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Ducgmjfct`s gf Mc·Ducgmjfct`s gf Mc·mbnsf Ducon`cmbmbnsf Ducon`cmb
Lfrmbg` I`tfba`, Gmcnfb Ufbbflrnc` f Fgumrg` WfnxfnrmLfrmbg` I`tfba`, Gmcnfb Ufbbflrnc` f Fgumrg` Wfnxfnrm
O`j O`j ?79 ?79 fxffxfroro··ĽonĽon`s`s
  
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@ @ tt··Ľtub` Ľtub` gfstf gfstf bnvr` bnvr` trmz trmz gums gums ncd`rjm±ncd`rjm±o˒o˒̀fs nlumbjfctf njp`rtmctfs. M prnjfnrm, f jmns`fs nlumbjfctf njp`rtmctfs. M prnjfnrm, f jmns
··̀ivnm, fstmifbfof quf sf trmtm gf uj bnvr` gf Mc·`ivnm, fstmifbfof quf sf trmtm gf uj bnvr` gf Mc·mbnsf Ducon`cmb. M sflucgm, c˒mbnsf Ducon`cmb. M sflucgm, cm˒` jfc`sm` jfc`s
njp`rtmctf, ncd`rjm quf sf trmtm gf uj tfxt` nctr`gut·njp`rtmctf, ncd`rjm quf sf trmtm gf uj tfxt` nctr`gut·̀rn`.`rn`.    ··F F cfofscfofss·s·mrn` mrn` quf quf nss` nss` ﬍quf﬍quf
obobmr` mr` gfgfsgf ` sgf ` prprnjnjfnfnr` r` o`co`ctmttmt̀ ` g` g` bfbfntnt`r `r o`j ` o`j ` bnbnvrvr`; `; trmtrmtm-tm-sf sf gf gf bnbnvr` fsornvr` fsornt` t` pmrpmrmm
fstugmctfs quffstugmctfs quf   c˒ c˒ m`m`  smifj Mc· smifj Mc·mbnsf Ducon`cmb f prftfcgfjmbnsf Ducon`cmb f prftfcgfj   mprfcgfr   mprfcgfr   m tf`rnm.  m tf`rnm.    ··F obmr`F obmr`
quf gfquf gfsfkmj`sfkmj`s quf ` s quf ` bnvr` sbnvr` sf t`rf t`rcf ujm cf ujm rfdfr˄rfdfrf˄conm fconm tmji·tmji·fj pmrfj pmrm bfntm bfnt`rfs j`rfs jmns fxpmns fxpfrnfctfrnfctfs,fs,
jms nss` ‟ sf mo`ctfofr ‟ sfr·jms nss` ‟ sf mo`ctfofr ‟ sfr·m uj lmca` mgnon`cmb. @ `ikftnv` ofctrmb fj t`rc` g` qumbm uj lmca` mgnon`cmb. @ `ikftnv` ofctrmb fj t`rc` g` qumb
` ` bnvr` bnvr` d`n d`n fsorfsornt` nt` ··f f sfrvnr sfrvnr pmrm pmrm m m ncstrncstru±u±o˒om˒` gf fstugmctfs quf c˒m` gf fstugmctfs quf cm˒` m` tfcamj o`cafonjfcttfcamj o`cafonjfct̀̀
pr·pr·fvn` fvn` g` g` mssuct`, f mssuct`, f sfrvnr sfrvnr tmji·tmji·fj fj gf gf lunm lunm f f r`tfnr` r`tfnr` pmrm pmrm uj uj ours` ours` nctr`gut·nctr`gut·̀rn`. Umrm`rn`. Umrm
` pr`dfss`r, sulfrnj`s mgnmctf r`tfnr`s pmrm ours`s gf g`ut`rmg` f gf jfstrmg`.` pr`dfss`r, sulfrnj`s mgnmctf r`tfnr`s pmrm ours`s gf g`ut`rmg` f gf jfstrmg`.
Fj sfu bnvr` X64Z, Fb`c B. Bnjm gf﬍cf, o`j juntm pr`prnfgmgf, mFj sfu bnvr` X64Z, Fb`c B. Bnjm gf﬍cf, o`j juntm pr`prnfgmgf, m    ··Mblfirm BncfmrMblfirm Bncfmr
o`j`o`j`  ` fstug` g`s fspm± ` fstug` g`s fspm±o`s vft`rnmns f gms trmcsd`rjm±o`s vft`rnmns f gms trmcsd`rjm±o˒ o˒ `fs bncfmrfs fctrf fbfs `fs bncfmrfs fctrf fbfs . _fluncg`. _fluncg`
fssm bncam, p`gfj`s gf﬍cnr m Mc·fssm bncam, p`gfj`s gf﬍cnr m Mc·mbnsf Ducon`cmb o`j` ` fstug` g`s fspm±mbnsf Ducon`cmb o`j` ` fstug` g`s fspm±o`s vft`rnmnso`s vft`rnmns
c`rjmg`s, fj fspfonmb `s fspm±c`rjmg`s, fj fspfonmb `s fspm±o`s go`s gf Imcf Imcmoa, f moa, f g`s `g`s `pfrmg`rfs pfrmg`rfs bncfmrfs bncfmrfs o`cto`ct··Ľcu`s fctrfĽcu`s fctrf
fbfbfs. fs. FssFssms gfms gf﬍c﬍cn±n±o˒o˒̀fs, `fs, mb·mb·fj fj gf gf prfonsmsprfonsms, , fvngfconfvngfconmj mj ms ms sfjfbasfjfbamc±mc±oms f ms gndfrfc±oms f ms gndfrfc±omsoms
fcfctrf trf fssfssms ms gumgums s tf`tf r̀nrnms ms jmjmtfjtfj··mtmtnonomsms. . MjMjimims trms trmtmtmj gf j gf fsfspmpm±±o`o`s s vvftft`r`rnmnmns ns f f gfgf
trmcsd`rjm±trmcsd`rjm±o˒o˒̀f`fs s bnbncfcfmrfmrfs, s, f f m m gngndfdfrfcrfc±±om om ofctrmb ·ofctrmb ·f f quf quf cm cm Mc·Mc·mbnsf mbnsf DDucon`cmb ucon`cmb fxnlnjfxnlnj`s`s
tmji·tmji·fj m fj m o`ctncungmgf. o`ctncungmgf. U`r quf U`r quf nss`0 nss`0 U`r quf U`r quf fjfj   ··Mblfirm Bncfmr c˒Mblfirm Bncfmr cm˒` c`s njp`rtmj`sm` c`s njp`rtmj`s
o`j o`ctncungmgf f fj Mc·o`j o`ctncungmgf f fj Mc·mbnsf Ducon`cmb gfbm dmzfj`s qufst˒mbnsf Ducon`cmb gfbm dmzfj`s qufstm˒`0 m`0 @ bfnt`@ bfnt r̀ fco`cr fco`ctrmrtrmr··mm
ms rfsp`stms bfcg` ` tfxt`, jms k·ms rfsp`stms bfcg` ` tfxt`, jms k·m m mgnmgnmctmctmj`s ` mj`s ` sflusfluncnctf; tf; ` ` p`ctp`ct` ` ofcofctrmb ·trmb ·f f quf quf mm
··Mblfirm Bncfmr, pmrm tfr m vmbn`sm dfrrmjfctm quf s˒Mblfirm Bncfmr, pmrm tfr m vmbn`sm dfrrmjfctm quf sm˒` ms jmtrnzfs em` ms jmtrnzfs em gnsp`sn±m gnsp`sn±o˒om˒`, trmtmm`, trmtm
fssfconmbjfctf gf fspm±fssfconmbjfctf gf fspm±o`s vft`rnmns gf gnjfcs˒o`s vft`rnmns gf gnjfcsm˒` ﬍cntm. F o`j` ` bfnt`r o`jpr`vmr·m` ﬍cntm. F o`j` ` bfnt`r o`jpr`vmr·m m m`m`
fstugmr fstf bnvr`, qumbqufr quf sfkm m jmcfnrm o`j quf uj fspm±fstugmr fstf bnvr`, qumbqufr quf sfkm m jmcfnrm o`j quf uj fspm±o` vft`rnmb gf o` vft`rnmb gf gnjfcs˒gnjfcsm˒`m`
﬍cn﬍cntm tm ··f f c`rjc`rjmg`, mg`, ms ms trmctrmcsd`sd`rjm±rjm±o˒o˒̀fs `fs bnbncfmcfmrfs rfs cfbcfbf f gfgf﬍cn﬍cngmgms s s˒sm˒` m` mut`jmtnmut`jmtnomjfcomjfctftf
o`cto`ct··Ľcums. Ľcums. @u @u sfkm, sfkm, fj fj gnjfcs˒gnjfcsm˒` m` ﬍cntm ﬍cntm m m o`ctncungo`ctncungmgf mgf ··f f o`csfo`csfqu˄quf˄conm fconm mut`mut`j·j·mtnom gmmtnom gm
bncfmrnbncfmrngmgf. gmgf. K·K·m fj gnjfcs˒m fj gnjfcsm˒` nc﬍cntm m sntum±m` nc﬍cntm m sntum±o˒om˒` ·m` ·f f ifj ifj gndfrfcgndfrfctf. tf. Cfj t`Cfj t`g` g` `pfrmg`r`pfrmg`r
bncfmr gf﬍cng` fj uj fspm±bncfmr gf﬍cng` fj uj fspm±o` gf gnjfcs˒o` gf gnjfcsm˒` m` nc﬍cntm nc﬍cntm ··f f o`cto`ct··Ľcu`, Ľcu`, f f ` ` bfnt`r bfnt`r tfr·tfr·m m v·v·mrnmsmrnms
`p`rtucngmgfs m` b`cl` g` tfxt` gf fxpb`rmr fssf dfc˄`p`rtucngmgfs m` b`cl` g` tfxt` gf fxpb`rmr fssf dfc˄̀jfc`. `jfc`. GmGm··Ľ Ľ m m cfofssncfofssngmgf gmgf gf gf sfsf
o`csngfrmr f fstugmr mqufbfs `pfrmg`rfs bncfmrfs quf s˒o`csngfrmr f fstugmr mqufbfs `pfrmg`rfs bncfmrfs quf sm˒` m` o`ct·o`ct·Ľcu`Ľcu`s, s, f f ··f f fxmfxmtmjtmjfctf fctf nss`nss`
` quf dmz m Mc·` quf dmz m Mc·mbnsf Dmbnsf Ducon`cmbucon`cmb. . O`j juntm bnifrgmgO`j juntm bnifrgmgf f f f uj p`uo` gf uj p`uo` gf pnomrgnpnomrgnm, p`m, p`gf-sfgf-sf
gnzfr quf mgnzfr quf m    ··Mblfirm Mblfirm Bncfmr Bncfmr ··f f ujm ujm Mc·Mc·mbnsf Ducon`cmb fj gnjfcs˒mbnsf Ducon`cmb fj gnjfcsm˒` ﬍cntm, m` jfsj`m` ﬍cntm, m` jfsj`
tfjp` quf m Mc·tfjp` quf m Mc·mbnsmbnsf f Ducon`Ducon`cmb cmb ··f f ujmujm    ··Mblfirm Bncfmr fj gnjfcs˒Mblfirm Bncfmr fj gnjfcsm˒` nc﬍cntm.m` nc﬍cntm.
F gf `cgf vfj fssm cfofssngmgf gf sf fstugmr `pfrmg`rfs bncfmrfs fctrf fspm±F gf `cgf vfj fssm cfofssngmgf gf sf fstugmr `pfrmg`rfs bncfmrfs fctrf fspm±o`s gfo`s gf
gnjfcs˒gnjfcsm˒` ncm` nc﬍c﬍cntntm0 m0 M jmtM jmtfjfj··mtnom pmss`u p`r uj gfsfcv`bvnjfct` vfrtnlnc`s` fctrfmtnom pmss`u p`r uj gfsfcv`bvnjfct` vfrtnlnc`s` fctrf
nn
  
`s `s s·s·foub`s foub`s =< =< f f =>, =>, f f ujm ujm gms gms o`csfquo`csfqu˄˄fconms d`n fconms d`n quf quf m m jmtfj·jmtfj·mtnom pmss`u m rfs`bvfrmtnom pmss`u m rfs`bvfr
ujm lrmcgf qumctngmgf (f vmrnfgmgf) gf pr`ibfjms quf mctfs c˒ujm lrmcgf qumctngmgf (f vmrnfgmgf) gf pr`ibfjms quf mctfs cm˒` p`gnmj sfr trmtmg`sm` p`gnmj sfr trmtmg`s
jmtfjmtnomjfctf gf d`rjm smtnsdmt·jmtfjmtnomjfctf gf d`rjm smtnsdmt·̀rnm. `rnm. Junt`s Junt`s gfssfs gfssfs pr`ibfjms pr`ibfjms t˄tf˄j fj duc±duc±o˒o˒̀fs ‟ f c˒`fs ‟ f cm˒`m`
c·c·ujfr`s ujfr`s `u `u ﬍lurms ﬍lurms lf`jlf`j··ftrnoms ftrnoms ‟ ‟ o`j` o`j` s`bu±s`bu±o˒o˒̀fs, p`r fxfjpb`, fqum±`fs, p`r fxfjpb`, fqum±o˒o˒̀fs gndfrfconmns`fs gndfrfconmns
ducon`cmducon`cmns, ns, `rgnc·`rgnc·mrnms `u pmronmns. Gfvfj`s fct˒mrnms `u pmronmns. Gfvfj`s fctm˒` iusomr ms s`bu±m` iusomr ms s`bu±o˒o˒̀fs gfssfs pr`ibfjms`fs gfssfs pr`ibfjms
fj o`ckuct`s gf duc±fj o`ckuct`s gf duc±o˒o˒̀fs. `fs. NbustrfjNbustrfj`s fssm sntum±`s fssm sntum±o˒om˒` o`j uj fxfjpb` anst·m` o`j uj fxfjpb` anst·̀rno` ob·`rno` ob·mssno`.mssno`.
GfcGfctrf trf `s `s v·v·mrnmrn`s `s pr`ipr`ibfjbfjms ms quf quf pmspmssmrmsmrmj j m m sfr sfr trmttrmtmg`s mg`s jmtfjmtfjmtnjmtnomjomjfctfctf f mp·mp·̀s `s `̀
mgvfcmgvfct` g` t` g` O·O·mboub`, uj mboub`, uj g`s jmns o·g`s jmns o·fbfirfs d`n fbfirfs d`n ` pr`ibfjm ` pr`ibfjm gm o`rgm gm o`rgm vnirmctf. vnirmctf. Oamjfj`sOamjfj`s
gfgf   uu((x,x, tt) ` gfsb`omjfct` vfrtnomb fj rfbm±) ` gfsb`omjfct` vfrtnomb fj rfbm±o˒om˒` m` eem p`sn±m p`sn±o˒om˒` gf rfp`us` s`drng` pfb` p`ct`m` gf rfp`us` s`drng` pfb` p`ct`
gf misonssmgf misonssm   xx  c` ncstmctf c` ncstmctf    tt   (gfsb`omjfct` p`sntnv` pmrm onjm f cflmtnv` pmrm imnx`).  (gfsb`omjfct` p`sntnv` pmrm onjm f cflmtnv` pmrm imnx`).
Fj =<7< ` fjncfctf jmtfj·Fj =<7< ` fjncfctf jmtfj·mtno` mtno` drmco˄drmcof˄s fs Kfmc-bf-Y`cg Kfmc-bf-Y`cg g„Mbfjifrt g„Mbfjifrt o`cobunu o`cobunu quf quf t`gmt`gm
s`bu±s`bu±o˒om˒`m` u u  g` pr`ibfjm gm o`rgmvnirmctf smtnsdmz cfofssmrnmjfctf m fqum± g` pr`ibfjm gm o`rgm vnirmctf smtnsdmz cfofssmrnmjfctf m fqum±o˒om˒` m` gndfrfcogndfrfconmbnmb
pmronmbpmronmb
∉ ∉ 99uu
∉t∉t99
  ∙∙ ∉  ∉ 
99uu
∉x∉x99
   : : 55,,
f, f, mb·mb·fj fj gnss`, gnss`, gnssf gnssf quf quf duc±duc±o˒o˒̀f`fs s g ̀g ̀ tntnpp`̀
uu((t,t, xx) ) :: d  d ((xx ++ tt) +) + l l((xx ∙∙ tt)),,
`cgf`cgf   d d    ff   ll   s˒sm˒` duc±m` duc±o˒o˒̀fs rfmns mrintr·`fs rfmns mrintr·mrnms gf ujm vmrn·mrnms gf ujm vmrn·mvfb rfmb, s˒mvfb rfmb, sm˒` s`bu±m` s`bu±o˒o˒̀fs gm fqum±`fs gm fqum±o˒om˒`m`
gndfrfconmb f p`rtmct` s˒gndfrfconmb f p`rtmct` sm˒` i`ms omcgngmtms m sfrfj s`bu±m` i`ms omcgngmtms m sfrfj s`bu±o˒o˒̀fs g` pr`ibfj`fs g` pr`ibfjm m `rn`rnlnclncmb. mb. CmCm
··fp`om fp`om gf gf g„Mbfjg„Mbfjifrt ifrt cfj cfj ` ` o`cofnt` gf o`cofnt` gf o`ctno`ctncungcungmgf mgf fxnstnfxnstnm, m, jms m` jms m` b`cl` g` b`cl` g` tfjp`tfjp`
`s jmtfj·`s jmtfj·mtno`s pfrofifrmj quf ms duc±mtno`s pfrofifrmj quf ms duc±o˒o˒̀fs`fs    d d    ff    ll   gfvfj sfr iusomgms c` o`ckuct`  gfvfj sfr iusomgms c` o`ckuct`
gms duc±gms duc±o˒o˒̀fs `fs quf quf t˄tf˄j fj pfb` pfb` jfc`s jfc`s gums gums gfrnvgfrnvmgms mgms o`cto`ct··ĽcuĽcums. ms. @o`rrf @o`rrf quf quf ` ` o`ckuco`ckuct`t`
d`rjmg` p`r fssms duc±d`rjmg` p`r fssms duc±o˒o˒̀fs, mssnj o`j` t`g`s `s `utr`s o`ckuct`s gf duc±`fs, mssnj o`j` t`g`s `s `utr`s o`ckuct`s gf duc±o˒o˒̀fs quf s˒`fs quf sm˒`m`
··utfns, s˒utfns, sm˒` fspm±m` fspm±o`s vft`rnmns gf gnjfcs˒o`s vft`rnmns gf gnjfcsm˒` nc﬍cntm` nc﬍cntm. m. ImsImstm `isfrvtm `isfrvmr quf mr quf `s p`bnc˄`s p`bnc˄̀jn`s k·`jn`s k·mm
lfrmj uj fspm±lfrmj uj fspm±o` gf gnjfcs˒o` gf gnjfcsm˒` nc﬍cntm. Vj sflucg` fxfjpb`; mssnj o`j` ujm s`bu±m` nc﬍cntm. Vj sflucg` fxfjpb`; mssnj o`j` ujm s`bu±o˒om˒`m`
gf uj snstfjm bncfmr o`jgf uj snstfjm bncfmr o`j   cc   nco·  nco·̀lcntms `lcntms m m o`f﬍onfctfs o`f﬍onfctfs rfmns rfmns ··f f ujmujm   cc-upbm (-upbm (xx==, . . . , x, . . . , xcc))
g`g`   YYcc, ujm s`bu±, ujm s`bu±o˒om˒` gf uj snstfjm bncfmr o`j nc﬍cntms nco·m` gf uj snstfjm bncfmr o`j nc﬍cntms nco·̀lcntms m o`f﬍onfctfs rfmns`lcntms m o`f﬍onfctfs rfmns
··f f ujujm m sfsfqu˄quf˄cfcononm m ((xxcc))
∓∓
cc:=:=   gf c·  gf c·ujfr`s ujfr`s rfmns8 rfmns8 f f o`ckuct`s o`ckuct`s gf gf sfqu˄sfquf˄conms fconms tmji·tmji·fj fj s˒sm˒`m`
fspm±fspm±o`s vft`rnmns gf gnjfcs˒o`s vft`rnmns gf gnjfcsm˒` m` nc﬍cntmnc﬍cntm. . Fj Fj rfsuj`, rfsuj`, fssm fssm c`vc`vm m mirmclmirmcl˄˄fconm fconm mbomc±mbomc±omgmomgm
pfbm jmtfj·pfbm jmtfj·mtnom o`b`o`u `s fspm±mtnom o`b`o`u `s fspm±o`s gf gnjfcs˒o`s gf gnjfcsm˒` m` nc﬍cntm fj nc﬍cntm fj v`lmv`lm, , p`ns p`ns ··f f cfbfs cfbfs qufquf
s˒sm˒` fco`ctrmgms ms s`bu±m` fco`ctrmgms ms s`bu±o˒o˒̀fs gfssm lrmcgf vmr`fs gfssm lrmcgf vmrnfgmgf gf pr`ibfjms. nfgmgf gf pr`ibfjms. F fspm±F fspm±o`s vft`rnmnso`s vft`rnmns
gf gnjfcs˒gf gnjfcsm˒` m` nc﬍cntm nc﬍cntm s˒sm˒`, m`, mctfs mctfs gf gf jmns jmns cmgm, cmgm, fspm±fspm±o`s o`s vft`rnvft`rnmns, mns, p`rtmct` p`rtmct` ms ms duc±duc±o˒o˒̀fs`fs
cmturmns quf gfvfj`s o`csngfrmr fctrf fbfs s˒cmturmns quf gfvfj`s o`csngfrmr fctrf fbfs sm˒` ms trmcsd`rjm±m` ms trmcsd`rjm±o˒o˒̀fs bn`fs bncfmcfmrfsrfs. . U`U r̀ ﬍j,r ﬍j,
o`j` k·o`j` k·m vnj`s, c` oms` gf trmcsd`rjm±m vnj`s, c` oms` gf trmcsd`rjm±o˒o˒̀fs bncfmrfs fctrf fspm±`fs bncfmrfs fctrf fspm±o`s gf gnjfcs˒o`s gf gnjfcsm˒` nc﬍cntmm` nc﬍cntm
gfvfj`s bfvmr fj o`ctm m o`ctncungmgf, o`jpbftmcg` m kustn﬍omtnvm pmrm ` fstug` g`sgfvfj`s bfvmr fj o`ctm m o`ctncungmgf, o`jpbftmcg` m kustn﬍omtnvm pmrm ` fstug` g`s
`pfrmg`r`pfrmg`rfs fs bncfmrfs bncfmrfs o`cto`ct··Ľcu`s Ľcu`s fctrf fctrf fspm±fspm±o`s gf gnjfcs˒o`s gf gnjfcsm˒` nc﬍cntm.m` nc﬍cntm.
M fso`bam g`s t·M fso`bam g`s t·̀pno`s `pno`s gf gf uj uj bnvr` bnvr` cucom ·cucom ·f f tmrfdm tmrfdm d·d·monb. monb. C` C` prfsfctf prfsfctf bnvr` bnvr` m m sntum±sntum±o˒om˒`m`
c˒cm˒` d`n gndfrfctf f mblucs tfjms nctfrfssmctfs tnvfrmj quf sfr prftfrng`s. _mifj`s qufm` d`n gndfrfctf f mblucs tfjms nctfrfssmctfs tnvfrmj quf sfr prftfrng`s. _mifj`s quf
nss` mi`rrfofr·nss` mi`rrfofr·m mblucs bfnt`rfs, cm vfrgmgf c`ssm `pncn˒m mblucs bfnt`rfs, cm vfrgmgf c`ssm `pncnm˒` m` ··f f quf quf gmg` gmg` uj uj bfnt`r bfnt`r fxnstffxnstf
(pfb` jfc`s) uj t·(pfb` jfc`s) uj t·̀pno` oukm `jnss˒`pno` oukm `jnssm˒` fstf bfnt`r c˒m` fstf bfnt`r cm˒` c`s pfrg`mr·m` c`s pfrg`mr·m. m. GntGnt` nss`, dmb` nss`, dmbfj`sfj`s
uj puj p`uo` g`uo` g`s tfjms `s tfjms ncobuncobu··Ľg`s. Ľg`s. C`ssm sfbf±C`ssm sfbf±o˒om˒` gf t·m` gf t·̀pno`s `pno`s pr`our`upr`our`u, , mb·mb·fj fj g`s g`s mssuct`smssuct`s
`irnlmt·`irnlmt·̀rn`rǹ s `s gf gf uj uj ourours` s` ncnctr`gtr`gut·ut·̀rn` gf Mc·`rn` gf Mc·mbnmbnsf sf DDuconucoǹ cmb`cmb, , nconcobunbunr r mblumblucs cs t·t·̀pno`s`pno`s
nnnn
  
stmcgmrg jms c˒stmcgmrg jms cm˒` t˒m` tm˒` `irnlmt·m` `irnlmt·̀rn`s `rn`s f f tmji·tmji·fj fj uj uj fsi`±fsi`±o` gf mblucs mvmc±o` gf mblucs mvmc±o`s rfofctfs,o`s rfofctfs,
junt`s g`s qumns njmlncmj`s sfrfj gfso`cafong`s ncobusnvf p`r junt`s pr`﬍ssn`cmnsjunt`s g`s qumns njmlncmj`s sfrfj gfso`cafong`s ncobusnvf p`r junt`s pr`﬍ssn`cmns
gm gm ··mrfmmrfm. . @s mssu@s mssuct`ct s̀ `irnls `irnlmt·mt·̀rn`s `rn`s fco`cfco`ctrmj-sf trmj-sf gfctr` gfctr` g`s g`s sftf sftf prnjfnrprnjfnr̀ s `s ompomp··Ľtub`sĽtub`s..
Mncgm Mncgm gfctr` g`s gfctr` g`s sftf sftf prnjfnrprnjfnr̀ s `s ompomp··Ľtub`sĽtub`s, , fbfcomj`fbfcomj s̀ s `s `s sfluncsflunctfs tfs t·t·̀pno`s o`j` c˒`pno`s o`j` cm˒`m`
`irnlmt·`irnlmt·̀rn`s, f quf, p`rtmct`, gfpfcgfcg` g`s `ikftnv`s g` pr`dfss`r f/`u g` fstugmctf,`rn`s, f quf, p`rtmct`, gfpfcgfcg` g`s `ikftnv`s g` pr`dfss`r f/`u g` fstugmctf,
p`gfj sfr `jntng`s fj ujm prnjfnrm bfnturm;p`gfj sfr `jntng`s fj ujm prnjfnrm bfnturm;
O`r`b·O`r`b·mrn` 9.?.6, _f±mrn` 9.?.6, _f±o˒om˒` m` ?.9?.9, , Ur`Ur`p`p`sn±sn±o˒om˒` m` ?.?.?, ?.?.?, WWf`rfjm f`rfjm 6.7.7, 6.7.7, WWf`rfjm f`rfjm 4.9.=9, 4.9.=9, _f±_f±o˒om˒`m`
4.6, _f±4.6, _f±o˒om˒` 4.4, Wf`rfjm <.9.< f Ur`p`sn±m` 4.4, Wf`rfjm <.9.< f Ur`p`sn±o˒om˒ ̀m ̀ <.<.9.9.1.1.
Mp·Mp·̀s `s fsfstutugmgmr r ` ` o`o`cctf·tf·ug` ug` i·i·msno` msno` gm gm gnsonpbncm, gnsonpbncm, ` ` fstugmctf fstugmctf tfj tfj tr˄trf˄s fs p`ssp`ss··ĽvfnsĽvfns
omjnca`s eomjnca`s em drfctf; m drfctf; (n) fstugmr ujm tf`rnm jmns lfrmb, (nn) mpbno(n) fstugmr ujm tf`rnm jmns lfrmb, (nn) mpbnomr `s o`ctf·mr `s o`ctf·ug`s fstugmg`s,ug`s fstugmg`s,
(nnn) mpr`ducg(nnn) mpr`ducgmr ` mr ` fstug` k·fstug` k·m rfm rfmbnzmgmbnzmg`. `. @s tr˄@s trf˄s omfs ompp··Ľtub`Ľtub`s ﬍cms ﬍cmns g` ns g` bnvr`, bnvr`, `s qu`s qumns pmns pmssmj`mssmj`ss
m gfsorfvfr fj sflungm, `dfrfofj m` bfnt`r ujm `p±m gfsorfvfr fj sflungm, `dfrfofj m` bfnt`r ujm `p±o˒om˒` fm` fj omj omgm ugm ujm gfjm gfssms ssms tr˄trf˄s gfs gnrf±nrf±o˒o˒̀fs.`fs.
C` C` OmpOmp··Ľtub` Ľtub` 1 1 mprfsfcmprfsfctmj`s tmj`s ujm ujm nctr`gu±nctr`gu±o˒om˒` m` m`s m`s Fspm±Fspm±o`s o`s TTft`rnmns ft`rnmns WW`p`b·`p`b·̀lno`s,`lno`s,
tf`rtf`rnm nm fstm fstm quf quf ··f f m m lfcflfcfrmbrmbnzm±nzm±o˒om˒` m` cmtcmturmurmb b g` g` fsfstugtug` ` g`g`s s FsFspm±pm±o`s C`rjo`s C`rjmg`mg s̀ s f f g`sg`s
Fspm±Fspm±o`s o`s gf gf ImcmoImcmoa. a. Cfstf ompCfstf omp··Ľtub` ` Ľtub` ` bfnt`r vfrn﬍omr·bfnt`r vfrn﬍omr·m quf junt` g` quf d`n fstugmg`m quf junt` g` quf d`n fstugmg`
mcmctfrntfrn`rj`rjfctfctf f vvmbf mbf fj fj uj uj mjmjinfcinfctf tf jmnjmns s mirmmirmclfcclfctf, tf, f f quf quf mblumblujms jms o`cso`cstru±tru±o˒o˒̀fs`fs
mctfrnmctfrǹ rfs `rfs s˒sm˒` m` oms`s oms`s pmrtnoubmrfs pmrtnoubmrfs gf gf o`cstru±o`cstru±o˒o˒̀fs jmns lfrmns.`fs jmns lfrmns.
@ @ OmpOmp··Ľtub` Ľtub` >, >, c` c` qumb qumb nctrnctr̀ guznj`s `guznj`s ` ` bfnt`r bfnt`r m m t·t·focnoms focnoms gf gf Mc·Mc·mbnsf C˒mbnsf Cm˒`m`-B-Bncfncfmrmr, , ··ff
ujujm m `p``p`rturtucncngmgmgf gf gf gf mpmpbnbnomomr r `s `s o`co`cafoafonjnjfcfct`t`s s gf gf McMc··mbnsf mbnsf DDucon`cmucon`cmb b rfofctrfofctfjfctfjfctff
mgqmgqununrnrng`g`s. s. Fj JmFj Jmtfjtfj··mtnomtnom m MpbMpbnomgnomgm, m, prnprnconpconpmbjmbjfcfctf tf cm cm J`gfbJ`gfbmlfj mlfj JmtfJmtfj·j·mtnom,mtnom,
c`rjmbjfctf m bncfmrngmgf c˒c`rjmbjfctf m bncfmrngmgf cm˒` fst·m` fst·m m prfsfctf8 prfsfctf8 f f cfstfcfstf ompomp··Ľtub` Ľtub` ` ` bfnt`r bfnt`r mprfcgfr·mprfcgfr·m m quf,quf,
jfsj` mirncg` j˒jfsj` mirncg` jm˒` gm m` gm bncfmrngmgf, junt` bncfmrngmgf, junt` p`p`gf sfr gf sfr dfnt` o`j dfnt` o`j `s o`cof`s o`cofnt`s f nt`s f ms t·ms t·focnomsfocnoms
gm Mc·gm Mc·mbnsf Ducon`cmb.mbnsf Ducon`cmb.
C` C` OmpOmp··ĽtuĽtub` b` =5 =5 mpr`mpr d̀ucgducgmj`s mj`s ` ` fstufstug` g` g`s g`s FspmFspm±±o`s o`s gf gf ImImcmocmoa a m` m` ncnctr`gtr`guznuznrr
mblucs tf`rfjms njp`rtmctfs quf c˒mblucs tf`rfjms njp`rtmctfs quf cm˒` s˒m` sm˒` mim` mi`rgm`rgmg`s fg`s fj ourj ours`s nctrs`s nctr`gu`gut·t·̀rn`s gf Mc·`rn`s gf Mc·mbnsfmbnsf
DDuconucoǹ cmb`cmb. . @s t·@s t·̀pno`s `pno`s mprfsfctmg`s mprfsfctmg`s cfstf cfstf ompomp··Ľtub` Ľtub` fxfjpbn﬍omj fxfjpbn﬍omj ` ` mbtmbt··Ľssnj` Ľssnj` lrmu lrmu gfgf
s`﬍stnom±s`﬍stnom±o˒om˒` mbomc±m` mbomc±omg` pfbm Wf`rnm g`s Fspm±omg` pfbm Wf`rnm g`s Fspm±o`s gf o`s gf Imcmoa mImcmoa m` b`cl` b`cl` g` s·` g` s·foub` foub` 95, p95, pr`or`ofss`fss`
fstf quf fstf quf oubjnc`u, c` oubjnc`u, c` ﬍cmb g` ﬍cmb g` s·s·foub`, o`j foub`, o`j m rfofctf m rfofctf tf`rnm gf tf`rnm gf fspm±fspm±o`s afrfgntmrnmjfctfo`s afrfgntmrnmjfctf
ncgfo`jp`cncgfo`jp`c··ĽvfnĽvfns, s, gfvngm, prnconpmbjfcgfvngm, prnconpmbjfctf, tf, m m \. \. W. W. L`wL`wfrs frs f f I. I. JmurfyJmurfy. . Fstm Fstm tf`rnm,tf`rnm,
quf rfcgfu m L`wfrs ujm Jfgmbam Dnfbgs fj =>>7, d`n o`cofingm o`j ` `ikftnv` gfquf rfcgfu m L`wfrs ujm Jfgmbam Dnfbgs fj =>>7, d`n o`cofingm o`j ` `ikftnv` gf
rfs`bvfr pr`ibfjms ofctrmns gm tf`rnm g`s fspm±rfs`bvfr pr`ibfjms ofctrmns gm tf`rnm g`s fspm±o`s gf Imcmoa, jms momi`u rfkuvfcfsofcg`o`s gf Imcmoa, jms momi`u rfkuvfcfsofcg`
f d`rtmbfofcg` t`gm m Mc·f d`rtmbfofcg` t`gm m Mc·mbnsf mbnsf DDucon`cmb ucon`cmb p`r p`r jfn` jfn` gms gms c`vc`vms ms t·t·focnoms focnoms nctr`guznctr`guzngmsngms
f, f, prncprnconponpmbjmbjfctfctf, f, p`r p`r jfnjfn` ` gms gms c`c`vvms ms o`cfo`cfx˒x˒̀fs `fs fsfstmiftmifbfbfongongms ms o`j o`j `ut`utrms rms ··mrfms gmmrfms gm
Jmtfj·Jmtfj·mtnom. mtnom. M M tf`rnm tf`rnm gf gf L`wL`wfrs‟Jmurfrs‟Jmurfy fy rfs`bvrfs`bvfu fu pr`ibfjmpr`ibfjms s o·o·fbfirfs fbfirfs gm gm Mc·Mc·mbnsfmbnsf
Ducon`cmb (gftmbafs p`gfj sfr fco`ctrmg`s fj X=5Z) f `s p`ct`s mbt`s gm tf`rnm s˒Ducon`cmb (gftmbafs p`gfj sfr fco`ctrmg`s fj X=5Z) f `s p`ct`s mbt`s gm tf`rnm s m˒`m`
` Wf`rfjm gm Gno`t`jnm gf L`wfrs X=5, Wf`rfjm 4.?Z f m o`cstru±` Wf`rfjm gm Gno`t`jnm gf L`wfrs X=5, Wf`rfjm 4.?Z f m o`cstru±o˒om˒` gf fspm±m` gf fspm±o`s gfo`s gf
Imcmoa quf l`zmj gf pr`prnfgmgfs ubtrm-fspfonmns.Imcmoa quf l`zmj gf pr`prnfgmgfs ubtrm-fspfonmns.
··F ifj sming` quf sf ujm ·F ifj sming` quf sf ujm ·mrfm gf pfsqunsm jmtfj·mrfm gf pfsqunsm jmtfj·mtnom pfrgf o`cfx˒mtnom pfrgf o`cfx˒̀fs o`j `utrms`fs o`j `utrms
··mrfms, o`rrf ` rnso` gf sf trmcsd`rjmr fj ‒mrtf pfbm mrtf‚ (‒mrt d`r mrt„s smhf‚ o`j`mrfms, o`rrf ` rnso` gf sf trmcsd`rjmr fj ‒mrtf pfbm mrtf‚ (‒mrt d`r mrt„s smhf‚ o`j`
fsorfsorfvfvfu fu v`v`c c CfuCfujmcc fj jmcc fj X4>X4>Z). Z). @ @ `ikftn`ikftnv` g`s v` g`s trtr˄˄fs fs ··ubtnj`ubtnj`s s ompomp··Ľtub`Ľtub`s s g` g` bnvr` bnvr` ··ff
o`cvo`cvfcofr fcofr ` ` bfnt`r bfnt`r gf gf quf quf fssf fssf ofrtmjfcofrtmjfctf tf c˒cm˒` m` ··f f ` ` oms` oms` gm gm Mc·Mc·mbnsf mbnsf DDucon`ucon`cmb. cmb. Mb·Mb·fjfj
gf sfrvnr gf imsf pmrm tf`rnms jmns lfrmns, fco`ctrmr mpbnom±gf sfrvnr gf imsf pmrm tf`rnms jmns lfrmns, fco`ctrmr mpbnom±o˒o˒̀fs `fs vvmrnmrnmgmsmgms, , f f `dfr`dfrfofrfofr
sustfctm±sustfctm±o˒om˒` m` pmrpmrm m gngnvvfrsfrsms ms ··mrfmrfmsms, , o`jo`j` ` FqFqum±um±o˒o˒̀fs `fs GndGndfrfcfrfconmnonmns s UmUmronmronmns, ns, O·O·mboumboub` b` gmsgms
nnnnnn
  
Tmrnm±Tmrnm±o˒o˒̀fs, Wf`rnm gm Mpr`xnjm±`fs, Wf`rnm gm Mpr`xnjm±o˒om˒` f Mc·m` f Mc·mbnmbnsf sf AmrAmrj˄j˄̀cn`cnom, om, m Mm Mc·c·mbnsf Dmbnsf Ducon`cmb tfj, a`ucon`cmb tfj, a`kf,kf,
sum isum ifbfzm fbfzm nctrnctr··ĽcsfoĽcsfom rfvmb`rnm rfvmb`rnzmgm f zmgm f sums osums o`cfx˒`cfx˒̀fs o`j `utrms ·`fs o`j `utrms ·mrfms d`rtmbfongmrfms d`rtmbfongms p`r ms p`r jfn`jfn`
gf sums mpbnom±gf sums mpbnom±o˒o˒̀fs f gm rfpfrouss˒`fs f gm rfpfroussm˒` gf sfus mvmc±m` gf sfus mvmc±o`s nctfrc`s, o`j` ifj fxfjpbn﬍om mo`s nctfrc`s, o`j` ifj fxfjpbn﬍om m
tf`rtf`rnm nm gf gf L`wfrs‟JmL`wfrs‟Jmurfy, urfy, oukms oukms t·t·focnofocnoms ms t˄tf˄j fj omr·omr·mtfr ucnvfrsmb.mtfr ucnvfrsmb.
Mncgm s`irf m `rlmcnzm±Mncgm s`irf m `rlmcnzm±o˒om˒` m` g` g` bnvr`, omgm bnvr`, omgm ompomp··Ľtub` tfrjncm Ľtub` tfrjncm o`j o`j gums gums sf±sf±o˒o˒̀fs, ujm`fs, ujm
gf o`jfct·gf o`jfct·mrn`s mrn`s f f c`tms c`tms anst·anst·̀rnoms f `rnoms f `utrm `utrm gf gf fxfrofxfro··Ľon`s. Ľon`s. M M sf±sf±o˒om˒` gf o`jfct·m` gf o`jfct·mrn`s fmrn`s f
c`tms anst·c`tms anst·̀rnoms tfj v·`rnoms tfj v·mrn`s `ikftnv`s; mrn`s `ikftnv`s; o`ctfo`ctfxtumbnzxtumbnzmr f mr f rfbmon`crfbmon`cmr ` mr ` o`ctf·o`ctf·ugug ̀g ̀g ̀o ̀omp·mp·ĽtĽtubub`̀
o`j `utr`s mssuct`s, o`jpbfjfctmr ` jmtfrnmb mprfsfctmg` o`j rfsubtmg`s f fxfjpb`so`j `utr`s mssuct`s, o`jpbfjfctmr ` jmtfrnmb mprfsfctmg` o`j rfsubtmg`s f fxfjpb`s
mgnon`cmns, o`jfctmr mspfot`s anst·mgnon`cmns, o`jfctmr mspfot`s anst·̀rn`rno`s gm o`s gm tf`rtf`rnm gfsfcnm gfsfcv`v b̀vnbvngm. gm. M M sf±sf±o˒om˒` m` gf gf fxffxfro·ro·Ľon`Ľon`ss
··f f pmrtf pmrtf nctflnctflrmctf rmctf f f gf gf mbtm mbtm njp`rt˄njp`rtm˄conm c` bnvr`.mconm c` bnvr`.    ··F F rfs`bvfcg` rfs`bvfcg` `s `s fxfrofxfro··Ľon`s Ľon`s quf quf `̀
fstugmctf p`gf mdfrnr sf mprfcgfu `u c˒fstugmctf p`gf mdfrnr sf mprfcgfu `u cm˒` m m` m tf`rnm ftf`rnm fstugmgm. stugmgm. Mb·Mb·fj gnss`, fj gnss`, rfs`bvfcg` `srfs`bvfcg` `s
fxfro·fxfro·Ľon`s Ľon`s ` ` fstufstugmctf gmctf trmvmr·trmvmr·m o`cafonjfct` o`j rfsubtmg`s f m o`cafonjfct` o`j rfsubtmg`s f fxfjpb`fxfjpb s̀ njp`rtmctfs qufs njp`rtmctfs quf
c˒cm˒` o`uifrmj c` tfxt`. U`r fxfjpb`, m tf`rnm gf fspm±m` o`uifrmj c` tfxt`. U`r fxfjpb`, m tf`rnm gf fspm±o`s qu`onfctfs fst·o`s qu`onfctfs fst·m gfsfcv`bvngmm gfsfcv`bvngm
c`s c`s fxfrfxfroo··Ľon`s Ľon`s m` m` b`cl` b`cl` g`s g`s ompomp··Ľtub`sĽtub`s. . @s @s fxfro·fxfro·Ľon`s Ľon`s jmrojmromg`s mg`s o`j o`j * * s˒sm˒` m` fspfspfonmbjfonmbjfctffctf
gndgnd··Ľofns f/`u Ľofns f/`u trmimba`strmimba`s`s. `s. Mblujms (p`uoms) Mblujms (p`uoms) vfzfs usmrfj`s rfsubtmg`s o`ctng`s c`svfzfs usmrfj`s rfsubtmg`s o`ctng`s c`s
fxffxfro·ro·ĽonĽon`s. `s. C` ﬍C` ﬍cmb cmb g` bg` bnvrnvr` a·` a·m ujm ujm sfm sf±±o˒om˒` ‟ m` ‟ Mp˄Mpf˄cgfcgnof nof G ‟ gG ‟ gf gnf gnoms oms pmrpmrm mbm mbluclucs fxs fxfro·fro·Ľon`Ľon`ss
sfbsfbfon`fon`cmg`cmg s̀. s. O`j` gf O`j` gf prmxprmxf f mbfmbfrtmjrtmj`s `s quf ` quf ` fstufstugmcgmctf tf gfvgfvf f o`cso`csubtubtmr mr m m gnom gf gnom gf ujuj
fxffxfro·ro·ĽonĽon` ` mpmpfcmfcms s mp·mp·̀s `s v·v·mrnmrnms ms tfcttfctmtnmtnvms vms gf gf rfsrfs`bv˄`bvf˄-b`f-b`, , omsoms` ` o`cto`ctr·r·mrn` ` lmca` mudfrng`mrn` ` lmca` mudfrng`
cm rfs`bu±cm rfs`bu±o˒om˒` m` g`s g`s fxfro·fxfro·Ľon`s Ľon`s fstmrfstmr··m o`jpr`jftng`.m o`jpr`jftng`.
McMctfs gf tfs gf sulfsulfrnr r`tfnrrnr r`tfnr`s pmrm `s pmrm oursours`s, `s, omif uj omif uj fsobfsobmrfomrfonjfnjfctct`. `. M` fsorfvM` fsorfvfr fstffr fstf
bnvr` pr`ourmj`s c`s gnstmconmr m` j·bnvr` pr`ourmj`s c`s gnstmconmr m` j·mxnj` g` j`gfb`mxnj` g` j`gfb`   c`tms gf mubm o`jpnbmgms   c`tms gf mubm o`jpnbmgms ..    ··FF
obmr` quf c`s ifcf﬍onmj`s g`s ours`s gf Mc·obmr` quf c`s ifcf﬍onmj`s g`s ours`s gf Mc·mbnsf Dumbnsf Ducon`cmb quf con`cmb quf jncnstjncnstrmj`s fj rmj`s fj c`ssmsc`ssms
ucnvucnvfrsngmgfrsngmgfs, jms fs, jms ` d`` d`o` o` gfstf bnvr` ·gfstf bnvr` ·f f ` ` fstugmcfstugmctf. tf. O`c﬍mj`O`c﬍mj s̀ s quf ` quf ` pr`dfsspr`dfss`r `r tfcamtfcam
jmturngmgf pmrm mgmptmr ` tfxt` ejmturngmgf pmrm mgmptmr ` tfxt` ems sums mubms8 fcqumct` quf ` fstugmctf, cm jmn`rnmms sums mubms8 fcqumct` quf ` fstugmctf, cm jmn`rnm
gms vfzfs, mncgm c˒gms vfzfs, mncgm cm˒` fst·m` fst·m pr`ct` pmrm sflunr uj tfxt` jmns ·m pr`ct` pmrm sflunr uj tfxt` jmns ·mrng` o`j` c`rjmbjfctfmrng` o`j` c`rjmbjfctf
‟ `u tmbvfz d`r±‟ `u tmbvfz d`r±o`smjfctf ‟ s˒o`smjfctf ‟ sm˒` m` ms c`tms ms c`tms gf gf mubm. mubm. GmGm··Ľ Ľ c`ssm `p±c`ssm `p±o˒om˒` p`r uj tfxt` jmnsm` p`r uj tfxt` jmns
fxpbnomfxpbnomtnvtnv`` f f rfoafrfoafmg` mg` gf gf o`jfco`jfct·t·mrn`s fj gftrnjfct` gf `utr` jmns gnrft` f fo`c˄mrn`s fj gftrnjfct` gf `utr` jmns gnrft` f fo`c˄̀jno`.`jno`.
Gnt` nss`, pmrm uj prnjfnr` ours` gf Mc·Gnt` nss`, pmrm uj prnjfnr` ours` gf Mc·mbnsf mbnsf DDucon`ucon`cmb fcmb fj c·j c·Ľvfb gf Ľvfb gf g`ut`g`ut`rmg` rmg` sulfrsulfrnj`snj`s
quf ` quf ` pr`dfss`r pr`dfss`r snlm `s snlm `s sftf prnjfnrsftf prnjfnr`s omp`s omp··Ľtub`s, fvfctumbjfctf `jntncg` Ľtub`s, fvfctumbjfctf `jntncg` `s ntfcs `s ntfcs bnstmg`sbnstmg`s
monjm o`j` c˒monjm o`j` cm˒` `irnlmt·m` `irnlmt·̀rn`s. `rn`s. Umrm uj Umrm uj ours` gf ours` gf jfstrmg`jfstrmg ,̀ , sulfrnjsulfrnj`s `s `s omp`s omp··Ľtub`s =Ľtub`s =
(fx(fxoftoft` ` _f±_f±o˒om˒` =.4), 9, ? (fxoft` _f±m` =.4), 9, ? (fxoft` _f±o˒o˒̀fs ?.9 `fs ?.9 f f ?.?)?.?), , 7, 7, 6 6 f f ms ms tr˄trf˄s fs prnprnjfnjfnrms sf±rms sf±o˒o˒̀fs g``fs g`
OmpOmp··Ľtub` Ľtub` 4, c4, cmturmmturmbjfctf bjfctf `jntnc`jntncg` `s g` `s t·t·̀pno`s bnstmg`s monjm o`j` c˒`pno`s bnstmg`s monjm o`j` cm˒` `irnlmt·m` `irnlmt·̀rn`s. @s`rn`s. @s
ompomp··Ľtub`s 1, Ľtub`s 1, > f > f =5 p=5 p`gfj `gfj sfrvnr gsfrvnr gf cf c··uobf` uobf` pmrm pmrm uj uj sflucg` sflucg` ours` ours` gf gf Mc·Mc·mbnsf mbnsf DDucon`cmbucon`cmb
fj fj cc··Ľvfb Ľvfb gf gf g`utg`ut`rmg``rmg`, , f, f, mb·mb·fj fj gnss`, gnss`, s˒sm˒` mgfqumg`s pmrm sfrvnr gf imsf pmrm pr`kft`sm` mgfqumg`s pmrm sfrvnr gf imsf pmrm pr`kft`s
o`j o`j fstugmcfstugmctfs.tfs.
DDmbfj`s uj mbfj`s uj p`uo` p`uo` g`s pr·g`s pr·f-rfqunsnt`s pmrf-rfqunsnt`s pmrm fstugmm fstugmr fstf r fstf bnvr`. bnvr`. _up˒_up˒̀f-sf quf ` bfnt`r`f-sf quf ` bfnt`r
tfcam o`cafonjfct`s s·tfcam o`cafonjfct`s s·̀bng`s gf Mc·`bng`s gf Mc·mbnsf cm mbnsf cm Yftm Yftm c` c·c` c·Ľvfb gf Ľvfb gf X?5, 67X?5, 67Z f Z f gfgf   ··Mblfirm BncfmrMblfirm Bncfmr
c` c` cc··Ľvfb gf Ľvfb gf X=4, X=4, 64Z. 64Z. C` C` Mp˄Mpf˄cgnof fcgnof M M ` ` bfnt`r bfnt`r fco`cfco`ctrmr·trmr·m ` quf sfr·m ` quf sfr·m cfofss·m cfofss·mrn` s`irfmrn` s`irf
` Bfjm gf [`rc` Bfjm gf [`rc. . Fj rfbFj rfbm±m±o˒om˒` m` eem W`p`b`lnm, ` bfnt`r gfvf g`jncmr m tf`rnm g`s Fspm±m W`p`b`lnm, ` bfnt`r gfvf g`jncmr m tf`rnm g`s Fspm±o`so`s
J·J·ftrnoftrno`s pmr`s pmrm fstum fstugmr `gmr `s prnjs prnjfnr`s fnr`s ompomp··Ľtub`s Ľtub`s f rugf rugnjfct`s njfct`s gf W`p`b`lgf W`p`b`lnm Lfrmnm Lfrmb m pmrb m pmrtnr g`tnr g`
OmpOmp··Ľtub` Ľtub` 4. 4. O`cafO`cafonjfct`onjfct`s jmns s jmns mvmc±mvmc±omg`s gf W`p`b`lnm mpfcms s˒omg`s gf W`p`b`lnm mpfcms sm˒` cfofss·m` cfofss·mrn`s pmrm mmrn`s pmrm m
i`m i`m o`jprffco`jprffcs˒sm˒` m` g`s g`s omp·omp·Ľtub`Ľtub`s 1s 1, , > f > f =5. =5. @ Mp˄@ Mpf˄cgnofcgnof I f I ··f uf uj rj rfsuj` fsuj` g` g` quf quf sf sf cfofscfofssntmsntm
gf gf WW`p`b``p`b`lnm. lnm. FxfFxfjpbjpb`s `s njp`njp`rtmcrtmctfs tfs f f mblmblucs ucs tf`rtf`rfjms gfjmcgfjms gfjmcgmj mj o`cao`cafonfonjfcjfct` t` gfgf
nvnv
  
Wf`rnm gm Jfgngm f Nctflrm±Wf`rnm gm Jfgngm f Nctflrm±o˒om˒`, m`, c` c` cc··Ľvfb f Ľvfb f cm cm fxtfcfxtfcs˒sm˒` m` g` g` quf quf sf sf fco`ctrm fco`ctrm c` c` Mp˄Mpf˄cgnoffcgnof
O.O.
Mlrmgfonjfct`s ...Mlrmgfonjfct`s ...
Vifrb˄Vifrbm˄cgnm, K`˒mcgnm, K`m˒` Ufss`m, D`rtmbfzmm` Ufss`m, D`rtmbfzm
Gfzfjir` gf 95==.Gfzfjir` gf 95==.
vv
  
__uujj··mmrrnn`̀
_uj·_uj·mmrrnn` ` vvnn
= = FFssppmm±±oo`̀s s TTfftt`̀rrnnmmnns s CC`̀rrjjmmgg`̀s s ==
=.=.= = GfGf﬍c﬍cn±n±o˒o˒̀̀ffs s f f pprrnnjjffnnrr`̀s s ffxxffjjppbb`̀s s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..    ==
=.=.9 9 @s @s fsfspmpm±±o`so`s  B B p p((Q,Q, ίί,, ¶¶) ) . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . .    44
=.=.? ? @ @ ffsspmpm±±o`o`  B B∓∓((Q,Q, ίί,, ¶¶) ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . .    >>
=.=.7 7 FsFspmpm±±oo`̀s s ggf f ssffqquu˄˄ffccoonnmms s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..    ====
=.6 =.6 O`ckO`ckucuct`s t`s o`jo`jpmotpmot`s `s fj fj fspmfspm±±oo`s `s vvfft`t r̀nrnmnmns s cc`̀rjrjmmgg`s `s . . . . . . . . . . . . . . . . . . ..    =9=9
=.=.4 4 FsFspmpm±±o`s c`rjmg`s sfpmr·o`s c`rjmg`s sfpmr·mmvvffnns s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . .    =6=6
=.=.< < O`O`jfjfcct·t·mrn`s f c`tms anst·mrn`s f c`tms anst·̀̀rrnnoomms . . . . s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..    =1=1
==..1 1 FFxxffrroo··ĽĽoonn`̀s s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..    9595
9 9 @@ppffrrmmgg`̀rrffs s BBnnccffmmrrffs s OO`̀cctt··ĽĽccuu`̀s s 9977
9.9.= = OmrOmrmotmotfrnfrnzm±zm±o˒o˒̀fs `fs g`g`s `s `pfrmpfrmg`rg`rfs fs bnbncfcfmrfmrfs s o`co`ctt··ĽcĽcu`s u`s . . . . . . . . . . . . . . . . . . . . . . ..    9696
99..9 9 FFxxffjjppbb`̀s s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . .    9<9<
99..? ? @ @ WWff`̀rrffjjm m ggf f IImmccmmooaa‟‟__ttffnnccaammuus s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. .    9191
9.9.7 7 @ W@ Wf`f`rfjrfjm gm m gm MpMpbnbnom±om±o˒om˒ ` M i f r t m . . . . . . . . . . . . . . . . . . . . . .m ` M i f r t m . . . . . . . . . . . . . . . . . . . . . .    ?9?9
99..6 6 @ @ WWff`̀rrffjjm m gg` ` LLrr··mm﬍﬍oo` ` DDffooaammgg` ` . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..    ?7?7
9.9.4 4 O`O`jfjfcct·t·mrn`s f c`tms anst·mrn`s f c`tms anst·̀̀rrnnoomms . . . . s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..    ?4?4
99..< < FFxxffrroo··ĽĽoonn`̀s s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..    ?1?1
? ? WWff`̀rrffjjmms s ggf f AAmmaacc‟‟IImmccmmooa a 77??
?.?.= = @ t@ tf`rf`rfjfjm gm gm fm fxtxtfcsfcs˒˒mm` ` ggf f AAmmaacc‟‟IImmccmmooa a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. .    7?7?
?.?.9 9 TTfrfrs˒s˒̀fs vft`rnmns g` Wf`rfjm gf Amac‟Imcmoa . . . . . . . . . . . . . .`fs vft`rnmns g` Wf`rfjm gf Amac‟Imcmoa . . . . . . . . . . . . . .    7<7<
?.?.? ? MpMpbnbnomom±±o˒o˒̀fs g` Wf`rfjm gf Amac‟Imcmoa pmrm fspm±`fs g` Wf`rfjm gf Amac‟Imcmoa pmrm fspm±o`s sfpmr·o`s sfpmr·mvfns . . . .mvfns . . . .    6565
?.7 ?.7 DD`rjm`rjms lf`js lf`j··ftrnftrnoms goms g` W` Wf`rff`rfjm gf Ajm gf Amac‟mac‟ImcImcmoa moa (oms(oms` rfmb` rfmb) ) . . . . . . . . . . ..    6969
?.?.6 6 O`O`jfjfcct·t·mrn`s f c`tms anst·mrn`s f c`tms anst·̀̀rrnnoomms . . . . s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..    6161
??..4 4 FFxxffrroo··ĽĽoonn`̀s s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..    6>6>
7 7 GuGumbmbngngmgmgf f f f FsFspmpm±±oo`̀s s YYff﬉﬉ffxxnnvv`̀s s 44??
7.7.= = @ g@ gumumb gb g`s `s fsfspmpm±±o`so`s   BB p p    . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . .. . . . . . . . . . . . . . . . . . . . .    4?4?
7.7.9 9 @s @s gugumnmns gs g`s `s fspfspm±m±o`so`s    p p   ff   oo55    . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . .    4444
7.7.? ? InIngugumb mb f f fsfspmpm±±oo`̀s s rrff﬉﬉ffxxnnvv`̀s s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..    4<4<
7.7.7 7 O`O`jfjfcct·t·mrn`s f c`tms anst·mrn`s f c`tms anst·̀̀rrnnoomms . . . . s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..    <9<9
vnvn
  
77..6 6 FFxxffrroo··ĽĽoonn`̀s s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..    <?<?
6 6 FFssppmm±±oo`̀s s ggf f AAnnbbiiffrrt t <<<<
6.6.= = FsFspmpm±±oo`̀s s oo`̀j j pprr`g`guutt` ` nnccttffrrcc` ` . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..    <<<<
66..9 9 @@rrtt`̀ll`̀ccmmbbnnggmmggf f . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. .    1=1=6.? 6.? O`ckO`ckucuct`s t`s `rt``rt`c`rjc`rjmns mns fj ffj fspm±spm±oo`̀s s ggf f AAnnbbiiffrrt t . . . . . . . . . . . . . . . . . . . . . . . . . .. .    1717
6.7 6.7 Ur`ofsUr`ofss` s` gf gf `rt``rt`l`cml`cmbnzbnzm±m±o˒om˒` m` f f susums ms o`o`cscsfqfquu˄˄fcfcononms ms . . . . . . . . . . . . . . . . . . . . . . ..    >=>=
6.6 6.6 DDuconucoǹ cmn`cmns bns bncfmrcfmrfs ffs fj fsj fspm±pm±o`s o`s gf gf Anbifrt; Anbifrt; ` ` WWf`rfjm f`rfjm gf gf Ynfsz‟DYnfsz‟Dr·r·foaftfoaft    >7>7
6.6.4 4 O`O`jfjfcct·t·mrn`s f c`tms anst·mrn`s f c`tms anst·̀̀rrnnoomms . . . . s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..    ><><
66..< < FFxxffrroo··ĽĽoonn`̀s s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..    >1>1
4 4 WW`̀pp`̀bb`̀llnnmms s DDrrmmoomms s ==55??
4.= 4.= M t`p`b`lnm lfrmgM t`p`b`lnm lfrmgm p`r ujm dmjm p`r ujm dmj··Ľbnm gf duc±Ľbnm gf duc±o˒o˒̀̀ffs s . . . . . . . . . . . . . . . . . . . . . . . . . .. .    =5?=5?
4.4.9 9 M M t`t`p`p`b`b`lnlnm dm drmrmom om fj fj uj uj fsfspmpm±±o` o` c`c r̀jrjmgmg` ` . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..    =56=56
44..? ? M M tt`̀p`p b̀b`̀llnnm m ddrrmmoomm--ffssttrrffbbm m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. .    ======
4.4.7 7 OO`j`jppmmonongmgmggf f ddrmrmom om f f rrf﬉f﬉ffxxnnvvnngmgmggf . f . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..    ==6==6
4.4.6 6 JfJftrtrnnzzmiminnbbnnggmmgf gf f f ssffppmrmrmiminnbbnnggmmgf . . gf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..    ==>==>
4.4.4 4 FsFspmpm±±oo`̀s s uuccnndd`̀rrjjffjjffccttf f oo`̀ccvvffxx`̀s s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..    =99=994.4.< < O`O`jfjfcct·t·mrn`s f c`tms anst·mrn`s f c`tms anst·̀̀rrnnoomms . . . . s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..    =91=91
44..1 1 FFxxffrroo··ĽĽoonn`̀s s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..    =?5=?5
< < WWff`r`rnm nm FsFspfpfototrrmb mb gf gf @p@pfrfrmgmg`̀rfrfs s OO`j`jpmpmotot`s `s f f MuMutt`m`mggkukucct`t`s s =?=?77
<.<.= = FsFspfpfototr` r` gf gf uj uj `p`pfrfrmgmg`r `r bnbncfcfmr mr o`o`cctt··ĽcĽcu` u` . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..    =?6=?6
<<..9 9 @@ppffrrmmgg`̀rrffs s oo`̀jjppmmoott`̀s s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..    =?<=?<
<.<.? ? WWf`f`rnrnm m fsfspfpfototrmrmb b gf gf `p`pfrfrmgmg`r`rfs fs o`o`jpjpmomot`t`s s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..    =7==7=
<.7 <.7 @pfrm@pfrmg`rfg`rfs mus mut`mgt`mgkuckuct`s t`s fj ffj fspmspm±±oo`̀s s ggf f AAnnbbififrrt t . . . . . . . . . . . . . . . . . . . . . . . . ..    =74=74
<.<.6 6 WWf`f`rnrnm m fsfspfpfototrmrmb b gf gf `p`pfrfrmgmg`r`rfs fs mumut`t`mgmgkukucct`t`s s . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..    =7>=7>
<.<.4 4 O`O`jfjfcct·t·mrn`s f c`tms anst·mrn`s f c`tms anst·̀̀rrnnoomms . . . . s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..    =67=67
<<..< < FFxxffrroo··ĽĽoonn`̀s s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..    =66=66
1 1 FFssppmm±±o`s Tft`rnmns W`p`b·o`s Tft`rnmns W`p`b·̀̀llnnoo`̀s s ==66>>
11..= = FFxxffjjppbb`̀s s f f pprrnnjjffnnrrmms s pprr`̀pprrnnffggmmggffs s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..    =45=45
1.1.9 9 FsFspmpm±±o`s b`ombjfctf o`cvfx`s . . . . . . . . . . . . . . . . . . . . . . . .o`s b`ombjfctf o`cvfx`s . . . . . . . . . . . . . . . . . . . . . . . .    =44=44
11..? ? __ffjjnncc`̀rrjjmms s f f tt`̀p`p b̀b`̀llnnmms s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. .    =<==<=
1.1.7 7 YfYfvvnnssnntmtmccg` g` AAmmacac‟‟IImcmcmmooa a f f LL`̀bbggssttnnccf f . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..    =<4=<4
1.1.6 6 O`O`jfjfcct·t·mrn`s f c`tms anst·mrn`s f c`tms anst·̀̀rrnnoomms . . . . s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..    =1==1=
11..4 4 FFxxffrroo··ĽĽoonn`̀s s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..    =19=19
> > NcNctrtr`gu`gu±±oo˒˒mm ̀̀ eem m MMcc··mmbbnnssf f CC˒˒mm`̀--BBnnccffmmr r ==11<<
>>..= = OO`̀ccttnnccuunnggmmggf f ccm m tt`̀p`p b̀b`̀llnnm m ggm m cc`̀rrjjm m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..    =1<=1<
>>..9 9 OO`̀ccttnnccuunnggmmggf f ccm m tt`̀p`p b̀b`̀llnnm m ddrrmmoom m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..    =1>=1>
>.? >.? Ur`iUr`ibfjbfjms ms gf gf jncjncnjnnjnzm±zm±o˒om˒m` . . ` . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . .    =>9=>9
>>..7 7 WWff`̀rrffjjmms s gg` ` pp`̀cctt` ` ﬍﬍xx` ` . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..    =>7=>7>.>.6 6 @pf@pfrmrmg`g`rfrfs s c˒cm˒m`̀--bbnnccffmmrrffs s oo`̀jjppmmoott`̀s s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. .    =><=><
vnnvnn
  
>.>.4 4 FbFbfjffjfctct`s `s g` g` o·o·mboub` gndfrfconmb fj fspm±mboub` gndfrfconmb fj fspm±oo`̀s s ggf f IImmccmmooa a . . . . . . . . . . . . . . . . ..    959959
>.>.< < NcNctftflrlrm±m±o˒om˒m` ` vvfftt`̀rrnnmmb b . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. .    954954
>.>.1 1 O`O`jfjfcct·t·mrn`s f c`tms anst·mrn`s f c`tms anst·̀̀rrnnoomms . . . . s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..    9==9==
>>..> > FFxxffrroo··ĽĽoonn`̀s s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..    9=?9=?
=5 Fbfjfct`s gm Wf`rnm g`s Fspm±=5 Fbfjfct`s gm Wf`rnm g`s Fspm±oo`̀s s ggf f IImmccmmooa a 99==<<
=5=5.= .= __··frfrnfnfs s f ` f ` WWf`f`rfrfjm jm gf gf GvGv`r`rftftzhzhy‟y‟Y`Y`lflfrs rs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..    9=<9=<
=5.9 M gfsnlumbgmgf gf Lr`tafcgnfoh . . . . . . . . . . . . . . . . . . . . . .=5.9 M gfsnlumbgmgf gf Lr`tafcgnfoh . . . . . . . . . . . . . . . . . . . . . .    99<99<
=5.? =5.? Imsfs gImsfs gf _oamugfr f _oamugfr f sfqu˄f sfquf˄conms fconms i·i·mmssnnoomms s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. .    9?99?9
=5.7 =5.7 @ @ prncoprnco··Ľpn` Ľpn` gf gf sfbf±sfbf±o˒om˒` gf Ifssmlm‟Ufm` gf Ifssmlm‟Ufbozy·bozy·c s h n . . . . . . . . . . . . . . .c s h n . . . . . . . . . . . . . . .    979979
=5.6 =5.6 O`jfO`jfct·ct·mrn`s f c`tms anst·mrn`s f c`tms anst·̀̀rrnnoomms s . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . .    965965
==55..4 Fx4 Fxffrroo··ĽĽoonn`̀s s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. .    96=96=
M M @ @ BBffjjm m ggf f [[`̀rrc c 996666
I I CC`̀±±o˒o˒̀̀ffs s ggf f WW`̀pp`̀bb`̀llnnm m LLffrrmmb b 996611
O O CC`̀±±o˒o˒̀fs gf Jfgngm f Nctflrm±`fs gf Jfgngm f Nctflrm±oo˒˒mm` ` 994466
G G YYffsspp`̀ssttmmss//ggnnoomms s ppmmrrm m ffxxffrroo··ĽĽoonn`̀s s ssffbbffoonn`̀ccmmgg`̀s s 99<<99
YYffddffrr˄˄ffccoonnmms s IInniibbnn`̀llrr··mm﬍﬍oomms s 99>>77
··NNccggnnoof f YYffjjnnssssnnvv` ` ??5555
vnnnvnnn
  
OOmmpp··ĽĽttuubb ̀̀ ==
Fspm±Fspm±o`s Tft`rnmns C`rjmg`so`s Tft`rnmns C`rjmg`s
Fj Fj fspm±fspm±o`s o`s vft`rnvft`rnmns mns smifj`s smifj`s s`jmr s`jmr sfus sfus fbfjfcfbfjfct`s t`s f f jubtjubtnpbnomnpbnomr r sfus sfus fbfjfcfbfjfct`s t`s p`rp`r
fsombmrfs, fcqumct` quf fj fspm±fsombmrfs, fcqumct` quf fj fspm±o`s o`s j·j·ftrno`s ftrno`s smifj`s smifj`s omboubmr omboubmr m m gnst˄gnstm˄conm fctrf g`ns gfmconm fctrf g`ns gf
sfus fbfjfcsfus fbfjfct`s. t`s. Fspm±Fspm±o` o` c`rjmg` c`rjmg` ··f f m m fstruturm fstruturm jmtfj·jmtfj·mtnom cm qumb tug` nss` p`gf sfrmtnom cm qumb tug` nss` p`gf sfr
dfnt` fj uj jfsj` mjinfctf.dfnt` fj uj jfsj` mjinfctf.
==..= = GGff﬍﬍ccnn±±o˒o˒̀fs f prnjfnr`s fxfjpb`s`fs f prnjfnr`s fxfjpb`s
@ @ s·s·ĽjĽjii b̀̀b`̀  H H  gfc`tmr· gfc`tmr·m, ncgnstnctmjfctf, ` o`rp`m, ncgnstnctmjfctf, ` o`rp`  Y Y  g`s c· g`s c·ujfujfr`s rfmns `u r`s rfmns `u ` ` o`rp`o`rp` O O  g g`s`s
c·c·ujfr`s o`jpbujfr`s o`jpbfx`s. fx`s. @s fbfjfc@s fbfjfct`s gft`s gf   HH   s˒sm˒` oamjmm` oamjmg`s gf g`s gf fsomfsombmrbmrfs. fs. M M ducduc±±o˒om˒` j·m` j·̀gub``gub`
‟ `u vmb`r mis`but` ‟‟ `u vmb`r mis`but` ‟
mm ∁∁ HH ↙↙ ||mm|| ∁∁ YY
smtnsdmz ms sflunctfs o`cgn±smtnsdmz ms sflunctfs o`cgn±o˒o˒̀fs;`fs;
• |• |mm|| ≦≦ 5 pmrm t`g` 5 pmrm t`g`   mm  ff | |mm|| : : 55   ⇕⇕⇙⇙    mm :  : 55..
• |• |mm ¼¼ ii|| :: ||mm|| ¼¼ ||ii||  pmrm t`g`s pmrm t`g`s   mm  f f  i i..
• |• |mm ++ ii|| ≧≧ ||mm|| ++ ||ii||  pmrm t`g`s pmrm t`g`s  m m   ff   ii..
U`r nss`, ` c·U`r nss`, ` c·ujfr`ujfr` | |mm ∙ ∙ i i||   ·  ·f f nctfrprftmg` nctfrprftmg` o`j` o`j` m m gnst˄gnstm˄conm fctrfmconm fctrf    mm    ff    ii. . __ffkmkm   F F    ujuj
fspm±fspm±o` vft`rnmb s`irfo` vft`rnmb s`irf   HH. . ]um]umbqbqufr dufr duc±uc±o˒om˒` quf rfpr`guzm ms o`cgn±m` quf rfpr`guzm ms o`cgn±o˒o˒̀fs monjm sfrvnr·`fs monjm sfrvnr·mmpmrm ncguznr ujm c`±pmrm ncguznr ujm c`±o˒om˒` gf gnst˄m` gf gnstm˄conm fjmconm fj   F F . Jmns prfonsmjfctf, ujm duc±. Jmns prfonsmjfctf, ujm duc±o˒om˒`m`
 ¼¼  ;;   F F  ∙ ∙↙↙ YY
··f f uujjmm  c`rjm  c`rjm  sf ms sflunctfs pr`prnfgmgfs fstnvfrfj smtnsdfntms; sf ms sflunctfs pr`prnfgmgfs fstnvfrfj smtnsdfntms;
(C=)(C=)  xx ≦≦ 5 pmrm t`g` 5 pmrm t`g`   xx ∁∁ F F    ff  xx : : 55   ⇕⇕⇙⇙    xx :  : 55..
(C9)(C9)  mxmx :: ||mm|| ¼¼ xx  pmrm t`g` fsombmr pmrm t`g` fsombmr   mm f t`g` f t`g`  x x ∁∁ F F ..
(C?)(C?)  xx ++ yy ≧≧ xx ++ yy  pmrm qumnsqufr pmrm qumnsqufr   x,x, yy ∁ ∁ F F ..
Vj fspm±Vj fspm±o` vft`rnmb jucng` gf ujm c`rjm sfr·o` vft`rnmb jucng` gf ujm c`rjm sfr·m oamjmg` gfm oamjmg` gf  fspm± fspm±o` vft`rnmb c`rjmg`o` vft`rnmb c`rjmg`,,
`u snjpbfsjfctf`u snjpbfsjfctf   fspm±  fspm±o` c`rjmg`o` c`rjmg`. . MsMssnsnj o`j` c` oms` g` o`rp` g`s fsoj o`j` c` oms` g` o`rp` g`s fsombmbmrfmrfs, ujs, uj
fspm±fspm±o` o` c`rjmc`rjmg` g` ··f f uj uj fspm±fspm±o` o` j·j·ftrftrno` no` o`j o`j m m j·j·ftrftrnom nom gmggmgm m p`p`rr
gg((x,x, yy) ) ::

xx
∙∙
yy

..   (=.=)  (=.=)
==
  
Cfssf Cfssf oms` oms` gnzfjgnzfj`s q`s quf uf m j·m j·ftrnoftrnomm g g  · ·f f ncguzncguzngm ngm pfbm pfbm c`rjmc`rjm   ¼¼ . . U`U`rtmcrtmct` t`gm m t` t`gm m tf`rtf`rnmnm
gf fspm±gf fspm±o`s o`s j·j·ftrnoftrno`s `s (vfkm (vfkm Mp˄Mpf˄cgnofcgnof f 9) 9) sf sf mpbnom mpbnom m`s m`s fspm±fspm±o`s c`rjmg`s. Fj pmrtnoubmr,o`s c`rjmg`s. Fj pmrtnoubmr,
ujm ujm sfqsfqu˄uf˄cofconm nm ((xxcc))
∓∓
cc:=:=  fj uj fspm± fj uj fspm±o` c`rjmg`o` c`rjmg`   F F  o`cvfrlf pmrm ` vft`r o`cvfrlf pmrm ` vft`r   xx ∁∁ F F    sfsf
bnjbnjcc↙∓↙∓ xxcc ∙∙ xx : : 55..
Cfssf oms` gnzfj`s qufCfssf oms` gnzfj`s quf   xx  · ·f f ` ` bnjbnjntf ntf gm gm sfqsfqu˄uf˄cofconm nm ((xxcc))
∓∓
cc:=:=  f fsorfvfj`s f fsorfvfj`s   xx : bnj : bnj
cc
xxcc   `u`u
xxcc ∙ ∙↙↙ xx..
@ q@ quf uf t`rcm t`rcm o`jpmo`jpmtt··Ľvfns Ľvfns ms ms fstrfstruturmuturms ms mbl·bl·firnofirnom f m f t`pt`p`b·`b·̀lnom gf uj fspm±`lnom gf uj fspm±o` c`rjmg`o` c`rjmg`
··f f ` ` dmt` dmt` gf gf quf, quf, fj ufj uj fspj fspm±m±o` c`rjmg`, ms `pfrm±o` c`rjmg`, ms `pfrm±o˒o˒̀fs `fs mbl·mbl·firfirnomnoms s s˒sm˒` duc±m` duc±o˒o˒̀f`fs s o`o`ct·ct·ĽcuĽcumsms
(vfkm (vfkm FxfrFxfroo··Ľon` Ľon` =.1.=?=.1.=?).).
Vj fspm±Vj fspm±o` c`rjmg`o` c`rjmg` F  F  · ·f f oaoamjmjmmg`g`  fspm± fspm±o` gf Imcmoa o` gf Imcmoa  qumcg` d`r uj fspm± qumcg` d`r uj fspm±o ̀o ̀ j·j·ftftrnrno`o`
o`jpbft` o`j o`jpbft` o`j m m j·j·ftrnom ncguzngm ftrnom ncguzngm pfbm pfbm c`rjm. c`rjm. G`s G`s ours`s fbfjfctmrfs gf ours`s fbfjfctmrfs gf Mc·Mc·mbnsfmbnsf
smifj`s quf (smifj`s quf (YY,, || ¼¼ ||) ) f f ((OO,, || ¼¼ ||) ) s˒sm˒` fspm±m` fspm±o`s gf Imcmoa.o`s gf Imcmoa.
M pr`p`sn±M pr`p`sn±o˒om˒` mimnx` fvngfconm m njp`rt˄m` mimnx` fvngfconm m njp`rtm˄conm mconm g`s g`s suifspm±suifspm±o`s dfo`s dfoamg`s gf oamg`s gf uj uj fspm±fspm±o`o`
gf Imcmoa.gf Imcmoa.
Ur`p`sn±Ur`p`sn±o˒om˒ ̀m ̀ =.=.=.=.==   _fkmj   _fkmj    F F   uj fspm±  uj fspm±o` gf Imcmoa f o` gf Imcmoa f    D D   uj suifspm±  uj suifspm±o` vft`rnmb gf o` vft`rnmb gf    F F ..Fct˒ Fct˒ m`m`   D D  · ·f f uj uj fsfspm±pm±o` gf Imcmoa, o`j m c`rjm ncguzngm gf o` gf Imcmoa, o`j m c`rjm ncguzngm gf   F F , sf, f s`jfctf sf,, sf, f s`jfctf sf,D D    ··f f 
 dfoamg`  dfoamg` fj fj   F F ..
Gfj`cstrm±Gfj`cstrm±oo˒˒mm`̀..   _up`cam  _up`cam    D D   Imcmoa f t`jf (  Imcmoa f t`jf (xxcc))
∓∓
cc:=:=   uj  ujm m sfqu˄sfquf˄conm fconm fjfj    D D   tmb quf  tmb quf
xxcc  ∙∙↙↙    xx ∁ ∁    F F . . FcFctt˒˒m` m` ((xxcc))∓∓cc:=:=  · ·f f gf gf Omuoay Omuoay fjfj    D D , f p`rtmct` o`cvfrlfctf p`ns, f p`rtmct` o`cvfrlfctf p`ns    D D    ··ff
o`jpbft` p`r anp·o`jpbft` p`r anp·̀tfs`tfsf. f. FxnFxnstf fctstf fct˒˒m`m`   yy ∁ ∁   D D   tmb quf  tmb quf   xxcc ∙ ∙↙↙   yy. . Gm ucnoGm ucnongmngmgf g` bnjntgf g` bnjntff
tfj`stfj`s   xx : :  y y ∁ ∁ D D , pr`vmcg` quf, pr`vmcg` quf   D D  · ·f f dfdfoamoamg ̀g ̀ fjfj  F  F ..
Yfonpr`omjfcYfonpr`omjfctf, tf, sup`camsup`cam D  D  dfoamg` fj dfoamg` fj F  F  f sfkm ( f sfkm (xxcc))
∓∓
cc:=:=  ujm  ujm sfqsfqu˄uf˄cofconm nm gf gf OmuOmuoayoay
fjfj   D D . . B`B`l` l` ((xxcc))
∓∓
cc:=:=  · ·f f gf gf Omuoay Omuoay fjfj   F F , f p`rtmct` fxnstf, f p`rtmct` fxnstf    xx ∁ ∁   F F   tmb quf  tmb quf   xxcc ∙ ∙↙↙   xx..
O`j`O`j`   D D  · ·f f dfoamg` dfoamg` sfluf sfluf qufquf   xx ∁∁ D D , ` quf pr`vm quf, ` quf pr`vm quf  D  D  · ·f f o`o`jpjpbfbft`t`..
Fxfjpb` =.=.9Fxfjpb` =.=.9   _fkm  _fkm   Q Q   uj o`ckuct` c˒  uj o`ckuct` cm˒`-vm`-vmzn`. mzn`. Vjm duc±Vjm duc±o˒om˒`m`   d d ;;   Q Q  ∙ ∙↙↙   HH  ··ff   bnjntmgm  bnjntmgm 
sf sum njmlfj d`r uj suio`ckuct` bnjntmg` gfsf sum njmlfj d`r uj suio`ckuct` bnjntmg` gf   HH, `u sfkm, sf fxnstf, `u sfkm, sf fxnstf    J J   ≦≦   5 tmb quf  5 tmb quf||d d ((xx))| | ≧≧    J J   pmrm t`g`  pmrm t`g`    xx   ∁∁    Q Q . . @ o@ o`c`ckukucct`t`    II((Q Q ) gf t`gms ms duc±) gf t`gms ms duc±o˒o˒̀fs `fs bnjntmbnjntmgmsgms
d d ;;  Q Q  ∙ ∙↙↙ HH, , quf quf ··f f uj uj fspfspm±m±o` vft`rnmb o`j ms `pfrm±o` vft`rnmb o`j ms `pfrm±o˒o˒̀fs usumns gf duc±`fs usumns gf duc±o˒o˒̀fs, t`rcm-sf uj`fs, t`rcm-sf uj
fspm±fspm±o` gf Imcmoa o`j m c`rjmo` gf Imcmoa o`j m c`rjm
d d ∓∓  : sup : sup
xx∁∁Q Q 
||d d ((xx))|| ..
Gf Gf dmt`, dmt`, ··f f d·d·monb vfr qufmonb vfr quf  ¼   ¼ ∓∓   ·  ·f f ujm c`rjm. ujm c`rjm. Umrm j`strmr m Umrm j`strmr m o`jpbfto`jpbftugf, ugf, t`jm-t`jm-
sf sf ujm ujm sfqu˄sfquf˄conm fconm ((d d cc))
∓∓
cc:=:=  gf Omuoay fj gf Omuoay fj   II((Q Q ) f vfrn﬍om-sf quf, pmrm t`g`) f vfrn﬍om-sf quf, pmrm t`g`   xx ∁ ∁   Q Q , , mm
sfqu˄sfquf˄conm fconm o`rro`rrfspfsp`cgfct`cgfctf f ((d d cc((xx))))
∓∓
cc:=:=  · ·f f gf gf OmuOmuoay oay fjfj   HH..  @ rfsubtmg` sfluf dmonbjfctf @ rfsubtmg` sfluf dmonbjfctf
gm o`jpbftugf gfgm o`jpbftugf gf   HH..
99
  
Fxfjpb` =.=.?Fxfjpb` =.=.?  O`j` X O`j` Xm,m, iiZ ·Z ·f f o`jo`jpmopmot` t` fjfj   YY, ` o`ckuct`, ` o`ckuct`   O O XXm,m, iiZ gf t`gms ms duc±Z gf t`gms ms duc±o˒o˒̀fs`fs
o`o`ctct··ĽcuĽcums ms gf gf XXm,m, iiZ fjZ fj HH ··f uf uj sj suiuifsfspmpm±±o` vft`rnmb g` fspm±o` vft`rnmb g` fspm±o` gf Imcmoao` gf Imcmoa I IXXm,m, iiZ, f p`rtmct`Z, f p`rtmct`
··f f uj uj fsfspmpm±±o` c`rjmg` o`j m c`rjmo` c`rjmg` o`j m c`rjm
d d ∓∓ : sup : sup{|{|d d ((xx))|| ;; x x ∁∁ XXm,m, iiZZ}} : jmx: jmx{|{|d d ((xx))|| ;; x x ∁∁ XXm,m, iiZZ}}..
Jmns mncgm,Jmns mncgm, O  O XXm,m, iiZ Z ··f f uj uj fsfspmpm±±o` gf Imcmoa. Gf dmt`, pfbm Ur`p`sn±o` gf Imcmoa. Gf dmt`, pfbm Ur`p`sn±o˒om˒` =.=.= imstm pr`vmrm` =.=.= imstm pr`vmr
qufquf   O O XXm,m, iiZ ·Z ·f f uj uj suisuifspfspm±m±o` dfoamg` gfo` dfoamg` gf  I I XXm,m, iiZ. Umrm tmct`, `isfrvf quf sfZ. Umrm tmct`, `isfrvf quf sf  d  d cc ∙ ∙↙↙ d d    fjfj
O O XXm,m, iiZ, fct˒Z, fctm˒` m` ((d d cc))
∓∓
cc:=:=  o`cvfrlf ucnd`rjfjfctf pmrm o`cvfrlf ucnd`rjfjfctf pmrm   d d . _fluf quf. _fluf quf   d d  · ·f f o`o`ct·ct·ĽcuĽcum m pp`r `r sfsfrr
` bnjntf ucnd`rjf gf duc±` bnjntf ucnd`rjf gf duc±o˒o˒̀f`fs s o`o`ct·ct·ĽcuĽcumsms..
G`s sfus fstug`s gf O·G`s sfus fstug`s gf O·mbmbououb`b`, , ` ` bfbfnt`nt r̀ r smif quf smif quf ducduc±±o˒o˒̀fs o`j ofrtms pr`prnfgmgfs`fs o`j ofrtms pr`prnfgmgfs
gf gndfrfconminbngmgf s˒gf gndfrfconminbngmgf sm˒` junt` ·m` junt` ·utfns cms mpbnom±utfns cms mpbnom±o˒o˒̀fs`fs. . Vj g`s prncoVj g`s prnconpmnnpmns `is `ikftnkftnv`v`s g`s g`
O·O·mboub` ·mboub` ·f f ` ` fstug` fstug` gm gm tmxm tmxm gf gf vmrnm±vmrnm±o˒om˒` ncstmct˄m` ncstmctm˄cfmcfm m gf gf ujm ujm ducduc±±o˒om˒` rfmbm` rfmb d  d . Fssm tmxm gf. Fssm tmxm gf
vmrnm±vmrnm±o˒om˒` m` ··f f oamjmgm oamjmgm gf gf gfrnvmgm gfrnvmgm gfgf   d d  f gfc`tmgm p`r f gfc`tmgm p`r   d d . . MprMprfcgffcgfj`s c` O·j`s c` O·mboub` qufmboub` quf
sfsf d  d    ff  l l   s˒sm˒` m` gndfrfcgndfrfcon·on·mvfns, fct˒mvfns, fctm˒` m` ((d d  + + ll) ) ··f f ujujm m duduc±c±o˒om˒` gndfrfcon·m` gndfrfcon·mvfb f (mvfb f (d d  + + ll))   :: d  d  + + ll
f f sfsf   θθ ∁ ∁ Y Y, , ((θd θd ))   :: θd  θd . Fj c`ssm bnclumlfj nss` snlcn﬍om quf ` o`ckuct` gf t`gms ms. Fj c`ssm bnclumlfj nss` snlcn﬍om quf ` o`ckuct` gf t`gms ms
duc±duc±o˒o˒̀fs gndfrfcon·`fs gndfrfcon·mvfns p`ssun ujm fstruturm cmturmb gf fspm±mvfns p`ssun ujm fstruturm cmturmb gf fspm±o` vft`rno` vft`rnmb. mb. Fstugmj`Fstugmj s̀ c`s c`
pr·pr·̀xnj` fxfjpb` `s fspm±`xnj` fxfjpb` `s fspm±o`s o`s gf gf ducduc±±o˒o˒̀fs o`ctncumjfctf gndfrfcon·`fs o`ctncumjfctf gndfrfcon·mvfns.mvfns.
Fxfjpb` =.=.7Fxfjpb` =.=.7    Gf﬍cnj`s Gf﬍cnj`s ` ` fspm±fspm±o` o` gms gms duc±duc±o˒o˒̀fs o`ctncumjfctf gndfrfcon·`fs o`ctncumjfctf gndfrfcon·mvfns p`r ;mvfns p`r ;
O O ==XXm,m, iiZ Z ;:;: {{d d ; ; XXm,m, iiZZ ∙∙↙↙ YY ;; d  d  · ·f f gngndfdfrfrfcocon·n·mvfb fj Xmvfb fj Xm,m, iiZ Z ff   d d  ∁ ∁ O O XXm,m, iiZZ}} ..
Gf Gf d`d r̀jrjm m ususumumbb, , c`c s̀ s fxfxtrtrfjfj`s `s g` g` ncnctftfrvrvmbmb̀ ` o`o`cscsngngfrfrmjmj`s `s `s `s bbnjnjntntfs fs bmbmtftfrmrmnsns
o`o`rrrrfsfsp`cp`cgfgfcctftfs. s. O`O`j` j` vnvnj`j`s s momonjnjm,m,    O O ==XXm,m, iiZ Z ··f f suifspm±suifspm±o` o` vvftft`r`rnmnmb b gfgf    O O XXm,m, iiZ.Z.
FctrftFctrftmct`, mct`, o`cd`rjf o`jpr`vo`cd`rjf o`jpr`vmrfj`s c` mrfj`s c` FxfjpbFxfjpb̀ ` 9.7.7,9.7.7,    O O ==XXm,m, iiZ Z c˒cm˒` m` ··f f dfoamg` dfoamg` fjfj
O O XXm,m, iiZ, f p`rtmct` c˒Z, f p`rtmct` cm˒` m` ··f f o`jpbft` o`jpbft` cm cm c`rjmc`rjm  ¼   ¼ ∓∓  pfbm Ur`p`sn± pfbm Ur`p`sn±o˒om˒` =.=.m` =.=.=. =. TTfkmjfkmj`s`s
quf m c`rjmquf m c`rjm
d d O O ==   :: d d ∓∓ + + d d ∓∓,,
dmz gfdmz gf  O  O 
==
XXm,m, iiZ uj fspm±Z uj fspm±o` gf Imcmoa. Gfnxmj`s pmrm ` bfnt`r m vfrn﬍om±o` gf Imcmoa. Gfnxmj`s pmrm ` bfnt`r m vfrn﬍om±o˒om˒` g`s mxn`jmsm` g`s mxn`jmsgf c`rjmgf c`rjm. . Ur`Ur`vfvfj`s quf (j`s quf (O O ==XXm,m, iiZZ,,  ¼¼ O O ==) ) ··f f o`jpbft`. o`jpbft`. Gf Gf dmt`, dmt`, gmgm gmgm ujm ujm sfqu˄sfquf˄conmfconm
gf Omuoay (gf Omuoay (d d cc))
∓∓
cc:=:=    fj fj ((O O 
==XXm,m, iiZZ,,  ¼¼ O O ==), o`j`), o`j`  d d ∓∓ ≧  ≧ d d O O ==    ff  d d ∓∓ ≧  ≧ d d O O ==, sfluf, sfluf
quf quf ms ms sfqu˄sfquf˄conmfconms s ((d d cc))
∓∓
cc:=:=    f f ((d d 

cc))
∓∓
cc:=:=    s˒sm˒` gf Omuoay fjm` gf Omuoay fj    O O XXm,m, iiZ. Z. G` fxfG` fxfjpbjpb` mct` mctfrnfrn`r`r
smifj`s qufsmifj`s quf O  O XXm,m, iiZ ·Z ·f f o`jpbfo`jpbft`, t`, b`l` b`l` fxnstffxnstfjj d d,, ll ∁ ∁ O O XXm,m, iiZ tmns qufZ tmns quf d  d cc ∙ ∙↙↙ d d    ff d  d cc ∙ ∙↙↙ ll
ucnd`rjfjfctf. ucnd`rjfjfctf. Ufb` Wf`rfjm Ufb` Wf`rfjm DDucgmjfctmb ucgmjfctmb g` g` O·O·mboub` mboub` p`gp`gfj`s fj`s fsorfvfrfsorfvfr
d d cc((xx)) ∙∙ d d cc((mm) ) ::
     xx
mm
d d cc((tt))gtgt
pmrm t`g`spmrm t`g`s    xx ∁ ∁    XXm,m, iiZ Z ff    cc ∁ ∁   CC. . WW`jm`jmcg` ` bnjncg` ` bnjntf qumcgtf qumcg`̀    cc ∙ ∙↙ ↙ ∓∓   o`c  o`cobobuu··Ľj`Ľj`s s ququff
d d  ∁ ∁ O O ==XXm,m, iiZ f qufZ f quf   d d   : :  l l. _fluf quf. _fluf quf   d d cc ∙ ∙↙↙ d d    fjfj  O  O ==XXm,m, iiZ.Z.
_m_mifjifj`s `s mnmncgcgm m g` g` O·O·mbmbouboub̀ ` ququf f m m sflsflucgucgm m gfgfrnrnvvmgm mgm d`d`rcfrcfof of ncncd`rd`rjm±jm±o˒o˒̀fs `fs s`s`irfirf
o`comvngmgf `u o`cvfxngmgf b`omb gm duc±o`comvngmgf `u o`cvfxngmgf b`omb gm duc±o˒om˒`. m`. U`gfj`s tmji·U`gfj`s tmji·fj fj trmtmr trmtmr duc±duc±o˒o˒̀fs gums`fs gums
??
  
vfzfs vfzfs gndfrfcgndfrfcon·on·mvfns cm bnclumlfj gf fspm±mvfns cm bnclumlfj gf fspm±o`s c`rjmg`s. Gf﬍cnj`smssnj ` fspm±o`s c`rjmg`s. Gf﬍cnj`s mssnj ` fspm±o` gmso` gms
duc±duc±o˒o˒̀fs gums vfzfs o`ctncumjfctf gndfrfcon·`fs gums vfzfs o`ctncumjfctf gndfrfcon·mvfns p`r;mvfns p`r;
O O 99XXm,m, iiZ Z ;:;:

d d ; ; XXm,m, iiZZ ∙∙↙↙ YY ;; d  d  · ·f f gngndfdfrfrfcocon·n·mvfb fj Xmvfb fj Xm,m, iiZ Z ff   d d  ∁ ∁ O O ==XXm,m, iiZZ

..
Gf d`rjm mc·Gf d`rjm mc·mb`lm,mb`lm,   O O 99XXm,m, iiZ sf t`rcm uj fspm±Z sf t`rcm uj fspm±o` gf Imcmoa o`j m c`rjmo` gf Imcmoa o`j m c`rjm
d d O O 99   :: d d ∓∓ + + d d ∓∓ + + d d ∓∓..
Ncgutnvmjfctf, gf﬍cnj`s ` fspm±Ncgutnvmjfctf, gf﬍cnj`s ` fspm±o` gms duc±o` gms duc±o˒o˒̀fs`fs    hh  vfzfs o`ctncumjfctf gndfrfcon· vfzfs o`ctncumjfctf gndfrfcon·mvfns,mvfns,
hh ∁ ∁ CC, p`r, p`r
O O hhXXm,m, iiZ Z ;:;:

d d ; ; XXm,m, iiZZ ∙∙↙↙ YY ;; d  d  · ·f f gngndfdfrfrfcocon·n·mvfb fj Xmvfb fj Xm,m, iiZ Z ff   d d  ∁ ∁ O O hh∙∙==XXm,m, iiZZ

..
O`j` ` bfnt`r k·O`j` ` bfnt`r k·m m pfrofifpfrofifu,u,  O O hhXXm,m, iiZ sf t`rcm uj fspm±Z sf t`rcm uj fspm±o` gf Imcmoa o`j m c`rjmo` gf Imcmoa o`j m c`rjm
d d O O hh   :: d d ∓∓ + + d d ∓∓ + + ¼¼ ¼¼ ¼¼ ++ d d ((hh))∓∓,,
`cgf`cgf   d d (( k k)) gfc`tm mgfc`tm m   kk-·-·fsnfsnjm jm gfrgfrnvmgm nvmgm gfgf   d d ..
]um]umctct̀ ` m`m`s s fsfspm±pm±o`s o`s c`c`rjmrjmg`s g`s gf gf gngnjfjfcscs˒˒m` m` ﬍cn﬍cntm, tm, ` ` bfbfntnt`r `r pr`pr v̀vmvmvfbfbjfcjfctf tf fstfst··mm
dmjnbnmrnzmg` o`j ms c`rjms usumns g`s fspm±dmjnbnmrnzmg` o`j ms c`rjms usumns g`s fspm±o`so`s  H Hcc,,  cc ∁∁ CC;;
((mm==, . . . , m, . . . , mcc))==  :: ||mm==|| ++ ¼¼ ¼¼ ¼¼ ++ ||mmcc||,,
((mm==, . . . , m, . . . , mcc))99  : : ((||mm==||99 ++ ¼¼ ¼¼ ¼¼ ++ ||mmcc||99))
==
99 , , ff
((mm==, . . . , m, . . . , mcc))∓∓ : jmx : jmx{|{|mm==||, . . . ,, . . . , ||mmcc|}|}..
WWmbvfz mbvfz smnim smnim tmji·tmji·fj fj quf quf fssfs fssfs fspm±fspm±o`s s˒o`s sm˒` o`jpbft`s. m` o`jpbft`s. C`ss` `ikftnv` njfC`ss` `ikftnv` njfgnmt` ml`rmgnmt` ml`rm
··f f j`strj`strmr mr quf quf nss`, nss`, fj fj fss˄fssf˄conmfconm, , ··f f gnzfr gnzfr quf quf t`t`g` g` fspm±fspm±o` c`rjmg` gf gnjfcs˒o` c`rjmg` gf gnjfcsm˒ ̀m ̀ ﬍c﬍cntntm m ··ff
o`jpbft`. Umrm nss` prfonsmj`s g`o`jpbft`. Umrm nss` prfonsmj`s g`
Bfjm =.=.6Bfjm =.=.6   _fkm   _fkm   II  :: {{xx==, . . . , x, . . . , xcc}}  uj o`ckuct` gf vft`rfs bncfmrjfctf ncgfpfcgfctfs  uj o`ckuct` gf vft`rfs bncfmrjfctf ncgfpfcgfctfs 
gf uj fspm±gf uj fspm±o` c`rjmg`o` c`rjmg`   F F . . Fct˒Fctm˒` m` fxnstf fxnstf ujm ujm o`cstmctf o`cstmctf   o o 33 5 5, quf gfpfcgf g` o`ckuct`, quf gfpfcgf g` o`ckuct`
II, tmb quf , tmb quf 
mm==xx== + + ¼¼ ¼¼ ¼¼ ++ mmccxxcc ≦≦ o o ((||mm==|| ++ ¼¼ ¼¼ ¼¼ ++ ||mmcc||)) ,,
pmrm qumnsqufr fsombmrfs pmrm qumnsqufr fsombmrfs   mm==, . . . , m, . . . , mcc..
Gfj`cstrm±Gfj`cstrm±oo˒˒mm`̀..   Urnjfnr` rfbfjirfj`s gm Mc·  Urnjfnr` rfbfjirfj`s gm Mc·mbnsf c`mbnsf c`   YYcc quf, gmgms gums c`rjmsquf, gmgms gums c`rjms
 ¼  ¼ mm    ff  ¼   ¼ ii    fjfj   YYcc, fxnstf ujm o`cstmctf, fxnstf ujm o`cstmctf    o o 33   5, quf gfpfcgf gms c`rjms, tmb quf  5, quf gfpfcgf gms c`rjms, tmb quf
xxmm ≧ ≧ ooxxii pmrm t`g` pmrm t`g` x x ∁∁ YYcc. . @ jfsj`@ jfsj`, o`j g, o`j gfj`csfj`cstrm±trm±o˒om˒` t`tmbjfctf mc·m` t`tmbjfctf mc·mb`lm, vmbfmb`lm, vmbf
pmrmpmrm OOcc. T`btfj`s fct˒. T`btfj`s fctm˒` pmrm ` oms` fj qufm` pmrm ` oms` fj quf HH :: YY `u`u OO. @ bfnt`r c˒. @ bfnt`r cm˒` tfr·m` tfr·m m gn﬍oubggn﬍oubgmgfmgf
fj fj vfrn﬍omr vfrn﬍omr quf quf m m o`rrfsp`o`rrfsp`cg˄cgf˄conmfconm
((mm==, . . . , m, . . . , mcc)) ∁∁ HHcc ∙∙↙↙

cc
 k k:=:=
mm k kxx k k
 ∁∁ YY
··f f ujm ujm c`rjm c`rjm fjfj   HHcc. . @ @ rfsrfsububtmgtmg` ` sfsfluf p`nsluf p`ns, , o`co`cd`d`rjf vnjrjf vnj`s monj`s monjm, m, m m fxpfxprfsrfss˒sm˒`m`

((mm
==
, . . . , m, . . . , m
cc
))
==
 : :
||
mm
==||
++
¼¼ ¼¼ ¼¼
++
||
mm
cc||
 tm tmji·ji·fj fj ··f f ujujm m c`c`rjrjm m fjfj   HHcc..
77
  
Wf`rfjm =.=.4Wf`rfjm =.=.4   W  W`g` `g` fspm±fspm±o` c`rjmg` gf gnjfcs˒ o` c`rjmg` gf gnjfcs˒ m` m` ﬍cntm ﬍cntm ··f f uj uj fspm±fspm±o` gf Imcmoa.o` gf Imcmoa.
O`csfqufO`csfqufctfjfctf, ctfjfctf, t`g` suifspm±t`g` suifspm±o` o` gf gf gnjgnjfcfcs˒ s˒ m` ﬍cntm gf uj fspm±m` ﬍cntm gf uj fspm±o` o` c`rc`rjmgjmg`̀    F F    ··f f 
 dfoamg`  dfoamg` fj fj   F F ..
Gfj`cstrm±Gfj`cstrm±oo˒˒mm`̀..   _fkmj  _fkmj    F F   uj   uj fspm±fspm±o` o` c`rjmg` c`rjmg` gf gf gnjfcs˒gnjfcsm˒`m`    cc    ff  {{ι ι ==, . . . , ι  , . . . , ι  cc}}    ujmujmimsf imsf c`rjmbnc`rjmbnzmgm zmgm (nst` (nst` ··f, f, `s `s vft`rvft`rfs fs t˄tf˄j fj c`rjm c`rjm =) =) gfgf    F F . . Gmgm ujm Gmgm ujm sfqusfqu˄˄fconm gffconm gf
Omuoay (Omuoay (xxhh))
∓∓
hh:=:=    fjfj    F F , pmrm omgm, pmrm omgm    hh ∁ ∁   CC   fxnstfj · fxnstfj ·ucno`s fsombmrfsucno`s fsombmrfs    mmhh==, . . . , m, . . . , mhhcc   tmns quf  tmns quf
xxhh   :: m m
hh
==ι ι == + + ¼¼ ¼¼ ¼¼ ++ m mhhccι ι cc. Gmg`. Gmg`   ζ ζ 33 5, p`gfj`s t`jmr 5, p`gfj`s t`jmr   cc55 ∁ ∁ CC tmb quf tmb quf  xxhh ∙ ∙ xxjj 2  2 oo ¼¼ ζζ
sfjprf qufsfjprf quf h h,, jj ≦≦ cc55, `cgf, `cgf o o  · ·f m f m o`cstmctf o`cstmctf g` g` Bfjm Bfjm =.=.6 =.=.6 pmrm pmrm ` ` o`ckuct` bncfmro`ckuct` bncfmrjfctfjfctf
ncgfpfcgfctfncgfpfcgfctf { {ι ι ==, . . . , ι  , . . . , ι  cc}}. _fluf fct˒. _fluf fctm˒` qufm` quf
cc
 k k:=:=
||mmhh k k ∙ ∙ mmjj k k | | ≧≧
  ==
oo

cc
 k k:=:=
((mmhh k k ∙ ∙ mmjj k k   ))ι ι  k k
 ::
  ==
oo
xxhh ∙ ∙ xxjj 2 2 ζ,ζ,
sfjprf qufsfjprf quf    h,h, jj ≦ ≦    cc55. . _fl_fluf quuf quf, pmrf, pmrm omgmm omgm   kk    : : ==, . . . , c, . . . , c, , m m sfqu˄sfquf˄conm fconm gf gf fsombmrfsfsombmrfs
((mmhh k k ))
∓∓
hh:=:=  · ·f f gf gf OmuoayOmuoay, , p`rtmct` p`rtmct` o`cvfrlfctf. o`cvfrlfctf. Gnlmj`sGnlmj`s   ii k k  : bnj : bnj
hh
mmhh k k ,,   kk   : : ==, . . . , c, . . . , c. . CfCfssssff
oms` tfj`s bnjoms` tfj`s bnj
hh

cc
 k k:=:=
||
mmhh k k
 ∙ ∙
ii k k
||
: 5. Gf﬍cncg`: 5. Gf﬍cncg`   xx : :  i i==ι ι == + +
¼¼ ¼¼ ¼¼
++ iiccι ι cc   tfj`s tfj`s   xx
∁∁
F F   ff
bnjbnj
hh
xxhh ∙ ∙ xx : bnj: bnj
hh

cc
 k k:=:=
((mmhh k k ∙ ∙ ii k k))ι ι  k k
 ≧≧ bnjbnjhh
cc
 k k:=:=
||mmhh k k ∙ ∙ ii k k|| : : 55
pr`vmcg` qufpr`vmcg` quf  x xhh ∙ ∙↙↙ xx f o`jpbftmcg` m gfj`cstrm± f o`jpbftmcg` m gfj`cstrm±o˒om˒` gf qufm` gf quf   F F  · ·f f Imcmoa. Imcmoa. M M sflucsflucgmgm
m﬍rjm±m﬍rjm±o˒om˒` sfluf gm prnjfnrm f gm Ur`p`sn±m` sfluf gm prnjfnrm f gm Ur`p`sn±o˒om˒` =.=.=.m` =.=.=.
O`j` k·O`j` k·m frm gf sf fspfrmr, fxnstfj fspm±m frm gf sf fspfrmr, fxnstfj fspm±o`s c`rjmg`s, gf gnjfcs˒o`s c`rjmg`s, gf gnjfcsm˒` m` nc﬍nc﬍cntcntm, ·m, ·f of obmrbmr`,`,
quf c˒quf cm˒` s˒m` sm˒` o`jpbft`s. Tfkmj`s uj fxfjpb`;m` o`jpbft`s. Tfkmj`s uj fxfjpb`;
Fxfjpb` =.=.<Fxfjpb` =.=.<    U`rU`r  oo55  gfc`tmj`s  gfc`tmj`s ` o`` o`ckuct` gf ckuct` gf t`gms t`gms ms sfqu˄ms sfquf˄conms gfconms gf fsombmrf fsombmrfs quffs quf
o`cvfrlfj pmrm zfr`, `u sfkm, ﬍xmg`o`cvfrlfj pmrm zfr`, `u sfkm, ﬍xmg`   HH :: YY  ` `uu   OO,,
oo55  : : {{((mmhh))∓∓hh:=:=  ; ;  m mhh ∁ ∁ HH pmrm t`g` pmrm t`g`   hh ∁ ∁ CC  ff   mmhh ∙ ∙↙↙ 55}}..
··F obmr` qufF obmr` quf   oo55  · ·f f uj uj fsfspmpm±±o` vft`rnmb o`j ms `pfrm±o` vft`rnmb o`j ms `pfrm±o˒o˒̀fs `fs usumns usumns gf gf sfqu˄sfquf˄conms fconms (`pfrm±(`pfrm±o˒o˒̀fs`fs
o``rgfcmgm m o``rgfcmgm).o``rgfcmgm m o``rgfcmgm).    ··F F d·d·monb o`jpr`vmr quf m fxprfss˒monb o`jpr`vmr quf m fxprfssm˒`m`
((mmhh))∓∓hh:=:=∓∓  : sup : sup{|{|mmhh|| ;; h h ∁ ∁ CC}}
t`rcmt`rcm o o55  uj fspm± uj fspm±o` c`rjmg`. @ j`tnv` g` suisornt`o` c`rjmg`. @ j`tnv` g` suisornt` ∓ ∓ ﬍omr· ﬍omr·m obmr` cm _f±m obmr` cm _f±o˒om˒` =.7 . _fkmm` =.7 . _fkm
((xxcc))
∓∓
cc:=:=  ujm  ujm sfqu˄sfquf˄conm fconm gf gf Omuoay Omuoay fjfj   oo55. Gnlmj`s quf. Gnlmj`s quf   xxcc   : : ((mm
hh
cc))
∓∓
hh:=:=  pmrm omgm pmrm omgm   cc ∁ ∁ CC..
Umrm omgmUmrm omgm   kk ∁ ∁ CC, m gfsnlumbgmgf, m gfsnlumbgmgf
||mm k kcc ∙∙ mm k kjj|| ≧≧ supsup{|{|mmhhcc ∙∙ mmhhjj|| ;; h h ∁ ∁ CC}} :: xxcc ∙∙ xxjj∓∓
gfnxm gfnxm obmr` obmr` quf quf m m sfqu˄sfquf˄conmfconm gf gf fsombmrfs fsombmrfs ((mm k kcc))
∓∓
cc:=:=  · ·f f gf gf OmuOmuoay oay fjfj   HH, b`l` o`cvfrlfctf., b`l` o`cvfrlfctf.
Gnlmj`sGnlmj`s   mm k k
cc
cc↙∓↙∓
∙∙↙↙
   mm
 k k
  pmrm omgm  pmrm omgm   kk
 ∁ ∁
  CC. . OamOamjmcgjmcg`̀   xx   : : ((mm
 k k
))∓∓
 k k:=:=
   c˒cm˒ ̀m ̀ ··f f gngndd··ĽoĽonb nb oafoafomomr r ququff
66
  
xx ∁∁ oo55  f quf f quf   xxcc ∙ ∙↙↙ xx  fjfj   oo55. . O`cO`cobu·obu·Ľj`Ľj`s fs fct˒ctm˒` qufm` quf   oo55  · ·f f uj uj fsfspmpm±±o` gf Imcmoa.o` gf Imcmoa.
Oamjf ml`rm gfOamjf ml`rm gf o o5555 ` suifspm± ` suifspm±o` gfo` gf o o55 d`r d`rjmg` pjmg` pfbms sfbms sfqu˄fquf˄conmfconms fvfctums fvfctumbjfctf cubbjfctf cubms,ms,
nnsstt ̀̀ ··ff,,
oo5555  : : {{((mmhh))∓∓hh:=:= ∁ ∁ oo55 ; fxnstf ; fxnstf   hh55 ∁ ∁ CC tmb quf tmb quf   mmhh : 5 pmrm t`g` : 5 pmrm t`g`  h h ≦ ≦ hh55}}..
O`csngfrf `s sflunctfs vft`rfs gfO`csngfrf `s sflunctfs vft`rfs gf   oo5555;;
xx==  : : (=(=,, 55,, 55,, 55, . . . ,, . . . , )), , xx99  ::

==,,
 = =
99
,, 55,, 55, . . ., . . .

, . . . , x, . . . , xcc  : :

==,,
 = =
99
, . . . ,, . . . ,
  ==
cc
,, 55,, 55 . . . ,. . . ,

, . . . ., . . . .
··F F oobbmmr` r` qquuf f ((xxcc))
∓∓
cc:=:=   ·  ·f f ujm ujm sfqsfqu˄uf˄cofconm nm fjfj    oo5555. . WW`j`jmcmcgg`̀    xx    : : ((
==
hh
))∓∓hh:=:=    ∁∁    oo55, , gfgf
xxcc ∙ ∙   xx∓∓    ::    ==cc+=+=   ∙∙↙↙   5,   5, o`coo`cobubu··Ľj`s Ľj`s qufquf    xxcc   ∙∙↙↙    xx    fjfj    oo55. . OO`̀jj`̀    x x //∁∁    oo5555   sfluf  sfluf
qufquf   oo5555  · ·f f uj uj suisuifspfspm±m±o` c˒o` cm˒`-dfoamg` gfm`-dfoamg` gf   oo55. . Gm UrGm Ur`p``p`snsn±±o˒om˒` =.=.= rfsubtm qufm` =.=.= rfsubtm quf   oo5555  · ·f f uujj
fspm±fspm±o` c`rjmg` nco`jpbft`.o` c`rjmg` nco`jpbft`.
==..9 9 @@s s ffssppmm±±o`so`s   BB p p((Q,Q, ίί,, ¶¶))
Cfstm sf±Cfstm sf±o˒om˒` f m` f cm pr·cm pr·̀xnjm usmrfj`s `xnjm usmrfj`s `s o`cofnt`s o`cofnt`s f `s f m tfrjnc`b`lnm m tfrjnc`b`lnm gm Wf`rnm gm Jfggm Wf`rnm gm Jfgngm,ngm,
`s `s ququmnmns s ` ` bfnbfnt`r p`gf t`r p`gf fcofco`c`ctrmtrmr r c` c` MpMp˄˄fcgfcgnonof f O. O. @s @s pr·pr·̀xnj`s `xnj`s fxfjpbfxfjpb̀ s `s fcvfcv`bv`bvfjfj
duc±duc±o˒o˒̀fs nctflr·`fs nctflr·mvfns, f mblucs rfsubtmg`s prfbnjncmrfs s˒mvfns, f mblucs rfsubtmg`s prfbnjncmrfs sm˒` m` cfofss·cfofss·mrn`s. mrn`s. FstmifbfofrFstmifbfofrfj`sfj`s
ms gfsnlumbgmgfs gf A¨ms gfsnlumbgmgfs gf A῭bgfr f Jnch`wshn pmrm nctflrmns, quf sfr˒`bgfr f Jnch`wshn pmrm nctflrmns, quf sfrm˒` m` ··utfns utfns mgnmcmgnmctf.tf.
_fkm (_fkm (Q,Q, ίί,, ¶¶) uj fspm±) uj fspm±o` gf jfgngm f =o` gf jfgngm f = ≧ ≧ p  p 22 ∓ ∓. @ o`ckuct` gf t`gms ms duc±. @ o`ckuct` gf t`gms ms duc±o˒o˒̀fs`fs
jfcsur·jfcsur·mvfns gfmvfns gf   Q Q    fjfj   HH tmns quf tmns quf
d d  p p  ;:;:
  
Q Q 
||d d || p p g¶g¶
 ==
pp
22 ∓∓
sfr·sfr·m gfc`tmg` p`rm gfc`tmg` p`r B B p p((Q,Q, ίί,, ¶¶).).
Wf`rfjm =.9.=Wf`rfjm =.9.=   (Gf  (Gfsnlsnlumbgumbgmgf mgf gf gf A¨A῭bg`bgfr fr pmrm pmrm ncnctflrtflrmnsmns))   _fkmj   _fkmj   p , q 3  p , q 3    ==   tmn  tmns s ququf f ==
 p p
  ++    ==
qq
   : : ==    f f    ((Q,Q, ίί,, ¶¶))  uj fspm± uj fspm±o` gf jfo` gf jfgngngm. gm. _f _f    d d   ∁ ∁ BB p p((Q,Q, ίί,, ¶¶))    f f    ll ∁  ∁ BBqq((Q,Q, ίί,, ¶¶)),,
fct˒ fct˒ m`m`   dd ll ∁ ∁ BB==((Q,Q, ίί,, ¶¶))   f f 
dd ll== ≧ ≧ d d  p p ¼¼ llqq . .
Gfj`cstrm±Gfj`cstrm±oo˒˒mm`̀..  @ oms` @ oms`  d d  p p  : 5 `u : 5 `u  llqq  : 5  : 5 ··f f snjpbfs. snjpbfs. _up`cam _up`cam fct˒fctm˒`m`  d d  p p  : : 55  ::
llqq . .  Urnjfnr`  Urnjfnr` ··f f o`cvfcnfctf j`strmr o`cvfcnfctf j`strmr quf, quf, pmrm pmrm qumnsqufrqumnsqufr   mm  ff   ii   c˒cm˒`-cflmtnv`s, tfj`sm`-cflmtnv`s, tfj`s
mm
==
pp ¼¼ ii ==qq ≧≧   mm
 p p
  ++
  ii
q q 
..   (=.9)  (=.9)
Umrm tmct`, o`csngfrf, pmrm omgm 5Umrm tmct`, o`csngfrf, pmrm omgm 5   2 2 ν ν 22  ==,,  m duc± m duc±o˒om˒`m`   d d    ::  d d νν ; ; (5(5,, ∓∓)) ∙ ∙↙↙  YY  gmgm p`r gmgm p`r
d d ((tt) ) :: t tνν ∙∙ νtνt. C`tf quf. C`tf quf  d  d  tfj uj j· tfj uj j·mxnj` fjmxnj` fj t t  : = f p`rtmct` : = f p`rtmct`  t tνν ≧≧ νtνt ++ (=(= ∙∙ νν) pmrm) pmrm
t`g`t`g`   t t 33 5. Dmzfcg` 5. Dmzfcg`   tt : :    mm
ii
   ff   νν : :    ==
 p p
 ` `it·it·fj-sf fj-sf (=.9)(=.9). . WW`jmcg`jmcg`̀
44
  
mm : :
 | |d d ((xx))|| p p
d d  p p p p
ff    ii : :
 | |ll((xx))||qq
llqqqq
fj (=.9), ` rfsubtmg` sfluf.fj (=.9), ` rfsubtmg` sfluf.
Wf`rfjm =.9.9Wf`rfjm =.9.9   (Gfsn  (Gfsnlumbgmgf gf lumbgmgf gf JnchJnch`wsh`wshn n pmrm pmrm nctflnctflrmns)rmns)   _fkmj   _fkmj    ==  ≧≧    p p 22  ∓∓    f f 
((Q,Q, ίί,, ¶¶))  uj fspm± uj fspm±o` gf jfgngm. _f o` gf jfgngm. _f   d,d, ll ∁ ∁ BB p p((Q,Q, ίί,, ¶¶)), fct˒ , fct˒ m`m`   d d   ++ ll ∁ ∁ BB p p((Q,Q, ίί,, ¶¶))   f f 
d d   ++ ll p p ≧ ≧ d d  p p + + ll p p ..   (=.?)  (=.?)
Gfj`cstrm±Gfj`cstrm±oo˒˒mm`̀..    _f_f   pp : = `u : = `u  d d   ++ ll p p  : : 55,, `  ` rfsubtmg` rfsubtmg` ··f f obmr`. obmr`. U`gfj`s U`gfj`s fct˒fctm˒` sup`rm` sup`r
d d   ++ ll p p  : : 5 5 ff   p p 33 = =.. Ufrofim quf, pmrm t`g` Ufrofim quf, pmrm t`g`   xx ∁∁ Q Q , tfj`s, tfj`s
||d d ((xx) +) + ll((xx))|| p p ≧≧ ((||d d ((xx))|| ++ ||ll((xx))||)) p p ≧≧   (9jmx  (9jmx{|{|d d ((xx))||,, ||ll((xx))|}|})) p p
: : 99 p p(jmx(jmx{|{|d d ((xx))||,, ||ll((xx))|}|})) p p ≧≧    99 p p ((||d d ((xx))|| p p ++ ||ll((xx))|| p p)) ,,
f f gm·gm·Ľ Ľ sflsfluf uf qufquf   d d   ++ ll ∁ ∁ BB p p((Q,Q, ίί,, ¶¶). Ml`rm vmj`s pr`vmr (=.?). @isfrvf prnjfnr` quf). Ml`rm vmj`s pr`vmr (=.?). @isfrvf prnjfnr` quf
||d d ((xx) +) + l l((xx))|| p p :: ||d d ((xx) +) + ll((xx))|| ¼|¼|d d ((xx) +) + ll((xx))|| p p∙∙==
≧≧ ||d d ((xx) +) + l l((xx))|| p p∙∙== ¼¼ ((||d d ((xx))|| ++ ||ll((xx))||))
:: ||d d ((xx))|| ¼|¼|d d ((xx) +) + ll((xx))|| p p∙∙== ++ ||ll((xx))|| ¼|¼|d d ((xx) +) + ll((xx))|| p p∙∙== (=.7)(=.7)
pmrm t`g`pmrm t`g` x x ∁ ∁ Q Q . W`jmcg`. W`jmcg` q  q 33 = tmb quf = = tmb quf =/p/p ++ ==/q /q  : = tfj`s ( : = tfj`s ( p p ∙∙ =)=)q q  : :  p p, f p`rtmct`, f p`rtmct`
||d d   ++ ll|| p p∙∙== :: ||d d   ++ ll||
pp
qq ∁∁ BBqq((Q,Q, ίί,, ¶¶)).. Gm gfsnlumbgmgf gf A¨ Gm gfsnlumbgmgf gf A῭bgfr, tfj`s`bgfr, tfj`s
  
Q Q 
||d d || ¼¼ ||d d  + + ll|| p p∙∙== g¶g¶ ≧≧
  
Q Q 
||d d || p p g¶g¶
 ==
pp
  
Q Q 
||d d   ++ ll||(( p p∙∙=)=)qq g¶g¶
 ==
qq
,,
  
Q Q 
||ll|| ¼¼ ||d d  + + ll|| p p∙∙== g¶g¶ ≧≧
  
Q Q 
||ll|| p p g¶g¶

==
pp
  
Q Q 
||d d   ++ ll||(( p p∙∙=)=)qq g¶g¶

==
qq
..
_`jmcg` ms gums gfsnlumbgmgfs monjm f o`jincmcg` o`j (=.7) tfj`s_`jmcg` ms gums gfsnlumbgmgfs monjm f o`jincmcg` o`j (=.7) tfj`s
  
Q Q 
||d d   ++ ll|| p p g¶g¶ ≧ ≧
  
Q Q 
||d d   ++ ll|| p p g¶g¶
 ==
qq
  
Q Q 
||d d || p p g¶g¶
 ==
pp
++
  
Q Q 
||ll|| p p g¶g¶
 ==
pp

,,
f gnvngncg` mji`s `s jfjir`s p`rf gnvngncg` mji`s `s jfjir`s p`r
  
Q Q 
||d d   ++ ll|| p p g¶g¶
 ==
qq
,, ` rfsubtmg` sfluf. ` rfsubtmg` sfluf.
C`tf qufC`tf quf   ¼¼  p p    c˒cm˒` m` ··f, f, fj fj lfrmb, lfrmb, ujm ujm c`rjm c`rjm fjfj B B p p((Q,Q, ίί,, ¶¶), p`ns p`gf mo`ctfofr), p`ns p`gf mo`ctfofr
d d  p p   :   : 5 pm5 pmrrmm    d d    c˒cm˒` m` nngfgfcctntnomomjfjfcctf tf ccububm. m. Gf Gf j`j`g` g` lflfrmrmb, b, sf sf ((Q,Q, ίί,, ¶¶) ) ··f f ujuj
fspm±fspm±o` gf jfgngm, nctr`guznj`s ujm rfbm±o` gf jfgngm, nctr`guznj`s ujm rfbm±o˒om˒` m` gf gf fqunvmbfqunvmb˄˄fconm fconm gnzfcg` gnzfcg` quf quf gums gums duc±duc±o˒o˒̀fs`fs
d,d, ll ;;   Q Q 
 ∙ ∙↙↙
 H H   s˒sm˒` fqunvmbfctfs sfm` fqunvmbfctfs sf   d d    :: l  l ¶¶-qumsf -qumsf sfjprf, sfjprf, nst` ·nst` ·f, f, sf sf fxnstf fxnstf uj uj o`ckuct`o`ckuct`
<<
  
MM  ∁∁   ί tmb quf  ί tmb quf    ¶¶((MM) : 5 f) : 5 f    d d ((xx) ) ::    ll((xx) pmrm t`g`) pmrm t`g`    x x //∁∁    MM. . GfGfc`tc`tmcgmcg` m ` m obmobmssssf gff gf
fqunvmb˄fqunvmbf˄conm fconm gf gf ujm ujm duc±duc±o˒om˒`m`   d d  p`r X p`r Xd d ZZ,, · ·f f njfgnmt` njfgnmt` quf quf c` c` o`ckuct` o`ckuct` qu`onfctfqu`onfctf
BB p p((Q,Q, ίί,, ¶¶) ) ;:;: {{XXd d Z Z ;; d  d  ∁ ∁ BB p p((Q,Q, ίί,, ¶¶))}},,
ms`pfrm±ms `pfrm±o˒o˒̀fs`fs
XXd d Z + XZ + XllZ Z : : XXd d   ++ llZ Z ff    ooXXd d Z Z : : XXod od ZZ
fst˒fstm˒` ifj gf﬍cngms f t`rcmjm` ifj gf﬍cngms f t`rcmj   BB p p((Q,Q, ίί,, ¶¶) uj fspm±) uj fspm±o` vft`o` vft`rnmb. rnmb. Mb·Mb·fj gnfj gnss`, gss`, gf﬍cncf﬍cncg`g`
XXd d ZZ p p ;: ;: d d  p p ,,
o`rrnlnj`s ` quf dmbtmvm pmrmo`rrnlnj`s ` quf dmbtmvm pmrm ¼¼ p p sfr ujm c`rjm sfr ujm c`rjm. . Mssnj Mssnj ((BB p p((Q,Q, ίί,, ¶¶)),, ¼¼ p p)) ··f uf uj fj fspmspm±±o`o`
vft`rnmb c`rjmg`. M sflunr, j`strmrfj`s qufvft`rnmb c`rjmg`. M sflunr, j`strmrfj`s quf   BB p p((Q,Q, ίί,, ¶¶) ) ··f f uj uj fsfspmpm±±o` gf Imcmoa.o` gf Imcmoa.
Gfvfj`s tfr fj jfctf quf `s vft`rfs gfGfvfj`s tfr fj jfctf quf `s vft`rfs gf    BB p p((Q,Q, ίί,, ¶¶) ) s˒sm˒` m` obmssfobmssfs s gf gf fqunvmb˄fqunvmbf˄conmfconm,,
jms jms c` c` o`tngnmc` o`tngnmc` ms ms obmssfs obmssfs gf gf fqunvmbfqunvmb˄˄fconms fconms s˒sm˒` `jntngms f fsorfvfj`sm` `jntngms f fsorfvfj`s   d d   c` bulmr gf  c` bulmr gf
XXd d Z. Z. DDmrfj`s, pr`p`sntmbjmrfj`s, pr`p`sntmbjfctf, m gfj`cstrm±fctf, m gfj`cstrm±o˒om˒` m sflunr fsorfvfcg` Xm` m sflunr fsorfvfcg` Xd d Z pmrm quf ` bfnt`rZ pmrm quf ` bfnt`r
pfrofim pfrofim quf quf fstf ·fstf ·f f uj uj prfon`snsj` prfon`snsj` gfscfofss·gfscfofss·mrn`.mrn`.
Wf`rfjm =.9.?Wf`rfjm =.9.?    _f _f    ==  ≧≧    p p 22  ∓∓, fct˒ , fct˒ m`m`    BB p p((Q,Q, ίί,, ¶¶))   ·  ·f f uj uj fsfspm±pm±o` gf Imcmoa o`j m o` gf Imcmoa o`j m 
c`rjm c`rjm 
XXd d ZZ p p  ::
  
Q Q 
||d d || p p g¶g¶
 ==
pp
..
Gfj`cstrm±Gfj`cstrm±oo˒˒mm`̀..   K·K·m smifj`s qufm smifj`s quf B B p p((Q,Q, ίί,, ¶¶) ·) ·f f uj uj fsfspmpm±±o` c`o` c`rjmg`rjmg`. . Yfstm Yfstm pr`vmr quf ·pr`vmr quf ·ff
uj fspm±uj fspm±o` o`jpbft`. Umrm nss` sfkm (Xo` o`jpbft`. Umrm nss` sfkm (Xd d ccZ)Z)
∓∓
cc:=:=  ujm  ujm sfqu˄sfquf˄conm fconm gf gf Omuoay Omuoay fjfj B B p p((Q,Q, ίί,, ¶¶))..
Fct˒Fctm˒`, gmg`m`, gmg`   ζ ζ 33 5 fxnstf 5 fxnstf   J J   :: J  J ((ζζ)) ∁∁ CC tmb quf tmb quf
  
Q Q 
||d d cc ∙∙ d d jj|| p p g¶g¶ : : d d cc ∙∙ d d jj p p p p 2  2 ζζ p p sfjprf qufsfjprf quf   j,j, cc  ≦≦ J.J.
_fkm (_fkm (llhh))
∓∓
hh:=:=  ujm  ujm suisfsuisfqu˄quf˄conm fconm gf gf ((d d cc))
∓∓
cc:=:=  tmb quf tmb quf  llhh+=+= ∙∙ llhh p p 2 2  9 9∙∙hh pmrm t`g`pmrm t`g`  h h ∁ ∁ CC..
O`csngfrf m duc±O`csngfrf m duc±o˒om˒`m`
ll ;;  Q Q  ∙ ∙↙↙ YY ∫{∓}∫{∓}   , , ll((xx) ) :: ||ll==((xx))|| ++
∓∓
hh:=:=
||llhh+=+=((xx)) ∙∙ llhh((xx))|| ..   (=.6)  (=.6)
_mifj`s quf_mifj`s quf   ll  · ·f f jfjfcscsurur··mvfb mvfb f f c˒cm˒`-m`-cflcflmtnmtnvm. vm. Mb·Mb·fj fj gnsgnss`,s`,
||ll((xx))|| p p : bnj: bnj
cc↙∓↙∓

||ll==((xx))|| ++
cc
hh:=:=
||llhh+=+=((xx)) ∙∙ llhh((xx))||
 p p
..
Ufb` Bfjm gf Dmt`u (vfkm Wf`rfjm O.=<), sfluf qufUfb` Bfjm gf Dmt`u (vfkm Wf`rfjm O.=<), sfluf quf
  Q Q 
||ll|| p p g¶g¶ ≧≧ bnbnjj ncncd d 
cc
↙∓↙∓   Q Q 
||ll==|| ++
cc
hh:=:=
||llhh+=+= ∙∙ llhh||

 p p
g¶.g¶.
11
  
Fbfvmcg` mji`s `s jfjir`s mFbfvmcg` mji`s `s jfjir`s m    ==
 p p
 f usmcg` m gfsnlumbgmgf gf Jnch`wshn, `itfj`s f usmcg` m gfsnlumbgmgf gf Jnch`wshn, `itfj`s
  
Q Q 
||ll|| p p g¶g¶

==
pp
≧≧ bnbnjj ncncd d 
cc↙∓↙∓

ll== p p + +
cc
hh:=:=
llhh+=+= ∙∙ llhh p p

 ≧ ≧ ll== p p + = + =..   (=.4)  (=.4)
Fct˒Fctm˒`, m`, gf﬍cncggf﬍cncg`̀ M M  : : {{xx ∁∁ Q Q    ;; l l((xx)) 2 2 ∓}∓} ,, gf (=.4) p`gfj`s o`cobunr quf gf (=.4) p`gfj`s o`cobunr quf  ¶ ¶((Q Q ∙∙ MM) ) ::
5. 5. B`l` m B`l` m s·s·frnf fj frnf fj (=.6) o`cvfrlf fxoft` (=.6) o`cvfrlf fxoft` tmbvfz c` o`ckutmbvfz c` o`ckuct` gf jfgngm ct` gf jfgngm cubmcubm Q  Q ∙∙MM, nst`, nst`
··f, f, m m s·s·frfrnf nf o`o`cvfcvfrlrlff   ¶¶-qu-qumsf sfjpmsf sfjprf. rf. _flu_fluf quf f quf m duc±m duc±o˒om˒`m`   ll ¼¼ όόMM ∁ B ∁ B p p((Q,Q, ίί,, ¶¶), `cgf), `cgf   όόMM
··f f m m duducc±±o˒om˒` m` omrmoomrmotfrtfr··Ľstnom Ľstnom gfgf   MM (vfkm  (vfkm ` ` Mp˄Mpf˄cgnof fcgnof O). O). Gf﬍cm Gf﬍cm fct˒fctm˒`m`  d  d ;;   Q Q  ∙ ∙↙↙ HH  p`rp`r
d d ((xx) ) ::



ll==((xx) +) +
∓∓
hh:=:=
((llhh+=+=((xx)) ∙∙ llhh((xx)))) ,,    sfsf   xx ∁∁ M,M,
55,,    sfsf   x x //∁∁ M.M.
O`j`O`j`   llhh   :: l l== + ( + (ll99 ∙∙ ll==) + () + (ll?? ∙∙ ll99) +) + ¼¼ ¼¼ ¼¼ + (+ (llhh  ∙∙ llhh∙∙==)),, tfj`s tfj`s
||llhh((xx))|| ≧≧ ||ll==((xx))|| ++
hh∙∙==

 k k:=:=
||ll k k+=+=((xx)) ∙∙ ll k k((xx))|| ≧≧ l l((xx))
pmrm t`g`pmrm t`g`    xx    ff    llhh((xx)) ∙ ∙↙↙    d d ((xx) pmrm t`g`) pmrm t`g`    xx ∁ ∁    MM, , nstnst` ` ··f, f, m m sfqsfqu˄uf˄cofconm nm ((llhh))∓∓hh:=:=   o`cvfrlf o`cvfrlf
pmrmpmrm d  d ¶¶-qumsf -qumsf sfjprf. sfjprf. Ufb` Wf`rfjm gm Ufb` Wf`rfjm gm O`cvfrlO`cvfrl˄˄fconm fconm G`jncmgm G`jncmgm (vfkm Wf`rfjm O.9=),(vfkm Wf`rfjm O.9=),
sfluf qufsfluf quf  d  d  ∁ ∁ BB p p((Q,Q, ίί,, ¶¶). O`j`). O`j`
||d d  ∙ ∙ llhh|| p p ≧≧ ((||d d || ++ ||llhh||)) p p ≧≧ (9(9ll)) p p ¼¼ όόMM   ¶¶-qumsf sfjprf f-qumsf sfjprf f
bnjbnj
hh↙∓↙∓
||d d  ∙ ∙ llhh|| p p : : 55   ¶¶-qumsf sfjprf,-qumsf sfjprf,
c`vc`vmjfctf mjfctf pfb` pfb` WWf`rfjm f`rfjm gm gm O`cvfrlO`cvfrl˄˄fconm fconm G`jncmgm G`jncmgm tfj`stfj`s
bnjbnj
hh↙∓↙∓   Q Q  ||
d d 
 ∙ ∙
llhh
||
 p p g¶g¶ : :
  Q Q 
55   g¶g¶ :  : 55..
Gm·Gm·Ľ Ľ o`o`cocobu·bu·ĽjĽj`s `s ququff   llhh ∙ ∙↙↙   d d    fjfj B B p p((Q,Q, ίί,, ¶¶), f p`rtmct` X), f p`rtmct` XllhhZZ ∙ ∙↙↙    XXd d Z Z fjfj   BB p p((Q,Q, ίί,, ¶¶).).
Mssnj (XMssnj (Xd d ccZ)Z)
∓∓
cc:=:=  · ·f f ujm ujm sfqu˄sfquf˄conm fconm gf gf Omuoay Omuoay quf quf tfj tfj ujm ujm suisfsuisfqu˄quf˄conm fconm (X(XllhhZ)Z)
∓∓
hh:=:=    qufquf
o`cvfrlf pmrm Xo`cvfrlf pmrm Xd d Z. _fluf njfgnmtmjfctf quf XZ. _fluf njfgnmtmjfctf quf Xd d ccZZ ∙∙↙↙ XXd d Z Z fjfj   BB p p((Q,Q, ίί,, ¶¶).).
=.? =.? @ @ fspm±fspm±o`o`   BB∓∓((Q,Q, ίί,, ¶¶))
_fkm_fkm BB∓∓((Q,Q, ίί,, ¶¶) ` ) ` o`ckuo`ckuct` gf ct` gf t`t`gms mgms ms ducs duc±±o˒o˒̀fs jfcsur·`fs jfcsur·mvfns quf s˒mvfns quf sm˒` bnjntmgmsm` bnjntmgms ¶ ¶-qumsf-qumsf
sfjprf, sfjprf, nst` nst` ··f, f, fxnstfj fxnstfj uj uj o`ckuct`o`ckuct`   C C  ∁ ∁  ί f uj c· ί f uj c·ujfr` rfmbujfr` rfmb   H H   tmns quf tmns quf   ¶¶((C C ) : 5 f) : 5 f
||d d ((xx))| ≧| ≧  H H  pmrm t`g` pmrm t`g`   x x //∁∁  C C . . _f_f   d d  ∁ B ∁ B∓∓((Q,Q, ίί,, ¶¶) ) ff   C C  ∁ ∁  ί  ί ··f f uj uj o`ckuct` o`ckuct` gf gf jfgngmjfgngm
cubmcubm, , gf﬍cnj`sgf﬍cnj`s
_ _ d d ((C C ) : sup) : sup
{|{|
d d ((xx))
||
;; x  x //
∁∁
C C 
}}
   ff
>>
  
d d ∓∓ : ncd  : ncd {{_ _ d d ((C C ) ) ;; C  C  ∁ ∁ ί ί ff   ¶¶((C C ) ) : : 55}} ..   (=.<)  (=.<)
C`tf quf, c`vmjfctf, p`gf mo`ctfofrC`tf quf, c`vmjfctf, p`gf mo`ctfofr  d d ∓∓  : 5 o`j : 5 o`j   d d    c˒cm˒` ngfctnom` ngfctnomjfctmjfctf cubm. f cubm. UmrmUmrm
o`ct`ro`ct`rcmr cmr fssf fssf pr`ibfjpr`ibfjm, m, tmjitmji··fj fj rfo`rrfj`s rfo`rrfj`s eems ms obmssfobmssfs s gf gf fqunvfqunvmb˄mbf˄conm. fconm. O`j` cmO`j` cm
sf±sf±o˒om˒` mctfrn`r, gnzfj`s quf gums duc±m` mctfrn`r, gnzfj`s quf gums duc±o˒o˒̀fs s˒`fs sm˒` fqunvmbfctfs (pfrtfcofj em` fqunvmbfctfs (pfrtfcofj em jfsjm obmssfm jfsjm obmssf
gf gf fqunvmb˄fqunvmbf˄conmfconm) ) sf sf o`ncoo`ncongfjngfj  ¶ ¶-qumsf sfjprf. @ o`ckuct`-qumsf sfjprf. @ o`ckuct`  B B∓∓((Q,Q, ίί,, ¶¶) ·) ·f f ` ` o`co`ckuctkuct` ` gfgf
t`gms t`gms ms ms obmssfs obmssfs gf gf fqunvmbfqunvmb˄˄fconm fconm gms gms duc±duc±o˒o˒̀fs jfcsur·`fs jfcsur·mvfnsmvfns d  d ;;   Q Q  ∙ ∙↙↙ HH qu quf f s˒sm˒` m` bnjnbnjntmgtmgmsms
¶¶-qumsf sfjprf.-qumsf sfjprf.    ··F obmr` qufF obmr` quf B B∓∓((Q,Q, ίί,, ¶¶) ·) ·f f uj uj fsfspmpm±±o` vft`rnmb o`j ms jfsjms `pfrm±o` vft`rnmb o`j ms jfsjms `pfrm±o˒o˒̀fs`fs
gm sf±gm sf±o˒om˒` mctfrn`r. _f Xm` mctfrn`r. _f Xd d ZZ ∁∁ BB∓∓((Q,Q, ίί,, ¶¶)),, gf﬍cnj`s gf﬍cnj`s
XXd d ZZ∓∓  : : d d ∓∓ ..   (=.1)  (=.1)
C` C` FxfroFxfro··Ľon` Ľon` =.1.9? =.1.9? ` ` bfnt`r bfnt`r pr`vpr`vmr·mr·m quf sf m jfgngmm quf sf m jfgngm   ¶¶ · ·f f ﬍c﬍cntntm m ff   d d  ∁ ∁   BB∓∓((Q,Q, ίί,, ¶¶),),
fct˒fctm˒`m`   d d   ∁∁   BB p p((Q,Q, ίί,, ¶¶) pmrm t`g`) pmrm t`g`   pp ≦ ≦   = f bnj  = f bnj
 p p↙∓↙∓
d d  p p    ::  d d ∓∓. . NsNss` kuss` kustntn﬍om m c`tm﬍om m c`tm±±o˒om˒`m`
BB∓∓((Q,Q,