• No results found

The role of noise in optimisation and diffusion limited aggregation

N/A
N/A
Protected

Academic year: 2020

Share "The role of noise in optimisation and diffusion limited aggregation"

Copied!
129
0
0

Loading.... (view fulltext now)

Full text

(1)

University of Warwick institutional repository: http://go.warwick.ac.uk/wrap

A Thesis Submitted for the Degree of PhD at the University of Warwick

http://go.warwick.ac.uk/wrap/4502

This thesis is made available online and is protected by original copyright.

Please scroll down to view the document itself.

(2)

#'(*))*+),

-/.

0 + 2143654+78 9

:<;'=?>A@B>

CEDE-EF!GIHHJLK%HM!HONPJRQSPGIT/JAU#VGIH#.WMYX[Z]\^U#_*GI`a

XbMcU+HONPJKEJLdPU#JLJ&MeX

fhg9iPj/gklg?mono;9@bpBgq>rgqs';qt

u

#v9#Pwxzy ,78{|W}~€'!,,*‚hƒ„*w{9

(3)
(4)

!"$#&%(')*

+-,/.01*23$*4&#65708 9

:/0;0(4&*4=<>%(?,/*8606 9

+-8'),/ 9

:/08')64@%6,/860 9 ?

ACB1<6*'EDGFIHC<68J5K?L*8$&0M10646,NBGL8?,OG<&8J5P*860 D

‰RQ‰ SUTVB`LMcF …XWJLHJASPJLVVYQQQZQQQ[QQQZQQZQQQ[QQQZQQQ[QQQZQQQZQQ[Q ‰

‰RQ‡ CPGF!DXW\eHJLK]\&SESPJA\/WIGSPdYQZQQQ[QQQZQQZQQQ[QQQZQQQ[QQQZQQQZQQ[Q ‡

‰RQ_^ `HONPJAU NPJADEU#GIVHGI`LVaQ[QQQZQQQ[QQQZQQZQQQ[QQQZQQQ[QQQZQQQZQQ[Q b

‰RQdc CPHMP`LNE\eVHGI`Z`&…PHGF!GIVO\eHGMPSeQQ[QQQZQQZQQQ[QQQZQQQ[QQQZQQQZQQ[Q f

ACB1<6*'hg6FIi16,RB18,7100GXj01# D;D

‡*Q‰ CPHMP`LNE\eVHGI`\/SESPJA\/WIGSPdhVk9GSEa\/SPK]l,\/WmWMQQQ[QQQZQQQ[QQQZQQQZQQ[Q ‰/‰

‡*Q‡ S\noIT/J\&…E…EU#Mc\e`LN]Q[QQQZQQQ[QQQZQQZQQQ[QQQZQQQ[QQQZQQQZQQ[Q ‰8^

‡*Q_^ CPGF!DXW\eHGSPd6\ HONPJAUFo\/WV.PVHJAF QQZQQZQQQ[QQQZQQQ[QQQZQQQZQQ[Q ‰c

‡*Qdc pJLVODXWIHV QQZQQQ[QQQZQQQ[QQQZQQZQQQ[QQQZQQQ[QQQZQQQZQQ[Q ‰8q

‡*Q_b r*NPJ&_ GKEHNMeX9HONPJ&JAUU#MPU'KEGIVHU#G-EDPHGIMcSsQZQQQ[QQQZQQQ[QQQZQQQZQQ[Q ‡/‡

(5)

^*Q‰ S/HOU McK DP`LHGIMcS7QQQ[QQQZQQQ[QQQZQQZQQQ[QQQZQQQ[QQQZQQQZQQ[Q ‡Rt

^*Q‡ k?McUF ` X r*NPJZ`&…PHGFo\/WrMcDEU CP`A\/WIGSPd3\&U#dcDEF JAS/HVQQ[QQQZQQQZQQ[Q ^/ˆ

^*Q_^ u,MPFo…EDPHO\eHGIMcSE\ WGpJLVODXWIHV"k?McU"r*NPJZT r*C*T Q[QQQZQQQ[QQQZQQQZQQ[Q ^/‡

^*Qdc r*NPJ GF GHGSPd]u \eVJ ˆ Q[QQQZQQZQQQ[QQQZQQQ[QQQZQQQZQQ[Q ^R^

^*Q_b S M/HJLV*MPSh\ WId/McU GHNEFBFo…XWIJAF!JAS^HO\eHGMPS QQQ[QQQZQQQ[QQQZQQQZQQ[Q ^Rb

^*Q_t u,MPSP`WDPVGMPS QQQ[QQQZQQQ[QQQZQQZQQQ[QQQZQQQ[QQQZQQQZQQ[Q ^Rt

^*Q_q \&…E…EJASPKEG QZQQQ[QQQZQQQ[QQQZQQZQQQ[QQQZQQQ[QQQZQQQZQQ[Q ^Rq

ACB1<6*' FIHC? !"$*4 HC<&8j5K?L*8$&0

cQ‰ S/HOU McK DP`LHGIMcS7QQQ[QQQZQQQ[QQQZQQZQQQ[QQQZQQQ[QQQZQQQZQQ[Q ^Rf

cQ‡ QSP`LJAU#HO\eGS^H#.QQQ[QQQZQQQ[QQQZQQZQQQ[QQQZQQQ[QQQZQQQZQQ[Q ccˆ

cQ_^ r*NPJ&M^GW JWKhF!MPKEJW QQZQQQ[QQQZQQZQQQ[QQQZQQQ[QQQZQQQZQQ[Q c ^

cQdc pJLVODXWIHV"QQQZQQQ[QQQZQQQ[QQQZQQZQQQ[QQQZQQQ[QQQZQQQZQQ[Q c t

cQ_b u,MPSP`WDPVGMPSPV QQQ[QQQZQQQ[QQQZQQZQQQ[QQQZQQQ[QQQZQQQZQQ[Q bE‰

ACB1<6*'6F- %6860=j5K$*4*#;#G') #;*8$&0

b*Q‰ S/HOU McK DP`LHGIMcS7QQQ[QQQZQQQ[QQQZQQZQQQ[QQQZQQQ[QQQZQQQZQQ[Q bR^

b*Q‡ ` 1V W\eHHGI`LJ!SPM/GIVJRU#JLK DP`LJLK%KEGqDPVGMPS-WIGF!GIHJLKh\ed/dcU#JLdP\eHGIMcS QQZQQQZQQ[Q tR^

b*Q_^ pJW\eHGIMcS<HMoHNPJF!JLHONPMPK QQQZQQZQQQ[QQQZQQQ[QQQZQQQZQQ[Q tRq

b*Qdc u,MPUU#JL`LHGIMcS<HM!V`A\ WGSPdJE…EMcSPJAS^HV QZQQZQQQ[QQQZQQQ[QQQZQQQZQQ[Q qE‰

b*Q_b pJLVODXWIHV QQZQQQ[QQQZQQQ[QQQZQQZQQQ[QQQZQQQ[QQQZQQQZQQ[Q qRb

b*Q_t u,MPSP`WDPVGMPS QQQ[QQQZQQQ[QQQZQQZQQQ[QQQZQQQ[QQQZQQQZQQ[Q qRf

ACB1<6*'6F 01G'N57?*8601064!- + j0" #

t*Q‰ S/HOU McK DP`LHGIMcS HMoU#JASPMcUF \/WIGV\eHGIMcS QZQQZQQQ[QQQZQQQ[QQQZQQQZQQ[Q f/ˆ

t*Q‡ $2JA\eVODEU JAF!JAS/HV MYX[HONPJ%"PJLK …†M/GS/H&SPM/GIVJ!U#JLK DP`LHGIMcS-QQQ[QQQZQQQZQQ[Q f/c

t*Q_^ &' JL`LHGT^JSPM/GIVJRU#JLK DP`LHGMPS QQ[QQQZQQZQQQ[QQQZQQQ[QQQZQQQZQQ[Q fRq

t*Qdc (1\ `WDPVHJAU#VdcU#Me_S GS]^)( Q[QQQZQQZQQQ[QQQZQQQ[QQQZQQQZQQ[Q *E‰

t*Q_b u,MPUU#JL`LHGIMcS<HM!V`A\ WGSPdJE…EMcSPJAS^HV*GSO^)( QQ[QQQZQQQ[QQQZQQQZQQ[Q */‡

t*Q_t k9GPJLK …EM^GS^HSPM/GIVJ!U#JLK DP`LHGIMcS GSO^)(YQQZQQQ[QQQZQQQ[QQQZQQQZQQ[Q */c

t*Q_q SDEF!-EJAU,MeX'\/UF V*GS]^)( QQQ[QQQZQQZQQQ[QQQZQQQ[QQQZQQQZQQ[Q *Rf

(6)

q*Q‰ CPHMP`LNE\eVHGI`\/SESPJA\/WIGSPd QZQQQ[QQQZQQZQQQ[QQQZQQQ[QQQZQQQZQQ[Q ‰AˆR^

q*Q‡ T+U#Mc-E\/-PGmWIGIVHGI`!HOU\ T/JWmWIGSPd<VO\/WIJLVOF \/S…EU#Mc-XWIJAF QQQZQQQ[QQQZQQQZQQ[Q ‰Aˆ/c

q*Q_^ `GmW JWIK2Mc…PHGF!GIVO\eHGMPS QZQQQ[QQQZQQZQQQ[QQQZQQQ[QQQZQQQZQQ[Q ‰Aˆ/c

q*Qdc S M/GIVJU#JLK DP`LHGIMcS%GS ((\ QQ[QQQZQQZQQQ[QQQZQQQ[QQQZQQQZQQ[Q ‰AˆRb

q*Q_b u,MPUU#JL`LHGIMcS<HM!V`A\ WGSPdJE…EMcSPJAS^HV QZQQZQQQ[QQQZQQQ[QQQZQQQZQQ[Q ‰AˆRb

q*Q_t pJASPMcUFo\ WGIVO\YHGIMcS<MeX (1\ QQ[QQQZQQZQQQ[QQQZQQQ[QQQZQQQZQQ[Q ‰AˆRt

(7)

‡*Q‰ r*NPJ Mc…PHGFo\/WE`LNPM/GI`LJMYXHONPJ\Y`L`LJA…PHO\/SP`LJ&…EU#Mc-E\/-PGmWIGIH . X McU JAUU#McU9VGFaV

…XWIJKEGIVHU#G-EDPHGIMcSPV8QQQQZQQQ[QQQZQQZQQQ[QQQZQQQ[QQQZQQQZQQ[Q ‰8f

‡*Q‡ `&…PHGFo\ W \e`L`LJA…PHO\/SP`LJ!UDXWIJLV*XbMcU,-PGF!MPK \/W KEGIVHU#G-EDPHGIMcSPV8Q6Q[QQQZQQQZQQ[Q ‡E‰

(8)

‰RQ‰ r*NPJUW\/DE-EJAU\/SPK $2JLHOU#Mc…†MRWIGIV \e`L`LJA…PH\/SP`LJ&XBDESP`LHGIMcSPV8Q Q[QQQZQQQZQQ[Q c

‰RQ‡ JLHHGSPdoVHODP`La GS%\IWIMc`A\ W F!GSPGFRDEFEQ QZQQQ[QQQZQQQ[QQQZQQQZQQ[Q q

‡*Q‰ r*NPJ…EU GSP`LG…XWIJW-†JANPGSPK2VHMc`LNE\eVHGI`!\/SESPJA\/WIGSPdQQQQZQQQ[QQQZQQQZQQ[Q ‰8^

‡*Q‡ \ VOaeJLH`LN<MeX9HONPJF!MPKEG JLK&\/DPVVG\/S<KEGIVHU#G-EDPHGIMcS1Q]QQQ[QQQZQQQZQQ[Q ‡/ˆ

‡*Q_^ \VaeJLH`LNhMeXHONPJWHNPJAUFo\/W,\e`L`LJA…PHO\/SP`LJlUDXWIJ<XbMcU HONPJF!McKEG JLK \/DPVVG\^S

KEGIVHOU G-EDPHGMPS1Q QQQ[QQQZQQQ[QQQZQQZQQQ[QQQZQQQ[QQQZQQQZQQ[Q ‡/‡

^*Q‰ r+.E…PG`A\ W SPJA\/U,Mc…PHGF \/W \o…EU#GIMcU G \/SPKl…EUDESPJLK7Tr*C*T HMcDEU#V8QQQZQQQZQQ[Q ‡Rf

^*Q‡ r*NPJ9JE…EJL`LHJLKR…EUDESPJLK!HMcDEU;WIJASPd/HONKEGITcGIKEJLKo-/.HONPJ+J …†JL`LHJLKRU#J VBMP…PHGF!GIVJLK

HMPDEUWIJASPd/HN QQQ[QQQZQQQ[QQQZQQZQQQ[QQQZQQQ[QQQZQQQZQQ[Q ‡ *

^*Q_^ r*NPJ,Fo\eVHJAU `ADEU T/J9XbMcUHONPJ,…EUDESPJLKHMcDEU1WIJASPd/HON KEGITcGIKEJLKo-^. …EUDESPJLK

6QZ^/‡

^*Qdc r*NPJ,Fo\eVHJAU `ADEU T/J9XbMcUHONPJ,\*…EU#GIMcU#GPHMcDEU(WJASPd^HON&KEGITPGKEJLK<-/.

\o…EU#GIMcU G Q^R^

^*Q_b CENPGXbH&XB\e`LHMcU#V XbMcU&\…EU#GIMcU#G9HMPDEU#V`LMPFo…E\/U#JLKhHMWHONPJ J DPGITr\ WJAS^HoF!JA\eVODEU J V

F!JAS^H*MeXco`LGIH .HMcDEU#V8Q QZQQQ[QQQZQQZQQQ[QQQZQQQ[QQQZQQQZQQ[Q ^/c

^*Q_t $2JLHONPMPK!XbMcU9Mc-PH\eGSPGSPd!JLVHGF \eHJLV'MYX HNPJ*`NE\^SPd/J*GSoHNPJJE…EJL`LHJLK…EUDESPJLK

HMPDEUWIJASPd/HN QQQ[QQQZQQQ[QQQZQQZQQQ[QQQZQQQ[QQQZQQQZQQ[Q ^Rb

^*Q_q r*NPJ!JE…EJL`LHJLK…EUDESPJLKhHMcDEU WIJASPd/HON%XbMcU\/SESPJA\/WIGSPd/V_NPJASo\/SPK

\/U J

GSP`AU#JA\eVJLK F!McSPM^HMcSPGI`A\/WmWI.RQ3QQ[QQQZQQZQQQ[QQQZQQQ[QQQZQQQZQQ[Q ^Rt

cQ‰ r*NPJ&H#.E…PGI`A\/WVJLHODE…6XMPU \ j^)( +VJLGIVOF!GI`!VODEU#T/JL.RQOQQZQQQ[QQQZQQQZQQ[Q ccˆ

cQ‡ r*NPJ…EU G`LJ MeX9`AUDPKEJM/GmW GS/-E\/UU#JW+X U McF ‰ *Rq/c!HMW‡^ˆ/ˆ/ˆ*QOQQZQQQZQQ[Q cc‡

cQ_^ k@WIMe_ KEG\YdcU\/F€VNPMr_ GSPd<HONPJ…EU#MP`LJLVV*MYX[Mc…PHGF GV\eHGIMcS Q[QQQZQQQZQQ[Q c ^

cQdc \ H#. …PGI`A\/W …EU McK DP`LHGIMcS%U\eHJXbMcU,\oU#JLVJAU#T/M/GUQ6Q[QQQZQQQ[QQQZQQQZQQ[Q cRc

(9)

SPJA\/WIGSPdQ QQZQQQ[QQQZQQQ[QQQZQQZQQQ[QQQZQQQ[QQQZQQQZQQ[Q c q

cQ_q r*NPJ&`LMPS dcDEU\eHGIMcS MeX HONPJ!-E\eVJ&`A\eVJRQ]QZQQQ[QQQZQQQ[QQQZQQQZQQ[Q c f

cQ_f r*NPJ`LMcS dcDEU\eHGIMcS MeX HONPJ&…EMRWIGI`L.`LNPM/VJAS<-/.<Mc…PHGF!GIVO\eHGMPSo_NPJAS<HNPJVG! LJ

MeX JA\e`N%U#JLVJAU#T^M/GU,GIV\eVVODEF JLKlHMo-EJ!acSPMe_S1Q QQQZQQQ[QQQZQQQZQQ[Q c *

cQ* r*NPJ&`LMPS dcDEU\eHGIMcS MeX9HONPJ…EMRWIGI`L.l`LNPM/VJAS%-/.<VHMP`LNE\eVHGI`R\^SESPJA\/WIGSPd_NPJAS

\o…†JASE\/WIH#.WGIVd/GIT/JASGX \!U#JLVJAU#T^M/GU,HMW-†JKEJLT/JWIMc…†JLK GIV*JAF …PH.RQQQQZQQ[Q b/‡

b*Q‰ & \/F …XWJLV MeX[KEG DPVGIMcSOWIGF!GIHJLK2dcU#Me_*HON!GS%U#JA\/WV.PVHJAF V8Q QQQZQQQZQQ[Q b/c

b*Q‡ r*NPJ&`LMPFo…EDPHO\eHGIMcSE\ W F!JLHONPMPK<XMPU (1\ McSNV W\eHHGI`LJRQKQQQ[QQQZQQQZQQ[Q bRt

b*Q_^ \ VOFo\/WmW ((\‚`WDPVHJAU dcU#Me_S McSNVW\eHHG`LJRQ QQ[QQQZQQQ[QQQZQQQZQQ[Q bRq

b*Qdc CPGIJAU…PGSPVOa^GV&dc\eVOaYJLH8Q-QQZQQQ[QQQZQQZQQQ[QQQZQQQ[QQQZQQQZQQ[Q bRf

b*Q_b \ SPM/GIVJRU JLK DP`LJLK (1\ `WDPVHJAUdcU#Me_S<McSNV W\eHHG`LJRQ QQQ[QQQZQQQZQQ[Q tE‰

b*Q_t \€SPM/GIVJ V U#JLK DP`LJLK (1\ `WDPVHJAU[b)VbXbMRWIKz\^SPGVM/HOU#Mc…^. `LMcS^HO\eGSPJLKh_*GIHONPGS2GIHV

JAS^T/MRWIMc…†JRQ QZQQQ[QQQZQQQ[QQQZQQZQQQ[QQQZQQQ[QQQZQQQZQQ[Q t/‡

b*Q_q r*NPJSPM^GVJ V U#JLK DP`LJLK (1\ \/WId/McU#GIHONEFEQ]QZQQQ[QQQZQQQ[QQQZQQQZQQ[Q t/c

b*Q_f & \/F …XWJSPM^GVJRU#JLK DP`LJLK2M)1V W\eHHGI`LJ (1\ `WDPVHJAU#VsQQQ[QQQZQQQZQQ[Q tRb

b*Q* r*NPJF JA\eVODEU#JLK XU\Y`LHO\/W KEGF!JASPVGMPSlMYX'(1\ GSl‡)( QQQ[QQQZQQQZQQ[Q tRt

b*Q‰Aˆ=r*NPJU JW\YHGIT/JR…†JASPJLHOU\eHGIMcS KEJA…PHON \YV\ XBDESP`LHGIMcSMeX8Q QQZQQ[Q tRq

b*Q‰/‰ r*NPJ…EU GF GHGT^J F \/… XMPU'HNPJ %hF!JLHONPMPK6Q QQ[QQQZQQQ[QQQZQQQZQQ[Q tRf

b*Q‰A‡=r*NPJ…EJASPJLHOU\YHGIMcSoKEJA…PHN XMcU %%`WDPVHJAU#V*\/SPKSPM/GIVJ&U#JLK DP`LJLK (1\ `WDPVHJAU#V8QIt *

b*Q‰8^ \ …ENE\eVJ&VO…E\Y`LJR…XWIM/H*MeX9HONPJ&KEG…EMNWJo…†Mr_,JAU \edc\YGSPVH*HONPJU#JW\eHGIT/J!…EJASNV

JLHOU\eHGIMcS<KEJA…PHON

QOQZQQQ[QQQZQQZQQQ[QQQZQQQ[QQQZQQQZQQ[Q q/c

b*Q‰c(&GJAU#JAS^HG \/W …XWIM/H,XbMcU,HONPJU JW\YHGIT/JR…†JASPJLHOU\eHGIMcS<KEJA…PHN

Q QQQZQQ[Q qRt

b*Q‰8b (&GJAU#JAS^HG\/W*…XWM^HV!XbMcU!HONPJU#VH!XbMcDEU bSPMcSNVBHU#GITcG\/W<FRDXWIHG…†MRWIJzF McF!JAS^HV

QCQZQQQ[QQQZQQQ[QQQZQQZQQQ[QQQZQQQ[QQQZQQQZQQ[Q qRq

b*Q‰8t=r*NPJ `LMcUU#JL`LHGIMcSzHMV`A\/WIGSPd2J …†McSPJAS/HV&XbMcUHONPJWF!DXWIHG…EMRWIJ…EMe_,JAU#V

*\/SPKHONPJ!U#JW\eHGIT/JR…†JASPJLHOU\eHGMPS<KEJA…PHON1Q Q[QQQZQQQ[QQQZQQQZQQ[Q qRf

b*Q‰8q=r*NPJ2\eV.EFo…PHM/HG`%Tr\/WDPJLV XMcU HONPJhFRDXWIHG…†MRWIJ]F!McF!JAS^HV "!

#

o_NPJAS SPMcU V

Fo\ WGIVJLK2-/.

#

Q6QQ[QQQZQQQ[QQQZQQZQQQ[QQQZQQQ[QQQZQQQZQQ[Q q *

t*Q‰ \ VGFo…XWIJoU#JASPMcUFo\ WGIVO\YHGIMcS<V`LNPJAF!JRQ"QQZQQQ[QQQZQQQ[QQQZQQQZQQ[Q fE‰

(10)

U\YKEGDPV)QZQQQZQQQ[QQQZQQQ[QQQZQQZQQQ[QQQZQQQ[QQQZQQQZQQ[Q fRq

t*Qdc '

\edc\eGSPVH QXQQZQQQ[QQQZQQZQQQ[QQQZQQQ[QQQZQQQZQQ[Q fRf

t*Q_b u,MPFo…E\e`LH,dcU#Me_*HON MeX9H#_ MoSPJLGIdcNP-EMcDEU GSPdl\^UF!V8Q QQZQQQ[QQQZQQQZQQ[Q f *

t*Q_t r*NPJo‡…†M/GS/H `LMcUU#JW\eHGIMcS XDESP`LHGIMcS FRDXWIHG…XWIGIJLK -/.2\/S%J JL`LHGIT/JWSPM/GIVJWU J V

K DP`LHGIMcS QKQQ[QQQZQQQ[QQQZQQZQQQ[QQQZQQQ[QQQZQQQZQQ[Q */ˆ

t*Q_q Me_4HNPJZWIJASPd/HON<MeX'-EU\/SP`LNPJLV&\/U#J&`A\/WI`ADXW\eHJLK6QQQQZQQQ[QQQZQQQZQQ[Q *E‰

t*Q_f r*NPJ-EU\/SP`N KEGIVHOU#G-EDPHGIMcS1QQQ[QQQZQQZQQQ[QQQZQQQ[QQQZQQQZQQ[Q */‡

t*Q* \ ^)( (1\ `WDPVHJAU _*GIHON ‰Aˆ\/SPK ˆ‰RQ*QZQQQ[QQQZQQQZQQ[Q *R^

t*Q‰Aˆ=r*NPJF JA\eVODEU#JLK XU\Y`LHO\/W KEGF!JASPVGMPSlMYX[`WDPVHJAU VdcU Mr_SoGS]^ (CQ@QZQQQZQQ[Q */c

t*Q‰/‰(&GJAU#JAS^HG \/W …XWIM/H*MeX

XbMcU,`WDPVHJAU#VdcU#Me_S GS]^)(CQ7QQ[QQQZQQQZQQ[Q *Rb

t*Q‰A‡ (&GJAU#JAS^HG\/W*…XWM^HV!XbMcU!HONPJU#VH!XbMcDEU bSPMcSNVBHU#GITcG\/W<FRDXWIHG…†MRWIJzF McF!JAS^HV

+X MPU'`WDPVHJAU#VdPU#Me_SoGSO^)(CQCQZQQQ[QQQZQQQ[QQQZQQQZQQ[Q *Rt

t*Q‰8^ (&GJAU#JAS^HG\/W …XWIM/H*MeX

XMcU`WDPVHJAU#VdcU#Me_S GSO^)(CQ6QQ[QQQZQQQZQQ[Q *Rq

t*Q‰cMr*NPJ&`LMPUU#JL`LHGIMcS HM!V`A\/WIGSPdWJE…EMcSPJAS^HV&F!JA\eVODEU JLK<X McU ^)( (1\ `WDPVHJAU#V8QQ *Rf

t*Q‰8b=r*NPJ‡ …EM/GS^H`LMcUU#JW\eHGIMcS XBDESP`LHGIMcS<XbMcU,`WDPVHJAU VdcU Mr_SoGS]^ KEGF JASPVGIMcSPV8Q1Q * *

t*Q‰8t=r*NPJ-EU\/SP`N KEGIVHOU#G-EDPHGIMcS6X McU`WDPVHJAU#VdcU#Me_S GSO^!KEGF!JASPVGIMcSPV8Q QQZQQ[Q ‰Aˆ/ˆ

t*Q‰8q=r*NPJU JW\YHGIT/JTe\/U#G\/-PGmWIGH#.%MeX'

GSl‡)( \/SPK%GSO^)(CQYQQ[QQQZQQQZQQ[Q ‰AˆE‰

(11)

9_,McDXWIKhWIGaYJHM!HONE\/SEa pMc-PGShl,\/WmWXMPU NPGIV…E\YHGIJASP`LJ&GS<KEGU#JL`LHGSPdF!JHONEU#McDPdPNoHONPGIV*VFo\/WmW

\/U#JA\ MeX[…EN^.cVGI`LV8Q

'\e`LaPSPMr_ WJLKEd^JJASCE\/SPKEJAU\/SPK &@WmW\/a<CPMcF XB\eG?XMcU+`LMRWmW\/-†McU\eHGIMcS GSHONPJ&_,McUa MeX

`LNE\/…PHJAU"b*Q r*NPJLGU,GSE…EDPH&GS HONPGIV\/U#JA\ _\YVF M/VH,XBUDPGIH#XBDXW Q

*_,McDXWIKKWIGaeJ<HMHONE\/SEa2CPHJLT^J McMPKE_*GS @T JLHJAU GSPdh\/SPK l+Mc-hCEF!GIHON XMcUHNPJLGU

NPJW…_*GIHON HONPJ&_,McUa MeX `LNE\/…PHJAU c;Q

E_*GIVONHM*JE…EU#JLVV HONE\/SEa^VHM,JLT/JAU#.^McSPJ'_NPMNE\eV NPJW…EJLKoF!J+GS F .VHDPKEGJLV)Q JLMcU#d/J

pMr_ W\^SPKEV_NPM/VJ,\eVVGIVH\/SP`LJ'GS!\/WmWcHNPGSPd^V[Fo\YHONPJAFo\eHGI`A\/WPNE\eV[-†JLJAS TcGIHO\/W /…E\/U HGI`ADXW\^U WI.XbMcU

GSE…EDPH&GSl`LNE\/…PHJAU‡*Q pMPKEKE.'\/SES2\/SPK/GF \edPDPJ&XbMcU*\eKETPGI`LJRMPS`LMcFo…EDPHGSPdFo\eHHJAU#V8Q

$h\/UaCl,\/UU#Me_ XbMcU*NPGIV&\eVVGIVH\/SP`LJ&GS dcDEU#J!…EU#JA…E\/U\YHGIMcS\/SPKlFo\/S^.<M/HONPJAU*\/UJA\eV8Q

r*NE\/SEa ./McD HMl T \/SPKHONPJ &T+CXp u XbMcUSE\/SP`LG\/W VODE…E…EMPU#H2K DEU#GSPd4F . VHODNV

KEJAS^HVONPG…1Q

k9GSE\/WmW. _*\/S^H HM&HONE\/SEa F.WIMeTPGSPd _*GXJ PCED1Q *JAUWIMeT/J*\/SPKo…E\eHGIJASP`LJ&NE\AT^J -EJLJAS

\eH,HGF!JLV †HONPJ&McSXWI.HONPGSPdoHONE\eH*NE\eVaeJA…PHF!J&d/M^GSPd;Q l+JANPGSPKlJLT/JAU#.<VODP`L`LJLVV XDXW F \/S<GIV\

(12)

r*NPJXbMRWmWIMr_ GSPd<VJL`LHGIMcSPVMeX9HONPGIV*HONPJLVGIV&\/U#J&GS^HJASPKEJLK%XMcU,…EDE-XWIGI`A\eHGIMcS1Q

u*NE\/…PHJAU ‡ CPHMc`LNE\eVHG`!\/SESPJA\/WIGSPd 1pMc-PGS]uQ1l,\/WmW' S*JLGmWmW &"Q1l+Me_"WIJAU GrqMW-†J

VODE-EF!GIHHJLKlHMCT'N^.PVGI`A\/W pJLTPGJL_ JLHHJAU#V ``LHMc-†JAU*‡/ˆ/ˆE‰NQ

u*NE\/…PHJAU[^ u*NE\/U\Y`LHJAU#G! A\eHGIMcS2MYXHNPJ6…EU#MP-E\/-PGmWGIVHG`%HOU\ T/JWIGSPdlVO\ WJLVFo\/S…EU#Mc-NV

WIJAF S JLGmWmW&"Ql'Me_"WIJAU r*NPMcFo\eV $ Q6k9GSEa pMc-PGS uQl,\/WmW ,CEDE-EF!GIHHJLK HM T'N^.PVGI`A\/W

pJLTcGIJL_ & ‡/ˆ/ˆE‰NQ

u*NE\/…PHJAU[c `GmW JWIK Mc…PHGF!G! A\eHGMPS DESPKEJAU<DESP`LJAU#HO\eGS^H . S*JLGmWmW &"Ql+Me_"WIJAU

pMc-PGS-uQl,\/WmW Sl…EU#JA…E\/U\eHGIMcS ‡/ˆ^ˆE‰RQ

u*NE\/…PHJAUUb ` 1V W\eHHGI`LJoSPM/GIVJWU JLK DP`LHGIMcSh\/SPK2HONPJoDXWIHGFo\eHJ!V`A\ WGSPd%MeX (1\ GS

H#_,M*KEGF!JASPVGIMcSPV /pMc-PGSIuQ8l,\/WmW RS*JLGmWmW & Q8l+Me_"WIJAU JLMcSE\/U#K $ QACE\/SPKEJAU &>WW\/a CPMcF XB\eG

CEDE-EF!GIHHJLK%HM3T+N/.PVGI`A\/W6pJLTcGIJL_ JLHHJAU#V ?‡/ˆ/ˆE‰RQ

u*NE\/…PHJAUt S M/GIVJoU#JLK DP`LHGIMcSh\^SPKhHONPJ "cJLK…EM/GS^H&MeX (1\ GSK^<KEGF!JASPVGIMcSPV

(13)

r*NPGIV'HONPJLVGV XMP`ADPVJLV9McS HONPJ*U#MRWIJ…XW\ ./JLKW-^."$DP`LHODE\eHGMPSPV'GSo-EM^HON!HONPJAUF \/WcMP…PHGF!G_V

VO\eHGMPS<HJL`NESPG!EDPJLV \^SPKKEG DPVGIMcSEWIGF GHJLKz\Yd/dcU#JLdc\YHGIMcS1Q@r*NPJaeJL.<GKEJA\ GIV*HONE\eH*-^.WHDESPGSPd

HONPJ"WIJLT/JWMeXGSE…EDPH*SPM/GIVJ&\eV.EFo…PHM/HGI`U JLVODXWIHV Fo\ .o-EJ&\eHH\eGSPJLKWF!McU#JU\/…PGIK*WI.RQ CPHMP`LNE\eVHGI`

Mc…PHGF!GIVO\YHGIMcSo…EU#Mc-XWIJAF!V'\^U#J,`LMcSPVGIKEJAU#JLK /_NPJAU#J*HONPJ+XBDESP`LHGIMcS!-EJLGSPd!Mc…PHGF!GIVJLKo`A\/SESPM^H[-†J

aPSPMr_S!J \Y`LH WI. \^SPKWFo\ .!McSXWI. -†JJLVHGFo\eHJLK6Q@r*NPJ&SPM/GIVJ*GS HONPJ*JLVHGFo\YHJLV'GIV*DPVJLKW\eV+HONPJ

\/SE\/WIM/dPDPJ MYXHONPJAUF \/W $DP`LHDE\eHGIMcSPV9GS!VGF!DXW\eHJLKW\/SESPJA\ WGSPdQ r*NPGIV \/SE\/WIM/d^.GIV'Fo\eKEJ,JE\e`LH

-^.RDPVJ*MYX \/S!\e`L`LJA…PHO\^SP`LJ,XDESP`LHGMPS c\/SPKoVHMc`LNE\eVHG`*\/SESPJA\ WGSPd!GIV'VJLJAS HM -†J*HONPJ*d^JASPJAU\/WIG_V

VO\eHGMPSMYXVGF!DXW\eHJLKo\/SESPJA\/WIGSPd!HMVHMc`LNE\eVHG`,Mc…PHGF GV\eHGIMcSo…EU#MP-XWJAF V8Q@r*NPJ…EU#Mc-E\^-PGWIGIVHGI`

HOU\ T/JWmWIGSPdVO\/WIJLVOFo\^Sl…EU#Mc-XWIJAFjT r*C*T GV&DPVJLKh\eV&\!HJLVH V -EJLK%XMPU,VHMP`NE\YVHGI`R\/SESPJA\ WGSPd

\/SPK!VGIdcSPG `A\/S^H'SPJL_ U#JLVODXWIHV'\/U J9XbMcDESPK6Q\ d/MPMcK `NE\^U\e`LHJAU#GIVO\YHGIMcS GV XbMcDESPK XbMcUHONPJ"Tr C*T

\/SPK!V`A\/WIGSPdo\/U#dcDEF JAS/HV9\/U#J,VONPMe_S HM&-EJ\e`L`ADEU\eHJ+XbMcU KEJLHJAUF GSPGSPd!HONPJ,JE…EJL`LHJLKCWIJASPd/HN

MeX*HONPJW…EUDESPJLK \/SPK]\2…EU#GIMcU#G+HMcDEU#V qVO…EJL`LG `A\/WmWI. …EUDESPJLK

…EUDESPJLK

6Q-\&S

M/GmW JWIK …EU#MJL`LH P\eV*\ `LMcFo…XWIJW`LMPFoF!JAU#`LG\/W…EU#Mc-XWIJAF †GV `LMcSPVGIKEJAU#JLKl\/SPKVHMc`LNE\eVHGI`&\/SNV

SPJA\/WIGSPdGIV*VJLJASHMWFo\/aeJ&\IW\/U#d^J&GF …EU#MeT/JAF!JAS^H*GSHONPJ J …†JL`LHJLK U#JLHDEUS<MeX9HONPJ…EU#M#JL`LH8Q

S*M^GVJU JLK DP`LHGIMcS!GS KEG DPVGIMcS WGF!GIHJLK\ed/dPU#JLdc\eHGIMcS (1\ qGIV aPSPMe_S&HM&-EJ `AUDP`LG\/W

HM<McDEU&DESPKEJAU#VH\/SPKEGSPdQ\ d^JASPJAU\/WIGIVO\eHGIMcS2MYX SPM/GIVJoU#JLK DP`LHGIMcS2M)1V W\eHHG`LJ!GIV&GS/HU#McK DP`LJLK

\/SPK SPM/GIVJU#JLK DP`LHGIMcSGIV!VONPMe_S2HM2-EJl\`LJAS^HOU\ W …E\/U\^F!JLHJAU&`LMPS/HOU MRWmWGSPd2HONPJ<dcU#Me_*HON%MeX

(1\IQ BS ‡,KEGF!JASPVGIMcSPV /\ WW DE\/S^HGIHGIJLV9\/…E…EJA\/UqHM*-EJ+GS $DPJASP`LJLKW-/.HONPJ9V WIMe_,JLVH`LMcUU#JL`LHGIMcS

HM<V`A\/WIGSPd _*GIHON%JE…EMcSPJAS^H ˆ^*Q%BS7^ KEGF JASPVGIMcSPV VMcF!J DE\^S/HGIHGIJLV\/UJRSPM/H&\ JL`LHJLK

-^.RHNPJ*V WIMe_ JLVH'`LMcUU JL`LHGIMcS!HM V`A\/WIGSPd PJ …†McSPJAS/H ˆ ‡*Q r NPJ U#JASPMPUFo\/WIGIVO\eHGIMcS MeX (1\ GV

`LMcSPVGIKEJAU#JLK ?\/SPKWHNPJSPM/GIVJU#JLK DP`LHGIMcS<\eH+HONPJ"PJLK…EM/GS^H GIV*F!JA\eVDEU#JLK6Q@r*NPJ SPM^GVJ cd^GT^JAS

\eV*HNPJU#JW\eHGIT/JTe\/U#G\/-PGmWIGIH .%GSlJPHOU#JAF \/W`WDPVHJAUU\eKEGDPVGIV XMcDESPKHMW-EJ

ˆˆ/ˆR^!\/SPK

(14)

S M/GIVJGIV\/S<GFo…†McU#HO\/S^H\eV…EJL`LHMeX …EN/.PVGI`LV ?`AU#JA\eHGSPd<GFo…EU#JL`LGIVGIMcS2GSlF JA\eVODEU#JAF!JAS^HV&\/SPK

Mc-PV`ADEU GSPd2DESPKEJAU WI.cGSPd2HOU#JASPKEV)QCr*NPGIVHONPJLVGIV J \/F GSPJLV HONPJoU#MRWIJoMeX*SPM/GIVJoGS2H#_,M6\^U#JA\eVUV

Mc…PHGF!GIVO\YHGIMcS2\/SPK2KEG DPVGIMcSOWIGF!GIHJLK \ed/dcU#JLdP\eHGIMcS1QBU McSPGI`A\/WmW. HONPJIW\eHHJAU*MeX'HNPJLVJoU#JWIGJLV

McS<HONPJ…EU JLVJASP`LJ!MeX[SPM/GIVJXMcUHONPJ\/…E…EJA\^U\/SP`LJMYX[HONPJ…ENPJASPMPF!JASPMcS1Q

BS Mc…PHGF!GIVO\eHGMPS SPM^GVJ*GIV\^S!GFo…EMcU HO\/S^H9XbJA\eHODEU#J*HMJASPVODEU#JHONE\eH'\^S!Mc…PHGF!GIVO\eHGMPS

SPKEV&HONPJ dRWIMc-E\/WMc…PHGF!DEF-QX'_,JR\/U#J&MPSl\<UDPd/d/JLK%HJAUU\eGS F MrTPGSPdWMPSXW.%KEMe_SENPGmWW _,J

Fo\ .h-EJL`LMcF JVHODP`LalGS \ WIMc`A\/WhF!GSPGF!DEF DESE\^-XWJHM SPKHONPJWHUDPJ3WIMe_,JLVHR…†M/GS/H!MeX

HONPJCW\/SPKEV`A\/…EJRQ*Me_,JLT/JAU GX*_,JRVMcF!JLHGF!JLV!F!MeT/JRDE…ENPGmWmW 'HONPJASz_ JoFo\ .2\AT^M/GIK d^JLHHGSPd

VHODP`La<\/SPK SPK dRWIMc-E\ W<F!GSPGF!DEF-QUr*NPGIV \Y`L`LJA…PHO\/SP`LJMYX DE…ENPGmWmW F!MeT/JLV*Fo\^SPGXbJLVHVGIHVJWX

\eV*HNPJAUFo\/W$DP`LHODE\eHGMPSPVGSVGF!DXW\eHJLK2\/SESPJA\/WIGSPdQ

(&GqDPVGMPSZWIGF!GIHJLK\ed/dcU JLdc\eHGIMcS (1\ GIV \F!MPKEJWEMeX \/…XW\e`LG\/S dcU#Me_*HN1Q U#Me_*HON

MeX'VOF McM/HNMc- #JL`LHVGS \ q\/…XW\e`LG\/S JWIKhGIVDESPVHO\/-XWIJo\/SPK2HONPJX U\Y`LHO\/W9…E\eHHJAUSPVMeX (1\

\/U#JHNPJlU#JLVODXWIHWMYX HONPGIV<GSPVHO\/-PGmWIGIH.RQ r NPJVHO\/SPK \/U K GF …XWJAF JAS/HO\YHGIMcS MeX(1\GIVSPM/GIV.

\/SPK2\!_*\A.<MeX'U#JLK DP`LGSPdHONPJGS $DPJASP`LJ!MeX SPM/GIVJ!NE\eV-†JLJASl`LJAS^HOU\ WHMoHONPJ!DESPKEJAU#VHO\^SPKEGSPd

MeX'GIHV&JLT/JAS^HODE\/W9-EJANE\ TPGIMcDEUGS7W\eHHGI`LJ V -E\eVJLKF!McKEJWIV8QZ NE\eH&NE\/…E…EJASPV _NPJAS%HONPJAU#J!GIVSPM

DESPKEJAU WI.PGSPdhW\eHHGI`LJNE\eVU#JAFo\eGSPJLK`LMcS^HOU#MeT/JAU#VG\ W PGShW\/U#d/J&…E\/U#H*-†JL`A\/DPVJR\ VODPGIHO\^-XWJSPM^GVJ

U#JLK DP`LHGIMcS HJL`LNESPGEDPJoNE\ T/J SPM^H-EJLJAS2\ Tr\eGmW\/-XWIJRQ

u*NE\/…PHJAU ‰,d/GIT/JLV*\/S!GS^HOU#MPK DP`LHGIMcSoHM&Mc…PHGF!GIVO\eHGIMcS<\/SPK<VHMP`NE\YVHGI`*Mc…PHGF!GIVO\YHGIMcS1Q

ZhJ`LMcSPVGIKEJAU+HONPJ&DPVJ*MeX NPJADEU#GIVHGI`LV \/…E…EU#M)cGFo\YHJF!JLHONPMPKEV HMVMNWT^J NE\/U KoMc…PHGF!GIVO\eHGMPS

…EU#Mc-XWIJAF!V)QCPHMc`LNE\eVHG`2Mc…PHGF GV\eHGIMcS …EU#MP-XWJAF V \/U#J2GS^HOU#MPK DP`LJLK &GS _NPGI`LN HONPJ2JE\e`LH

Tr\ WDPJ MeX HONPJ XBDESP`LHGIMcSh-†JLGSPd2Mc…PHGF GVJLKhGIVRSPM/H&acSPMe_ JE\e`LH WI. \^SPK]F!DPVH -†JoJLVHGFo\YHJLK

\/SPKl…EU#JLTPGIMcDPVVMRWDPHGIMcSlF JLHONPMcKEV&\/U#JU JLTcGIJL_,JLK6Q

u*NE\/…PHJAU‡ GV&`LJAS^HOU#JLK2\/U#McDESPKh\<…E\/U#HGI`ADXW\/UVHMc`LNE\eVHGI` Mc…PHGF!GIVO\eHGIMcS%HJL`LNESPG! DPJ

(15)

\/SESPJA\/WIGSPdQ r*NPGIV<GIVW\e`LNPGJLT^JLK„-^.`NPMPM/VGSPd \/S \e`L`LJA…PHO\/SP`LJ XDESP`LHGMPS *\/SPK HONPJl\^…E…EU#M/V

…EU#G\eHJo\e`L`LJA…PH\/SP`LJ XDESP`LHGIMcSzGVXbMcDESPK2XbMcU\WSPDEF!-EJAUMeX,KEGIVHOU#G-EDPHGIMcSPV MeX HONPJ!JLVHGF \eHJLV8Q

JASP`LJ EVHMP`NE\YVHGI` \^SESPJA\/WIGSPd GV HO\/aeJAS!HMR-EJ&HNPJd/JASPJAU\/WIGV\eHGIMcS<MeX VGF!DXW\eHJLKl\^SESPJA\/WIGSPd

HMoVHMc`LNE\eVHG`&MP…PHGF!GIVO\eHGIMcS%…EU#Mc-XWIJAF!V8Q

u*NE\/…PHJAU V ^ co\/U#J&`LMPSP`LJAUSPJLK_*GIHON<H#_,MR\/…E…XWIGI`A\eHGIMcSPVMeX9VHMP`NE\eVHGI` \/SESPJA\ WGSPdQ

r*NPJ&…EU#Mc-E\/-PGmWIGIVHGI`HU\ T/JWmWGSPd!VO\/WIJLVOF \/SW…EU Mc-XWIJAF GIV*\F McKEJWMeX \e`A\eKEJAF G`Tr\ WDPJ&_NPG`LN GV

aPSPMr_S<HM6-†JRNE\/U K PXMPU*_NPGI`LNl\ d/MPMcK`LNE\/U\e`LHJAU#GIVO\eHGIMcS GV&…EU#MeTPGKEJLK6Q"`GmW JWIKh…EU#M#JL`LHV

\/U#J2`LMcF …XWJ …EU#MJL`LHV_NPGI`N …EU MrTPGIKEJ]\KEJAF!McSPVHOU\eHGIMcS MeX!HONPJ]DPHGmWIGIH#.4MYXRVHMP`LNE\eVHGI`

\/SESPJA\/WIGSPdGSl\ `LMcFoF JAU#`LG\/WVJLHHGSPd;Q

u*NE\/…PHJAU V b t!`LMcSP`LJAS^HOU\eHJMcS(1\ \/SPK%HONPJ&GFo…EMcU HO\/SP`LJ&MeX'SPM/GIVJU#JLK DP`LHGIMcS1Q

\‚SPJL_ \/SPK<JW`LGIJAS/H*SPM/GIVJU JLK DP`LHGIMcS<HJL`LNESPGEDPJ&GIV,KEJLV`AU#G-EJLK XMcU M)1VW\eHHGI`LJ%(1\ _NPGI`LN

Fo\ . -EJ\/…E…XWIGJLK GS \/S^. KEGF!JASPVGIMcS1Q \V. F …PHM/HGI`2Tr\/WDPJLV2\/SPK `LMcUU#JL`LHGIMcS„HM]V`A\/WIGSPd

JE…EMcSPJAS^HVXbMcU,GIVM/HOU Mc…PGI` ((\ \/U#JXMPDESPK6Q r*NPJoU#JASPMcUFo\/WIGIVO\eHGIMcS MeX (1\ GIV&`LMcSPVGKEJAUJLK

\/SPK HONPJ "PJLK …EM/GS^H6SPM/GIVJhU#JLK DP`LHGMPS GIV<X MPDESPK ,_NPGI`NWIJA\eKEV<HM]`LMPSPVGIKEJAU\eHGIMcS MeX\/S

J JL`LHGIT/JSPM/GIVJU#JLK DP`LHGIMcS1Qr*NPJ U#VH'JLT^JAU GF …XWJAF JAS/HO\YHGIMcSoMYXqSPM^GVJU#JLK DP`LHGIMcS M)1V W\eHHGI`LJ

(16)

! #"$%&'(*)+-,/.0123456,/7*8-29:;<=>,?:;.@

23.BAC,/0D0929:<==,?:;%2FE$'G0H7I<+J)<7I8K0L,NM87/7O:=0L'GP

QBR

2TS"%,D2VU+A!T7/7

`&…PHGF!GIVO\eHGIMcS Fo\ .6-†JoKEJLV`AU#G-EJLK\eVSPKEGSPd%HONPJ!JcHOU JAFo\/WTe\/WDPJoMeX \ X DESP`LHGIMcS1Q

X'MPSPJ`LMcSPVGIKEJAU#V\ VGFo…XWIJ X DESP`LHGMPS VO\ . /W XW

†HONPJoF!GSPGF!DEF MeX9VODP`LNl\ X DESP`LHGMPS

Fo\ . -EJ%XbMcDESPK T/JAU#. DPGI`LaRWI.RQ *Me_,JLT/JAU ,HONPJ2F!McU#J2`LMcF …XWJ HONPJ%X DESP`LHGMPS HONPJ F McU#J

KEGW`ADXWIHWHNPGVo\/SE\ W.PVGIVW-†JL`LMcF!JLV '…E\^U#HGI`ADXW\/U WI. GSHONPJW`A\YVJWMYX S T VB`LMcFo…XWIJLHJl…EU#MP-XWJAF V8Q

kDEU HONPJAUF!McU J /GSoVMcF!J`A\eVJLV'MPSPJ Fo\ .!SPM/H*-EJ&\/-XWIJHM J \Y`LH WI.oJLTr\/WDE\eHJHONPJ XBDESP`LHGIMcS hQ

BS HONPGIV*`LNE\/…PHJAU+HONPJHMc…PGI`&MeX S T VB`LMcFo…XWIJLHJASPJLVVGIV,GS^HOU#MPK DP`LJLKl\eV*\/U#JVMcF JHJL`LNNV

SPG! DPJLV&_NPG`LNl\^U#J&KEJLVGIdcSPJLK2HMoVMNWT^JZS T VB`LMcFo…XWIJLHJR…EU Mc-XWIJAF!V8Q*CPHMP`NE\YVHGI`&Mc…PHGF!GIVO\eHGMPS

GIV*HONPJAS KEJLV`AU#G-EJLK \/SPKHONPJ&VO…†JL`LG `R`LNE\/WmWIJASPd/JR…†M/VJLK2-/.oVHMc`LNE\eVHG`&MP…PHG F!GIVO\eHGIMcS%…EU#Mc-NV

WIJAF!VGV `LMcSPVGIKEJAU#JLK6Q

Y 3 Y 0 ƒ[Z/{9&R 9'

\^]_\^]`\ a ;!b?jl@bj%@B>

ZhJFo\ .2`LMcSPVGIKEJAUo…E\/U#HGHGMPSPGSPd Mc…PHGF!GIVO\eHGIMcS…EU#Mc-XWIJAF!V GS^HM%H_,M%VJLHV _NPGI`LN_ J _*GmWW

KEJLV`AU#G-†J6\eV JA\eV. \/SPK ONE\^U#K Q &'\eV. …EU#Mc-XWIJAF!V\^U#J!McSPJLVXMPU_NPGI`N%_,J!`A\/S SPKhHONPJ

VMRWDPHGIMcSzGS \<HGF!JW…EMNW.ESPMcF!G\/W+GS2HONPJoVG! LJoMYX HONPJW…EU Mc-XWIJAF-Q \^U#Kh…EU#Mc-XWIJAF!VR\^U#J!McSPJLV

(17)

MeX*HONPJW…EU Mc-XWIJAF GjQJRQ<HONPJoHGF JWU#JEDPGU JLK]HMSPKHONPJoVMRWDPHGIMcShdPU#Me_*VJE…EMPSPJAS/HG\/WmWI.McU

XB\eVHJAU+_*GIHON HONPJ…EU#Mc-XWIJAF VG! LJ Q

S T VB`LMcFo…XWIJLHJGIV\^S!GFo…EMcU HO\/S^H'`W\eVV+MeX …EU#Mc-XWIJAF!V,_NPGI`N<\/U#J 0,/:4. HM!-EJ&NE\/U#K

VJLJ‡E‰ E`LNE\/…PHJAU%*Q $h\/S^.W`LMcF!-PGSE\eHMPU#G\/WMc…PHGF!GIVO\YHGIMcS2…EU#Mc-XWIJAF!V&\/U#JZSUTVB`LMcFo…XWIJLHJ

VODP`LN2\eV&HONPJ!HOU\ T/JWmWIGSPdlVO\/WIJLVOF \/S2…EU#Mc-XWIJAFjr*C*T &\/SPKh…EU#GF!J!XB\e`LHMcU#GIVO\YHGIMcS1QU\ WIHONPMcDPdPN

HONPJLVJ…EU Mc-XWIJAF!V \^U#J-EJWIGIJLT/J!HMW-†JRNE\/U#K †HONPGIVNE\eVSPJLT/JAU-EJLJAS%…EU#MeT/JLK6Q

S T VB`LMcFo…XWIJLHJh…EU#Mc-XWIJAF!Vl\^U#J SPM/H<HONPJhNE\/U#KEJLVH6…EU#MP-XWJAF Vl…EM/VVG-XWIJRQ r*NPJAU#J2\/U#J

…EU#Mc-XWIJAF!V!_NPM^VJoVMRWDPHGIMcSU#JEDPGU#JLVR\/S JE…EMcSPJAS^HG\/W'HGF JWHM_U#GIHJRQ *Me_,JLT/JAU HONPJ GFaV

…EMPU#HO\/SP`LJ!MeX,HONPJo`W\eVV&MeX"S T VB`LMcFo…XWIJLHJW…EU Mc-XWIJAF!V VGSP`LJ<HONPJL.2\/U#J!VGFo…XWIJWHMWVH\eHJo-EDPH

KEGW`ADXWIH&HM!VMRWIT/J EVNPMcDXWIK SPM/H-†JRSPJLdRWIJL`LHJLK6Q

\^]_\^] a

;t @bj >@ s*g k/j=bqj+= =„mOgkh;'=9k/@B> je@BiE>

r*NPJ*`W\eVV'MeXNE\/U#KW…EU#MP-XWJAF V'GIV,GF …EMcU#H\/S/H eVGSP`LJGIH'KEJ SPJLV,_NPJAS GH+GIV SPMZWIMcSPd^JAU XbJA\eVG-XWIJ

HM%VJA\/U#`LN%XbMcU&JE\e`LH!VMRWDPHGIMcSPV8QO\UWHNPMcDPdcN GH!GIVo…EM/VVG-XWJHM SPK \/ShJE\e`LH VMRWDPHGIMcSHM

\hNE\/U K …EU#MP-XWJAF ,HONPJVG! LJlMeXHONPJl…EU#Mc-XWIJAF HONE\eHoFo\ .]-EJ%VMRWIT/JLK _*GmWmW&\/WI_\ .PVo-EJT/JAU#.

WIGF!GIHJLK6Q *JASP`LJ\…EMc…EDXW\/U \/…E…EU#Mc\Y`N2GIV HMVJLH\"PJLKhHGF!J XMcU&\^S \ Wd^McU#GIHONEFHMlUDES

\/SPK2HO\/aYJRHNPJ -†JLVHVMRWDPHGIMcS XMcDESPK _*GIHONPGS2HONE\eH&HGF!JRQIr*NPGIVR\^…E…EU#Mc\e`LN%GV&MPSXW.2VJASPVG-XWIJ

_*GIHON%…EU#Mc-XWIJAF!VXbMcU,_NPGI`LNl\<SPJA\/U VBMc…PHGFo\ WVMRWDPHGIMcS%GIV DPVJXDXWjQCPMPF!JR…EU#MP-XWJAF V VDP`N%\eV

…EU#GF!JXB\e`LHMcU GV\eHGIMcS cU J DPGU#J\/S JE\e`LH'VMNWDPHGMPS \/SPK<GH,GIV,HONPGIVU#JEDPGU#JAF!JAS^H HNE\eH*Fo\/aeJLV

GS/HJAUSPJLHVJL`ADEU#GIH.<VMoKEGW`ADXWIH&HMo-EU#JA\^aQ

BS HONPJ&SPJcH*H#_,M&VJL`LHGIMcSPV _ J `LMcSPVGIKEJAU,VMcF J NPJADEU#GIVHG`LV E`LMcSP`LJAS^HOU\eHGSPd!McS VGF!DNV

W\eHJLKl\/SESPJA\ WGSPd \eV*HNPGVNE\eV-EJLJAS HONPJ&VODE- #JL`LH*MeX9HONPJ&_,McUa …EU#JLVJAS^HJLKGSHONPJXbMRWmWIMr_ GSPd

`LNE\/…PHJAU#V8Q

Y 3 & $ ,+€!'* )

\&SESPJA\/WIGSPdGIV\o…EU Mc`LJLVV&-^._NPGI`N%\oFo\eHJAU G\ W VODP`LNl\eV dRW\eVV GIV NPJA\eHJLKlHMW\ F!McKEJAU\eHJ

HJAFo…†JAU\eHODEU JHM]U#JAF!MeT/JKEJXbJL`LHVW\^SPK VMeXbHJASHONPJ2Fo\eHJAU G\ W'XbMcU!_,McUa/GSPdQ GUaP…E\eHOU G`La

JLH&\/WjQbN^9DPVJLK2HONPGIV…EU#Mc`LJLVV \eV GSPV…PGU\eHGIMcS%XbMcU*\/S Mc…PHGF!GIVO\eHGMPSlHJL`NESPG!EDPJ VGFRDXW\eHJLK

\/SESPJA\/WIGSPdQ"`&SPJ `LMcSPVGIKEJAU#V&HONPJ&Mc…PHGF GV\eHGIMcS2…EU#Mc-XWIJAF \eV\!HONPJAUFo\/WV.PVHJAF GS J DPGmWIG-NV

U#GDEF\eH&\od^GT^JAS2HJAFo…†JAU\eHODEU#JRQ X'HONPGIV&V.cVHJAF GIV\/WmWMe_,JLK2HM<`LMcMRW V WIMe_"WI. HONPJAS2\eH LJAU#M

(18)

VMRWIT/JLK6Q*CPGF!DXW\eHJLKh\/SESPJA\/WIGSPdlGIV&\W…†Me_ JAUX DXW ?d/JASPJAU\/WMP…PHGF!GIVO\eHGIMcS%HJL`LNESPG! DPJ!HONE\eHNE\eV

XbMcDESPK \/…E…XWIGI`A\eHGIMcS GSlFo\/S^.…EU#Mc-XWIJAF!V8Q

\^] 6]`\

:<;'=†k g 9t *b @BiE> b?kg W;*b?@'>

\€V.PVHJAFNPJWIK \eH\"PJLK]HJAF …EJAU\eHDEU#J _*GmWmW,Mc`L`ADE…^.h\VHO\YHJ _*GIHON2JASPJAU#d/.

\e`L`LMcU KEGSPd<HM!HONPJ!…EU#Mc-E\/-PGmWIGIH .

#‰RQ‰

_NPJAU#J

#

GIVHONPJ&GS/T^JAU#VJHJAFo…EJAU\YHODEU#J \/SPK"!$# GIV l+MRWIH AF \/SES V`LMcSPVHO\/S^H8Q BS HONPJ

XbMRWmWMe_*GSPd%_ J!`LNPM/VJ!$# ‰ XbMcU`LMcS^T/JASPGIJASP`LJRQ!ZhJ!_*GmWmW \eHHJAFo…PHHMVGF!DXW\YHJW\ V.cVHJAF

XbMcU,_NPGI`LNHONPJ&VH\eHJLV*\/U#J&`LNPM/VJAS%\e`L`LMcU#KEGSPdoHM HONPGIVUDXWIJRQ

Z NPJASlDPVGSPd $2McS^HJZu*\/U WIMWF JLHONPMcKEV _,J_*\/S^H*HM JW`LGIJAS/H WI.%VO\/Fo…XWIJ!HONPJKEGIVHU#G_V

-EDPHGIMcS%MeX'VH\eHJLV*MeX'HNPJV.PVHJAF VMoHONE\YH_,JRF \A.<JLVHGFo\YHJR\ T/JAU\Yd/J DE\^S/HGIHGIJLV VDP`N%\eV

HONPJ&\ T/JAU\ed/J,JASPJAU#d^.oMeXHONPJV.cVHJAF-Q bH+_ MPDXWKSPM/H,-EJ&JW`LGIJAS^H HMR\eHHJAFo…PH'HM GFo…XWIJAF!JAS/H

J1Q ‰RQ‰,KEGU#JL`LH W. †GSPVHJA\eK<GH+GIV `LMPS/T/JASPGIJAS^H,HMDPVJ&\%$z\^UaeMeTR…EU#MP`LJLVV8Q@r*NPJGSPGIHG\/WVHO\eHJ,MeX

HONPJ&V.PVHJAF GIV*`LNPM/VJASU\^SPKEMcFIWI.6\^SPKHONPJ&V.PVHJAF GIVHONPJAS\ WWIMe_,JLKHMoFo\/aeJ&\!HOU\/SPVGIHGIMcS

MeXbHJAS `A\/WmWJLKh\<F!MeT/J XU#MPF€McSPJVHO\eHJ HM6\^SPM/HONPJAU E_ GHN \<…EU#Mc-E\^-PGWIGIH#.l_NPGI`LN%KEJA…EJASPKEV

McSXWI.McS<HONPJ&VHO\eHJLV,GS/T^MRWIT/JLKlGS%HONPJF!MeT/JRQ

k McU+JA\e`LN<F!MeT/JHNE\eH,_ J&…EU#Mc…†M/VJ E_,J`NPMPM/VJ&HM \Y`L`LJA…PH*HONE\eH,F!MeT/J_ GHN\d^GT^JAS

…EU#Mc-E\^-PGWIGIH#. &% ',XbMcU*\<F!MeT/JXBU#McF VHO\eHJ(% HMWVHO\eHJ)Q,Z JGFo…EM^VJ!HONPJ`LMcUU#JL`LH

JEDPGWIG-EU#GDEFKEGIVHU#G-EDPHGIMcS%TcG\ HONPJ&`LMcSPKEGIHGIMcSMYX[KEJLHO\YGWIJLKh-E\/W\/SP`LJ ?_NPGI`N GIV

&% &% * +% #‰RQ‡

& 1Q‰RQ‰GIVHONPJAS<VO\eHGV JLK …EU#MeTPGKEJLK

&% ,

-%

. 0/1 #‰RQ^

_NPJAU#J)2 GIV*HNPJJASPJAU#d/.<KEGJAU#JASP`LJR-†JLH _,JLJAS HONPJ&VHO\eHJLV3%\/SPK4%@Q

\^] 6]

=Ejekcgs*g pb@B> b 652p`b 87*=†k b?p9 g k/@bj/; >

Z NPGmWVHoJGQ,‰RQ_^lJASPVDEU#JLVoHONPJAUF \/W'JEDPGmWG-EU#GD?F \eH!GS^T/JAU#VJWHJAF …EJAU\eHDEU#J 9GIHoKEMPJLVoSPM/H

XBDXWWI. KEJLHJAUF!GSPJlHONPJ<XbMcUF MYX ZQr*NPGIVWF!JA\/SPVoHNE\eH!HONPJAU#Jl\/U#Jl\ ScDEF!-EJAU MeX KEGJAU#JAS^H

(19)

1

0

Metropolis: Min(1,e )

-

β

x

Glauber: 1

1+e

x

β

x

k9GIdcDEU#Jo‰RQ‰ r*NPJUW\/DE-EJAU*\^SPK $ JLHU#Mc…EMNWGIV\Y`L`LJA…PHO\/SP`LJXBDESP`LHGIMcSPV8Q X[_,J&\/U#JVGFRDXW\eHGSPd

\ HONPJAUFo\/WqV.cVHJAF\eH*GS^T/JAU#VJ!HJAF …EJAU\eHDEU#J \^SPK F!JA\eVDEU#JHONPJ J JL`LHMeX'\<F!MeT/J&HMW-†J

2 HONPJAS _ JFo\ .2`NPMPM/VJ<HMh\e`L`LJA…PHoF!MeT/JLV _*GIHON \l…EU#MP-E\/-PGmWGIH#. d/GIT/JAS-/. HONPJ6\^-EMeT/J

XBDESP`LHGIMcSPV

W\/U#d/J*\eV9…EM^VVG-XWIJ 9GIV9HM`NPMPM/VJ 2 ‰9XbMcU 2

e_*GIHON \eV VOFo\/WmWE\eV'…EM/VVG-XWJRQ

r*NPGIVGV HONPJ$2JLHOU#MP…EMRWIGIV\/WId/McUGHNEF \/SPK%GV KEJ SPJLKz-^.

2

‰ XbMcU 2

ˆ

0/1 XbMcU 2 ˆ

#‰RQc

VGSP`LJ!JGQq‰RQ^!JASPVODEU JLVHONE\YH ˆ*Q \ KEG JAU JAS/H`NPM^G`LJ!MeX GIVHMoGSPVGIVH&HONE\eH 2

2 ‰RQr*NPGIVUWJA\YKEVHM!HONPJ W\/DE-†JAU*\e`L`LJA…PHO\/SP`LJ!UDXWIJ d^GT^JASl-^.

2

‰

‰ 0/1

#‰RQb

_NPGI`LN NE\eV!HONPJl-†JASPJ HoMeX-EJLGSPd \^SE\/WI.cHGI`RQ l+M/HON\e`L`LJA…PHO\/SP`LJlUDXWJLV\/U#J<GmWmWDPVHOU\eHJLK GS

dcDEU#Jo‰NQ‰RQ

\^] 6]

:= s*=†k8b?j 9kc=h>ei/;+= 9pB= >

X&_,Jl\/…E…XWI. HONPJVGF!DXW\eHJLK4\^SESPJA\/WIGSPd \/WId/McU GHNEF '_,JlNPMc…†JlHM2VGF!DXW\eHJ2\2V.PVHJAF GS

HONPJAUF \/W,J DPGmWIG-EU#GDEFx\eH!VMcF JHJAFo…EJAU\eHODEU#JRQZhJ`LMcSPVGKEJAU VHO\/U#HGSPd HONPJV.PVHJAF \eHo\

(20)

HJAFo…†JAU\eHODEU J HONPJ!V.PVHJAF _*GmWmW9Mc`L`ADE…^.lHNPJRdPU#McDESPKlVHO\eHJ McU J DPGITr\ WJAS^H WI.hNE\ T/J!VMRWIT/JLK

HONPJ Mc…PHGF!GIVO\eHGIMcS2…EU#MP-XWJAF …EU#MeTPGIKEJLK _,J!NE\AT^JU#JAFo\eGSPJLK2GSlJEDPGmWIG-EU GD?F-Q Me_ JLT^JAU EHM

VHO\ .GS2JEDPGmWG-EU#GDEF KEMe_S%HM LJAU#MHJAF …EJAU\eHDEU#JoU#J DPGU#JLV!\/S%GS SPGHJ6\/F McDES/HMeX,HGF!JRQ

BS6X \Y`LH JAF \/S JAFo\/S _^Rf VONPMe_,JLKHONE\eH*GX'McSPJ Tr\/U#GIJLVHONPJHJAFo…EJAU\YHODEU#J\e`L`LMcU KEGSPd

HM

WS

#‰RQt

_NPJAU#J GIV\`LMPSPVHO\/S^H9_NPGI`N<F!DPVH'-EJ`LNPM/VJAS!HM-†J*VOD W`LGIJAS^H WI.CW\/U#d/J ^HONPJAS!HONPJ V.cVHJAF

_*GmWmWMc`L`ADE…^.oHONPJ*dPU#McDESPKoVH\eHJ*_*GIHON …EU#Mc-E\/-PGmWIGIH .2‰ /\eV+HONPJHGF!J*HONEU#MPDPdcN!HONPJ*VGF!DXW\eHGIMcS

MQ

QS^XbMcU#HDESE\eHJWI. EHONPGIV&GIV SPM^HF!DP`N%DPVJ EVGSP`LJ!_,JRSPJLT/JAU*NE\ T/J!\/S<GS SPGIHJW\/F!MPDES/H

MeX*HGF!JRQ7r*NPJAU#JNE\ T/J6-†JLJAS \lSPDEF!-EJAU MeX*VODPd/d/JLVHGIMcSPV X McUd/MPMcKV`LNPJLK DXWJLV _NPJASHGF!J

GIV3WIGF!GIHJLK * * @fRb +‡/ˆ Q $ M^VHoSPM/HO\/-XWIJlGIV<HONPJV`LNPJLK DXWIJ _NPG`LN Fo\eGS^HO\eGSPV\2`LMcSPVHO\^S/H

HONPJAUF McKE.ESE\/F!GI`lV…EJLJLK _fRt *Rb Q r*NPJl…EU#GSP`LG…XWIJhGIV6\2VGF …XWJEVGXMcSPJ`A\/SESPM/HoVHO\ .hGS

JEDPGWIG-EU#GDEF McSPJ VONPMcDXWIKh\eHHJAFo…PH HMoVHO\ .\eH&\ "cJLK HNPJAUF!MPKE. SE\/F G` KEGIVHO\/SP`LJ XBU#McF

JEDPGWIG-EU#GDEF-QUr*NPGIV WIJA\eKEVHM!HONPJ&VODPd/d^JLVHGIMcSHM DPVJ

$

#‰RQq

_NPJAU#J \^SPK 8\^U#J*HONPJU JW\ \eHGMPS!HGF!J&\/SPKWNPJA\eH+`A\/…E\e`LGIH#.RMYX HNPJ*V.PVHJAF /U#JLV…EJL`LHGIT/JWI.RQ

GIV9HONPJ,HONPJAUF McKE.ESE\/F!GI`,VO…EJLJLK P\*`LMPSPVHO\/S^H MPSPJ'`LNPMcM/VJLV HMKEJLHJAUF GSPJ HONPJ'MeT/JAU\ WWPVO…†JLJLK

MeX HNPJ&`LMcMRWIGSPdQ BS…EU#GSP`LG…XWIJoHONPGIV*GIV*HONPJ&MP…PHG Fo\/WV`LNPJLK DXWJ!HMoDPVJRQ *Me_,JLT/JAU ^GHU#J DPGU#JLV

HONE\eH _ JaPSPMe_ HONPJU#JW\)E\eHGIMcS HGF!J&MeX9HONPJV.cVHJAF _NPG`LNGIV&SPM/H*VGFo…XWIJ!HMoF!JA\eVDEU#JRQ

pJLJLV l,\/WmW *†‰ 9\/WIVMWDPVJLK2\V`NPJLK DXWIJ!_NPGI`N%DPHGmWIGVJLKhHONPJU#JW\)E\eHGIMcS<HGF!JRQ

BS HONPGIV!HONPJLVGIVWF!DP`N F!McU#J<VGFo…XWIJV`LNPJLK DXWJLV<NE\AT^J6-†JLJAS DPVJLK '-E\eVJLK F!McU JoMcS

GS/HDPGHGMPS \^SPK_NE\eH\/…E…†JA\/U#V*HMR_,McUaoHONE\^S…EN/.PVGI`A\/W …EU#GSP`LG…XWIJLV8Q

Y

3 9€,$&?{9

CPGF!DXW\YHJLK4\/SESPJA\ WGSPdGIVWSPM/HoHONPJMPSXW. NPJADEU#GIVHGI`2\/WId/McU#GIHONEF ,\/SPK GIVW-^.]SPM]F!JA\^SPVoHONPJ

VGFo…XWIJLVH8Q `HONPJAU'\/WId/MPU#GIHONEF!V NE\ T/J…EU#MeT/JLKoDPVJX DXW?GSoKEGJAU#JAS^H'VGIHODE\eHGMPSPV c\/SPK HONPJ*`LNPM/GI`LJ

MeX'_NPGI`LNhNPJADEU#GIVHGI`oHMlDPVJ!GV\^SlGFo…†McU#HO\^S/HMcSPJRQZrqMd/GIT/Jo\ $\ T/MPDEU*MeX,HONPJ!KEGIT/JAU#VGIH#. \

(21)

\^]C]`\ a

gp b p9 gk/@bj/;

r*NPJ%ZhMRW8\/WId/McU GHNEF ‰^‰8t ‰AˆNf&GIVl\2Tr\^U#G\/S/HMYX VGF!DXW\eHJLK \/SESPJA\/WIGSPd KEJLVGIdcSPJLK XbMcU

JW`LGIJAS^HVGF!DXW\eHGIMcS2MeX,HONPJ bVGSPd2F!McKEJWjQIr*NPJ VGSPd F!MPKEJW9GIV\F!GI`AU#M/V`LMc…PGI`WF McKEJW9MeX

\ XbJAUU#McFo\YdcSPJLH_NPJAU#J!JA\e`N%VGIHJ MeX\W\eHHGI`LJW…†M/VVJLVVJLVR\ VO…PGSh_NPGI`LN2`A\/S2-EJ JLGHNPJAU&DE…

McU+KEMe_S1Q\K \e`LJAS^H*VO…PGSPV…EU#JXbJAU,HMo-†J…E\/U\/WmWIJW _*GIHON\ "PJLKJASPJAU#d/.…EJASE\/WIH#. X McU,-†JLGSPd

\/S^HG_V …E\/U\/WmWIJWjQZ\ H WIMe_ HJAFo…†JAU\eHODEU#JLV VGFRDXW\eHJLKz\^SESPJA\/WIGSPd%GIV&GSPJ<`LGJAS^Ho\eHVGFRDXW\eHGSPd

HONPJ bVGSPdF!McKEJWjQ X'_,J&`LMcSPVGIKEJAU&\oF MrT^JHM $ G…2\!VGSPdNWJ VO…PGS HONPGIV*_*GmWmW'F!M/VH WIGaeJWI. -†J

U#J#JL`LHJLK VGSP`LJF!M/VHVO…PGSPV _*GmWW,-EJ<VODEUU#MPDESPKEJLK]-/.2Fo\/S^.lM/HNPJAUVO…PGSPVR…EM^GS^HGSPd2GS2HONPJ

VO\/F J&KEGU JL`LHGIMcS1Q

r*NPJ,ZhMRW6\/WId/MPU#GIHONEF \ T/M/GIKEV9HONPGIV@WIMr_ HJAFo…EJAU\YHODEU#J …EU Mc-XWIJAF -^."$ G…E…PGSPd`WDPVHJAU#V

MeX'VO…PGSPV!\eHHNPJRV\/F!JHGF JRQ r NPJ \ Wd^McU#GIHONEFFo\eGS^HO\eGSPV\ HONPJAUFo\/W VJWIJL`LHGIMcShUDXWJ q_*GIHON

JA\e`LNF!MeT/J-EJLGSPd<VJWIJL`LHJLK2GSl\oFo\^SESPJAU,VMoHONE\YH JLT^JAU#.6…EU Mc…EM/VJLK F!MeT/JGIV \e`L`LJA…PHJLK6Q

CPGFRDXW\eHJLK \/SESPJA\/WIGSPd GIVoGSPJW`LGIJAS^HW\eH[WMe_ HJAFo…EJAU\eHODEU#JLV VH\A.PGSPdlGSMcSPJ<`LMcSNV

dcDEU\YHGIMcS XMPUR\OWIMcSPdHGF!JRQ r*NPGIVoGIVoVGF!GmW\/U HM2HONPJ`LMcSP`LJA…PHoMeX\OWIMc`A\/WF!GSPGF!DEF GS

Mc…PHGF!GIVO\YHGIMcS1Q BFo\ed^GSPJ!HONE\YH_,JR\^U#JVJLJAa^GSPdHMWF!GSPGF!GIVJ /W +VONPMe_S GS dPDEU#J‰RQ^*Q‰NQ

r*NPJ*V.cVHJAF VHO\^U#HV9GSoVMcF!J*GSPGIHG\/W`LMcS dPDEU\eHGIMcS (W

P\/SPK\ XHJAU9\SPDEFR-†JAU+MeX F!MeT/JLV+HONPJ

V.PVHJAF GIVGS%VHO\eHJ<W Q"r NPGV&GIV&\CWIMP`A\/W9F!GSPGF!DEF \^SPKlHONPJ V.cVHJAF€_*GmWmW'NE\ T/J&HM\e`L`LJA…PH

\hSPDEF!-EJAU!MeXDES^XB\AT^McDEU\/-XWIJlF!MeT/JLV!HM2JLV`A\/…†JlHONPGIVF!GSPGFRDEFEQX&HONPJlHJAFo…EJAU\YHODEU#J<GV

WIMe_ GIH_*GmWmW HO\^aeJ\CWIMcSPd<HGF!J HMoJLV`A\/…†J!HONPGIV`LMcS dPDEU\eHGIMcS1Q SlHNPJ bVGSPdF!MPKEJW HONPJAU#J

GIVRSPM%KEGJAU#JASP`LJ<-EJLH#_,JLJAS2HONPJoV.PVHJAFNE\ TPGSPd \/WmW'V…PGSPV!DE…2McU&\/WmW+VO…PGSPVKEMe_S1QI\*HWIMe_

HJAFo…†JAU\eHODEU JLV /VGF!DXW\eHJLKW\^SESPJA\/WIGSPd!_*GmWmWSPM^H'-EJ\/-XWIJHM&VO\/F …XWJ-EM/HON HONPJLVJ*VHO\eHJLV9VGSP`LJ

HONPJL.l\/U JVJA…E\/U\eHJLKl-^.l\/SJASPJAU#d^.l-E\/UU#GIJAUQ%bHGV XbMcU*HNPGV&U#JA\eVMcSHONE\YH*HONPJHJAFo…EJAU\YHODEU#J

V`LNPJLK DXWIJRGS VGFRDXW\eHJLK \/SESPJA\/WIGSPd<GVVMoGFo…†McU#HO\^S/H8Q

\^]C]

: b 7 >r=Fb?kci/;

\ HO\/-ED VJA\/U#`LN ^ * Nccˆ GIV*\/So\eHHJAFo…PH9HM!\AT^M/GIKoHONPJ…EU#MP-XWJAF MeXd/JLHHGSPdVHODP`La GSCWIMP`A\/W

F!GSPGFo\*Q `&SPJ`LMcSPVHOUDP`LHV*\IWIGIVHMeX HONPJ!…EU#JLTcGIMcDPV 2`LMcS dcDEU\YHGIMcSPV*MeX HNPJ&V.cVHJAF€_NPGI`LN

\/U#JHO\/-†McM \^SPKlFo\ .WSPM/H*-EJ&TPGIVGIHJLKlGSHNPJ SPJPHF!MeT/JRQ@r*NPGIV*_*\ .o_,J F \A.<-EJXbMcU#`LJLKHM

(22)

y

2

y

1

y

0

y

E(y)

!#"%$&'"($)*)*)+-, +&. '/&0)*21+3546*78

"( '0&$8:9.&<;=>?$$@A)B&<CD E3F5G H& I;> >%. 'J '"( K> *C("($ E&(LM1*6INFLM>

8 ')O"DP"QR4TS=> 'UV;M$?$W V*+$&'"Q$X)BY)*)+Z[4D\'Z["D.E;M$$]&M7^ H"7$

&*_`.+a> H$&7"($b)*Y)*)+Zc4<d(

egfFhfih j=klmcnpo(qcm[rls

,t> 'K)K0$ ')* 'DU)u$"( .v" '"($[Z;V p0J&0& p"*)*&(C VLw&)x> =8 ')*V'zy

'DV&D_["(?&K&*"* ?>7^&K&D_["(&{, I" |Lw &H>& =;> >U80

&(LM)*&(C E3F'"($$ .}a> B)B&<CD ayw 6=&U&. 'Z".a>I;V K)K"T8v>&&D K> *

&(LI> +$&'"Q$ >7^&>&&.g~"D8$&'"($ '">€> '"D v7"( .-&‚> v>&0^

>"(/;P '"‚>&& v> %)*&(C ayw K&‚"(E& ' >"(K> ƒL„?&:>"Q"%YD$? Z

$&7"(${)*)u)ƒZ…"D.&V)K"D8H$&'"($g)*Y)/"†NFLb>†'"U7 I"(>? CD .gZa> 'E"=)K0$

"($&J?a>)‡;> >"( '0=&$8KLw"TC&"D7$? H)B&<CD M;M$$g&$C ˆ> H0&7$ ')+

‰M>M"00&"(>/;R"Q†C '8U0&0$"V;=> 'U&)K0 'Š)* =;M"(P .v".U>

&(LpD$?&7"($ '">€> '+;M"(E&/L‹ '"Q7$? ~"D8€>"'CD A"00$ .$?&'"($= '">

> 'M;|?a>E '"Q&"7$ I IŒŽDZ(Z’‘“‰M> = >&D? I&QLW)B&<CD ayw †"($&K"BY)uy

0&"D†"(0^ P;=> 'E*)*$"( .+" '"($[Z["(Š>R> $0M&u)K"Q Pa> p"($&>)

(23)

Y 3 { *?{ ‚ ,&

T'U JLTcGIMcDPV WI. E_,JNE\ T/J&\eVVODEF JLKWHONE\YH HNPJ*XBDESP`LHGIMcS_,J_*GIVON<HM!Mc…PHGF!GIVJ W eGIVaPSPMe_S

JE\e`LH WI.RQ@\ F!McU Jd/JASPJAU\/WVGIHODE\eHGMPS acSPMe_S\YV*VHMP`NE\YVHGI`&Mc…PHGF!GIVO\eHGMPS EGIV_NPJAU J /W

GIVoSPM/HRaPSPMe_S 9-EDPH F \A.h-†JWJLVHGF \eHJLK6Q7r*NPJAU#J<GV!VMPF!JWDESPKEJAU WI.PGSPd XBDESP`LHGIMcShHNE\eH_,J

_*GIVON HMWMP…PHGF!GIVJ \^SPKlMcDEU JLVHGFo\eHJLVMeX9HONPGIV&\/U#JKEGIVHU#G-EDPHJLKh\/U#McDESPK%HONPGIV&Tr\/WDPJ /W

-EJLGSPd2HONPJWF JA\/S%Tr\/WDPJ<MeX*HONPJ!JLVHGFo\eHJLV)Q X*_,JW\eVVODEF JRJA\Y`N2JLVHGFo\eHJoHMl-EJoVH\eHGIVHG_V

`A\/WmWI.oGSPKEJA…EJASPKEJAS^H EHONPJAS _,J*Fo\ .R`LMPSPVGIKEJAU'JA\Y`NoF MrT^J GSPKEJA…†JASPKEJAS/H WI. \/SPKW\/WIVM!\ T/JAU\ed/J

U#JA…†JA\eHJLKlJLVHGFo\eHJLV*HMRdP\eGSlF!McU J…EU#JL`LGIVJRaPSPMe_"WIJLKEd/JMeX W Q

&'\e`LN2JLVHGF \eHJo`A\/S -EJ<HONPMcDPdcN^H&MeX\YVd/GITPGSPd2HONPJ<DESPKEJAUWI.cGSPdhTe\/WDPJVODE- #JL`LHHM

VMcF JRSPM/GIVJRQr*NPJoSPM/GIVJ!GS%HONPJ!JLVHGFo\eHJLV&SPJLJLKEV&HM-EJoaeJA…PHVOFo\/WmW GX WIMe_‚HJAF …EJAU\eHDEU#JLV

\/U#JHMo-EJ&VGF!DXW\eHJLK †_NPGmWJ!\eH NPGIdcN HJAFo…†JAU\eHODEU#JLV WIMe_ DE\/WIGIH#.WJLVHGFo\eHJLV*Fo\ .W-†JDPVJLK6Q

r*NPJoKEGW`ADXWIH#.hGS2HONPJ<…EU#Mc-XWIJAF GV!SPMe_ HOU\/SPV#XbJAUU#JLK%XBU#McF HONPJ `LMcFo…XWIJcGIH#.hMeXHMl\/WIVM

`LMcSPVGIKEJAU#GSPdlNPMe_ …EU#JL`LGIVJR\^S<JLVHGFo\eHJ&GIV&SPJLJLKEJLK VMW\eV,HMW\ T/M/GIK<VO…EJASPKEGSPdlDESESPJL`LJLVV\/U#.

J McU#H8Q

\^]C]`\

n kc=9@Bg +>lgs je@ @B>b?je@Bg =†jY;+g '>

CPHMP`NE\YVHGI`!Mc…PHGF!GIVO\eHGIMcS%_*\eV&-EMcUSPJRMPDPHMeX \^SlGIKEJA\-^.-pMP-E-PGSPV $2McSEU#M */‡ Qr*NPJL.

`LMcSPVGIKEJAU#JLK2VMNWTPGSPdWHNPJ…EU#Mc-XWIJAFMYX SPKEGSPd

/W #‰RQf

_NPJAU#J GIVVMcF!J2F!McSPM/HMPSPG`%XBDESP`LHGIMcS MeX W *HONPJ%VMRWDPHGIMcS HM -EJ%XMPDESPK \/SPK GIVl\

…E\/U\^F!JLHJAUQ GIVSPM/H XMPDESPK KEGU#JL`LH W. -EDPH&`A\/SMPSXW.2-EJ!JLVHGFo\YHJLK6Qr*NPJLGUHJL`NESPG!EDPJoHM

VMRWIT/J HONPGIV …EU Mc-XWIJAFGIV`A\/WmWJLK2VHMc`LNE\eVHGI`\/…E…EUM PGFo\eHGIMcS1Q \ ScDEF!-EJAU MeX9Tr\/U G\^S/HV MeX'HONPGIV

V`LNPJAF!JNE\ T/JVGSP`LJR-EJLJAS KEJLT/JWIMc…EJLK ‰A‡ *bRt b * cGSP`WDPKEGSPd \ d/JASPJAU\ WGIVO\YHGIMcSHMo-†J\/-XWIJ

HMoH\e`aNWIJ&VHMP`NE\YVHGI`&Mc…PHGF!GIVO\eHGMPSl…EU#Mc-XWIJAF!VbRb Q

CPHMP`NE\eVHGI`WMc…PHGF GV\eHGIMcS GVo\%F!McU#Jod^JASPJAU\/W'VGHDE\eHGIMcS 9_NPJAU JW_,Jo_*GIVON HM SPK

HONPJF!GSPGFRDEF MeXVMcF!J!XBDESP`LHGIMcS '-EDPH!_NPJAU JWHONPGIV XBDESP`LHGIMcS]F \A.hNE\ T/JWF \/S/.hF!GSPGFo\*Q

cDPVH*\eV,_,J&Fo\ .o`LMcSPVGIKEJAU,McU KEGSE\^U#.oMc…PHGF!GIVO\YHGIMcSF!JLHONPMPKEV,HM X \/WmWGS/HMRH#_,M`A\YHJLd/McU#GIJLV V

JE\e`LHF!JLHONPMPKEV\/SPKlNPJADEU GVHGI`LV"V'VHMc`LNE\eVHGI`Mc…PHGF!GIVO\eHGMPSlF!JLHONPMPKEV*`A\/S\ WVMW-EJ&`W\eVVJLK

\eV*KEJLT^JWMP…EF!JAS/HVXU#MPF€HONPJLVJ H _,M!`A\eHJLd^McU#GIJLV*MeX F!JLHNPMcK6Q r,_,MoMeX9HONPJKEJLT^JWMP…EF!JAS/HVMeX

(24)

\ScDEF!-EJAUMeXNPJADEU GVHGI`LVoJcGIVH HMHO\e`LaNWJ VHMP`NE\eVHGI`oMc…PHGF!G! A\eHGIMcS…EU#Mc-XWIJAF!V‡

‡ * Xc ‰ ‰/‰8q ‰/‰)f /_*GIHON Tr\/U .cGSPdW\eKETe\/S^HO\ed/JLV,\/SPKWKEGIVO\YKETr\/S^HO\ed/JLV)Q k?McU9JE\/Fo…XWIJ&`LMcSPVGKEJAU

HONPJoVHMc`LNE\eVHG` `LMcFo…E\/U#GIVMPSh\/WId/McU#GIHONEF c ‰ q_NPGI`N GIVR\KEJLT^JWMP…EF!JAS/H!MeX*\/Sh\/WId/MPU#GIHONEF

-^.'\/S\/SPK $hDEa \eG ‰^‰8f QZ GIHONhHNPGVo\/WId/MPU#GIHONEF McSPJFo\/aeJLV!\ SPDEF!-EJAU MeX*JLVHGFo\YHJLV

MeX*HONPJoJASPJAU d/.hMeX,HONPJW`ADEUU#JAS/H VHO\eHJ

\/SPKMeX*HONPJWSPJL_ VHO\eHJ

Q X\/S/.2MeX

HONPJ%JLVHGFo\eHJLV MeX HNPJlJASPJAU#d/. MYX HONPJlSPJL_ VHO\YHJl\/U#JdcU JA\eHJAU!HONE\^SHONPJl`LMcUU#JLVO…EMPSPKEGSPd

JLVHGFo\YHJ!MeX HONPJ!JASPJAU#d^. MYX'HONPJo`ADEUU#JAS^H&VHO\YHJ HNPJAS2HONPJ!`ADEUU#JAS^H&VHO\eHJRGIVaeJA…PH\^SPKhHONPJ

…EU#Mc…†M/VJLK F!MeT/JoU#J#JL`LHJLK6Q bHGIVEDPGHJW`WIJA\/U HONE\eH&HNPGV!GIVR\^S2JW`LGIJAS/HoF!JLHNPMcK XMPU&NPGdPN

HJAFo…†JAU\eHODEU JLV -EDPH*_*GmWmW9U#J JL`LHFo\/S^.oXB\ T/McDEU\/-XWIJF!MeT/JLV*\eH WIMe_‚HJAFo…EJAU\YHODEU#JRQ

$h\/S/. MeXHONPJNPJADEU#GIVHG`LV\/U#J*KEJLT/JWIMc…EF JAS/HV+XBU#McF VGF!DXW\eHJLK\/SESPJA\/WIGSPd‰ ‰)f Nc ^

b/‡ Q&CPGF!DXW\YHJLKh\/SESPJA\/WIGSPd c ^ 'NE\eV&-EJLJAS2VONPMe_S cc‡ 9HMWVMRWIT/J!VHMP`NE\eVHGI`Mc…PHGF!G! A\eHGMPS

…EU#Mc-XWIJAF!V _*GIHON]…EU Mc-E\/-PGmWIGH#. ‰…EU#MeTcGIKEJLK HONPJSPM/GIVJGShHNPJJLVHGFo\eHJLV!\eH!HGF!J<VHJA… &GV

aeJA…PH-†JWIMr_ ,_NPJAU#J ‰RQ

$2McKEG JLK VGF!DXW\eHJLK \/SESPJA\ WGSPd \/WId/McU#GIHONEF V ‰ *‰)f b/‡ HO\^aeJWGS^HM \e`L`LMcDES^HoHONPJ

_*GIKEHON MeXHONPJ<KEGIVHOU#G-EDPHGIMcSMYXHONPJ<JLVHGFo\eHJLV HM F \/aeJ<VODEU#J<HONE\eHHNPJWJLVHGF \eHJLVMYXHONPJ

JASPJAU#d/.o\/U J VGdPSPG `A\^S/H WI.oKEGJAU#JAS^H -†JXMcU J\Y`L`LJA…PHGSPdRMPU9U#J JL`LHGSPd!\&F!MeT/JRQ CPMPF!J*MeXHONPJLVJ

F!JLHONPMPKEV*NE\ T/J -†JLJASVONPMe_S HM!`LMcS^T/JAU#d/JHM HONPJ&dRWIMc-E\/WF!GSPGF!DEF€MeX HONPJ XBDESP`LHGIMcSl-†JLGSPd

Mc…PHGF!GIVJLK2DESPKEJAU*d^GT^JAS`LMcSPKEGIHGIMcSPV^Rq b/‡ Q

CPHMP`NE\eVHGI`R\/SESPJA\/WIGSPd _^/c [GIV\/S%JPHJASPVGIMcS%MeX'VGF!DXW\eHJLK]\/SESPJA\ WGSPd%HMoVHMP`LNE\eV V

HGI`<Mc…PHGF!GIVO\eHGIMcS …EU#Mc-XWIJAF!V8Q HKEGJAU#V&XBU#McF HONPJW\/-†MeT/Jo\/…E…EU#Mc\e`LNPJLVGS H#_ MlaYJL.z\^U#JA\eV8Q

k9GU#VH WI. HONPJ!d/JASPJAU\/W `LMPSP`LJA…PH&MeX'HONPJ!M/HONPJAU\/…E…EU#Mc\Y`NPJLVGIVMcSPJ MeXF!GSPGF!GIVGSPd2HONPJRSPM^GVJ

…EU#JLVJAS^HGS%HONPJ JLVHGFo\eHJLV8QbH&GIVJAS/HGU#JWI.2SE\eHODEU\/WqHONE\eHHONPJRSPM/GIVJo…EU#JLVJAS/HGSlHNPJJASPJAU#d/.

JLTr\ WDE\YHGIMcSPVoVONPMcDXWIK -EJlDPVJLK \eVo\2…EM/VGIHGIT/JJ JL`LH VGSP`LJMcSPJlFo\ .2HONPGSEahMeXVGFRDXW\eHJLK

\/SESPJA\/WIGSPd\eV*\oSPM/GIV. DPJASP`LN1Q'CPJL`LMcSPK*WI. HONPGIV\/…E…EU Mc\e`LNo`A\/S-†JF!MPKEG JLK%HM!d/GIT/J&JE\e`LH

VGF!DXW\eHGIMcS2MeX*\<HONPJAUFo\ W9V.cVHJAF-Q[\ WIHONPMcDPdPNlHNPGV GIVRSPM/H&VO…†JL`LG `A\/WmW. UDXWIJLKMcDPH-/.%HONPJ

M/HONPJAU \/…E…EU#Mc\e`LNPJLV ESPM\eHHJAFo…PH*NE\eV-EJLJASlF \eKEJ&HMoVO\YHGIV#X.<HONPGIV*`LMcSPKEGIHGIMcS1Q

\^]C]

iEiE=?s9j=b 'iE=s9kcg7*b 79@bpb@bje@B= >2@ > jYg9i^;!b> jr@Bi> t'> jY= >

BSHNPJWVHMc`LNE\eVHGI`Mc…PHGF GV\eHGIMcS`LMcS^HJPHHONPJl…EU#MP-E\/-PGmWGIH#.

2 MeX \Y`L`LJA…PHGSPd]\

F!MeT/J9_NPM^VJ'JASPJAU#d^.&`NE\/SPd^J9GV82 GIV'SPM/HHONPJ+VO\/F!J'\YV HONPJ,…EU#Mc-E\/-PGmWIGIH#. MYX†\e`L`LJA…PHGSPd

F!MeT/JLV _NPM/VJ JASPJAU#d/.`LNE\/SPd/J*\/…E…†JA\/U#V+HM -†J 6Q@r*NPJLVJ\^U#J,H _,MKEGIVHGSP`LH DE\/S^HGIHGIJLV ^VGSP`LJ

(25)

MeX&JLVHGFo\eHGSPd HONE\eHoHONPJJASPJAU d/. `NE\^SPd/JGIV _NPJAS GIHVoHUDPJlTe\/WDPJlGIV 2 3Q r*NPJAS HONPJ

…EU#Mc-E\^-PGWIGIH#.MeX'\e`L`LJA…PHGSPd6\ F!MeT/J&_NPM/VJ&HUDPJ&JASPJAU#d/.<`NE\^SPd/J&GIV 2 GIVd/GIT/JAS%-/.

2

2 #‰RQ*

r*NPGIV,XbMcUF!DXW\!GIV\/…E…XWIGI`A\/-XWIJ!HM!VHMP`NE\eVHGI`&Mc…PHGF!GIVO\eHGIMcS<GS d/JASPJAU\/W -EDPH,VM!XB\/U,NE\eV,McSXWI.

-EJLJAS `LMcSPVGIKEJAU#JLKl-^.Ik9GSEa l,\/WmW ^/c Q X _,J*`A\^S!`NPMPM/VJ 9VODP`LN HONE\eH

XbMRWmWIMe_*V

J1Q‰RQ^ †HONPJAS_,J&_*GmWmW[-†J!GF …XWJAF JAS/HGSPdl\!d^JASPJAU\/WIGIVJLK2T/JAU#VGIMcS MeX'VGF!DXW\YHJLK \/SESPJA\ WGSPd

(26)

e

N)-A! 23A AC8=7*,/7CA!-A!%A! .>+,D2M@C,/7CA!+'0H. 23+=7 %=0I01.

=@CA!+'0H. ,?7

Q

-0 7!,I2 7/,I2

CPHMP`NE\eVHGI`Mc…PHGF!GIVH\eHGIMcSl…EU Mc-XWIJAF!V\/U#J McSPJLVGS%_NPGI`LNl_,J&_*GIVON%HM<…EJAU XbMcUF \/S

Mc…PHGF!GIVO\YHGIMcSo_NPJAU#J*JE\e`LH'acSPMe_"WIJLKEd/JMeXHONPJ,XBDESP`LHGIMcSoHM-EJMc…PHGF!GIVJLK<GV,SPM/H \ Te\eGmW\^-XWJRQ

bH&GIV!\eVVODEF!JLK2HONE\YH&GSP`AU JA\eVGSPdRWI.]…EU#JL`LGIVJoJLVHGFo\eHJLV&MeX,HONPJ XDESP`LHGMPSh\/U#Jo\ Tr\eGmW\/-XWIJ 9-EDPH

HONE\eHHONPJLVJ JLVHGFo\eHJLV \/U JGSP`AU#JA\eVGSPdRWI.hJE…EJASPVGT^JRQZr*NPJH\eVOa<GIV&HM…EJAUXbMcUF \/SMP…PHGF!G_V

VO\eHGMPS<HONE\eH*GIVSPM/H…EU#McNPG-PGIHGIT/JWI. JE…EJASPVGT^JRQ

r*NPJ&\/…E…EU#Mc\e`LN!MeXVHMc`LNE\eVHGI`\^SESPJA\/WIGSPd GV*\ d/JASPJAU\ WGIVO\YHGIMcSoVGF!DXW\eHJLKl\^SESPJA\/WIGSPd

HM<VHMP`NE\YVHGI`!Mc…PHGF!GIVO\eHGIMcS2…EU#MP-XWJAF V8Qar NPJRMPU#GId/GSE\/W@k9GSEa l,\/WmW ^/c \/WId/McU#GIHONEF McSXWI.

d/GIT/JLV&\/S2\/…E…EU#M)PGF \eHGIMcSHMoHOUDPJ!HONPJAUF \/WVO\/F …XWGSPdQr*NPGIV&HJL`LNESPG! DPJoFo\ .l-†J \YK \/…PHJLK

HMod^GT^J&J \e`LH W.HNPJAUFo\/WVO\/F …XWGSPd \/SPKHONPGIVGIV*KEJAF!McSPVHU\eHJLKGSl\ U\/SPd/J&MeX9`A\eVJLV)Q

3 Y { *?{ !+* ) Z 2&|x 5 *

ZhJ,NE\AT^J \/WU#JA\eKE.&VJLJAS VJL`LHGIMcSo‰NQcQ‡ HONE\eHHONPJ,\e`L`LJA…PHO\/SP`LJ*…EU#MP-E\/-PGmWGIH#.MYX\F!MeT/J9_NPM/VJ

DESPKEJAU WI.PGSPdJASPJAU d/.W`LNE\/SPd/J GIV 2 GIV

2

2 b‡*Q‰

GIVoHONPJl…EU#MP-E\/-PGmWGIH#.HONE\eH!_,J`NPMPM/VJ<HMh\e`L`LJA…PHW\2F!MeT/J<_NPM/VJ<JASPJAU#d/.`LNE\/SPd/J<GV

JLVHGFo\YHJLKlHM<-EJ 2\/SPK 2 +GVHONPJ!…EU#Mc-E\/-PGmWIGIH .2KEJASPVGIH#.MeX'JLVHGF \eHGSPdWHNPJJASPJAU#d/.

`LNE\/SPd/JlMYX\hF MrT^JHM]-EJ _NPJAS GHVWHOUDPJTr\/WDPJ2GIV 2 hQk9GSEa \/SPKMl,\/WmW^/c `LNPM/VJ

‰!XbMcU

ˆ%\/SPK €ˆ<M/HONPJAU#_ GVJRQ3r*NPGIV`LNPM/GI`LJ<VHJAF!VXBU#McF HONPJ!`LMcSP`LJA…PH

MeX'\!…EU#Mc-E\/-PGmWIGIVHGI` DPJASP`LNhV'HONE\YH*_,J \e`L`LJA…PH&\/WmW F!MeT/JLV,_NPGI`LN

G

(27)

X*_,JoNE\ T/JRF!MPU#Jo…EU#JL`LGIVJWJLVHGFo\eHJLV&MeX,HONPJ!JASPJAU#d/.2`LNE\/SPd/J HONPJAS2_,J!_*GmWmW \e`L`LJA…PHXbJL_,JAU

DES^XB\AT^McDEU\/-XWIJoF!MeT/JLVUV*J DPGITe\/WIJAS/HoHM3WIMe_,JAU#GSPdlHONPJ!HJAFo…†JAU\eHODEUJRQCr*NcDPV `NE\/SPd^GSPd%HONPJ

…EU#JL`LGIVGIMcS%MeX9McDEU+JLVHGFo\eHJLV*MeX82 \/WmWIMe_*V&DPV*HMW…†JAUX MPUF\/S\/SESPJA\ WGSPdQ

k9GSEaW\/SPK-l,\/WmW^/c `LMcSPVGIKEJAU#JLK%HONPJ&…E\/U#HGI`ADXW\/U,`A\eVJMeX -†JLGSPdW\ \/DPVVG\^S<KEGVU#G_V

-EDPHGIMcS

2

‡

b‡*Q‡

CEDE-PVHGIHODPHGMPSMeX9HONPGIVGS^HMoJ1Q ‡*Q‰_*GIHON HONPJ&`LNPM/GI`LJMeX ‚\^-EMeT/J&d/GIT/JLV

2

‰

‡

‰ hJAU X

2

‡

b‡*Q^

r*NPGIV*GIV*VOaYJLH`NPJLKGS dQ?‡*Q‰ ^_NPGI`N<GmWmWDPVHOU\eHJLVHONPJ…EUGSP`LG…XWIJo-EJANPGSPK2VHMP`NE\YVHGI` \^SESPJA\/W_V

GSPdQrM \eVVODEU JHONPJAUFo\/W J DPGmWIG-EU#GDEF HONPJ \e`L`LJA…PH\/SP`LJ …EU#MP-E\/-PGmWGIHGIJLV2FRDPVHoMc-EJL. JGQ

‰RQ^

2

2

/ b‡*Qc

_NPJAU#J

#

GV HONPJGS^T/JAU#VJ&HJAFo…EJAU\YHODEU#J&\/SPK"2 GIV*HONPJ&JASPJAU#d^.oKEG JAU JASP`LJR-EJLH#_,JLJAS

HONPJ%VHO\eHJLV!GS HNPJ F!MeT/JRQ Q VGSPdHONPJ]k9GSEa sl,\/WmW`LNPM/GI`LJlMYX KEMcJLVSPM/Hod/GIT/J2\/S

\e`L`LJA…PHO\^SP`LJW…EU#Mc-E\/-PGmWIGIH#.]_NPGI`LN2VO\eHGV JLVJ1Q ‡*QcJE\e`LH WI. 9-EDPH`WIM/VJW.\/…E…EU#M)cGFo\YHJLV&GIH8Q

Q VGSPdWJ1Q ‡*Q^!\/SPKJE…E\/SPKEGSPdGS"2 _,J&Mc-PHO\eGS

/10

/10!

"

$#&%('

)

#&%('

)

*2

,+

*02

b‡*Qb

_NPJAU#J

*

f

/

b‡*Qt

GIKEJAS/HG JLV<HONPJJ JL`LHGIT/JGS^T/JAU#VJHJAF …EJAU\eHDEU#JRQKr*NPJVOFo\ WW,`LMPJW`LGIJAS/H

ˆˆ/‡&MYXHONPJ

`ADE-PGI`&HJAUF GS<JGQE‡*Q_bFo\/aYJLV'HONPGIV*\!U\eHONPJAU+d/MPMcK<\/…E…EU#M)cGFo\YHGIMcSoHMRHUDPJHNPJAUFo\/W?VJWIJL` V

HGIMcS1Q

rM%`LMcS^HOU#MRW HONPJ<HJAFo…EJAU\eHODEU#Jo_,JoSPJLJLKHM -†J6\/-XWIJ<HM`LMcS^HOU#MRW0>Q3r*NPGIVoFo\ . -†J

\e`LNPGIJLT/JLK4-^. F \/a/GSPd]\ ScDEF!-EJAU MeX VHO\eHGIVHGI`A\/WmWI. GSPKEJA…EJASPKEJAS^HJLVHGFo\eHJLVoMeX)2 XBU#McF

HONPJ &\/DPVVG\/S KEGIVHOU#G-EDPHGIMcS1Q=r NPJlF!JA\/STr\ WDPJ%MeX \2VJLHRMYX %JLVHGFo\eHJLV!GIV &\/DPVVG\/SXWI.

KEGIVHOU#G-EDPHJLKh\/U McDESPK"2 P_*GIHON<VHO\/SPK \^U#KKEJLTcG\eHGMPS

(28)

0

0

1

0

1

convolution

f(x-

E)

A(x)

P (

accept

E)

x

x-

E

1

β

1

β

G

,

E

Glauber function

Fink & Ball rule

!#"%$%&'("%$%)' '(*,+- &&."% "$%&/'0 $21&(3 "

4

$56')&'$7/8:9<;*=?> @;A"B'A "%$%1C'($D &(EFHGJILKNMPOQOQRTSUV9<FCG=W"$5 &3&.X$D&

&(EB8:9<;W=

4

$PC$5 - D&!"%$P$%&ZY[9<;\ZFHG/=VB ^]_X'. )'( $P' _E`$D&a"

'("%&C"P&

4

HE<&@&1C'"%&b

4

_c_"d$5 )"5'.1C )"%e gfhE

4

)"% _$5"2'("&[ "%$%1C'($D 2&0EiFHGJI($P 2 PjN $%3 _3. "%

$% 1C %'($5k ^ &1 "

lbmZnpo q

c

o rNs

9<

q

=

'!t"$5' u.7v3('7Xwcv"H"5'1Cx1 $5&!y'("%&v'("$5 Z'(!3('.$5'( a$5'($

EQ&-'. zc{$5 J .$5%'0g1$C$5 & 1|!*3 "$5 a!"%$5$%&}&(E$5 #'L3 %'(. /&0E$5

"%$%1C'0$% ")$%&

4

'!")'a]-'"%"%'{ 3 #~E$_"&.$)"%&EQ&)c

n

.2€&$% -$5'0$,E<&2'H"51&&$5

' '("% !* a& "5&!v$P'‚( ' $%&Z1 '0"5

s

'!v&.$5&. lbm %'($P _$5'

ƒ%"%$gc'0& B"d$5%"d&$$D&^ )1C&$5'.$?EQ&?$5 )&'"%$%-$5%'L3 ^"5'0 "P1C'

& 1#IW"% -"% W„†…

‡6ˆ‰‡ ŠŒ‹[Ž’‘”““6•—–_‹)˜?™

fhE

4

@1H'‚( 2'!~jN $& )&(Eš8:9<;W=N$%&-‚6'!C›2'0Nœ~„(Lž‰I

4

, '& 2$D&"5'0$%"E<76$5

! $5'( !C'(' g&!$D&bI( Ÿš  *I ¡'($%7?¢v g‚&

4

$5'($bEQ&W"%1X'0$% !C' '(

4

'#1C'.‚( '1 ,&(E?!†jN .$,£&. "_&(EB'0 $5' _E¤$%&bI"5#'(",$5 6]'

&¥ $5&&"'( $5' C% "{9T"% $%&w††=_fhE

4

^"%"%$^$P'($K MPOQOQRTSU EQ&&

4

(29)

MeX HNPJLVJUDXWIJLV EHONPJAS _,J_ GWmW NE\ T/JVO\eHGV JLKJGQE‡*Qc;Q k?McU+J \^Fo…XWIJ&JGQE‡*Q‰_,McDXWIKhWJA\YKlHM

*

2

2 b‡*Q*

BS HONPJ`A\eVJ*_NPJAU#J 2 "42 HONPGIV GIV*\&KEJ VB`LMPS/T/MNWDPHGMPS…EU#Mc-XWIJAF X McU Q

r \/a/GSPdhk McDEU GJAU V \/…XW\e`LJHU\/SPV#XbMcUF!V,MeX'-EM^HON<VGIKEJLV ?_ JNE\ T/J

*

b‡*Q‰Aˆ

_NPJAU#J&HNPJZk MPDEU#GIJAUV \/…XW\e`LJ HOU\/SPV#XbMcUF GIVKEJ SPJLKz-^.

b‡*Q‰/‰

BS HONPJ`A\YVJ&MeX9HONPJUW\^DE-EJAU\e`L`LJA…PHO\/SP`LJ!UDXWIJ J1Q‰RQb P_*GIHON h\YV*HONPJ \^DPVVG\/SKEGIVHU#G_V

-EDPHGIMcS JGQ?‡*Q‡ ^HONPGIV -†JL`LMcF!JLV

‡ &

b‡*Q‰A‡

r*NPJ&KEGW`ADXWIH#.GIV*HONE\eH F!DPVH*-†J\o…EU#Mc-E\/-PGmWIGIH#. \/SPKHNcDPV"WIGIJGS<HONPJU\/SPd/Jˆ ‰ XbMcU

\/WmW &Q3r*NPJAU JoGIVRSPMlMP-/TPGIMcDPV_*\ .lHMJASPVDEU#JoHONE\eH HONPGIV`LMcSPKEGIHGIMcS GIVVO\YHGIV JLK VG SP`LJWJ1Q

‡*Q‰ ‡,XBDXWmW. VO…EJL`LG JLV Q ZhJ*DPVJ\UWIJLVV,U#JLVHOU G`LHGT^J\/…E…EU#Mc\Y`N!-EJWIMe_ _NPG`LN<\/WmWMe_*V9VMcF!J

XBU#JLJLKEMcF€GSMPDEU,`NPM^G`LJ MeX \^SPKlNPJASP`LJ_,JFo\ .W`LNPMPM/VJ&GIH \YV\ …EU#Mc-E\/-PGmWIGIH .RQ

3 & $ ,) * 2?w ,

6]C]`\

5h=+=Ek8b?p&ib>r=

BS HONPJXbMRWmWMe_*GSPd\/SE\/WI.PVGIV_,J&_*GmWmW[-†JR\eHHJAFo…PHGSPdoHM!`LNPMcM^VJR\/S 'VDP`N HONE\eH

VO\eHGV JLVWKEJLHO\YGWIJLK -E\/W\/SP`LJ ,J1Q'‡*Qc;Q ZhJ_*GmWmW \YVVODEF!JHONE\YHoHONPJXBDESP`LHGIMcS 2 GV

aPSPMr_S<J \e`LH W.RQ u,McF!-PGSPGSPd<J1Q ‡*Qco_ GHN<JGQ ‡ Q‰_ JNE\ T/J

2

/1

2 b‡*Q‰8^

ZhJFo\/aeJ<HONPJ<VODE-PVHGIHODPHGMPSPV 2

/ 2 \/SPK

d/GITPGSPd

2 2 ˆ b‡*Q‰c

\V\!VMRWDPHGIMcS ?_,J&HOU#.

(30)

_NPJAU#J l\/SPK \/U#JXBDESP`LHGIMcSPVHMW-EJ `NPM/VJAS1Q CEDE-PVHGHDPHGSPdWHONPGIVGS/HM<JEDE\eHGIMcSl‡ Q‰co\/SPK

U#J V \/UU\/SPd/GSPdo_,J%SPK

2

! 2 2 ! 2 2 2 ˆ b‡*Q‰8t

_NPJAU#J

! 2 2

2 2 b‡*Q‰8q

& 1Q‡*Q‰)t!GIVVO\eHGV JLKh_NPJAS 2

2

&\^SPK ! 2 2

6! 2 2

\/U#JJLT/JAS X DESP`LHGMPSPV8Q'Z J NE\ T/JSPM^H VDP`L`LJLJLKEJLKlGS<HO\/a^GSPdRHNPGV*F!M^VH,d/JASPJAU\/W?`A\eVJ X DEU#HNPJAU

HONPJNE\/U Kl…E\/U#H-†JLGSPdHMRJASPVDEU#JHONE\YHˆ ‰ XMcU,\o…EU Mc-E\/-PGmWIGH#.RQ

6]C] s*= iP@ ,i]ib>e=

ZhJSPMe_ Fo\/aYJoHONPJ<HOU\/SPVGIHGIMcSzHM \OWIJLVV!d/JASPJAU\/W+`A\eVJ XMPU&HONPJoJAUU#McU&KEGIVHU#G-EDPHGIMcS1Q2ZhJ

`LMcSPVGIKEJAU,HNE\eH 2 GIV\ XDESP`LHGMPS<MeX "42 \/WIMcSPJ EF!JA\^SPGSPd!HONE\YH'HONPJKEGIVHOU#G-EDPHGIMcS

MeX JAUU#McU 2 ^GIVGSPKEJA…EJASPKEJAS^H&MeX 2 GIHVJWX Q'ZhJ&`LNPMPM/VJ&HONPGIV*GSl…E\^U#H*-EJL`A\/DPVJ GX[_,J

\/WmWIMe_ HONPJ!XbMcUF MYX HM%KEJA…EJASPKMcS 2 HONPJAS2HNPJAU#JoGIVRSPM2U#JA\eVMcS XbMcUSPM^H\/WmWMe_*GSPd2GIH

HMoKEJA…†JASPKhMcS VMcF!J M/HONPJAU Tr\/U#G\/-XWIJ HONPJ `A\eVJo\/-†MrT^J-EJLGSPdlJLVVJAS/HG\/WmWI.lGIKEJAS^HGI`A\/W9HM<HONPJ

F!M/VH d^JASPJAU\/W`A\eVJ…EM/VVG-XWIJRQ

k McU HONPGIV,`A\eVJ P_,J*`NPMPM/VJHONE\eH 2

"42

+\/SPKWVDE-PVHGIHODPHJ&GS^HMJ1Q

‡*Q‰)qHMod/JLH

! 2 2

2 2 b‡*Q‰8f

r*NPGIV&GIV \^SJLT/JAS6XDESP`LHGIMcS%GS%HONPJVJASPVJHONE\YHGX'_,JVODE-PVHGIHODPHJ W 2 2

GS^HM

HONPGIV E_,J%SPK

! 2 2

/W 2 /W 2 8W b‡*Q‰ *

! 2 2 b‡*Q‡/ˆ

r*NPJAU XbMcU#J

2 2 2 b‡*Q‡E‰

GIVR\<VMNWDPHGMPShHMJ1Q‡*Q‰Lc6…EU#MeTPGIKEJLK GIVJLT/JAS1QCr \/a^GSPdOk McDEU GJAUV \/…XW\e`LJ<HOU\/SPV#XbMcUF VMeX

HONPGIV_ JNE\ T/J

(31)

ZhJlVHGmWWNE\ T/J2VMPF!Jl`LNPM/GI`LJhGS

*VGSP`LJ HONPJ2McSXWI.4U#JLVHU#GI`LHGIMcS _ J2NE\ T/JlVMXB\/U

VO…†JL`LG JLK GIV HONE\eHGH GIVJLT/JAS

hF!DPVH-EJoJLT^JAS VGSP`LJ 2GIVJLT/JASQ GIT/JAS2GIH&GIVKEJLVGU\/-XWIJ

HMoU#J#JL`LH\eV+XbJL_‚F!MeT/JLV*\eV*…EM/VVG-XWJ _,J`NPMPM/VJ

9HM -†JGIKEJAS^HGI`A\/WmW.‰

-EJWIMe_ VMcF!J&HONEU JLVONPMRWIKXbMcU 6Q JASP`LJ_,J&KEJ SPJ

b‡*Q‡R^ _NPJAU#J

XMPU \^SPK ˆ XbMcU

Q r*NcDPV _ JNE\ T/J

XMPU '(

b‡*Q‡/c ZhJ'SPJLJLKHM JASPVDEU#J'HONE\YH 2

2 `LMPUU#JLVO…†McSPKEV

HMW\<…EU#Mc-E\^-PGWIGIH#.RQr*NPGIVF!JA\/SPV*HONE\YH

2 ‚‰X MPU*\/WmW 2 \^SPKlVGSP`LJo_,J\/U#J GSNV

VGIVHGSPd<McS<J1Q ‡*Qc!HONPJAS

2 / b‡*Q‡Rb

XbMcU*\/WmW02 hQ JASP`LJ_,JNE\ T/J

2 / b‡*Q‡Rt

r*NPGIV"WIJA\eKEV,HMoU#JLVHU#GI`LHGIMcSPV,McS \^SPK "2 ^VGSP`LJ-EM/HON<\/U#J&…EM^VGIHGIT/JRQ CPGSP`LJ&_,J

\/U#J&`LNPMPM/VGSPd ‰ XMcU

_,JNE\ T/J 2 /1 b‡*Q‡Rq

_NPGI`LNRF JA\/SPV DPGIHJ,U#JA\eVMcSE\^-XW.qHONE\eH <F!DPVHXB\/WmWcM)W\eHWIJA\eVH9\eVXB\eVH9\eVHONPJ+KEGIVHOU#G-EDPHGIMcS

MeX9VHO\eHJLV _,J\/U J\eHHJAFo…PHGSPd HM!VGFRDXW\eHJRQkU#McF JGQ?‡*Q‡Rt _,JNE\ T/J

/1

/W 2 /W (W /1 b‡*Q‡Rf

CPGSP`LJ ˆo\eV PGIHGV&U#JA\eVMcSE\/-XWIJ&HMW\eVVODEF!J&HONE\eH GIV*`LMcSP`A\ T/J†HONE\eH,GIV

/W 2 2 W 2 b‡*Q‡ *

u,McF!-PGSPGSPdWHNPGV_*GIHON<J1Q ‡*Q‡Nfd^GT^JLV

2

/1

X McU 2 ˆ b‡*Q^/ˆ

rqMo`A\/WI`ADXW\eHJHNPJZWIGF!GIHHONPGIV&…XW\e`LJLVMPS

(32)

_NPGI`LN<XbMRWmWMe_*VXU#MPF€JGQ?‡*Q‡R^*Q&kU#MPF€HONPGIV E_,J%SPK b‡*Q^/‡

_NPJAU#J GIVVMcF!J&`LMcSPVHO\/S^H P_NPGI`NEWJA\YKEVHM

'( b‡*Q^R^ JASP`LJ

&GIV6-†McDESPKEJLK XbMcU!\/WmW '(

Q]CPGSP`LJ

GIVWSPM/H!dcDE\/U\/S/HJLJLK]HMh-†J

-EMPDESPKEJLKoXbMcU9\/WmW <FRDPVH'-EJ*`LNPM/VJAS HM&JASPVODEU#J HONE\eH9HONPGIV'`LMPSPKEGHGMPS<McS

lGIV*F!JLH8Q Q*VGSPd

HONPJ&Z GIJASPJAU V Mc…^X F!JLHONPMPK q * _ JF \A. KEJ SPJ

b‡*Q^/c

_NPJAU#J ,GIVSPMcSNV LJAU M6\/SPKl-†McDESPKEJLK%XMPU

ˆ E\/SPK%VGF!GmW\^U WI.XbMcU

XbMcU ˆ Q

CPGSP`LJ GIVW-EMPDESPKEJLK XbMcU '(

_,JlSPJLJLK HMhHO\/aeJ\e`L`LMcDES^HoMeX&HONPJl…†MRWIJ

`LMcS^HO\eGSPJLKJE…XWGI`LGIH WI.lGS<J1QE‡*Q‡/cQqZhJ&`LNPM/VJ

+HM `LMcS^HO\eGS\!…EMRWIJ&MeX MP…E…EM/VGIHJU JLVGIK DPJ

\eH

E_,J&`A\/S`A\^SP`LJWHONPJJ JL`LH*MYX[HONPJ!…EMRWIJ GS JGQ‡*Q‡ cQ X

_,J&JASPVODEU J HONE\eH ,GIV-EMPDESPKEJLKlXbMcU '( Q'CPGSP`LJ

,F!DPVH-†JJLT/JAS †_ JNE\ T/J

b‡*Q^Rb _NPGI`LNd/GIT/JLV b‡*Q^Rt

CPMRWDPHGIMcS MeXHONPGIV,J DE\eHGMPSCWIJA\eKEV,HM&HONPJMc…PHGFo\ W \e`L`LJA…PHO\/SP`LJ&UDXWIJXMPU Q bH,Fo\ .!VJLJAF

HONE\eH+HONPJAU#J&GIV,VHGmWmWVMcF J*XBU#JLJLKEMcF GS<HONPJ`NPM/GI`LJ&MeX Q@r*NPGIV*GIVSPM^H HNPJ&`A\eVJ PVGSP`LJ&HONPJ

`LMcSPKEGIHGIMcS McS J1Q ‡*Q^N^+_*GmWWqKEJ SPJ Q

ZhJSPMe_ d/M!McS<HM KEGV`ADPVVVMcF!J…E\/U#HG`ADXW\/UXBDESP`LHGIMcSPV Q

3 +$ #

(33)

S\/F!J

2

rqMc… \YH

XbMcU 2

ˆ M/HONPJAU _*GIVJ

‰ XbMcU XbMcU XbMcU

u,McS^T/MRWIT/JLK & …†McSPJAS^HG\/WIV

‰ XbMcU ˆ ‰ ‰

XbMcU ˆ

r \/-XWIJ‡*Q‰ @r*NPJ&Mc…PHGF \/W`NPM/GI`LJ&MeX HONPJ\e`L`LJA…PHO\/SP`LJ…EU Mc-E\/-PGmWIGH#. YXMcU,\!d^GT^JAS<JAUU#McU

KEGIVHOU#G-EDPHGIMcS DPVGSPdlMPDEU&F!JLHONPMPK6QRCPMcF JWF!McU#J `LMcFo…XWIGI`A\eHJLKJ \^Fo…XWIJLVR\/U#J XbMcDESPK2GS

HO\/-XWIJ!‡*Q‡

MeX,HONPJoJE…EMcSPJAS^HG\/W'KEGIVHU#G-EDPHGIMcS '\/SPKhHNPJ \/DPVVG\/S%KEGVHOU#G-EDPHGIMcSGS2VJL`LHGIMcS‡*Qc;Q‡*Q S

VJL`LHGIMcS‡*QcQ^z_ J2`LMcSPVGKEJAU HONPJ OF!MPKEG JLK \/DPVVG\^S KEGIVHU#G-EDPHGIMcS &_NPGI`LN …EU#MeT/JLVHM

-EJ2F!DP`N F!MPU#JWHU#McDE-XWIJLVMcF!JRQ pJLVODXWIHV XMcU HONPGIVoHJL`LNESPGEDPJ2_NPJAS \/…E…XWIGIJLK HM]-PGF!MPK \/W

KEGIVHOU#G-EDPHGIMcSPV\^U#J&VODEFoFo\^U#GIVJLKGSHO\/-XWIJ!‡*Q‡*Q

6]C]`\ !

s*g '=qje@`b?p @B>Ajek/@ 7 9je@Bg

ZhJ&KEJ SPJ!HONPJ&JE…EMcSPJAS^HG\/W KEGIVHOU#G-EDPHGIMcS%\eV

2

‡

/1 b‡*Q^Rq

_NPGI`LN-WIJA\eKEVHM b‡*Q^Rf ZhJ&`LNPMcM^VJ

b‡*Q^ *

b‡*Qccˆ r*NPGIVUWJA\YKEVHM

(34)

\&…E…XW.PGSPdJGQ?‡*Q^R^ ˆ*Q@pJ V \/UU\/SPd/JAF!JAS^H,d/GIT/JLV b‡*Qcc‡

r*NPJ&GS^T/JAU#VJ HOU\/SPV#XbMcUF \eHGIMcS `A\/S-EJ!…EJAU XbMcUF!JLK2-/.<GSPVO…EJL`LHGMPSlHMRd^GT^J

‰ XbMcU ˆ

XbMcU ˆ

b‡*Qc ^

ZhJoSPM/HJ!HONE\YHHONPGIV!\e`L`LJA…PHO\/SP`LJ!XBDESP`LHGIMcS2MPSXW.2U#JAFo\eGSPV!…EM/VGIHGIT/J XMcU\/WmW 2_NPJAS

\eV\^S/HGI`LG…E\eHJLK J1Q ‡*Q‡Rq Q

6]C]

5b '>e> @`b 9@B> jek/@ 7 9je@Bg

ZhJl\eHHJAFo…PH!HM SPK4\2HNPJAUFo\/W*\e`L`LJA…PHO\^SP`LJ UDXWIJ<XbMcU!HONPJ &\/DPVVG\/SKEGIVHOU G-EDPHGMPS JGQ

‡*Q‡ QkU#McF HONPGIV_,J%SPK

b‡*QcRc

r*NPGIVXBDESP`LHGIMcSlNE\eV&\/S JLVVJAS^HG \/W VGSPdPDXW\^U#GIH.2\eH EVM *\^SPK

,`A\/SESPM/H VHOU#GI`LH WI.

-EJ `NPM^VJAS1Q bdPSPMcU#GSPdWHNPGV KEG<`ADXWH#. _,J`NPMPM/VJ

b‡*Qc b

_NPGI`LN-WIJA\eKEVHM

b‡*Qc t

CPGSP`LJ FRDPVHKEJL`A\ .]\eV

\YV JGQ9‡*Q^N^ _,JF!DPVHRNE\ T/J _NPGI`LN-WIJA\eKEVHM ‰ XbMcU XbMcU

b‡*Qc q

\ WIHONPMPDPdcN

*GIV&SPM/H-†McDESPKEJLK]\eH -^.l\ XMPU#HODPGIHMcDPV`A\/SP`LJWmW\eHGIMcS%MeX'HJAUF V\

HONPJAUF \/W \Y`L`LJA…PHO\/SP`LJ*UDXWIJ,XbMcUHNPJ &\/DPVVG\/S `A\eVJ NE\eV'-†JLJAS XMPDESPK6Q r NPGV9`A\^SP`LJWW\eHGMPSo_*\eV

_NE\eH&\/WmWIMe_ JLK k9GSEa l \ WW _^/c 'HMWV…EM/HHONPGIVVMNWDPHGMPS1QZr*NPGIVR\e`L`LJA…PH\/SP`LJoUDXWIJW\/…E…EJA\^U#V

T/JAU#.VGF GW\/UHMWHONPJ $2JLHOU#MP…EMRWIGIV UDXWJ J1Q ‰RQc ?\/SPK2JAFo…ENE\eVGIVJLV&HONPJ \/DPVVG\/S2-EJLGSPdl\

(35)

0

2

− 2

σ

σ

x− E

f(x− E)

k9GIdcDEU#J2‡*Q‡-\ VOaeJLH`N MeX HNPJlF!McKEG JLK \^DPVVG\/S KEGIVHU#G-EDPHGIMcS1Q S M/HJHONE\YHoHONPGIVWGIV\

-PGF!MPK \/W KEGIVHU#G-EDPHGIMcS_*GIHON \/DPVVG\/S<HO\eGmWIV8Q

6]C]

g 9@ =65 b +>r> @`b @B> jrk^@ 7 9je@Bg

S Mr_HONE\eHq_ J*NE\ T/J,\F!JLHNPMcK _NPGI`N!_,McUa/V_,JWmWPXbMcU\ScDEF!-EJAU MYX JAUU#McUqKEGIVHOU#G-EDPHGIMcSPV VJLJ

HO\/-XWIJLV*‡*Q‰ c‡*Q‡ /HONPJ EDPJLVHGIMcS\/U#GIVJLV cKEMPJLV,GIH _,McUa XMPU'\/WmW`NPM^G`LJLV MeX 7r*NPJ&\/SPV_,JAU+HM

HONPGIV,GIV\ UF OSPM P-/.oTPGU#HODPJMeX[\ KEGIVHOU#G-EDPHGIMcS<_,J&NE\ T/J`A\ WWIJLKHONPJ&F!MPKEG JLK \^DPVVG\/S

KEGIVHOU#G-EDPHGIMcS1Q \UWHNPMcDPdcN HONPGIVKEGIVHOU#G-EDPHGIMcS _ GWmWSPM/H-EJDPVJLK X McU<\ dcU#JA\eHWSPDEF!-EJAU<MeX

\/…E…XWIGI`A\eHGIMcSPV ?GIH`A\^DPVJLV…EU#Mc-XWIJAF!V XbMcU,McDEU*F JLHONPMcK6Q bH*GIVd/GIT/JAS-/.

2

‡

2

b‡*Qc f

_NPJAU#J GV\2`LMcSPVHO\/S^H8Q r*NPGIV6-PGF!MPK \/WKEGVHOU#G-EDPHGIMcS GIVWVOaYJLH`NPJLK GS dcDEU#J ‡ Q‡*Q r*NPJ

k MPDEU#GIJAU V \/…XW\e`LJ!HOU\^SPV#X MPUF MeX J1Q'‡*Qc f GIV

.

‡

‡

b‡*Qc *

r*NPGIVUWJA\YKEVHM

‰

b‡*Qb/ˆ

(36)

S\/F!J

2

l+GF!McK \ W

2

2

‰ X McU

X McU X McU

l+GF!McK \ W & E…EMcSPJAS^HG\/WIV

/ /1 ‰ XbMcU XbMcU XbMcU

r \/-XWIJ‡*Q‡@`&…PHGFo\/W\e`L`LJA…PH\/SP`LJUDXWIJLV,XbMcU'-PGF!MPK \/WKEGIVHOU G-EDPHGMPSPV"V9VGF!GmW\/U*HMHONPJF!MPKXV

G JLK&\/DPVVG\/S KEGVHOU#G-EDPHGIMcS JGQ ‡ Qc f P-EDPH*XbMcU+_NPGI`LNHONPJF!JLHNPMcK%McDPH WIGSPJLKlGSVJL`LHGMPS

‡*Q^ Q‡ GV VODP`LJLVV#XBDXW

MeX LJAU#McJLV X MPU '(

ˆ_NPGI`N„GV DPVHJASPMcDPdcN HM V`ADE…E…EJAUMPDEUF!JLHONPMPK6Q \*HHONPGIV

…EM^GS^H_,JoSPM/HJ!HONE\YHHONPJ \/DPVVG\^SlKEGIVHU#G-EDPHGIMcS]…EU MrT^JLK HU#McDE-XWIJLVMcF!J VMGH GIV…EJAUNE\/…PV

SPM/H VODEU…EU#GIVGSPdHONE\eHGIH!GIV!HONPJ6F McKEG JLK \^DPVVG\/S KEGVHOU#G-EDPHGIMcS HNE\eH&XB\/WmWIV!X MPDXWQ7r \/-XWIJ

‡*Q‡z`LMcS/H\eGSPV6\2VDEFoFo\/U#.MeX&M/HONPJAU<-PGF!McK \ WKEGIVHOU G-EDPHGMPSPVhVRVGF!GmW\/U<HMhHNPJ F!MPKEG JLK

\^DPVVG\/S<_NPGI`LNKEMWSPM/H VOD JAU,HONPJ&VO\/F J…EU#Mc-XWIJAF!V8Q

\HONPJAUFo\ W \e`L`LJA…PH\/SP`LJhUDXWIJlXbMcU HONPJ F McKEG JLK &\/DPVVG\/S KEGIVHOU#G-EDPHGIMcS„`A\/S -†J

XbMcDESPK6Q Me_,JLT/JAU P_,JFRDPVHU#JWIG S DPGIVON2HNPJGSPVGIVHO\/SP`LJ HONE\eH ‰XbMcU %-EJWIMe_‚VMcF!J

HONEU#JLVNPMRWIK6Q X9_,J`LNPM/VJ&HONE\YH ‰

P_NPJAU#J *\/SPK \/U J`LMcSPVHO\/S^HV PHONPJAS

_,J%SPK ‰ XbMcU XbMcU b‡*QbE‰

r*NPGIVGV&VaeJLH`LNPJLKlGS dcDEUJR‡*Q^ Q S M/HJ&HONE\YH*HONPGIV&GIV \<U\eHONPJAUMcKEK2\e`L`LJA…PHO\/SP`LJ!UDXWIJ VGSP`LJ

GIH*GIV SPM^HF!McSPM/HMcSPGI`A\/WmW.KEJL`AU JA\eVGSPd \eV*GIV*HONPJ `A\eVJXbMcU+M/HONPJAU,`LNPM/GI`LJLVMYX Q

\ WIHONPMPDPdcNGIH*GIVVO\eHGIV#Xb.cGSPd<HONE\eH*\<UDXWIJ`A\/S-†J&XbMcDESPK HNPGV GV&\IWIJLVVHONE\/S%…EJAU XbJL`LH

VGIHODE\eHGMPS1QlZhJNE\ T/JW\/S\/…E…EU#MP\e`N _NPGI`LNh_,McUa/V _ JWmW,GS]F \/S/.2VGIHODE\YHGIMcSPV8Q *Me_,JLT/JAU

HONPJAU#J&GIV`WIJA\/U WI. \!VGHDE\eHGIMcS<GS_NPGI`LN%GH KEMcJLV&SPM/H*_,McUa;Q@r*NPJU#JLVODXWIHV&U#JA…†McU#HJLKGS%HO\/-XWIJ

‡*Q‡ VONPMe_ `WIJA\/U WI.lHNE\eH*-PGF McK \/WIGIH#.MeX9HONPJ&KEGIVHGU#G-EDPHGIMcS%GIV SPM/H,GSGIHVJWX'HNPJ&`A\/DPVJ&MeX9HONPJ

KEGW`ADXWIH#.RQCu*DEUU#JAS^H WI. SPM_*\A.2NE\eV&-EJLJASzXMcDESPK2HMKEJLHJAUF!GSPJo_NPGI`LN2KEGIVHOU#G-EDPHGIMcSPV _*GmWW

(37)

−βσ 0

x

2

2

0

1

A(x)

k9GIdcDEU#J*‡*Q^6\ VaeJLH`LN&MeXHONPJ,HONPJAUFo\/WE\e`L`LJA…PHO\^SP`LJUDXWIJ,XbMcUHNPJF McKEG JLK \/DPVVG\^SKEGIVHU#G_V

-EDPHGIMcS1Q r*NPGIV\e`L`LJA…PHO\^SP`LJRUDXWIJGIV&…EJAU#`ADXWIG\/UGSHNE\eH,GHGIV SPM^HF McSPM/HMcSPGI`A\/WmWI.lKEJL`AU JA\eVGSPd

3 !#"$% &'

r(*)+-,-./(-+/,-0 *0&(-124335,*687-0:9;+-<=(*243?>@1A1B<6824C*10&(*240+/<-1$9D3$2E>-FD1G0:+H6#2EI417*)+E.-)1A339DJE1AFDK

6+/)L1G7*)1AMA9D3181A3:09N682401A3+4O 2 QPSR +-)05(-1MB243:1+4OT05(-11AU*7*+/<-1B<V09N24FC*9D305)L9W>*,-0:9;+-< 05(-9D39;3

MAFD1B2E)FDKX<-1AMA1A3352E)LK=39N<-MA18YS182E)18<-+E0$2E>-FD180+Q39N6,-FW2Z01[2#05(-1B)6824F3:K/30:1B6\9]O

^31A1

052E>-FD1[_@PN` aY?(-1B)1 b9D3$05(-1#Y9DC*05(+4OT05(-1#1B))+/)aC*9D305)9N>*,-09D+/<c2E<-Cd9D30&(-1G9N<VJE1B)31#01B6e

7*1B):2405,*)1EPfg<#05(-1 h2E,-3:39N2E<MB243105(-9D3T9D3<-+E0S<-1AMA1A3352E)K -39N<-MA105(-1 h2E,-33:9W2V<C*9D305)9N>*,-09D+/<

Og24FDF;3a+4i=Og24301B)j05(*2E<Q1AU*7*+/<-1B<V09N24FkPSl+4YS1AJE1B) V+/,*)S61A05(-+-CmC*1A3MB)9N>*1AC2E>*+4JE1Y9DFDFn>*1AMA+/6H1

FD1A33$2E<-CmFD1A33?1QMA9;1B<V0$,*<-F;1A3:3 C*1AMB)L1B2431A3$240SF;1B2Z30h2Z3TOg2430243 X9N<-MB)1B243:1A3BP

o

1MA+/<-3:9;C*1B)j05(-1$2E,-339N2E<C*9D30&)9N>*,-09D+/< p1%P-_*Pq_.E1B<-1B)240:1ACOr)L+/6)2E<-C*+/6s3&2E6e

7-FD9N<-. ^1%P_*PqtAPuR9N<*ImvwS24FDF ]x4y z24C*C )1A33:1AC=05(-19;3:35,-1+4OMA(-+/+E3:9W<-.Q05(-1352E687-FD13:9 A1>EK

69N<-9N69D39N<-.m05(-1$MA+/687*,-0&2409D+/<*24FMA+E30S+4O24MAMA1B7-09N<-.[<-1B,-05)2ZF{6H+|JV1A3 }Op+/)~Y?(-9DMA(42 €AP

 (-,-3 *05(-1AKQMA+/<-39DC*1B)1ACƒ‚%<-C*9N<-.

„Y(-9;MA(=6H9W<-9N69D31A3

A… E†

…

_*Pq‡E_V†

 (-1$3+VFW,-0:9;+-<m+4O05(-9D3?9D3.E9DJE1B<m>VK

€N`ˆtE‰

…

_*Pq‡ExV†

References

Related documents

Our results demonstrated that the prevalence of uterine lesions among cases with 2 recurrent miscarriages was 42.1% (22.1% Acquired, and 20% congenital), meanwhile, for subjects

This paper shows a design of automatic irrigation system based on android appli- cation using Raspberry Pi microcontroller, soil moisture and temperature sensor to help a famer

A canonical basis for the morphisms between string complexes and (one- dimensional) band complexes in D b (Λ) was given in [4] in terms of three types of morphism: graph maps ,

Considering an embedded graph representation of G in a torus R/ ∼, with the detached disc represented by an open subset of R , it follows that in fact any torus with hole graph

The study is undertaken with the objectives to review various literatures and understand Pradhan Mantri Awas Yojana (PMAY), an affordable housing scheme for all

There are two elements at play here, understanding both of which is essential in understanding the feelings of hopelessness in depressed patients in Iran and the emphasis placed