_\gh\vihemqbeboZjZdl_jgmx^ey\k_oaZ^ZqDhrbaZ\bkbfhklv Hp_gdb wnn_dlb\ghc ehdZebaZpbb nmgdpb_c lbiZ [m^_f gZau\Zlv Zijbhjgufb hp_gdZfb lZd dZd bo fh`gh ihemqblv [_a bgnhjfZpbb h gZqZev ghfmkeh\bbbj_r_gbbaZ^ZqbDhrbbkihevamyebrvij_^iheh`_gbyhihklh ygkl\_ gme_\h]h fhf_glZ l_fi_jZlmju 0 W FRQVW b kdhjhklb m[u\Zgby l_fi_jZlmjgh]hiheyijb_ [_

[[7

7 [ W ] = [ - [

= -





H[hkgh\Zlv fh`gh lhevdh Zihkl_jbhjgh l_ ihemqb\ y\guc \b^ j_r_ gby h^gZdh \ [hevrbgkl\_ kemqZ_\ ^_ckl\bl_evgh bf__l f_klh Mkeh\b_ 0 = FRQVW kijZ\_^eb\h ^ey aZ^Zq Dhrb ijb hlkmlkl\bb \ kbkl_f_ \gmlj_g gbo bklhqgbdh\klhdh\ l_ieZ < ijhlb\ghf kemqZ_ dZd b \ djZ_\uo aZ^ZqZo 0 0 W b hp_gdm ihemqblv g_evay Ba mkeh\bc b kihkh[Z ihemq_gby


hp_gdb [ ke_^m_lqlhhgZh^bgZdh\Z^ey\k_oaZ^ZqDhrb

[

G W

DW



:ihkl_jbhjgu_hp_gdbl_bkihevamxsb_bgnhjfZpbxhj_r_gbbaZ^Zqb Dhrb gZijhlb\ bg^b\b^mZevgu GZc^_f bo ihimlgh mlhqgb\ kfuke Zijbhj guo hp_ghd b bo fh^bnbdZpbc± hp_ghd©ª \\_^_gguo ih ZgZeh]bb k l_hjb_c\_jhylghkl_c

G V
Ih^klZ\bf [

GW

D W

DW



G

DW \7 [ W bihemqbf 7 W 7 W 7 / = SDW 7 W 7G 7 W # W
V



Ijb [

G V

DW gZoh^bf


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7 G W 7

W W = / D



Hp_gdZ W = / + DW ijbih^klZgh\d_\ ^Z_l

7 G V W ¹ W ! 7
>ey 7 [ W ihemqbf



7 G W

7 7 7 7 G V W

7 ¹ 7 7

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Zihkl_jbhjgu_ g_klZpbhgZjgmx b klZpbhgZjgmx hp_gdm khhl\_lkl\_ggh A^_kv 7 ± l_fi_jZlmjZ \ lhqd_ [ \ fhf_gl \j_f_gb W = W / = = / D / D GZoh^bf

[

[G lZdbf qlh 7 [G W 7

Hp_gdb [G b [G ij_^klZ\eyxl kh[hc

[ DW [G W G [ /


DW



W W [ = - OQ õ G ü W W é / ç W / / / Ijb W W / W W / I nhjfmeZ hibku\Z_l ^\b`_gb_ l_fi_jZlmjgh]h ©njhglZª gZijZ\h \ klhjhgm [ ! Nmgdpby [G / \hajZklZ_l kh \j_f_g_f ^hklb]Zy fZdkbfmfZ ijb W W I Z aZl_f ijb W W / I W W / W W / 6 ± m[u
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[ W I õ W ü # õ W ü # [ = / éW/ ç I éW/ ç 6 / PD[ >ey nmgdpbb 7 [ W hp_gdm [G W hij_^_ey_f ba mkeh\by 7 [G W 7 / W Ba ijb[eb`_gghgZoh^bf [G [G W # / DW Lh]^Z
>eyhij_^_e_gby [ W ihlj_[m_f 7 [G W 7

[G W # / DW W !
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lh


[ # / DW W ! G

< kemqZ_ 7 [ W hp_gdb [G W g_l Z [G hij_^_ey_lky ba mkeh\by 7 [G W 7 7 7 dhlhjh_^Z_l

[G
Ijbmkeh\bb 7 [ W G

DW W ! DW W !



7 7 7 ihemqZ_f [ G





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klZpbhgZjgu_ ke_^mxl ba mkeh\bc hkeZ[e_gby ihey gZ©njhgl_ª \ jZa Z ijbm`_klhq_gbbwlh]hmkeh\by\ fh^bnbpbjmxlkyg_kms_kl\_ggh :ihkl_jbhjgu_ hp_gdb dZd b Zijbhjgu_ ijhihjpbhgZevgu W qlh ijb\h^bl d ZghfZebb± [_kdhg_qghc gZqZevghc kdhjhklb ^\b`_gby l_fi_jZ lmjgh]h©njhglZª

jhflZd`_ [ DW Hp_gdb wnn_dlb\ghc ehdZebaZpbb l_fi_jZlmjgh]h ihey ih^jZa^_eyxl ky gZ Zijbhjgu_ b Zihkl_jbhjgu_ I_j\u_ h^bgZdh\u ^ey \k_o aZ^Zq Dhrb GW DW G V W DW ijbq_fwnn_dlb\gZlhevdhhp_gdZG V W :ihkl_jbhjgu_ hp_gdb h[hbo \b^h\ [G W g_klZpbhgZjgu_ b [G W




VDjZ_\u_aZ^Zqb Ijb bkke_^h\Zgbb djZ_\uo aZ^ZqdZghgbq_kdbo fh^_e_c ± mqblu\Zy [Oevr__ \ebygb_ bo ki_pbnbdb q_f m aZ^Zq Dhrb h[s_]h ih^oh^Z dZd \ V gZf_qZlvg_[m^_f Fh^_evBaj_r_gby ihemqZ_f

7 -7 3 [ W = F DW

[ H[S - [ DW





kmlkl\b_ ZghfZebc >\hcgu_ ij_^_eu \_^ml k_[y ZgZeh]bqgh ij_^u^msbf fh ^_eyfaZ^ZqbDhrb

[ (7 - 7 ) T [ W = F H[S - DW DW >ey nmgdpbb 7 [ W \uqbke_gb_ h^ghdjZlguo ij_^_eh\ ^_fhgkljbjm_l hl

/ /
/ 7. /

/ /
7 /

/ /

7 7F


H ° í7 ÿ î7. 7. 7 HUI 1 H ÿ H ! ï7.



/

7 /

7.

/

H ° í7 ÿ. 7. 7. 7 HUI 1 H î ÿ7 H ! ï

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[G # DW



Fh^_ev Ih\_^_gb_ nmgdpbc 7 [ W 3 [ W T [ W gZ ]jZgbpZo ZgZ eh]bqgh ih\_^_gbx khhl\_lkl\mxsbo nmgdpbc \ fh^_eyo b

\_lkl\m_l jZkkfhlj_gbx ijhf_`mldh\ \j_f_gb W / D ihwlhfm \k_ ij_^_ euijb W mfh^_ebkh\iZ^ZxlklZdh\ufb\fh^_eyobIhwlhfmh]jZ gbqbfky jZkkfhlj_gb_f kemqZ_\ _ [_ W _ [_ / W >ey [hevrboagZq_gbc W j_r_gb_ klhqghklvx±aZibr_f\\b^_

lb 7 [ W baf_gy_lky fhghlhggh Z 3 [ W bf__l wdklj_fmf \ dhlhjhf DW :ijbhjghc hp_gdb ehdZebaZpbb g_l lZd dZd 0 0 W :ihkl_ [ jbhjgZyklZpbhgZjgZyhp_gdZehdZebaZpbbkh\iZ^Z_lk Fh^_ev ?keb aZnbdkbjh\Z\ e_\mx ]jZgbpm h[eZklb : [ / m^Zeblv ijZ\mx [ = / gZ [_kdhg_qghklv lh ihemqbf Fh^_ev Wlh khhl

a a 7 [ W 7F + (7 - 7k ) [ W T [ W ) DW /

õ S ü S[ü ) û FRVõ H [ Sè S é / ç é ç





W / D







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[ °:

: klZpbhgZjgh]h khklhygby k l_fi_ \j_fy 7F 7 [ Ijb [ / T6 ( L0 2 , t ) = min T6 ( x, t ) =
x

m]h^gh fZehf i_j_iZ^_ l_fi_jZlmj

7F Kmq_lhf ihemqZ_f

P7 = PD[ 7 [ W - PLQ 7 [ W = 7 W - 7F [ [ 7 = OLP P7 = 7 - 7F = 7 - 7F W P7 7 W - 7F a ) = = W = H[S - 7 7 - 7F



J_amevlZlujZkq_lh\ihihke_^g_cbanhjfme ijb\_^_gu\lZ[ebp_ LZ[ebpZ

>bgZfbdZ[_ajZaf_jghcl_fi_jZlmju

a T W )

±

±

±

±

±





Mf_gvr_gb_ T W gZ ihjy^hdhl T ± ^h T ± h[_ki_qb\Z_l kydZd\b^ghbalZ[ebpu\hajZklZgb_f\j_f_gbijhp_kkZ\ a jZaZgh^hklb`_gb_dhg_qgh]hagZq_gby T lj_[m_l W Fh^_ev JZkkfhljbf \gZqZe_ Ix djZ_\mx aZ^Zqm ^ey ^\mokehcghc kbk l_fu ^: : ` =jZgbqgu_ mkeh\by ± Z mjZ\g_gb_ l_iehijh\h^ghklb± J_r_gb_bf__l\b^

a

a

a

7 [ 7 [

õ7 7 I/ 7 è F O é /

ü I [ û [ O ç



dh\l_ieZgZ]jZgbp_dhglZdlZ

õ7 7 I/ü I 7 è F û [ [ / O ç O é ]^_ l_fi_jZlmjZ gZ dhglZdl_ keh_\± 7 ± g_ba\_klgZy \_ebqbgZ >ey hij_^_e_ gby 7 bkihevam_lky \lhjh_ ba ]jZgbqguo mkeh\bc IV]h jh^Z± kde_cdb ihlh T [


[

T [

[





TL

O L

7L L [L





Ih^klZgh\dZ\ \ujZ`_gbc b ^Z_l

7

õ è é

O O ü û / / ç



õO 7 è F é /

ü O I / I / 7 F û / ç



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T [ [ = / + T [ [ = / =



?keb \ ih^klZ\blv j_r_gby b ihemqbf mjZ\g_gb_ dhlhjh_ kh\f_klgh k y\eyxsbfky ke_^kl\b_f h[jZam_l kbkl_fm ^\mo Ze ]_[jZbq_kdbo mjZ\g_gbc hlghkbl_evgh ^\mo g_ba\_klguo± 7 b 7F Hij_^_eb l_evwlhckbkl_fu '




'

D D

D D

D

O O / /



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fh`_lh[jZsZlvky\lZddZd T a

^\b`_gby njhglZ [ W a W kdhjhklv njhglZ \ gZqZevguc fhf_gl \j_f_gb lZd`_h[jZsZ_lky\ ey dhg_qguo h[eZkl_cdZd b ^ey [_kdhg_qguo i_j_oh^ hl gZqZevghc l_fi_jZlmju 7 d dhg_qghc 7F lj_[m_l [_kdhg_qgh]h \j_f_gb ijb ex[hf 7 7F ! >ey aZ^Zqb k pbdebq_kdbfb ]jZgbqgufb mkeh\byfb hlkmlkl\m_l klZpbh gZjgh_ijb W j_r_gb_ < aZ^Zq_ Kl_nZgZ bf_xlky ZghfZebb ZgZeh]bqgu_ lZdh\uf ^ey ebg_c guoaZ^ZqDhrbbdjZ_\uo jZkkfhlj_gguojZg__

LZddZdaZdhg W




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fZdjhkdhibq_kdh_©[_kdhg_qgh fZeh_ª l_ nbdkbjm_fh_ k ihfhsvx fZdjh kdhibq_kdbo ijb[hjh\ baf_g_gb_ wlhc \_ebqbgu ) ijb i_j_oh^_ kbkl_fu ba

G X 3 G - P G 1 $ G D \ dhlhjhf agZd ^bnn_j_gpbZeZ G ) m nb]mjbjmxsbo \_ebqbg ihgbfZ_lky dZd

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LN ±©\yadbcª l_gahj gZijy`_gbc < e_\Zy qZklv ihke_ ^_e_gby gZ U7 _klv ihegZy km[klZgpbhgZevgZy ijhba\h^gZy m^_evghc wgljhibb ih \j_f_gb ijZ\hc qZk lb fh`_l [ulv ijb^Zg \b^ kmffu ^b\_j]_gpbb ihlhdZ wgljhibbk h[jZlguf agZdhf b nmgdpbb iehlghklb bklhqgbdh\ wgljhibb lZd qlh ijbf_l \b^ < [17] ba mjZ\g_gby ^Ze__ ihemq_gh mjZ\g_gb_ l_iehi_j_ghkZ \ g_ k`bfZ_fhc `b^dhklbNmjv_Dbjo]hnZ Kh\iZ^_gb_ M;W k mjZ\g_gb_f l_i ehi_j_ghkZ h[gZjm`b\Z_lky b ^ey [_a^bkkbiZlb\ghc k\_jol_dmq_c `b^dhklb [17]

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AZl_f \ ih^klZ\eyxlky qZklgu_ [ZeZgku lbiZ b hkms_kl\ey_lky kjZ\g_gb_e_\uobijZ\uoqZkl_cwlbo^\momjZ\g_gbchlghkbl_evghwgljhibb
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LZdbf h[jZahf 'V T 'V T Q l_ ihlhdh\Zy wgljhiby aZ\bkbl hl oh^Z ijh p_kkZqlhijhlb\hj_qbll_jfhklZlbq_kdhfmhij_^_e_gbxwgljhibbIhlhdh\Zy wgljhiby hij_^_ey_fZy q_j_a g_ij_ju\gh baf_gyxsmxky l_fi_jZlmjm ^bkkb a a iZlhjZ 70 70 W g_ kms_kl\m_l qlh kha^Z_l ^be_ffm eb[h ihlhdh\hcl_ kms_kl\_ggh l_jfh^bgZfbq_kdhc wgljhibb \ hlebqb_ hl l_jfhklZlbq_kdhc Zddmfmebjm_fhc g_ kms_kl\m_l eb[h hgZ kms_kl\m_l gh lhevdh ijb ^bk dj_lghfbaf_g_gbbl_fi_jZlmju IjbgbfZy \lhjmx lhqdm aj_gby jZkkfhljbf \hafh`ghklv kh]eZkh\Zgby 'V $ b 'V T Ij_^iheZ]Z_fqlhl_fi_jZlmjZ^bkkbiZlhjZbaf_gy_lky^bkdj_l ghl_ijb W W F ijhbkoh^blkdZqhd 7 76 W F °> W @ Bf__f




7 [ F ) a 70 = 76 [F ]
Lh]^Zba b ke_^m_l



T dT A = VM cV = S0l0cV 0 T T0 0 T0

TS



, T0 T c . 0

q =
IjbjZ\gb\Zy 'V $

4S 4S0 TS - T0 d = 0 T l l0 0 0 0





c





'V T ihemqZ_f
WF W < W

ijb ebg_cghf baf_g_gbb 70 W d W W U >bkdj_lgh_ kdZqdhh[jZagh_ baf_g_gb_ l_fi_jZlmju ^bkkbiZlhjZ iha\hey_l ©e_]blbfbabjh\Zlvª l_jfh^bgZfbq_kdmx wgljhibx h^gZdh baf_gy_l _]h ojh gh]_g_a Ojhghbwgljhih]_g_jZpby^bkkbiZlhjh\ Ijh^he`bf gZqZlh_ \ ij_^u^ms_f iZjZ]jZn_ bkke_^h\Zgb_ ojhgh b wg ljhih]_g_jZpbb^bkkbiZlhjZfb\gZqZe_keZ[ufbZaZl_fbkbevgufb :KeZ[uc^bkkbiZlhj AZ^ZqZ . fh^_ev ^bkkbiZlhjZ JZkkfhljbf ij_^_evguc i_j_oh^ d g_ ij_ju\ghcfh^_eb^bkkbiZlhjZdh]^Z '4 G 4 '4 G 4 Bf__f


O Q D





a

G 4

a 6OF9 G 70 '4
a 70 a 70 W

6 T W G W 6

IjbjZ\gb\Zy ^bnn_j_gpbZeu ihemq_ggh]h G 4 b i_j_^Zggh]h G 4 l_i eZihemqZ_fmjZ\g_gb_

G 76


GW W W

O D




Bgl_]jbjmy gZoh^bf

a 70 = 7 + 76 - 7 [ - H[S(- W )] a 70

a 7 OLP 70 W
W

76



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6 bf__f

76 G7 ]^_ G7 ±fZeZyih]j_rghklvaZf_gu 76 gZ 7
6 Ba

a 70 W
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IheZ]Z_f qlh W khhl\_lkl\m_l ' i_jbh^m W Q
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kl\m_l ' i_jbh^m W # W Ihkdhevdm ih]j_rghklv G7 H '7 fZeZ lZddZdfZehb '7 fh`gh kqblZlv qlh W W W W U ^Z_l m^h\e_l\h jbl_evgh_ kh]eZkh\Zgb_ ' b . fh^_e_c < wlhf aZdexqZ_lky \lhjhc Zj]m f_gl \ ihevam \\_^_gby kh[kl\_ggh]h \j_f_gb ^bkkbiZlhjZ W W U l_ ' i_jbh^Zhij_^_ey_fh]hijb
Ijb G7 H '7 '7 bf__f W W W W U l_ kh\iZ^_gb_ kh[kl\_ggh]h \j_f_gb keZ[h]h ^bkkbiZlhjZ k©nbgblbabjh\Zggufª \j_f_g_f W khhl\_lkl\mxsbf i_j_oh^m 7 7
6 76 G7 76 '7 \ . fh^_eb M\_ebq_gb_ lhqghklb©nbgblbaZpbbª gZ ^\Z ihjy^dZ l_ mf_gvr_gb_ ih]j_rghklb G7 76 7
6 ^h \_kvfZ fZeh]h agZq_gby G7 '7 khhl\_l




AZ^ZqZ G_klZpbhgZjguc ^bkkbiZlhj LZdh\uf [m^_f gZau\Zlv h^gh f_jguc ^bkkbiZlhj \k_ iZjZf_lju dhlhjh]h O F9 O baf_gy_lky kh \j_f_ g_f±\bjlmZevguf\j_f_g_f W °> W @ Wlhbaf_g_gb_hibku\Z_fZgZeh]bqgh a baf_g_gbx 70 W gh k g_dhlhjufb ^jm]bfb agZq_gbyfb bg^_dkZ ojhghg_eb g_cghklb

O F9 O

O W

W O O6 O õ ü D °> é W ç E W F9 W F9 F96 F9 õ ü E °> é W ç W O O 6 O õ ü é W ç
J

D

O W

J °>

<\_^_fh[hagZq_gby

÷ ÿ ÿ ÿ ú ÿ ÿ ÿ Þ



'OF96 F9 'O 'F H O 9 O F9 < kbem fZehklb W [m^_f kqblZlv qlh H \\ujZ`_gb_^ey '4 ihemqbf

O6 O

'F9 O 6 O 'O 'O HO H F O O H F H O Ih^klZ\eyy


a 4 = 4 = 6 O F9 G 70








]^_ 70 W aZ^Zgh kh]eZkgh HimkdZy g_keh`gu_ ij_h[jZah\Zgby gZoh ^bf '4 W 6OF9 '7 ¹ ) D E J

a

( ) = + µ( Q )O + µ( Q )F
]^_ P D Q kbiZlhjm



Q D Q PE Q
4
+

Q E Q L_iehijblhdba;hj]ZjlhgZd^bk
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= 6 T T
6

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O W a 7 70 W O W 6

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J Q HO Q D Q HO






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Q D D Q

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= a Q = QW ( Q )W ) D E J Q



) D E J ) D E J

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)
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= +
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Q ba ke_^m_l
) õ H O H F ü õ H O H O ü é çé ç


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PD Q °>@PE Q °>@Q J Q °>@QD Q °>@ ) °>) ) @ ) °>) ) @ ) ) ) H O ) HO HF H H H O ) °> a a@ a = + O + F

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O 7 = O + O è F9 7 = F9 + F9 è ë7 = ë + ë è



A^_kv D E J ± ijhba\hevgu_ iheh`bl_evgu_ ihklhyggu_ Z T jZaf_jgZyl_fi_jZlmjZ^bkkbiZlhjZ

T W ± [_a

a 70 - 7 è = T W °>@ 76 - 7
Ba ke_^m_lqlh

3

T W

M Q K

KQ K

W W



Ih^klZ\b\ \ gZoh^bf

O 7 = O + O F9 7 = F9 + F9 ë7 = ë + ë
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W

WQQ


)D E Q ) D J Q

]^_ ) L L

hij_^_e_gu ih b Ijb Q á = áö = ä = ä b j_r_gby aZ^Zq b iheghklvx kh\iZ^Zxl l_ ojhgh]_g_jZpby =ö

g_ebg_cgh]h^bkkbiZlhjZlh`^_kl\_ggZlZdh\hc^eyg_klZpbhgZjgh]h^bkkbiZ lhjZWlhy\ey_lkylj_lvbfZj]mf_glhf\u^_e_gghklbkemqZy Q AZ^ZqZ Wgljhih]_g_jZpby ^\mo g_\aZbfh^_ckl\mxsbo ^bkkbiZlh jh\Ihf_klbf\;hj]Zjlhg^\Zh^bgZdh\uo^bkkbiZlhjZkjZaebqgufb l_fi_ jZlmjZfb 7 76 '7 7 76 '7 JZkkfhljbf i_j_oh^u 7 76 b 7 76 \ ^bkkbiZlhjZo 0 b 0 Wgljhiby 0 [m^_l \hajZklZlv


'V ! Z 0 ± mf_gvrZlvky 'V GZk bgl_j_km_l ihegh_ baf_g_ gb_ wgljhibb \ kbkl_f_©;hj]Zjlhg±^bkkbiZlhjª Hij_^_ey_f l_jfhklZlbq_ kdmxZddmfmebjm_fmx wgljhibx



$

= 6OF9

7 G7 =6OF9 OQ 6 7 7 7

76









$

= 6OF9

7 G7 = - 6OF9 OQ 7 7 6 7

76





Ihkdhevdm baf_g_gb_ wgljhibb ;hj]ZjlhgZ jZ\gh gmex ih hij_^_e_gbx ihe gZywgljhibykbkl_fu[m^_l
6 'V$ 'V$ 'V$

õ 7 6OF9 OQè 6 é 77

ü û ç



6 Mkeh\b_ 'V$ [m^_l kh[ex^_gh _keb 7 77 Ijh\_jbf ihke_^ 6 g__g_jZ\_gkl\h





7 6 77

7 6 76 '7 76 '7



7 '7
6

7 6





6 ! 'V$ !

Ih\lhjy_fjZkq_l^ey^bkdj_lghcihlhdh\hcl_jfh^bgZfbq_kdhc wgljhibb
'V$ 6 'V$

'7 6OF9 õ ü 'V$ é 7 ç
'V$ 'V$

'7 ü 6OF9 õ é 7 ç


ü ! 6OF9 '7 õ é 7 7 ç

ihkdhevdm 7 76 '7 7 '7 7 '7 ! LZdbf h[jZahf h[Z f_lh^Z jZkq_lZ ijb\h^yl d h^ghfm j_amevlZlm h[s__ baf_g_gb_ wgljhibb kbk l_fu ©;hj]Zjlhg±^bkkbiZlhjuª ± iheh`bl_evgh qlh y\ey_lky nZdlbq_kdb \u\h^hf \lhjh]h gZqZeZ l_jfh^bgZfbdb \ deZkkbq_kdhc nhjfmebjh\d_ gZ hk gh\_jZg__ijbgylhcwd\b\Ze_glghc_c

lby l_fi_jZlmju ^bkkbiZlhjZ gZ '7 l_ bgl_j\Zeu \j_f_gb W N g_h[oh^b fu_^eyi_j_oh^Z 7N 7N N 1 Bf__f

7 7N
a 7N W

7 '77

7 '7

7 '7 76


7 N'771

7 1'7

7N '7M Q W W N Q N = Q = Q '4N 6OF9 '7 '4


E_\ZyqZklvM>kf kh\iZ^Z_lklZdh\hc^eykeZ[h]h^bkkbiZlhjZ IjZ\ZyqZklvaZibku\Z_lkyZgZeh]bqgh ]^_

'4 N T
+ N

6 T W N N O a 7 7N W O 6

= N






N

a a T + ( )G T W N N


>

@



Ih^klZ\b\ \ gZoh^bf

T

+ N

7 Q - = O 7 (1 - N ) + - G = O (1 - N ) + Q Q
[

(

)]

[

]

IjbjZ\gb\Zy b kmq_lhf ihemqbf

W N Q

NhjfmeZ hibku\Z_l ojhgh]_g_jZpbx kbevgh]h ^bkkbiZlhjZ ijbq_f iZ jZf_lju
>

O

WQ




. N

= . + . N N- N

1




A^_kv\_jogbcbg^_dk© . ªh[hagZqZ_l . fh^_ev^eydhlhjhckh]eZkgh

7N

7 76 7 H[S W .6 W . = W OQ + N N 1 -N

>



@

>ey ' fh^_eb ij_h[jZamy ihemqZ_f h[hagZqZy lbi fh^_eb \_jogbf bg^_dkhf



' N

QW W - - = = - Q + Q (1 - N )Q + 1 - N 1 - N 1 - N

+ - - N (1 - N ) (1 - N ) >eyijb[eb`_ggh]hjZ\_gkl\Z\j_f_g N ]hi_jbh^Z\ ' b . fh^_eyol_^ey \uiheg_gby mkeh\by W . # W ' g_h[oh^bfh \ jy^Zo b m^_j N N `Zlvlhevdhi_j\u_^\Zqe_gZ iheh`blv \b_ [m^_l kh[ex^_gh _keb (1 - N

Mkeh\b_f jZaeh`_gby \ y\ey_lky lj_[h\Zgb_


-





- N ) Wlh ^Z_l 1 N BlZd ^ey \k_o N 1 - b Q W . # W ' qlh ^Z_l q_l\_jluc Zj]mf_gl \ ihevam \u N N [hjZagZq_gby Q H[jZlguc i_j_oh^ hl ' fh^_eb b . fh^_eb we_f_glZj_g \ ^_eZ a a a xlky aZf_gu W N G W '7 G 7 1 N 76 7 W G 7 7N 7 W
-

( ((1

))

((

)

))

<< qlh dhgdj_lbabjm_f \ \b^_

qlh^Z_l

GW W

a G7 a 76 7 W



\ khhl\_lkl\bb k Hij_^_ebf ihegu_ \j_f_gZ j_eZdkZpbbaZiheg_gby ©yfuª 76 7 \ ' fh^_eyo b . fh^_eyo LZdh_ \j_fy i_j_oh^Z 7 76 \ . fh^_eb dZd b \ kemqZ_ keZ[h]h ^bkkbiZlhjZ jZ\gh

= . = W OQ 1H = W + OQ 1

(

)



Ijb 1 W . # W Ihegh_ \j_fy i_j_oh^Z 7 76 \ ' fh^_eb 6 fh`gh gZclb kmffbjh\Zgb_f ih N \ gh wlh ^hklZlhqgh ]jhfha^dh Ih


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W ' 6
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1 W OQõ ü N é ç

1



ba ke_^m_l W ' W Ihegh_\j_fy W ' 6 6 W ' 6

W ' W ' 6 6

N 1

ã

1

W ' N W ' 6

W õ ü # W é ç W W . W 6


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76 7

1'77 76

1 '7

[bebabjm_lky lZd dZd ijblhd l_ieZ d ^bkkbiZlhjm hl ;hj]ZjlhgZ q_j_a ih\_jo ghklv k l_fi_jZlmjhc 76 kjZ\gy_lky k hllhdhf l_ieZ hl ^bkkbiZlhjZ q_j_a ih \_joghklvkl_fi_jZlmjhc 76 :gZeh]bqghij_^u^ms_fmihemqZ_f

76 76 1 1 1 7 7 6 a L_fi_jZlmjZ ^bkkbiZlhjZ 7 W [m^_l \hajZklZlv hl 7 ^h 7 ihke_ q_]h klZ 6

ë 7 ë - (1 - N ) + ¹Q T N- = 7 (1 + N ) - ¹- T + = Q N O O W
N
W 1 N
[

]

[

]

>

@



N

1 1

I_j_oh^ d . fh^_eb ^ey hq_\b^_g hgZ kh\iZ^Z_l k jZg__ jZkkfhlj_g ghc k aZf_ghc 76 7 Ihke_^gbc ' i_jbh^ W
N ijb N 1 kh\iZ^Z_l k 6 W Q ^ey keZ[h]h ^bkkbiZlhjZ JZkkfhljbf khhlghr_gby f_`^m khhl\_lkl\mx sbfb N fb ' i_jbh^Zfb ^ey ^\mo b h^ghl_fi_jZlmjgh]h ;hj]Zjlhgh\ Ba b ijb
1 1


W
N WN

1 N 1 N



1 1 1 1 N



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W
W



1 1 W
W
W
N ! N 1 W N W N W1 1 1
Q

1 1



AZ^ZqZ >bkkbiZlhj k i_j_f_ggufb iZjZf_ljZfb JZkkfhljbf i_j_ oh^ 7N 7N \kbevghf^bkkbiZlhj_Bf__f

a 7N W

W 7N '7 õ ü é WN ç

7 N '7 '7KQ K

W WN

Baf_g_gb_iZjZf_ljh\kh\j_f_g_fhibku\Z_lkynhjfmeZfb

O N W

O N H O N KD F9 N W O N W





F9 N H F N KE

O N H O N K J

E_\ZybijZ\ZyqZklbM>khhl\_lkl\_ggh












a 4 N = 6 O N F9 N 1 - N G 7N 1 - N '4 N




N

6 T ¹ W N N

]^_

=

N - a + N - O N (76 - 7N 1 - N ) O N -

N 1 - N a 76 - 7N = T+ = N O N 1 - N

[

]

(

)(

)



Ih^klZ\b\ \ nhjfmeu dhlhju_ ijbjZ\gb\Z_f ^jm] ^jm]m ± ihke_g_kdhevdh]jhfha^dbo\udeZ^hdgZoh^bf

a= N

O N [(1 - N )Q + ] DN N Q = + µ Q O

QW

N -



N W N - =

- - N





N

=

+ µ Q O N N Q = + N Q N + N Q O N

N N



PD Q

1 - N Q + Q + + Q - Q N Q = DQ (1 - N ) + - ( + ) Q

[

]






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PE Q N Q = N Q = Q lZddZdQ Q 1 N D

= = = PD Q

E

J


PE Q

Q > 1 N Q @ 1 N
>

@


IZjZf_lju Q N J Q b Q N D Q ijb Q

q_gby[ebadb_d AZ^ZqZ >bkkbiZlhj k g_ebg_cgufb iZjZf_ljZfb A^_kv lZd`_ bf_ _lkyZgZeh]bykg_ebg_cgufkeZ[uf^bkkbiZlhjhfIjb Q bf__lkyihegZy wd\b\Ze_glghklv g_klZpbhgZjghfm ^bkkbiZlhjm JZkkfhljbf qZklgu_ kemqZb ebg_cguc jhkl iZjZf_ljh\ k l_fi_jZlmjhc kdZqhd iZjZf_ljh\ \ dhgp_ pbdeZgZijhly`_gbbdhlhjh]hhgb©aZfhjh`_guªBf__f D E J Q
baf_gy_lky \ ij_^_eZo
Q N J Q ) N

Q N D Q

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H O N H F N ) N

H O N H O N Q 1 N ) N

) N ) N

) N °> aN aN @ aN H H H

H O N H F N H ON Q N D Q


D

E J PD Q PE Q Q N J Q Q ) N ) N ) N

LZdbfh[jZahfkemqZc ± kdZqdhh[jZagh]h baf_g_gby iZjZf_ljh\ ^bkkbiZlh jZ \ dhgp_ pbdeZ [ aN ] ijb Q ijb\h^bl d kh\iZ^_gbx ' i_jbh^h\ g_ebg_cgh]h b ebg_cgh]h ^bkkbiZlhjh\ Wlh ^Z_l r_klhc Zj]mf_gl ©\u^_e_gghklbªkemqZy Q


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\j_f_gb[_ajZaf_jgufl_fi_jZlmjZf\lhqdZo [ = O O O G_\yadZ ^\mo fh^_e_c hdZaZeZkv m[u\Zxs_c kh \j_f_g_f b ih f_j_ m^Ze_gby hl hkb kbff_ljbb kbkl_fu >ey lj_o i_j\uo ^bkkbiZlhjh\ 0 N N b fh f_glh\\j_f_gb MW U M [_ajZaf_jgu_l_fi_jZlmju^bkkbiZlhjh\ N M b khhl\_lkl\mxsbfbgZc^_ggufbih . fh^_eb N M ijb\_^_gu\lZ[e LZ[ebpZ
KjZ\g_gb_

Kbkl_fmmjZ\g_gbcj_rZ_fbkdexq_gb_fg_ba\_klguo 7N M gZqbgZyk N KjZ\g_gb_ ihemq_gguo j_amevlZlh\ k . fh^_evx± aZ^Zq_c Dhrb [43] hkms_kl\ey_fke_^mxsbfh[jZahf;_ajZaf_jgu_l_fi_jZlmju^bkkbiZlhjh\\ fhf_glu \j_f_gb WM6 MW U khihklZ\ey_f \uqbke_gguf ^ey l_o `_ fhf_glh\

a

' b . fh^_e_c

N


N M a N M N M a N M N M a N M N M a N M

WM6
WU 31,94 32,22 12,46 11,85 WU 44,29 31,27 31,34 16,27 16,27 WU

MW U
WU 34,80 28,28 28,55 18,96 19,20 WU 31,90 26,86 26,94 19,25 19,35

38,70 29,82 30,16 18,44 18,27




ghfm \j_f_gb keZ[h]h ^bkkbiZlhjZ \ebygb_ ^bkdj_lghklb g_kms_kl\_ggh b fh`gh bkihevah\Zlv . fh^_ev KjZ\g_gb_ ih l_ieh\uf ihlhdZf [ueh ijh\_ ^_gh ^ey lhq_d [ OdhglZdl 0 b 0 b [ OdhglZdl 0 b 0 b l_o`_fhf_glh\\j_f_gb IjZdlbq_kdh_ kh\iZ^_gb_ ihlhdh\ jZkkqblZgguo ih ' b . fh^_eyf[uehihemq_ghk H lZd`_m`_^ey W WU

Ba lZ[e ke_^m_l qlh ijb W W U ih]j_rghklv aZf_gu ' fh^_eb . fh^_evx g_kms_kl\_ggZ l_ ^ey \j_f_g jZ\guo mq_l\_j_gghfm kh[kl\_g




=E:<:P?IHQDB>BKKBI:LHJH<
Fu g_ gZ[ex^Z_f g_ihkj_^kl\_ggh ^bnn_ j_gpbZevguo mjZ\g_gbc fu gZ[ex^Z_f ebrv dh g_qgu_ mjZ\g_gby dhlhju_ y\eyxlky g_ihkj_^kl \_gguf\ujZ`_gb_fgZ[ex^Z_fuoy\e_gbc :ImZgdZj_ Ml\_j`^Zxl qlh mki_o f_lh^Z =_ca_g[_j]Z fh`_l [ulv ijb\_^_g d qbklh Ze]_[jZbq_kdhfm f_lh^m hibkZgby ijbjh^u l_ bkdexq_gbx ba nb abdbg_ij_ju\guonmgdpbcGhlh]^Zgm`gh[m^_l \ ijbgpbi_ hldZaZlvky hl ijhkljZgkl\_ggh \j_f_ggh]hdhglbgmmfZ :Wgrl_cg

Ebg_cgu_h^ghjh^gu_p_ihqdb < gZklhys_c ]eZ\_ jZkkfZljb\Zxlky kbkl_fu \aZbfh^_ckl\mxsbo ^bkkb iZlhjh\± p_ihqdb ba 1 h^ghf_jguo iehlgh dhglZdlbjmxsbo ^jm] k ^jm]hf h^ghjh^guo ^bkkbiZlhjh\ 0 N N 1 =jZgbqgu_ ^bkkbiZlhju 0 b 0 1 bf_xl l_fi_jZlmjm ;hj]ZjlhgZ± 76 < gZklhys_f iZjZ]jZn_ jZkkfZljb \Zxlky h^ghjh^gu_ ebg_cgu_k ihklhyggufb b h^bgZdh\ufb iZjZf_ljZfb p_ihqdb ^bkkbiZlhjh\ Z ^Ze__± bo h[h[s_gby g_h^ghjh^gu_ p_ihqdb g_klZpbhgZjgu_ bg_ebg_cgu_p_ihqdb >ey \u\h^Z mjZ\g_gby ^bkkbiZlhjghc p_ihqdbM>P \u^_ebf \ g_c ljb ijhba\hevguo ihke_^h\Zl_evguo ^bkkbiZlhjZ ^ 0 N 0 N 0 N ` k mq_lhf \aZbfh^_ckl\by k 0 N b 0 N ^ey M ]h pbdeZ ' i_jbh^Z dh]^Z W ° W M Ijbjhklm[uev dhebq_kl\Z l_ ieZ\ 0 N

4N M = 6OF9 7N M '7N M

7N M 7N M



Iehlghklv ihlhdZ l_ieZ ba 0 N b 0 N beb \ h[jZlghf gZijZ\e_gbb \ M f pbde_

a a M T N - N ó = 7N - M - 7N M N - O

[

]

1

>ey ]jZgbqguo ^bkkbiZlhjh\ 0 b 0 1 h[f_gb\Zxsboky l_iehf k ;hj]Zj


lhghf
T M W %

a O 7 7 W T 1M % W O 6 M

>

@

a O W 76 7 O 1 M

>

@

pbdeZ± 7N M N 1 M - 7N 7 \ dhgp_ M ]h pbdeZ± 7N M Baf_ g_gb_l_fi_jZlmju^bkkbiZlhjZ 0 N \oh^_ M ]hpbdeZ

]^_© % ª h[hagZqZ_l ;hj]Zjlhg 76 ± l_fi_jZlmju ih^^_j`b\Z_fu_ ;hj]Zjlh ghfgZ\g_rgbo]jZgbpZop_ihqdbL L_fi_jZlmju\ 0 N \gZqZe_ M ]h
L

a 7N M W


õWü 7N M '7N M è û éW M ç 7N M a 7N M (ó)Gó = 7N M - +



Kj_^gyyaZ M cpbdel_fi_jZlmjZ\ 0 N

7N M = óM




M




T M- = % M óM
M N - - N M =






M

ë 76 - 7 M T M- (ó)Gó = % O

(

)



T

óM óM






M

ë M T N - N (ó)Gó = 7N - M - 7N M N - O ë T M N + (ó)Gó = 7N M - 7N + M N N- O óM


(

)

1



T

M N - N + M =




M

(

)

1

T

M 1 - % M =




M

ë 71 M - 76 T M - % (ó)Gó = 1 O

(

)



J_amevlbjmxsbcl_iehijblhd\ 0 N

' 4 N M


êM ø 6 W M T N N T M N N ë M Mý ì ù


O 6 W M 7N M 7N M 7N M N O
Ijb N

1



bf__f
' 4M

ê ø M 6 W M T M T ë% M Mý ì ù

O 7 7 7 7 M 6 W M O 6 M M

>

@



Ijb N

1

M 4 + M = 6 M T 1 - - 1 - T M - 6 = 1 1 M M O 6 W M 7 71 M 71 M 71 M O 6

>

@



Bamkeh\byl_ieh\h]h[ZeZgkZbkmq_lhf b ihemqZ_f

7N M D = 7N - M - 7N M + 7N + M N M O

(

)

1



Ihemq_ggh_ mjZ\g_gb_ hlebqZ_lky hl \klj_qZxsboky \ ebl_jZlmj_ dh g_qghjZaghklguo ZiijhdkbfZpbc mjZ\g_gby l_iehijh\h^ghklb l_f qlh \_eb qbgu 7Q M Q N N N \g_f± mkj_^g_ggu_ ih M fm \j_f_gghfm pbdem Z g_ hlg_k_ggu_ d M fm fhf_glm \j_f_gb Wlh b]jZ_l j_rZxsmx jhev lZd dZd iha\hey_l ba ihemqblv p_euc deZkk mjZ\g_gbc \ qZklghklb b ke_^mxs__ ba ijb W M G W 7N M G 7 O G [ h[uqgh_mjZ\g_gb_Nmjv_

7 7 =D [



>Ze__ [m^_l ihdZaZgh qlh ± g_ [he__ q_f gme_\h_ ijb[eb`_gb_ . fh^_e_cke_^mxsboba ' fh^_eb >ey ijZdlbq_kdbo jZkq_lh\ mjZ\g_gbx p_e_khh[jZagh ijb^Zlv bgmx g_ kh^_j`Zsmx kj_^gbo l_fi_jZlmj nhjfm >ey wlh]h gZ hkgh\_ aZibr_f

7N - M = 7N - M - +

Ih^klZgh\dZ \ ihke_ijhkluoij_h[jZah\Zgbc^Z_lM>P

7N - M '7N M 7N M 7N M '7N M 7N M 7N M



- ; N - M + 5 M ; N M - ; N + M = EN M N

1




]^_

;N M

'7N M 5 M '7
N

õW è U é WM 7N M - 7N - M

(

)

ü û EN M '7 '7 N N M ç - - 7N M - + 7N + M -

:gZeh]bqgu_ mjZ\g_gby ^ey ]jZgbqguo ^bkkbiZlhjh\ p_ihqdb 0 b 0 1 ke_^mxlba



5 M ; M ; M

E M E M =

; 1 M 5 M ; 1 M





76 - 7 M - + 7 M - 7 E1 M

(

)

E1 M

7 71 M 71 M '7 6



M>P kh\f_klgh k h[jZamxl kbkl_fm ba 1 g_ba\_klguo ; N M N 1 b 5 M Wlmkbkl_fmfh`ghaZibkZlv\fZljbqghf\b^_

$M ; M
]^_ ; M

%M



lhjklhe[_p ijZ\uo qZkl_c hij_^_ey_fuo kh]eZkgh ± FZljbpZ $ ± lj_o^bZ]hgZevgZybkbff_ljbqgZy

^; N M `N

1

\_dlhjklhe[_pg_ba\_klguo % M

^EN M `N

1

± \_d

$M

M = M D N D N Q N 1
D MQ N

(

- = N - N + < N - > N + 5 M N N = 1 - 5 M + N N = 1

)

N = D MN N = Q N



JZkkfhljbf kihkh[u j_r_gby kbkl_fu I_j\uc aZdexqZ_lky \ aZ^Z gbb ^ey dZ`^h]h M ]h pbdeZ \_ebqbgu W M Z ke_^h\Zl_evgh b 5 M Qbkeh g_ba \_klguo lh]^Z kh\iZ^Z_l k qbkehf mjZ\g_gbc J_r_gb_ fh`_l [ulv gZc ^_ghf_lh^hfh[jZlghcfZljbpu

;M

$ % M ; N M =

M

- (DN M6 ) 6 =

1

E6 M






]^_ D M ±we_f_gluh[jZlghcfZljbpu $Ijb\lhjhfkihkh[_aZ^Zxlky M N 6 \k_ \_ebqbgu ; 1 M qlh iha\hey_l ba ihke_^g_]h mjZ\g_gby kbkl_fu gZclb



qZ_lky \ hlg_k_gbb 5 M d g_ba\_klguf b \ aZ^Zgbb h^gh]h ba©\gmlj_ggboª ; N M Imklv aZ^Zgh ; 6 M ; M Lh]^Z kbkl_fZ jZkiZ^Z_lky gZ ^\_ 6 kbkl_fumjZ\g_gbclZdh]h`_\b^Zihjy^hdh^ghcbaih^kbkl_f± 6 Z^jm ]hc± 1 6 DZ`^mxba^\moihemq_gguo©mdhjhq_gguoªkbkl_ffh`gh j_rZlv\lhjufkihkh[hfQ_l\_jluckihkh[ij_^klZ\ey_lkh[hc\lhjhcj_Zeb am_fuc©k\_jom\gbaªaZ^Z_lky ; M b^Ze__ZgZeh]bqgh AZ^ZqZ>\mf_jgu_kbkl_fuKbkl_fZ e_]dhh[h[sZ_lkygZ kem qZc ^\mf_jguo d\Z^jZlguo ^bkkbiZlhjh\©dm[bdh\ª aZihegyxsbo ijyfh m]hevgmx h[eZklv lhesbghc O Ijhba\hevguc \gmlj_ggbc ^bkkbiZlhj kbkl_ fu fh`gh jZkkfZljb\Zlv dZd h[sbc we_f_gl ^\mo i_j_k_dZxsboky p_ihq_d \aZbfh^_ckl\mxsbc k q_lujvfy©khk_^yfbª Imklv lZdhc ^bkkbiZlhj± 0 N P \oh^bl \ dZq_kl\_ N ]h \ i_j\mx p_ihqdm b \ dZq_kl\_ P]h± \h \lhjmx _c i_ji_g^bdmeyjgmx N 1 P 1 ?]h©khk_^yfbª [m^ml ih i_j\hc p_ihqd_± ^bkkbiZlhju 0 N P b 0 N P ih \lhjhc± 0 N P b 0 N P Dhebq_kl\Zl_ieZi_j_r_^r_]h\^bkkbiZlhj 0 N P hl©khk_^_cªgZ M fpbde_

; 1 M b l^ BkdexqZy lZd ; N M ijboh^bf d mjZ\g_gbyf kl_i_gb 1 hl ghkbl_evgh 5 M Imklv wlh mjZ\g_gb_ j_r_gh Lh]^Z ih^klZgh\dhc 5 M \h \k_ mjZ\g_gby gZqbgZy k ihke_^g_]h gZoh^bf \k_ ; N M Lj_lbc kihkh[ aZdex

^

`

^

`

^

`

'4N P M

O F9 '7N P M '7N P M

7N P M 7N P M



7H¡R¡S¢R 0 N P bl_iehhllhdbbag_]hh[mkeh\e_ggu_\aZbfh^_ckl\b_f k©khk_^yfbªfh`ghij_^klZ\blv\\b^_

' 4 N M ' 4 N M
]^_

+ + O W M T N M 4 P M = O M T P M O W M T N M ' 4P M O W M T P M



T N M T N M

O 7 7N P M T P M O N P M

O 7 7N P M O N P M

ë O 7 7N P M T - M = 7N P M - 7N P - M P O N P M O

(

)

KhklZ\eyy [ZeZgk l_ieZ \ 0 N P gZ M f pbde_ gZ hkgh\_ ± ihemqZ _f




'7N P M WM
]^_
N

ø Dê ë ' 7N P M ' 7N P M ý O ì N ý P ë ù



7N P M = 7N - P M - 7N P M + 7N + P M 7N P M = 7N P - M - 7N P M + 7N P + M


P

MjZ\g_gb_ y\ey_lky ^\mf_jguf ZgZeh]hf mjZ\g_gby b ijb mijh s_gghf j_^mpbjh\Zgghf ij_^_evghf i_j_oh^_ W M G W O G [ G \ '7N P M G 7 [ \ W ^Z_l^\mf_jgh_mjZ\g_gb_Nmjv_

7 W

õ 7 7 ü Dè û é [ \ ç



dhlhjh_lZd`_dZdb y\ey_lkylhevdhgme_\ufijb[eb`_gb_fiheghc . fh^_eb khhl\_lkl\mxs_c Ij_h[jZah\Zgb_ ZgZeh]bqgh hkms_kl\ e_gghfmk biha\hey_lihemqblv^\mf_jgucZgZeh]mjZ\g_gbc

; N P M ; N P M 5 M ; N P M ; N P M ; N P M

EN P M


; DE M

FZljbpZ $M hl\_qZxsZy kbkl_f_ mjZ\g_gbc iylb^bZ]hgZevgZy ih jy^dZ 1 1 H[h[s_gb_ gZ lj_of_jguc kemqZc hq_\b^gh hgh ijb\h^bl d k_fb^bZ]hgZevghc fZljbp_ $M k ^bZ]hgZevgufb we_f_glZfb 5 M Ihjy ^hd wlhc fZljbpu± 1 1 1 ]^_ 1 ± qbkeh©^bkkbiZlhjguo iehkdh kl_cª 1 1

'7DE M EN P M = 7N P M - + 7N P M - '7 7 N P

(

)

(

)

G_h^ghjh^gu_p_ihqdb G_h^ghjh^gufb gZau\Z_f p_ihqdb ^bkkbiZlhjh\ hlebqZxsboky ^jm] hl ^jm]Z \k_fb iZjZf_ljZfb O N O N F9 N F9 N N N +

W U N W U N N = 1 - JZkkfhljbf [ZeZgk l_ieZ \ ^bkkbiZlhj_ 0 N gZ M f pbde_ ' i_jbh^_ HimkdZy h^bgZdh\mx ^ey \k_o ^bkkbiZlhjh\ iehsZ^v l_iehi_j_^Zxsboih\_joghkl_c 6bf__f
M 4N M = F9 N O N 7N M 4 + = ó M T N - N - T M N + N M - N-

[

]




T
]^_

M N N

WM

W

ò

M

>

a 7N M W 5N

a 7N M W
N

@

GW

7 7N M 5N N N M O N ON

5N N

U N U N U N

IjbjZ\gb\Zy Zddmfmebjm_fh_ dhebq_kl\h l_ieZ '4N M ihlhdh\hfm '4N M ihke_g_keh`guoij_h[jZah\ZgbcgZoh^bf

7N M DN = N 7N M M O N N

(

)



DN

ON J N F9 N

UN J N 5N N

UN N = N + N + 5N N

MjZ\g_gb_ y\ey_lky ZgZeh]hf mjZ\g_gby b i_j_oh^bl \ g_]h ijb \ujh`^_gbb g_h^ghjh^ghklb D N D O N O J N J N J N I_ j_oh^ \ hl kj_^gbo aZ pbde l_fi_jZlmj ^bkkbiZlhjh\ d gZqZevguf^ey ^Zggh]hpbdeZ bdijbjZs_gbyfboijb\h^bldM>P\b^Z
D M N & N M D M & N M D M & N M N NN N N

EN M



&N M

'7N M D M N N '7

J N D M J N N N


óUN + äN D MN = N óM Ihkdhevdm ^ey ]jZgbqguo ^bkkbiZlhjh\ p_ihqdb N N
fZljbqghf \b^_ ZgZeh]bqgZ fZljbpZ $ M

D MQ lj_o^bZ]hgZevgZy b N i_j_oh^bl \ fZljbpm $ M ijb \ujh`^_gbb g_h^ghjh^ghklb N - = N = = N + = F_lh^uj_r_gby khhl\_lkl\mxllZdh\uf^ey

1 [ZeZgkh\u_ mjZ\g_gby ZgZeh]bqgu b ihegZy kbkl_fZ ba 1]h mjZ\g_gby \
a




G_klZpbhgZjgu_p_ihqdb JZkkfhljbf g_h^ghjh^gmx p_ihqdm ^bkkbiZlhjh\ ^ 0 N ` N = 1 - iZjZf_lju dhlhjuo baf_gyxlky \h \j_f_gb >ey M ]h pbdeZ M = - wlh ba f_g_gb_hibku\Z_lkynhjfmeZfb ON M - ON M - O N = ON M = ON M - + O M O M = N N ON M - M F9 N M - F9 N M - F9 N = F9 N M = F9 N M - + F M F M = N N F9 N M - M N M - N M - N = N M = N M - + = N M N M N M - M Ihkdhevdm gZ g_[hevrbo ijhf_`mldZo \j_f_gb W M hlghkbl_evgh_ baf_g_gb_ iZjZf_ljh\fZeh^Ze__iheh`bfqlh

HNQ M
a 7N M W



HQ M N



¹ HP Q P N M



O F O
ü û ç


Baf_g_gb_\h\j_f_gbl_fi_jZlmju^bkkbiZlhjZ 0 N

õWü 7N M 7N M 7N M è û éW M ç a G 7N M


õW 7N M '7N M è éW M

GW '7N M WM

:ddmfmebjm_fh_l_jy_fh_ 0 N gZ M fpbde_l_ieh

a 4N M = 6 ON M F9 N M G 7N M = 4 M - MN N




M

'

4 M N

6ON M F9 N M '7N M )MN HO M HF M N N





Ihlhdh\h_l_ieh ]^_

4 + = 6 M T + - T - N M N M N M T
N M

[

]



WM

W

ò

a a 7N M W 7N M W G W 5N N W
M

>

@




T

N M

WM

W

a a 5N - N (ó) = aN - M (ó) + aN M (ó) UN M W
Kmq_lhf -

(

ò

M

>

a a 7N M W 7N M W G W 5N N W

@

)

ON M W O N M W



7N + M - 7N M 7N + M - - 7N M - 7N + M - 7N M M N + - + T+ = N M - 5 MN + N 7N M - 7N - M 7N M - - 7N - M - 7N M - 7N - - + T - = N M 5 M -N N -
]^_
M

M N -

)

M N

U N M HO M HO U N M HO M HO M N N M N N 5 M N N





N - M - O- M - M + N M - O M - N N- N N M MN - = M - 5 N - N

(



(

)


)









(

)

5 M -N = ðN - M - + ðN M - N -

(

)



Ih^klZ\eyygZc^_ggu_\ujZ`_gby\mjZ\g_gb_[ZeZgkZ\ 0 N '4N M '4 N M ihemqbf

õ W UN M M ü ) N û '7N M èW ç é M
A^_kvh[hagZq_gh

'

N

>J

M N

( M 7N M ' ( M 7N M N N N

@





W UN M J N (

N M - O M N M - = J N - D N M 5 M -N N - M J M J M ( M = N N N N M M M N U N M ) N ( N ( N

M N M -

U N M 5 M N N

MN -

M

( M N




?keb iZjZf_lju ^bkkbiZlhjZ \h \j_fy M ]h pbdeZ ihklhyggu gh kdZqdhh[jZa ghf_gyxlkyijb W W M ijbgbfZyagZq_gby Q N M = O F9 lh\_ebqbgu \_ebqbgu

HQ M h[jZsZxlky \ gmeb Lh]^Z kh]eZkgh b \ gmeb h[jZsZxlky b N
) MN ) MN ( M N ( M N



DZd ke_^m_l ba lh]^Z )MN b mjZ\g_gb_ i_j_oh^yl \ mjZ\g_ gb_ hlebqZykvebrvbg^_dkhf© M ªmiZjZf_ljh\ ?keb ijb W W M iZjZf_lju kdZqdhh[jZagh g_ baf_gyxlky hklZ\Zykv ih klhyggufb lh bg^_dk© M ª fh`_l [ulv hims_g b kh\iZ^_gb_ j_^mpbjh\Zggh]h mjZ\g_gby kmjZ\g_gb_f^eyg_h^ghjh^ghcp_ihqdb [m^_liheguf MjZ\g_gb_ fh`_l [ulv dZd wlh m`_ [ueh k^_eZgh jZg__ ^ey h^ghjh^guo bg_h^ghjh^guop_ihq_dijb\_^_ghd\b^mkh^_j`Zs_fm g_ kj_^gb_ l_fi_jZ lmjuZijbjZs_gbyl_fi_jZlmjgZ M fpbde_

a a a a D MN - N - M + D MN N M + D MN + N + M = EN M N N N
]^_
M M - a a D MN - = - N - - ( M- D M J N ( M N N N N N

(

)





óUN M - a M + ä M - - ( M D MN = N N N N ó M a EN M ' êõ J M ( M ü 7N M ø ý 7 N ëé N Nç ' ù ì


(

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õ O ü è DW M û '7N M é ç

' 7N M ' 7N M
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7N M 7N M 7N M

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(

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W 1 N

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IhdZ`_f qlh ijbgpbi = Pb]e_jZ± fZdkbfZevghc kdhjhklb wgljhih]_g_ jZpbb - wd\b\Ze_gl_g ijbf_gbl_evgh d ^bkkbiZlhjm ijbgpbim gZbf_gvr_]h ^_ckl\by Kdhjhklv wgljhih]_g_jZpbb gZ N f ' i_jbh^_ \ kbem ihklhygkl\Z \_ebqbgu ijbjZs_gby wgljhibbgh jZaebqgh]h ^ey ebg_cgh]h b g_ebg_cgh]h ^bkkbiZlhjh\ hij_^_ey_lky hlghr_gb_f 'V N W N Q Kh]eZkgh ijbgpbiZ =Pb]e_jZ

'V N W N Q

PD[ W N Q

PLQ



LZdbf h[jZahf h[Z kjZ\gb\Z_fuo ijbgpbiZ wd\b\Ze_glgu lZd dZd k\h^ylky d ijbgpbim fbgbfZevgh]h ojhgh]_g_aZ - fbgbfZevghklb N ]h ' i_jbh^Z dhlh jucmkljZgy_lZjl_nZdl±gZebqb_fgh]hqbke_gguo_ckl\bl_evgh ba ke_^m_l qlh \ ebg_cghf kemqZ_ ijb Q wlh agZq_gb_ Q [ueh h[hkgh\Zgh k_fvx jZaebqgufbkihkh[Zfb

>
67 N

1 N @W N

W




'8 1 N W N

67

'8 W

PLQ N

1



7 8 1 - N = 90 F9 7V - 7N +

'8 9 0 F9 '7

A^_kv '8 1 N bf__l kfuke©^_n_dlZª ijb 7V ! 7 beb ba[uldZ 7V < 7 \gmlj_gg_c wg_j]bb ^bkkbiZlhjZ ih hlghr_gbx d ;hj]Zjlhgm gZ N f ' i_jbh^_ Z '8 ± fbgbfZevgh \hafh`gh_ baf_g_gb_ _]h \gmlj_gg_c wg_j ]bb LZdbf h[jZahf khhlghr_gby bf_xl kfuke l_jfh^bgZfbq_kdh_


^_ckl\b_ gZ dZ`^hf N f rZ]_ N 1 h^bgZdh\h b jZ\gh k\h_fm fbgbfZev ghfmagZq_gbx < g_ebg_cghf kemqZ_ Q 6 7C = 8 C- N N = 6 7C = 8 C W = PLQ 1 N



]^_bg^_dkhf©gª\\_jomh[hagZq_gg_ebg_cguckemqZcZ
8 C- N = 90 F9 [7V - 7N + 7 ] 8 C = 8 1



< h[hbo kemqZyo± ebg_cghf b g_ebg_cghf fbgbfZevgu_ agZq_gby l_jfh^bgZ fbq_kdh]h ^_ckl\by kh\iZ^Zxl Z hlebqby \ e_\uo qZklyo b kdZ au\Zxlky \ hlebqbb ojhgh]_g_jZpbc g_ebg_cgh]h b ebg_cgh]h ^bkkbiZlhjh\ Ijb wlhf ' i_jbh^u g_ebg_cguo ^bkkbiZlhjh\ f_gvr_ q_f m ebg_cguo W N W N qlh k\b^_l_evkl\m_l h [hevr_c bgl_gkb\ghklb ijhl_dZgby g_eb g_cguoijhp_kkh\i_j_ghkZ Ex[hiulgZ gZ gZr \a]ey^ ke_^mxsZy ZgZeh]by < we_f_glZjghc d\Zglh \hcl_hjbbba\_klgZnhjfmeZ

(Q (P

KQQP

'(QP



hibku\ZxsZy baemq_gb_ d\ZglZ wg_j]bb ijb i_j_oh^_ we_dljhgZ k mjh\gy gZ [he__ gbadbc mjh\_gv ?keb iheh`b\ Q QP '7 QP ]^_ '7 QP ± i_jbh^ baem qZ_fhc\hegui_j_ibkZlv \\b^_

'(QP '7 QP

K

PLQ 6N



]^_ 6N ±©d\Zglh\h_ª ^_ckl\b_ lh \b^gZ ZgZeh]by f_`^m b \ dh lhjhc fbgbfZevghfm©d\Zglh\hfmª ^_ckl\bx 6N K khhl\_lkl\b_ fbgbfZev gh_l_jfh^bgZfbq_kdh_^_ckl\b_ 67 DZd ba\_klgh nmgdpby ^_ckl\by ^himkdZ_l ij_^klZ\e_gby± nZdlhjbaZpbb gZ wg_j]bx b \j_fy b gZ bfimevk b dhhj^bgZlm ?keb qZklhl_ QQP fu khihklZ \beb fbgbfZevgh_ oZjZdl_jgh_ \j_fy '7 QP i_jbh^ dhe_[Zgbc lh ^ebg_ \hegu O _kl_kl\_ggh khihklZ\blv fbgbfZevgh_ jZkklhygb_ '[O Lh]^Z ba nhjfmeuE>_;jhceyke_^m_l

'[O 'S

K

PLQ 6N



l_nhjfmeu b _klvke_^kl\byfbgbfmfZd\Zglh\h]h^_ckl\by Khhlghr_gby \aZbfghklb jZkkfhljbf gZ ijbf_j_ kh\f_klgh ijhl_dZx sbo ijhp_kkh\ l_iehijh\h^ghklb b ^bnnmabb [113] Dhgklblmlb\gu_ mjZ\g_ gbybf_xl\b^

,T

/TT ;T /TP ;P , P = /PT ; T + /PP ; P



]^_ , ; ± ihlhdb b l_jfh^bgZfbq_kdb_ kbeu khhl\_lkl\_ggh /LN ± dbg_lbq_ kdb_ dhwnnbpb_glu Mfgh`bf i_j\h_ ba mjZ\g_gbc gZ ;T Z \lhjh_± gZ ;P bkeh`bfbo

, T ;T , P ; P

/TT ;T /PP ;P /TP /PT ;T ;P



A^_kv e_\Zy qZklv _klv kdhjhklv wgljhih]_g_jZpbb [113] dhlhjZy \ kbem ijbg pbiZ=Pb]e_jZ^he`gZ[ulvfZdkbfZevgZKe_^h\Zl_evgh

/TT ;T /PP ;P /TP /PT ;T ;P PD[ Imklv \oh^ysb_ \ dhwnnbpb_glu /LN fh]ml ijbgbfZlv jZaebqgu_ agZ PD q_gby kj_^b dhlhjuo _klv fZdkbfZevgu_± /LN [ Ihkdhevdm ;T b ;P \ h[
s_fkemqZ_ijhba\hevgumkeh\b_ lj_[m_l\uiheg_gbymkeh\bc

/TT = / PD[ /PP = / PD[ /TP + /PT PD[ = /TP PD[ + /PT PD[ PP TT
Ihke_^g__bakhhlghr_gbc kijZ\_^eb\h_keb

/TP PD[ = /PT PD[



LZdbf h[jZahf ba ijbgpbiZ = Pb]e_jZ ke_^m_l qlh Z dbg_lbq_kdb_ dhwnnb pb_glu ijbgbfZxl fZdkbfZevgu_ agZq_gby [ ©i_j_dj_klgu_ª dhwnnbpb_glu jZ\gu /TP /PT Ih^l\_j`^Z_lky ]bihl_aZ Ljmk^_eeZ ©?keb khhlghr_gby \aZbfghklb\_jgulh^he`gZkms_kl\h\Zlvb\hafh`ghklvboqbklhn_ghf_gh eh]bq_kdh]h \u\h^Zª pbl ih [113]) Wdklj_fZevghklvmax) dbg_lbq_kdbo dh wnnbpb_glh\[ueZjZg__ihdZaZgZiml_fijbf_g_gbyZgZeh]ZijbgpbiZPb]e_ jZdmjZ\g_gbx;hevpfZgZ[19,69,70] Bghc g_aZ\bkbfuc b n_ghf_gheh]bq_kdbc ih kms_kl\m \u\h^ khhlghr_ gbc \aZbfghklb ihemqZ_f f_lh^Zfb we_f_glZjghc fhe_dmeyjghdbg_lbq_kdhc l_hjbb bkihevamxs_c ebrv kZf nZdl kms_kl\h\Zgby qZklbpghkbl_e_c wg_j ]bbbfZkku kh\_jrZxsboh^ghf_jgu_kemqZcgu_[em`^Zgby[61,63,206] JZkkfhljbf ^bnnmabx qZklbp fZkkhc P \ h^ghf_jghc l_fi_jZlmjgh g_h^ghjh^ghc kj_^_ Imklv \ k_q_gbyo _^bgbqghc iehsZ^b 6 kj_^u ± [ - K [ [ + K khkj_^hlhq_gu khhl\_lkl\_ggh 1 [ K 1 [ 1 [ K qZklbp Lh]^Z \ we_f_glZjghf h[t_f_ 9 6K [m^ml gZoh^blvky qZklbpu k kmffZjghc fZkkhc 0 [ K 0 [ 0 [ K ]^_ 0 [ 1 [ P <\_^_fiehlghklbqZklbp

U [ K


0 [ K U [ 9

0 [ U [ K 9

0 [ K 9


Wnn_dlb\guc ihlhd qZklbp d k_q_gbx [ hij_^_ey_lky jZagbp_c ihlhdh\ qZk lbphlk_q_gbc [ K b [ K

T1

- 1 - 1

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=

P = N7 -

õNü é P ç



7

- 7

->7 [ @

- U -U JZaeZ]Zy nmgdpbb \ jy^u L_cehjZ ih kl_i_gyf K b h]jZgbqb\Zykv qe_gZfb eb g_cgufbih K ihemqbf TU

]^_ N ± ihklhyggZy ;hevpfZgZ 7 ± l_fi_jZlmjZ Hl ihlhdZ qZklbp T 1 i_j_c ^_fdihlhdmfZkku TU

T 1 P 9

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ð ð 7 T = - K [ + [ 7 [
Ba ke_^m_lqlh



- - 7
HdhgqZl_evghgZoh^bf

ø ê õNü OQ ë é P ç OQ 7 ý 7 ì ù

7



TU

U U 7 ü '7 '7 õ é [ 7 [ ç

K- 7



Ihemq_gh mjZ\g_gb_ ^ey ihlhdZ fZkku h^gh ba mjZ\g_gbc
1 1 7 7 = 7 7 = FRQVW 7 U U [
]^_ H7

H7 1 7 9

H

N7 ±kj_^gyywg_j]byghkbl_e_cwg_j]bbIhemqZ_f ðX ðX 7 7 + [ 7 7 [


TX =

[(7 ðX )- - (7 ðX )+ ] = - K7


?keb \ kh\_jrblv ij_h[jZah\Zgb_ ZgZeh]bqgh_ b \hkihevah \Zlvkykhhlghr_gb_f

lhihemqbf

8 ô1 = F9 7 ðX = 7 7 = 9 9 7 O 7 [

TX = - ë7

Qlh[u mq_klv \ebygb_ gZ ihlhd l_ieZ h^gh\j_f_ggh b^ms_]h ijhp_kkZ ^bnnm abb d TX g_h[oh^bfh ^h[Z\blv khklZ\eyxsmx h[mkeh\e_ggmx fZkkhi_j_gh khf

K- 7 F9 7



Ihegucihlhdl_ieZ[m^_l Ba ± ke_^m_l

T X' = T P



T X = T X + T X'



O 6 7

Nhjfmehc ^Z_lky \lhjh_ dhgklblmlb\gh_ mjZ\g_gb_ hij_^_eyxs__ iehlghklvihlhdZl_ieZh[mkeh\e_ggh]hkh\f_klghijhl_dZxsbfbijhp_kkZfb l_iehijh\h^ghklb b ^bnnmabb ?keb khihklZ\blv ihemq_ggu_ mjZ\g_gby b k bo©dZghgbq_kdbfbª ij_^klZ\e_gbyfb b \hkihevah \Zlvky\ujZ`_gbyfb^eyl_jfh^bgZfbq_kdbokbe[211]

7 - '7 7 T X = - 7 [ [ U' 7 H '7 '7 7 O 7 7 P 7

;T
lhg_f_^e_gghihemqbf

7 ;P 7 [

N7 U UP [



/TP /PT LZdbf h[jZahf khhlghr_gb_ \aZbfghklb HgaZ]_jZ ihemq_gu ^\mfy n_gh f_gheh]bq_kdbfb kihkh[Zfb qlh mkljZgy_l Zjl_nZdl± \u\h^ bo gZ hkgh\_ ©fbdjhh[jZlbfhklb \j_f_gbª MkljZg_gb_ kljmdlmjgh]h Zjl_nZdlZ \ p_ehf aZ dexqZ_lky \ mkljZg_gbb _]h khklZ\eyxsbo± Zjl_nZdlh\ y^jZ iZjZ^b]fu qlh gZfb b hkms_kl\e_gh < dZq_kl\_ gh\h]h y^jZ iZjZ^b]fu ebr_ggh]h kljmd lmjgh]hb \k_o qZklguo Zjl_nZdlh\ ij_^eZ]Z_lky jZa\blb_ l_o we_f_glh\ ^bk dj_lghcl_jfh^bgZfbdb;hj]Zjlhgbdb dhlhju_baeh`_gu\QZklbII < ihke_^g_c c ]eZ\_ jZkkfZljb\Z_lky k\yaZgguc kh \k_f ij_^u^msbf gh bf_xsbc b kZfhklhyl_evgh_ h[s_nbabq_kdh_ agZq_gb_ \hijhk h kms_kl\h \Zgbb b mkljZg_gbb nmg^Zf_glZevgh]h ojhghZjl_nZdlZ± iZjZ^b]fu \j_f_ gb\nbabd_



=E:<: NMG>:F?GL:EVGUCOJHGH:JL?N:DL
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+

hkb 2[ jZ\guo [ [ [ [ = / lh _klv ijhreZ imlv [W + = [ = / ?keb wlh ^\b`_gb_ ©h[jZlblvª lh ^\b]Zykv \ ©h[jZlghfª \j_f_gb

( ( jhgm [ ! < fhf_glu \j_f_gb t 1 + ) = 0 t ( + ) = 20 t 3+ ) = 30 t ( + ) = 40 2 4 lh _klv ijb W ° W lhqdZ hdZau\ZeZkv gZ jZkklhygbyo hl gZqZeZ [

= W + = gZoh^beZkv \ Imklv fZl_jbZevgZy lhqdZ \ fhf_gl \j_f_gb W lhqd_ [ hkb Ox \^hev dhlhjhc b ^\b]ZeZkv ^Zevr_ \ijZ\h lh _klv \ klh

>

@

( t ( -) [0,- t ( + )] ijb t1-) = -0 fZl_jbZevgZy lhqdZ ^he`gZ bf_lv dhhj^bgZlm 4

[0,-40 ] Ijh\_jbf h^gZdh dZd k\yaZgu©ijyfh_ª b©h[jZlgh_ª \j_fy ^ey dZ`^hc lhqdb fZjrjmlZ [ [ / b h[jZlgh >ey x = x0 = 0
( [ t1+ l^K\yavijyfh]hbh[jZlgh]h\j_fzg^eyex[h]h [ =

ihkljh_gh lhqdZ \ha\jZsZ_lky hl [ f_gb t ( - )

[ = [ ijb t [ = [ b ijb

( -) = -2 ­ dhhj^bgZlm [ [ ijb ( -) = -3 ­ dhhj^bgZlm t3 0 0 2 ( -) = -4 ­ [ = [ = DZaZehkv [u ©h[jZlgh_ª ^\b`_gb_ t4 0

/ d [

^\b]Zykv \©h[jZlghfª \j_

t ( + ) = t ( + ) = 0 Z t ( -) = - t ( + ) = -40 >ey [ 0 4

)

= 0 Z t ( -) = -30 b

[L L =


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Ijb^jm]boL wlhckbff_ljbb (t
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= -W

- + L lh ihemqbf qlh W L

= W + = = - W - lh_klv L

i = 2.



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= =

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+

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