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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
  
UUrrffdd··mmoonn`̀
@ @ 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,ίί,, ¶¶).).
Gfnxmj`s m omrl` g` bfnt`r m gfj`cstrm±Gfnxmj`s m omrl` g` bfnt`r m gfj`cstrm±o˒om˒` gf qufm` gf quf   ¼¼ ∓∓  fst· fst·m m ifj ifj gf﬍cngf﬍cngm gm f ·f ·f uf ujmjm
c`rjm fjc`rjm fj  BB∓∓((Q,Q, ίί,, ¶¶))..  Tfkmj`s quf sf X Tfkmj`s quf sf Xd d ZZ ∁∁ BB∓∓((Q,Q, ίί,, ¶¶)),,  fct˒ fctm˒`m`
||d d ((xx))|| ≧≧ d d ∓∓    ¶¶--qquummssf f ssffjjpprrff. . ((==..>>))
O`j fdfnt`O`j fdfnt ,̀ , pfbm gf﬍cn±pfbm gf﬍cn±o˒om˒` gfm` gf  d d ∓∓   fxnstf   fxnstf ujm ujm sfqu˄sfquf˄conm fconm ((C C cc))∓∓cc:=:=   gf o`ckuct`s gf  gf o`ckuct`s gf
jfgngm cubm tmns qufjfgngm cubm tmns quf
bnjbnj
cc↙∓↙∓
_ _ d d ((C C cc) ) :: d d ∓∓    ff  ||d d ((xx))|| ≧≧ _ _ d d ((C C cc) pmrm t`g`) pmrm t`g`  x  x //∁∁ C C cc..
B`l`, t`jmcg`B`l`, t`jmcg`   C C   ::
∓∓
cc:=:=
C C cc  rfsubtm quf rfsubtm quf  C  C   tfj jfgngm cubm f tfj jfgngm cubm f
||d d ((xx))|| ≧≧ _ _ d d ((C C cc) pmrm t`g`) pmrm t`g`   x x //∁∁ C.C.
Dmzfcg` ` bnjntf o`jDmzfcg` ` bnjntf o`j  cc  ∙∙↙↙ ∓∓  `itfj`s (=.>). `itfj`s (=.>).
Wf`rfjm =.?.=Wf`rfjm =.?.=    BB∓∓((Q,Q, ίί,, ¶¶))  · ·f f uj uj fsfspm±pm±o` gf Imcmoa.o` gf Imcmoa.
Gfj`cstrm±Gfj`cstrm±oo˒˒mm`̀..    _·_·̀ rfstm j`strmr quf ` fspm±` rfstm j`strmr quf ` fspm±o` ·o` ·f f o`jpbft`o`jpbft .̀ . Umrm nss`, Umrm nss`, sfkm (Xsfkm (Xd d ccZ)Z)
∓∓
cc:=:=
ujm ujm sfqu˄sfquf˄conm fconm gf gf Omuoay Omuoay fjfj   BB∓∓((Q,Q, ίί,, ¶¶))..  U`r (=.>), pmrm omgm U`r (=.>), pmrm omgm   kk ∁ ∁  C C  fxnstf fxnstf   J J  k k ∁ ∁   ίί
tmb quftmb quf  ¶ ¶((J J  k k) : 5 f) : 5 f
||d d  k k((xx))|| ≧≧ d d  k k∓∓  pmrm t`g` pmrm t`g`   x x //∁∁ J J  k k..
W`jmcg`W`jmcg`   J J 55  ::
∓∓
 k k:=:=
J J  k k ,, tfj`s tfj`s
||d d  k k((xx))|| ≧≧ d d  k k∓∓  pmrm t`g` pmrm t`g`   x x //∁∁ J J 55..
Jmns mncgm, pmrm omgm pmr (Jmns mncgm, pmrm omgm pmr (c,c, jj)) ∁∁ CC99 fxnstffxnstf   J J c,jc,j ∁ ∁ ί o`jί o`j  ¶ ¶((J J c,jc,j) : 5 f) : 5 f
||d d cc((xx)) ∙∙ d d jj((xx))|| ≧≧ d d cc ∙∙ d d jj∓∓  pmrm t`g` pmrm t`g`   x x //∁∁ J J c,jc,j..
=5=5
  
_fkm_fkm   J J   :: J  J 55 ∫∫
    ∓∓
c,jc,j:=:=
J J c,jc,j

. Fct˒. Fctm˒`m`  ¶ ¶((J J ) : 5 f pmrm qumnsqufr) : 5 f pmrm qumnsqufr   j,j, cc ∁ ∁CC,,

   ||d d cc((xx))|| ≧≧ d d cc∓∓  pmrm t`g` pmrm t`g`   x x //∁∁ J J    ff
||
d d 
cc
((xx))
∙∙
d d 
jj
((xx))
|| ≧≧ 
d d 
cc ∙∙
d d 
jj∓∓
 pmrm t`g` pmrm t`g`   x x //
∁∁
J.J.
  (=.=5)  (=.=5)
Fct˒Fctm˒`m`, , pmpmrm rm omomgmgm    x x //∁∁    J J , , m m sfqusfqu˄˄fconm fconm ((d d cc((xx))))∓∓cc:=:=   ·  ·f f gf gf Omuoay Omuoay fjfj    HH, , p`rp`rtmctmct`t`
o`cvfrlfctf. U`gfj`s fct˒o`cvfrlfctf. U`gfj`s fctm˒` gf﬍cnrm` gf﬍cnr
d d ;;   Q Q  ∙ ∙↙↙ HH  , , d d ((xx) ) ::

   bnjbnj
cc↙∓↙∓
d d cc((xx)),,    sfsf   x x //∁∁ J J 
55,,    sfsf  x x ∁∁ J.J.
_fluf quf_fluf quf   d d  · ·f f jfjfcscsurur··mvfb. U`r (=.=5) f o`j` (Xmvfb. U`r (=.=5) f o`j` (Xd d ccZ)Z)
∓∓
cc:=:=  · ·f f gf gf OmuOmuoay, gmoay, gmg`g`   ζ ζ 33 5 fxnstf 5 fxnstf
cc55 ∁ ∁ CC tmb quf suptmb quf sup
xx//∁∁J J 
||d d cc((xx)) ∙∙ d d jj((xx))|| ≧≧ ζζ sfjprf quf sfjprf quf j, j, cc  ≦≦ cc55.. Dmzfcg` Dmzfcg` j j ∙∙↙↙ ∓∓ sfluf sfluf
qufquf
supsup
xx//∁∁J J 
||d d cc((xx)) ∙∙ d d ((xx))|| ≧≧ ζζ   (=.==)  (=.==)
sfjprf qufsfjprf quf   cc ≦ ≦   cc55..  Mssnj, ( Mssnj, (d d cc))∓∓cc:=:=   ·  ·f f ucnd`rucnd`rjfjfctf jfjfctf o`cvfrlfcto`cvfrlfctf f pmrmpmrm   d d    fjfj   Q Q  ∙ ∙ J.J.
Gf (=.==) sfluf qufGf (=.==) sfluf quf   d d cc ∙ ∙ d d   ∁ B∁ B∓∓((Q,Q, ίί,, ¶¶) pmrm) pmrm   cc  su﬍onfctfjfctf  su﬍onfctfjfctf lrmcgf. lrmcgf. GmGm··Ľ, Ľ, o`j`o`j`d d    ::    d d cc ∙ ∙ ( (d d cc ∙ ∙ d  d )),,   sfluf quf X  sfluf quf Xd d ZZ ∁ ∁    BB∓∓((Q,Q, ίί,, ¶¶))..   Mssnj, p`gfj`s rffsorfvfr (=.==),  Mssnj, p`gfj`s rffsorfvfr (=.==),
o`cobuncg` qufo`cobuncg` quf
XXd d ccZZ ∙∙ XXd d ZZ∓∓  : : d d cc ∙∙ d d ∓∓ ≧ ≧  supsup
xx//∁∁J J 
||d d cc((xx)) ∙∙ d d ((xx))|| ≧≧ ζζ
sfjprf qufsfjprf quf   cc ≦ ≦ cc55..  U`rtmct` (X U`rtmct` (Xd d ccZ)Z)∓∓cc:=:=  o`cvfrlf pmrm X o`cvfrlf pmrm Xd d Z Z fjfj   BB∓∓((Q,Q, ίί,, ¶¶))
C`tm±C`tm±o˒om˒`.m`.    M M `o`rr˄`o`rrf˄conm fconm jmns jmns drfqufctf drfqufctf g`s g`s fspm±fspm±o`so`s    BB p p((Q,Q, ίί,, ¶¶) sf g·) sf g·m qumcg`m qumcg`    Q Q    ::
XXm,m, iiZZ ⊄ ⊄   YY, , ί ί ··f f m m snljm-·snljm-·mblfirm g`s o`ckuct`s Bfifsluf-jfcsur·mblfirm g`s o`ckuct`s Bfifsluf-jfcsur·mvfns mvfns (quf (quf o`ct·o`ct·fj fj `s`s
i`rfbnmc`s) fi`rfbnmc`s) f   ¶¶ · ·f m f m jfgngm gf jfgngm gf Bfifsluf. Bfifsluf. Cfssf oms` Cfssf oms` fsorfvfj`sfsorfvfj`s   BB p pXXm,m, iiZ.Z.
==..7 7 FFssppmm±±o`o`s s gf gf sfsfqu˄quf˄cfcononmsms
K·K·m fstugmj`s `s fspm±m fstugmj`s `s fspm±o`s o`s gf gf sfqsfqu˄uf˄cofconmsnms  o o55 f f  o o5555. Cfstm sf±. Cfstm sf±o˒om˒` fstugmrfj`s `utr`s fspm±m` fstugmrfj`s `utr`s fspm±o`so`s
c`t·c`t·mvfns gf mvfns gf sfqu˄sfquf˄conms. fconms. Umrm oUmrm omgm mgm c·c·ujfr` rfmbujfr` rfmb   pp ≦≦ =, gf﬍cnj`s=, gf﬍cnj`s
 p p : :

((mm k k))
∓∓
 k k:=:=  ;; m m k k ∁ ∁ HH pmrm t`g` pmrm t`g`   kk ∁ ∁ CC    ff
∓∓
 k k:=:=
||mm k k|| p p 22 ∓∓

..
O`csngfrmcg` ` o`ckuct`O`csngfrmcg` ` o`ckuct` U  U ((CC) gms pmrtfs gf) gms pmrtfs gf   CC  f m jfgngm gf o`ctmlfj f m jfgngm gf o`ctmlfj   ¶¶oo    fjfj U  U ((CC),),
c˒cm˒ ̀m ̀ ··f f gngndd··ĽoĽonb nb vfrvfrn﬍n﬍omomr r ququff    p p  · ·f f cm cm vfrgmgvfrgmgf f ` ` fspm±fspm±o`o`   BB p p((CC,, U U ((CC)),, ¶¶oo). ). MncMncgm jmns, cfsgm jmns, cfssfsf
oms` ms `pfrm±oms` ms `pfrm±o˒o˒̀fs usumns gf duc±`fs usumns gf duc±o˒o˒̀fs sf trmcsd`rjmj cms `pfrm±`fs sf trmcsd`rjmj cms `pfrm±o˒o˒̀f`fs s ususumumns ns gf gf sfsfqu˄quf˄cfcononmsms
f m c`rjmf m c`rjm   ¼¼  p p  sf trmcsd`rjm fj sf trmcsd`rjm fj

((mm k k))
∓∓
 k k:=:=

 p p
  ::

  ∓∓
 k k:=:=
||mm k k|| p p

==
pp
..
====
  
Gfssm d`rjm rfsubtm qufGfssm d`rjm rfsubtm quf   p p · ·f f uj uj ffspspm±m±o` gf Imcmoa o`j ms `pfrm±o` gf Imcmoa o`j ms `pfrm±o˒o˒̀fs `fs usuusumns gmns gf sff sfqu˄quf˄cofconmsnms
f o`j m c`rjmf o`j m c`rjm   ¼¼  p p. Fj pmrtnoubmr, g`s Wf`rfjms =.9.= f =.9.9 tfj`s;. Fj pmrtnoubmr, g`s Wf`rfjms =.9.= f =.9.9 tfj`s;
Ur`p`sn±Ur`p`sn±o˒om˒ ̀m ̀ =.=.7.7.==   (Gfsnlumbgmgf gf A¨ (Gfsnlumbgmgf gf A῭bgfr `bgfr pmrm pmrm sfqu˄sfquf˄conmfconms)s)  _fkmj  _fkmj   cc ∁ ∁ C C   f f  p, q 3 p, q 3  = =
tmns quf tmns quf 
   ==
 p p  ++
   ==
qq    : : ==. Fct˒ . Fct˒ m`m`
cc
 k k:=:=
||mm k kii k k || ≧≧

   cc
 k k:=:=
||mm k k|| p p
 ==
pp
¼¼

   cc
 k k:=:=
||ii k k ||qq
 ==
qq
pmrm qumnsqufr fsombmrfs pmrm qumnsqufr fsombmrfs   mm==, . . . , m, . . . , mcc,, ii==, . . . , i, . . . , icc..
Ur`p`sn±Ur`p`sn±o˒om˒ ̀m ̀ =.=.7.7.99   (Gf  (Gfsnlumsnlumbgmgf bgmgf gf Jncgf Jnch`wshn pmrm h`wshn pmrm sfqu˄sfquf˄conmfconms)s)  Umrm  Umrm   pp ≦≦ ==,,  tfj`s  tfj`s 

   cc
 k k:=:=
||mm k k + + ii k k|| p p
 ==
pp
≧≧

   cc
 k k:=:=
||mm k k|| p p
 ==
pp
++

   cc
 k k:=:=
||ii k k|| p p
 ==
pp
pmrm qumnsqufr pmrm qumnsqufr   cc ∁ ∁CC  f fsombmrfs  f fsombmrfs   mm==, . . . , m, . . . , mcc,, ii==, . . . , i, . . . , icc..
UmrmUmrm p p  : : ∓∓, gf﬍cnj`s, gf﬍cnj`s  ∓∓  o`j` ` fspm± o`j` ` fspm±o`s o`s gms gms sfqu˄sfquf˄conms fconms bnjntmgms bnjntmgms gf gf fsombmrfs, fsombmrfs, `u`u
sfkm;sfkm;
∓∓  : :

((mm k k))
∓∓
 k k:=:=  ;; m m k k ∁ ∁ HH pmrm t`g` pmrm t`g`   kk ∁ ∁ CC   f sup  f sup
 k k∁∁CC
||mm k k|| 22 ∓∓

..
U`gfj`s o`cobunr qufU`gfj`s o`cobunr quf   ∓∓  · ·f f uj uj fsfspmpm±±o` gf Imcmoa o`j ms `pfrm±o` gf Imcmoa o`j ms `pfrm±o˒o˒̀fs usumns gf duc±`fs usumns gf duc±o˒o˒̀fs`fs
f o`j m c`rjmf o`j m c`rjm
((mm k k))∓∓ k k:=:=∓∓  : sup : sup{|{|mm k k|| ;; k k ∁ ∁ CC}}
p`r gums jmcfnrms; p`r gums jmcfnrms; `isfrv`isfrvmcg` qufmcg` quf  ∓∓  : :  B B∓∓((CC,, U U ((CC)),, ¶¶oo) `u quf) `u quf  ∓∓   :: I I((CC) (Fxfjpb`) (Fxfjpb`
=.=.9). =.=.9). @ @ dmt` gf dmt` gf t`gm t`gm sfqu˄sfquf˄conm o`cvfrlfconm o`cvfrlfctf sfr fctf sfr bnjntbnjntmgm mgm njpbnnjpbnom om qufquf   oo55  · ·f f susuiifsfspmpm±±o`o`
gfgf   ∓∓8 f p`r sfr Imcmoa,8 f p`r sfr Imcmoa,   oo55  · ·ff susuiifsfspmpm±±o` dfoamg` gfo` dfoamg` gf   ∓∓  pfbm Ur`p`sn± pfbm Ur`p`sn±o˒om˒` =.=.=.m` =.=.=.
==..6 6 OO`̀cckkuucctt`̀s s oo`̀jjppmmoott`̀s s ffj j ffssppmm±±o`o`s s vvftft`r`rnmnmnsns
c`rjmg`sc`rjmg`s
M M o`jo`jpmopmongngmgfmgf, , mbmb··fj fj gf gf sfsfr r uj uj g`g`s s ncncvvmrnmrnmcmctfs tfs t`p`t`p b̀·b·̀ln`lno`s o`s jmnjmns s njp`rnjp`rtmctmctfs,tfs,
gfsfgfsfjpfcajpfcam m pmpfb pmpfb ducgducgmjfcmjfctmb tmb cm cm Mc·Mc·mbnmbnsf. sf. U`r U`r fxffxfjpbjpb`, `, uj uj rfsurfsubtmgbtmg` ` ofcofctrmb ‟ trmb ‟ ff
junt` ·junt` ·utnb ‟ gm mc·utnb ‟ gm mc·mbnsf fbfjfcmbnsf fbfjfctmr gnz quf tmr gnz quf t`gm duc±t`gm duc±o˒om˒` m` o`cto`ct··Ľcum gĽcum gf﬍cngf﬍cngm fm fj uj uj o`j o`jpmotjpmot`̀
··f f bnjntmbnjntmgm gm f f mssujmssujf f j·j·mxnmxnj` j` f f jj··ĽcnĽcnj`.j`.
Mb·Mb·fj fj gf gf sum sum njpnjp`rt`rt˄˄mconm cmturmb, cm Mc·mconm cmturmb, cm Mc·mbnsf Dumbnsf Ducon`cmb m con`cmb m o`jpmongo`jpmongmgf mgf lmcam mncgmlmcam mncgm
jmns rfbfv` p`ns rfprfsfctm ujm d`rtf rupturm fctrf ms gnjfcs˒jmns rfbfv` p`ns rfprfsfctm ujm d`rtf rupturm fctrf ms gnjfcs˒̀fs ﬍cntm f nc﬍cntm.`fs ﬍cntm f nc﬍cntm.
TTfrffrfj`j`s s mgnmgnmcmctf tf ququf f m m i`bi`bm m ucnucnt·t·mrnmrnm m dfodfoamgamgm m fj fj uj uj fsfspm±pm±o` o` vfvft`rt`rnmnmb b c`rc`rjmjmg` g` gfgf
gnjfcs˒gnjfcsm˒` m` ﬍cn﬍cntm ·tm ·f f sfjpsfjprf rf o`jpo`jpmotmmotm. . K·K·m fj gnjfcs˒m fj gnjfcsm˒` m` nc﬍nc﬍cntcntm, m, sfr·sfr·m pr`vmg` quf mm pr`vmg` quf m
i`bm cucom ·i`bm cucom ·f f o`jpo`jpmotm cm motm cm t`p`bt`p`b̀ lnm gm `lnm gm c`rjc`rjm. m. UmrUmrm m o`co`ct`rct`rcmr mr fssfssf f pr`ipr`ibfjbfjm m gmgm
=9=9
  
c˒cm˒` o`jpmongmgf gm i`bm ucnt·m` o`jpmongmgf gm i`bm ucnt·mrnm fj gnjfcs˒mrnm fj gnjfcsm˒` m` nc﬍cntm, nc﬍cntm, fstugmrfj`s fstugmrfj`s c` c` OmpOmp··Ľtub` Ľtub` 44
ms ms t`p`b`lnms drmom f t`p`b`lnms drmom f drmom fstrfbm, quf, drmom fstrfbm, quf, p`r p`r tfrfj jfc`s tfrfj jfc`s mifrt`s, dmonbntmj m mifrt`s, dmonbntmj m `o`rr˄`o`rrf˄conmfconm
gf o`ckuct`s o`jpmot`s.gf o`ckuct`s o`jpmot`s.
U`r gf﬍cn±U`r gf﬍cn±o˒om˒`, m`, uj uj suio`suio`ckuckuct`ct`    H H   gf uj fspm±  gf uj fspm±o` o` t`p`b·t`p`b·̀lno` `lno` ··f f o`jpmot` o`jpmot` sf sf t`gmt`gm
o`ifrturm mifrtm gfo`ifrturm mifrtm gf H  H  mgjntf suio`ifrturm ﬍cntm. Fj fspm± mgjntf suio`ifrturm ﬍cntm. Fj fspm±o`s o`s j·j·ftrftrno`no`s, vmbf s, vmbf m sfm sflunlunctfctf
omrmotfrnzm±omrmotfrnzm±o˒om˒`; uj suio`ckuct`m`; uj suio`ckuct` H  H  g gf f uj uj fsfspmpm±±o ̀o ̀ j·j·ftftrnrno ̀o ̀ ··f f o`o`jpjpmomot ̀t ̀ sfsf, , f f s`s`jfjfctf ctf sfsf, ·, ·ff
sfqufconmbjfctf sfqufconmbjfctf o`jpmot`, o`jpmot`, nst` nst` ··f, f, t`gm t`gm sfqu˄sfquf˄conm fconm d`rjmgm d`rjmgm p`r p`r fbfjfct`s fbfjfct`s gfgf   H H   mgjntf  mgjntf
ujm ujm suisfqu˄suisfquf˄conm fconm quf quf o`cvfrlf pmro`cvfrlf pmrm m uj uj fbfjfct` fbfjfct` gfgf   H H  (vfkm X6?, Ur`p`sn± (vfkm X6?, Ur`p`sn±o˒om˒` 1.6.<Z).m` 1.6.<Z).
Ur`p`sn±Ur`p`sn±o˒om˒ ̀m ̀ =.=.6.6.==    _f _f    F F   ·  ·f f uj uj fsfspm±pm±o` o` vft`rnmb vft`rnmb c`rjmg` c`rjmg` gf gf gnjfcs˒gnjfcsm˒` m` ﬍cntm, ﬍cntm, fct˒ fct˒ m` `s m` `s 
o`jpmot`s fj o`jpmot`s fj   F F    s˒ s˒ m` prfonsmjfctf `s o`ckuct`s bnjntmg`s f dfoamg`s.m` prfonsmjfctf `s o`ckuct`s bnjntmg`s f dfoamg`s.
Gfj`cstrm±Gfj`cstrm±oo˒˒mm`̀..   O`ckuct`s o`jpmot`s fj fspm±  O`ckuct`s o`jpmot`s fj fspm±o`s o`s j·j·ftrnoftrno`s `s s˒sm˒` sfjprf dfoamg`s fm` sfjprf dfoamg`s f
bnjntmg`s (vfkm X6?, Ur`p`sn±bnjntmg`s (vfkm X6?, Ur`p`sn±o˒o˒̀fs 1.=.`fs 1.=.= f = f 1.=.1.=.9Z). 9Z). ImsImstm fct˒tm fctm˒` pr`vmr quf t`g` o`ckuct`m` pr`vmr quf t`g` o`ckuct`
bnjntmg` f dfoamg`bnjntmg` f dfoamg` H  H  ⊄ ⊄ F F  · ·f o`jf o`jpmot`pmot`. . _up_up`cam `cam quf gnquf gnjj F F   :: c c  f sfkm f sfkm {{ff==, . . . , f, . . . , fcc}} ujmujm
imsf c`rjmbnzmgm gfimsf c`rjmbnzmgm gf   F F . . O`j` fsO`j` fstmj`tmj s̀ fj fspm±s fj fspm±o`s o`s j·j·ftrno`s, ftrno`s, imstm imstm j`strmr j`strmr quf quf t`gmt`gm
sfsfqu˄quf˄cfcononm m fjfj   H H  mgjn mgjntf tf suisfsuisfqu˄quf˄conm fconm o`cvfrlfo`cvfrlfctf ctf fjfj   H H . . _fkm p`r_fkm p`rtmctmct` (t` (xxjj))
∓∓
jj:=:=   ujmujm
sfsfqu˄quf˄cfcononm m fjfj  H  H . Umrm omgm. Umrm omgm j j  fxnstfj fsombmrfs fxnstfj fsombmrfs  m m((jj))==   , . . . , m  , . . . , m((jj))cc    tmns tmns qufquf x xjj : :
cc

 k k:=:=
mm((jj)) k k    ff k k..
O`j`O`j` H  H  · ·f f bnjnbnjntmgtmg`, `, fxnfxnstfstf  B  B 33 5 tmb quf 5 tmb quf  xxjj ≧≧ B B  pmrm t`g` pmrm t`g` j j. O`csngfrfj`s fj. O`csngfrfj`s fj  H Hcc
m c`rjm gm s`jmm c`rjm gm s`jm   ¼¼ ==. Ufb` Bfjm =.=.6 fxnstf. Ufb` Bfjm =.=.6 fxnstf   o o 33 5 tmb quf 5 tmb quf
BB ≦≦ xxjj ::

cc
 k k:=:=
mm
((jj))
 k k    ff k k
 ≦≦ oo

   cc
 k k:=:=
mm((jj)) k k


:: o o ¼¼ ((mm((jj))==   , . . . , m  , . . . , m((jj))cc    ))==
pmpmrm rm t`gt`g`̀    jj    ∁∁    CC. . MsMssnsnj j m m sfsfququ˄˄fcfcononmm

((mm
((jj))
==   , . . . , m  , . . . , m
((jj))
cc    ))
∓∓
jj:=:=
··f f bnjntmgm bnjntmgm fjfj
HHcc. . Ufb` Wf`rfjm Ufb` Wf`rfjm gf gf I`bzmc`‟\I`bzmc`‟\fnfrstrmss fnfrstrmss fstm fstm sfqu˄sfquf˄conm fconm p`ssun p`ssun ujm ujm suisfqu˄suisfquf˄conmfconm
((mm
((jjhh))
==   , . . . , m  , . . . , m
((jjhh))
cc    ))
∓∓
hh:=:=
quf o`cvfrlf pmrm uj ofrt`quf o`cvfrlf pmrm uj ofrt`   ii :  : ((ii==, . . . , i, . . . , icc)) ∁∁ HHcc. . GfGf

cc
 k k:=:=
mm
((jjhh))
 k k    ff k k ∙ ∙
cc
 k k:=:=
ii k kff k k
 ≧≧
cc
 k k:=:=
mm((jjhh)) k k    ∙∙ ii k k
 :: mm((jjhh))==   , . . . , m  , . . . , m((jjhh))cc
∙∙ ii
==
∙∙↙↙ 55,,
o`o`cocobu·bu·Ľj`Ľj`s s ququff   xxjjhh ∙ ∙↙↙
cc
 k k:=:=
ii k kff k k. U`r. U`r   H H  sfr dfoamg` smifj`s quf sfr dfoamg` smifj`s quf
cc
 k k:=:=
ii k kff k k ∁ ∁ H H ..
_fkm_fkm   F F  uj fspm± uj fspm±o` c`rjmg`. @ o`ckuct`o` c`rjmg`. @ o`ckuct`
IIF F   :: {{xx ∁∁ F F   ;; xx ≧≧ = =}}
··f f oamoamjmjmg ̀g ̀ gfgf  i`bm ucnt·  i`bm ucnt· mrnm dfoamgm mrnm dfoamgm    gfgf   F F ..
O`O`r`br`b··mrnmrn ̀̀ =.=.6.6.99  M i`bm ucnt·  M i`bm ucnt· mrnm dfoamgm fj uj fspm±mrnm dfoamgm fj uj fspm±o` c`rjmg` gf gnjfcs˒ o` c`rjmg` gf gnjfcs˒ m ̀m ̀ ﬍c﬍cntntm m ··f f 
o`jpmotm.o`jpmotm.
=?=?
  
M sflunr mprfsfctmj`s uj dmj`s` rfsubtmg`, gfvng` m D. Ynfsz, quf sfr·M sflunr mprfsfctmj`s uj dmj`s` rfsubtmg`, gfvng` m D. Ynfsz, quf sfr·m m ··utnb pmrmutnb pmrm
j`strmr quf m i`bm ucnt·j`strmr quf m i`bm ucnt·mrnm dfoamgm fj fspm±mrnm dfoamgm fj fspm±o`s gf gnjfcs˒o`s gf gnjfcsm˒` m` nc﬍cntnc﬍cntm m cucom cucom ··f f o`jpmo`jpmotm.otm.
Bfjm =.6.?Bfjm =.6.?   (Bfjm gf Ynfsz)  (Bfjm gf Ynfsz)   _fkm   _fkm    J J   uj   uj suifspsuifspm±m±o` dfoamg` pr· o` dfoamg` pr· `prn` gf uj fspm±`prn` gf uj fspm±o`o`
c`rjmg`c`rjmg`   F F   f sfkm   f sfkm   αα  uj c·  uj c· ujfr` rfmb tmb quf ujfr` rfmb tmb quf   55   2 2 α α 22   ==. . FcFct˒ t˒ m` fxnstf m` fxnstf   yy ∁ ∁   F F  ∙ ∙ J J    tmb tmb quf quf   yy : : ==   f f   yy ∙∙ xx ≦≦ α α  pmrm t`g` pmrm t`g`   xx   fj fj   J J ..
Gfj`cstrm±Gfj`cstrm±oo˒˒mm`̀..   _fkm  _fkm   yy55 ∁ ∁ F F  ∙ ∙ J J  f o`csngfrf ` c· f o`csngfrf ` c·ujfr`ujfr`
gg : gnst( : gnst(yy55,, J J ) ;: ncd ) ;: ncd 
xx∁∁J J 
yy55 ∙∙ xx ..
O`j`O`j`    J J   ·  ·f f dfoadfoamg` mg` ff    yy55    //∁∁    J J , tfj`s, tfj`s    g g 33    55..   Gf dmt`, sup`cg`  Gf dmt`, sup`cg`    gg  : 5 fxnstnrnm ujm : 5 fxnstnrnm ujm
sfqu˄sfquf˄conm fconm gf gf fbfjffbfjfct`s ct`s gfgf   J J   o`cvfrlncg` pmrm  o`cvfrlncg` pmrm   yy55, , f f cfssf cfssf oms` oms` tfr·tfr·Ľmj`sĽmj`s   yy55 ∁ ∁   J J   p`ns  p`ns
J J  · ·f df dfoamfoamg`. g`. O`jO`j`̀    gg
αα
   3 3 gg, p`gfj`s fso`bafr, p`gfj`s fso`bafr   xx55 ∁ ∁ J J   tmb quf tmb quf
yy55 ∙∙ xx55 ≧≧
  gg
αα
..
Tfkmj`s quf fso`bafcg`Tfkmj`s quf fso`bafcg`
yy  : :    yy
55 ∙∙ xx55
yy55 ∙∙ xx55
,,
ms o`cgn±ms o`cgn±o˒o˒̀fs rfqufrngms fst˒`fs rfqufrngms fstm˒` smtnsdfntms.m` smtnsdfntms.    ··F obmr` qufF obmr` quf   yy   tfj  tfj c`rjm c`rjm = = f f c˒cm˒` m` pfrtfcof pfrtfcof mm
J J . . _f_fkmkm  xx ∁ ∁   J J . . O`O`j`j`  J J  · ·ff susuiifsfspmpm±±o` vft`rnmb, (o` vft`rnmb, (xx55 + + yy55 ∙∙ xx55 xx)) ∁ ∁   J J , f p`rtmct`, f p`rtmct`
gg ≧≧ yy55 ∙∙ ((xx55 + + yy55 ∙∙ xx55 xx)). U`r ﬍j. U`r ﬍j
yy ∙∙ xx ::

yy55 ∙∙ xx55
yy55 ∙∙ xx55
 ∙ ∙ xx
 ::
  yy55 ∙∙ ((xx55 + + yy55 ∙∙ xx55 xx))
yy55 ∙∙ xx55
   ≦≦    ggyy55 ∙∙ xx55
 ≦ ≦ α.α.
Wf`rfjm =.6.7Wf`rfjm =.6.7   Vj fspm±  Vj fspm±o` c`rjmg`o` c`rjmg`   F F   tfj   tfj gnjfcs˒gnjfcsm˒` m` ﬍cntm ﬍cntm sf, sf, f f s`jfctf sf, s`jfctf sf, m m i`bm i`bm 
ucnt· ucnt· mrnm dfoamgm gf mrnm dfoamgm gf   F F   ·  ·f f o`o`jpmjpmototm.m.
Gfj`cstrm±Gfj`cstrm±oo˒˒mm`̀..   @ O`r`b·  @ O`r`b·mrn` =.6.9 lmrmctf ujm gms njpbnom±mrn` =.6.9 lmrmctf ujm gms njpbnom±o˒o˒̀fs. `fs. YfsYfstm-ctm-c`s pr`v`s pr`vmrmr
quf quf sf sf m m i`bm i`bm ··f f o`jpmotm, o`jpmotm, fct˒fctm˒` m` ` ` fspm±fspm±o` o` tfj tfj gnjfcs˒gnjfcsm˒` ﬍cntm` ﬍cntm. m. _up`c_up`cam qufam quf   F F  tfcam tfcam
gnjfcs˒gnjfcsm˒` nc﬍m` nc﬍cntcntm. m. Fso`Fso b̀ambam   xx== ∁ ∁   F F   o`j c`rjm = o`j c`rjm =..  O`j` gnj O`j` gnj F F    :: ∓ ∓,,  suifspm± suifspm±o` o` XXxx==ZZ
lfrmlfrmg` g` p`rp`r   xx==  · ·f f uj uj suisuifspfspm±m±o` pr·o` pr·̀prn` gf`prn` gf   F F . U`r tfr gnjfcs˒. U`r tfr gnjfcsm˒` ﬍cntm, Xm` ﬍cntm, Xxx==Z Z ··f f susuiifsfspmpm±±o`o`
dfoamg` gfdfoamg` gf  F F  pfb` Wf`rfjm =.=.4. Ufb` Bfjm gf Ynfsz, o`j pfb` Wf`rfjm =.=.4. Ufb` Bfjm gf Ynfsz, o`j  α α  : :    ==
99
, fxnstf, fxnstf  xx99 ∁ ∁ F F ∙∙ XXxx==ZZ
gf c`rjm = tmb qufgf c`rjm = tmb quf
xx99 ∙∙ xx== ≦≦
  ==
99
..
MpbMpbnomnomcg` cg` c`c`vvmjfcmjfctf tf ` ` mrlumrlujfcjfct` t` pmrm pmrm ` ` suifssuifspm±pm±o` o` XXxx==,, xx99Z Z lflfrmrmg` g` p`rp`r  {{xx==,, xx99}},,
quf o`ctncum sfcg` uj suifspm±quf o`ctncum sfcg` uj suifspm±o` dfoamg` pr·o` dfoamg` pr·̀prn` pfbms jfsjms rmz˒`prn` pfbms jfsjms rmz˒̀fs, fxnstf`fs, fxnstf    xx??  ∁∁
F F   ∙∙ XXxx==,, xx99Z gf c`rjm = tmb qufZ gf c`rjm = tmb quf

xx??
∙∙
xx k k
 ≦≦
  ==
99
  pmrm  pmrm   kk   : : ==,, 9.9.
=7=7
  
U`gfj`s o`ctncumr fssf pr`ofgnjfct` ncgf﬍cngmjfctf p`ns fj t`gms ms ftmpms tfrfj`sU`gfj`s o`ctncumr fssf pr`ofgnjfct` ncgf﬍cngmjfctf p`ns fj t`gms ms ftmpms tfrfj`s
uj suifspm±uj suifspm±o` gf o` gf gnjfcs˒gnjfcsm˒` ﬍cntm gfm` ﬍cntm gf F  F , b`l` dfoamg` f pr·, b`l` dfoamg` f pr·̀prn`. Ur`ofgfcg` gfssm d`rjm,`prn`. Ur`ofgfcg` gfssm d`rjm,
o`o`cscstrtruu··Ľj`Ľj`s s ujujm m sfsfqu˄quf˄cfcononm m ((xxcc))
∓∓
cc:=:=   fjfj   IIF F  tmb quf tmb quf
xxjj ∙∙ xxcc ≦≦   ==99
sfjprf qufsfjprf quf j j  :: c. c.  Mssnj, ( Mssnj, (xxcc))∓∓cc:=:= · ·f f ujujm m sfsfququ˄˄fcfcoonm nm fjfj  I IF F  quf c˒ quf cm˒` pm` p`ss`ssun suun suisfisfqu˄quf˄cofconmnm
o`cvfrlfctf, ` quf njpfgf qufo`cvfrlfctf, ` quf njpfgf quf   IIF F  sfkm o`jpmotm. sfkm o`jpmotm.
==..4 4 FFssppmm±±o`s c`rjmg`s sfpmr·o`s c`rjmg`s sfpmr·mvfnsmvfns
M` M` b`b`clcl` ` g` g` sfu sfu trftrfncncmjmjfcfct` t` jmjmtfjtfj··mtnomtno`, `, ` ` bfnbfnt`r t`r ofrtofrtmjfcmjfctf tf pmsspmss`u `u p`r p`r gnvgnvfrsmfrsmss
sntum±sntum±o˒o˒̀fs `fs fj fj quf quf m m gfcsngmgfcsngmgf gf g` g` o`ckuco`ckuct` t` g`s g`s c·c·ujfr`s rmon`cmns cm rftm gfsfjpfcamujfr`s rmon`cmns cm rftm gfsfjpfcam
pmpfpmpfb b oruoruononmbmb. . Gm jfsjGm jfsjm m d`d`rjmrjm, , m m prfprfsfcsfc±±om gf uj suio`ckuct` fcujfr·om gf uj suio`ckuct` fcujfr·mvfb f gfcs`mvfb f gfcs`
fj uj fspm±fj uj fspm±o` o` c`rjmc`rjmg` g` ··f f ··utnb fj gnvfrsms sntum±utnb fj gnvfrsms sntum±o˒o˒̀fs. `fs. @o`@o`rrf quf, o`jrrf quf, o`j` vfrf` vfrfj`s fjj`s fj
irfvf, cfj t`g` fspm±irfvf, cfj t`g` fspm±o` c`o` c`rjmg` rjmg` o`ct·o`ct·fj uj fj uj suisuio`ckuo`ckuct` fcujfct` fcujfr·r·mvfb f gfcs`. U`r nss`mvfb f gfcs`. U`r nss`
gfstmomj`s m obmssf g`s fspm±gfstmomj`s m obmssf g`s fspm±o`s quf l`zmj gfssm pr`prnfgmgf;o`s quf l`zmj gfssm pr`prnfgmgf;
Gf﬍cn±Gf﬍cn±o˒om˒ ̀m ̀ =.=.4.4.==   Vj fspm±  Vj fspm±o` c`rjmg`o` c`rjmg`    F F   quf   quf o`ct·o`ct·fj fj uj uj suio`ckuct` suio`ckuct` fcujfr·fcujfr·mvfb fmvfb f
gfcs` fjgfcs` fj    F F   ·  ·f f ggnntt`̀   sfpmr·   sfpmr· mvfb mvfb . . Jmns Jmns lfrmbjfctf, lfrmbjfctf, uj uj fspm±fspm±o` o` j·j·ftrno`ftrno`    J J   ·  ·f f sfsfpmpmr·r·mvfbmvfb
qumcg` qumcg` o`ct·o`ct·fj fj uj uj suio`ckuct` suio`ckuct` gfcs` gfcs` f f fcujfr·fcujfr·mvfb.mvfb.
Fxfjpb` =.4.9Fxfjpb` =.4.9   Fspm±  Fspm±o`s o`s c`rjc`rjmg`s gf mg`s gf gnjgnjfcs˒fcsm˒` ﬍cntm s˒m` ﬍cntm sm˒` sfpmr·m` sfpmr·mvmvfnfns. s. _f_fkmkm    F F    ujuj
fspm±fspm±o` c`rjmg` gf gnjfcs˒o` c`rjmg` gf gnjfcsm˒`m`   cc. . Fso`Fso b̀ambam { {xx==, . . . , x, . . . , xcc}}  ujm imsf gf ujm imsf gf   F F  f, c` oms` rfmb, f, c` oms` rfmb,
o`csngfrf ` o`ckuct`o`csngfrf ` o`ckuct`
MM : :

   cc
 k k:=:=
mm k kxx k k    ;; m m==, . . . , m, . . . , mcc ∁ ∁ ]]

,,
gms o`jincm±gms o`jincm±o˒o˒̀fs bncfmrfs g`s vft`rfs gm imsf o`j fsombmrfs rmon`cmns. `fs bncfmrfs g`s vft`rfs gm imsf o`j fsombmrfs rmon`cmns. C` oms` o`jpbfx`C` oms` o`jpbfx`
suistntumsuistntum   ]]  p`rp`r   ]] ++ n n]] :: { { p p + + nq  nq   ;; p, p, q q  ∁ ∁ YY}}. @ bfnt`r p`gf o`jpr`vmr dmonbjfctf quf. @ bfnt`r p`gf o`jpr`vmr dmonbjfctf qufMM · ·f f fcfcujujfrfr··mvfb f gfcs` fjmvfb f gfcs` fj   F F ..
@ Bfjm m sflunr sfr·@ Bfjm m sflunr sfr·m junt` ·m junt` ·utnb cm utnb cm gfj`cstrgfj`cstrm±m±o˒om˒` gf quf v·m` gf quf v·mrn`s g`s fspm±mrn`s g`s fspm±o`s o`j qufo`s o`j quf
fstmj`s trmimbamcg` s˒fstmj`s trmimbamcg` sm˒` sfpm` sfpmr·mr·mvfns. mvfns. Gmg` uGmg` uj suij suio`cko`ckuct`uct`  M M  gf uj fspm± gf uj fspm±o` vft`rnmbo` vft`rnmb F  F ,,
p`r Xp`r XMMZ fstmj`s gfc`tmcg` ` suifspm±Z fstmj`s gfc`tmcg` ` suifspm±o` gfo` gf F  F  lfrmg` p`r lfrmg` p`r  M M, , nst` ·nst` ·f, f, ` ` o`ckuct` o`ckuct` gf gf t`gmst`gms
ms o`jincm±ms o`jincm±o˒o˒̀fs bncfmrfs (﬍cntms) gf fbfjfct`s gf`fs bncfmrfs (﬍cntms) gf fbfjfct`s gf   MM..
Bfjm =.4.?Bfjm =.4.?  Vj fspm± Vj fspm±o` c`rjmg`o` c`rjmg` F  F  · ·f f ssfpfpmmr· r· mvfb sf, f s`jfctf sf, fxnstf uj suio`ckuct`mvfb sf, f s`jfctf sf, fxnstf uj suio`ckuct`
fcujfr· fcujfr· mvfb mvfb   MM ⊄⊄ F F   tmb quf  tmb quf    XXMMZZ  · ·f f gfgfcscs ̀̀ fj fj   F F ..
Gfj`cstrm±Gfj`cstrm±oo˒˒mm`̀..    _f_f    MM   d`r fcujfr·  d`r fcujfr·mvfb f gfcs` fjmvfb f gfcs` fj    F F , fct˒, fctm˒`m`    F F    ::    MM  ⊄⊄    XXMMZZ  ⊄⊄    F F , , ff
p`rtmct` Xp`rtmct` XMMZ Z ··f f gfcgfcs` s` fjfj   F F . . Yfonpr`omjfcYfonpr`omjfctf, sup`camj`s quf fxnstm uj suio`ckuctf, sup`camj`s quf fxnstm uj suio`ckuct`t`
fcujfr·fcujfr·mvfbmvfb  M M
  ⊄⊄
F F  tmb quf X tmb quf XMMZ Z :: F  F . Oamjfj`s gf. Oamjfj`s gf   II  ` o`ckuct` d`rjmg` p`r t`gms ms ` o`ckuct` d`rjmg` p`r t`gms ms
=6=6
  
o`jincm±o`jincm±o˒o˒̀fs bncfmrfs ﬍cntms gf fbfjfct`s gf`fs bncfmrfs ﬍cntms gf fbfjfct`s gf   MM  o`j o`f﬍onfctfs fj o`j o`f﬍onfctfs fj   ]]HH, `cgf, `cgf   ]]YY   :: ] ]
ff  ] ]OO : : ]] ++ nn]], `u sfkm;, `u sfkm;
II  :: {{mm==xx== + + ¼¼ ¼¼ ¼¼ ++ mmccxxcc  ;; x x==, . . . , x, . . . , xcc ∁ ∁ M, M, mm==, . . . , m, . . . , mcc ∁ ∁ ]]HH    ff   cc ∁∁ CC}}..
··F d·F d·monb vfr qufmonb vfr quf I I · ·f f ffcucujfjfrr··mvfb; o`j`mvfb; o`j` ]]HH · ·f f fcfcujujffr·r·mvfb, pmrm omgmmvfb, pmrm omgm c c ∁ ∁CC, ` , ` o`ckuco`ckuct` gmst` gms
o`jincm±o`jincm±o˒o˒̀fs bncfmrfs gf`fs bncfmrfs gf c c  fbfjfct`s gf fbfjfct`s gf M M  o`j o`f﬍onfctfs fj o`j o`f﬍onfctfs fj  ] ]HH · ·f f fcfcuujfjfr·r·mvfb. _flufmvfb. _fluf
qufquf I I  · ·f f fcfcuujfjfrr··mvfb p`r sfr m ucn˒mvfb p`r sfr m ucnm˒` fcujfr·m` fcujfr·mvfb gf o`ckuct`s fcujfr·mvfb gf o`ckuct`s fcujfr·mvfns. Ur`vmrfj`smvfns. Ur`vmrfj`s
ml`rm qufml`rm quf   II   ·  ·f f gfgfcscs ̀̀ fjfj   F F . . UmUmrm nsrm nss` sfks` sfkmjmj   xx ∁ ∁   F F    ff   ζ ζ 33   5  5. . O`O`jj` X` XMMZ Z ::   F F , fxnstf, fxnstf
yy55 ∁ ∁ XXMMZ tmb qufZ tmb quf xx∙∙yy55 22    ζζ99 . Gnlmj`s. Gnlmj`s y y55  :: i i==xx==++¼¼ ¼¼ ¼¼++iihhxxhh, `cgf, `cgf h h ∁ ∁ CC,, i i==, . . . , i, . . . , ihh ∁ ∁ HH
ff   xx==, . . . , x, . . . , xhh ∁ ∁ MM. O`j`. O`j`  ] ]HH  · ·f f gfgfcscs ̀̀ fjfj   HH, fxnstfj, fxnstfj   mm==, . . . , m, . . . , mhh ∁ ∁ ]]HH  tmns quf tmns quf
||mm k k ∙ ∙ ii k k|| 22
   ζζ
99
= += +
hh
nn:=:=
xxnn
  pmrm t`g` pmrm t`g`   kk   : : ==, . . . , h ., . . . , h .
W`jmcg`W`jmcg`   yy  :: m m==xx== + + ¼¼ ¼¼ ¼¼ ++ mmhhxxhh   tfj`s tfj`s   yy ∁ ∁ II   ff
xx ∙∙ yy :: xx ∙∙ yy55 + + yy55 ∙∙ yy ≧≧ xx ∙∙ yy55 ++ yy55 ∙∙ yy
22
  ζζ
99 + + ((ii== ∙∙ mm==))xx== + + ¼¼ ¼¼ ¼¼ + (+ (iihh  ∙∙ mmhh))xxhh
≧≧   ζζ
99
  + + jjmmxx
 k k:=:=,...,h,...,h
||ii k k ∙ ∙ mm k k|| ((xx== ++ ¼¼ ¼¼ ¼¼ ++ xxhh))
22
  ζζ
99
 + +
   ζζ
99

= += +
hh
nn:=:=
xxnn
 ¼ ¼
hh
nn:=:=
xxnn ≧≧
  ζζ
99
 + +
  ζζ
99
  :: ζ, ζ,
` ` ququf f prpr`̀vvm m ququff    II    ::    F.F.   Mssnj  Mssnj    II    ⊄⊄    F F   ·  ·f f fcufcujfjfr·r·mvmvfb fb f f gfcgfcs`, s`, o`o`jpbjpbftmftmcgcg` ` mm
gfj`cstrm±gfj`cstrm±o˒om˒` gf qufm` gf quf   F F  · ·f f sfsfpmpmr·r·mvfb.mvfb.
Fxfjpb` =.4.7Fxfjpb` =.4.7    oo55   ff    p p, , == ≧≧  p  p 22 ∓∓, , s˒sm˒` sfpmr·m` sfpmr·mvfns. Umrm omgmmvfns. Umrm omgm   cc  ∁∁ CC, o`csngfrf, o`csngfrf
ffcc  :  : (5(5,, 55, . . . ,, . . . , 55,, ==,, 55,, 55, . . ., . . .))
m m sfqusfqu˄˄fconm fconm d`rjmgm d`rjmgm p`r p`r = = cmcm    cc-·-·fsnjm o``rgfcmgm f fsnjm o``rgfcmgm f 5 5 cms gfjmns o`cms gfjmns o``rgfcmgms`rgfcmgms. . @s@s
vft`rfsvft`rfs f f==,, ff99, . . ., . . . s˒ sm˒` oamjmg`s gfm` oamjmg`s gf vft`rfs ucnt·  vft`rfs ucnt· mrn`s omc˄ mrn`s omc˄ `cno`s `cno`s  g`s fspm± g`s fspm±o`s o`s gf gf sfqsfqu˄uf˄cofconmsnms..
Gmg`Gmg`   xx :  : ((mm k k))
∓∓
 k k:=:= ∁ ∁ oo55, tfj`s, tfj`s
bnjbnj
hh
xx ∙∙
hh
 k k:=:=
mm k kff k k

∓∓
: bnj: bnj
hh
(5(5,, 55, . . . ,, . . . , 55,, mmhh+=+=,, mmhh+9+9, . . ., . . .))∓∓  : bnj : bnj
hh
supsup
 k3h k3h
||mm k k|| : : 55,,
p`nsp`ns   mm k k
 k k↙∓↙∓∙∙↙↙  5, f sfluf quf 5, f sfluf quf
hh
 k k:=:=
mm k kff k k
hh↙∓↙∓∙∙↙↙   xx  fjfj   oo55. O`j`. O`j`
hh
 k k:=:=
mm k kff k k ∁ ∁ X Xff==,, ff99, . . ., . . .Z pmrm t`g`Z pmrm t`g`
hh, g` Bfjm =.4.? sfluf quf, g` Bfjm =.4.? sfluf quf   oo55  · ·f f sfsfpmpmr·r·mvfb. Gmg`mvfb. Gmg`   xx :  : ((mm k k))
∓∓
 k k:=:= ∁ ∁  p p, o`j`, o`j`
bnjbnj
hh


xx ∙∙
hh
 k k:=:=
mm k kff k k

 p p
: bnj: bnj
hh
(5(5,, 55, . . . ,, . . . , 55,, mmhh+=+=,, mmhh+9+9, . . ., . . .)) p p : bnj : bnj
hh
∓∓
 k k::hh+=+=
||mm k k|| p p : : 55,,
=4=4
  
pp`n`ns s m m s·s·frfrnfnf
∓∓
 k k:=:=
||mm k k || p p ··f f o`cvfrlfctf, ` o`cvfrlfctf, ` jfsj` jfsj` mrlujfct` mrlujfct` j`strm j`strm qufquf    p p  · ·f f ssffppmmrr··mmvfvfbb..
Fxfjpb` =.4.6Fxfjpb` =.4.6    
∓∓
  c˒cm˒ ̀m ̀ ··f f sfsfpmpmr·r·mvfb. _up`cam qufmvfb. _up`cam quf  
∓∓
 o o`ctfca`ctfcam m ujm ujm sfqu˄sfquf˄conm fconm ((xxcc))
∓∓
cc:=:=
gfcsm. Umrm omgmgfcsm. Umrm omgm c c ∁∁ CC, fsorfvmj`s, fsorfvmj`s x xcc  :  : ((mmcc k k ))∓∓ k k:=:=. _fkm. _fkm y y   : : ((ii k k))∓∓ k k:=:= m  m sfqsfqu˄uf˄cofconm nm gf﬍gf﬍cngcngmm
p`rp`r   ii k k  : 5 sf : 5 sf | |mm k k k k| ≦| ≦  = = ff   ii k k   ::  m m k k k k + = sf + = sf | |mm k k k k||  2 2   =.=.    ··F obmr` qufF obmr` quf   yy ∁ ∁   ∓∓  p`ns p`ns | |ii k k||  22 9 pmrm 9 pmrm
t`g`t`g`   kk. . Gm gfcsngmGm gfcsngmgf gm sfqu˄gf gm sfquf˄conm fconm ((xxcc))
∓∓
cc:=:=   fxnstf  fxnstf    cc55 ∁ ∁   CC   tmb quf tmb quf  yy ∙ ∙ xxcc55∓∓    22   =.=.
JmsJms
yy ∙∙ xxcc∓∓  : sup : sup{|{|ii== ∙∙ mmcc== ||,, ||ii99 ∙∙ mmcc99 ||, . . . ,, . . . , ||iicc ∙∙ mmcccc||,, ||iicc+=+= ∙∙ mmcccc+=+=||, . . ., . . .}} ≦≦ ||iicc ∙∙ mmcccc||
::

   ||55 ∙∙ mmcccc||   sfsf | |mmcccc|| ≦≦ = =
||mmcccc + = + = ∙∙ mmcccc||   sfsf | |mmcccc|| 22 = =
   ::

 | |mmcccc||   sfsf | |mmcccc|| ≦≦ ==
= = sfsf | |mmcccc|| 22 = =
   ≦≦ ==
pmrm t`g`pmrm t`g`   cc ∁∁ CC ‟ o`ctrmgn± ‟ o`ctrmgn±o˒om˒`. _fluf qufm`. _fluf quf   ∓∓   c˒cm˒ ̀m ̀ ··f f sfsfpmpmr·r·mvfb.mvfb.
Umrm pr`vmr m sfpmrminbngmgf g` fspm±Umrm pr`vmr m sfpmrminbngmgf g` fspm±o`o`   O O XXm,m, iiZ prfonsmj`s gf uj tf`rfjm ob·Z prfonsmj`s gf uj tf`rfjm ob·mssno`mssno`
ouk` ouk` oms` oms` rfmb ·rfmb ·f f gfj`cstrmgfj`cstrmg` g` fj fj bnvr`s gf bnvr`s gf Mc·Mc·mbnsf cm Yftm (vfkm, p`r fxfjpb`, X?5,mbnsf cm Yftm (vfkm, p`r fxfjpb`, X?5,
WWf`rfjm >.=<Z)f`rfjm >.=<Z). . Cm bnclumlfj gm Mc·Cm bnclumlfj gm Mc·mbnsf cm mbnsf cm Yftm Yftm fssf rfsubtmg` fssf rfsubtmg` gnz gnz quf quf t`gm t`gm duc±duc±o˒om˒`m`o`ct·o`ct·Ľcum Ľcum fj fj XXm,m, iiZ Z p`gf p`gf sfr sfr ucnucnd`d`rjrjfjffjfctctf f mprmpr̀ x`xnjnjmgmgm m p`r p`r p`bp`bncnc˄̀˄j`jn`n`s. s. Cm Cm c`c`ssssmm
bnclumlfj gnz quf ` o`ckuct` g`s p`bnc˄bnclumlfj gnz quf ` o`ckuct` g`s p`bnc˄̀jn`jn`s `s ··f f gfcgfcs` s` fjfj   O O XXm,m, iiZ;Z;
Wf`rfjm =.4.4Wf`rfjm =.4.4  (Wf`rfjm gm Mpr`xnjm± (Wf`rfjm gm Mpr`xnjm±o˒om˒` gf \fnfrstrmss)m` gf \fnfrstrmss)  _fkm  _fkm   d d ; ; XXm,m, iiZZ ∙ ∙↙↙   HH   ujm ujm 
 duc± duc±o˒ o˒ m` m` o`cto`ct··Ľcum. Ľcum. Fct˒ Fct˒ m` pmrm t`g`m` pmrm t`g`    ζ ζ 33    55   fxnstf uj p`bnc˄   fxnstf uj p`bnc˄ `jn``jn`    U U  ; ;   HH  ∙∙↙↙   HH   tmb quf   tmb quf 
||U U ((xx)) ∙∙ d d ((xx))|| 2 2 ζζ  pmrm t`g` pmrm t`g`   xx ∁∁ XXm,m, iiZZ..
Fxfjpb` =.4.<Fxfjpb` =.4.<    O O XXm,m, iiZ Z ··f f sfpsfpmr·mr·mvmvfbfb. . UUmrmrm omgm omgmm    cc  ∁∁    CC   o`csngfrfj`s m duc±  o`csngfrfj`s m duc±o˒om˒`m`    tt  ∁∁
XXm,m, iiZZ  ↙↙    d d cc((tt) ) ::    ttcc ∁∁   HH. . WW`j`jmcgmcg`̀    MM    ::  {{d d cc  ;; c c ∁∁ CC}}   tfj`s  tfj`s    MM  ⊄⊄    O O XXm,m, iiZ Z f f XXMMZ Z ··f f `̀
o`ckuct` gf t`g`s `s p`bnc˄o`ckuct` gf t`g`s `s p`bnc˄̀jn`s. _fluf g` Wf`rfjm =.4.4 quf X`jn`s. _fluf g` Wf`rfjm =.4.4 quf XMMZ Z :: O  O XXm,m, iiZ f p`rtmct`Z f p`rtmct`
O O XXm,m, iiZ Z ··f f sfsfpmpmr·r·mvfb pfb` Bfjm =.4.?.mvfb pfb` Bfjm =.4.?.
Fxfjpb` =.4.1Fxfjpb` =.4.1    BB p pXXm,m, iiZ, =Z, = ≧≧  p  p 22 ∓∓, ·, ·f sf sfpfpmrmr··mvfb. _fkmjmvfb. _fkmj d  d  ∁ ∁ BB p pXXm,m, iiZ fZ f ζ  ζ 33 5. O`cd`rjf 5. O`cd`rjf
vfrfj`s c` Wf`rfjm =.<.=, ` o`ckuct` gms duc±vfrfj`s c` Wf`rfjm =.<.=, ` o`ckuct` gms duc±o˒o˒̀fs `fs o`ct·o`ct·ĽcumĽcums s ··f f gfcgfcs` s` fjfj    BB p pXXm,m, iiZ, Z, ff
p`rtmct` fxnstfp`rtmct` fxnstf l l ∁ ∁ O O XXm,m, iiZ tmb qufZ tmb quf  d d ∙∙ll p p  22    ζζ99 . G` Wf`rfjm =.4.4 fxnstf uj p`bnc˄. G` Wf`rfjm =.4.4 fxnstf uj p`bnc˄̀jn``jn`
U U   tmb quf tmb quf  ll ∙∙ U U ∓∓  2 2    ζζ9(9(ii∙∙mm)) . Mssnj,. Mssnj,
d d  ∙ ∙ U U  p p ≧ ≧ d d  ∙ ∙ ll p p + + ll ∙∙ U U  p p  22
  ζζ
99
 +  + ((ii ∙∙ mm))ll ∙∙ U U ∓∓  2  2 ζ,ζ,
` quf pr`vm quf ` o`ckuct` g`s p`bnc˄` quf pr`vm quf ` o`ckuct` g`s p`bnc˄̀jn`s `jn`s ··f f gfcs` gfcs` fjfj   BB p pXXm,m, iiZ. Z. M sfpmrM sfpmrminbminbngmngmgf gfgf gf
BB p pXXm,m, iiZ sfluf ml`rm fxmtmjfctf o`j` c` Fxfjpb` =.4.<.Z sfluf ml`rm fxmtmjfctf o`j` c` Fxfjpb` =.4.<.
Gm tf`rnm g`s fspm±Gm tf`rnm g`s fspm±o`s j·o`s j·ftrno`s ftrno`s smif-sf smif-sf quf t`quf t`g` suig` suio`ckuct` gf o`ckuct` gf uj fspm±uj fspm±o ̀o ̀ j·j·ftftrnrno`o`
sfpmr·sfpmr·mvmvfb fb ··f f tmtmji·ji·fj fj sfsfpmpmr·r·mvfb (vfkm, p`r fxfjpb`, X6?, Fxfjpb` >.=.7Z). U`gfj`s fct˒mvfb (vfkm, p`r fxfjpb`, X6?, Fxfjpb` >.=.7Z). U`gfj`s fctm˒`m`
fcuconmr ` sflunctf;fcuconmr ` sflunctf;
Ur`p`sn±Ur`p`sn±o˒om˒ ̀m ̀ =.=.4.4.>>    WW`g` `g` suifsuifspm±spm±oo` ` ggf f uuj j ffssppmm±±o` o` c`c`rjrjmgmg` ` sfsfppmrmr· · mvmvfb fb ··f f tmtmji·ji·fj fj 
sfpmr· sfpmr· mvfb.mvfb.
=<=<
  
=.< =.< O`jfcO`jfct·t·mrn`s f mrn`s f c`tms anst·c`tms anst·̀rnoms`rnoms
@s mxn`jms gf fspm±@s mxn`jms gf fspm±o` vft`rnmb, s`irf `s c·o` vft`rnmb, s`irf `s c·ujfr`s rfmns, d`rmj ujfr`s rfmns, d`rmj fstmifbfonfstmifbfong`s pg`s pfbm prnjfnrmfbm prnjfnrm
vfvfz pfb` z pfb` ntmntmbnmbnmc` L. Ufmc` fj =111. c` L. Ufmc` fj =111. M gf﬍cn±M gf﬍cn±o˒om˒` mistrmtm gf o`rp` mpmrfofu o`j A.m` mistrmtm gf o`rp` mpmrfofu o`j A.
\fifr fj =1>?. @ prnjfnr` bnvr` s`irf fspm±\fifr fj =1>?. @ prnjfnr` bnvr` s`irf fspm±o`s o`s vft`rnmns, vft`rnmns, fsornt` fsornt` pfb` pfb` tmji·tmji·fj fj ntmbnmc`ntmbnmc`
_. Uncoafrbf fj o`bmi`rm±_. Uncoafrbf fj o`bmi`rm±o˒om˒` o`j sfu mbuc` V. Mjmbgn fj =>5=, k·m` o`j sfu mbuc` V. Mjmbgn fj =>5=, k·m trmimbamvm o`jm trmimbamvm o`j
mblucs fspm±mblucs fspm±o`sgf duc±o`s gf duc±o˒o˒̀f`fs. s. O`O`j` ` O·j` ` O·mboub` trmtm gf duc±mboub` trmtm gf duc±o˒o˒̀fs gf﬍cngms fj o`ckuct`s`fs gf﬍cngms fj o`ckuct`s
cuj·cuj·frno`s frno`s f f sfu sfu bnvr` bnvr` trmtmvm gf trmtmvm gf duc±duc±o˒o˒̀fs gf﬍cngms fj o`ckuct`s gf duc±`fs gf﬍cngms fj o`ckuct`s gf duc±o˒o˒̀fs, Uncoafrbf`fs, Uncoafrbf
oamj`u sfu fstug` gfoamj`u sfu fstug` gf    O· O· mboub` Ducon`cmb mboub` Ducon`cmb . . Fj =>5?Fj =>5?, , K. AmgK. Amgmjmjmrg pmsmrg pmss`s`u u m m oaoamjmjmrmr
ujm trmcsd`rjm±ujm trmcsd`rjm±o˒om˒` bncfmr m vmb`rfs c` o`rp` gf fsombmrfs gfm` bncfmr m vmb`rfs c` o`rp` gf fsombmrfs gf   ducon`cmb   ducon`cmb , f ` c`jf gf, f ` c`jf gf
c`ssm gnsonpbncm,c`ssm gnsonpbncm, Mc·  Mc· mbnsf Ducon`cmb mbnsf Ducon`cmb , mpm, mpmrfofu rfofu pfbm pfbm prnjfnrm prnjfnrm vfz fj vfz fj uj bnvr` uj bnvr` gf U. B·gf U. B·fvyfvy
puibnomg` fj =>99.puibnomg` fj =>99.
@s mxn`jms gf c`rjm mpmrfofrmj c`s mc`s =>95-=>99 fj trmimba`s gf C. \nfcfr,@s mxn`jms gf c`rjm mpmrfofrmj c`s mc`s =>95-=>99 fj trmimba`s gf C. \nfcfr,
F. Afbby, _. Imcmoa f A. Amac. @s mxn`jms gf \nfcfr frmj o`cdus`s, Afbby c˒F. Afbby, _. Imcmoa f A. Amac. @s mxn`jms gf \nfcfr frmj o`cdus`s, Afbby cm˒` fxnlnmm` fxnlnm
quf m c`rjm fstnvfssf gf﬍cngm fj t`g`s `s fbfjfct`s g` fspm±quf m c`rjm fstnvfssf gf﬍cngm fj t`g`s `s fbfjfct`s g` fspm±o` f Amac ncobunu juntmso` f Amac ncobunu juntms
fxnlfxnl˄˄fconms mgnon`cmns, fconms mgnon`cmns, ncobusncobusnvf m nvf m o`jpbftuo`jpbftugf. gf. Fj Fj uj uj c`t·c`t·mvfb mvfb trmimba` trmimba` gf gf mistrm±mistrm±o˒om˒`m`
m m pmrtpmrtnr nr gf gf pr`ipr`ibfjbfjms ms jmnjmns s o`coo`corft`rft s̀ s fstufstugmg`gmg s̀ s p`r p`r TT`btf`btfrrm rrm f f AmgAmgmjmmjmrg, rg, ImcmImcmoaoa
mprfsfct`u `s mxn`jms o`j` `s o`cafofj`s a`kf fj sum tfsf X4Z, gfdfcgngm fj =>95mprfsfct`u `s mxn`jms o`j` `s o`cafofj`s a`kf fj sum tfsf X4Z, gfdfcgngm fj =>95
f f puibpuibnomgnomgm m fj =>99. fj =>99. ImcmImcmoa f oa f sfus o`jpsfus o`jpmcafmcafnr`s p`b`cnr`s p`b`cfsfs sf fsfs sf rfdfrfdfrnmj m`s fspm±rnmj m`s fspm±o`so`s
c`rjmg`s o`jpbft`s o`j`c`rjmg`s o`jpbft`s o`j`  fspm± fspm±o`s gf tnp` I o`s gf tnp` I . @ tfrj`. @ tfrj`  fspm± fspm±o` gf Imcmoa o` gf Imcmoa   d`n oucamg`  d`n oucamg`
pfb` jmtfj·pfb` jmtfj·mtno` mtno` drmcdrmco˄of˄s J. fs J. DDr·r·foaft foaft fj =>fj =>91. 91. Jmns Jmns gftmbgftmbafs pafs p`gf`gfj sfr j sfr fco`ctfco`ctrmg`s rmg`s fjfj
X97Z f X=6Z.X97Z f X=6Z.
M gfsnlumbgmgf gf A¨M gfsnlumbgmgf gf A῭bgfr, `bgfr, pmrm pmrm sfqusfqu˄˄fconms, fconms, d`n d`n cm cm vfrgmvfrgmgf gf pr`vpr`vmgm mgm pfbm pfbm prnjfnrprnjfnrmm
vfvfz z p`r p`r B. B. O. O. Y`lY`lfrs fj frs fj =11=111. 1. M M gfjgfj`cs`cstrmtrm±±o˒om˒` gf A¨m` gf A῭b`bgfgfr r mpmpmrmrfofofu fj =11fu fj =11>. >. MM
gfsnlgfsnlumbgmumbgmgf gf (=.9) (=.9) ··f oamf oamjmgm jmgm gfgf  Gfsnlumbgmgf gf ^`ucl  Gfsnlumbgmgf gf ^`ucl ..
O`jO`j` ` ducduc±±o˒o˒̀fs `fs o`cto`ct··Ľcums Ľcums fj fj XXm,m, iiZ Z s˒sm˒` m` Ynfjmcc-Ynfjmcc-nctflnctflr·r·mvmvfnfns s f f duduc±c±o˒o˒̀fs `fs YnfYnfjmccjmcc--
nctflr·nctflr·mvfns s˒mvfns sm˒` Bfifsluf-nctflr·m` Bfifsluf-nctflr·mvfnmvfns, s, ··f f obmr` obmr` quf quf ` ` o`ckuco`ckuct` t` gms gms duc±duc±o˒o˒̀̀fs fs oo`c`ct·t·ĽcĽcuums ms ··ff
suifspm±suifspm±o` gfo` gf   BB p pXXm,m, iiZ. Jmns mncgm;Z. Jmns mncgm;
Wf`rfjm =.<.=Wf`rfjm =.<.=   Umrm   Umrm   == ≧≧  p  p 22  ∓∓,,   O O XXm,m, iiZZ  · ·f f gfgfcscs ̀̀ fj fj   BB p pXXm,m, iiZZ..
Umrm ujm gfj`cstrm±Umrm ujm gfj`cstrm±o˒om˒` vfkm X15, Waf`rfj ?.=7Z.m` vfkm X15, Waf`rfj ?.=7Z.
Umrm 5Umrm 5   2 2 p p 22  = ms gf﬍cn± = ms gf﬍cn±o˒o˒̀fs gf`fs gf   BB p p((Q,Q, ίί,, ¶¶) ) ff   ¼¼  p p    s˒sm˒` ms jfsjms g` oms`m` ms jfsjms g` oms`   pp ≦ ≦  =.=.
Cfstf oms`Cfstf oms` B B p p((Q,Q, ίί,, ¶¶) o`ctncum sfcg` fspm±) o`ctncum sfcg` fspm±o` vft`rnmb, m gndfrfc±o` vft`rnmb, m gndfrfc±oom m ··f f qquuff ¼¼ p p c˒ cm˒ ̀m ̀ ··f f jmjmnsns
ujm c`rjm, p`ns c˒ujm c`rjm, p`ns cm˒` vmbf m gfsnlumbgmgf trnmclubmr. m` vmbf m gfsnlumbgmgf trnmclubmr. @ nctfrfssmctf@ nctfrfssmctf ··f quf m gfsnlumbgmgff quf m gfsnlumbgmgf
trnmclubmr c˒trnmclubmr cm˒` mpfcms c˒m` mpfcms cm˒` vmbf o`j` vmbf m` o`ctr·m` vmbf o`j` vmbf m` o`ctr·mrn` pmrm duc±mrn` pmrm duc±o˒o˒̀fs c˒`fs cm˒`-cflmtnvms, `m`-cflmtnvms, `
quf c`s g·quf c`s g·m m ` ` quf quf ··f f o`cafong` o`cafong` o`j`o`j`  gfsnlumbgmgf gf Jnch`wshn ncvfrtngm  gfsnlumbgmgf gf Jnch`wshn ncvfrtngm ;;
Wf`rfjm =.<.9Wf`rfjm =.<.9   _fkmj   _fkmj    55    2 2 p p 22    ==    f f    ((Q,Q, ίί,, ¶¶))   uj   uj fspfspm±m±o` go` gf jff jfgngngmgm. . _f _f    d,d, ll    ∁∁
BB p p((Q,Q, ίί,, ¶¶))   f f   d,d, ll ≦ ≦ 55, fct˒ , fct˒ m`m`
d d  p p + + ll p p ≧ ≧ d d   ++ ll p p ..
Umrm Umrm m gm gfj`fj`cstcstrm±rm±o˒om˒` vfkm X=7, p·m` vfkm X=7, p·mlncm <4Z `u X?9, Ur`p`sntn`c 6.?.=Z.mlncm <4Z `u X?9, Ur`p`sntn`c 6.?.=Z.
=1=1
  
@s fspm±@s fspm±o`s o`s gf gf duc±duc±o˒o˒̀fs gndfrfcon·`fs gndfrfcon·mvfns, nctr`guzng`s c` Fxfjpb` =.=.7, ifj o`j`mvfns, nctr`guzng`s c` Fxfjpb` =.=.7, ifj o`j`
sums vfrs˒sums vfrs˒̀fs mc·`fs mc·mb`lms pmrm duc±mb`lms pmrm duc±o˒o˒̀fs gf v·`fs gf v·mrnms vmrn·mrnms vmrn·mvfnsmvfns d  d ; ; ℤℤ  ⊄⊄ YYcc ∙∙↙↙ YY, d`rcfofj ujm, d`rcfofj ujm
bnclumlfj mpr`prnmgm pmrm ` fstug` gf fqum±bnclumlfj mpr`prnmgm pmrm ` fstug` gf fqum±o˒o˒̀fs gndfrfconmns pmronmns.`fs gndfrfconmns pmronmns.
U`r `utr` U`r `utr` bmg`, fj bmg`, fj junt`s oms`s junt`s oms`s c`s o`c`s o`cdr`ctmj`s o`j cdr`ctmj`s o`j duc±duc±o˒o˒̀fs quf, mpfsmr gf c˒`fs quf, mpfsmr gf cm˒`m`
sfrfj gndfrfcon·sfrfj gndfrfcon·mvfns, l`zmj gf uj j·mvfns, l`zmj gf uj j·̀gub` `gub` gf gf o`ctncungmgf mpr`pro`ctncungmgf mpr`prnmg`, nmg`, o`j` ·o`j` ·f f ` ` oms`oms`
gmsgms  duc± duc±o˒ o˒ `fs A¨ `fs A¨ b̀̀bgfgfr r o`ct·o`ct·ĽcĽcumums s ;;
Gf﬍cn±Gf﬍cn±o˒om˒ ̀m ̀ =.=.<.<.??  Gmg` uj c· Gmg` uj c·ujfr` rfmb 5ujfr` rfmb 5 2  2 ν ν 22 =, gf﬍cnj`s =, gf﬍cnj`s
O O 55,ν,νXXm,m, iiZ Z ;:;: {{d d  ∁ ∁ O O XXm,m, iiZ Z ;;  ∏∏ O O 33 5 tmb quf 5 tmb quf | |d d ((xx)) ∙∙ d d ((yy))|| ≧≧ O O ||xx ∙∙ yy||νν,, ∅∅ x,x, yy ∁ ∁ XXm,m, iiZZ}} ..
UmrmUmrm  d d  ∁ ∁ O O 55,ν,νX5X5,, iiZ gf﬍cnj`sZ gf﬍cnj`s
d d O O 55,ν,ν   : sup  : sup
tt∁∁XXm,im,iZZ
||d d ((tt))|| + + ssuupp
x,yx,y∁∁XXm,im,iZZ,x,x::yy
||d d ((xx)) ∙∙ d d ((yy))||
||xx ∙∙ yy||νν    ..
Tfrn﬍om-sf qufTfrn﬍om-sf quf   ¼¼ O O 55,ν,ν  gf﬍cf ujm c`rjm fj gf﬍cf ujm c`rjm fj  O  O 55,ν,νXXm,m, iiZ f, o`j mbluj trmimba`, p`gf-sfZ f, o`j mbluj trmimba`, p`gf-sf
o`cstmtmr quf (o`cstmtmr quf (O O 55,ν,νXXm,m, iiZZ,,  ¼¼ O O 55,ν,ν) ) ··f f uj uj fsfspmpm±±o` gf Imcmoa.o` gf Imcmoa.
@ oms` bnjntf@ oms` bnjntf ν ν : = rfprfsfctm ` fspm± : = rfprfsfctm ` fspm±o` gmso` gms duc± duc±o˒ o˒ `fs `fs BnpBnpsoasoantz ntz o`cto`ct··ĽcuĽcums ms  f  f ··f f gfgfc`c`tmtmg`g`
p`r BnpXp`r BnpXm,m, iiZ. @ oms`Z. @ oms`   ν ν 33 =  = ··f f gfsnctfrfssmctf gfsnctfrfssmctf p`ns, p`ns, cfstf cfstf oms`,oms`,   O O 55,ν,νXXm,m, iiZ tfj gnjfcs˒Z tfj gnjfcsm˒`m`
=.=.
JunJuntms tms sntsntum±um±o˒o˒̀fs pr·`fs pr·mtnoms c`s o`cguzfj m` fstug` g`s fspm±mtnoms c`s o`cguzfj m` fstug` g`s fspm±o`so`s   O O h,νh,ν ;:;: { {d d   ∁∁   O O hh ;;
GGhhd d  ∁ ∁   O O 55,ν,ν}}. . O`cd`rjO`cd`rjf gfnxm f gfnxm obmr` ` obmr` ` tf`rfjm m tf`rfjm m sflunr, `s fspm±sflunr, `s fspm±o`s gf o`s gf duc±duc±o˒o˒̀f`fs s A¨A῭b̀bgfgfrr
o`o`ctct··ĽcuĽcums ms s˒sm˒` m` pmpmrtnrtnouboubmrjmrjfcfctf tf njnjp`rp`rtmctmctfs tfs pmpmrm rm m m tf`tf r̀nrnm m gmgms s fqfqum±um±o˒o˒̀fs `fs gndfrfcogndfrfconmnsnmns
pmrpmronmonmns ns fbfb··ĽptĽptnomnoms;s;
Wf`rfjm =.<.7Wf`rfjm =.<.7   (Wf`rfjm gf Yflubmrngmgf gf _oamugfr)  (Wf`rfjm gf Yflubmrngmgf gf _oamugfr)   _fkmj   _fkmj    II==    f f    II==//99   ms i`bms   ms i`bms 
mifrtms c`mifrtms c` YYcc gf rmn`s gf rmn`s  = =  f  f  = =//99, rfspfotnvmjfctf, ofctrmgms cm `rnlfj. _fkm , rfspfotnvmjfctf, ofctrmgms cm `rnlfj. _fkm  u u ;;   II== ∙ ∙↙↙ YY
ujm duc±ujm duc±o˒ o˒ m` smtnsdmzfcg`m` smtnsdmzfcg`
∄∄uu ;: ;:
cc
nn:=:=
∉ ∉ 99uu
∉x∉x99nn
:: d  d    fj fj   II==..
_f _f   d d   ∁∁   O O 55,ν,ν((II==)), fct˒ , fct˒ m`m`   uu ∁ ∁   O O 99,ν,ν((II==//99))    f f   uuO O 99,ν,ν((II==//99)) ≧ ≧O O 

uuBB∓∓((II==)) + + d d O O 55,ν,ν((II==))

,,
`cgf `cgf   O O  gfpfcgf mpfcms gm gnjfcs˒  gfpfcgf mpfcms gm gnjfcs˒ m`m`   cc..
@ tf`rfjm monjm, oukm gfj`cstrm±@ tf`rfjm monjm, oukm gfj`cstrm±o˒om˒` p`gf sfr fco`ctrmgm fj X?7, Waf`rfj 7.4Z,m` p`gf sfr fco`ctrmgm fj X?7, Waf`rfj 7.4Z,
tfj omr·tfj omr·mtfr ducgmjfctmb c` fstug` gf fqum±mtfr ducgmjfctmb c` fstug` gf fqum±o˒o˒̀fs `fs gndfrfcgndfrfconmns onmns pmronmnpmronmns s f f ··f f ··̀tnj` fj`tnj` fj
v·v·mrn`s mspfot`s. U`r fxfjpb`, sfmrn`s mspfot`s. U`r fxfjpb`, sf   d d  d d`r `r jfrmjjfrmjfctf fctf o`cto`ct··Ľcum, Ľcum, c˒cm˒` p`gfj`s lmrmctnr qufm` p`gfj`s lmrmctnr quf
uu ∁ ∁   O O 99((II==//99), ` quf j`strm quf ` fspm±), ` quf j`strm quf ` fspm±o` gms duc±o` gms duc±o˒o˒̀f`fs s A¨A῭b`bgfgfr r o`o`ctct··ĽcuĽcums ms ··f f ̀̀ mjimjinfnfctfctf
mpr`prnmgmpr`prnmg` ` pmrm pmrm m m tf`rnm gf tf`rnm gf rflubmrnrflubmrngmgf gmgf fj fj pr`ibfjpr`ibfjms ms fbfb··Ľptno`Ľptno s̀ s gf gf sflucgm `rgfj.sflucgm `rgfj.
@ @ BfjBfjm m =.=.6.6.? ? d`d`n n prpr`v`vmg` mg` p`r p`r D. D. YnYnfsfsz z fj fj =>==>=1. 1. C` C` FxFxfrofro··ĽoĽon` n` =.=.1.1.?? ?? ` ` bfbfntnt`r`r
o`jpr`vmr·o`jpr`vmr·m quf ` bfjm sf fstfcgf pmrmm quf ` bfjm sf fstfcgf pmrm α α  :  : = c` oms` gf fspm±= c` oms` gf fspm±o`s c`rjmg`s gf gnjfcs˒o`s c`rjmg`s gf gnjfcsm˒`m`
﬍cn﬍cntm. tm. F vfF vfr·r·m m c` c` FxfroFxfro··Ľon` Ľon` =.1.?7 quf =.1.?7 quf ` ` jfsj` jfsj` c˒cm˒` `o`rrf, fj lfrmb, fj fspm±m` `o`rrf, fj lfrmb, fj fspm±o`s gfo`s gf
gnjfcs˒gnjfcsm˒` nc﬍cntm.m` nc﬍cntm.
@ tfrj`@ tfrj`   fspm±  fspm±o` sfpmr· o` sfpmr· mvfb mvfb   d`n   d`n oucoucamamg`, g`, c` c` o`co`ctfxtfxt` t` gf gf fsfspm±pm±o`s t`p`b·o`s t`p`b·̀ln`lno`s, o`s, p`rp`r
DDr·r·foafoaft ft fj fj =>54=>54, , c` c` jfsjfsj` j` mrtnmrtnl` l` fj fj quf quf ncnctr`guztr`guznu nu `s `s fspfspm±m±o`s j·o`s j·ftrno`s. ftrno`s. C˒Cm˒`m`
o`csflunj`s mpurmr o`j ﬍gfgnlcngmgf ` j`tnv` quf ` bfv`u m fso`bafr fxmtmjfctfo`csflunj`s mpurmr o`j ﬍gfgnlcngmgf ` j`tnv` quf ` bfv`u m fso`bafr fxmtmjfctf
fstf tfrj`.fstf tfrj`.
=>=>
  
=.1 =.1 FxfroFxfro··Ľon`sĽon`s
FxFxfro·fro·ĽoĽon ̀n ̀ =.=.1.1.==  O`jpbftf `s gftmbafs g` Fxfjpb` =.=.9. O`jpbftf `s gftmbafs g` Fxfjpb` =.=.9.
FxFxfro·fro·ĽoĽon ̀n ̀ =.=.1.1.99  J`strf quf ` o`ckuct` J`strf quf ` o`ckuct`
 { {
d d 
 ∁ ∁
O O XXm,m, iiZ Z ;; d  d ((xx)) 3 3  5 pmrm t`g` 5 pmrm t`g`   xx
∁∁
XXm,m, iiZZ
}}
 · ·ff
mifrt` fjmifrt` fj   O O XXm,m, iiZ.Z.
FxFxfro·fro·ĽoĽon ̀n ̀ =.=.1.1.??   Gum  Gums s c`rjc`rjmsms   ¼  ¼ ==    ff   ¼  ¼ 99   fj   fj uj uj fsfspmpm±±o` o` vfvft`rnt`rnmbmb    F F    s˒sm˒` m` gngntmstms
fqunvmbfctfs fqunvmbfctfs  sf fxnstnrfj o`cstmctfs p`sntnvms sf fxnstnrfj o`cstmctfs p`sntnvms  o o==   ff   oo99  tmns quf tmns quf
oo== xx== ≧ ≧ xx99 ≧ ≧ oo99 xx==  pmrm t`g` pmrm t`g`   xx ∁∁ F.F.
Ur`vf quf sfUr`vf quf sf  F  F  · ·f f uj uj fsfspmpm±±o` c`rjmg` gf gnjfcs˒o` c`rjmg` gf gnjfcsm˒` ﬍cntm, fct˒m` ﬍cntm, fctm˒` qumnsqufr gums c`rjmsm` qumnsqufr gums c`rjms
fjfj   F F    s˒sm˒` fqunvmbfctfs.m` fqunvmbfctfs.
FxFxfro·fro·ĽoĽon ̀n ̀ =.=.1.1.77   Ur`vf   Ur`vf quf quf m m o`rrfo`rrfsp`sp`cg˄cgf˄conmfconm   d d   ∁∁   O O X5X5,, =Z=Z  ↙↙
  ==
55
 | |d d ((tt))|| gtgt ∁ ∁   YY  · ·f f uujjmm
c`rjm fjc`rjm fj  O O X5X5,, =Z quf c˒=Z quf cm˒` m` ··f f fqufqunvmbfctnvmbfctf f eem c`rjmm c`rjm   ¼¼ ∓∓..
FxFxfro·fro·ĽoĽon ̀n ̀ =.=.1.1.66  G˄ Gf˄ f fxfjpfxfjpb` b` gf gf uj uj fspm±fspm±o` vft`rnmbo` vft`rnmb   F F   f  f gf gf ujm ujm j·j·ftrftrnomnom   gg   fjfj   F F    qufquf
c˒cm˒` fst·m` fst·m mss`onmgm m ujm c`rjm pfbm nlumbgmgf (=.=).m mss`onmgm m ujm c`rjm pfbm nlumbgmgf (=.=).
FxFxfro·fro·ĽoĽon ̀n ̀ =.=.1.1.44   J`strf quf sf 5  J`strf quf sf 5    2 2 νν==    2 2 νν99   ≧≧   =,   =, fcfct˒tm˒`m`    O O ==XXm,m, iiZZ        O O νν99XXm,m, iiZZ    
O O νν== XXm,m, iiZZ  O O XXm,m, iiZ.Z.
FxFxfro·fro·ĽoĽon ̀n ̀ =.=.1.1.<<    J`strJ`strf quf quf ·f ·f pf p`ss`ss··Ľvfb gĽvfb gf﬍cnr f﬍cnr ujm ujm c`rjm c`rjm fj qfj qumbqufumbqufr fr fspm±spm±o` vft`rnmb.o` vft`rnmb.
FxFxfro·fro·ĽoĽon ̀n ̀ =.=.1.1.11    _f_f    MM    ff    II    s˒sm˒` suio`ckuct`s gf uj fspm±m` suio`ckuct`s gf uj fspm±o` c`rjmg`o` c`rjmg`    F F , j`strf quf, j`strf quf
mMmM : :  mM mM  pmrm t`g` fsombmr pmrm t`g` fsombmr   mm   ff   MM ++ II ⊄ ⊄ MM ++ II..
FxFxfro·fro·ĽoĽon ̀n ̀ =.=.1.1.>>   _fkm  _fkm   F F  uj fspm± uj fspm±o` c`rjmg`. Vj suio`ckuct`o` c`rjmg`. Vj suio`ckuct`   MM ⊄ ⊄ F  F  · ·f f ggnntt`̀  bnjntmg` bnjntmg`
sf fxnstnrsf fxnstnr   J J 33  5 tmb quf 5 tmb quf  xx ≧ ≧  J J  pmrm t`g` pmrm t`g`   xx ∁ ∁  M.M.   _f_f   MM  d`r bnjntmg` fj uj fspm± d`r bnjntmg` fj uj fspm±o`o`
c`rjmg`c`rjmg`   F F , j`strf quf, j`strf quf   MM t tmjmji·i·fj fj ··f f bnbnjnjntmtmg`g`..
FxFxfro·fro·ĽoĽon ̀n ̀ =.=.1.1.=5=5    _f_f   F F  · ·f f uj uj fsfspmpm±±o` gf Imcmoa fo` gf Imcmoa f   LL  · ·f f susuiifsfspmpm±±o` gfo` gf   F F , j`strf quf `, j`strf quf `
dfoa`dfoa`  LL   gfgf   LL  t tmjmji·i·fj fj ··f f susuiifsfspmpm±±o` gfo` gf   F F ..
FxFxfro·fro·ĽoĽon ̀n ̀ =.=.1.1.====   Ur`vf quf suifspm±  Ur`vf quf suifspm±o`s pr·o`s pr·̀prn`s gf fspm±`prn`s gf fspm±o`s o`s c`rjmg`s c`rjmg`s t˄tf˄j fj nctfrn`rnctfrn`rvmzn`.vmzn`.
FxFxfro·fro·ĽoĽon ̀n ̀ =.=.1.1.=9=9   (Fspm±  (Fspm±o` pr`gut`) _fkmjo` pr`gut`) _fkmj   F F ==, . . . , F  , . . . , F  cc  fspm± fspm±o`s c`rjmg`s.o`s c`rjmg`s.
(m) Ur`vf quf ms fxprfss˒(m) Ur`vf quf ms fxprfss˒̀fs`fs
((xx==, . . . , x, . . . , xcc))==  : : xx== ++ ¼¼ ¼¼ ¼¼ ++ xxcc,,
((xx==, . . . , x, . . . , xcc))99  :  : ((xx==99 ++ ¼¼ ¼¼ ¼¼ ++ xxcc99))
==
99 , , ff
((xx==, . . . , x, . . . , xcc))∓∓  : jmx : jmx{{xx==, . . . ,, . . . , xxcc}},,
gf﬍cfj c`rjms fqunvmbfctfs c` pr`gut` omrtfsnmc`gf﬍cfj c`rjms fqunvmbfctfs c` pr`gut` omrtfsnmc`   F F == Ü ¼ ¼ ¼ ÜÜ ¼ ¼ ¼ Ü F F cc..
(i) _fkmj ((i) _fkmj (xx== k k))
∓∓
 k k:=:=, . . . ,, . . . , ((xx
cc
 k k ))
∓∓
 k k:=:=   sf  sfqu˄quf˄cfcononms ms fjfj   F F ==, . . . , F  , . . . , F  cc, rfspfotnv, rfspfotnvmjfcmjfctf. tf. Ur`vUr`vf quff quf
xx== k k ∙ ∙↙↙   xx== ∁ ∁   F F ==, . . . , x, . . . , xcc k k  ∙∙↙↙  xxcc ∁ ∁   F F cc  sf, f s`jfctf sf, sf, f s`jfctf sf,

((xx== k k , . . . , x, . . . , x
cc
 k k ))
∓∓
 k k:=:=
 o`cvfrlf pmrm o`cvfrlf pmrm
((xx==, . . . , x, . . . , xcc) ) fjfj F  F == Ü ¼ ¼ ¼ÜÜ ¼ ¼ ¼Ü F F cc  jucng` gf mblujm (f p`rtmct` t`gms) gms c`rjms monjm. jucng` gf mblujm (f p`rtmct` t`gms) gms c`rjms monjm.
(o) Ur`vf quf(o) Ur`vf quf   F F == Ü ¼ ¼ ¼ Ü Ü ¼ ¼ ¼ Ü F F cc  jucng jucng` ` gf gf ujm ujm (f (f p`rtmct` t`gms) p`rtmct` t`gms) gms gms c`rjms monjm c`rjms monjm ··ff
Imcmoa sf, f s`jfctf sf,Imcmoa sf, f s`jfctf sf,   F F ==, . . . , F  , . . . , F  cc   s˒sm˒` Imcmoa.m` Imcmoa.
9595
  
FxFxfro·fro·ĽoĽon ̀n ̀ =.=.1.1.=?=?   _fkm  _fkm    F F   uj   uj fspfspm±m±o` c`rjo` c`rjmg`mg .̀ . C` pr`gC` pr`gut` omrut` omrtfstfsnmnmc` o`csc` o`csngngfrffrf
qumbqufr qumbqufr ujm ujm gms gms c`rjms c`rjms g` g` FxfroFxfro··Ľon` Ľon` =.1.=9.=.1.=9.
(m) Ur`vf quf ms `pfrm±(m) Ur`vf quf ms `pfrm±o˒o˒̀fs `fs mbl·mbl·firfirnomnoms s gfgf   F F ;;
((x,x, yy)) ∁∁ F F   ÜÜ F F   ↙↙ xx + + yy ∁ ∁ F F ff    ((x,x, mm)) ∁∁ F F   ÜÜHH ↙↙ mxmx  ∁∁ F,F,
s˒sm˒` duc±m` duc±o˒o˒̀f`fs s o`o`ctct··ĽcuĽcumsms..
(i) Ur`vf quf m c`rjm(i) Ur`vf quf m c`rjm   xx ∁∁ F F   ↙↙ xx ∁∁ YY · ·f f ujujm m duduc±c±o˒om˒ ̀m ̀ o`o`ct·ct·ĽcĽcumum..
FxFxfro·fro·ĽoĽon ̀n ̀ =.=.1.1.=7=7* (Fspm±* (Fspm±o` qu`onfctf) _fkmjo` qu`onfctf) _fkmj F  F  uj  uj fspfspm±m±o` co` c`rj`rjmg` mg` ff J  J  uj suifspm± uj suifspm±o`o`
gfgf   F F . Gmg`s. Gmg`s   x,x, yy ∁ ∁  F  F , gnzfj`s quf, gnzfj`s quf   xx ∲ ∲ y y   sfsf   xx ∙∙ yy ∁ ∁  J  J . Umrm omgm. Umrm omgm   xx ∁ ∁ F  F , gf﬍cnj`s, gf﬍cnj`s
XXxxZ Z :: {{yy ∁ ∁ F F   ;; x x ∲∲ yy}}. . Gf﬍Gf﬍cnjcnj`s `s tmji·tmji·fjfj   F/J F/J   :: {{XXxxZ Z ;; x x ∁∁ J J }}. Ur`vf quf. Ur`vf quf
(m)(m) ∲ ∲ · ·f f ujujm m rfrfbmbm±±o˒om˒` m` gf gf fqufqunvmb˄nvmbf˄cofconm.nm.
(i) Ms `pfrm±(i) Ms `pfrm±o˒o˒̀fs X`fs XxxZ + Z + XXyyZ Z : : XXxx + + y yZZ ff   mmXXxxZ Z : : XXmxmxZ fst˒Z fstm˒` ifj gf﬍cngms f t`rcmjm` ifj gf﬍cngms f t`rcmj   F/J F/J 
uj fspm±uj fspm±o` o` vft`rnvft`rnmb.mb.
(o) _f(o) _f   J J  · ·f f dfdfoamoamg ̀g ̀ fjfj   F F , fct˒, fctm˒` m` m m fxpfxprfsrfss˒sm˒`m`

XXxxZZ

: gnst(: gnst(x,x, J J ) : ncd ) : ncd 
{{
xx
∙∙
yy

;; y y
 ∁ ∁
J J 
}}
gf﬍cf ujm c`rjm fjgf﬍cf ujm c`rjm fj   F/J F/J ..
(g) _f(g) _f  F  F  · ·f f ffssppmm±±o` gf Imcmoa, fct˒o` gf Imcmoa, fctm˒`m`   F/J F/J   t tmmjiji··ffj j ··f f ffssppmm±±o` gf Imcmoa.o` gf Imcmoa.
FxFxfro·fro·ĽoĽon ̀n ̀ =.=.1.1.=6=6   _fkm  _fkm   F F  uj fspm± uj fspm±o` vft`rnmb. Vj suio`ckuct`o` vft`rnmb. Vj suio`ckuct`   MM ⊄ ⊄ F  F  · ·f f ggnntt`̀  o`cvfx` o`cvfx`
sf sfjprf qufsf sfjprf quf   x,x, yy ∁ ∁ MM, ` ‒sfljfct` dfoamg`‚, ` ‒sfljfct` dfoamg`‚
{{mxmx + (= + (= ∙∙ mm))yy  ;  ; 55 ≧≧ mm ≧≧ ==}}
fstnvfr nctfnrmjfctf o`ctng` fjfstnvfr nctfnrmjfctf o`ctng` fj M M. J`strf quf ` o`ckuct`. J`strf quf ` o`ckuct` { {xx ∁∁ F F   ;; xx ≧≧ r r}}, oamjmg`, oamjmg`
gf i`bm dfoamgm ofctrmgm cm `rnlfj gf rmn`gf i`bm dfoamgm ofctrmgm cm `rnlfj gf rmn`   r r 33 5,  5, ··f f o`o`cvfxcvfx`.`.
FxFxfro·fro·ĽoĽon ̀n ̀ =.=.1.1.=4=4  J`strf quf uj suio`ckuct` J`strf quf uj suio`ckuct`  O  O  gf uj fspm± gf uj fspm±o` o` vft`rnmvft`rnmb ·b ·f f o`cvfx` o`cvfx` sf, sf, ff
s`jfctf, sfs`jfctf, sf
cc

nn:=:=
mmnnxxnn ∁ ∁ O O  sfjprf quf sfjprf quf x x==, . . . , x, . . . , xcc ∁ ∁ O O   ff m m==, . . . , m, . . . , mcc ≦ ≦ 5 smtnsdmzfj5 smtnsdmzfj
cc

nn:=:=
mmnn  : : ==..
FxFxfro·fro·ĽoĽon ̀n ̀ =.=.1.1.=<=<   _fkm  _fkm    MM   uj   uj susuio`cio`ckukuctct` ` gf gf uj uj fsfspm±pm±o` o` vfvft`rnt`rnmbmb    F F . . @ o@ o`c`ckkuucct`t`
o`cv(o`cv(MM), oamjmg` gf), oamjmg` gf  fcv`bt·  fcv`bt· `rnm o`cvfxm gf `rnm o`cvfxm gf   MM, , ··f f gf﬍cng` gf﬍cng` o`j` o`j` m m nctfrsf±nctfrsf±o˒om˒` gf t`g`sm` gf t`g`s
`s suio`ckuct`s o`cvfx`s gf`s suio`ckuct`s o`cvfx`s gf   F F  q quf uf o`o`ct·ct·fjfj   MM. J`strf quf. J`strf quf
(m) M fcv`bt·(m) M fcv`bt·̀rnm o`cvfxm gf qumbqufr suio`ckuct` gf`rnm o`cvfxm gf qumbqufr suio`ckuct` gf   F F  · ·f f uj uj o`ckuo`ckuct` ct` o`cvfx`.o`cvfx`.
(i) Umrm qumbqufr(i) Umrm qumbqufr   MM ⊄⊄ F F ,,
o`cv(o`cv(MM) ) ::

   cc
nn:=:=
θθnnxxnn  ;;
cc
nn:=:=
θθnn : =, o`j : =, o`j   θθcc ≦ ≦ 55, , xxnn ∁ ∁ M, M, nn :  : ==, . . . , c, . . . , c  ff   cc  ∁∁ CC

..
FxFxfro·fro·ĽoĽon ̀n ̀ =.=.1.1.=1=1   _fkmj  _fkmj M M f f I I  suio`ckuct`s o`cvfx`s f o`jpmot`s g` fspm± suio`ckuct`s o`cvfx`s f o`jpmot`s g` fspm±o` c`rjmg`o` c`rjmg`
F F . Ur`vf quf o`cv(. Ur`vf quf o`cv(MM
∫∫
II) ·) ·f f o`o`jpjpmomot`t`..
9=9=
  
FxFxfro·fro·ĽoĽon ̀n ̀ =.=.1.1.=>=>    UrUr`̀vvf f ququf f ` ` dfdfooa` a` gf gf uj uj susuio`io`ckckucuct` t` o`o`ccvvfxfx` ` gf gf uj uj fsfspmpm±±o`o`
c`rc`rjmgjmg` ` ··f f o`cvfo`cvfx`.x`.
FxFxfro·fro·ĽoĽon ̀n ̀ =.=.1.1.9595  _fkm = _fkm = ≧≧  p  p 22 ∓∓..
(m) Ur`vf quf m fxprfss˒(m) Ur`vf quf m fxprfssm˒`m`
d d  p p  ::
     ii
mm
||d d ((xx))|| p p gxgx
 ==
pp
··f f ujm ujm c`rjm c`rjm c` c` fspm±fspm±o` vft`rnmb gms duc±o` vft`rnmb gms duc±o˒o˒̀f`fs s o`o`ctct··ĽcuĽcumsms   d d ; ; XXm,m, iiZZ ∙∙↙↙ HH..
(i) J`strf quf fspm±(i) J`strf quf fspm±o` o` c`rjmc`rjmg` g` g` g` ntfj ntfj (m) (m) c˒cm˒` m` ··f f o`jpbfo`jpbft`.t`.
FxFxfro·fro·ĽoĽon ̀n ̀ =.=.1.1.9=9=  J`strf quf J`strf quf  B B∓∓((Q,Q, ίί,, ¶¶)) ⊄⊄ BB==((Q,Q, ίί,, ¶¶) sf, f s`jfctf sf,) sf, f s`jfctf sf, ¶ ¶((Q Q )) 2 2 ∓∓..
FxFxfro·fro·ĽoĽon ̀n ̀ =.=.1.1.9999    _f _f ((Q,Q, ίί,, ¶¶) ) ··f f uj uj fspm±fspm±o` o` gf gf jfgngm jfgngm ﬍cntm ﬍cntm f f ==  ≧≧    rr   ≧≧    pp, , fcfct˒tm˒`m`
BB p p((Q,Q, ίί,, ¶¶)) ⊄⊄ BBrr((Q,Q, ίί,, ¶¶) ) ff  d d rr ≧ ≧ d d  p p ¶¶((Q Q ))ss `cgf`cgf  s s  : :    ==rr ∙ ∙    == p p ..
FxFxfro·fro·ĽoĽon ̀n ̀ =.=.1.1.9?9?   _fkmj (  _fkmj (Q,Q, ίί,, ¶¶) uj fspm±) uj fspm±o` gf jfgngm ﬍cntm fo` gf jfgngm ﬍cntm f    d d   ∁∁    BB∓∓((Q,Q, ίί,, ¶¶).).
J`strf qufJ`strf quf   d d  ∁ ∁ BB p p((Q,Q, ίί,, ¶¶) pmrm t`g`) pmrm t`g`   pp ≦≦ = f qu= f quf f bnbnjj
 p p↙∓↙∓
d d  p p : : d d ∓∓..
FxFxfro·fro·ĽoĽon ̀n ̀ =.=.1.1.9797  (Gfsnlumbgmgf gf A¨ (Gfsnlumbgmgf gf A῭bgfr lfcfrmbnzmgm) _fkmj`bgfr lfcfrmbnzmgm) _fkmj  c c  ∁∁ CC,, p p,, q,s 3q,s 3 5 tmns 5 tmns
qufquf    ==
 p p
  ++    ==
qq
  ::    ==
ss
.. Ur`vf quf Ur`vf quf

   cc
 k k:=:=
||mm k kii k k||ss
==
ss
≧≧

   cc
 k k:=:=
||mm k k|| p p
 ==
pp
¼¼

   cc
 k k:=:=
||ii k k||qq
==
qq
pmrm qumnsqufr fsombmrfspmrm qumnsqufr fsombmrfs   mm==, . . . , m, . . . , mcc,, ii==, . . . , i, . . . , icc..
FxFxfro·fro·ĽoĽon ̀n ̀ =.=.1.1.9696  J`strf, sfj pmssmr pfb` fspm± J`strf, sfj pmssmr pfb` fspm±o`o`   BB∓∓((Q,Q, ίί,, ¶¶), quf), quf   ∓∓  · ·f f uj uj fsfspmpm±±o`o`
gf Imcmoa.gf Imcmoa.
FxFxfro·fro·ĽoĽon ̀n ̀ =.=.1.1.9494   V  Vj j fbfbfjfjfcfct` t` ((mm k k))
∓∓
 k k:=:=    ∁∁    ∓∓   ·  ·f f oamjmoamjmg` g` gf gf duc±duc±o˒om˒` m` snsnjpjpbfbfs s sfsf
{{mm k k    ;; k k ∁ ∁ CC}} d`r ﬍cnt`. d`r ﬍cnt`. J`strf quf ` o`ckucJ`strf quf ` o`ckuct` gms duc±t` gms duc±o˒o˒̀fs snjpbfs d`rjm uj suio`ckuct``fs snjpbfs d`rjm uj suio`ckuct`
gfcs` gfgfcs` gf   ∓∓..
FxFxfro·fro·ĽoĽon ̀n ̀ =.=.1.1.9<9<   O`csngfrf  O`csngfrf
oo : :

((mm k k))
∓∓
 k k:=:=  ; ;  m m k k ∁ ∁ HH pmrm t`g` pmrm t`g`   kk ∁ ∁ CC f bnj f bnj
 k k↙∓↙∓
mm k k  fxnstf fj fxnstf fj  H H

,,
` o`` o`ckuct` gf ckuct` gf t`gms t`gms ms sfqu˄ms sfquf˄conms ofconms o`cvfrlfctfs d`rjmgms `cvfrlfctfs d`rjmgms p`r p`r fbfjfct`s gffbfjfct`s gf   HH. J`strf. J`strf
qufquf  o o  · ·f f susuiifsfspmpm±±o` dfoamg` gfo` dfoamg` gf   ∓∓..
FxFxfro·fro·ĽoĽon ̀n ̀ =.=.1.1.9191* * _f_fkmkm    H H   uj   uj suio`ckucsuio`ckuct` t` rfbmtnvrfbmtnvmjfctmjfctf f o`jpmot`, o`jpmot`, nst` nst` ··ff    H H    ··ff
o`jpmot`, gfo`jpmot`, gf    ==. . UrUr`̀vvf quf quff    H H  · ·f f bnjnbnjntmg` tmg` f f quf quf pmrm pmrm t`g`t`g`    ζ ζ 33   5 fxnstf  5 fxnstf    ccζζ  ∁∁   CC    tmbtmb
qufquf
supsup
((mmcc))
∓∓
cc:=:=
∁∁
H H 


cc
≦≦
ccζζ
||mmcc||

≧≧ ζ.ζ.
9999
  
FxFxfro·fro·ĽoĽon ̀n ̀ =.=.1.1.9>9>   Gftfrjncf  Gftfrjncf   oo5555   fjfj  o o55..
FxFxfro·fro·ĽoĽon ̀n ̀ =.=.1.1.?5?5   Gftfrjncf  Gftfrjncf   oo5555   fjfj    p p..
FxFxfro·fro·ĽoĽon ̀n ̀ =.=.1.1.?=?=   J`s  J`strf trf qufquf  {{ff==,, ff99, . . ., . . .}}    c˒cm˒` m` ··f f imsf imsf mbl·mbl·firnom firnom (`u (`u gf gf Amjfb) Amjfb) gfgf
 p p,, == ≧≧  p p ≧≧ ∓∓..
FxFxfro·fro·ĽoĽon ̀n ̀ =.=.1.1.?9?9  Ur`vf sf  Ur`vf sf vfrgmgvfrgmgfnr` `u fnr` `u g˄gf˄ f uj uj o`ctrmo`ctrmfxfjpb` sf fxfjpb` sf dmbs`; dmbs`; _f_f   H H  · ·f f uujj
suio`ckuct` dfoamg` gf uj fspm±suio`ckuct` dfoamg` gf uj fspm±o` o` c`rjc`rjmg`, mg`, fcfct˒tm˒` ` suifspm±m` ` suifspm±o` o` XXH H Z lfrmg` p`rZ lfrmg` p`r    H H 
ttmjmji·i·fj fj ··f f dfdfoaoamgmg`̀..
FxFxfro·fro·ĽoĽon ̀n ̀ =.=.1.1.????   _fkmj  _fkmj   F F  uj fspm± uj fspm±o` vft`rnmb c`rjmg` gf gnjfcs˒o` vft`rnmb c`rjmg` gf gnjfcsm˒` ﬍cntm fm` ﬍cntm f   J J    ujuj
suifspm±suifspm±o ̀o ̀ prpr··̀p`prnrn ̀̀ gfgf   F F . J`strf quf fxnstf. J`strf quf fxnstf   yy55 ∁ ∁  F  F    o`jo`j  yy55  :  : = = ff  yy55 ∙∙ xx ≦ ≦ = pmrm = pmrm
t`g`t`g`   xx ∁∁ J J ..
FxFxfro·fro·ĽoĽon ̀n ̀ =.=.1.1.?7?7   (Bnjntm±  (Bnjntm±o˒om˒` g` Bfjm gf Ynfsz) _fkmjm` g` Bfjm gf Ynfsz) _fkmj   F F   :: { {d d  ∁ ∁  O  O X5X5,, =Z =Z ;;  d  d (5) : 5(5) : 5}}
ff   D D    ::

d d  ∁ ∁ F F   ;;
  
==
55
  d d ((tt))gtgt :  : 55

..
(m) Ur`vf quf(m) Ur`vf quf  F  F  · ·f f susuiifsfspmpm±±o` dfoamg` gfo` dfoamg` gf   O O X5X5,, =Z.=Z.
(i) Ur`vf quf(i) Ur`vf quf  D  D  · ·f f susuiifsfspmpm±±o` dfoamg` gfo` dfoamg` gf   F F ..
(o) J`strf quf c˒(o) J`strf quf cm˒` fxnstfm` fxnstf   ll ∁ ∁ F F  tmb quf tmb quf  ll : : = = ff  ll ∙∙ d d  ≦≦ = pmrm t`gm = pmrm t`gm   d d  ∁ ∁ D D ..
FxFxfro·fro·ĽoĽon ̀n ̀ =.=.1.1.?6?6* (@ oui` gf Anbifrt) Oamjf gf* (@ oui` gf Anbifrt) Oamjf gf  oui` gf Anbifrt  oui` gf Anbifrt  ` suio`ckuct` gf ` suio`ckuct` gf   99
d`rjmd`rjmg` g` pfbpfbms mssfqu˄sfquf˄conmfconms (s (mm k k))
∓∓
 k k:=:=   tmns quf tmns quf | |mm k k| ≧| ≧    == k k  pmrm t`g` pmrm t`g`   kk ∁ ∁  C C. Ur`vf quf ` oui`. Ur`vf quf ` oui`
gf gf AnbiAnbifrt frt ··f f o`jpmo`jpmot` ot` fjfj   99..
FxFxfro·fro·ĽoĽon ̀n ̀ =.=.1.1.?4?4  J`strf quf s˒ J`strf quf sm˒` fqunvmbfctfs pmrm uj fspm±m` fqunvmbfctfs pmrm uj fspm±o` gf Imcmoao` gf Imcmoa   F F ;;
(m)(m)   F F  · ·f f sfsfpmpmr·r·mvfb.mvfb.
(i) M i`bm ucnt·(i) M i`bm ucnt·mrnm mrnm dfoamdfoamgmgm   IIF F   :: {{xx ∁∁ F F   ;; xx ≧≧ ==}} · ·f f sfsfppmrmr··mvfb.mvfb.
(o) M fsdfrm ucnt·(o) M fsdfrm ucnt·mrnmmrnm   _ _ F F   :: {{xx ∁∁ F F    ;; xx : : ==}} · ·f f sfsfpmpmrr··mvfb.mvfb.
FxFxfro·fro·ĽoĽon ̀n ̀ =.=.1.1.?<?<   Ur`vf quf  Ur`vf quf  B B∓∓XXm,m, iiZ Z c˒cm˒ ̀m ̀ ··f f sfsfpmpmr·r·mvfb.mvfb.
FxFxfro·fro·ĽoĽon ̀n ̀ =.=.1.1.?1?1   _fkmj  _fkmj F  F  uj fspm± uj fspm±o` gf Imcmoa sfpmr·o` gf Imcmoa sfpmr·mvfb fmvfb f D  D  uj suifspm± uj suifspm±o` o` dfoamdfoamg`g`
gfgf   F F . Ur`vf quf. Ur`vf quf   F/D F/D  · ·f f sfsfpmpmr·r·mvfb.mvfb.
FxFxfro·fro·ĽoĽon ̀n ̀ =.=.1.1.?>?>* * (Yfo(Yfo··Ľpr`om Ľpr`om gm gm gfsnlumbgmgf gfsnlumbgmgf gf gf A¨A῭bgfr) _fkmj (`bgfr) _fkmj (Q,Q, ίί,, ¶¶) uj fspm±) uj fspm±o`o`
gf jfgngmgf jfgngm   ϏϏ-﬍cntm f-﬍cntm f   p,p, q q  ≦ ≦  = tmns quf = tmns quf    ==
 p p
  ++    ==
qq
  : =. Oamjf gf  : =. Oamjf gf   _ _  ` o`ckuct` gf t`gms ms ` o`ckuct` gf t`gms ms
duc±duc±o˒o˒̀fs snjpbfs jfcsur·`fs snjpbfs jfcsur·mvfnsmvfns   d d   quf sf mcubmj d`rm gf uj o`ckuct` gf jfgngm ﬍cntm,  quf sf mcubmj d`rm gf uj o`ckuct` gf jfgngm ﬍cntm,
nst` nst` ··f, f, fxnstfxnstf f uj uj o`ckuo`ckuct`ct` M M ∁ ∁ ί tmb qufί tmb quf ¶ ¶((MM)) 2 2 ∓∓ f f  d  d ((xx) : 5 pmrm t`g`) : 5 pmrm t`g` x x ∁∁ MMoo . _fkm. _fkm
ll ;;   Q Q  ∙ ∙↙↙ YY ujm duc± ujm duc±o˒om˒` jfcsur·m` jfcsur·mvfb tmb quf;mvfb tmb quf;
(n)(n)   dd ll ∁ ∁ BB==((Q,Q, ίί,, ¶¶) pmrm t`gm) pmrm t`gm   d d  ∁ ∁ _ _ ,,
(nn) sup(nn) sup


  
Q Q 
 d d ll


;; d  d  ∁ ∁ _ _ ff  d d  p p  : : ==

22 ∓∓..
Ur`vf qufUr`vf quf   ll ∁ ∁ BBqq((Q,Q, ίί,, ¶¶) ) ff  llqq  : sup : sup
  
Q Q 
 d d ll

;; d  d  ∁ ∁ _ _ ff  d d  p p  : : ==

..
9?9?
  
OOmmpp··ĽĽttuubb ̀̀ 99
@p@pfrfrmg`mg`rfrfs s BnBncfmcfmrfrfs s O`O`ctct··Ľcu`sĽcu`s
@s fspm±@s fspm±o`s o`s c`rjmg`s c`rjmg`s t˄tf˄j fj ujm ujm fstruturm fstruturm mbl·mbl·firnom firnom gf gf fspm±fspm±o` vft`rnmb, eo` vft`rnmb, em qumb fst˒m qumb fstm˒`m`
mss`onmgms ms trmcsd`rjm±mss`onmgms ms trmcsd`rjm±o˒o˒̀fs bncfmrfs, f ujm fstruturm t`p`b·`fs bncfmrfs, f ujm fstruturm t`p`b·̀lnom gf fspm±`lnom gf fspm±o ̀o ̀ j·j·ftftrnrno`o`,,
eem m qumb qumb fstfst˒˒m` m` mssmss`onmg`onmgms ms ms ms duc±duc±o˒o˒̀fs `fs o`co`ctt··ĽcĽcums. ums. MssMssnj nj `s `s j`r﬍j`r﬍sj`sj`s s fcfctrf trf fspfspm±m±o`so`s
c`rjmg`s s˒c`rjmg`s sm˒` ms duc±m` ms duc±o˒o˒̀fs quf s˒`fs quf sm˒` snjubtm` snjubtmcfmmcfmjfctf bncjfctf bncfmrffmrfs f o`ct·s f o`ct·Ľcums. Ľcums. WWmns ducmns duc±±o˒o˒̀fs s˒`fs sm˒`m`
c`rjmbjc`rjmbjfctf oamjmgms fctf oamjmgms gf gf `pfrmg`rfs bncfmrfs `pfrmg`rfs bncfmrfs o`cto`ct··ĽcuĽcu`s. `s. _fcg` _fcg` mssnj, ujmssnj, uj  `pfrmg`r  `pfrmg`r 
bnbncfmr cfmr o`ct·o`ct·ĽcĽcu`u`   g` fspm±  g` fspm±o` c`rjmg`o` c`rjmg`    F F   c` fspm±  c` fspm±o` c`rjmg`o` c`rjmg`    D D , mji`s s`irf ` jfsj`, mji`s s`irf ` jfsj`
o`rp`o`rp`  H H, ·, ·f f ujujm m duduc±c±o˒om˒`m`   W W  ; ;   F F  ∙ ∙↙↙ D D  q quf uf ··f f bnbncfcfmrmr, , nsnst ̀t ̀ ··ff
••   W W ((xx + + yy) ) :: W  W ((xx) +) + W W ((yy) pmrm qumnsqufr) pmrm qumnsqufr  x, x, yy ∁ ∁ F F   ff
••   W W ((mxmx) ) :: mW  mW ((xx) pmrm t`g`) pmrm t`g`  m m ∁∁ HH f qumbqufr f qumbqufr   xx  fjfj   F F 88
f f o`ct·o`ct·ĽcumĽcum, , nstnst` ` ··f, f, pmrpmrm m t`t`g`sg`s   xx55 ∁ ∁  F  F    ff   ζ ζ 33 5 fxnstf 5 fxnstf   η η 33 5 tmb quf 5 tmb quf  W W ((xx)) ∙∙ W W ((xx55)) 2  2 ζζ
sfjprf qufsfjprf quf   xx ∁∁ F F   ff  xx ∙∙ xx55 2 2 η η ..
@ o`c@ o`ckuct` gf kuct` gf t`g`s t`g`s `s `p`s `pfrmg`rfs frmg`rfs bncfmrfs obncfmrfs o`ct`ct··Ľcu`s gfĽcu`s gf  F  F   fjfj D  D   sfr·  sfr·m gfc`tmg` p`rm gfc`tmg` p`r
BB((F,F, D D ).).    ··F obmr` qufF obmr` quf B B((F,F, D D ) ) ··f f uj uj fspfspm±m±o` vft`rnmb s`irfo` vft`rnmb s`irf   HH  o`j ms `pfrm± o`j ms `pfrm±o˒o˒̀fs usumns`fs usumns
gf duc±gf duc±o˒o˒̀fs`fs. . ]um]umcg`cg`   D D    ··f f ` ` o`rp` o`rp` g`s g`s fsombmrffsombmrfs, s, fsorfvfsorfvfj`sfj`s    F F   c` bulmr gf c` bulmr gf B B((F,F, HH),),
oamjmj`s fssf fspm±oamjmj`s fssf fspm±o` gfo` gf  gumb t`p`b·  gumb t`p`b· `lno` gf `lno` gf   F F , `u snjpbfsjfctf, `u snjpbfsjfctf  gumb gf  gumb gf   F F , f gnzfj`s, f gnzfj`s
quf sfus fbfjfct`s s˒quf sfus fbfjfct`s sm˒`m`   ducon  ducon`cm`cmns ns bncbncfmrfs o`cfmrfs o`ctt··Ľcu`Ľcu`s s ..
_fluncg` m bncam gf quf `s j`r﬍sj`s fctrf fspm±_fluncg` m bncam gf quf `s j`r﬍sj`s fctrf fspm±o`s c`rjmg`s s˒o`s c`rjmg`s sm˒` `s `pfrmg`rfsm` `s `pfrmg`rfs
bncfmrfs bncfmrfs o`cto`ct··Ľcu`Ľcu s̀, s, gnzfj`s gnzfj`s quf quf g`ns g`ns fspm±fspm±o`s c`rjmg`so`s c`rjmg`s    F F    ff    D D    s˒sm˒`m`   t`p`b`lnomjfctf   t`p`b`lnomjfctf 
ns`j`rd`s ns`j`rd`s , `u snjpbfsjfctf, `u snjpbfsjfctf   ns`j`rd`s   ns`j`rd`s , , sf sf fxnstnfxnstnr r uj uj `pfrmg`r `pfrmg`r bncfmr bncfmr o`cto`ct··Ľcu` Ľcu` inkft`rinkft`r
W W  ; ;   F F  ∙ ∙↙↙   D D   ouk` `pfrmg`r ncvfrs`  ouk` `pfrmg`r ncvfrs`   W W ∙∙== ;;   D D   ∙∙↙↙   F F   ‟   ‟ ququf f ··f f sfsfjpjprf rf bncbncfmfmr r ‟ ‟ ··f f tmtmji·ji·fjfj
o`cto`ct··Ľcu`. Ľcu`. WWmb `pmb `pfrmg`rfrmg`r    W W   ·  ·f f oamjmg` oamjmg` gfgf   ns`j`r﬍s  ns`j`r﬍sj` j` t`pt`p`b· `b· `lno``lno`, , `u `u snjsnjpbfspbfsjfcjfctftf
ns`j`r﬍sj`ns`j`r﬍sj`..
Vj Vj duduc±c±o˒om˒`m`   d d ;;   F F  ∙ ∙↙↙  D D    ‟ ‟ c˒cm˒` cfofssmrnmjfctf bncfmr ‟ tmb qufm` cfofssmrnmjfctf bncfmr ‟ tmb quf  d d ((xx))   ::  xx  pmrm pmrm
t`g`t`g`    xx  ∁∁    F F   ·  ·f f oamjoamjmgm mgm gfgf   ns`jftrnm   ns`jftrnm . . Vj `pfVj `pfrmgrmg`r bn`r bncfcfmrmr    W W  ; ;   F F   ∙∙↙↙    D D   q  quf uf ··f f ujujmm
ns`jfns`jftrnm trnm ··f f oamjmg` oamjmg` gfgf   ns`jftrnm bncfmr  ns`jftrnm bncfmr . . @isfrvf quf @isfrvf quf t`gm t`gm ns`jftrnm bncfmns`jftrnm bncfmr ·r ·f nckftf nckft`rm`rm
f f o`cto`ct··Ľcum. Ľcum. Vj Vj ns`j`rns`j`r﬍sj` ﬍sj` quf quf ··f f tmji·tmji·fj fj ujm ujm ns`jfns`jftrnm trnm ··f f oamjmgoamjmg` ` gfgf   ns`j`r﬍sj` ns`j`r﬍sj`
nsns`j`j··ftftrnrno`o`, f cfssf oms` gnzfj`s quf `s fspm±, f cfssf oms` gnzfj`s quf `s fspm±o`s s˒o`s sm˒`m`  ns`j`rd`s ns`jftrnomjfctf. ns`j`rd`s ns`jftrnomjfctf.
9797
  
9.9.= = OmOmrmrmototfrfrnznzm±m±o˒o˒̀̀ffs s gg`̀s s `̀ppffrrmmgg`̀rrffs s bbnnccffmmrrffss
oo`̀cctt··ĽĽcucu`̀ss
Yfbfjirfj`s gums obmssfs njp`rtmctfs gf duc±Yfbfjirfj`s gums obmssfs njp`rtmctfs gf duc±o˒o˒̀fs c` o`ctfxt` gf fspm±`fs c` o`ctfxt` gf fspm±o`s o`s j·j·ftrftrno`no`s; s; ujmujm
duc±duc±o˒om˒`m`   d d ;;   J J  ∙ ∙↙↙ C C  fctrf fspm± fctrf fspm±oo s̀ s̀ j·j·fftrtrnono s̀ s̀ ··ff
••   bnpsoantznmcm   bnpsoantznmcm   sf fxnstf ujm o`cstmctf sf fxnstf ujm o`cstmctf   B B 33  5 tmb quf 5 tmb quf   gg((d d ((xx)),, d d ((yy)))) ≧ ≧   BB ¼¼ gg((x,x, yy) pmrm) pmrm
t`g`st`g`s   x,x, yy ∁ ∁ J J 88
••  ucn ucnd`rd`rjfjfjfjfctf ctf o`cto`ct··Ľcum Ľcum   sf pmrm t`g`  sf pmrm t`g`   ζ ζ 33  5 fxnstf 5 fxnstf   η η 33  5 tmb quf 5 tmb quf   gg((d d ((xx)),, d d ((yy))))   2 2 ζζ
sfjprf qufsfjprf quf   x,x, yy ∁ ∁ J J    ff   gg((x,x, yy)) 2  2 η η ..
_mif-sf quf pmrm duc±_mif-sf quf pmrm duc±o˒o˒̀fs fctrf fspm±`fs fctrf fspm±o`s o`s j·j·ftrnoftrno`s, `s, ms ms njpbnonjpbnom±m±o˒o˒̀fs`fs
bnpsoantznmcm :bnpsoantznmcm :⇙⇙  ucnd` ucnd`rjfrjfjfctjfctf of o`ct·`ct·Ľcum Ľcum ::⇙⇙ o` o`ct·ct·ĽcĽcum um ::⇙⇙ o`ct· o`ct·Ľcum Ľcum fj fj uj uj p`p`ct`ct`
s˒sm˒` vfrgmgfnrms f quf, fj lfrmb, t`gms ms njpbnom±m` vfrgmgfnrms f quf, fj lfrmb, t`gms ms njpbnom±o˒o˒̀fs ncvfrsms s˒`fs ncvfrsms sm˒` m` dmbsms. dmbsms. @ @ pr·pr·̀xnj``xnj`
rfsubtmg`, quf rfj`ctm m`s trmimba`s gf D. Ynfsz, j`strm quf t`g`s fssfs o`cofnt`s s˒rfsubtmg`, quf rfj`ctm m`s trmimba`s gf D. Ynfsz, j`strm quf t`g`s fssfs o`cofnt`s sm˒`m`
fqunvmbfctfs c` o`ctfxt` gf `pfrmg`rfs bncfmrfs fctrf fspm±fqunvmbfctfs c` o`ctfxt` gf `pfrmg`rfs bncfmrfs fctrf fspm±o`s c`o`s c r̀jmrjmg`sg`s. . @u sfk@u sfkm, mm, m
bncfmrngmgf, `u fj ·bncfmrngmgf, `u fj ·ubtnjm ncst˄ubtnjm ncstm˄conm mconm m m fstruturm fstruturm mbl·mbl·firnom, firnom, snjpbn﬍om snjpbn﬍om ` ` o`jp`rtmo`jp`rtmjfct`jfct`
t`p`b·t`p`b·̀lno`. `lno`. Mb·Mb·fj fj gnss` gnss` pr`vpr`vmrfj`s mrfj`s `utrms `utrms fqunvfqunvmb˄mbf˄conms fconms quf quf juntjunt̀ ` ··utfnutfns s sfr˒sfrm˒` c`m` c`
gfo`rrfr g` tfxt`.gfo`rrfr g` tfxt`.
Wf`rfjm 9.=.=Wf`rfjm 9.=.=   _fkmj   _fkmj    F F    f f    D D   fspm±  fspm±o`s c`rjmg`s s`irf o`s c`rjmg`s s`irf   HH    f f    W W  ; ;   F F   ∙∙↙↙    D D   bn  bncfcfmr. mr. Ms Ms 
sflunctfs o`cgn±sflunctfs o`cgn±o˒ o˒ `fs s˒ `fs s˒ m` fqunvmbfctfs;m` fqunvmbfctfs;
(m)(m)   W W  · ·f f bnpbnpsoasoantzntznmcnmc`.`.
(i)(i)   W W    ··f f ucnucnd`d`rjfrjfjfjfctf ctf o`ct·o`ct·ĽcĽcu`u`..
(o)(o)   W W   ·  ·f f òò cct·t·ĽĽccuu`̀..
(g)(g)   W W  · ·f f o`ct·o`ct·ĽcĽcu ̀u ̀ fj fj mbmbluj luj p`ctp`ct ̀̀ gf gf   F F ..
(f)(f)   W W  · ·f f o`co`ct·t·ĽcĽcu ̀u ̀ cm cm `r`rnlnlfjfj..
(d) sup(d) sup{{W W ((xx)) ;; x x ∁∁ F F    ff  xx ≧≧ = =}} 22 ∓∓..
(l)(l)  Fxnstf ujm o`cstmctf  Fxnstf ujm o`cstmctf   O O 
 ≦ ≦
55  tmb quf  tmb quf 
  
W W ((xx))
 ≧≧
 O  O 

xx

 pmrm t`g` pmrm t`g`   xx
∁∁
F F ..
Gfj`cstrm±Gfj`cstrm±oo˒˒mm`̀..   Ms njpbnom±  Ms njpbnom±o˒o˒̀fs (m):`fs (m):⇙⇙(i) :(i) :⇙⇙(o) :(o) :⇙⇙(g) s˒(g) sm˒` v·m` v·mbngms c` o`ctfxt` gfmbngms c` o`ctfxt` gf
fspm±fspm±o`o`s s j·j·ftftrnrno`o`s, s, nstnst ̀̀ ··f, f, c˒cm˒` gfpfcgfj gm bncfmrngmgf gfm` gfpfcgfj gm bncfmrngmgf gf   W W ..
(g):(g):⇙⇙(f) _up`cam(f) _up`cam W  W  o o`ct·`ct·ĽcuĽcu ̀c ̀c ̀p ̀p`ct`ct`̀ x x55 ∁ ∁ F F . _fkm. _fkm ζ  ζ 33 5. Fct˒ 5. Fctm˒` fxnstfm` fxnstf η  η 33 5 tmb quf 5 tmb quf
W W ((xx)) ∙∙ W W ((xx55)) 2 2 ζζ sfjprf quf sfjprf quf  xx ∙∙ xx55 2 2 η η . W`jf. W`jf x x ∁∁ F F  tmb quf tmb quf  xx ∙∙ 55 :: xx 2 2 η η ..
Fct˒Fctm˒`m`  ((xx ++ xx55)) ∙∙ xx55 :: xx 2 2 η η . U`rtmct`. U`rtmct`
W W ((xx)) ∙∙ W W (5)(5) :: W W ((xx)) ∙∙ 55 :: W W ((xx)) :: W W ((xx) +) + W  W ((xx55)) ∙∙ W W ((xx55))
:: W W ((xx ++ xx55)) ∙∙ W W ((xx55)) 2 2 ζ,ζ,
pr`vmcg` qufpr`vmcg` quf   W W  · ·f f o`o`ct·ct·ĽcuĽcu ̀̀ cm cm `r`rnlnlfjfj..
(f):(f):⇙⇙(d) Gm o`ctncungmgf gf(d) Gm o`ctncungmgf gf  W  W  cm `rnlfj fxnstf cm `rnlfj fxnstf   η η 33 5 tmb quf 5 tmb quf  W W ((xx)) 22 = sfjprf quf = sfjprf quf
xx    2 2 η η . . _f_f  xx  ≧≧    =,=,   ηη
99
xx    2 2 η η , f fct˒, f fctm˒`m`    ηη
99
W W ((xx))    ::  W W (( ηη
99
xx))    22   =.   =. NsNss` pr`s` pr`vvm qufm quf
supsup{{W W ((xx)) ;; x x ∁∁ F F   ff  xx ≧≧ ==}} ≧≧
   99
ηη    22 ∓∓..
9696
  
(d) :(d) :⇙⇙(l) Umrm(l) Umrm  x x ∁∁ F F ,,  xx : 5, tfj`s: 5, tfj`s
W W ((xx))
xx    ::


W W 

   xx
xx


≧≧ supsup{{W W ((yy)) ;; yy ≧≧ = =}},,
f p`rtmct`f p`rtmct`  W W ((xx)) ≧ ≧ (sup (sup{{W W ((yy)) ; ;  yy ≧ ≧ = =}}))xx  pmrm t`g` pmrm t`g`   xx  : 5. @ rfsubtmg` sfluf: 5. @ rfsubtmg` sfluf
p`ns p`ns fssm fssm gfsnlumbgmgf gfsnlumbgmgf ··f f trnvnmbjfctf trnvnmbjfctf vfrn﬍omgm vfrn﬍omgm pmrmpmrm   xx :  : 5.5.
(l) :(l) :⇙⇙(m) Gmg`s(m) Gmg`s   xx==,, xx99 ∁ ∁ F,F,
W W ((xx==)) ∙∙ W W ((xx99)) :: W W ((xx== ∙∙ xx99)) ≧≧ O O xx== ∙∙ xx99 ,,
f p`rtmct`f p`rtmct`   W W  · ·f f bnpsoantbnpsoantznmc` znmc` o`j o`j o`csto`cstmctfmctf   O O ..
O`O`r`br`b··mrnmrn ̀̀ 9.9.=.=.99   _fkm   _fkm  W  W  ; ;   F F  ∙ ∙↙↙ D D  uj `pfrmg`r bncfmr inkft`r fctrf fspm± uj `pfrmg`r bncfmr inkft`r fctrf fspm±o`s c`rjmg`s.o`s c`rjmg`s.
Fct˒ Fct˒ m`m`    W W    ··f f uj uj ns`j`rns`j`r﬍sj` ﬍sj` sf, sf, f f s`jfcs`jfctf, tf, sf sf fxnstfxnstfj fj oo`cstmc`cstmctfs tfs    O O ==,, O O 99    33   55   tmns quf  tmns quf 
O O ==xx ≧≧ W W ((xx)) ≧≧ O  O 99xx  pmrm t`g` pmrm t`g`   xx ∁∁ F F ..
@isfrvm±@isfrvm±o˒om˒ ̀m ̀ 9.9.=.=.??  Gm o`cgn± Gm o`cgn±o˒om˒` (d) g` Wf`rfjm 9.=.= sfluf quf `s `pfrmg`rfs bncfmrfsm` (d) g` Wf`rfjm 9.=.= sfluf quf `s `pfrmg`rfs bncfmrfs
o`cto`ct··Ľcu`s Ľcu`s s˒sm˒` m` fxmtmjfctf fxmtmjfctf mqufbfs mqufbfs quf quf trmcsd`rjmj trmcsd`rjmj o`ckuct`s o`ckuct`s bnjntmg`s bnjntmg`s c` c` g`jg`j··Ľcn`Ľcn`
fj fj o`ckuct`s o`ckuct`s bnjntmg`s bnjntmg`s c` c` o`ctrmg`jo`ctrmg`j··Ľcn`. Ľcn`. U`r nss` U`r nss` `s `s `pfrmg`r`pfrmg`rfs fs bncfmrfs bncfmrfs o`cto`ct··Ľcu`sĽcu`s
s˒sm˒` m` jjununtms tms vvfzffzfs s oaoamjmjmg`mg s̀ s gfgf   `p  `pfrmg`rfrmg`rfs fs bncfmrbncfmrfs fs bnjntmg`s bnjntmg`s . . MMpfpfssmmr gf sfr gf sfr ujr ujmm
tfrjnc`btfrjnc`b̀ lnm mjpbmjfctf gnducgngm, `lnm mjpbmjfctf gnducgngm, c˒cm˒` m mg`tmrfj`s p`r sfr nco`csnstfctf o`j mm` m mg`tmrfj`s p`r sfr nco`csnstfctf o`j m
c`±c`±o˒om˒` gf duc±m` gf duc±o˒om˒` bnjntmgm m` bnjntmgm g` Fxfjpg` Fxfjpb` =.=.9. b` =.=.9. B·B·m ujm m ujm duc±duc±o˒om˒` m` ··f f bnjnbnjntmgtmgm m sf sf sum sum njmlnjmlfj fj ··ff
bnjntmgm8 mqun bnjntmgm8 mqun uj `puj `pfrmg`r ·frmg`r ·f bnjntmg` f bnjntmg` sf trmcsf trmcsd`rjm o`ckusd`rjm o`ckuct`s bnjntmg`s fj ct`s bnjntmg`s fj o`ckuct`so`ckuct`s
bnjntmbnjntmg`s. g`s. Umrm fvntmr Umrm fvntmr mjinmjinlungmgf sflunrfj`s lungmgf sflunrfj`s o`j o`j ` ` tfrj` tfrj` `pfrmg`r `pfrmg`r bncfmr o`ctbncfmr o`ct··ĽcuĽcu`̀
f o`j m c`±f o`j m c`±o˒om˒` gf duc±m` gf duc±o˒om˒` bnjntmgm o`j` mqufbm quf tfj njmlfj bnjntmgm.m` bnjntmgm o`j` mqufbm quf tfj njmlfj bnjntmgm.
U`r `utr` bmg`, m jfsjm o`cgn±U`r `utr` bmg`, m jfsjm o`cgn±o˒om˒` (d) c`s fcsncm o`j` c`rjmr ` fspm±m` (d) c`s fcsncm o`j` c`rjmr ` fspm±o`o` B B((F,F, D D ))
g`s g`s `pfrmg`r`pfrmg`rfs fs bncfmrfs bncfmrfs o`cto`ct··Ľcu`s Ľcu`s gfgf   F F    fjfj   D D ;;
Ur`p`sn±Ur`p`sn±o˒om˒ ̀m ̀ 9.9.=.=.77   _fkmj   _fkmj   F F    f f   D D   fspm±  fspm±o`s c`rjmg`s.o`s c`rjmg`s.
(m)(m)  M fxprfss˒  M fxprfss˒ m`m`

W W 

: sup: sup
{{
W W ((xx))

;; x x
∁∁
F F    ff
  
xx
 ≧≧
 = =
}}
gf﬍cf ujm c`rjm c` fspm±gf﬍cf ujm c`rjm c` fspm±o`o` B B((F,F, D D ))..
(i)(i)  W W ((xx)) ≧≧ W W  ¼¼ xx  pmrm t`g`s  pmrm t`g`s   W W  ∁ ∁ BB((F,F, D D ))   f f   xx ∁∁ F F ..
(o)(o)   _f _f   D D   d`r Imcmoa, fct˒   d`r Imcmoa, fct˒ m`m` B B((F,F, D D ))  tm tmjiji··fj fj ··f f IImcmcmomoa.a.
Gfj`cstrm±Gfj`cstrm±oo˒˒mm`̀..    GfGfnxnxmj`mj s̀ (m) f s (m) f (i) m omrl(i) m omrl` g` ` g` bfbfntnt`r. `r. Ur`Ur v̀fvfj`j`s (o). s (o). _fk_fkm (m (W W cc))
∓∓
cc:=:=
ujm ujm sfqu˄sfquf˄conm fconm gf gf Omuoay Omuoay fjfj BB((F,F, D D ). Gmg`). Gmg` ζ  ζ 33 5 fxnstf 5 fxnstf c c55 ∁ ∁ CC tmb quftmb quf W W cc∙∙W W jj ≧≧ ζ ζ
sfjprf qufsfjprf quf   c,c, jj  ≦≦ cc55. B`l`. B`l`
W W cc((xx)) ∙∙ W W jj((xx)) :: ((W W cc ∙∙ W W jj)()(xx)) ≧≧ W W cc ∙∙ W W jj ¼¼ xx ≧≧ ζ ζxx   (9.=)  (9.=)
pmrm t`g`spmrm t`g`s   xx ∁ ∁   F F    ff   c,c, jj ≦ ≦  cc55. . _flu_fluf quf pmrm omf quf pmrm omgmgm   xx ∁ ∁   F F , , m m sfqsfqu˄uf˄cofconm nm ((W W cc((xx))))∓∓cc:=:=   ··ff
gf Omuoay fjgf Omuoay fj  D  D , b`l` o`cvfrlfctf p`ns, b`l` o`cvfrlfctf p`ns   D D  · ·f Imf Imcmoa. cmoa. U`gfjU`gfj`s `s fct˒fctm˒` gf﬍cnrm` gf﬍cnr
W W  ; ;   F F 
 ∙ ∙↙↙
D D , , W W ((xx) : bnj) : bnj
cc↙∓↙∓
W W 
cc
((xx))..
9494
  
M bncfmrngmgf gfM bncfmrngmgf gf    W W    sflusfluf f gms pr`prngms pr`prnfgmgfgmgfs g`s fs g`s bnjbnjntfntfs. s. DDmzfcmzfcg`g`   jj ∙ ∙↙ ↙ ∓∓  fj (9.=) fj (9.=)
`itfj`s`itfj`s
((W W cc ∙∙ W W )()(xx)) :: W W cc((xx)) ∙∙ W W ((xx)) ≧≧ ζζxx   (9.9)  (9.9)
pmrm t`g`spmrm t`g`s   xx
∁∁
F F    ff   cc
≦≦
cc
55
. Fj pmrtnoubmr,. Fj pmrtnoubmr,
((W W cc55 ∙ ∙ W W )()(xx)) :: W W cc55((xx)) ∙∙ W W ((xx)) ≧≧ ζζxx
pmrpmrm m t`g`t`g`    xx   ∁∁    F F , ` quf c`s lmrmctf quf (, ` quf c`s lmrmctf quf (W W   ∙∙   W W cc55))   ∁ ∁ BB((F,F, D D ). ). UU`r`rtmtmcct`t`    W W     ::
((W W  ∙ ∙  W  W cc55) ) ++ W  W cc55  ∁ ∁ BB((F,F, D D ). ). Gf (9.9) Gf (9.9) sfluf tmji·sfluf tmji·fj fj qufquf  W W cc ∙ ∙ W  W   ≧≧    ζζ   pmrm t`g`  pmrmt`g`
cc ≦≦ cc55, f mssnj sfluf quf, f mssnj sfluf quf   W W cc ∙ ∙↙↙ W W    fjfj B B((F,F, D D ).).
C` C` FxfroFxfro··Ľon` Ľon` 9.<.1 9.<.1 ` ` bfnt`r bfnt`r pr`vpr`vmr·mr·m m v·v·mrnms fxprfss˒mrnms fxprfss˒̀fs mbtfrcmtnvms pmrm`fs mbtfrcmtnvms pmrm  W W . . VVjjmm
quf sfr·quf sfr·m fspfonmbjfctf ·m fspfonmbjfctf ·utnutnb ·b ·f f m m sflsflunctunctf;f;
W W  : ncd : ncd {{O O    ;;  W W ((xx)) ≧≧ O  O xx  pmrm t`g` pmrm t`g`   xx ∁∁ F F }}..
C` oms` gf ducon`cmns bncfmrfs, m c`rjm gf `pfrmg`rfs sf trmcsd`rjm fjC` oms` gf ducon`cmns bncfmrfs, m c`rjm gf `pfrmg`rfs sf trmcsd`rjm fj
ϖϖ : sup: sup{|{|ϖϖ((xx))|| ;; x x ∁∁ F F    ff  xx ≧≧    ==}}
pmrm t`g` fspm±pmrm t`g` fspm±o` c`rjmg`o` c`rjmg`   F F  f t`g` ducon`cmb f t`g` ducon`cmb   ϖϖ ∁ ∁ F F . Gm Ur`p`sn±. Gm Ur`p`sn±o˒om˒` 9.=.7(o) tfj`sm` 9.=.7(o) tfj`s
O`O`r`br`b··mrnmrn ̀̀ 9.9.=.=.66  @ gumb  @ gumb   F F   gf qumbqufr fspm± gf qumbqufr fspm±o` c`rjmg`o` c`rjmg`   F F  · ·f f uj uj fsfspm±pm±o` gf Imcmoa.o` gf Imcmoa.
99..9 9 FFxxffjjppbb`̀ss
··F obmr` quf, fj qumbqufr fspm±F obmr` quf, fj qumbqufr fspm±o` c`rjmg`o` c`rjmg`   F F , m ngfctngmgf fj, m ngfctngmgf fj   F F  · ·f f uj uj `pf`pfrmg`r rmg`r bncfmrbncfmr
o`cto`ct··Ľcu` gf Ľcu` gf c`rjm c`rjm =, =, f f quf, quf, fctrf g`ns fctrf g`ns fspm±fspm±o`s c`rjmg`s qumnsqufr, ` `pfrmg`r cub`o`s c`rjmg`s qumnsqufr, ` `pfrmg`r cub`
··f f uj uj `pf`pfrmg`rmg`r r bncfmbncfmr r o`cto`ct··Ľcu` Ľcu` gf gf c`rjm c`rjm 5. 5. WWmji·mji·fj fj ··f f junt` junt` snjpbfsnjpbfs s vfrn﬍omvfrn﬍omr r quf quf sfsf
ϖϖ ∁∁ F F    ff   yy ∁ ∁ D D , fct˒, fctm˒`m`
ϖϖ ⊜⊜ yy ;;   F F  ∙ ∙↙↙ D D , , ϖϖ ⊜⊜ yy((xx) ) :: ϖ ϖ((xx))y,y,
··f f uj uj `p`pfrmfrmg`r g`r bncfbncfmr mr o`ct·o`ct·Ľcu` Ľcu` gf gf c`rc`rjmjm ϖϖ¼¼yy. . TTfkmj`s mblucs fxfjpfkmj`s mblucs fxfjpb`s fj fspm±b`s fj fspm±o`so`s
fsfsppfo·fo·Ľ﬍Ľ﬍o`o`s;s;
Fxfjpb` 9.9.=Fxfjpb` 9.9.=   _fkm =  _fkm = ≧ ≧   p p 22 ∓ ∓. . Fso`bam Fso`bam ujm ujm sfqu˄sfquf˄conm fconm ((ii k k))∓∓ k k:=:= ∁ ∁    p p   f o`csngfrf ` f o`csngfrf `
`pfrmg`r`pfrmg`r
W W  ; ;   ∓∓ ∙ ∙↙↙  p p    , , W W ((((mm k k))∓∓ k k:=:=) ) : : ((mm k kii k k))∓∓ k k:=:=..
M bncfmrngmgf gfM bncfmrngmgf gf   W W  · ·f nf njfgjfgnmtnmtm. m. GfGf
W W ((((mm k k))∓∓ k k:=:=))

 p p
 : :
   ∓∓
 k k:=:=
||mm k kii k k|| p p
==/p/p
≧≧
   ∓∓
 k k:=:=
||ii k k|| p p
==/p/p
¼¼ supsup
 k k
||mm k k||
::

((ii k k))
∓∓
 k k:=:=

 p p
¼¼ 
((mm k k))
∓∓
 k k:=:=
∓∓
9<9<
  
pmrm pmrm t`gt`gm m sfqu˄sfquf˄conm fconm ((mm k k))
∓∓
 k k:=:= ∁ ∁ ∓∓, , o`co`cobu·obu·Ľj`Ľj`s s qufquf  W  W   ··f f oo`c`ct·t·ĽcĽcu ̀u ̀ f f ququff  W W  ≧≧ ((ii k k))∓∓ k k:=:= p p..
W`jmcg`W`jmcg`    xx    : : (=(=,, ==,, ==, . . . ,, . . . , ))   ∁∁    ∓∓   tfj`s  tfj`s  xx∓∓    : : = = ff  W W ((xx))    ::    ((ii k k))∓∓ k k:=:= p p, , b`b`l`l`
W W     ::  ((ii k k))∓∓ k k:=:= p p. . FstFstf `pfrf `pfrmg`mg r̀r    W W   ·  ·f f oamjoamjmg` mg` gfgf   `pfrmg`r gnml`cmb   `pfrmg`r gnml`cmb   p  pfbm fbm sfqsfqu˄uf˄cofconmnm
((ii k k))
∓∓
 k k:=:=..
Fxfjpb` 9.9.9Fxfjpb` 9.9.9    WWrmcsp`rtmcg` ` fxfjpb` monjm pmrm `s fspm±rmcsp`rtmcg` ` fxfjpb` monjm pmrm `s fspm±o`s gf duc±o`s gf duc±o˒o˒̀fs, `fs, o`csngfro`csngfrff
== ≧≧  p  p 22 ∓∓  f fso`bam ujm duc± f fso`bam ujm duc±o˒om˒`m`   ll ∁ ∁ BB p pX5X5,, =Z. Mssnj o`j` mctfs, ` `pfrmg`r=Z. Mssnj o`j` mctfs, ` `pfrmg`r
W W  ; ;   O O X5X5,, =Z=Z  ∙∙↙↙ BB p pX5X5,, =Z=Z  , , W W ((d d ) ) :: d d l,l,
··f f bncbncfmfmr, r, o`o`ct·ct·ĽcuĽcu ̀̀ ff  W W    ::  ll p p. . CfCfststf omsf oms`̀   W W  · ·f f oamjoamjmg` mg` gfgf  `pfrmg`r jubtnpbnom± `pfrmg`r jubtnpbnom±o˒ o˒ m`m`
pfbm duc±pfbm duc±o˒om˒`m`  l l ..
C` C` FxfroFxfro··Ľon` Ľon` 9.<.9 9.<.9 ` ` bfnt`r bfnt`r o`jpr`vo`jpr`vmr·mr·m quf t`g` `pfrmg`r bncfmr gf﬍cng` fj ujm quf t`g` `pfrmg`r bncfmr gf﬍cng` fj uj
fspm±fspm±o` c`rjmg` gf gnjfcs˒o` c`rjmg` gf gnjfcsm˒` m` ﬍cntm ﬍cntm ··f f o`cto`ct··Ľcu`. Ľcu`. TTfrfj`frfj`s s m m sflunsflunr, r, f f o`j o`j jmns jmns ˄˄fcdmsfcdmsff
mncgm c` mncgm c` FxfroFxfro··Ľon` 9.<.?, Ľon` 9.<.?, quf cfj quf cfj sfjprf `psfjprf `pfrmg`rfs frmg`rfs bncfmrfs fj bncfmrfs fj fspm±fspm±o`s gf gnjfcs˒o`s gf gnjfcsm˒`m`
nc﬍nc﬍cntm cntm s˒sm˒` m` o`cto`ct··Ľcu`Ľcu s̀. s. Fssm Fssm ··f f m m gndfrfcgndfrfc±±om i·om i·msnmsnom om fcfctrftrf    ··MbMblfilfirm rm BnBncfmcfmr r f f McMc··mbnsfmbnsf
DDucon`cmucon`cmb. b. _f _f ` ` bfnt`r njmlncm quf bfnt`r njmlncm quf pmrm uj pmrm uj `pfrmg`r bncfmr `pfrmg`r bncfmr sfr gfso`ctsfr gfso`ct··Ľcu` fbf Ľcu` fbf gfvfgfvf
sfr mbl` junt` o`jpbnomg`, ofrtmjfctf sf surprffcgfr·sfr mbl` junt` o`jpbnomg`, ofrtmjfctf sf surprffcgfr·m o`j m snjpbnongmgf g` fxfjpb`m o`j m snjpbnongmgf g` fxfjpb`
m sflunr.m sflunr.
Fxfjpb` 9.9.?Fxfjpb` 9.9.?   O`c  O`csngsngfrf frf ` ` suio`suio`ckucckuct`t`  U U X5X5,, =Z gf=Z gf    O O X5X5,, =Z =Z d`rd`rjmgjmg` ` pfbpfbms ms duduc±c±o˒o˒̀fs`fs
p`bnc`jnmns (`u p`bnc˄p`bnc`jnmns (`u p`bnc˄̀jn`s).`jn`s).    ··F obmr` qufF obmr` quf U  U X5X5,, =Z =Z ··f f uj uj suisuifspmfspm±±o` vft`rnmb gfo` vft`rnmb gf    O O X5X5,, =Z,=Z,
b`lb`l` ` ··f f uj uj fspfspm±m±o` c`rjmg` o`j m c`rjmo` c`rjmg` o`j m c`rjm  ¼¼ ∓∓  afrgmgm gf afrgmgm gf O  O X5X5,, =Z.=Z.    ··F F obmobmr` r` tmji·tmji·fj fj qufquf
` `pfrmg`r gfrnvm±` `pfrmg`r gfrnvm±o˒om˒`m`
W W  ; ; U  U X5X5,, =Z=Z  ∙∙↙↙ U U X5X5,, =Z=Z  , , W W ((d d ) ) :: d  d   : gfrnvmgm gf : gfrnvmgm gf   d,d,
··f bncf bncfmr. fmr. _up_up`cam `cam qufquf  W  W  sf sfkm o`km o`ctct··Ľcu`. Ľcu`. Cfssf Cfssf oms` oms` fxnstffxnstf  O  O  tmb quf tmb quf  W W ((d d ))∓∓ ≧ ≧ O O d d ∓∓
pmrm t`g` p`bnc˄pmrm t`g` p`bnc˄̀jn``jn` d  d  ∁ ∁ U U X5X5,, =Z. Umrm omgm=Z. Umrm omgm  c c  ∁∁ CC t`jf t`jf  d  d cc ∁ ∁ U U X5X5,, =Z =Z m m duduc±c±o˒om˒` gmgm p`rm` gmgm p`r
d d cc((tt) ) :: t t
cc. Fct˒. Fctm˒`m`
cc : : d d cc∓∓   :: W W ((d d cc))∓∓ ≧ ≧ O O d d cc∓∓  : :  O  O 
pmrm t`g`pmrm t`g`   cc ∁∁ CC, f nss` o`c﬍lurm ujm o`ctrmgn±, f nss` o`c﬍lurm ujm o`ctrmgn±o˒om˒`, b`l`m`, b`l`   W W  · ·f f ggffssoo`̀ct·ct·ĽĽcucu`̀..
9.? 9.? @ W@ Wf`rf`rfjm fjm gf gf ImcImcmomoa‟_a‟_tfntfncamcamusus
Vj Vj g`s g`s prnprnconconpmpmns ns j`j`tntnv`v s̀ s g` g` susuofsofss` s` gm gm McMc··mbnsf mbnsf DDucon`cmb ucon`cmb ··f f m m fxnst˄fxnstf˄conm fconm gf gf ujuj
o`ckuct` gf tf`rfjms fstruturmns quf, m` jfsj` tfjp` fj quf ﬉fxninbnzmj m tf`rnmo`ckuct` gf tf`rfjms fstruturmns quf, m` jfsj` tfjp` fj quf ﬉fxninbnzmj m tf`rnm
f pfrjntfj sum nctfrm±f pfrjntfj sum nctfrm±o˒om˒` m` o`j o`j gnvfgnvfrsms rsms `utrms ·`utrms ·mrfms, mrfms, m m t`rcmj rfpbftm t`rcmj rfpbftm gf gf fxfjpbfxfjpb̀ s`s
ncnctfrftfrfssmssmctctfs. fs. O`jfO`jf±±omrfj`s m gfsorfvfr f gfj`cstrmr fssfs tf`rfjms cfstm sf±omrfj`s m gfsorfvfr f gfj`cstrmr fssfs tf`rfjms cfstm sf±o˒om˒` m` ff
tfrjncmrfj`s mpfcms c` pr·tfrjncmrfj`s mpfcms c` pr·̀xnj` `xnj` ompomp··Ľtub` Ľtub` o`j o`j ` ` WWf`rfjm f`rfjm gf gf Amac‟ImAmac‟Imcmoa cmoa f f sumssums
mpbnom±mpbnom±o˒o˒̀fs.`fs.
Junt`s g`s prnconpmns tf`rfjms fj mc·Junt`s g`s prnconpmns tf`rfjms fj mc·mbnsf fst˒mbnsf fstm˒` mss`onmg`s m mbluj tnp` gf o`ctr`bfm` mss`onmg`s m mbluj tnp` gf o`ctr`bf
ucnd`rjf imsfmg` fj anp·ucnd`rjf imsfmg` fj anp·̀tfsfs p`ctumns. Vj fxfjpb` ob·`tfsfs p`ctumns. Vj fxfjpb` ob·mssno` ·mssno` ·f f ` ` dmt` dmt` gf gf quf quf duc±duc±o˒o˒̀fs`fs
9191
  
o`cto`ct··Ľcums Ľcums fj fj o`ckuo`ckuct`s ct`s o`jpmo`jpmot`s ot`s s˒sm˒` m` ucnd`rjfjfctf ucnd`rjfjfctf o`cto`ct··Ľcums. Ľcums. @ @ prnconpmb prnconpmb rfsubtmg`rfsubtmg`
gfstm sf±gfstm sf±o˒om˒`, ` m`, ` WWf`rfjm f`rfjm gf Imcmgf Imcmoa‟_tfncmamus, lmroa‟_tfncmamus, lmrmctf quf mctf quf ujm dmj·ujm dmj·Ľbnm gf Ľbnm gf `pfrmg`rf`pfrmg`rfss
bncfmrfs bncfmrfs o`cto`ct··Ľcu`s Ľcu`s ··f f ucnd`rjfjfctf ucnd`rjfjfctf bnjntmgm bnjntmgm sfjprf sfjprf quf quf d`r d`r p`ctumbjfctf p`ctumbjfctf bnjntmgm.bnjntmgm.
Njp`rtmctfs Njp`rtmctfs o`csfqu˄o`csfquf˄conms fconms gfstf gfstf tf`rfjm tf`rfjm sfr˒sfrm˒` m` vnstms m vnstms m sflunr f sflunr f tmjitmji··fj fj fj fj v·v·mrn`smrn`s`ut`utr`s p`ctr`s p`ct`s g` `s g` tfxtfxt`t`. . UmUmrm rm susum m gfjgfj`c`cstrstrm±m±o˒om˒` prfonsmj`s g` sflunctf ob·m` prfonsmj`s g` sflunctf ob·mssno` gmmssno` gm
t`p`b`lnm g`s fspm±t`p`b`lnm g`s fspm±o`o`s s j·j·ftftrnrno`o`s;s;
Wf`rfjm 9.?.=Wf`rfjm 9.?.=   (W  (Wf`rff`rfjm jm gf gf ImnImnrf)rf)   _fkmj   _fkmj    ((J,J, gg))   uj   uj fspfspm±m±o` o` j·j·ftrnoftrno` ` o`jpbfo`jpbft` t` f f 
((D D cc))
∓∓
cc:=:=  ujm  ujm sfqusfqu˄˄fconm fconm gf gf suiosuiò ckuct`s `ckuct`s dfoamg`s dfoamg`s gf gf    J J   tmns quf   tmns quf    J J    ::
∓∓
cc:=:=
D D cc. . FcFctt˒ ˒ m`m`
fxnstf fxnstf   cc55 ∁ ∁ CC  tmb quf  tmb quf   D D cc55  tfj nctfrn`r c˒  tfj nctfrn`r c˒ m`-vmzn`.m`-vmzn`.
Gfj`cstrm±Gfj`cstrm±oo˒˒mm`̀..  Gfc`tfj`s p`r nct( Gfc`tfj`s p`r nct(MM) ` nctfrn`r gf uj suio`ckuct`) ` nctfrn`r gf uj suio`ckuct` MM gfgf J J . Dm±. Dm±omj`somj`s
p`r p`r mismisurgurg`, `, f f pmpmrm rm nsnss` s` supsup`ca`camj`mj s̀ s ququf f ncnct(t(D D cc) ) ::    ∆∆   pmr  pmrm m t`g`t`g`    cc. . OaOamjmjmcmcg`g`
MMcc  : : ((D D cc))
oo : : ((J J  ∙ ∙ D D cc), omgm), omgm  M Mcc  · ·f f mimifrfrt ̀t ̀ ff
MMcc  : : ((D D cc))oo : (nct(: (nct(D D cc))))
oo :: ∆∆oo :: J  J 
pmrm t`g`pmrm t`g` c c. Fj pmrtnoubmr, omgm. Fj pmrtnoubmr, omgm M Mcc · ·f f cc˒˒m`-vmzn`. Fso`bamm`-vmzn`. Fso`bam x x== ∁ ∁ MM== f usf ` dmt` gf f usf ` dmt` gf M M== sfr sfr
mifmifrt` prt` pmrm mrm lmrmctlmrmctnr m fnr m fxnst˄xnstf˄conm fconm gf 5gf 5  2  2 η η ==  22 = tmb quf m i`bm dfoamgm gf ofctr` = tmb quf m i`bm dfoamgm gf ofctr` x x== f rmn` f rmn`η η ==, gfc`tmgm p`r, gfc`tmgm p`r   IIXXxx==,, η η ==Z, fst·Z, fst·m o`ctngm fjm o`ctngm fj   MM==. . GfGf   MM99   :: J  J   tfj`s  tfj`s   MM99 ∮∮ II((xx==,, η η ==))  :: ∆ ∆,,
`cgf`cgf   II((xx==,, η η ==) gfc`tm m i`bm mifrtm gf ofctr`) gfc`tm m i`bm mifrtm gf ofctr`   xx==  f rmn` f rmn`    η η ==. . F oF o`̀jj`̀   MM99 ∮ ∮ II((xx==,, η η ==) ) ··ff
mifrt`, fxnstfjmifrt`, fxnstfj x x99 ∁ ∁ MM99 f  f 55 2  2 η η 99  22    ==99  tmns quf tmns quf I IXXxx99,, η η 99ZZ ⊄⊄ MM99∮∮II((xx==,, η η ==)) ⊄⊄ MM99∮∮IIXXxx==,, η η ==Z.Z.
O`ctncumcg` O`ctncumcg` ` ` pr`ofss` pr`ofss` o`cstruo`cstru··Ľj`s Ľj`s ujm ujm sfqu˄sfquf˄conm fconm ((xxcc))
∓∓
cc:=:=    fjfj    J J   f   f ujm ujm sfqsfqu˄uf˄cofconmnm
((η η cc))
∓∓
cc:=:=  gf c· gf c·ujfr`s rfmns tmns qufujfr`s rfmns tmns quf
55 2  2 η η cc 2 2
   ==
cc
   ff    IIXXxxcc+=+=,, η η cc+=+=ZZ ⊄⊄ MMcc+=+= ∮∮ IIXXxxcc,, η η ccZZ
pmrm t`g`pmrm t`g`    cc. . GGmmgg`̀    ζ ζ 33   5 fso`bam  5 fso`bam    cc55   ∁∁    CC   tmb quf  tmb quf    cc55    33    99ζζ . . _f_f    j,j, cc  ≦≦    cc55, o`j`, o`j`
IIXXxxcc,, η η ccZZ ∮∮ IIXXxxjj,, η η jjZZ ⊄⊄ IIXXxxcc55,, η η cc55Z, tfj`sZ, tfj`s
gg((xxjj,, xxcc)) ≧≧ gg((xxjj,, xxcc55) +) + gg((xxcc,, xxcc55)) ≧≧ 99η η cc55   22
   99
cc55
2 2 ζ.ζ.
Nss` j`strm quf (Nss` j`strm quf (xxcc))∓∓cc:=:= · ·f f gf gf OmOmuoay uoay fjfj  J  J , b`l` o`cvfrlfctf. Gnlmj`s, b`l` o`cvfrlfctf. Gnlmj`s  x xcc ∙ ∙↙↙ xx ∁∁ J J ..
_fkm_fkm    cc ∁ ∁   CC. . OO`j`j`̀    xxjj  ∁∁    IIXXxxjj,, η η jjZZ ⊄ ⊄    IIXXxxcc,, η η ccZ pmrm t`g`Z pmrm t`g`    jj ≦ ≦    cc    ff    IIXXxxcc,, η η ccZ Z ··f f uujj
o`ckuct` dfoamg`, sfluf qufo`ckuct` dfoamg`, sfluf quf   xx ∁∁ IIXXxxcc,, η η ccZZ ⊄⊄ MMcc. U`rtmct`. U`rtmct`
xx ∁∁
∓∓
cc:=:=
MMcc  ::
∓∓
cc:=:=
((D D cc))
oo ::
 ∓ ∓
cc:=:=
D D cc
oo
:: J  J oo :: ∆∆,,
o`ctrmgn±o`ctrmgn±o˒om˒` fstm quf o`cobun m gfj`cstrm±m` fstm quf o`cobun m gfj`cstrm±o˒om˒`.m`.
Wf`rfjm 9.?.9Wf`rfjm 9.?.9  (Wf`rfjm gf Imcmoa‟_tfncamus) (Wf`rfjm gf Imcmoa‟_tfncamus)   _fkmj   _fkmj   F F   uj fspm±  uj fspm±o` gf Imcmoa,o` gf Imcmoa,    D D 
uj fspm±uj fspm±o` c`rjmg` f o` c`rjmg` f    ((W W nn))nn∁∁N N   ujm   ujm dmjdmj··Ľbnm Ľbnm gf gf `pfrmg`r`pfrmg`rfs fs fj fj  B B((F F 88 D D ))   smtnsdmzfcg` m   smtnsdmzfcg` m 
o`cgn±o`cgn±o˒ o˒ m` gf quf pmrm omgm m` gf quf pmrm omgm   xx ∁∁ F F   fxnstf  fxnstf   O O xx  2 2 ∓∓  tmb quf  tmb quf 
supsup
nn∁∁N N 
W W nn((xx)) 2 2 O O xx..   (9.?)  (9.?)
Fct˒ Fct˒ m`m`   supsupnn∁∁N N W W nn 22 ∓∓..
9>9>
  
Gfj`cstrm±Gfj`cstrm±oo˒˒mm`̀..   Gm o`ctncungmgf gf  Gm o`ctncungmgf gf   W W nn   rfsubtm quf ` o`ckuct` rfsubtm quf ` o`ckuct` { {xx ∁ ∁   F F    ;;  W W nn((xx))  ≧≧
cc}}    : : (( ¼  ◤ ¼  ◤ W W nn))∙∙== (X5(X5,, ccZ) Z) ··f f dfoamg` dfoamg` pmrm pmrm omgmomgm    cc ∁ ∁   CC   f omgm  f omgm    nn ∁ ∁    N N . . GfGfssssm d`rm d`rjmjm, `, `
o`ckuct`o`ckuct`
MMcc ;: ;: xx ∁∁ F F  ; sup ; sup
nn∁∁N N 
W W nn((xx)) ≧≧ c c ::  
nn∁∁N N 
{{xx ∁∁ F F    ;; W W nn((xx)) ≧≧ c c}}
··f f dfodfoamg` amg` p`r p`r sfr sfr ujm ujm ncnctfrstfrsf±f±o˒om˒` gf m` gf dfdfooamamg`g s̀. s. Gf (9Gf (9.?.?) ) sfsfluluf f ququff    F F     ::
∓∓
cc:=:=
MMcc
f, f, pfbpfb` ` WWf`rf`rfjfjm m 9.9.?.?.=, =, mbmblulujj    MMcc   p`ssu  p`ssun n ncnctfrntfrǹ r `r c˒cm˒`-m`-vvmzmzn`n`. . _fk_fkmm    cc55   uj c·  uj c·ujfr`ujfr`
cmturmb tmb qufcmturmb tmb quf    MMcc55   tfcam nctfrn`r c˒ tfcam nctfrn`r cm˒`-vmzn`, f sfkmjm`-vmzn`, f sfkmj    mm ∁ ∁   nct(  nct(MMcc55) ) ff    r r 33   5 tmns quf  5 tmns quf
{{xx ∁ ∁   F F    ;;  xx ∙∙ mm ≧ ≧   rr} ⊄} ⊄   nct(  nct(MMcc55))..   _fkm  _fkm   yy ∁ ∁   F F    o`jo`j  yy ≧ ≧   ==. . __ff   xx   ::   mm + +  ry ry , fct˒, fctm˒`m`
xx ∙∙ mm :: ryry ≧≧ r r  f p`rtmct` f p`rtmct`   xx ∁∁ MMcc55. Mssnj. Mssnj
W W nn((xx ∙∙ mm)) ≧≧ W W nn((xx)) ++ W W nn((mm)) ≧≧ cc55 + + cc55
pmrm t`g`pmrm t`g`  n n ∁∁ N N . B`l`. B`l`  W W nn((ryry)) :: W W nn((xx ∙∙ mm)) ≧≧ 9 9cc55   ff  W W nn((yy)) ≧≧    99cc55rr   pmrm t`g`  pmrm t`g`   nn  f fjj  N N ..
U`rtmct` supU`rtmct` supnn
∁∁
N N 

W W nn
 ≧≧
   99cc55
rr
   ..
Gm Gm Mc·Mc·mbnsf cm mbnsf cm Yftm smifj`s Yftm smifj`s quf quf ` ` bnjntbnjntf f ucnd`rjucnd`rjf f gf gf duc±duc±o˒o˒̀fs `fs o`cto`ct··Ľcums Ľcums gf﬍cngf﬍cngmsgms
fj fj uj uj ncnctfrvtfrvmb` mb` gm gm rftm rftm t`jmt`jmcg` cg` vvmb`mb`rfs rfs cm cm rftm ·rftm ·f f ujm ujm duc±duc±o˒om˒` m` o`cto`ct··Ľcum, Ľcum, fcqumfcqumct`ct`
quf ` bnjntf p`ctumb gf duc±quf ` bnjntf p`ctumb gf duc±o˒o˒̀fs `fs o`cto`ct··Ľcums Ľcums p`p`gf gf c˒cm˒` m` sfr sfr ujm ujm ducduc±±o˒om˒` m` o`ct·o`ct·ĽcumĽcum. . TTfrffrfj`sj`s
mimnxmimnx` ` quf, quf, cm cm prfsfprfsfc±c±om om gm gm bncfmrbncfmrngmgfngmgf, , m m o`cvfrl˄o`cvfrlf˄conm fconm p`ctp`ctumb umb ··f f su﬍onsu﬍onfctf.fctf.
O`O`r`br`b··mrnmrn ̀̀ 9.9.?.?.??   _fkmj   _fkmj    F F   uj fspm±  uj fspm±o` gf Imcmoa,o` gf Imcmoa,    D D   uj fspm±  uj fspm±o` c`rjmg` f o` c`rjmg` f    ((W W cc))
∓∓
cc:=:=
ujm ujm sfqu˄sfquf˄confconm m fj fj  B B((F F 88 D D ))   tmb quf  tmb quf    ((W W cc((xx))))∓∓cc:=:=  · ·f f o`cvo`cvfrlffrlfctf ctf fj fj   D D   pmrm t`g`  pmrm t`g`   xx   fj fj   F F ..
_f gf﬍cnrj`s _f gf﬍cnrj`s 
W W  ; ;   F F  ∙ ∙↙↙ D D , , W W ((xx) : bnj) : bnj
cc↙∓↙∓
W W cc((xx)),,
fct˒ fct˒ m`m`   W W  · ·f f uj uj `pfrm`pfrmg`g`r r bncbncfmr fmr o`ct·o`ct·ĽcĽcu`u`..
Gfj`cstrm±Gfj`cstrm±oo˒˒mm`̀..   M bncfmrngmgf gf  M bncfmrngmgf gf   W W   sfluf   sfluf gms gms pr`prnfgmgfs pr`prnfgmgfs mrntj·mrntj·ftnoms ftnoms g`s g`s bnjntfs.bnjntfs.
U`r anp·U`r anp·̀tf`tfsf, sf, pmrpmrm m omgomgmm    xx   ∁∁    F F   m   m sfqu˄sfquf˄conm fconm ((W W cc((xx))))∓∓cc:=:=   ·  ·f f o`cvfrlfctf, o`cvfrlfctf, f f p`rtmct`p`rtmct`
bnbnjnjntmgtmgm. m. MsMssnsnj supj supcc∁∁CC W W cc((xx))    22  ∓∓ pmrm t`g` pmrm t`g`   xx  ∁∁    F.F.   Ufb` Wf`rfjm 9.?.9 fxnstf  Ufb` Wf`rfjm 9.?.9 fxnstf
o o 33 5 tmb quf sup 5 tmb quf sup
cc∁∁CC
W W cc ≧≧ o o. _fluf fct˒. _fluf fctm˒` qufm` quf
W W cc((xx)) ≧≧ W W cc ¼¼ xx ≧≧ o o xx
pmrm t`g`spmrm t`g`s    xx ∁ ∁    F F    ff    cc ∁ ∁   CC. . DDmzfmzfcg`cg`   cc ∙ ∙↙ ↙ ∓∓   `itfj`s  `itfj`s  W W ((xx))  ≧≧    oo xx   pmrm t`g`  pmrm t`g`
xx ∁∁ F F , ` quf o`jpbftm m gfj`cstrm±, ` quf o`jpbftm m gfj`cstrm±o˒om˒` gf qufm` gf quf   W W   ·  ·f f oo`̀cct·t·ĽĽccuu`̀..
Umrm mprfonmr m pr·Umrm mprfonmr m pr·̀xnjm mpbnom±`xnjm mpbnom±o˒om˒` g` Wf`rfjm gf Imcmoa‟_tfncamus, bfjirfj`sm` g` Wf`rfjm gf Imcmoa‟_tfncamus,bfjirfj`s
uj fxfjpb` ob·uj fxfjpb` ob·mssno` g` O·mssno` g` O·mboub`; m duc±mboub`; m duc±o˒om˒`m`
d d ;;   YY99
∙∙↙↙
YY  ,  , d d ((x,x, yy) ) ::
   xyxy
xx99 ++ yy99
, , d d (5(5,, 5) : 55) : 5,,
?5?5
  
··f f sfsfpmpmrmrmgmgmjfjfctf ctf o`o`ct·ct·ĽcuĽcum, m, nsnst ̀t ̀ ··f,f,    d d ((x,x, ¼¼)) ;;   YY  ∙∙↙↙    YY    ff    d d ((¼¼,, yy)) ;;   YY  ∙∙↙↙    YY    s˒sm˒` duc±m` duc±o˒o˒̀fs`fs
o`cto`ct··Ľcums Ľcums pmrm pmrm qumnsqqumnsqufrufr    x,x, yy   ﬍  ﬍x`sx`s. . FcFctrfttrftmcmct`t`    d d   c  c˒˒m ̀m ̀ ··f f o`o`ct·ct·ĽcĽcumum; ; t`t`jmjmcgcg`̀    yy    ::    txtx
o`jo`j    xx  : : 5,5,    d d ((x,txx,tx) ) ::    tt
=+=+tt99
, p`rtmct` c˒, p`rtmct` cm˒m` ` ffxxnnssttf f ` ` bbnnjjnnttf f bbnnjj
((x,yx,y))↙↙(5(5,,5)5)
d d ((x,x, yy). ). M sM sflflununrr
vfrn﬍omj`s quf pmrm mpbnom±vfrn﬍omj`s quf pmrm mpbnom±o˒o˒̀fs`fs   inbncfmrfs  inbncfmrfs , , ` ` dfc˄dfc˄̀jfc` `jfc` monjm monjm c˒cm˒` `o`rrf, jfsj` fjm` `o`rrf, jfsj` fjgnjfcs˒gnjfcsm˒` nc﬍cntm.m` nc﬍cntm.
Gf﬍cn±Gf﬍cn±o˒om˒ ̀m ̀ 9.9.?.?.77   _fkmj  _fkmj F  F ==,, F  F 99 f f  D  D   fspm± fspm±o`s vfto`s vft`rnmns`rnmns. . Vjm mpbVjm mpbnom±nom±o˒om˒`m` I I ;;   F F ==ÜÜF F 99 ∙ ∙↙↙ D D 
··f f ggnnttmm  inbncfmr  inbncfmr   sfsf   II((xx==,, ¼¼)) ;;   F F 99 ∙ ∙↙↙ D  D    ff   II((¼¼,, xx99)) ;;  F F == ∙ ∙↙↙ D  D    s˒sm˒` `pfrmg`rfs bncfmrfs pmrmm` `pfrmg`rfs bncfmrfs pmrm
qumnsqufrqumnsqufr   xx== ∁ ∁ F F ==   ff   xx99 ∁ ∁ F F 99  ﬍xmg`s. ﬍xmg`s.
O`O`r`br`b··mrnmrn ̀̀ 9.9.?.?.66   _fkmj   _fkmj    F F ==,,    F F 99    f f    D D    fspm±  fspm±o`s o`s vftvft̀ rn`rnmns mns c`rc`rjmgjmg`s,`s,    F F 99   o`  o`jpbjpbft` ft` f f 
II ;;   F F == ÜÜ F F 99 ↙ ↙ D D  ujm mpbnom± ujm mpbnom±o˒ o˒ m` inbncfmr. _up`cam quf m` inbncfmr. _up`cam quf  I I  sfk sfkm m sfpmrmgmsfpmrmgmjfctjfctf f o`cto`ct··ĽcumĽcum,,
nnsst̀ t̀ ··ff,,    II((x,x, ¼¼)) ;;   F F 99 ∙ ∙↙↙   D D    f f   II((¼¼,, yy)) ;;   F F == ∙ ∙↙↙   D D    s˒ s˒ m` m` `pfrmg`rfs `pfrmg`rfs bncfmrfs bncfmrfs o`cto`ct··Ľcu`s Ľcu`s pmrpmrm m 
qumnsqufr qumnsqufr  x x ∁∁ F F ==   f f  y y ∁ ∁ F F 99  ﬍xmg`s. Fct˒  ﬍xmg`s. Fct˒ m`m` I I ;;   F F ==ÜÜF F 99 ∙ ∙↙↙ D D  · ·f f ujm ujm mpmpbnbnom±om±o˒ o˒ m ̀m ̀ o`co`ct·t·ĽcĽcumum..
Gfj`cstrm±Gfj`cstrm±oo˒˒mm`̀..   O`csngfrf  O`csngfrf   D D    ::    {{II((x,x, ¼¼) ) ;; x x ∁∁ F F ==,,  xx ≧≧ = =} } ⊄ ⊄ BB((F F 99,, D D )). . UU`̀rr
aanpnp··̀t̀tffsfsf,,    II((
¼¼
,, yy) ) ··f f bncfmbncfmr r f f o`cto`ct··Ľcu` Ľcu` pmrm pmrm omgmomgm    yy
  ∁∁
   F F 99, p`rtmct`, pmrm t`g`, p`rtmct`, pmrm t`g`    xx
 ∁ ∁
   F F ==,,
xx ≧≧ =, tfj`s=, tfj`s
II((x,x, yy)) ≧≧ II((¼¼,, yy))BB((F F ==,D ,D ))  :; :;  O  O yy..
MssMssnj nj m m dmj·dmj·ĽbnmĽbnm   D D   ·  ·f f p`ctumbjp`ctumbjfctf fctf bnjntmbnjntmgm gm f, f, pfb` pfb` WWf`rfjm f`rfjm gf gf ImcmoImcmoa‟_tfncama‟_tfncamus,us,
fxnstf ujm o`cstmctffxnstf ujm o`cstmctf   O O  tmb quf tmb quf
supsup
xx≧≧==
supsup
yy≧≧==
II((x,x, yy)) 2 2 O.O.
_fluf gm inbncfmrngmgf gf_fluf gm inbncfmrngmgf gf   II   qufquf  II((x,x, yy)) ≧ ≧  O O xx ¼¼ yy  pmrm t`g`s pmrm t`g`s   xx ∁ ∁ F  F ==    ff   yy ∁ ∁  F  F 99..
M o`ctncungmgf gfM o`ctncungmgf gf   II  sfluf  sfluf ml`rm ml`rm g` g` FxfrFxfroo··Ľon` Ľon` 9.<.969.<.96..
DncDncmbnmbnzmj`zmj s̀ s fstfstm m sf±sf±o˒om˒` m` vfvfrn﬍rn﬍omcgomcg` ` m m cfofcfofssnssngmgf gmgf gm gm o`jpo`jpbftubftugf gf g` g` g`jg`j··ĽcnĽcn` ` c`c`
fcuconmg` g` Wf`rfjm gf Imcmoa‟_tfncamus.fcuconmg` g` Wf`rfjm gf Imcmoa‟_tfncamus.
Fxfjpb` 9.?.4Fxfjpb` 9.?.4  Umrm omgm Umrm omgm   cc ∁∁ CC, , o`csngfrf o`csngfrf ` ` ducon`cmb ducon`cmb bncfmr bncfmr o`cto`ct··Ľcu`Ľcu`
ϖϖcc ;;   oo5555 ∙ ∙↙↙ HH  , , ϖϖcc((((mm k k))∓∓ k k:=:=) ) :: cm cmcc..
··F obmr` quf (F obmr` quf (ϖϖcc))
∓∓
cc:=:= ⊄ ⊄ ((oo5555)) f f ϖϖcc :: c c  pmrm t`g` pmrm t`g` c c. . Gmgm Gmgm ujm sujm sfqu˄fquf˄conm fconm ((mm k k))∓∓ k k:=:= ∁ ∁ oo5555,,
t`jmcg`t`jmcg` k k55 tmb quf tmb quf m m k k  : 5 pmrm t`g` : 5 pmrm t`g` k k ≦ ≦  k k55, sfluf quf, sfluf quf ϖ ϖcc((((mm k k))∓∓ k k:=:=) : 5 pmrm t`g`) : 5 pmrm t`g` c c ≦≦  k k55..
Mssnj, supMssnj, sup
cc
||ϖϖcc((xx))|| 22 ∓∓  pmrm t`g` pmrm t`g`   xx ∁∁ oo5555  jms sup jms sup
cc
ϖϖcc :: ∓∓..
?=?=
  
9.9.7 7 @ W@ Wf`f`rfrfjm jm gm gm MpMpbnbnomom±±o˒om˒m ̀̀ MMiiffrrttmm
@ Wf`rfjm gm Mpbnom±@ Wf`rfjm gm Mpbnom±o˒om˒` m` Mifrtm ·Mifrtm ·f f jmns uj jmns uj rfsubtmgrfsubtmg` ` dmj`s` gm dmj`s` gm Mc·Mc·mbnsf Ducon`cmb,mbnsf Ducon`cmb,
gfvng` m Imcmoa (=>9>), f lmrmctf quf sfgfvng` m Imcmoa (=>9>), f lmrmctf quf sf   F F    ff   D D    s˒sm˒` fspm±m` fspm±o`s gf Imcmoa, fct˒o`s gf Imcmoa, fctm˒` t`g`m` t`g`
`pf`pfrmg`r rmg`r bncfmr bncfmr o`cto`ct··Ľcu` f Ľcu` f s`irfs`irfkft`kft`rr W  W  ; ;   F F  ∙ ∙↙↙ D D  · ·f f uujjmm mpbnom± mpbnom±o˒ o˒ m` mifrtm m` mifrtm , , nsnstt ̀̀ ··ff,, W  W ((MM) ·) ·ff
mifrt` fjmifrt` fj D  D  sfjprf quf sfjprf quf M M d`r mifrt` fj d`r mifrt` fj F  F . Umrm gfj`cstr·. Umrm gfj`cstr·m-b`, prfonsmj`s g` sflunctfm-b`, prfonsmj`s g` sflunctf
bfjm. U`rbfjm. U`r   IIF F ((xx5588 rr) gfc`tmrfj`s m i`bm mifrtm fj) gfc`tmrfj`s m i`bm mifrtm fj   F F  ofctrmgm fj ofctrmgm fj  xx55  gf rmn` gf rmn`   rr, , nsnst ̀t ̀ ··f,f,
IIF F ((xx5588 rr) ) :: {{xx ∁∁ F F   ;; xx ∙∙ xx55 2 2 rr}}..
Bfjm 9.7.=Bfjm 9.7.=   _fkmj   _fkmj  F  F  uj fspm± uj fspm±o` gf Imcmoa,o` gf Imcmoa,  D  D  uj fspm± uj fspm±o` c`rjmg` f o` c`rjmg` f  W  W  ; ;   F F  ∙ ∙↙↙ D D    uj uj 
`pfrmg`r `pfrmg`r bncfmr bncfmr o`cto`ct··Ľcu`. Ľcu`. _f _f fxnstnrfj fxnstnrfj  Y Y,, r 3r 3 5 5  tmns quf  tmns quf   W W ((IIF F (58(58 YY)))) ⊌⊌ IID D (58(58 rr)),, f fcctt˒˒mm`̀
W W ((IIF F (58(58 YY)))) ⊌⊌ IID D 

5858    rr
99

..
Gfj`cstrm±Gfj`cstrm±oo˒˒mm`̀..  O`j` pmrm t`g` suio`ckuct` O`j` pmrm t`g` suio`ckuct`   J J   ⊄⊄   F F   f t`g` fsombmr f t`g` fsombmr   mm ∁ ∁   HH   tfj-sf tfj-sf
mJ mJ   :: mJ  mJ   (Fx (Fxfrofro··Ľon` Ľon` =.1.1)=.1.1), sf, sfluf luf gm agm anp·np·̀tfsf quf`tfsf quf
W W ((IIF F (58(58 mYmY)))) ⊌⊌ IID D (58(58 mrmr) ) ((99..77))
pmrm t`g`pmrm t`g`    mm  ∁∁   YY    p`snp`sntnvtnv`. `. _fkm_fkm    yy  ∁∁    IID D  5858   rr99 ..  U`r (9.7) fxnstf U`r (9.7) fxnstf    xx==  ∁∁    IIF F 5858   YY99     tmbtmb
qufquf  yy ∙∙ W W ((xx==))    22    rr77 , , nstnst` ` ··f,f,    yy ∙ ∙   W W ((xx==))  ∁∁    IID D 

5858   rr
77

. . C`C`vvmjmjfcfctf p`r tf p`r (9.(9.7), fxns7), fxnstftf
xx99 ∁ ∁ IIF F 

5858   YY
77

 tmb quf tmb quf
((yy ∙∙ W W ((xx==)))) ∙∙ W W ((xx99)) 22
  rr
11
..
O`j` (9.7) vmbf pmrm t`g`O`j` (9.7) vmbf pmrm t`g` m  m 33 5, p`gfj`s o`ctncumr fssf pr`ofgnjfct` ncgf﬍cngmjfctf 5, p`gfj`s o`ctncumr fssf pr`ofgnjfct` ncgf﬍cngmjfctf
gf gf d`rjm d`rjm m m o`cstrunr o`cstrunr ujm ujm sfqu˄sfquf˄conm fconm ((xxcc))
∓∓
cc:=:=   fjfj  F  F   tmb quf tmb quf   xxcc ∁ ∁ IIF F 

5858    YY
99cc

 f f
yy ∙∙ W W ((xx==)) ∙ ¼ ¼ ¼ ∙∙ ¼ ¼ ¼ ∙ W W ((xxcc)) 22
   rr
99cc+=+=
  (9.6)  (9.6)
pmrm t`g`pmrm t`g`   cc ∁∁ CC. . M M s·s·frfrnfnf
∓∓

cc:=:=
xxcc · ·f f o`cvfo`cvfrlfrlfctf ctf p`p`nsns  xxcc 22
   YY
99cc
  pmrm t`g`  pmrm t`g`   cc, b`l`, b`l`

jj
 k k::cc
xx k k
 ≧≧
jj
 k k::cc
xx k k j3cj3c↙∓↙∓∙∙↙↙    55,,
f pf p`rt`rtmct` mct` m sfm sfqu˄quf˄cofconmnm

   cc
 k k:=:=
xx k k
∓∓
cc:=:=
··f f gf gf OmOmuoay uoay fjfj F  F . O`j`. O`j` F  F  · ·f If Imcmcmoamoa, f, fxnxnststff x x ∁∁ F F 
pmrm pmrm ` qum` qumb fssm b fssm sfqu˄sfquf˄conm fconm o`cvfrlf. o`cvfrlf. _fluf _fluf fct˒fctm˒` qufm` quf
xx ::
 bnjbnjcc↙∓↙∓
cc
 k k:=:=
xx k k
 ≧≧    bnjbnjcc↙∓↙∓
cc
 k k:=:=
xx k k :: xx== ++
∓∓
cc:9:9
xxcc
22
  YY
99
   ++
∓∓
cc:9:9
xxcc ≧≧
  YY
99
   ++
∓∓
cc:9:9
YY
99cc
   ::
∓∓
cc:=:=
YY
99cc
   :: Y, Y,
?9?9
  
f p`rtmct`f p`rtmct`   xx ∁∁ IIF F (58(58 YY))..  Dmzfcg` Dmzfcg`   cc  ∙∙↙↙ ∓∓ fj (9.6) `itfj`s fj (9.6) `itfj`s
yy ∙∙ W W ((xx)) ::


yy ∙∙ W W 

bnjbnj
cc↙∓↙∓
cc

 k k:=:=
xx k k


: bnj: bnj
cc↙∓↙∓
yy∙∙ W W ((xx==)) ∙ ¼ ¼ ¼ ∙∙ ¼ ¼ ¼ ∙ W W ((xxcc))
≧≧    bnjbnj
cc↙∓↙∓
rr
99cc+=+=
   : : 55,,
f o`csfqufctfjfctff o`csfqufctfjfctf   yy  :: W  W ((xx))..  G Gmm··ĽĽ  yy ∁ ∁ W W ((IIF F (58(58 YY)) f ` rfsubtmg` fst·)) f ` rfsubtmg` fst·m pr`vmg`.m pr`vmg`.
Wf`rfjm 9.7.9Wf`rfjm 9.7.9   (Wf`rfjm gm Mpbnom±  (Wf`rfjm gm Mpbnom±o˒om˒` Mifrtm)m` Mifrtm)   _fkmj   _fkmj    F F    f f    D D   fspm±  fspm±o`s gf Imcmoa f o`s gf Imcmoa f 
W W  ; ;   F F   ∙∙↙↙    D D   bncfmr  bncfmr, , o`cto`ct··Ľcu` Ľcu` f f s`irfkft`rs`irfkft`r. . Fct˒ Fct˒ m`m`    W W   ·  ·f f ujm ujm mpbmpbnom±nom±o˒ o˒ m` mm` miifrfrtmtm. . Fj Fj 
pmpmrtnourtnoubmr, bmr, t`t`g` g` `pf`pfrrmg`r mg`r bncfbncfmr mr oo`ct`ct··Ľcu` Ľcu` f f inkftinkft̀ r `r fctrfctrf f fspfspm±m±o`s o`s gf gf ImcImcmoa moa ··f f uj uj 
ns`j`r﬍sj`.ns`j`r﬍sj`.
Gfj`cstrm±Gfj`cstrm±oo˒˒mm`̀..    GfGf   F F   ::
∓∓
cc:=:=
IIF F (58(58 cc) f gm s`irfkftnvngmgf gf) f gm s`irfkftnvngmgf gf   W W   sfluf quf sfluf quf
D D    :: W  W ((F F ) ) ::
∓∓
cc:=:=W W ((IIF F (58(58 cc)) )) ::
∓∓
cc:=:=W W ((IIF F (58(58 cc))))..
Ufb` Wf`rfjm 9.?.= fxnstfUfb` Wf`rfjm 9.?.= fxnstf   cc55 ∁ ∁  C C  tmb quf tmb quf   W W ((IIF F (58(58 cc55)) )) tfj ntfj ncctfrtfrn`n`r c˒r cm˒`-vm`-vmzn`. mzn`. MssnjMssnj
fxnstfjfxnstfj   ii ∁∁ D D    ff  r  r 33 5 tmns quf 5 tmns quf  I ID D ((ii88 rr)) ⊄⊄ W W ((IIF F (58(58 cc55)) . O`j`)) . O`j`
W W ((IIF F (58(58 cc55)) )) :: ∙∙W W ((IIF F (58(58 cc55)))),,
rfsubtm qufrfsubtm quf
IID D ((∙∙ii88 rr) ) :: ∙∙IID D ((ii88 rr)) ⊄⊄ W W ((IIF F (58(58 cc55))))..
O`j`O`j`   xx : :    ==
99
((ii ++ xx) +) +    ==
99
((∙∙ii ++ xx),),
IID D (58(58 rr)) ⊄ ⊄
  ==
99
IID D ((ii88 rr) +) +
  ==
99
IID D ((∙∙ii88 rr))
⊄⊄   ==
99
W W ((IIF F (58(58 cc55)) +)) +  ==
99
W W ((IIF F (58(58 cc55)) )) :: W  W ((IIF F (58(58 cc55)))),,
`cgf m ·`cgf m ·ubtnjm nlumbgmgf gfo`rrf gm o`cvfxngmgf g` o`ckuct`ubtnjm nlumbgmgf gfo`rrf gm o`cvfxngmgf g` o`ckuct`  W  W ((IIF F (58(58 cc55)). Ufb` Bfjm)). Ufb` Bfjm
9.7.= smifj`s quf9.7.= smifj`s quf    W W ((IIF F (58(58 cc55)))) ⊌ ⊌    IID D (58(58 ύύ) pmrm) pmrm   ύύ    ::    rr99 , f p`rtmct`, f p`rtmct`   W W ((IIF F (58(58 ococ55)))) ⊌ ⊌
IID D (58(58 oύoύ) pmrm t`g` rfmb p`sntnv`) pmrm t`g` rfmb p`sntnv`   o.o.  Tfkmj`s quf Tfkmj`s quf
W W ((IIF F ((xx88 ococ55)))) ⊌⊌ IID D ((W W ((xx)8)8 oύoύ))
pmrm t`g`pmrm t`g`   xx ∁∁ F F  f t`g` f t`g`   o o 33 5. Gf dmt`, 5. Gf dmt`,   IIF F ((xx88 ococ55) ) :: x x + + IIF F (5(5,, ococ55), f fct˒), f fctm˒`m`
W W ((IIF F ((xx88 ococ55)) )) :: W  W ((xx) +) + W  W ((IIF F (58(58 ococ55)))) ⊌⊌ W W ((xx) +) + IID D (58(58 oύoύ) ) :: I ID D ((W W ((xx)8)8 oύoύ))..
Ml`rm p`gfj`s pr`vmr qufMl`rm p`gfj`s pr`vmr quf    W W ((V V ) ) ··f f mifrt` mifrt` fjfj    D D   pmrm omgm  pmrm omgm    V V   mifrt` fj  mifrt` fj    F F . . _f_fkmkmjj
xx ∁ ∁ V  V    ff   o o 33 5 tmns quf 5 tmns quf   IIF F ((x,ocx,oc55)) ⊄ ⊄ V. V.  Fct˒ Fctm˒`m`   W W ((V V )) ⊌ ⊌  W W ((IIF F ((xx88 ococ55)))) ⊌ ⊌ I ID D ((W W ((xx)8)8 oύoύ),),
????
  
pr`vmcg` qufpr`vmcg` quf   W W ((V V ) ) ··f f mimifrtfrt` ` fjfj   D D ..
M sflucgm m﬍rjm±M sflucgm m﬍rjm±o˒om˒` m` ··f f o`csfo`csfqu˄quf˄conm fconm njfgnmnjfgnmtm tm gm gm prnjfprnjfnrm.nrm.
M s`iM s`irfkftrfkftnvngmgnvngmgf g` f g` `pf`pfrmg`rmg`r ·r ·f anp·f anp·̀tfsf fssfconmb c` Wf`rfjm gm Mpbnom±`tfsf fssfconmb c` Wf`rfjm gm Mpbnom±o˒om˒` Mifrtm,m` Mifrtm,
p`ns t`gm p`ns t`gm mpbnommpbnom±±o˒om˒` bncfmr mifrtmm` bncfmr mifrtm W  W  ; ;   F F  ∙ ∙↙↙ D D  fctrf fspm± fctrf fspm±o`s o`s c`rjmc`rjmg`s g`s ··f f cfofcfofssmrnmssmrnmjfctfjfctf
s`irs`irfkftfkft`rm. `rm. O`j fdfntO`j fdfnt`, gmgm `, gmgm ujm i`bm ujm i`bm mifrtmifrtm m ofcofctrmgtrmgm m cm `rnlfj gfcm `rnlfj gf   F F , sum njmlfj, sum njmlfj
p`rp`r   W W , , p`r p`r sfr misfr mifrtm, o`ct·frtm, o`ct·fj fj ujm iujm i`bm ofctrmgm cm `bm ofctrmgm cm `rnlfj gf`rnlfj gf   D D . . M s`irfkM s`irfkftnftnvngvngmgfmgf
sfluf dmonbjfctf gm bncfmrngmgf gfsfluf dmonbjfctf gm bncfmrngmgf gf   W W ..
@ pr·@ pr·̀xnj` fxfjpb` j`strm quf m anp·`xnj` fxfjpb` j`strm quf m anp·̀tfsf g`s fspm±`tfsf g`s fspm±o`s o`s sfrsfrfj fj o`jo`jpbfpbft`s t`s tmji·tmji·fj fj ··ff
fssfconmb.fssfconmb.
Fxfjpb` 9.7.?Fxfjpb` 9.7.?   _fkm  _fkm    W W  ; ;    oo5555    ∙∙↙↙    oo5555   `   ` `pf`pfrmrmg`g`r r bnbncfcfmr mr gmgmg` g` p`rp`r    W W ((((mmcc))∓∓cc:=:=) ) ::
((mm==,,
  mm99
99
 , ,   mm??
??
 , . . . , . . .)).. C`  C` Fxfro·Fxfro·Ľon` Ľon` 9.<.99.<.91 ` 1 ` bfnt`r bfnt`r o`jpro`jpr`vmr·`vmr·m qufm quf W  W  · ·f bf bncncfmfmr, r, inkinkftft`r `r f of o`ct·`ct·ĽcuĽcu`,`,
jms quf ` `pfrmg`r ncvfrs`jms quf ` `pfrmg`r ncvfrs`   W W ∙∙== c˒cm˒m ̀̀ ··f f oo`̀ct·ct·ĽcĽcuu`̀..
Fxfjpb` 9.7.7Fxfjpb` 9.7.7   (_uifspm±  (_uifspm±o`s dfoamg`s gfo`s dfoamg`s gf O  O X5X5,, =Z) Tfkmj`s quf t`g` suifspm±=Z) Tfkmj`s quf t`g` suifspm±o` o` dfoamdfoamg`g`
D D  gf gnjfcs˒ gf gnjfcsm˒` nc﬍cntm gfm` nc﬍cntm gf   O O X5X5,, =Z =Z o`ct·o`ct·fj fj ujm ujm ducduc±±o˒om˒` quf c˒m` quf cm˒` pfrtfcof mm` pfrtfcof m   O O ==X5X5,, =Z. Nss`=Z. Nss`
pr`vm, fj pmrtnoubmr, qufpr`vm, fj pmrtnoubmr, quf  O  O 
==
X5X5,, =Z c˒=Z cm˒` m` ··f f dfoadfoamg` mg` fjfj  O  O X5X5,, =Z. _up`cam, o`j pr`p·=Z. _up`cam, o`j pr`p·̀snt` gf`snt` gfo`ctrmgn±o`ctrmgn±o˒om˒`, qufm`, quf D  D  ⊄ ⊄ O O ==X5X5,, =Z. O`j`=Z. O`j` D  D  · ·f df dfofoamamg ̀g ̀ fjfj  O  O X5X5,, =Z, `s fspm±=Z, `s fspm±o`so`s F  F ==  ;:  ;: ((D,D, ¼¼∓∓))
ff    F F 99    ;: ;: ((D,D,  ¼  ¼ O O ==) ) s˒sm˒` mji`m` mji s̀ s o`jo`jpbpbft`ft s̀. s. M mpbnM mpbnom±om±o˒om˒` ngfctngmgfm` ngfctngmgf    nn ;;   F F 99   ∙∙↙↙    F F ==    ··ff
obmrmjfctf inkft`rm f bncfmr.obmrmjfctf inkft`rm f bncfmr.    ··F F tmji·tmji·fj fj o`ct·o`ct·Ľcum Ľcum p`p`ns, ns, pmrpmrm m t`t`gmgm   ll ∁ ∁ D D ,,
nn((ll))F F ==   :: ll∓∓ ≧ ≧ ll∓∓ + + ll∓∓  :: llF F 99 ..
Mssnj, ` Wf`rfjm gm Mpbnom±Mssnj, ` Wf`rfjm gm Mpbnom±o˒om˒` m` Mifrtm Mifrtm lmrmctf lmrmctf m m fxnst˄fxnstf˄conm fconm gf gf ujm ujm o`cstmctfo`cstmctf   O O 33   55
tmb quftmb quf
ll∓∓ ≧ ≧ O O ll∓∓
pmrm t`gm duc±pmrm t`gm duc±o˒om˒`m`   ll ∁ ∁   D D . . __fkfkm (m (d d cc))∓∓cc:=:=  ujm  ujm sfqu˄sfquf˄conm fconm cm cm i`bm i`bm ucnt·ucnt·mrnm gfmrnm gf   F F ==, , nstnst ̀̀ ··f,f,
jmxjmx
tt∁∁X5X5,,=Z=Z
||d d cc((tt))| ≧| ≧ = pmrm t`g` = pmrm t`g`   cc. . UfUfbm gfsnlbm gfsnlumbumbgmgf monjgmgf monjm, m, jmxjmx
tt∁∁X5X5,,=Z=Z
||d d cc((tt))| ≧| ≧  O O  pmrm t`g` pmrm t`g`   cc..
@ o`ckuct` d`rjmg` pfbms duc±@ o`ckuct` d`rjmg` pfbms duc±o˒o˒̀fs`fs   d d cc,,   cc
 ∁ ∁
  CC, smtnsdmz fct˒, smtnsdmz fctm˒` ms o`cgn±m` ms o`cgn±o˒o˒̀fs (m) f (i) g``fs (m) f (i) g`
WWf`rfjm f`rfjm gf gf Mso`bn Mso`bn (W(Wf`rfjm f`rfjm I.< I.< g` g` Mp˄Mpf˄cgnof fcgnof 9)8 9)8 f f p`rtmct` p`rtmct` sfu sfu dfoa` ·dfoa` ·f f o`jpmot` o`jpmot` fjfj
O O X5X5,, =Z. =Z. @itfj`s @itfj`s mssnj ujm mssnj ujm suisfqu˄suisfquf˄conm fconm ((d d cckk))
∓∓
 k k:=:=   f ujm duc± f ujm duc±o˒om˒`m`   d d   ∁∁   O O X5X5,, =Z tmns quf=Z tmns quf
d d cckk ∙ ∙↙↙ d d   fjfj O  O X5X5,, =Z. _mifj`s quf=Z. _mifj`s quf d  d  ∁ ∁ D D  p`ns p`ns D  D  · ·f dfoaf dfoamg`mg`. . Ur`vmjUr`vmj`s fct`s fct˒˒m` quf m i`bmm` quf m i`bm
dfoamgm ucnt·dfoamgm ucnt·mrnm gfmrnm gf   F F ==  · ·f f o`jpmotm. o`jpmotm. Ufb` Wf`rfUfb` Wf`rfjm jm gf gf Ynfsz Ynfsz (W(Wf`rfjm f`rfjm =.6.7),=.6.7),  D D   tfrnm  tfrnm
`irnlmt`rnmjfctf gnjfcs˒`irnlmt`rnmjfctf gnjfcsm˒` ﬍cntm, ` quf o`ctrmgnz c`ssm anp·m` ﬍cntm, ` quf o`ctrmgnz c`ssm anp·̀tfsf `tfsf ncnonmncnonmb.b.
9.6 9.6 @ W@ Wf`rfjm g` Lr·f`rfjm g` Lr·m﬍o` Dm﬍o` Dfoamfoamg`g`
_fkmj_fkmj   F F    ff   D D   fspm±  fspm±o`s c`rjmg`s fo`s c`rjmg`s f   W W  ; ;   F F  ∙ ∙↙↙ D  D  uj `pfrmg`r bncfmr. @ uj `pfrmg`r bncfmr. @   lr· lr· m﬍o`m﬍o`   gfgf   W W    ··f f `̀
o`ckuct`o`ckuct`
LL((W W ) ) :: {{((x,x, yy) ) ;; x x ∁∁ F F    ff   yy : :  W  W ((xx))}} :: {{((x,x, W W ((xx)) )) ;; x x ∁∁ F F }} ⊄⊄ F  F  Ü Ü D.D.
?7?7
  
C`tf qufC`tf quf L L((W W )) ··f uf uj sj suifuifspm±spm±o` vft`rnmb gfo` vft`rnmb gf F F ÜÜD D  (quf sfr· (quf sfr·m jucng` o`j qumbqufr ujmmjucng` o`j qumbqufr ujm
gms gms c`rjms c`rjms fqunvmbfcfqunvmbfctfs tfs g` g` FxfroFxfro··Ľon` =.1.=9)Ľon` =.1.=9). . Fj lFj lfrmbfrmb L L((W W ) p`gf sfr uj suio`ckuct`) p`gf sfr uj suio`ckuct`
dfoamg` gfdfoamg` gf    F F  Ü Ü  D  D   `u c˒  `u cm˒`. m`. TTfkmjfkmj`s quf, `s quf, pmrm `pfrmgpmrm `pfrmg`rfs bncf`rfs bncfmrfs fctmrfs fctrf fspm±rf fspm±o`s gfo`s gf
Imcmoa, m o`ctncungmgf gfImcmoa, m o`ctncungmgf gf   W W  · ·f f fqunvmbfctf fqunvmbfctf m` m` dmt` dmt` gfgf   LL((W W ) sfr dfoamg`.) sfr dfoamg`.
Wf`rfjm 9.6.=Wf`rfjm 9.6.=   (Wf`rfjm g` Lr·  (Wf`rfjm g` Lr·m﬍o` Dfoamg`)m﬍o` Dfoamg`)   _fkmj   _fkmj    F F    f f    D D   fspm±  fspm±o`s gf Imcmoa f o`s gf Imcmoa f 
W W  ; ;   F F  ∙ ∙↙↙ D D   uj `pfrmg`r bncfmr. Fct˒   uj `pfrmg`r bncfmr. Fct˒ m`m`   W W  · ·f f o`ct·o`ct·ĽcĽcu ̀u ̀ sfsf, , f f s`s`jfcjfctf tf sfsf,,   LL((W W )) · ·f f dfodfoamamgg`̀
fj fj   F F   ÜÜ D D ..
Gfj`cstrm±Gfj`cstrm±oo˒˒mm`̀..  _up`cam _up`cam   W W   o`  o`ctct··ĽcuĽcu`. `. FctFct˒˒m` m m` m ducduc±±o˒om˒`m`
d d ;;   F F   ÜÜ D D  ∙ ∙↙↙ YY  , , d d ((x,x, yy) ) :: W W ((xx)) ∙∙ yy
··f f o`o`ct·ct·ĽcuĽcum, m, f f pp`r`rtmtmctct`̀ L L((W W ) ) :: d  d ∙∙==(({{55}}) ·) ·f f dfoamg` pdfoamg` p`r `r sfr m sfr m njmlfj njmlfj ncvfrsm g` ncvfrsm g` dfoamg`dfoamg`
{{55}}  pfbm duc± pfbm duc±o˒om˒ ̀m ̀ o`o`ct·ct·ĽcĽcumum   d d ..
Yfonpr`omjfcYfonpr`omjfctf, tf, sup`camsup`cam L L((W W ) dfoamg`. ) dfoamg`. G` FxfG` Fxfroro··Ľon` =.1.=9 Ľon` =.1.=9 smifj`s smifj`s qufquf  F  F ÜÜ D D    ··ff
fspm±fspm±o` gf Imcmoa o`j m c`rjmo` gf Imcmoa o`j m c`rjm   ¼¼ ==. M duc±. M duc±o˒om˒`m`
υυ ;;   LL((W W )) ∙∙↙↙ F F , , υυ((x,x, W W ((xx)) )) :: x, x,
··f f obmrmobmrmjfctf jfctf bncfmbncfmr r f f inkftinkft`rm. `rm. Mb·Mb·fj fj gnss`,gnss`,   υυ  · ·f f o`o`ct·ct·ĽcĽcum um pp`̀nsns
υυ((x,x, W W ((xx)))) :: xx ≧≧ xx ++ W W ((xx)) :: ((x,x, W W ((xx))))==..
G` Wf`rfjm gm Mpbnom±G` Wf`rfjm gm Mpbnom±o˒om˒` Mifrtm sfluf fct˒m` Mifrtm sfluf fctm˒` qufm` quf    υυ∙∙== ··f f o`ct·o`ct·Ľcum Ľcum f f p`p`rtmrtmct` ct` fxnfxnstfstf
O O 33 5 tmb quf 5 tmb quf  ((x,x, W W ((xx))))== ≧ ≧ O O xx  pmrm t`g` pmrm t`g`   xx ∁∁ F F . B`l`. B`l`
W W ((xx)) ≧≧ W W ((xx)) ++ xx :: ((x,x, W W ((xx))))== ≧ ≧ O O xx
pmrm t`g`pmrm t`g`   xx ∁∁ F F . Nss` pr`vm quf. Nss` pr`vm quf   W W  · ·f f oo`̀ct·ct·ĽĽcucu`̀..
C` C` WWf`rfjm f`rfjm 9.4.= 9.4.= f f c` c` FxfroFxfro··Ľon` Ľon` 9.<.?7 9.<.?7 ` ` bfnt`r bfnt`r sf sf ofrtn﬍omr·ofrtn﬍omr·m m gf gf quf quf ujm ujm f f mpfcmsmpfcmsujm gms njpbnom±ujm gms njpbnom±o˒o˒̀fs g` Wf`rfjm g` Lr·`fs g` Wf`rfjm g` Lr·m﬍o` Dfoamg` o`ctncum v·m﬍o` Dfoamg` o`ctncum v·mbngm pmrm `pfrmg`rfsmbngm pmrm `pfrmg`rfs
fctrf fctrf fspm±fspm±o`s o`s c`rjmc`rjmg`s g`s c˒cm˒`-o`jpbft`s.m`-o`jpbft`s.
M lrmcgf utnbngmgf g` Wf`rfjm g` Lr·M lrmcgf utnbngmgf g` Wf`rfjm g` Lr·m﬍o` Dfoamg` rfsngf cm sflunctf `isfrvm±m﬍o` Dfoamg` rfsngf cm sflunctf `isfrvm±o˒om˒`;m`;
pmrm j`strmr m o`ctncungmgf gf uj `pfrmg`rpmrm j`strmr m o`ctncungmgf gf uj `pfrmg`r  W  W  ; ;   F F  ∙ ∙↙↙ D D    (bncfm(bncfmr r `u `u c˒cm˒`) m`) vnm vnm gf﬍cngf﬍cn±±o˒om˒`,m`,
gfvfj`s gfvfj`s pr`vpr`vmr mr quf, quf, pmrm pmrm t`gm t`gm sfqu˄sfquf˄conm fconm o`cvfrlfctfo`cvfrlfctf  xxcc ∙ ∙↙↙ xx  fjfj   F F , , ··f f vfrgvfrgmgf mgf qufquf;;
(m) ((m) (W W ((xxcc))))
∓∓
cc:=:=  · ·f f o`cvfo`cvfrlfrlfctf ctf fjfj   D D , , nstnst` ` ··f, f, fxnfxnstfstf   yy ∁ ∁ D D   tmb quf tmb quf   W W ((xxcc)) ∙∙↙↙ yy88
(i)(i)   yy  : :  W  W ((xx).).
Fj Fj lfrmlfrmb, b, pr`pr`vvmr mr (m), (m), nstnst` ` ··f, f, pr`pr`vvmr mr quf quf ((W W ((xxcc))))
∓∓
cc:=:=   o`  o`cvfrlf, cvfrlf, ··f f ujm ujm tmrftmrfdm dm gndgnd··Ľonb,Ľonb,
fspfonmbjfctf fj gnjfcs˒fspfonmbjfctf fj gnjfcsm˒` nc﬍cm` nc﬍cntntm. m. @ @ WWf`rf`rfjfjm m g` Lr·g` Lr·m﬍o` Dfoamg` c`s d`rcfof, c`m﬍o` Dfoamg` c`s d`rcfof, c`
oms` fj qufoms` fj quf   W W  · ·f f bncfmr bncfmr f f mtum mtum fctrf fctrf fspm±fspm±o`s gf Imcmoa, ` ntfj (m) lrmtuntmjfctf, `uo`s gf Imcmoa, ` ntfj (m) lrmtuntmjfctf, `u
sfkm, sfkm, p`gfj`s p`gfj`s sup`r sup`r m m o`cvo`cvfrl˄frlf˄conm fconm gf gf ((W W ((xxcc))))
∓∓
cc:=:=, , rfsrfstmctmcg` mpfcmg` mpfcms s vfvfrnrn﬍om﬍omr r (i)(i). . @@
fxfjpb` m sflunr nbustrm ` quf momimj`s gf gnzfr;fxfjpb` m sflunr nbustrm ` quf momimj`s gf gnzfr;
?6?6
  
Fxfjpb` 9.6.9Fxfjpb` 9.6.9   _fkmj  _fkmj    F F   uj fspm±  uj fspm±o` gf Imcmoa fo` gf Imcmoa f    W W  ; ;   F F   ∙∙↙↙    F F   uj `pfrmg`r bncfmr uj `pfrmg`r bncfmr
ssnnj·j·fftrtrnno`o`, , nsnst ̀t ̀ ··f,f,   W W  · ·f f bnbncfcfmr mr ff
W W ((xx)()(yy) ) :: W  W ((yy)()(xx) pmrm t`g`s) pmrm t`g`s  x,x, yy
 ∁ ∁
F.F.
J`strfj`s qufJ`strfj`s quf W  W  · ·f of o`ct·`ct·Ľcu`Ľcu`. . Fj Fj vnstvnstm gm g` Wf`r` Wf`rfjm fjm g` g` Lr·Lr·m﬍o` Dfoamg`, imstm vfrn﬍omrm﬍o` Dfoamg`, imstm vfrn﬍omr
qufquf  L L((W W ) ·) ·f f dfoamg`. dfoamg`. Umrm nst`, Umrm nst`, sfkm sfkm ((xxcc))
∓∓
cc:=:=  ujm  ujm sfqu˄sfquf˄conm fconm o`cvfrlnco`cvfrlncg` g` pmrmpmrm  x x  f fjj F  F   ff
sup`cam qufsup`cam quf   W W ((xxcc)) ∙∙↙↙ ϖϖ   fjfj   F F . Dnxmg`. Dnxmg`   yy ∁ ∁ F F , t`jmcg` ` bnjntf qumcg`, t`jmcg` ` bnjntf qumcg`   cc ∙ ∙↙↙ ∓∓,,
ϖϖ((yy)) ↕↕∙∙ W W ((xxcc)()(yy) ) :: W  W ((yy)()(xxcc)) ∙∙↙↙ W W ((yy)()(xx) ) :: W  W ((xx)()(yy)),,
` ` ququf f c`c`s s pfrpfrjnjntf tf o`o`cocobubunr nr ququff    W W ((xx)()(yy) ) ::    ϖϖ((yy)). . OO`̀jj`̀    yy    ∁∁    F F   d`n   d`n fso`fso b̀anbang`g`
mrintrmrnmrintrmrnmjfcmjfctf, tf, sfluf sfluf qufquf  ϖϖ : :  W  W ((xx), f p`rtmct`), f p`rtmct`   LL((W W ) ) ··f f dfdfoamoamg`g`..
9.4 9.4 O`jfcO`jfct·t·mrn`s f mrn`s f c`tms anst·c`tms anst·̀rnoms`rnoms
_fkm_fkm W  W  ; ;   F F 
 ∙ ∙↙↙
D D    uj `puj `pfrmg`r frmg`r bncfmr o`ctbncfmr o`ct··Ľcu`. Ľcu`. Gm Ur`pGm Ur`p`sn±`sn±o˒om˒` 9.=.7(i) smifj`s quf cmm` 9.=.7(i) smifj`s quf cm
fxprfss˒fxprfssm˒`m`
W W  : ncd : ncd {{O O    ;;  W W ((xx)) ≧≧ O  O xx  pmrm t`g` pmrm t`g`   xx ∁∁ F F }},,
 ̀· ̀·ĽcĽc﬍j﬍j ̀̀ ··f f mtmtncnclnlng`g`..    ··F cmturmb fct˒F cmturmb fctm˒` njmlncmr quf cm fxprfss˒m` njmlncmr quf cm fxprfssm˒`m`
W W  : sup: sup{{W W ((xx)) ;; x x ∁∁ F F   ff  xx : : ==}},,
` su` suprfj` prfj` tmji·tmji·fj ·fj ·f mtf mtnclng`nclng`. . FctrftFctrftmct`, mct`, o`j` o`j` ` bf` bfnt`r nt`r vfrn﬍omvfrn﬍omr·r·m m c` c` Fxfro·Fxfro·Ľon` Ľon` 9.<.=9.<.==,=,
nssnss` ` cfj sfjprf mo`ctfcfj sfjprf mo`ctfof of f f ` ` mssmssuctuct̀ ` ··f f ifj ifj jmnjmns s gfbgfbnomgnomg` ` g` g` quf pquf p`gf `gf pmrfpmrfofr ofr eemm
prnjfnrm prnjfnrm vnstm. vnstm. Gnz-sf quf Gnz-sf quf uj `puj `pfrmg`r frmg`r bncfmr bncfmr o`cto`ct··Ľcu`Ľcu`  W W  ; ;   F F  ∙ ∙↙↙ D D  mtnclf m c`rjm  mtnclf m c`rjm    sfsf
fxnstffxnstf   xx ∁ ∁  F F  o`j c`rjm = tmb quf o`j c`rjm = tmb quf  W W ((xx))   ::  W W  ..  Vj rfsubtmg` pr`ducg`, o`cafong` Vj rfsubtmg` pr`ducg`, o`cafong`
o`j` Wf`rfjm gf Insa`p‟Uafbps, lmrmctf quf, sfo`j` Wf`rfjm gf Insa`p‟Uafbps, lmrmctf quf, sf   F F    ff   D D    s˒sm˒` fspm±m` fspm±o`s c`rjmg`s (s`irfo`s c`rjmg`s (s`irf
`s rfmns), fct˒`s rfmns), fctm˒` ` o`ckuct` g`sm` ` o`ckuct` g`s   ducon`cmns   ducon`cmns   bnc  bncfmrfs fmrfs o`cto`ct··Ľcu`s Ľcu`s quf quf mtnclmtnclfj fj m m c`rjm c`rjm ··ff
gfcs` fjgfcs` fj   F F ..  M  M j`c`lrm﬍m j`c`lrm﬍m X9?Z X9?Z ··f f ujm ujm i`m i`m rfdfr˄rfdfrf˄conm fconm fj fj p`rtulu˄p`rtuluf˄s fs s`irf s`irf ` ` mssuct`.mssuct`.
Umrm `utrms vfrs˒Umrm `utrms vfrs˒̀fs g` Wf`rfjm gf Insa`p‟Uafbps f rfbm±`fs g` Wf`rfjm gf Insa`p‟Uafbps f rfbm±o˒o˒̀fs `fs o`j o`j j·j·ft`ft`g`s g`s vmrnmon`cvmrnmon`cmnsmns
f mpbnom±f mpbnom±o˒o˒̀fs e`fs ems fqum±ms fqum±o˒o˒̀fs `fs gndfrfconmns gndfrfconmns pmronmns pmronmns fbfb··Ľptnoms Ľptnoms sulfrnj`s sulfrnj`s ` ` bnvr` X9>bnvr` X9>Z.Z.
@ Wf`rfjm gf Imnrf tfj v·@ Wf`rfjm gf Imnrf tfj v·mrnms d`rjubm±mrnms d`rjubm±o˒o˒̀fs fqunvmbfctfs (vfkm, p`r fxfjpb` X6?,`fs fqunvmbfctfs (vfkm, p`r fxfjpb` X6?,
_f±_f±o˒om˒` m` <.<Z<.<Z). ). FcFcuconuconmj`mj`s s c` c` WWf`rff`rfjm jm 9.?9.?.= .= mqumqufbm fbm quf ·quf ·f f jmnjmns s o`co`cvfvfcnfcnfctctf f pmrm `spmrm `s
c`ss`s pr`p·c`ss`spr`p·̀snt`s. M gfj`cstrm±`snt`s. M gfj`cstrm±o˒om˒` m` quf mprquf mprfsfctmj`s ·fsfctmj`s ·f m f m quf mquf mpmrfof pmrfof fj X<6Z.fj X<6Z.
@ @ WWf`rf`rfjm fjm 9.9.?.?.9 9 d`d`n n pr`pr v̀vmgmg` ` p`r p`r ImImcmcmoa oa f f _tf_tfncncamamus us fj fj uj uj mrmrtnltnl` ` o`o`ckuckuctct̀̀
puipuibnbnomgomg` ` fj =>9<fj =>9<. . @ O`r`@ O`r`b·b·mrn` mrn` 9.?.? 9.?.? tmji·tmji·fj fj ··f f oamjmg` oamjmg` gf gf WWf`rfjm f`rfjm gf gf Imcmoa‟Imcmoa‟
_tfncamu_tfncamus, f s, f mji`s s˒mji`s sm˒` tm` tmji·mji·fj ofj o`ca`cafonfong`s g`s o`jo`j` Ur` Urnco·nco·Ľpn` Ľpn` gm Bgm Bnjntnjntm±m±o˒om˒` Vcnd`rjf. _˒m` Vcnd`rjf. _m˒`m`
o`cafongo`cafongms ms mblujms gfj`cstrm±mblujms gfj`cstrm±o˒o˒̀fs g` Wf`rfjm gf Imcmoa‟_tfncamus quf c˒`fs g` Wf`rfjm gf Imcmoa‟_tfncamus quf cm˒` utnbnzmjm` utnbnzmj
` Wf`rfjm gf Imnrf ‟ vfkm, p`r fxfjpb` X=<, =7.=Z f X?>Z.` Wf`rfjm gf Imnrf ‟ vfkm, p`r fxfjpb` X=<, =7.=Z f X?>Z.
_. Imcmoa f A. _tfncamus s˒_. Imcmoa f A. _tfncamus sm˒` pfrs`cmlfcs ofctrmns gm Mc·m` pfrs`cmlfcs ofctrmns gm Mc·mbnsf Ducon`cmb f gm k·mbnsf Ducon`cmb f gm k·mm
bfcg·bfcg·mrnm fso`bm p`b`cfsm gf jmtfj·mrnm fso`bm p`b`cfsm gf jmtfj·mtnom g` mtnom g` s·s·foub` 95. foub` 95. M pmrtf M pmrtf gm ·gm ·mrv`mrv`rf rf lfclfcfmbfmb··̀ln`lnomom
(jmtfjmtnomjfctf dmbmcg`) gf Imcmoa f _tfncamus fxningm mimnx`, fj quf t`g`s ‟(jmtfjmtnomjfctf dmbmcg`) gf Imcmoa f _tfncamus fxningm mimnx`, fj quf t`g`s ‟
?4?4
  
fxoft` fxoft` Anbifrt, Anbifrt, ··f f obmr` obmr` ‟ ‟ s˒sm˒` jmtfj·m` jmtfj·mtno`s mtno`s p`b`cfsfs p`b`cfsfs gf gf lrmcgf lrmcgf rfc`jf rfc`jf nctfrcmon`cmb, ·nctfrcmon`cmb, ·ff
pr`vm gpr`vm gm m nc﬉u˄nc﬉uf˄conm fconm gm gm jmtfjjmtfj··mtnom mtnom p`b`cfsm p`b`cfsm m` m` b`cl` b`cl` gf gf t`g` t`g` ` ` s·s·foub` foub` 95 95 f f gfstf gfstf ncnc··Ľon`Ľon`
gf gf s·s·fofoubub ̀̀ 9= 9= ( ̀( ̀ ss··ĽjiĽji`b`b`̀   MM ↙↙ II  ncgnom quf ncgnom quf   MM  d`n `rnfctmg`r gf d`n `rnfctmg`r gf   II););
AnbifrtAnbifrt ↙↙ _tfncamus_tfncamus  ↙↙


HmoHmo
_oamugfr_oamugfr
@rbnoz@rbnoz ↙↙ OnfsnfbshnOnfsnfbshn  ↙↙ HwmpnfcHwmpnfc
ImcmoaImcmoa ↙↙



VbmjVbmj
JmzurJmzur ↙↙



UfUfbozy·bozy·cshncshn ↙↙



W`jozmh-KmfrlfrjmcW`jozmh-Kmfrlfrjmc
DnlnfbDnlnfb
\`ktmszozyh\`ktmszozyh
IfssmlmIfssmlm
Y`bfwnozY`bfwnoz
[fbmzh`[fbmzh`
UmrUmrm m dmzfdmzfr r kuskustn±tn±om om eem m jmtjmtfj·fj·mtnmtnom om p`b`cp`b`cfsm fsm gfvgfvfj`s fj`s jfcjfcon`con`cmr mr quf quf m m ··mrvmrv`rf `rf monmonjmjm
nbnbusustrm mpfcmtrm mpfcms s uj bmg` gm uj bmg` gm fsfso`bo`bm m p`bp`b`c`cfsmfsm, , m m smsmifr, ifr, ` ` rmjrmj` ` gf gf BwBw··̀w, `u ` rmj``w, `u ` rmj`
gf gf Mc·Mc·mbnmbnsf sf DDuconucoǹ cmb`cmb. . W˒Wm˒` m` nnjpjp`r`rtmtmcctf tf ququmcmct` t` ` ` rmrmj` j` gf gf BwBw··̀w `w ··f f ` ` rmrmj` j` gfgf
Tmrs·Tmrs·̀vnm, `vnm, quf quf ncobunncobun, , fctrf `utr`s fctrf `utr`s c`jfs c`jfs nbustrfnbustrfs, s, `s `s sfluncsflunctfs tfs jmtfj·jmtfj·mtno`s; mtno`s; [mrfj[mrfjim,im,
_nfrpn·_nfrpn·cshn, Jmzurhnfwnoz, Hurmt`wshn, _mhs, I`rsuh, [yljucg, Wmrshn, Jmronchnfwnozcshn, Jmzurhnfwnoz, Hurmt`wshn, _mhs, I`rsuh, [yljucg, Wmrshn, Jmronchnfwnoz
f f KmcnszKmcnszfwshn.fwshn.
@ Wf`rfjm gm Mpbnom±@ Wf`rfjm gm Mpbnom±o˒om˒` Mifrtm f ` Wf`rfjm g` Lr·m` Mifrtm f ` Wf`rfjm g` Lr·m﬍o` Dfoamg` d`rmj puibnomg`sm﬍o` Dfoamg` d`rmj puibnomg`s
p`r p`r ImcImcmomoa a fj fj =>?9 c` =>?9 c` bnvbnvr` r` X<ZX<Z, , quf ·quf ·f f c`rjc`rjmbjmbjfcfctf tf o`cso`csngfngfrmg` rmg` o`jo`j` ` ` ` jmrjmro` o` gmgm
ducgm±ducgm±o˒om˒` gm Mc·m` gm Mc·mbnsf mbnsf DDucon`ucon`cmb ocmb o`j` `j` gnsonpgnsonpbncm bncm jmtfjjmtfj··mtnommtnom. . M prnM prnjfnrm jfnrm gfj`cgfj`cstrm±strm±o˒om˒`m`
g` Wf`rfjm gm Mpbnom±g` Wf`rfjm gm Mpbnom±o˒om˒` Mim` Mifrtm ·frtm ·f gff gfvngm m vngm m K. _oamugfr K. _oamugfr fj =fj =>?5.>?5.
C` Wf`rfjm g` Lr·C` Wf`rfjm g` Lr·m﬍o` Dfoamg`, m njpbnom±m﬍o` Dfoamg`, m njpbnom±o˒om˒` m` ‑‑W W  o o`c`ct·t·ĽcĽcu ̀u ̀ ::⇙⇙ lr·lr·m﬍o` gfm﬍o` gf W  W   dfoamg`„  dfoamg`„
vmbf fj uj o`ctfxt` junt` jmns lfrmb, quf sfr·vmbf fj uj o`ctfxt` junt` jmns lfrmb, quf sfr·m m gfj`cstrmgm gfj`cstrmgm f f utnbnzmgm utnbnzmgm c` c` OmpOmp··Ľtub` Ľtub` 44
(vfkm Bfjm 4.9.1);(vfkm Bfjm 4.9.1);
Wf`rfjm 9.4.=Wf`rfjm 9.4.=   _fkmj   _fkmj   Q Q    f f   ^ ^   fspm±  fspm±o`s t`p`b· o`s t`p`b· `lno`s o`j `lno`s o`j   ^ ^    gf Amusg`rst. gf Amusg`rst. _f m _f m duc±duc±o˒ o˒ m`m`
d d ;;  Q Q  ∙ ∙↙↙ ^ ^    ··f f o`o`ct·ct·ĽcĽcumum, , fcfct˒  t˒  m` ` lr· m` ` lr· m﬍o` gf m﬍o` gf   d d  · ·f f dfoadfoamgmg ̀̀ fj fj   Q Q  Ü Ü ^ ^ ..
Vjm gms bn±Vjm gms bn±o˒o˒̀fs `fs ofctofctrmnrmns gfs gfstf stf omp·omp·ĽtuĽtub` ·b` ·f ` dmf ` dmt` gt` gf quf f quf pr`pr`prnprnfgmfgmgfs mgfs mbl·bl·firfirnomnoms mdfs mdftmjtmj,,
gf d`rjm snlcn﬍omtnvm, ` o`jp`rtmjfct` t`p`b·gf d`rjm snlcn﬍omtnvm, ` o`jp`rtmjfct` t`p`b·̀lno` gms duc±`lno` gms duc±o˒o˒̀fs fctrf fspm±`fs fctrf fspm±o`s c`rjmg`s.o`s c`rjmg`s.
A·A·m uj fxfjpb` c`t·m uj fxfjpb` c`t·mvfb, gfvng` m _. Jmzur f _. Vbmj (=>?9), gf o`j` pr`prnfgmgfsmvfb, gfvng` m _. Jmzur f _. Vbmj (=>?9), gf o`j` pr`prnfgmgfs
t`p`b·t`p`b·̀lnom`lnoms ts tmji·mji·fj fj p`p`gfj gfj mdftmmdftmr r ` ` o`jpo`jp`rtm`rtmjfct` jfct` mbl·mbl·firnofirno`;`;
Wf`rfjm 9.4.9Wf`rfjm 9.4.9   _fkmj   _fkmj    F F    f f    D D   fspm±  fspm±o`s o`s c`rc`rjmg`jmg`s s rrffmns mns f f    d d ;;   F F   ∙∙↙↙    D D   ujm duc±  ujm duc±o˒ o˒ m`m`
s`irfkft`rm tmb quf s`irfkft`rm tmb quf   d d (5) : 5(5) : 5   f f 
d d ((xx)) ∙∙ d d ((yy)) :: xx ∙∙ yy
pmrm qumnsqufr pmrm qumnsqufr   x,x, yy    fj fj   F F . Fct˒ . Fct˒ m`m`   d d  · ·f f bnbncfmcfmr.r.
Vjm gfj`cstrm±Vjm gfj`cstrm±o˒om˒` p`gf sfr fco`ctrmgm fj X17, Oampntrf Q, Bfoturf =Z.m` p`gf sfr fco`ctrmgm fj X17, Oampntrf Q, Bfoturf =Z.
?<?<
  
9.< 9.< FxfroFxfro··Ľon`sĽon`s
FxFxfro·fro·ĽoĽon ̀n ̀ 9.9.<.<.==   (m) Ur`vf quf t`g` fspm± (m) Ur`vf quf t`g` fspm±o` c`rjmg` gf gnjfcs˒o` c`rjmg` gf gnjfcsm˒` ﬍cntmm` ﬍cntm   cc   s`irf s`irf   HH  ··ff
ns`j`rd` m` fspm±ns`j`rd` m` fspm±o` fuobngnmc` (o` fuobngnmc` (HHcc,,
 ¼¼ 
99).).
(i) Ur`vf quf sf(i) Ur`vf quf sf F  F  f f  D  D   s˒sm˒` fspm±m` fspm±o`s c`rjmg`s, s`irf ` o`s c`rjmg`s, s`irf ` jfsj` o`rp`, gf jfsj` o`rp`, gf jfsjm gnjfcsjfsjm gnjfcs˒˒m`m`
﬍cntm, fct˒﬍cntm, fctm˒`m`   F F   ff   D D    s˒sm˒` ns`j`rd`s.m` ns`j`rd`s.
FxFxfro·fro·ĽoĽon ̀n ̀ 9.9.<.<.99    W`g` `W`g` `pfpfrmgrmg`r b`r bncfncfmr omr ouk` uk` g`j·g`j·Ľcn` Ľcn` ··f uf uj fsj fspm±pm±o` c`rjmg` gf o` c`rjmg` gf gnjfcs˒gnjfcsm˒`m`
﬍﬍cncnttm m ··f f o`o`ct·ct·ĽcĽcu`u`..
FxFxfro·fro·ĽoĽon ̀n ̀ 9.9.<.<.??  J`strf quf pmrm t`g` fspm± J`strf quf pmrm t`g` fspm±o` c`rjmg` gf gnjfcs˒o` c`rjmg` gf gnjfcsm˒` nc﬍cntmm` nc﬍cntm F  F  f t`g` f t`g`
fspm±fspm±o` c`rjmg`o` c`rjmg`  D  D   :: {{55}},, fxnstf  fxnstf uj uj `pfrmg`r `pfrmg`r bncfmr bncfmr gfso`ctgfso`ct··Ľcu`Ľcu`  W W  ; ;   F F  ∙ ∙↙↙ D D ..
FxFxfro·fro·ĽoĽon ̀n ̀ 9.9.<.<.77   _fkm  _fkm    F F   uj fspm±  uj fspm±o` c`rjmg` s`irfo` c`rjmg` s`irf   OO. . _f_f    ϖϖ   ·  ·f f uj uj ducon`cmb ducon`cmb bncfmrbncfmr
gfgfsoso`ct·`ct·ĽcuĽcu ̀̀ fjfj   F F , j`strf quf, j`strf quf { {ϖϖ((xx) ) ;; x x ∁∁ F F    ff  xx ≧≧ = =}} :: OO..
FxFxfro·fro·ĽoĽon ̀n ̀ 9.9.<.<.66   _fkmj  _fkmj    F F   uj   uj fsfspm±pm±o` o` c`rjc`rjmg`,mg`,    D D   uj fspm±  uj fspm±o` gf Imcmoa fo` gf Imcmoa f    LL    ujuj
suifspm±suifspm±o` gfo` gf    F.F.    _f_f    W W  ; ;   LL
  ∙∙↙↙
   D D   ·  ·f f bncfmr bncfmr f f o`cto`ct··Ľcum, Ľcum, j`strf j`strf quf quf fxnstf fxnstf ujm ujm ··ucnomucnom
fxtfcs˒fxtfcsm˒` m` bncbncfmr fmr o`ct·o`ct·Ľcum Ľcum gfgf  W  W  m` dfoa` gf m` dfoa` gf  L L. . J`strf J`strf tmji·tmji·fj qufj quf m f m c`rjm c`rjm gm gm fxtfcs˒fxtfcsm˒`m`
o`ncongf o`j m c`rjm gfo`ncongf o`j m c`rjm gf W  W  f quf sf f quf sf  W  W  · ·f ujf ujm nsm ns`jftr`jftrnm bnnm bncfmr, cfmr, fct˒fctm˒` m fxtfcs˒m` m fxtfcsm˒m ̀̀ tmtmji·ji·ffjj
··f.f.
FxFxfro·fro·ĽoĽon ̀n ̀ 9.9.<.<.44   _fkmj  _fkmj F  F  uj f uj fspmspm±±o` go` gf If Imcmmcmoa,oa, D  D  uj fspm± uj fspm±o` c`rjmg` fo` c`rjmg`f W  W  ∁ ∁ BB((F,F, D D ))
ujm ns`jftrnm bncfmr. J`strf qufujm ns`jftrnm bncfmr. J`strf quf   W W ((F F ) ) ··f f dfoadfoamg` mg` fjfj  D  D ..
FxFxfro·fro·ĽoĽon ̀n ̀ 9.9.<.<.<<   (m) Ur`vf quf sf uj fspm± (m) Ur`vf quf sf uj fspm±o` c`rjmg`o` c`rjmg`   F F  · ·f f ns`j`rd` ns`j`rd` m m uj uj fspm±fspm±o` gfo` gf
Imcmoa, fct˒Imcmoa, fctm˒`m`   F F  · ·f f ffssppmm±±o` gf Imcmoa.o` gf Imcmoa.
(i) J`strf quf ` ntfj (m) c˒(i) J`strf quf ` ntfj (m) cm˒` vmbf fj fspm±m` vmbf fj fspm±o`s o`s j·j·ftrftrno`no`s, s, nstnst` ·` ·f, f, fxnfxnstfstfj j fspfspm±m±o`o`s j·s j·ftftrnrno`o`ss
a`jf`j`rd`sa`jf`j`rd`s  J  J    ff  C  C    o`jo`j   J J  o`jpbft` f o`jpbft` f   C C    c˒cm˒`-o`jpbft`.m`-o`jpbft`.
(o) (o) O`j` O`j` v`o˄v`of˄ f fxpbnom fxpbnom m m gnsorfp˄gnsorfpm˄conm fctrf `s ntfcs (m) f (i)0mconm fctrf `s ntfcs (m) f (i)0
FxFxfro·fro·ĽoĽon ̀n ̀ 9.9.<.<.11   (m) _fkmj  (m) _fkmj   F F    ff   D D   fspm±  fspm±o`s c`rjmg`s fo`s c`rjmg`s f   W W  ; ;   F F 
 ∙ ∙↙↙
 D  D   b  bncfncfmr mr f f o`ct·o`ct·Ľcu`Ľcu`..
J`strf qufJ`strf quf
W W  : sup: sup
xx:5:5
W W ((xx))
xx   : sup  : supxx:=:=
W W ((xx)) : sup: sup
xx22==
W W ((xx))
: ncd : ncd {{O O    ;; W W ((xx)) ≧≧ O  O xx   pmrm t`g`  pmrm t`g`  x x ∁∁ F F }} ..
(i) Ur`vf quf(i) Ur`vf quf  W  W   ↙↙ W W  · ·f f ujm ujm c`rc`rjm jm fjfj B B((F,F, D D ).).
FxFxfro·fro·ĽoĽon ̀n ̀ 9.9.<.<.>>   _fkmj  _fkmj    W W   ∁ ∁ BB((F,F, D D ) ) ff    YY ∁  ∁ BB((D,D, LL). ). UrUr`v`vf quff quf  YY  ◤◤ W W   ∁ ∁ BB((F,F, LL) ) ff
YY ◤◤ W W  ≧≧ YY ¼¼ W W . . G˄Gf˄ uf uj fj fxfjpb` xfjpb` fj fj qufquf  YY ◤◤ W W  22 YY ¼¼ W W ..
FxFxfro·fro·ĽoĽon ̀n ̀ 9.9.<.<.=5=5   _fkmj  _fkmj F  F  f f  D  D   fspm±  fspm±o`s c`rjmg`s s`irf ` jfsj` o`rp`o`s c`rjmg`s s`irf ` jfsj` o`rp` HH. J`strf quf. J`strf quf
F F  ÜÜ D D    f f ((F F  Ü Ü D D ))   s˒sm˒` ns`jftrnomjfctf ns`j`rd`s o`csngfrmcg`-sf fjm` ns`jftrnomjfctf ns`j`rd`s o`csngfrmcg`-sf fj   F F   ÜÜ D D   m c`rjm  m c`rjm
 ¼¼ ∓∓   f f fjfj   F F  ÜÜ D D   m c`rjm m c`rjm   ¼¼ ==  (vfkm ms gf﬍cn± (vfkm ms gf﬍cn±o˒o˒̀fs `fs gms gms c`rjms c`rjms c` c` FxfroFxfro··Ľon` Ľon` =.1.=9).=.1.=9).
?1?1
  
FxFxfro·fro·ĽoĽon ̀n ̀ 9.9.<.<.====   (  (Vj duVj ducon`ccon`cmb bnmb bncfmr cfmr o`cto`ct··Ľcu` qĽcu` quf cuf c˒˒m` mtnclf m c`rjm) _fkmm` mtnclf m c`rjm) _fkm
ϖϖ ;;   oo55 ∙ ∙↙↙ HH  , , ϖϖ((((mm k k))∓∓ k k:=:=) ) ::
∓∓

 k k:=:=
mm k k
99 k k
..
Ur`vf quf;Ur`vf quf;
(m)(m)   ϖϖ  ∁∁ ((oo55))   (c˒(cm˒` sf fsquf±m` sf fsquf±om gf pr`vmr, prnjfnr`, qufom gf pr`vmr, prnjfnr`, quf   ϖϖ  fst· fst·m ifj gf﬍cng`).m ifj gf﬍cng`).
(i)(i)  ϖϖ : : =.=.
(o) C˒(o) Cm˒` fxnstfm` fxnstf   xx ∁∁ oo55  tmb quf tmb quf  xx ≧≧ = = ff  ϖϖ :: ||ϖϖ((xx))||..
FxFxfro·fro·ĽoĽon ̀n ̀ 9.9.<.<.=9=9   _fkmj  _fkmj   F F    ff   D D   fspm±  fspm±o`s c`rjmg`s fo`s c`rjmg`s f   W W  ; ;   F F  ∙ ∙↙↙   D D   uj `pfrmg`r bncfmr  uj `pfrmg`r bncfmr
oo`c`ct·t·ĽcĽcu`u`..
(m) Ur`vf quf `(m) Ur`vf quf ` c·  c· uobf` gf uobf` gf  W  W , gf﬍cng` p`r hfr(, gf﬍cng` p`r hfr(W W ) ) :: {{xx ∁∁ F F    ;; W  W ((xx) ) : : 55}}, ·, ·f f uj uj suisuifspfspm±m±o`o`
dfoamg` gfdfoamg` gf   F F ..
(i) Ur`vf quf m(i) Ur`vf quf m   njmlfj gf  njmlfj gf   W W , gf﬍cngm p`r Nj(, gf﬍cngm p`r Nj(W W ) ) :: { {W W ((xx) ) ;; x x ∁ ∁  F F }}, , ··f f uj uj suisuifspm±fspm±o`o`
gfgf   D D . Dfoamg`0. Dfoamg`0
FxFxfro·fro·ĽoĽon ̀n ̀ 9.9.<.<.=?=?* O`csngfrf* O`csngfrf   Q Q    ::   YY99 jucng` o`j m c`rjmjucng` o`j m c`rjm
  
((x,x, yy))

   ;: ;: ((xx77 ++ y y77))==//77..
Omboubf fxpbnontmjfctf m c`rjm gf omgm ducon`cmbOmboubf fxpbnontmjfctf m c`rjm gf omgm ducon`cmb   ϖϖ  ∁∁   Q Q ..
FxFxfro·fro·ĽoĽon ̀n ̀ 9.9.<.<.=7=7* (Mpbnom±* (Mpbnom±o˒om˒` qu`onfctf) _fkmjm` qu`onfctf) _fkmj   J J  uj suifspm± uj suifspm±o` dfoamg` g` fspm±o` dfoamg` g` fspm±o`o`
c`rjmg`c`rjmg`   F F   ff   υυ ;;   F F  ∙ ∙↙↙ F/J F/J  gmgm p`r gmgm p`r   υυ((xx) ) : : XXxxZ Z (vfkm (vfkm FxfroFxfro··Ľon` Ľon` =.1.=7). =.1.=7). Ur`vf quf;Ur`vf quf;
(m)(m)  υυ((xx)) ≧≧ xx  pmrm t`g` pmrm t`g`   xx ∁∁ F F ..
(i) Gmg`s(i) Gmg`s  x x ∁∁ F F   ff   ζ ζ 33 5, fxnstf 5, fxnstf   yy ∁ ∁ F F  tmb quf tmb quf   υυ((xx) ) :: υ υ((yy) ) ff  yy ≧≧ υυ((xx)) ++ ζζ..
(o) M njmlfj gm i`bm ucnt·(o) M njmlfj gm i`bm ucnt·mrnm mifrtm gfmrnm mifrtm gf   F F   p`rp`r   υυ  · ·f f m m i`i`bm bm ucnucnt·t·mrnm mifrtm gfmrnm mifrtm gf   F/J F/J ..
(g)(g)   υυ ∁ ∁ BB((F,F/J F,F/J ), hfr(), hfr(υυ) ) :: J  J    ff   υυ  · ·f f ujujm m mpmpbnobnom±m±o˒om˒` mifrtm.m` mifrtm.
(f) _f(f) _f   J J   :: F  F , fct˒, fctm˒`m`  υυ : : =.=.
(d) Umrm t`g` fspm±(d) Umrm t`g` fspm±o` c`rjmg`o` c`rjmg`   D D    f f t`g` t`g` `pfrmg`r `pfrmg`r bncfmr bncfmr o`cto`ct··Ľcu`Ľcu`   W W  ; ;   F F  ∙ ∙↙↙   D D   fxnstf  fxnstf
uj uj ··ucno` ucno` `pfrmg`r `pfrmg`r bncfmr bncfmr o`cto`ct··Ľcu`Ľcu`    ˒˒W W  ; ;   F/J F/J   ∙∙↙↙    D D   tmb quf  tmb quf   W W    ::    ˒˒W W  ◤ ◤ υυ. . JmnJmns mns mncgmcgm,,
W W  ::  ˒˒W W ..
FxFxfro·fro·ĽoĽon ̀n ̀ 9.9.<.<.=6=6   (W  (Wf`rff`rfjm jm g` g` Ns`jNs`j`r﬍s`r﬍sj`) j`) _fkm_fkmjj    F F    ff    D D   fspm±  fspm±o`o`s s gf gf ImImcmcmooa a ffW W  ∁ ∁ BB((F,F, D D ). Ur`vf quf sf). Ur`vf quf sf   W W ((F F ) ) ··f f dfoadfoamg` mg` fjfj   D D   fct˒  fctm˒`m`  W  W ((F F ) ) ··f f ns`ns`j`rj`rd` d` mm   F/F/hfr(hfr(W W ).).
FxFxfro·fro·ĽoĽon ̀n ̀ 9.9.<.<.=4=4   _fkmj  _fkmj   F F    ff   D D   fspm±  fspm±o`s c`rjmg`s. _fo`s c`rjmg`s. _f   W W  ∁ B ∁ B((F F 88 D D ) f fxnstf) f fxnstf   o o 33 5 tmb 5 tmb
qufquf  W W ((xx))  ≦≦    oo xx  pmrm t`g` pmrm t`g`   xx ∁ ∁   F F , j`strf quf ` `pfrmg`r ncvfrs`, j`strf quf ` `pfrmg`r ncvfrs`   W W ∙∙== fxfxnsnstf tf f f ··ff
oo`c`ct·t·ĽcĽcu`u`..
FxFxfro·fro·ĽoĽon ̀n ̀ 9.9.<.<.=<=<   _fkmj  _fkmj    F F    ff    D D   fspm±  fspm±o`s c`rjo`s c`rjmg`smg`s. . J`stJ`strf quf uj `pfrmg`r bncrf quf uj `pfrmg`r bncfmrfmr
W W  ; ;   F F   ∙∙↙↙    D D   ·  ·f f o`cto`ct··Ľcu` Ľcu` sf, sf, f f s`jfctfs`jfctf, , sfsf    W W ((MM) ) ··f f bnjntmbnjntmg` g` fjfj    D D   sfjprf quf  sfjprf quf    MM    d`rd`r
bnjntmg` fjbnjntmg` fj  F F  (vfkm m gf﬍cn± (vfkm m gf﬍cn±o˒om˒` m` gf gf o`ckuct` o`ckuct` bnjntmg` bnjntmg` c` c` FxfroFxfro··Ľon` Ľon` =.1.>).=.1.>).
FxFxfro·fro·ĽoĽon ̀n ̀ 9.9.<.<.=1=1   _fkmj  _fkmj    F F    ff    D D   fspm±  fspm±o`s c`o`s c`rjrjmg`mg s̀. s. J`sJ`strf qutrf quf sff sf    W W  ; ;   F F   ∙∙↙↙    D D    ··ff
ns`j`ns`j`r﬍sj` r﬍sj` ns`j·ns`j·ftrnoftrno`, `, fct˒fctm˒`m`  W W  : =. U`r `utr` bmg`, fco`ctrf uj fspm±: =. U`r `utr` bmg`, fco`ctrf uj fspm±o` gf Imcmoao` gf Imcmoa
F F   f uj ns`j`r﬍sj` t`p`b·  f uj ns`j`r﬍sj` t`p`b·̀lno``lno`    W W  ; ;   F F   ∙∙↙↙    F F    o`jo`j  W W    : =, quf c˒  : =, quf cm˒` m` ··f f ns`j`rns`j`r﬍sj`﬍sj`
nsns`j`j··ftftrnrno`o`..
?>?>
  
FxFxfro·fro·ĽoĽon ̀n ̀ 9.9.<.<.=>=>   J`strf quf qumnsqufr g`ns fspm±  J`strf quf qumnsqufr g`ns fspm±o`s gfctrf `s fspm±o`s gfctrf `s fspm±o`s (o`s (YYcc,,  ¼  ¼ ==),),
((YYcc,,  ¼¼ 99) ) f f ((YYcc,,  ¼¼ ∓∓) ) c˒cm˒ ̀m ̀ s˒sm˒` m` ns`j`rdns`j`rd̀ s `s ns`jftrns`jftrnomjfcnomjfctf.tf.
FxFxfro·fro·ĽoĽon ̀n ̀ 9.9.<.<.9595    (@pfrmg`r(@pfrmg`rfs gf fs gf p`st` p`st` ﬍cnt`) Gnz-sf ﬍cnt`) Gnz-sf quf uj `quf uj `pfrmg`r pfrmg`r bncfmr o`ctbncfmr o`ct··Ľcu`Ľcu`
W W  ; ;   F F  ∙ ∙↙↙ D  D  · ·f f ggff  p`st` ﬍cnt` p`st` ﬍cnt`  sf m njmlfj gf sf m njmlfj gf   W W  · ·f f uj uj suisuifspfspm±m±o` gf gnjfcs˒o` gf gnjfcsm˒` ﬍cntm gfm` ﬍cntm gf
D D ..
(m) Ur`vf quf(m) Ur`vf quf   W W  · ·f f gf gf p`st` p`st` ﬍cnt` sf, ﬍cnt` sf, f f s`jfcs`jfctf, tf, sf sf fxnstfjfxnstfj  cc ∁ ∁   CC,,   ϖϖ==, . . . , ϖ, . . . , ϖcc ∁ ∁   F F     ff
yy==, . . . , y, . . . , ycc ∁ ∁ D D   tmns quf tmns quf  W  W    :: ϖ ϖ== ⊜⊜ yy== + + ¼¼ ¼¼ ¼¼ ++ ϖϖcc ⊜⊜ yycc..
(i) (i) Ur`vUr`vf f quf quf ` ` o`ckuco`ckuct` t` g`s `pg`s `pfrmg`rfs bncfmrfs o`ctfrmg`rfs bncfmrfs o`ct··Ľcu`Ľcu s̀ s gf gf p`st` p`st` ﬍cnt` gf﬍cnt` gf   F F    fjfj   D D 
··f f uj uj suisuifspfspm±m±o` vft`rnmb gfo` vft`rnmbgf B B((F,F, D D ). Dfoamg`0). Dfoamg`0
FxFxfro·fro·ĽoĽon ̀n ̀ 9.9.<.<.9=9=    Gnz-sf qGnz-sf quf uj suifspm±uf uj suifspm±o`o` J  J  gf uj fspm± gf uj fspm±o` vft`rnmbo` vft`rnmb F  F  tf tfjj g gf﬍f﬍onon˄˄fcfcononm m 
 ﬍cntm  ﬍cntm   sf fxnstnr uj c·  sf fxnstnr uj c·ujfr` ﬍cnt` gf vft`rfsujfr` ﬍cnt` gf vft`rfs    xx==, . . . , x, . . . , xcc    fjfj    F F  ∙ ∙   J J   tmns quf  tmns quf    F F    ::
XXJ J  ∫ ∫ {{xx==, . . . , x, . . . , xcc}}Z. J`strf quf ` c·Z. J`strf quf ` c·uobf` gf uj ducon`cmb bncfmr c˒uobf` gf uj ducon`cmb bncfmr cm˒`-cub`m`-cub`  W W  ; ;   F F  ∙ ∙↙↙ H H
tfj (fjtfj (fj  F F ) ) gf﬍ogf﬍on˄nf˄conm fconm ﬍cntm ﬍cntm o`jo`j   cc :  : ==..
FxFxfro·fro·ĽoĽon ̀n ̀ 9.9.<.<.9999   _fkmj =  _fkmj = ≧≧  p p ≧≧ q q  ≧ ≧ ∓∓..  J`strf quf J`strf quf    p p ⊄ ⊄ qq  f quf m ncobus˒ f quf m ncobusm˒`m`    p p  ↙↙  qq
··f f uj uj `pfrmg`r `pfrmg`r bncfmr bncfmr o`cto`ct··Ľcu`. Ľcu`. Gftfrjncf Gftfrjncf m m c`rjm c`rjm gfssm gfssm ncobus˒ncobusm˒`.m`.
FxFxfro·fro·ĽoĽon ̀n ̀ 9.9.<.<.9?9?  O`csngfrf ` o`ckuct` O`csngfrf ` o`ckuct`
osos : :

((mm k k))
∓∓
 k k:=:=  ; ;  m m k k ∁ ∁ HH pmrm t`g` pmrm t`g`   kk ∁ ∁ CC  ff
∓∓
 k k:=:=
mm k k  o`cvfrlf o`cvfrlf

..
(m) J`strf quf(m) J`strf quf   osos · ·f f uj uj fsfspmpm±±o` c`rjmg` o`j ms `pfrm±o` c`rjmg` o`j ms `pfrm±o˒o˒̀fs `fs usumns usumns gf gf sfqu˄sfquf˄conms fconms f f o`jo`j
m c`rjmm c`rjm
((mm k k))∓∓ k k:=:=
 : sup: sup
cc∁∁CC

cc
 k k:=:=
mm k k
 ..
(i) Ur`vf quf(i) Ur`vf quf   osos  · ·f f ns`j`rd` ns`j`rd` ns`jftrnomjfctf ns`jftrnomjfctf m` m` fspm±fspm±o`o`   oo  gms  gms sfqu˄sfquf˄conmfconms s o`cvfrlfo`cvfrlfctfsctfs
gf﬍cgf﬍cng` ng` c` c` Fxfro·Fxfro·Ľon` Ľon` =.1.9<=.1.9<..
FxFxfro·fro·ĽoĽon ̀n ̀ 9.9.<.<.9797  O`csngfrf ` o`ckuct` O`csngfrf ` o`ckuct`
isis : :

((mm k k))
∓∓
 k k:=:=  ;; m m k k ∁ ∁ HH  pmrm t`g` pmrm t`g`   kk ∁ ∁ CC  f sup  f sup
cc∁∁CC

cc
 k k:=:=
mm k k
 22 ∓∓

..
(m) J`strf quf(m) J`strf quf   isis · ·f f uj uj fsfspmpm±±o` c`rjmg` o`j ms `pfrm±o` c`rjmg` o`j ms `pfrm±o˒o˒̀fs `fs usumns usumns gf gf sfqu˄sfquf˄conms fconms f f o`jo`j
m c`rjmm c`rjm
((mm k k))∓∓ k k:=:=
 : sup: sup
cc∁∁CC

cc
 k k:=:=
mm k k
 ..
(i) Ur`vf quf(i) Ur`vf quf  is is  · ·f f ns`j`ns`j`rd` rd` ns`jfns`jftrnomtrnomjfctf jfctf mm   ∓∓..
FxFxfro·fro·ĽoĽon ̀n ̀ 9.9.<.<.9696   _fkmj  _fkmj  F, D F, D,, LL  fspm± fspm±o`s c`rjmg`s fo`s c`rjmg`s f   MM ;;   F F  Ü Ü D D  ∙ ∙↙↙ D  D  ujm mpbnom± ujm mpbnom±o˒om˒`m`
inbncfmr. O`csngfrf fjinbncfmr. O`csngfrf fj F  F ÜÜD D  qum qumbqufr ujm bqufr ujm gms c`rjms gms c`rjms g` Fxfro·g` Fxfro·Ľon` =.1.=9. Ľon` =.1.=9. Ur`vf qufUr`vf quf
ms sflunctfs o`cgn±ms sflunctfs o`cgn±o˒o˒̀fs s˒`fs sm˒` fqunvmbfctfs;m` fqunvmbfctfs;
7575
  
(m)(m)   MM · ·f f oo`̀ct·ct·ĽĽcucumm..
(i)(i)   MM · ·f f o`o`ct·ct·ĽcĽcum um cm cm `r`rnlnlfjfj..
(o) sup(o) sup{{MM((x,x, yy)) ;; x x ∁∁ IIF F ,, yy ∁ ∁ IIyy}} 22 ∓∓..
(g) Fxnstf(g) Fxnstf  O  O 
 ≦ ≦
5 tmb quf5 tmb quf
  
MM((x,x, yy))
 ≧≧
O O 

xx
 ¼¼ 
yy

 pmrm t`g`s pmrm t`g`s   xx
∁∁
F F    ff   yy
 ∁ ∁
D D ..
FxFxfro·fro·ĽoĽon ̀n ̀ 9.9.<.<.9494   _fkm =  _fkm =   ≧≧    p p 22  ∓∓..    _f _f ((ii k k))∓∓ k k:=:=   ·  ·f f ujm ujm sfqsfqu˄uf˄cofconm nm cuj·cuj·frnfrnom om tmb tmb qufquf
∓∓
 k k:=:=
mm k kii k k   22 ∓∓ pmrpmrm t`m t`gm sgm sfqu˄fquf˄cofconm (nm (mmcc))∓∓cc:=:= ∁ ∁  p p,, j`strf quf ( j`strf quf (iicc))∓∓cc:=:= ∁ ∁  p p∜∜,, `cgf `cgf    == p p++    == p p∜∜    : : ==..
FxFxfro·fro·ĽoĽon ̀n ̀ 9.9.<.<.9<9<   _fkmj  _fkmj    F,F, D D    fspm±  fspm±o`o`s s gf gf IImcmcmomoa a ff    WW,, W W ==,, W W 99, . . . ,, . . . ,   `pfr  `pfrmg`rmg`rfs fs fjfj
BB((F,F, D D ) tmns quf) tmns quf   W W cc((xx)) ∙ ∙↙↙   W W ((xx) pmrm t`g`) pmrm t`g`   xx ∁ ∁   F F . . J`stJ`strf quf, pmrm t`g` o`jrf quf, pmrm t`g` o`jpmotpmot̀̀
H H  ⊄ ⊄ F F , sup, sup
xx∁∁H H 
W W cc((xx)) ∙∙ W W ((xx)) ∙∙↙↙ 5.5.
FxFxfro·fro·ĽoĽon ̀n ̀ 9.9.<.<.9191  O`jpbftf `s gftmbafs g` Fxfjpb` 9.7.?. O`jpbftf `s gftmbafs g` Fxfjpb` 9.7.?.
FxFxfro·fro·ĽoĽon ̀n ̀ 9.9.<.<.9>9>* Ur`vf m sflunctf vfrs˒* Ur`vf m sflunctf vfrsm˒` jmns d`rtf g` Bfjm 9.7.=; _fkmjm` jmns d`rtf g` Bfjm 9.7.=; _fkmj  F  F   fspm± fspm±o`o`
gf gf ImcImcmoamoa,,    D D   fspm±  fspm±o` o` c`c`rjrjmgmg` ` ff    W W 
   ∁ ∁ BB
((F F 88 D D ))..   _f   _f fxnfxnstnstnrfjrfj   Y , r 3  Y , r 3   5   5 tmtmns qufns quf
W W ((IIF F (58(58 YY)))) ⊌⊌ IID D (58(58 rr)),, fct˒ fctm˒`m`   W W ((IIF F (58(58 YY)))) ⊌⊌ IID D (58(58 rr))..
FxFxfro·fro·ĽoĽon ̀n ̀ 9.9.<.<.?5?5   _fkmj  _fkmj   ¼¼ ==   ff   ¼¼ 99  gums c`rjms fj uj fspm± gums c`rjms fj uj fspm±o` vft`rnmbo` vft`rnmb  F F  tmns quf tmns quf
((F,F,  ¼¼ ==) ) f f ((F,F,  ¼¼ 99) ) s˒sm˒` o`jpbft`s.m` o`jpbft`s.
(m) (m) _up_up`ca`cam m ququff  xxcc==   ∙∙↙↙   5   5 sfsfjpjprf rf njnjpbpbnqnquf uf ququff  xxcc99   ∙∙↙↙   5  5. . UrUr̀̀ vvf quf quf msf ms
gums c`rjms s˒gums c`rjms sm˒` fqunvmbfctfs (vfkm gf﬍cn±m` fqunvmbfctfs (vfkm gf﬍cn±o˒om˒` m` c` c` FxfrFxfroo··ĽonĽon` ` =.1=.1.?). .?). Fj Fj pmrtpmrtnounoubmrbmr, , mm
o`cvfo`cvfrl˄rlf˄cofconm nm gf gf ujm ujm sfqsfqu˄uf˄cofconm nm (c˒(cm˒` cfofssmrnmjfctf pmrm zfr`) fj ujm gms c`rjmsm` cfofssmrnmjfctf pmrm zfr`) fj ujm gms c`rjms
njpbnom fj njpbnom fj o`cvfrlo`cvfrl˄˄fconm fconm (pmrm (pmrm ` jfsj` ` jfsj` bnjntf) cbnjntf) cm `utm `utrm c`rm c`rjm.rjm.
(i) _f fxnstf(i) _f fxnstf    oo   tmb quf  tmb quf  xx==  ≧≧    oo xx99   pmrm t`g`  pmrm t`g`    xx  ∁∁    F F , fct˒, fctm˒` ms gums c`rjms s˒m` ms gums c`rjms sm˒`m`
fqunvmbfctfs.fqunvmbfctfs.
(o) _f m t`p`b`lnm lfrmgm pfbm c`rjm(o) _f m t`p`b`lnm lfrmgm pfbm c`rjm   ¼¼ ==   fst· fst·m o`ctngm cm t`p`b`lnm lfrmgm p`rm o`ctngm cm t`p`b`lnm lfrmgm p`r   ¼¼ 99,,
fct˒fctm˒` ms gum` ms gums c`rjms s˒ms c`rjms sm˒` fqunvmbfctfs. m` fqunvmbfctfs. Fj pmrtnoubmr, Fj pmrtnoubmr, ms t`pms t`p`b`lnms lfrmgms `b`lnms lfrmgms pfbms gumpfbms gumss
c`rjms c`rjms o`ncongo`ncongfj.fj.
FxFxfro·fro·ĽoĽon ̀n ̀ 9.9.<.<.?=?=   _fkmj  _fkmj   F F   uj fspm±  uj fspm±o` gf Imcmoa fo` gf Imcmoa f   W W  ; ;   F F  ∙ ∙↙↙   F F   uj `pfrmg`r bncfmr uj `pfrmg`r bncfmr
tmb quftmb quf  W  W ((xx)()(yy) ) :: W  W ((yy)()(xx) pmrm t`g`s) pmrm t`g`s   x,x, yy ∁ ∁ F F . Ur`vf quf. Ur`vf quf  W  W  · ·f f oo`̀ct·ct·ĽĽcucu`̀..
FxFxfro·fro·ĽoĽon ̀n ̀ 9.9.<.<.?9?9* _fkmj* _fkmj   F F  uj fspm± uj fspm±o` gf Imcmoa fo` gf Imcmoa f   W W  ; ;   F F  ∙ ∙↙↙ F  F   uj `pfrmg`r bncfmr uj `pfrmg`r bncfmr
tmb quftmb quf  W  W ((xx)()(xx)) ≦≦ 5 pmrm t`g`5 pmrm t`g`   xx ∁∁ F F . Ur`vf quf. Ur`vf quf   W W  · ·f f oo`̀ct·ct·ĽĽcucu`̀..
FxFxfro·fro·ĽoĽon ̀n ̀ 9.9.<.<.????  Tfrn﬍quf quf, c` Wf`rfjm g` Lr· Tfrn﬍quf quf, c` Wf`rfjm g` Lr·m﬍o` Dfoamg`, m njpbnom±m﬍o` Dfoamg`, m njpbnom±o˒om˒`m`
W W   o`ct·  o`ct·ĽcuĽcu ̀̀ ::⇙⇙ LL((W W ) dfoamg` fj) dfoamg` fj   F F   ÜÜ D D 
o`ctncum v·o`ctncum v·mbngm pmrm `pfrmg`rfs bncfmrfs fctrf fspm±mbngm pmrm `pfrmg`rfs bncfmrfs fctrf fspm±o`s c`rjmg`s c˒o`s c`rjmg`s cm˒` m` cfofssmrncfofssmrnmjfctmjfctff
o`jpbft`s.o`jpbft`s.
7=7=
  
FxFxfro·fro·ĽoĽon ̀n ̀ 9.9.<.<.?7?7  U Ur`vf quf ` r`vf quf ` `pf`pfrmg`r rmg`r bncfmr bncfmr gfsogfso`ct`ct··Ľcu`Ľcu`   W W ∙∙== g` Fxfjpb` 9.7.? tfjg` Fxfjpb` 9.7.? tfj
lr·lr·m﬍o` m﬍o` dfoamg`. dfoamg`. O`cobum O`cobum quf, c` quf, c` WWf`rfjm f`rfjm g` Lr·g` Lr·m﬍o` m﬍o` DDfoamg`, m foamg`, m njpbnom±njpbnom±o˒om˒`m`
LL((W W ) dfoamg` fj) dfoamg` fj   F F 
  ÜÜ
D D    ::
⇙⇙
W W   o  o`̀ct·ct·ĽcĽcuu`̀
c˒cm˒m` ` o`o`ccttnnccuum m vv··mbmbngngm m pmpmrm rm `p`pfrfrmgmg`r`rfs fs bnbncfcfmrmrfs fs fcfctrtrf f fsfspmpm±±oo`s `s cc`̀rjrjmmgg`s `s cc˒˒m`m`
cfofssmrnmjfctf o`jpbft`s.cfofssmrnmjfctf o`jpbft`s.
FxFxfro·fro·ĽoĽon ̀n ̀ 9.9.<.<.?6?6   _fkmj  _fkmj F  F  uj fspm± uj fspm±o` gf Imcmoa,o` gf Imcmoa, D  D  uj fspm± uj fspm±o` c`rjmg` fo` c`rjmg` f W  W  ; ;   F F  ∙ ∙↙↙ D D 
uj `pfrmg`r bncfmr gf lr·uj `pfrmg`r bncfmr gf lr·m﬍o` dfom﬍o` dfoamg`amg .̀ . J`sJ`strf quf trfquf sf ` sf ` `pfrmg`pfrmg`r ncv`r ncvfrs`frs`  W W ∙∙== fxnstf ffxnstf f
··f f oo`c`ct·t·ĽcĽcu`u`, , fcfct˒t˒m` m njmlfj Nj(m` m njmlfj Nj(W W ) ·) ·f f susuiifsfspmpm±±o` dfoamg` gfo` dfoamg` gf   D D ..
FxFxfro·fro·ĽoĽon ̀n ̀ 9.9.<.<.?4?4   _  _ffkkm m ((xx k k))
∓∓
 k k:=:=   ujm   ujm sfqusfqu˄˄fconm fconm c` c` fspm±fspm±o` o` gf gf ImImcmcmooaa    F F    tm  tmb b ququff
∓∓
 k k:=:=
||ϖϖ((xx k k))|| 22 ∓∓  pmrm t`g` pmrm t`g`   ϖϖ  ∁∁ F F ..  J` J s̀trstrf quf f quf supsup
ϖϖ≧≧==
∓∓
 k k:=:=
||ϖϖ((xx k k))|| 22 ∓∓..
7979
  
OOmmpp··ĽĽttuubb ̀̀ ??
Wf`rfjms gf Amac‟ImcmoaWf`rfjms gf Amac‟Imcmoa
··F F gndgnd··Ľonb Ľonb jfcsurmr jfcsurmr m m njp`rt˄njp`rtm˄conm g` Wf`rfjm gf Amac‟Imcmoa cm Mc·mconm g` Wf`rfjm gf Amac‟Imcmoa cm Mc·mbnsf Ducon`cmb,mbnsf Ducon`cmb,
tmct`s s˒tmct`s sm˒` sfus o`r`b·m` sfus o`r`b·mrn`s f sums mpbnom±mrn`s f sums mpbnom±o˒o˒̀fs. `fs. Fco`ctrm Fco`ctrm tmji·tmji·fj fj mpbnom±mpbnom±o˒o˒̀fs fj `utrms`fs fj `utrms
··mrfms gm jmtfj·mrfms gm jmtfj·mtnom8 p`r fxfjpb`, Mc·mtnom8 p`r fxfjpb`, Mc·mbnsf O`jpbfxm, Wf`rnm gm Jfgngm, Wf`rnm g`mbnsf O`jpbfxm, Wf`rnm gm Jfgngm, Wf`rnm g`
O`ctr`O`ctr b̀f, bf, Ur`lrmjm±Ur`lrmjm±o˒om˒` m` O`cvO`cvfxm fxm f f WWf`rnm f`rnm g`s g`s K`l`s K`l`s (pmrm (pmrm rfdfrrfdfr˄˄fconms fconms prfonsms vfkmprfonsms vfkm
X4<, p·X4<, p·mlncm mlncm =>7Z)=>7Z). . Mpbnom±Mpbnom±o˒o˒̀fs rfofctfs e`fs rfofctfs em Mc·m Mc·mbnsf O`jincmt·mbnsf O`jincmt·̀rnm p`gfj sfr fco`ctrmgms`rnm p`gfj sfr fco`ctrmgms
fj X?4Zfj X?4Z. . Cm vfrCm vfrgmgmgf, gf, sfsfu u mbmbomomcof ubtrcof ubtrmpmpmssmssm m `s bnjn`s bnjntfs gm tfs gm jmtjmtfj·fj·mtnmtnom, om, vvfkmfkm, , p`rp`r
fxfjpb`, ujm mpbnom±fxfjpb`, ujm mpbnom±o˒om˒` m` eem Wfrj`gnc˄m Wfrj`gncm˄jnom fmjnom fj X9<Z. j X9<Z. Ur`vUr`vmrfj`s mrfj`s cfstf cfstf ompomp··Ľtub` v·Ľtub` v·mrnmsmrnms
d`rjms g` Wf`rfjm gm Amac‟Imcmoa, o`jf±d`rjms g` Wf`rfjm gm Amac‟Imcmoa, o`jf±omcg` omcg` o`j rfo`j rfsubtmgsubtmg`s pur`s purmjfctf mjfctf mbl·mbl·firnofirno`s,`s,
pmssmcg` p`r rfsubtmg`s t`p`b·pmssmcg` p`r rfsubtmg`s t`p`b·̀lno`s, prnjfnr` c` oms` rfmb f gfp`ns c` oms` o`jpbfx`,`lno`s, prnjfnr` c` oms` rfmb f gfp`ns c` oms` o`jpbfx`,
f f oubjncmcg` oubjncmcg` cms cms d`rjms d`rjms lf`j·lf`j·ftrnoms ftrnoms g` g` tf`rfjm, tf`rfjm, tmji·tmji·fj fj oamjmgms oamjmgms gf gf tf`rfjms tf`rfjms gfgf
sfpmrm±sfpmrm±o˒om˒`. m`. MblujmMblujms s prnjfnprnjfnrms rms o`csfquo`csfqu˄˄fconms gfssms fconms gfssms gnvgnvfrsms d`rjms frsms d`rjms g` g` WWf`rfjm gff`rfjm gf
AmaAmac‟Ic‟Imcmmcmoa toa tmji·mji·fj fj sfrsfr˒˒m` fxpb`rmgms.m` fxpb`rmgms.
?.= ?.= @ tf`rfjm gm fxtfcs@ tf`rfjm gm fxtfcs˒˒m` gf Amac‟Imm` gf Amac‟Imcmocmoaa
M M fss˄fssf˄conm fconm g` g` WWf`rfjm f`rfjm gf gf Amac‟Imcmoa, Amac‟Imcmoa, fj fj sum sum vfrs˒vfrsm˒` m` pmrpmrm m fspfspm±m±o`s o`s c`rc`rjmgjmg`s, `s, ··f f qufquf
ducon`cmns ducon`cmns bncfmrfs bncfmrfs o`cto`ct··Ľcu`s gĽcu`s gf﬍cng`s f﬍cng`s fj fj uj uj suifspm±suifspm±o`o`   LL  gf uj fspm± gf uj fspm±o` c`rjmg`o` c`rjmg`   F F p`gfj sfr fstfcgng`s m t`g` ` fspm±p`gfj sfr fstfcgng`s m t`g` ` fspm±o`o`    F F  prfsfr prfsfrvmcg` vmcg` bncfmrngmgf, bncfmrngmgf, o`ctncungmgf o`ctncungmgf f f mt·mt·ff
jfsj` jfsj` ` vmb`r g` vmb`r gm c`rm c`rjm. jm. M pM pf±f±om duom ducgmjfcgmjfctmb gm ctmb gm gfj`cgfj`cstrm±strm±o˒om˒` g` Wf`rfjm gf Amac‟m` g` Wf`rfjm gf Amac‟
Imcmoa Imcmoa ··f f uj uj mrlujfct` mrlujfct` mbl·mbl·firno` firno` quf quf j`strm j`strm quf quf tmb tmb fxtfcs˒fxtfcsm˒ ̀m ̀ ··f f pp`s`sss··ĽvfĽvfb b gfgf   LL   pmrm  pmrm
LL ⊖⊖ XXvvZ o`jZ o`j   v v //∁∁ L L. . Fj Fj sflungm sflungm ` ` Bfjm Bfjm gf gf [`rc [`rc ··f f mpbnomg` mpbnomg` o`j` o`j` ncstrujfct` ncstrujfct` t·t·focno`focno`
gf ncgu±gf ncgu±o˒om˒`, m`, quf quf p`gp`gf f sfr sfr suistntusuistntu··Ľg` Ľg` p`r p`r uj uj mrlujfct` mrlujfct` omc˄omc˄̀cno` `cno` gf gf ncgu±ncgu±o˒om˒` ﬍cntm c`m` ﬍cntm c`
oms` fj qufoms` fj quf   F F   d`r sfpmr· d`r sfpmr·mvfb.mvfb.
Umrm mo`jpmcamr m gfj`cstrm±Umrm mo`jpmcamr m gfj`cstrm±o˒om˒` g` tf`rfjm mimnx`, ` bfnt`r p`uo` dmjnbnzmrnmg`m` g` tf`rfjm mimnx`, ` bfnt`r p`uo` dmjnbnzmrnmg`
o`j ` o`j ` Bfjm Bfjm gf [gf [`rc g`rc gfvf o`csubtmr fvf o`csubtmr ` Mp˄` Mpf˄cgnof fcgnof M.M.
Wf`rfjm ?.=.=Wf`rfjm ?.=.=   (Wf`rfjm gf Amac‟Imcmoa ‟ oms` rfmb) (Wf`rfjm gf Amac‟Imcmoa ‟ oms` rfmb)  _fkmj  _fkmj   F F  uj fspm± uj fspm±o` vft`rnmb o` vft`rnmb 
s`irf ` o`rp` g`s rfmns f s`irf ` o`rp` g`s rfmns f   pp ;;   F F  ∙ ∙↙↙ YY  ujm duc± ujm duc±o˒ o˒ m` quf smtnsdmz m` quf smtnsdmz 
 p p((mxmx) ) :: mp mp((xx))  pmrm t`g` pmrm t`g`   m m 33 5 5  f t`g` f t`g`   xx ∁∁ F,F,    f f ((??..==))
7?7?
  
 p p((xx + + yy)) ≧≧  p p((xx) +) + p p((yy))  pmrm qumnsqufr  pmrm qumnsqufr   x,x, yy ∁ ∁ F.F.
_f_fkmkmj j tmtmji·ji·fj fj    LL   uj suifspm±  uj suifspm±o` vft`rnmb gf o` vft`rnmb gf    F F    f f    ϖϖ ;;   LL ∙ ∙↙↙   YY   uj ducon`cmb bncfmr tmb   uj ducon`cmb bncfmr tmb 
quf quf   ϖϖ((xx)) ≧ ≧   pp((xx))   pmrm t`g`  pmrm t`g`   xx ∁ ∁   LL. . FcFct˒ t˒ m` fxnstf uj ducon`cmb bncfmr m` fxnstf uj ducon`cmb bncfmr 
  
ϖϖ ;;   F F  ∙ ∙↙↙   YY    quf quf 
fstfcgf fstfcgf   ϖϖ, , nsnst ̀t ̀ ··f f   ϖϖ((xx) ) ::   ϖϖ((xx))   pmrm t`g`  pmrm t`g`   xx ∁ ∁   LL, f quf smtnsdmz , f quf smtnsdmz   ϖϖ((xx)) ≧ ≧   pp((xx))   pmrm t`g`  pmrm t`g`xx ∁∁ F F ..
Gfj`cstrm±Gfj`cstrm±oo˒˒mm`̀..    O`csngO`csngfrf frf m sm sflunctf flunctf dmjdmj··ĽbnmĽbnm U  U  gf ducon`cmns bncfmrfs gf﬍cng`s fj gf ducon`cmns bncfmrfs gf﬍cng`s fj
suifspm±suifspm±o`s gfo`s gf   F F  q quf uf o`o`ct·ct·fjfj   LL;;
U U   ::



ττ ;;   GG((ττ)) ⊄⊄ F F  ∙ ∙↙↙ YY ;; G G((ττ) ) ··f f suisuifspfspm±m±o` o` vft`vft`rnmrnmb b gfgf   F F ,,
ττ · ·f f bnbncfcfmrmr,,  LL ⊄⊄ GG((ττ)), , ττ((xx) ) :: ϖ ϖ((xx) pmrm t`g`) pmrm t`g`   xx ∁∁ LL
ff   ττ((xx)) ≧≧  p p((xx) pmrm t`g`) pmrm t`g`   xx  fjfj   GG((ττ))


 ..
FjFj U  U , gf﬍cnj`s m rfbm±, gf﬍cnj`s m rfbm±o˒om˒` gf `rgfj pmronmbm` gf `rgfj pmronmb
ττ== ≧ ≧ ττ99 ⇕ ⇕⇙⇙ GG((ττ==)) ⊄⊄ GG((ττ99) ) ff   ττ99   fstfcgf  fstfcgf   ττ==,,
nnsstt ̀̀ ··ff   ττ99((xx) ) :: τ τ==((xx) pmrm t`g`) pmrm t`g`  x x
∁∁
GG((ττ==))..
C`tf qufC`tf quf U U  · ·f f cc˒˒m`-vmzn` p`nsm`-vmzn` p`ns ϖ ϖ  ∁∁ U U . Tfkmj`s quf t`g` suio`ckuct` t`tmbjfctf `rgfcmg`. Tfkmj`s quf t`g` suio`ckuct` t`tmbjfctf `rgfcmg`
gfgf U  U  mgjntf ujm o`tm supfrn`r. O`j fdfnt`, gmg` mgjntf ujm o`tm supfrn`r. O`j fdfnt`, gmg` ] ] ⊄⊄ U U  t`tmbjfctf `rgfcmg`, gf﬍cm t`tmbjfctf `rgfcmg`, gf﬍cm
ττ ;;  GG((ττ)) ∙∙↙↙ YY  p`rp`r
GG((ττ) ) ::
  
αα∁]∁]
GG((αα) ) ff    ττ((xx) ) :: α α((xx) ) sfsf   xx ∁∁ GG((αα))..
C`tf quf m i`m gf﬍cn±C`tf quf m i`m gf﬍cn±o˒om˒` gfm` gf   ττ   gfo`rrf gm `rgfcm±  gfo`rrf gm `rgfcm±o˒om˒` t`tmb gfm` t`tmb gf ] ]..    ··F njfgnmt` qufF njfgnmt` quf   ττ ∁ ∁
U U  f quf f quf   ττ · ·f f o`tm o`tm supsupfrn`r frn`r pmrmpmrm ] ]. U`gfj`s fct˒. U`gfj`s fctm˒` usmr ` Bfjm gf [`rc f o`cobunr qufm` usmr ` Bfjm gf [`rc f o`cobunr quf
U U  mgjntf uj fbfjfct` jmxnjmb, quf sfr· mgjntf uj fbfjfct` jmxnjmb, quf sfr·m gfc`tmg` p`rm gfc`tmg` p`r  ϖϖ. . @is@isfrvfrvf quf pmrm `itf quf pmrm `itfr `fr `
rfsubtmg` imstm j`strmr qufrfsubtmg` imstm j`strmr quf   GG((ϖϖ) ) :: F F..  Umrm nss` sup`cam quf Umrm nss` sup`cam quf   GG((ϖϖ))  :: F  F . Cfssf oms`. Cfssf oms`
p`gfj`s fso`bafrp`gfj`s fso`bafr   xx55 ∁ ∁ F F   ∙∙ GG((

ϖϖ) f gf﬍cnr) f gf﬍cnr
  
ττ ;;   GG((

ττ)) ∙∙↙↙ YY  p`rp`r
GG((ττ) ) :: G G((ϖϖ) + X) + Xxx55Z Z ff  ττ((xx + + txtx55) ) ::  ϖϖ((xx) +) + tν,tν,
`cgf`cgf    νν   ·  ·f f ujm ujm o`cstmctf o`cstmctf quf quf sfr·sfr·m fso`bangm p`stfrn`rjfctf gf d`rjm m lmrmctnr qufm fso`bangm p`stfrn`rjfctf gf d`rjm m lmrmctnr quf
ττ ∁∁ U U . ]ufrfj`s, p`r fcqumct`, quf. ]ufrfj`s, p`r fcqumct`, quf   νν smtnsdm± smtnsdm±om ms sflunctfs gfsnlumbgmgfs;om ms sflunctfs gfsnlumbgmgfs;
ϖϖ((xx) +) + ν ν : :  ττ((xx + + xx55)) ≧≧  p p((xx ++ xx55) pmrm t`g`) pmrm t`g`   xx ∁∁ GG((ϖϖ) ) ff
ϖϖ((xx)) ∙∙ νν : :  ττ((xx ∙∙ xx55)) ≧≧  p p((xx ∙∙ xx55) pmrm t`g`) pmrm t`g`   xx ∁∁ GG((ϖϖ))..
Umrm tmct` imstm fso`bafrUmrm tmct` imstm fso`bafr   ννgf j`g` quf gf j`g` quf
supsup
xx∁∁GG((ϖϖ))
{{ϖϖ((xx)) ∙∙ p p((xx ∙∙ xx55))}} ≧≧ ν ν ≧ ≧    ncd ncd 
xx∁∁GG((ϖϖ))
{{ p p((xx ++ xx55)) ∙∙  ϖϖ((xx))}}..
DDfbnzjffbnzjfctf ctf tmb tmb fso`bafso`bam m ··f f p`sp`sss··Ľvfb Ľvfb p`p`ns ns pmrmpmrm   x,x, yy ∁ ∁ GG((

ϖϖ) tfj`s) tfj`s
ϖϖ((xx) +) +  ϖϖ((yy) ) ::  ϖϖ((xx ++ yy)) ≧≧  p p((xx ++ yy)) ≧≧  p p((xx + + xx55 + + yy ∙∙ xx55)) ≧≧  p p((xx + + xx55) +) + p p((yy ∙∙ xx55)),,
7777
  
f o`csfqufctfjfctff o`csfqufctfjfctf
ϖϖ((yy)) ∙∙ p p((yy ∙∙ xx55)) ≧≧  p p((xx + + xx55)) ∙∙  ϖϖ((xx))
pmrm qumnsqufrpmrm qumnsqufr x, x, yy ∁ ∁ GG((

ϖϖ)).. Fst· Fst·m fstmifbfong` fct˒m fstmifbfong` fctm˒ ̀m ̀ ququf f ··f f pp`s`sss··ĽvfĽvfb b fsfso`o`babafrfr  ν ν mtfcgfcg` mtfcgfcg`
eemqufbmqufbms ms gums gums fxnl˄fxnlf˄conmfconms ncs ncnonmnsnonmns. . DDfnt` fnt` nss`, nss`, o`coo`cobubu··Ľj`s Ľj`s quf;quf;••  Umrm Umrm   t t 33 5, 5,
ττ((xx + + txtx55) ) ::  ττ

tt
xx
tt
   ++ xx55

:: t tττ
xx
tt
   ++ xx55

:: t t

ϖϖ
xx
tt

++ νν

≧≧ tptp
xx
tt
   ++ xx55

:: p p((xx ++ txtx55))..
••  Umrm Umrm   t t 22 5, 5,
ττ((xx + + txtx55) ) ::  ττ

∙∙tt

  xx
∙∙tt ∙ ∙ xx55

:: ∙∙ttττ

  xx
∙∙tt ∙ ∙ xx55

:: ∙∙tt


ϖϖ
∙∙xx
tt

∙∙ νν

≧≧ ∙∙tptp
∙∙xx
tt
   ∙∙ xx55

:: p p((xx + + txtx55))..
••  Umrm Umrm   tt :  : 55,, ττ((xx + + txtx55) ) ::  ττ((xx) ) ::  ϖϖ((xx)) ≧≧  p p((xx) ) :: p p((xx ++ txtx55))..
_fluf fct˒_fluf fctm˒` qufm` quf  ττ ∁ U  ∁ U ,,  ϖϖ ≧ ≧  ττ   ff  ϖϖ  ::  ττ. . O`jO`j` nss` dfrf m ` nss` dfrf m jmxjmxnjmnjmbngbngmgf gfmgf gf  ϖϖ, tfj`s, tfj`s
GG((ϖϖ) ) :: F  F  f ` tf`rfjm fst· f ` tf`rfjm fst·m pr`vmg`.m pr`vmg`.
M sflunr gfj`cstrmrfj`s m vfrs˒M sflunr gfj`cstrmrfj`s m vfrsm˒` m` g` g` WWf`rfjm f`rfjm gf gf Amac‟Imcmoa Amac‟Imcmoa quf quf tmji·tmji·fj fj ··ff
v·v·mbngm pmrm fspm±mbngm pmrm fspm±o`s o`s vft`rnvft`rnmns mns o`jpbfxo`jpbfx`s;`s;
Wf`rfjm ?.=.9Wf`rfjm ?.=.9  (Wf`rfjm gf Amac‟Imcmoa) (Wf`rfjm gf Amac‟Imcmoa)  _fkmj  _fkmj  F  F  uj fspm± uj fspm±o` vft`rnmb s`irf ` o`rp`o` vft`rnmb s`irf ` o`rp`
HH :: YY   `u `u   OO   f f   pp ;;   F F  ∙ ∙↙↙ YY  ujm duc± ujm duc±o˒ o˒ m` quf smtnsdmz m` quf smtnsdmz 
 p p((mxmx) ) :: ||mm|| p p((xx))  pmrm t`g` pmrm t`g`   mm ∁∁ HH  f t`g` f t`g`   xx ∁∁ F,F,    f f ((??..99))
 p p((xx + + yy)) ≧≧  p p((xx) +) + p p((yy))  pmrm qumnsqufr  pmrm qumnsqufr   x,x, yy ∁ ∁ F.F.   (?.?)  (?.?)
_f _f    LL   ⊄⊄    F F   ·  ·f f uj uj suifspm±suifspm±o` o` vfvft`t`rnrnmb mb f f    ϖϖ ;;   LL   ∙∙↙↙    HH   ·  ·f f uj uj duconducoǹ cmb `cmb bncfbncfmr mr tmb tmb quf quf 
||ϖϖ((xx))| | ≧≧    pp((xx))   pmr  pmrm m t`g`t`g`    xx  ∁∁    LL, fct˒ , fct˒ m` fxnstf uj ducon`cmb bncfmr m` fxnstf uj ducon`cmb bncfmr   ϖϖ ;;   F F   ∙∙↙↙    HH    quf quf 
fstfcgf fstfcgf   ϖϖ   m m   F F  f quf smtnsdmz  f quf smtnsdmz  | |ϖϖ((xx))|| ≧≧  p p((xx))  pmrm t`g` pmrm t`g`   xx ∁∁ F F ..
Gfj`cstrm±Gfj`cstrm±oo˒˒mm`̀..  Mctfs gf tug`, vfkmj`s quf gf (?.9) f (?.?) sfluf quf Mctfs gf tug`, vfkmj`s quf gf (?.9) f (?.?) sfluf quf   pp((xx)) ≦≦ 5 pmrm5 pmrm
t`g`t`g`   xx ∁ ∁   F F , dmt` fstf quf sfr·, dmt` fstf quf sfr·m usmg` cm pmrtf ﬍cmb gm gfj`cstrm±m usmg` cm pmrtf ﬍cmb gm gfj`cstrm±o˒om˒`. m`. O`j fdO`j fdfntfnt`, gf`, gf
(?.?) rfsubtm quf(?.?) rfsubtm quf  p p(5) :(5) : p p(5+5)(5+5)  ≧≧ 99 p p(5), (5), f f o`csfqufo`csfqufctfjfctfjfctfctf p p(5)(5) ≦≦ 5. Gf (?.9) sfluf5. Gf (?.9) sfluf
qufquf  p p((xx) ) :: p p((∙∙xx) f mssnj, pmrm t`g`) f mssnj, pmrm t`g`   xx ∁∁ F F ,,
99 p p((xx) ) :: p p((xx) +) + p p((∙∙xx)) ≦≦  p p((xx + ( + (∙∙xx)) )) :: p p(5)(5) ≦≦ 55..
WWrmtfj`s rmtfj`s prnjfnr` prnjfnr` ` ` oms` oms` rfmb, rfmb, nst` nst` ··ff H H :: YY. Cfssf oms` m anp·. Cfssf oms` m anp·̀tfsf c`s lmrmctf quf`tfsf c`s lmrmctf quf
ϖϖ((xx)) ≧ ≧  pp((xx) pmrm t`g`) pmrm t`g`   xx ∁ ∁   LL. . @ @ WWf`rfjm ?.=.= f`rfjm ?.=.= lmrmctlmrmctf f m m fxnstfxnst˄˄fconm gf fconm gf uj uj ducon`cmducon`cmbb
7676
  
bncfmrbncfmr  ϖϖ ;;   F F   ∙∙↙↙   YY   quf fstfcgf  quf fstfcgf   ϖϖ   mm   F F   f quf smtnsdmz  f quf smtnsdmz  ϖϖ((xx)) ≧ ≧    pp((xx) pmrm t`g`) pmrm t`g`   xx ∁ ∁   F F ..
Gfssm gfsnlumbgmgf f gf (?.9) tfj`sGfssm gfsnlumbgmgf f gf (?.9) tfj`s
∙∙
ϖϖ((xx) ) ::
  
ϖϖ((
∙∙
xx))
≧≧
 p p((
∙∙
xx) ) ::
|∙|∙
==
||
 p p((xx) ) :: p p((xx)),,
pmrm t`g`pmrm t`g`   xx ∁∁ F F . B`l`. B`l` | |ϖϖ((xx))|| ≧≧  p p((xx) pmrm t`g`) pmrm t`g`   xx ∁∁ F F ..
DDm±m±omjomj`s `s ml`ml r̀m rm ` ` omsoms` ` o`jo`jpbpbfxfx`̀    HH    ::    OO. . CCffssssf of omsms`̀    F F   ·  ·f f uj uj fspm±fspm±o` o` vfvft`rnt`rnmbmb
o`jpbfx` fo`jpbfx` f   ϖϖ  t`jm vmb`rfs fj t`jm vmb`rfs fj   OO. . Gf﬍cGf﬍cncgncg`̀   ϖϖ==,, ϖϖ99 ;;   F F  ∙ ∙↙↙ Y Y   p`rp`r   ϖϖ==((xx) : Yf() : Yf(ϖϖ((xx)) )) ff
ϖϖ99((xx) : Nj() : Nj(ϖϖ((xx)))), ·, ·f f obmobmr` r` qufquf  ϖ ϖ==   ff ϖ ϖ99   s˒sm˒` bncfmrfs, t`jmj vmb`rfs rfmns fm` bncfmrfs, t`jmj vmb`rfs rfmns f  ϖ ϖ  : :  ϖ ϖ== ++ nϖnϖ99..
O`j` O`j` mrtnd·mrtnd·Ľon`, Ľon`, oamjfj`oamjfj`s s gfgf   F F YY    ff   LLYY  `s fspm± `s fspm±o`s vft`rnmns rfmns suikmofctfs mo`s vft`rnmns rfmns suikmofctfs m   F F    ff   LL
(nst` (nst` ··f, f, o`j` o`j` o`ckuct` o`ckuct` ` ` rfspfotnv` rfspfotnv` fspm±fspm±o` ·o` ·f f ` ` jfsj`, jfsj`, m m `pf`pfrm±rm±o˒om˒` gf mgn±m` gf mgn±o˒om˒ ̀m ̀ ··f f m m jfjfsjsjmm
f m jubtnpbnom±f m jubtnpbnom±o˒om˒` pm` p`r fsombmr ·`r fsombmr ·f f dfntm mpfcms dfntm mpfcms o`j fsombmrfs rfmns). o`j fsombmrfs rfmns). Fct˒Fctm˒`m`    ϖϖ==    ff    ϖϖ99    s˒sm˒`m`
ducon`cmns bncfmrfs s`irfducon`cmns bncfmrfs s`irf   LLYY.. Umrm Umrm   xx ∁∁ LLYY, `itfj`s, `itfj`s
ϖϖ==((xx)) ≧≧ ||ϖϖ==((xx))|| ≧≧ ||ϖϖ((xx))|| ≧≧  p p((xx))..
Ufb` Wf`rfjm ?.=.= fxnstf fct˒Ufb` Wf`rfjm ?.=.= fxnstf fctm˒` uj ducon`cmb bncfmrm` uj ducon`cmb bncfmr
  
ϖϖ== ;;  F F YY ∙ ∙↙↙ YY quf fstfcgf quf fstfcgf ϖ ϖ==   mm F  F YY
f quf smtnsdmzf quf smtnsdmz  ϖϖ==((xx)) ≧≧  p p((xx) pmrm t`g`) pmrm t`g` x x ∁∁ F F YY. . FstuFstugfj`gfj s̀ ml`rm ` oms` gfs ml`rm ` oms` gf ϖ ϖ99.. C`tf quf C`tf qufpmrmpmrm   xx ∁∁ L,L,
nn ((ϖϖ==((xx) +) + nϖnϖ99((xx)) )) :: nϖ nϖ((xx) ) :: ϖ ϖ((nxnx) ) :: ϖ ϖ==((nxnx) +) + nϖnϖ99((nxnx))..
U`rtmct`U`rtmct`  ϖϖ99((xx) ) :: ∙∙ϖϖ==((nxnx) pmrm t`g`) pmrm t`g`   xx ∁∁ LL. Gf﬍cncg` fct˒. Gf﬍cncg` fctm˒`m`
ϖϖ ;;   F F  ∙ ∙↙↙ OO  ,,  ϖϖ((xx) ) ::  ϖϖ==((xx)) ∙∙ nnϖϖ==((nxnx)),,
sfluf qufsfluf quf  ϖϖ((xx) ) ::    ϖϖ((xx) pmrm t`g`) pmrm t`g`    xx ∁ ∁    LL. . TTfkmfkmj`j`s qufs quf  ϖϖ   ·  ·f f uj uj ducon`cmb ducon`cmb bncfmr bncfmr c`c`
fspm±fspm±o` o`jpbfx`o` o`jpbfx`    F F ..    ··F njfgnmt` qufF njfgnmt` quf  ϖϖ((xx + +  y y) ) ::  ϖϖ((xx) ) ++  ϖϖ((yy) pmrm t`g`s) pmrm t`g`s    x,x, yy  ∁∁    F F ..
Gmg`s (Gmg`s (mm ++ inin)) ∁∁ OO  ff   xx ∁∁ F F ,,

ϖϖ((((mm ++ inin))xx) ) ::
  
ϖϖ==((mxmx ++ nixnix)) ∙∙ nn

ϖϖ== (( ((nn((mm ++ inin))xx))))
:: m mϖϖ==((xx) +) + iiϖϖ==((nxnx)) ∙∙ nn ((mmϖϖ==((nxnx)) ∙∙ iiϖϖ==((xx))))
: : ((mm ++ inin)) ((ϖϖ==((xx)) ∙∙ nnϖϖ==((nxnx)) : ()) : (mm ++ inin))ϖϖ((xx))..
J`strfj`s ﬍cmbjfctf qufJ`strfj`s ﬍cmbjfctf quf  ||ϖϖ((xx))| | ≧≧    pp((xx) pmrm t`g`) pmrm t`g`    xx  ∁∁    F.F.    _f_f  ϖϖ((xx) : 5, fct˒) : 5, fctm˒` m` mm
gfsgfsnlunlumbgmbgmgf mgf ··f f ··̀ivnm p`ns`ivnm p`ns   pp((xx)) ≦ ≦  5. 5. WW`jf`jf   xx ∁ ∁ F  F  tmb quf tmb quf  ϖϖ((xx))  : : 55..  Fct˒ Fctm˒` fxnstfm` fxnstf   αα    tmbtmb
qufquf  ϖϖ((xx) ) :: ||ϖϖ((xx))|| ffnαnα. _fluf quf. _fluf quf | |ϖϖ((xx))|| :: f f∙∙nαnα ϖϖ((xx) ) ::  ϖϖ((ff∙∙nαnαxx). O`j`). O`j` | |ϖϖ((xx))|| ··f f rfrfmbmb, , pp`r`r
(?.9), tfj`s(?.9), tfj`s
||ϖϖ((xx))|| ::  ϖϖ((ff∙∙nαnαxx) ) ::  ϖϖ==((ff∙∙nαnαxx)) ≧≧  p p((ff∙∙nαnαxx) ) ::
ff∙∙nαnα
 p p((xx) ) :: p p((xx))..
Umrm sfr o`frfctf o`j m tfrjnc`b`lnm o`csmlrmgm pfbm bntfrmturm, oamjmrfj`s mUmrm sfr o`frfctf o`j m tfrjnc`b`lnm o`csmlrmgm pfbm bntfrmturm, oamjmrfj`s m
sflunsflunctf ctf o`csfo`csfqu˄quf˄conm fconm njfgnmnjfgnmtm tm tmji·tmji·fj fj gf gf WWf`rff`rfjm jm gf gf Amac‟Amac‟Imcmoa.Imcmoa.7474
  
O`O`r`br`b··mrnmrn ̀̀ ?.?.=.=.??   (W  (Wf`rfjm f`rfjm gf gf Amac‟IAmac‟Imcmoa)mcmoa)   _fkm   _fkm    LL   uj   uj suisuifspfspm±m±o` gf uj fspm±o` gf uj fspm±o`o`
c`rjmg`c`rjmg`    F F   s`irf   s`irf    HH    ::    YY    `u `u    OO   f sfkm   f sfkm    ϖϖ ;;   LL  ∙∙↙↙    HH   uj   uj ducon`ducon`cmb cmb bncfmr bncfmr o`cto`ct··Ľcu`.Ľcu`.
Fct˒ Fct˒ m` m` fxnstf fxnstf uj uj ducon`ducon`cmb cmb bncfmr bncfmr o`cto`ct··Ľcu`Ľcu`
  
ϖϖ ;;   F F  ∙ ∙↙↙ HH oukm rfstrn± oukm rfstrn±o˒ o˒ m` m` m m  L L  o`ncongf o`j  o`ncongf o`j 
ϖϖ   f f 
  
ϖϖ

::

ϖϖ

..
Gfj`cstrm±Gfj`cstrm±oo˒˒mm`̀..  Imstm mpbnomr ` Wf`rfjm ?.=.9 o`j Imstm mpbnomr ` Wf`rfjm ?.=.9 o`j   pp((xx) ) :: ϖϖ ¼¼ xx ..
M pmrtnr gf ml`rm v`btmj`s m` pr`ofgnjfct` mctfrn`r c` qumb uj fspm±M pmrtnr gf ml`rm v`btmj`s m` pr`ofgnjfct` mctfrn`r c` qumb uj fspm±o` c`rjmg`o` c`rjmg`
snlcn﬍om uj fspm±snlcn﬍om uj fspm±o` c`rjmg` s`irfo` c`rjmg` s`irf   HH :: YY  `u`u  O O..
O`O`r`br`b··mrnmrn ̀̀ ?.?.=.=.77   _fkm   _fkm   F F   uj fspm±  uj fspm±o` c`rjmo` c`rjmg`. g`. UmrUmrm m t`t g̀`g`    xx55 ∁ ∁   F F ,,    xx55  : : 55, fxnstf uj , fxnstf uj 
 ducon`cmb  ducon`cmb bncfmr bncfmr   ϖϖ  ∁∁ F F    tmb quf  tmb quf   ϖϖ : : ==   f f   ϖϖ((xx55) ) :: xx55 ..
Gfj`cstrm±Gfj`cstrm±oo˒˒mm`̀..  Mpbnquf ` O`r`b· Mpbnquf ` O`r`b·mrn` ?.=.? pmrmmrn` ?.=.? pmrm   LL :  : XXxx55Z Z ff   ϖϖ((mxmx55) ) :: m mxx55..
M pr·M pr·̀xnjm mpbnom±`xnjm mpbnom±o˒om˒` g` Wf`rfjm gf Amac‟Imcmoa sfr·m` g` Wf`rfjm gf Amac‟Imcmoa sfr·m utnbnzmgm v·m utnbnzmgm v·mrnms vfzfs m`mrnms vfzfs m`
b`cl` g` tfxt`.b`cl` g` tfxt`.
O`O`r`br`b··mrnmrn ̀̀ ?.?.=.=.66   _fkmj   _fkmj   F F  uj fspm± uj fspm±o` c`rjmg`,o` c`rjmg`,   F F 
  
::
{{
55
}}
, , f f   xx
∁∁
F F . Fct˒ . Fct˒ m`m`
xx : sup: sup{|{|ϖϖ((xx))|| ;; ϖ ϖ ∁∁ F F     f f   ϖϖ ≧≧ = =}}
f f ` ` suprfj` suprfj` ··f f mtnclnmtnclng`.g`.
Gfj`cstrm±Gfj`cstrm±oo˒˒mm`̀..   Umrm omgm ducon`cmb  Umrm omgm ducon`cmb    ϖϖ  ∁∁    F F ,,  ϖϖ  ≧≧   =,   =, ··f f njfgnmt` njfgnmt` qufquf  ||ϖϖ((xx))| | ≧≧
ϖϖ¼¼xx ≧≧ xx. . Nss` j`strm Nss` j`strm quf ` quf ` suprfj` ·suprfj` ·f jfc`r f jfc`r `u nlumb `u nlumb mm xx. @ dmt` g` suprfj`. @ dmt` g` suprfj`
sfr sfr mtnclng` mtnclng` ··f f lmrmctng` lmrmctng` pfb` pfb` O`r`b·O`r`b·mrn` ?.=.7.mrn` ?.=.7.
??..9 9 TTffrrss˒̀˒fs vft`rnmns g` Wf`rfjm gf Amac‟Imcmoa`fs vft`rnmns g` Wf`rfjm gf Amac‟Imcmoa
Fj Fj vnsvnstm tm g` g` WWf`rff`rfjm jm gf gf AmaAmac‟Imc‟Imcmocmoa, a, ··f f cmtucmturmb rmb fspfofspfoubmr ubmr sf sf `pfrmg`pfrmg`rfs `rfs bncbncfmrffmrfss
o`cto`ct··Ľcu`s Ľcu`s p`gp`gfj fj sfr sfr fstfcgng`s fstfcgng`s pmrm pmrm fspm±fspm±o`s jmn`rfso`s jmn`rfs. . Jmns prfonsJmns prfonsmjfctmjfctf, sff, sf   LL  · ·f f uujj
suifspm±suifspm±o` g` fspm±o` g` fspm±o` c`rjmg`o` c`rjmg`   F F    ff   W W 
  ∁ B∁ B
((L,L, D D ), sfr·), sfr·m quf sfjprf fxnstf uj `pfrmg`rm quf sfjprf fxnstf uj `pfrmg`r
W W   ∁ ∁ BB((F,F, D D ) quf o`ncongf o`j) quf o`ncongf o`j    W W    fjfj    LL0 0 UmUmrm trmtrm trmtmr gfssmr gfssm m ququfsfst˒tm˒` prfonsmj`s g`sm` prfonsmj`s g`s
o`cofnt`s gf pr`kf±o`cofnt`s gf pr`kf±o˒om˒` f suifspm±m` f suifspm±o` o` o`jpbfjo`jpbfjfctmg`fctmg .̀.
Gf﬍cn±Gf﬍cn±o˒om˒ ̀m ̀ ?.?.9.9.==   _fkm  _fkm F  F  uj fspm± uj fspm±o` gf Imo` gf Imcmoa. cmoa. Vj `pVj `pfrmg`frmg`r bncfmr bncfmr o`ctr o`ct··Ľcu`Ľcu` U  U  ; ;   F F  ∙ ∙↙↙
F F  · ·f f uujjmm  pr`kf± pr`kf±o˒ o˒ m`m`    sfsf   U U 99 ;:;:   U U  ◤ ◤ U U    ::   U U ..    ··F obmr` quf sfF obmr` quf sf   U U   : : 5 5 ··f f ujm ujm pr`pr`kf±kf±o˒om˒`, fct˒m`, fctm˒`m`
U U  ≦≦ =. =.
Ur`p`sn±Ur`p`sn±o˒om˒ ̀m ̀ ?.?.9.9.99   _fkm   _fkm    D D    uj   uj suisuifspfspm±m±o` o` g` g` fsfsppm±m±o` o` gf gf ImImcmcmoa oa    F F . . Ms Ms sfsfluluncncttfs fs 
m﬍rjm±m﬍rjm±o˒ o˒ `fs `fs s˒sm˒` m` fqunvmfqunvmbfctbfctfs;fs;
(m)(m)  Fxnstf ujm pr`kf± Fxnstf ujm pr`kf±o˒ o˒ m`m` U  U  ; ;   F F  ∙ ∙↙↙ F F  oukm njmlfj o`ncongf o`j  oukm njmlfj o`ncongf o`j  D  D . Cfstf oms` gnzfj`s . Cfstf oms` gnzfj`s 
quf quf   U U  · ·f f ujm ujm pr`kpr`kf±f±o˒ o˒ m` gf m` gf   F F   s`irf  s`irf   D D ..
(i)(i)    D D   ·  ·f f dfoamg` dfoamg` f f fxnstf fxnstf uj uj suifspsuifspm±m±o` dfoamg`o` dfoamg`    LL    gf gf    F F   tmb quf   tmb quf    F F    ::    D D  ⊖ ⊖  L L, , nsnst ̀t ̀ ··f,f,
F F   :: D  D   ++ LL   f f   D D  ∮ ∮ LL : : {{55}}..
Cfstf oms`Cfstf oms`   D D    :: {{xx ∁∁ F F    ;; U  U ((xx) ) :: x x}}   f f   LL : hfr( : hfr(U U ))..
7<7<
  
Gfj`cstrm±Gfj`cstrm±oo˒˒mm`̀..  (m) : (m) :⇙⇙ (i) Ur`vfj`s prnjfnr` m nlumbgmgf (i) Ur`vfj`s prnjfnr` m nlumbgmgf D  D    :: {{xx ∁∁ F F    ;; U  U ((xx) ) :: x x}}..
_f_f   xx ∁ ∁   D D , t`jmcg`, t`jmcg`   yy ∁ ∁   F F  tmb quf tmb quf   U U ((yy) ) ::   xx  sfluf quf sfluf quf   xx   ::   U U ((yy) ) ::   U U ((U U ((yy)) )) ::   U U ((xx).).
_f_f   xx : :  U  U ((xx), fct˒), fctm˒` obmrmjfctfm` obmrmjfctf   xx ∁ ∁ Nj( Nj(U U ) ) :: D  D . O`csfqufctfjfctf, oamjmcg` gf ng. O`csfqufctfjfctf, oamjmcg` gf ngF F 
` `pfrmg`r ngfctngmgf fj` `pfrmg`r ngfctngmgf fj   F F , , ··f f vfrgvfrgmgf mgf qufquf
D D    :: {{xx ∁∁ F F    ;; U  U ((xx) ) :: x x}} :: {{xx ∁∁ F F    ; ; ((U U  ∙ ∙ ngngF F )()(xx) ) : : 55}} : : ((U U  ∙ ∙ ngngF F ))∙∙==(({{55}})),,
quf ·quf ·f f dfoamg` dfoamg` o`j` o`j` m m njmlfj njmlfj ncvncvfrsm frsm g` g` dfoamg`dfoamg` { {55}}  pfbm duc± pfbm duc±o˒om˒ ̀m ̀ o`o`ct·ct·ĽcuĽcumm   U U  ∙ ∙ ngngF F ..
W`jf ml`rmW`jf ml`rm L L  : hfr( : hfr(U U ).).    ··F obmr` qufF obmr` quf L L  · ·f f susuiifsfspmpm±±o` dfoamg` gfo` dfoamg` gf F  F . Umrm t`g`. Umrm t`g` x x ∁∁ F F 
vmbf quf (vmbf quf (xx ∙∙ U U ((xx)))) ∁ ∁   LL,,   U U ((xx)) ∁ ∁   D D    ff   xx   : : ((xx ∙∙ U U ((xx)) )) ++ U  U ((xx))..   _f_f   xx ∁ ∁   LL ∮∮ D D , tfj`s, tfj`s
xx : :  U  U ((xx) ) p`np`nss   xx ∁∁ D D    ff  U  U ((xx) : 5 ) : 5 p`np`nss   xx ∁∁ LL. _fluf quf. _fluf quf   xx :  : 5.5.
(i) :(i) :⇙⇙  (m) Umrm omgm (m) Umrm omgm   xx ∁ ∁   F F , fxnstfj ·, fxnstfj ·ucno`sucno`s   xx== ∁ ∁   D D    ff   xx99 ∁ ∁   LL   tmns quf tmns quf   xx   ::   xx== + +  x x99..
··F njfgnmt` quf ` `pfrmg`rF njfgnmt` quf ` `pfrmg`r    U U  ; ;   F F   ∙∙↙↙    F F   gmg` p`r  gmg` p`r    U U ((xx) ) ::    xx==   fst·  fst·m m ifj ifj gf﬍cng`, gf﬍cng`, ··ff
bncfmr,bncfmr,    U U 99 ::   U U , Nj(, Nj(U U ) ) ::   D D , hfr(, hfr(U U ) ) ::   LL   ff   D D    :: { {xx ∁ ∁   F F    ;;   U U ((xx) ) ::   xx}}. . YfsYfstm pr`tm pr`vvmrmr
qufquf    U U   ·  ·f f o`cto`ct··Ľcu`. Ľcu`. Umrm Umrm nss` nss` sfkm sfkm ((xxcc))
∓∓
cc:=:=  uj ujm m sfqu˄sfquf˄conm fconm fjfj    F F   tmb quf  tmb quf    xxcc  ∙∙↙↙    xx    ff
U U ((xxcc))  ∙∙↙↙    yy. . UUmrmrm om omgmgmm    cc, , fsorfvfsorfvmm    xxcc    ::    yycc   ++  z  z cc    o`jo`j    yycc  ∁∁    D D    ff    z z cc  ∁∁    LL. . FFcctt˒˒m`m`
z z cc    ::    xxcc
 ∙ ∙
 y ycc    ::    xxcc
 ∙ ∙
 U  U ((xxcc))
 ∙ ∙↙↙
   xx
 ∙ ∙
 y y. . OO`̀jj`̀    LL   ·  ·f f dfdfoaoamgmg ,̀̀,    xx
  ∙∙
 y y
  ∁∁
   LL, f p`rtmct`, f p`rtmct`
U U ((xx) ) ::   U U ((yy). ). U`U`r `utrr `utr` bmg`` bmg ,̀,   yycc   ::   U U ((xxcc)) ∙ ∙↙↙   yy. . O`O`j`j`  D D   ·  ·f f dfdfoaoamgmg ,̀̀,   yy ∁ ∁   D D . . MsMssnsnjj
yy  : :  U  U ((yy) ) :: U  U ((xx). Ufb` Wf`rfjm g` Lr·). Ufb` Wf`rfjm g` Lr·m﬍o` Dfoamg` rfsubtm quf ` `pfrmg`r bncfmrm﬍o` Dfoamg` rfsubtm quf ` `pfrmg`r bncfmr   U U    ··ff
oo`c`ct·t·ĽcĽcu`u`..
Gf﬍cn±Gf﬍cn±o˒om˒ ̀m ̀ ?.?.9.9.??  Vj suifspm± Vj suifspm±o`o` D  D  g` fspm± g` fspm±o` gf Imcmoao` gf Imcmoa F  F  · ·ff  o`jpbfjo`jpbfjfctmg`fctmg`  sf smtnsdmz sf smtnsdmz
ms o`cgn±ms o`cgn±o˒o˒̀fs fqunvmbfctfs gm Ur`p`sn±`fs fqunvmbfctfs gm Ur`p`sn±o˒om˒` ?.9m` ?.9.9.9. . GnGnzfzfj`s quj`s quff    D D    ··ff    θθ-o`jpbfjfctmg`,-o`jpbfjfctmg`,
θθ ≦≦ =, sf=, sf  D  D    ··f f o`jpbfjfctmg` o`jpbfjfctmg` p`r p`r ujm ujm pr`pr`kf±kf±o˒om˒` gf c`rjm nlumb mm` gf c`rjm nlumb m   θθ..
_fluf gm Ur`p`sn±_fluf gm Ur`p`sn±o˒om˒` ?.9.9 quf t`g` suifspm±m` ?.9.9 quf t`g` suifspm±o` o`jpbfjfctmg` gf uj fspm±o` o`jpbfjfctmg` gf uj fspm±o` gfo` gf
ImcImcmoamoa ··f f dfoadfoamg`mg`..
Fxfjpb` ?.9.7Fxfjpb` ?.9.7   (m) W`g` suifspm±  (m) W`g` suifspm±o` o` gf gf gnjfcsgnjfcs˒˒m` m` ﬍cntm ﬍cntm gf gf uj uj fspm±fspm±o` o` gf gf Imcmoa Imcmoa ··ff
o`jpo`jpbfjbfjfctfctmg`. mg`. _fkm_fkm    D D   uj   uj suifssuifspm±pm±o` o` gf gf gngnjfjfcs˒csm˒`m`    cc   g` fspm±  g` fspm±o` gf Imcmoao` gf Imcmoa    F F    ff
{{
ff==, . . . , f, . . . , fcc
}}
  ujm imsf pmrm  ujm imsf pmrm    D D . . UUmrmrm omm omgmgm    kk    : : ==, . . . , c, . . . , c, o`csngfrf ` ducon`cmb bncfmr, o`csngfrf ` ducon`cmb bncfmr
oo`̀ct·ct·ĽĽcucu`̀   ϖϖ k k ∁ ∁  D  D   gmg` p`r gmg` p`r   ϖϖ k k     cc
hh:=:=
mmhhffhh : :  m m k k. . Ufb` WUfb` Wf`rfjm gf Amac‟Imcmof`rfjm gf Amac‟Imcmoa fxnstfa fxnstf
ϖϖ k k ∁ ∁  F  F    fxtfcs˒  fxtfcsm˒` gfm` gf   ϖϖ k k    mm   F F ,,   kk   : : ==, . . . , c, . . . , c. @ bfnt`r c˒. @ bfnt`r cm˒` tfr·m` tfr·m gn﬍oubgmgf fj oafomr qufm gn﬍oubgmgf fj oafomr quf
` `pfrmg`r` `pfrmg`r   U U    ;:;:
cc
 k k:=:=
ϖϖ k k ⊜ ⊜ ff k k  · ·f f ujm ujm pr`pr` kf±kf±o˒om˒` gfm` gf   F F   s`irf  s`irf   D D ..
(i) _fkmj(i) _fkmj   F F    ff   D D   fspm±  fspm±o`s gf Imcmoa. U`r jfn` gm pr`kf±o`s gf Imcmoa. U`r jfn` gm pr`kf±o˒om˒` m` ((x,x, yy)) ∁ ∁ F  F  Ü Ü D D   ↙↙ ( (x,x, 5)5) ∁ ∁
F F   ÜÜ D D , vfj`s quf, vfj`s quf   F F   :: F  F   ÜÜ {{55}} · ·f f =-o=-o`jp`jpbfjbfjfctmfctmg` g` fjfj   F F  Ü Ü D D ..
(o(o) ) K·K·m m smsmifjifj`s `s ququf f uj uj susuifsifspmpm±±o` c˒o` cm˒`-m`-dfodfoamamg` g` gf gf uj uj fsfspm±pm±o` o` gf gf ImImcmcmooa a c˒cm˒m ̀̀ ··ff
o`jpo`jpbfjbfjfctfctmg`. mg`. U`U`r r fxfjfxfjpb`pb ,̀,    oo5555    c˒cm˒` m` ··f f o`jpbfjfctmg` o`jpbfjfctmg` fjfj    oo55. . MoMorfrfgngntmtm-s-sf quf mf quf m
fxnst˄fxnstf˄conm fconm gf gf suisuifspm±fspm±o`s dfoamg`s c˒o`s dfoamg`s cm˒`-o`jpbfjfctmg`s d`n pr`vmgm pfbm prnjfnrm vfzm`-o`jpbfjfctmg`s d`n pr`vmgm pfbm prnjfnrm vfz
fj =>?< p`r Jurrmy X46Z, quf pr`v`u quf pmrm t`g` 9fj =>?< p`r Jurrmy X46Z, quf pr`v`u quf pmrm t`g` 9  :: p  p 33 =, =,    p p o` o`ct·ct·fj fj uj uj suisuifspfspm±m±o`o`
dfoamg` c˒dfoamg` cm˒`-o`jpbfjfctmg`. Fj =>75, Uanbbnps X<9Z pr`v`u qufm`-o`jpbfjfctmg`. Fj =>75, Uanbbnps X<9Z pr`v`u quf o o55 c˒ cm˒`m` ··f f o`jpbo`jpbfjfctmgfjfctmg`̀
fjfj   ∓∓..
7171
  
··F F d·d·monb vfr quf `s `pfrmg`rfs gf﬍cng`s fj suifspm±monb vfr quf `s `pfrmg`rfs gf﬍cng`s fj suifspm±o`s o`jpbfjfctmg`s p`gfj sfro`s o`jpbfjfctmg`s p`gfj sfr
fstfcgng`s (jmctfcg` m o`ctncungmgf) m` fspm±fstfcgng`s (jmctfcg` m o`ctncungmgf) m` fspm±o` t`g`;o` t`g`;
Ur`p`sn±Ur`p`sn±o˒om˒ ̀m ̀ ?.?.9.9.66   _fkmj   _fkmj    LL   uj fspm±  uj fspm±o` gf Imcmoa,o` gf Imcmoa,    D D   uj suifspm±  uj suifspm±o` o`jpbfjfctmg`o` o`jpbfjfctmg`
g` fspm±g` fspm±o` gf Imcmoa o` gf Imcmoa   F F    f f   W W  ∁ ∁ BB((D,D, LL)). Fct˒ . Fct˒ m` fxnstf m` fxnstf   W W  ∁ ∁ BB((F,F, LL))  fxtfcs˒  fxtfcs˒ m` gf m` gf   W W    m m   F F ..
Gfj`cstrm±Gfj`cstrm±oo˒˒mm`̀..   W`jf  W`jf   U U  ujm pr`kf± ujm pr`kf±o˒om˒` gfm` gf   F F   s`irf  s`irf   D D   f gf﬍cm  f gf﬍cm  W W    :: W  W  ◤ ◤ U U ..
Fctrftmct`, qumcg` ` suifspm±Fctrftmct`, qumcg` ` suifspm±o` c˒o` cm˒` m` ··f f o`jpbfjfctmg` o`jpbfjfctmg` m m sntum±sntum±o˒om˒` p`gf sf mbtfrmrm` p`gf sf mbtfrmr
rmgnombrmgnombjfctf. jfctf. TTfkmj`s quf, m` fkmj`s quf, m` o`ctr·o`ctr·mrn` mrn` g`s g`s ducon`cmns ducon`cmns bncfmrfs bncfmrfs o`cto`ct··Ľcu`s, Ľcu`s, `pfrmg`rf`pfrmg`rfss
bncfmrfs o`ctbncfmrfs o`ct··Ľcu`Ľcu s̀ s cfj cfj sfjprf psfjprf p`gfj `gfj sfr sfr fstfcgngfstfcgng`s. `s. @u @u sfkm, c˒sfkm, cm˒` fxnstf ujm vfrs˒m` fxnstf ujm vfrsm˒`m`
vft`rnmb g` Wf`rfjm gf Amac‟Imcmoa;vft`rnmb g` Wf`rfjm gf Amac‟Imcmoa;
Ur`p`sn±Ur`p`sn±o˒om˒ ̀m ̀ ?.?.9.9.44   _fkmj   _fkmj     F F    uj   uj fsfsppm±m±oo` ` ggf f IImmccmmooa a f  f     D D    uj   uj suisuifspfspm±m±oo` ` cc˒  ˒  m`-m`-
o`jpbfjfctmg` gf o`jpbfjfctmg` gf   F F . . FcFct˒ t˒ m` c˒ m` c˒ m` m` fxnstf fxnstf `pfr`pfrmg`r mg`r bncfmr bncfmr o`cto`ct··Ľcu`Ľcu`   W W  ; ;   F F  ∙ ∙↙↙   D D   tmb quf   tmb quf 
W W ((xx) ) :: x x  pmrm t`g` pmrm t`g`  x x ∁∁ D D , `u sfkm, ` `pfrmg`r ngfctngmgf fj , `u sfkm, ` `pfrmg`r ngfctngmgf fj  D  D    c˒ c˒ m` p`gf sfr fstfcgng`m` p`gf sfr fstfcgng`
o`ctncumjfctf m o`ctncumjfctf m   F F ..
Gfj`cstrm±Gfj`cstrm±oo˒˒mm`̀..    _up`cam _up`cam quf fquf fxnstm uj xnstm uj `pfrmg`r `pfrmg`r bncfmr bncfmr o`cto`ct··Ľcu`Ľcu` W  W  ; ;   F F  ∙ ∙↙↙ D D  tmb quf tmb quf
W W ((xx) ) :: x x  pmrm t`g` pmrm t`g` x x ∁ ∁ D D ..    ··F obmr` quf m ncobus˒F obmr` quf m ncobusm˒`m` n nD D  ; ;   D D  ∙ ∙↙↙ F F  · ·f f bnbncfcfmr mr f f o`o`ct·ct·ĽcuĽcum, m, b`b`l`l`
nnD D  ◤ ◤ W W    ··f f uj uj `pf`pfrmg`r rmg`r bncfmbncfmr r o`cto`ct··Ľcu` Ľcu` gfgf   F F    fjfj   F F . . O`O`j`j`  W W ((xx)) ∁ ∁   D D  pmrm t`g` pmrm t`g`   xx ∁ ∁   F F ,,
sfluf qufsfluf quf   W W 99((xx) ) :: W  W ((W W ((xx)) )) :: W  W ((xx) pmrm t`g`) pmrm t`g`   xx ∁∁ F F . Mssnj. Mssnj   nnD D  ◤ ◤ W W   sfrnm ujm pr`kf±  sfrnm ujm pr`kf±o˒om˒`m`
s`irfs`irf  D  D , ` quf o`ctrmgnz ` dmt` gf, ` quf o`ctrmgnz ` dmt` gf   D D    c˒cm˒` sfr o`jpbfjfctmg` fjm` sfr o`jpbfjfctmg` fj   F F ..
JfsJfsj` j` c˒cm˒` m` amamvfvfcg` cg` vfvfrs˒rsm˒` m` vft`vft r̀nmb rnmb g` g` WWf`rfjm f`rfjm gf gf Amac‟ImAmac‟Imcmoa, cmoa, fst·fst·m m mifrtmmifrtm
m m p`ssnp`ssninbinbngmgngmgf f gf gf `pfrm`pfrmg`rfg`rfs s bncbncfmrffmrfs s o`co`ctt··ĽcĽcu`s u`s fj fj gftfgftfrjnrjncmg`cmg`s s fspmfspm±±o`s o`s sfrsfrfjfj
fstffstfcgcg··Ľvfns. Ľvfns. U`r fxfU`r fxfjpb`, jpb`, `pf`pfrmg`rmg`rfs bnrfs bncfmrcfmrfs o`fs o`ctct··Ľcu`s tĽcu`s t`jmcg`jmcg` vmb`rfs ` vmb`rfs fjfj   ∓∓  p`gfj p`gfj
sfjprf sfr fstfcgng`s;sfjprf sfr fstfcgng`s;
Wf`rfjm ?.9.<Wf`rfjm ?.9.<   (Wf`rfjm gf Uanbbnps) (Wf`rfjm gf Uanbbnps)  _fkmj  _fkmj  D  D   uj suifspm±  uj suifspm±o` g` fspm±o` g` fspm±o` c`rjmg`o` c`rjmg`   F F   f f 
W W  ∁ ∁ BB((D,D, ∓∓)).. Fct˒  Fct˒ m` fxnstf m` fxnstf   W W  ∁ ∁ BB((F,F, ∓∓))  fxtfcs˒  fxtfcs˒ m` gf m` gf   W W    m m   F F . Jmns mncgm,. Jmns mncgm,  W W  :: W W ..
Gfj`cstrm±Gfj`cstrm±oo˒˒mm`̀..   Umrm omgm  Umrm omgm   cc ∁ ∁   CC, , o`csngfrf o`csngfrf ` ` ducon`cmb ducon`cmb bncfmr bncfmr o`cto`ct··Ľcu` gf Ľcu` gf c`rjm c`rjm ==
gmg` p`rgmg` p`r
ϖϖcc ;;  ∓∓ ∙ ∙↙↙ HH  , , ϖϖcc

((mm k k))
∓∓
 k k:=:=

:: m mcc..
Fct˒Fctm˒`m` ϖ ϖcc◤◤W W  ∁ ∁ D D  pmrm t`g` pmrm t`g` c c f f W  W ((xx) : (() : ((ϖϖcc◤◤W W )()(xx))))∓∓cc:=:= pmrm t`g` pmrm t`g` x x ∁∁ D D . Ufb` Wf`rfjm. Ufb` Wf`rfjm
gf Amac‟Imcmoa, pmrm omgmgf Amac‟Imcmoa, pmrm omgm  c c  fxnstf fxnstf  ϖϖcc  fxtfcs˒ fxtfcsm˒` gfm` gf ϖ ϖcc ◤◤ W W    mm F  F   o`jo`j  ϖϖcc ◤◤ W W  :: ϖϖcc..
··F F d·d·monb vfr quf ` `pfrmg`rmonb vfr quf ` `pfrmg`r
W W  ; ;   F F  ∙ ∙↙↙ ∓∓    ,,  W W ((xx) ) : : ((ϖϖcc((xx))))∓∓cc:=:=,,
7>7>
  
··f f bncbncfmrfmr, , o`ct·o`ct·Ľcu` Ľcu` f f fstfstfcgfcgff   W W . . U`r ﬍jU`r ﬍j, rfbfj, rfbfjirmcirmcg` qufg` quf   IILL   gfc`tm m i`bm ucnt·  gfc`tm m i`bm ucnt·mrnmmrnm
dfoamgm g` fspm±dfoamgm g` fspm±o` c`rjmg`o` c`rjmg`   LL, tfj`s, tfj`s


W W  : sup: sup
xx
∁∁
II
FF


W W ((xx)) : sup: sup
xx
∁∁
II
FF
supsup
cc
||

ϖϖcc((xx))|| : sup: sup
cc
supsup
xx
∁∁
II
FF
||

ϖϖcc((xx))||
: sup: sup
cc
ϖϖcc : sup: sup
cc
ϖϖcc ◤◤ W W  : sup: sup
cc
supsup
xx∁∁IID D 
||ϖϖcc((W W ((xx))))||
: sup: sup
xx∁∁IID D 
supsup
cc
||ϖϖcc((W W ((xx))))|| : sup: sup
xx∁∁IID D 
W W ((xx)) :: W W ..
O`O`r`br`b··mrnmrn ̀̀ ?.?.9.9.11    _f _f    ∓∓   ·  ·f f susuifsifspmpm±±o` dfoamg` gf uj fspm±o` dfoamg` gf uj fspm±o` gf Imcmoa o` gf Imcmoa    F F , fct˒ , fct˒ mm ̀̀ ··f f    ==--
o`jpbfjfctmg` fj o`jpbfjfctmg` fj   F F ..
Gfj`cstrm±Gfj`cstrm±oo˒˒mm`̀..    MpbMpbnqunquf f ` ` WWf`rff`rfjm jm ?.9?.9.< .< pmrm ` pmrm ` `pfrmg`pfrmg`r `r ngfngfctctngmgngmgf f fjfj    ∓∓   pmrm  pmrm
`itf`itfr r uj uj `pfrmg`pfrmg`r`r  W W  ; ;   F F  ∙ ∙↙↙  ∓∓  quf fstfcgf m ngfctngmgf f quf fstfcgf m ngfctngmgf f  W W  : =. Oamjmcg` gf : =. Oamjmcg` gf   nn
m ncobus˒m ncobusm˒` gfm` gf   ∓∓   fjfj   F F  sfluf quf sfluf qufnn ◤◤ W W  · ·f f ujm ujm pr`pr` kf±kf±o˒om˒` gfm` gf   F F   s`irf  s`irf   ∓∓  gf c`rjm =. gf c`rjm =.
??..? ? MMppbbnnoomm±±o˒o˒̀fs g` Wf`rfjm gf Amac‟Imcmoa pmrm`fs g` Wf`rfjm gf Amac‟Imcmoa pmrm
fspm±fspm±o`o`s s sfsfpmpmr·r·mvfmvfnsns
O`j` k·O`j` k·m m amvamv··Ľmj`s Ľmj`s mgnmctmg`, mgnmctmg`, `s `s fspm±fspm±o`s sfpmr·o`s sfpmr·mvfns l`zmj gf pr`prnfgmgfs fspfonmnsmvfns l`zmj gf pr`prnfgmgfs fspfonmns
quf t`rcmj m tf`rnm g`s fspm±quf t`rcmj m tf`rnm g`s fspm±o`s sfpmr·o`s sfpmr·mvfns ifj mvfns ifj jmns rnom jmns rnom quf m quf m tf`rnm lftf`rnm lfrmb. rmb. TTfrfj`sfrfj`s
cfstm sf±cfstm sf±o˒om˒` `m` `s prs prnjfnr`njfnr`s ncs ncgg··Ľon`s Ľon`s gfstf gfstf dfc˄dfc˄̀jfc`. Yfbfjirf quf sf`jfc`. Yfbfjirf quf sf  M M  · ·f f uj uj suisuio`co`ckuctkuct`̀
gf uj fspm±gf uj fspm±o` c`rjmg`o` c`rjmg`   F F   ff   xx ∁∁ F F , fct˒, fctm˒` gnst(m` gnst(x,x, MM) : ncd ) : ncd {{xx ∙∙ yy ;; y y ∁ ∁ MM}}..
Ur`p`sn±Ur`p`sn±o˒om˒ ̀m ̀ ?.?.?.?.==   _fkmj   _fkmj    F F   uj   uj fspfspm±m±o` o` c`rc`rjmgjmg`,`,    J J   uj   uj suisuifspfspm±m±o` o` dfdfoaoamg` mg` gf gf    F,F,
yy55   ∁∁    F F  ∙ ∙   J J    f f    gg   :   : gngnst(st(yy55,, J J ))..   Fct˒   Fct˒ m` m` fxnfxnstf stf uj uj ducducon`on`cmb cmb bncbncffmr mr    ϖϖ  ∁∁    F F    tmb quf   tmb quf 
ϖϖ : : ==, , ϖϖ((yy55) ) :: g g   f f   ϖϖ((xx) ) : : 55  pmrm t`g` pmrm t`g`   xx ∁∁ J J ..
Gfj`cstrm±Gfj`cstrm±oo˒˒mm`̀..   _fkm  _fkm C  C   :: J  J ++XXyy55Z. Fct˒Z. Fctm˒`, pmrmm`, pmrm z  z 
 ∁ ∁
C C  fxnstfj · fxnstfj ·ucno`sucno`s m m
∁∁
HH ff x x
∁∁
J J 
tmns quftmns quf  z  z   :: x x ++ mymy55. Gf﬍cm. Gf﬍cm
ϖϖ55 ;;   C C  ∙ ∙↙↙ HH  , , ϖϖ55((xx + + mymy55) ) :: mg. mg.
··F obmr` qufF obmr` quf   ϖϖ55  · ·f f bnbncfcfmrmr,,   ϖϖ55((J J ) ) :: { {55}}  f quf f quf   ϖϖ55((yy55) ) ::  g.g.  Ur`vfj`s quf Ur`vfj`s quf  ϖϖ55  : : ==..  _fkm _fkm
z z   :: x x ++ mymy55 ∁ ∁ C C . Umrm. Umrm  m m  : : 5,5,
z z  :: xx + + mymy55 :: ||mm|| ¼¼

xx
∙∙mm ∙ ∙ yy55
 ≦≦ gg ||mm|| :: ||ϖϖ55((z z ))|| ,,
f pmrmf pmrm   mm : 5 m gfsnlumbgmgf : 5 m gfsnlumbgmgf  z z  ≦ | ≦ |ϖϖ55((z z ))||  ·  ·f f ··̀i`ivnvnm. m. _f_fluluf f ququff  ϖϖ55 ≧ ≧ =. Gmg` =. Gmg`   ζ ζ 33  5 5,,
fxnstffxnstf    xxζζ ∁ ∁    J J   tmb quf  tmb quf    gg ≧  ≧ yy55 ∙∙ xxζζ  ≧≧   gg + +  ζ ζ. . __fkfkmm    z z ζζ    ::
   yy55 ∙∙ xxζζ
yy55 ∙∙ xxζζ
..   Fct˒  Fctm˒`m`    z z ζζ ∁ ∁    C,C,
z z ζζ : : = = ff
ϖϖ55((z z ζζ) ) ::
   gg
yy55 ∙∙ xxζζ ≦ ≦
   gg
gg ++ ζζ ..
6565
  
O`j`O`j`   ζ ζ 33  5  5 ··f f mrmrinintrtr··mrn`, sfluf qufmrn`, sfluf quf  ϖϖ55 ≦ ≦  =, f p`rtmct` =, f p`rtmct`  ϖϖ55   : : ==..  Ufb` Wf`rfjm gf Ufb` Wf`rfjm gf
Amac‟Imcmoa fxnstfAmac‟Imcmoa fxnstf   ϖϖ  ∁∁ F F   quf fstfcgf quf fstfcgf   ϖϖ55   mm  F  F   tmb quf tmb quf  ϖϖ :: ϖϖ55 : : ==..
Wf`rfjm ?.?.9Wf`rfjm ?.?.9    _f _f   F F   · ·f f ssfpfpmmr· r· mvfb, fct˒ mvfb, fct˒ m`m`   F F   t  tmmjji·i·ffj j ··f f ssffpmpmr· r· mvfb.mvfb.
Gfj`cstrm±Gfj`cstrm±oo˒˒mm`̀..   _fkm  _fkm   _ _ F F   m fsdfrm ucnt· m fsdfrm ucnt·mrnm gfmrnm gf   F F 
, `u sfkm,, `u sfkm,   _ _ F F    :: { {ϖϖ ∁ ∁ F  F    ;;  ϖϖ :  : ==}}..
O`cd`rjf k·O`cd`rjf k·m jfcon`cmj`s mctfs, suio`ckuct` gf fspm±m jfcon`cmj`s mctfs, suio`ckuct` gf fspm±o` o` j·j·ftrnftrno` o` sfpmrsfpmr··mmvfvfb b ··f f ttmmjiji··ffjj
sfpmr·sfpmr·mvfb, b`l`mvfb, b`l`  _  _ F F   · ·f f sfsfpmpmr·r·mvfmvfb. b. _f_fkmkm { {ϖϖcc  ;; c c  ∁∁ CC}} uj suio`ckuct` fcujfr· uj suio`ckuct` fcujfr·mvfb f gfcs`mvfb f gfcs`
gfgf   _ _ F F ..  Umrm omgm Umrm omgm   cc ∁ ∁   CC   p`gfj`s t`jmr  p`gfj`s t`jmr   xxcc ∁ ∁   _ _ F F   tmb quf  tmb quf | |ϖϖcc((xxcc))| ≦| ≦    ==99 . . OamOamjfj`jfj s̀s
J J   : : XXxx==,, xx99, . . ., . . .Z f pr`vfj`s qufZ f pr`vfj`s quf  J  J   :: F F.. Umrm tmct`, sup`camj`s quf Umrm tmct`, sup`camj`s quf  J  J   sfkm gndfrfctf sfkm gndfrfctf
gfgf  F  F  f fso`bamj`s f fso`bamj`s  y y55 ∁ ∁ F F ∙∙ J.J.  Ufb Ufbm m Ur`Ur`p`p`sn±sn±o˒om˒` ?.?.= fxnstf uj ducon`cmbm` ?.?.= fxnstf uj ducon`cmb   ϖϖ  ∁∁ F F    o`jo`j
ϖϖ : = tmb quf: = tmb quf  ϖ ϖ((yy55) ) :: g g  : gnst( : gnst(yy55,, J J ) ) ff   ϖϖ((xx) : 5 pmrm t`g`) : 5 pmrm t`g`   xx ∁∁ J J . Fct˒. Fctm˒`m`
ϖϖ ∙∙ ϖϖcc : sup: sup
xx∁∁IIFF
||((ϖϖ ∙∙ ϖϖcc)()(xx))|| ≦≦ ||((ϖϖ ∙∙ ϖϖcc)()(xxcc))|| :: ||ϖϖcc((xxcc))|| ≦≦
  ==
99
..
Jms Jms nss` nss` ··f f uj uj misurg` misurg` p`nsp`ns { {ϖϖcc    ;;    cc  ∁∁    CC}}   ·  ·f f gfcgfcs` s` fjfj    _ _ F F ..   U`rtmct`  U`rtmct`    J J    ::    F F    f f mm
sfpmrminsfpmrminbngmgf bngmgf gfgf   F F  sfluf g` Bfjjm =.4.?. sfluf g` Bfjjm =.4.?.
Tfrfj`s cm @isfrvm±Tfrfj`s cm @isfrvm±o˒om˒` m` 7.9.9 7.9.9 quf quf m m rforfo··Ľpr`om Ľpr`om g` g` WWf`rfjm f`rfjm ?.?.9 ?.?.9 c˒cm˒` m` ··f f vfrgvfrgmgfmgfnrmnrm..
··F F surprffcgsurprffcgfctf fctf m m fxnstfxnst˄˄fconm fconm gf gf uj uj fspm±fspm±o` gf Imcmoa gfctr` g` qumb p`gfj`so` gf Imcmoa gfctr` g` qumb p`gfj`s
fco`ctrmr t`g`s `s fspm±fco`ctrmr t`g`s `s fspm±o`s sfpmr·o`s sfpmr·mvfnmvfns. s. Jmns surprffcgfcJmns surprffcgfctf tf mncgm ·mncgm ·f f ` ` dmt` gf dmt` gf quf quf fssffssf
supfr-fspm±supfr-fspm±o` o` ··f f uj uj g`s g`s c`ss`s c`ss`s vfba`s ovfba`s o`cafong`s;`cafong`s;
Ur`p`sn±Ur`p`sn±o˒om˒ ̀m ̀ ?.?.?.?.??   W`g` fspm±  W`g` fspm±o` c`rjmg` sfpmr· o` c`rjmg` sfpmr· mvfb mvfb ··f f ns`j`rd` ns`j`rd` ns`jftrnons`jftrnomjfctf mjfctf m m uj uj 
suifspm±suifspm±o` gf o` gf   ∓∓..
Gfj`cstrm±Gfj`cstrm±oo˒˒mm`̀..   _fkmj  _fkmj    F F  u uj j fspm±fspm±o` o` c`rjmc`rjmg` g` sfpmrsfpmr··mvfb mvfb ff G G    ::  {{xxcc    ;;    cc ∁ ∁   CC}}    ujuj
suio`ckuct` fcujfr·suio`ckuct` fcujfr·mvfb gfcs` fjmvfb gfcs` fj    F.F.   U`gfj`s obmrmjfctf sup`r quf 5  U`gfj`s obmrmjfctf sup`r quf 5    //∁ ∁ GG. . UUffbb`̀
O`r`b·O`r`b·mrn` ?.=.7, pmrm omgmmrn` ?.=.7, pmrm omgm   cc ∁ ∁ C C  fxnstf uj ducon`cmb bncfmr fxnstf uj ducon`cmb bncfmr   ϖϖcc ∁ ∁  F  F    tmb quf tmb quf  ϖϖcc :  : ==
ff   ϖϖcc((xxcc) ) :: xxcc ..  O`csngfrf ` `pfrmg`r O`csngfrf ` `pfrmg`r
W W  ; ;   F F  ∙ ∙↙↙ ∓∓    , , W W ((xx) ) : : ((ϖϖcc((xx))))∓∓cc:=:=..
··F obmr` qufF obmr` quf | |ϖϖcc((xx))| | ≧ ≧ ϖϖcc xx    ::  xx  pmrm t`g`s pmrm t`g`s   cc ∁ ∁   CC    ff    xx ∁ ∁    F F , p`rtmct`, p`rtmct`   W W   fst·  fst·mm
ifj gf﬍cng` c` sfctng` gf quf (ifj gf﬍cng` c` sfctng` gf quf (ϖϖcc((xx))))
∓∓
cc:=:= ∁ ∁    ∓∓   pmrm t`g`  pmrm t`g`   xx ∁ ∁   F F . . Mb·Mb·fj gfj gnssnss`,`,   W W    ··ff
obmrmjfctf bncfmr fobmrmjfctf bncfmr f
W W ((xx)) : sup: sup{|{|ϖϖcc((xx))|| ;; c c  ∁∁ CC}} ≧≧ xx ,,   (?.7)  (?.7)
` quf pr`vm, fj pmrtnoubmr, quf` quf pr`vm, fj pmrtnoubmr, quf   W W   ·  ·f f o`o`ct·ct·ĽcĽcu`u`. . C`C`tf tf ququff
W W ((xxhh)) : sup: sup{|{|ϖϖcc((xxhh))|| ;; c c  ∁∁ CC}} ≦≦ ||ϖϖhh((xxhh))|| :: xxhh   (?.6)  (?.6)
pmrm t`g`pmrm t`g`   hh ∁ ∁   CC. . Gf (?.Gf (?.7) f 7) f (?.(?.6) rfs6) rfsububtm quftm quf  W W ((xxhh))   ::  xxhh  pmrm t`g` pmrm t`g`   hh ∁ ∁   CC. . GmGm
gfcsngmgf g` o`ckuct`gfcsngmgf g` o`ckuct` { {xxcc  ;; c c ∁ ∁CC}} f gm o`ctncungmgf gm duc± f gm o`ctncungmgf gm duc±o˒om˒`m`  x x ∁∁ F F   ↙↙ W W ((xx)) ∁∁ YY
sfluf qufsfluf quf W W ((xx)) :: xx pmrm t`g`pmrm t`g` x x  fj fj  F  F . Mssnj. Mssnj W  W   ··f f uj uj ns`ns`j`rj`r﬍sj﬍sj` ` ns`ns`j·j·ftrftrno` no` fctrfctrff
F F    ff   W W ((F F )) ⊄⊄ ∓∓..
6=6=
  
?.?.7 7 DD`r`rjmjms s lflf`j`j··ftftrnrnomoms s g` g` WWf`f`rfrfjm jm gf gf AmAmacac‟‟
Imcmoa (oms` rfmb)Imcmoa (oms` rfmb)
CfCfststm sm sf±f±o˒om˒` gfj`cstrmrfj`s g`ns tf`rfjms, o`cafong`s o`j` prnjfnrm f m` gfj`cstrmrfj`s g`ns tf`rfjms, o`cafong`s o`j` prnjfnrm f sflucgm d`rjmssflucgm d`rjmslf`j·lf`j·ftrnoms ftrnoms g` tf`rfg` tf`rfjm gf jm gf Amac‟Imcmoa. Amac‟Imcmoa. Fssfs rfsuFssfs rfsubtmg`s gnzfj, btmg`s gnzfj, fssfconmbjfctf, quffssfconmbjfctf, quf
sfsf   MM   ff   II  s˒ sm˒` m` suisuio`ckuo`ckuct`s ct`s g` g` fspm±fspm±o` o` c`rjmc`rjmg` g` rfmbrfmb   F F  quf c˒ quf cm˒` s˒m` sm˒` junt` fctrfbm±m` junt` fctrfbm±omg`somg`s
(fj uj sfctng` quf ﬍omr·(fj uj sfctng` quf ﬍omr·mobmr` c` j`jfct` mpr`prnmg`), fct˒m obmr` c` j`jfct` mpr`prnmg`), fctm˒` fxnstf uj ducon`cmbm` fxnstf uj ducon`cmb
bncbncfmr fmr o`ct·o`ct·Ľcu` Ľcu` fjfj  F  F  tmb quf tmb quf
ϖϖ((xx)) 2  2 ϖϖ((yy) pmrm t`g`s) pmrm t`g`s   xx ∁∁ MM  ff   yy ∁ ∁ I.I.
_`i gftfrjncmgms o`cgn±_`i gftfrjncmgms o`cgn±o˒o˒̀fs s`irf`fs s`irf   MM   ff   II  p`gf-sf nr jmns b`clf f lmrmctnr quf p`gf-sf nr jmns b`clf f lmrmctnr quf
supsup{{ϖϖ((xx) ) ;; x x ∁∁ MM}} 22 ncd  ncd {{ϖϖ((xx) ) ;; x x ∁∁ II}}..
U`gfj`s nctfrprftmr fssfs rfsubtmg`s s`i m pfrspfotnvm gf quf ` ducon`cmbU`gfj`s nctfrprftmr fssfs rfsubtmg`s s`i m pfrspfotnvm gf quf ` ducon`cmb    ϖϖ   sfpmrm  sfpmrm
`s o`ckuct`s`s o`ckuct`s   MM   ff   II. . U`U`r nss`, fssfr nss`, fssfs rfsubts rfsubtmg`s s˒mg`s sm˒` m` tmji·tmji·fj fj oamjmg`oamjmg`s s gfgf   tf`rfjms gf  tf`rfjms gf 
sfpmrm±sfpmrm±o˒ o˒ m`m`. . M M mi`rgmi`rgmlfj gfssmlfj gfssfs rfsubtfs rfsubtmg`s gfjmcmg`s gfjmcgm mblujmgm mblujms s gf﬍cgf﬍cn±n±o˒o˒̀fs f rfsubtmg`s`fs f rfsubtmg`s
prfprfbnbnjnjncmrcmrfs. fs. U`U r̀ r snsnjpjpbnbnonongmgmgf gf trmtrmtmrtmrfjfj`s `s mpfcmpfcms ms ` ` omoms` s` rfmrfmb, b, fjfji`rm fxnsi`rm fxnstmjtmj
rfsubtmgrfsubtmg`s `s pmrfongpmrfong`s `s pmrm ` pmrm ` oms` o`jpbfx` (vfkm Fxfrooms` o`jpbfx` (vfkm Fxfro··Ľon` ?.4.9<). Ľon` ?.4.9<). Mssnj, m` Mssnj, m` b`cl`b`cl`
gfstm sf±gfstm sf±o˒om˒` t`g`s `s fspm±m` t`g`s `s fspm±o`s vft`rnmns s˒o`s vft`rnmns sm˒` m` rfmrfmns, ns, nstnst` ·` ·f,f,  H H :: YY. . C` OmC` Ompp··Ľtub` Ľtub` 1 ms 1 ms d`rjmd`rjmss
lf`j·lf`j·ftrnoftrnoms ms gf gf Amac‟Amac‟Imcmoa Imcmoa sfr˒sfrm˒` trmtmgms cuj o`ctfxt` jmns lfrmb.m` trmtmgms cuj o`ctfxt` jmns lfrmb.
Gf﬍cn±Gf﬍cn±o˒om˒ ̀m ̀ ?.?.7.7.==   _fkm  _fkm    T T   uj fspm±  uj fspm±o` vft`rnmb c˒o` vft`rnmb cm˒`-m`-cucub`b`. . VjVj  anpfrpbmc` anpfrpbmc`    gfgf    T T   ·  ·f f uujj
suifspm±suifspm±o`o`    \ \   ::    T T   tmb quf sf  tmb quf sf    \ \ ==   ·  ·f f susuiifsfspmpm±±o` gfo` gf    T T    ff    \ \   ⊄⊄    \ \ ==, fct˒, fctm˒`m`    \ \ ==    ::    T T    `u`u
\ \ ==  :: \. \.
Tfkmj`s quf `s anpfrpbmc`s s˒Tfkmj`s quf `s anpfrpbmc`s sm˒` fxmtmjfctf `s c·m` fxmtmjfctf `s c·uobf`s g`s ducon`cmns bncfmrfs;uobf`s g`s ducon`cmns bncfmrfs;
Ur`p`sn±Ur`p`sn±o˒om˒ ̀m ̀ ?.?.7.7.99   _fkmj   _fkmj   T T   uj fspm±  uj fspm±o` vft`rnmb,o` vft`rnmb,    T T   :: { {55}}, , f f   \ \   uj suifspm±  uj suifspm±o` gf o` gf   T T ..
Fct˒ Fct˒ m`m`   \ \  · ·f f uj uj anpfranpfrpbmpbmc` c` gf gf   T T   sf, f s`jfctf sf, fxnstf uj ducon`cmb bncfmr c˒   sf, f s`jfctf sf, fxnstf uj ducon`cmb bncfmr c˒ m`-cub`m`-cub`
ϖϖ ;;   T T   ∙∙↙↙ YY  tmb quf  tmb quf  hfr( hfr(ϖϖ) ) :: \  \ ..
Gfj`cstrm±Gfj`cstrm±oo˒˒mm`̀..   _up`cam quf  _up`cam quf    \ \   sfkm anpfrpbmc` gf  sfkm anpfrpbmc` gf    T T . . OO`̀jj`̀    \ \   ::    T T , p`gfj`s, p`gfj`s
fso`bafrfso`bafr   vv55 ∁ ∁   T T   ∙∙ \.\. W`jmcg` W`jmcg`   \ \ ==   : : XX\ \  ∫ ∫ {{vv55}}Z, o`j`Z, o`j`   \ \   ⊄⊄   \ \ ==    ff   \ \   ::   \ \ ==, tfj`s, tfj`s
\ \ ==    ::    T.T.   Mssnj sfluf dmonbjfctf quf omgm  Mssnj sfluf dmonbjfctf quf omgm    vv  ∁∁    T T   sf fsorfvf gf j`g` ·  sf fsorfvf gf j`g` ·ucno` o`j`ucno` o`j`
vv  :: u u ++ mvmv55   o`jo`j   uu ∁∁ \ \    ff   mm  ∁∁ YY..    ··F njfgnmt` quf ` ducon`cmbF njfgnmt` quf ` ducon`cmb
ϖϖ ;;  T T   ∙∙↙↙ YY    , , ϖϖ((uu + + mvmv55) ) :: m, m,
··f f c˒cm˒`m`-cu-cub`b`, , bncbncfmfmr r f f hfrhfr((ϖϖ) ) :: \  \ ..
Yfonpr`omjfcYfonpr`omjfctf, tf, sfkmsfkm ϖ ϖ ;;   T T   ∙∙↙↙ YY uj ducon`cmb bncfmr c˒uj ducon`cmb bncfmr cm˒`-cub` tmb quf hfr(m`-cub` tmb quf hfr(ϖϖ) ) :: \  \ ..
··F obmr` quf hfr(F obmr` quf hfr(ϖϖ)) :: T  T   p`ns  p`ns ϖ ϖ  : 5. _fkm: 5. _fkm \  \  suifspm± suifspm±o` gfo` gf T  T  tmb quf hfr( tmb quf hfr(ϖϖ)) ⊄⊄ \ \ . Imstm. Imstm
j`strmr quf sf hfr(j`strmr quf sf hfr(ϖϖ))

:: \  \ , fct˒, fctm˒`m` \  \    :: T T..   ··F fxmtmjfctF fxmtmjfctf nss` quf vmj`s dmzff nss` quf vmj`s dmzfr! r! _up`cam_up`cam
6969
  
fct˒fctm˒` hfr(m` hfr(ϖϖ))  ::   \ \   f fso`bam  f fso`bam   vv55 ∁ ∁   \ \  ∙ ∙ hfr(hfr(ϖϖ))..  Gmg` Gmg`   vv ∁ ∁   T T , t`jmcg`, t`jmcg`   uu   ::   vv ∙ ∙    ϖϖ((vv))ϖϖ((vv55))vv55
tfj`stfj`s   uu ∁∁ hfr(hfr(ϖϖ)) ⊄⊄ \.\. Ml`rm sfluf quf Ml`rm sfluf quf   vv : :  u u + +    ϖϖ((vv))
ϖϖ((vv55))
vv55 ∁ ∁ \ \ ..
_f_f   A A  · ·f f uj uj anpanpfrpbmfrpbmc` c` gfgf   T T    ff   vv55
 ∁ ∁
T,T,  ` o`ckuct` ` o`ckuct`
vv55 + + A A   :: {{vv55 + + vv ; ;  v v ∁ ∁ A A }}
··f f oaoamjmjmgmg`̀   anpfrpbmc` m﬍j   anpfrpbmc` m﬍j    gfgf   T T . . Gm UrGm Ur`p``p`snsn±±o˒om˒` ?.7.9 sfluf quf `s anpfrpbmc`s m﬍csm` ?.7.9 sfluf quf `s anpfrpbmc`s m﬍cs
gfgf   T T    s˒sm˒` prfonsmjfctf `s o`ckuct`s gm d`rjmm` prfonsmjfctf `s o`ckuct`s gm d`rjm
{{vv ∁ ∁ T T    ;; ϖ ϖ((vv) ) :: m m}},,
`cgf`cgf  ϖ ϖ  · ·f f uj uj duconducon`cmb `cmb bncfmr bncfmr c˒cm˒`-cub` fjm`-cub` fj T  T    ff m m  · ·f f uuj j cc··ujfr` rfmb. Gf ml`rm fj gnmctfujfr` rfmb. Gf ml`rm fj gnmctf
oamjmrfj`s `s anpfrpbmc`s m﬍cs snjpbfsjfctf gf anpfrpbmc`s.oamjmrfj`s `s anpfrpbmc`s m﬍cs snjpbfsjfctf gf anpfrpbmc`s.
Ur`p`sn±Ur`p`sn±o˒om˒ ̀m ̀ ?.?.7.7.??   _fkm   _fkm    A A    ::   {{xx  ∁∁    F F    ;;    ϖϖ((xx) ) ::    mm}}   uj   uj anpanpfrpfrpbmcbmc` ` gf gf uj uj fspfspm±m±o`o`
c`rjmg`c`rjmg`    F F , `cgf , `cgf    ϖϖ  · ·f f uj uj ducon`cmb ducon`cmb bncfmr bncfmr fj fj    F F    f f    mm ∁ ∁   YY. . FFcctt˒ ˒ m` ` anpfrpbmc`m` ` anpfrpbmc`    A A    ··f f 
 dfoamg`  dfoamg` sf, sf, f f s`jfctf s`jfctf sf, sf, ` ` ducon`cmb ducon`cmb bncfmr bncfmr   ϖϖ  · ·f f òò cct·t·ĽĽccuu`̀..
Gfj`cstrm±Gfj`cstrm±oo˒˒mm`̀..   _up`cg`  _up`cg`    ϖϖ   o`ct  o`ct··Ľcu`, Ľcu`, ` ` anpanpfrpbmfrpbmc`c`    A A    ::    ϖϖ∙∙==(({{mm}}) ) ··f f dfoamgdfoamg` ` o`j`o`j`
njmlfj njmlfj ncvncvfrsm frsm gf gf o`ckuco`ckuct` t` dfoamdfoamg` g` p`r p`r duc±duc±o˒om˒` o`m` o`ctct··Ľcum. Ľcum. Yfonpr`omjfYfonpr`omjfctf, supctf, sup`cam`cam
qufquf   A A   sfk  sfkm dfoamm dfoamg`. g`. FcFct˒tm˒`m`   F F  ∙ ∙ A A  · ·f f mifmifrt` rt` f f c˒cm˒`-vmzn`, b`l` fxnstfjm`-vmzn`, b`l` fxnstfj   xx55 ∁ ∁   F F  ∙ ∙ A A    ff
r r 33   5 tmns quf  5 tmns quf   IIF F ((xx5588 rr)) ⊄ ⊄   F F  ∙ ∙ A A . . O`O`j`j`  ϖϖ((xx55))  ::   mm, p`gfj`s sup`r, sfj pfrgm gf, p`gfj`s sup`r, sfj pfrgm gf
lfcfrmbnlfcfrmbngmgf, gmgf, qufquf  ϖϖ((xx55)) 2  2 m.m.
M﬍rjm±M﬍rjm±o˒om˒`.m`.    ϖϖ((xx)) 2  2 mm  pmrm t`g` pmrm t`g`   xx ∁∁ II((xx5588 rr).).
Gf dmt`, sf fxnstnssfGf dmt`, sf fxnstnssf   xx== ∁ ∁ I I((xx5588 rr) o`j) o`j   ϖϖ((xx==)) 3  3 mm, t`jmcg`, t`jmcg`   tt : :    ϖϖ((xx==))∙∙mmϖϖ((xx==))∙∙ϖϖ((xx55))   t tfr·fr·ĽmĽmj`j`ss
txtx55 + (= + (= ∙∙ tt))xx== ∁ ∁ II((xx5588 rr) ) ff
ϖϖ((txtx55 + ( + (== ∙∙ tt))xx==) ) :: tϖ tϖ((xx55) + (=) + (= ∙∙ tt))ϖϖ((xx==))
::
   ϖϖ((xx==)) ∙∙ mm
ϖϖ((xx==)) ∙∙ ϖϖ((xx55))
ϖϖ((xx55) +) +

==
∙∙
   ϖϖ((xx==)) ∙∙ mm
ϖϖ((xx==)) ∙∙ ϖϖ((xx55))
ϖϖ((xx==) ) :: m. m.
M m﬍rjm±M m﬍rjm±o˒om˒` sfluf p`ns nss` o`ctrmgnz ` dmt` gf qufm` sfluf p`ns nss` o`ctrmgnz ` dmt` gf quf    IIF F ((xx5588 rr)) ⊄ ⊄    F F  ∙ ∙ A  A . . U`U`rtmcrtmct`,t`,
ϖϖ((xx55) +) +  rϖ rϖ((z z ) ) ::   ϖϖ((xx55 + +  rz  rz ))   2 2 mm  pmrm t`g` pmrm t`g`   z z  ∁ ∁   F F    o`jo`j  z z    22   =  =. . O`O`j`j`   ∙∙ z z    ::  z z ,,
sfluf qufsfluf quf
ϖϖ((xx55)) ∙∙ mm
rr
   2 2 ϖϖ((z z )) 2 2
  mm ∙∙ ϖϖ((xx55))
rr
   ,,
pmrm t`g` pmrm t`g`pmrm t`g` pmrm t`g`  z  z  ∁ ∁ F F   o`jo`j  z z  22 =. Nss` njpbnom quf =. Nss` njpbnom quf  ϖϖ ≧≧    mm∙∙ϖϖ((xx55))
rr
   , , ` ` quf quf lmrmclmrmctftf
m o`ctncungmgf gfm o`ctncungmgf gf   ϖϖ..
NcNctr`gtr`guznuznrfjrfj`s `s m m sflsflununr r ` ` duduconcoǹ cm`cmb b gf gf JnJnchch`w`wshshn, n, ququf f c` c` j`jj`jfcfct` t` c`s c`s sfrsfr··mm
··utnb cm `itfc±utnb cm `itfc±o˒om˒` m` gms gms d`rjms lf`j·d`rjms lf`j·ftrnoms g` ftrnoms g` WWf`rfjm gf f`rfjm gf Amac‟ImAmac‟Imcmoa, jms cmoa, jms quf quf c`c`
Omp·Omp·ĽtuĽtub` 1 gfb` 1 gfsfjsfjpfpfcamcamr·r·m uj pmpm uj pmpfb ofctrmb cm o`cstru±fb ofctrmb cm o`cstru±o˒om˒` gm tf`rnm gf Fspm±m` gm tf`rnm gf Fspm±o`s o`s TTft`rnmnsft`rnmns
W`p`b·W`p`b·̀lno`s.`lno`s.
6?6?
  
Gf﬍cn±Gf﬍cn±o˒om˒̀m ̀ ?.?.7.7.77   _fkm  _fkm   O O  uj  uj suio`ckucsuio`ckuct` t` o`cvo`cvfx`, fx`, mifrt` mifrt` f f quf quf o`cto`ct··fj fj m m `rnlfj g``rnlfj g`
fspm±fspm±o` c`rjmg`o` c`rjmg`  F  F . M mpbnom±. M mpbnom±o˒om˒`m`
 p pO O ;;   F F 
 ∙ ∙↙↙
YY    , , ppO O ((xx) : ncd ) : ncd 

m m 33 5  5 ;;
  xx
mm ∁ ∁
O O 

,,
··f f oaoamjmjmgmgmm  ducon`cmb gf Jnch`wshn gf  ducon`cmb gf Jnch`wshn gf   O O ..
Ur`p`sn±Ur`p`sn±o˒om˒ ̀m ̀ ?.?.7.7.66  @ ducon`cmb gf Jnch`wshn l`zm gms sflunctfs pr`prnfgmgfs; @ ducon`cmb gf Jnch`wshn l`zm gms sflunctfs pr`prnfgmgfs;
(m)(m)   ppO O ((ixix) ) :: ip ipO O ((xx))  pmrm t`g` pmrm t`g`   i i 33 5 5  f t`g` f t`g`   xx ∁∁ F.F.
(i)(i)   O O   :: {{xx ∁∁ F F   ;; p pO O ((xx)) 2 2  = =}}..
(o)(o)  Fxnstf  Fxnstf   J J 33 5 5  tmb quf  tmb quf   55 ≧≧  p pO O ((xx)) ≧≧ J J xx  pmrm t`g` pmrm t`g`   xx   fj fj   F F ..
(g)(g)   ppO O ((xx ++ yy)) ≧≧  p pO O ((xx) +) + p pO O ((yy))  pmrm qumnsqufr  pmrm qumnsqufr   x,x, yy ∁ ∁ F.F.
Gfj`cstrm±Gfj`cstrm±oo˒˒mm`̀..    (m)(m)   ppO O ((ixix) : ncd ) : ncd {{m m 33 5  5 ;;    ixixmm  ∁∁ O O }} :: i i ¼¼ ncd ncd {{m m 33 5  5 ;;    xxmm ∁ ∁ O O }} :: ip ipO O ((xx))..
(i) O`j`(i) O`j`    O O   ·  ·f f mifrt`, mifrt`, pmrm pmrm t`g`t`g`    xx  ∁∁    O O   fxnstf  fxnstf    ζ ζ 33   5 tmb quf (= +  5 tmb quf (= +  ζ ζ))xx ∁ ∁    O.O.   Mssnj  Mssnj
xx
(=+(=+ζζ))∙∙==
  ∁∁   O O  f p`rtmct` f p`rtmct`   ppO O ((xx)) ≧ ≧    (= (= ++ ζ ζ))∙∙== 22  =.  =. YfonYfonpr`omjpr`omjfcfctf, sftf, sf   ppO O ((xx))   22  =, fct˒ =, fctm˒`m`
gm gf﬍cn±gm gf﬍cn±o˒om˒` m` gf ·gf ·Ľc﬍j` Ľc﬍j` fxnstf fxnstf 55    2 2 m m 22   = tmb quf  = tmb quf    xx
mm
  ∁∁    O.O.   O`j`  O`j`    O O   ·  ·f f o`cvfo`cvfx` x` tfjtfj`s`s
xx : :  m m 
xx
mm+ (=+ (= ∙∙ mm)) ¼¼ 55 ∁∁ O O ..(o) _fkm(o) _fkm   r r 33 5 tmb quf 5 tmb quf   II(58(58 rr)) ⊄ ⊄ O  O . Umrm 5. Umrm 5  2 s 2 r 2 s 2 r  tfj`s tfj`s   ss    xxxx ∁ ∁  O  O  pmrm t`g` pmrm t`g`   xx ∁ ∁ F  F ,,
xx  : 5. : 5. Gf (mGf (m) f ) f (i(i) sfl) sfluf quuf quff   ppO O ((xx)) ≧ ≧    xxss   pmrm t`g`  pmrm t`g`   xx ∁ ∁   F F    c˒cm˒`-cub`.m`-cub`.    ··F obmr` qufF obmr` quf
fssm ·fssm ·ubtnjm ubtnjm gfsnlugfsnlumbgmgf mbgmgf tmji·tmji·fj fj vmbf pmvmbf pmrmrm  x x  : 5, f  : 5, f o`csfo`csfqufctfjqufctfjfctf fctf ` rf` rfsubtmsubtmg` fg` fst·st·mm
pr`vmg` o`jpr`vmg` o`j   J J   : : ==/s./s.
(g) _fkmj(g) _fkmj   x,x, yy ∁ ∁ F F    ff   ζ ζ 33 5 gmg`s. Tfkmj`s quf 5 gmg`s. Tfkmj`s quf    xx
 p pO O ((xx)+)+ζζ
 ∁ ∁ O O . O`j fdfnt`, p`r (m),. O`j fdfnt`, p`r (m),
 p pO O 

   xx
 p pO O ((xx) +) + ζζ

::
   ==
 p pO O ((xx) +) + ζζ
 p pO O ((xx)) 2 2  = =,,
f p`r (i) sfluf quff p`r (i) sfluf quf    xx
 p pO O ((xx)+)+ζζ
 ∁ ∁   O O . . Gm jfsGm jfsjm d`rjjm d`rjm,m,    yy
 p pO O ((yy)+)+ζζ
 ∁ ∁   O O . . O`O`j`j`  O O  · ·f f o`o`cvfcvfx`x`,,
t`jmcg` 5t`jmcg` 5 2  2 tt ;: ;:    ppO O ((xx)+)+ζζ
 p pO O ((xx)+)+ p pO O ((yy)+9)+9ζζ
  22 =, tfj`s =, tfj`s
xx ++ yy
 p pO O ((xx) +) + p pO O ((yy) + 9) + 9ζζ   :: t t
   xx
 p pO O ((xx) +) + ζ ζ   + (=+ (= ∙∙ tt))
   yy
 p pO O ((yy) +) + ζζ ∁ ∁ O.O.
Gf Gf (m) (m) f f (i) (i) o`cobuo`cobu··Ľj`s Ľj`s qufquf
==
 p pO O ((xx) +) + p pO O ((yy) + 9) + 9ζζ
 p pO O ((xx + + yy) ) :: p pO O 

   xx + + yy
 p pO O ((xx) +) + p pO O ((yy) + 9) + 9ζζ

22 = =,,
f p`rtmct`f p`rtmct`   ppO O ((xx + + yy)) 2  2 ppO O ((xx) +) + p pO O ((yy) + 9) + 9ζζ. @ rfsubtmg` sfluf dmzfcg`. @ rfsubtmg` sfluf dmzfcg`   ζζ  ∙∙↙↙ 5.5.
M fssm mbturm ` bfnt`r k·M fssm mbturm ` bfnt`r k·m m gfvgfvf f tfr tfr pfrofipfrofing` ng` quf quf m m ngfnngfnm m ··f f mpbmpbnomnomr r ` ` WWf`rff`rfjm jm gfgf
Amac‟Imcmoa o`j ` ducon`cmb gf Jnch`wshn dmzfcg` ` pmpfb g` ducon`cmbAmac‟Imcmoa o`j ` ducon`cmb gf Jnch`wshn dmzfcg` ` pmpfb g` ducon`cmb   pp..
Bfjm ?.7.4Bfjm ?.7.4   _fkm   _fkm    O O   uj suio`ckuct` o`cvfx`, mifrt`, pr·   uj suio`ckuct` o`cvfx`, mifrt`, pr· `prn` f c˒ `prn` f c˒ m`-vmzn` g` fspm±m`-vmzn` g` fspm±o`o`
c`rjmg`c`rjmg`    F F   f sfkm   f sfkm    xx55
  ∁∁
   F F 
 ∙ ∙
  O.O.   Fct˒   Fct˒ m` fxnstf uj ducon`cmb bncfmr m` fxnstf uj ducon`cmb bncfmr    ϖϖ
  ∁∁
   F F    tmb quf   tmb quf 
ϖϖ((xx)) 2  2 ϖϖ((xx55))  pmrm t`g` pmrm t`g`   xx ∁∁ O O ..
6767
  
Gfj`cstrm±Gfj`cstrm±oo˒˒mm`̀..  _up`cam quf 5 _up`cam quf 5   //∁∁ O O . Fso`bam. Fso`bam z  z 55 ∁ ∁ O O , o`csngfrf, o`csngfrf G G  : : {{xx ∙∙ z z 55  ;; x x ∁∁ O O }}
f oamjff oamjf    yy55    ::    xx55 ∙ ∙ z  z 55..  Wfj`s mssnj quf Wfj`s mssnj quf    yy55    //∁∁    GG, , 55  ∁∁    GG    ff    GG   ·  ·f f o`cvfx` o`cvfx` f f mifrt`.mifrt`.
U`gfj`s fct˒U`gfj`s fctm˒` o`csngfrmr ` ducon`cmb gf Jnch`wshnm` o`csngfrmr ` ducon`cmb gf Jnch`wshn   ppGG    gfgf   GG. . Gf﬍Gf﬍cncncgcg`̀   LL   : : XXyy55Z Z ff
ll((tyty55) ) :: t t
¼¼
 p pGG((yy55) pmrm t`g`) pmrm t`g`  t t
∁∁
YY tfj`s tfj`s l l((xx))
  ≧≧
 p pGG((xx) pmrm t`g`) pmrm t`g`  x x
∁∁
LL. Gf dmt`, pmrm. Gf dmt`, pmrm
t t 33 5, 5,
ll((tyty55) ) :: t t ¼¼ p pGG((yy55) ) :: p pGG((tyty55))
pfbm Ur`p`sn±pfbm Ur`p`sn±o˒om˒` ?.7.6(m). F pmrmm` ?.7.6(m). F pmrm   tt ≧≧ 5,5,
ll((tyty55) ) :: t t ¼¼ p pGG((yy55)) ≧≧ 55 ≧≧  p pGG((tyty55)),,
p`nsp`ns   ppGG((xx)) ≦ ≦ 5 pmrm t`g` 5 pmrm t`g`   xx ∁ ∁  F F . . Ufb` WUfb` Wf`rfjm gf Amac‟Imcmof`rfjm gf Amac‟Imcmoa fxnstf uj a fxnstf uj ducon`cmducon`cmbb
bncfmrbncfmr   ϖϖ ;;   F F  ∙ ∙↙↙   YY   tmb quf tmb quf   ϖϖ((xx) ) ::    ll((xx) pmrm t`g`) pmrm t`g`   xx ∁ ∁   LL    ff   ϖϖ((xx)) ≧ ≧   ppGG((xx) pmrm t`g`) pmrm t`g`
xx ∁ ∁   F F . . UfUfbm Urbm Ur`p`sn`p`sn±±o˒om˒` ?.7.6(o), fxnstfm` ?.7.6(o), fxnstf   J J 33  5 tmb quf 5 tmb quf   ϖϖ((xx)) ≧ ≧   ppGG((xx)) ≧ ≧   J J xx   pmrm  pmrm
t`g`t`g`    xx ∁ ∁    F F , ` quf lmrmctf m o`ctncungmgf gf, ` quf lmrmctf m o`ctncungmgf gf    ϖϖ. . Gm UGm Ur`r`p`sp`sn±n±o˒om˒` m` ?.7.6(i) ?.7.6(i) sfluf sfluf qufquf
ϖϖ((yy)) ≧≧  p pGG((yy)) 2 2  = pmrm t`g` = pmrm t`g`   yy ∁ ∁ GG. O`j`. O`j`   ppGG((yy55)) ≦≦ = = p`np`nss   yy55    //∁∁ GG,,
ϖϖ((yy)) 2 2  = =
≧≧
 p pGG((yy55) ) :: l l((yy55) ) :: ϖ ϖ((yy55) ) :: ϖ ϖ((xx55
∙∙
z z 55))
pmrm t`g`pmrm t`g`   yy ∁ ∁ GG. Gm gfsnlumbgmgf mctfrn`r f gm gf﬍cn±. Gm gfsnlumbgmgf mctfrn`r f gm gf﬍cn±o˒om˒` gfm` gf   GG  sfluf quf sfluf quf
ϖϖ((xx) ) :: ϖ ϖ((xx ∙∙ z z 55) +) + ϖ ϖ((z z 55)) 2  2 ϖϖ((xx55 ∙∙ z z 55) +) + ϖϖ((z z 55) ) :: ϖ ϖ((xx55))
pmrm t`g`pmrm t`g`   xx  fjfj   O O . _f 5. _f 5  ∁∁ O O , imstm t`jmr, imstm t`jmr   z z 55  :  : 5.5.
@ @ ··ubtnj` ubtnj` nclrfgnnclrfgnfctf fctf quf quf c`s c`s dmbtm ·dmbtm ·f f ujm ujm pr`prnfgpr`prnfgmgf mgf nctfnctfrfssmcrfssmctf tf g`s g`s ducon`cmducon`cmnsns
bncfmbncfmrfs rfs o`cto`ct··Ľcu`s Ľcu`s fj fj fspm±fspm±o`s c`rjmg`s;o`s c`rjmg`s;
Bfjm ?.7.<Bfjm ?.7.<   _fkmj   _fkmj    F F   uj fspm±  uj fspm±o` c`rjmg`,o` c`rjmg`,    ϖϖ  ∁∁    F F    uj ducon`cmb c˒   uj ducon`cmb c˒ m`-cub` f m`-cub` f    MM    uj uj 
suio`ckuct` o`cvfx`, mifrt` f c˒ suio`ckuct` o`cvfx`, mifrt` f c˒ m`-vmzn` gf m`-vmzn` gf   F F . . FcFct˒t˒m`m`   ϖϖ((MM))  · ·f f uj uj nctfrvmnctfrvmb` b` mifrt` mifrt` (c˒ (c˒ m`m`
cfofssmrnmjfctf bnjntmg`).cfofssmrnmjfctf bnjntmg`).
Gfj`cstrm±Gfj`cstrm±oo˒˒mm`̀..    Fj prnjfnr` bulmr `isfrvFj prnjfnr` bulmr `isfrvf quf o`j`f quf o`j`  ϖ ϖ

: 5, fct˒: 5, fctm˒`m` ϖ ϖ · ·f sf s`ir`irfkffkft`rt`r, p, p`ns`ns
sum sum njmlfnjmlfj ·j ·f f uj uj suisuifspm±fspm±o` vft`rnmb gfo` vft`rnmb gf  Y Y. Gm bncfmrngmgf gf. Gm bncfmrngmgf gf  ϖ ϖ  f gm o`cvfxngmgf gf f gm o`cvfxngmgf gf  MM
sfluf dmonbjfctf qufsfluf dmonbjfctf quf    ϖϖ((MM) ) ··f f uj uj suio`ckucsuio`ckuct` t` o`cvo`cvfx` fx` gfgf   YY, `u sfkm, uj nctfrvmb`., `u sfkm, uj nctfrvmb`.
C` oms` fj quf ` nctfrvmb`C` oms` fj quf ` nctfrvmb`    ϖϖ((MM) ) d`r d`r bnjbnjntmntmg` g` supfrsupfrn`rjn`rjfcfctf, tf, oaoamjfjmjfj`s `s gfgf    mm    sumsum
fxtrfjngmgf supfrn`r. _up`camj`s quffxtrfjngmgf supfrn`r. _up`camj`s quf  m m ∁∁ ϖϖ((MM). Fct˒). Fctm˒` fxnstfm` fxnstf x x ∁∁ MM tmb quf tmb quf ϖ ϖ((xx) ) :: m m
ff   ϖϖ((yy)) ≧ ≧   mm  pmrm t`g` pmrm t`g`   yy ∁ ∁   MM. . O`O`j`j`  MM  · ·f f mimifrtfrt` ` fxnfxnstfstf   ζ ζ 33  5 tmbquf m i`bm mifrtm gf 5 tmb quf m i`bm mifrtm gf
ofctr`ofctr`   xx f rmn` f rmn`   ζζ  fst· fst·m o`ctngm fjm o`ctngm fj   MM. _fkm. _fkm   z z  ∁ ∁ F F  uj vft`r c˒ uj vft`r cm˒`-cub`. Gfm`-cub`. Gf
xx ++
   ζζ
99z z  ¼ ¼ z z ∙∙ xx
 ::
   ζζ
99z z  ¼ ¼ z z  ::
  ζζ
99
  2 2 ζζ
sfluf qufsfluf quf  x x  + +
   ζζ
99z z  ¼ ¼ z z  pfrtfcof m pfrtfcof m   MM. Cfssf oms`,. Cfssf oms`,
mm  ≦≦ ϖϖxx + +
   ζζ
99z z  ¼ ¼ z z  :: ϖ ϖ((xx) +) +
   ζζ
99z z  ¼ ¼ ϖϖ((z z ) ) :: m m ++
   ζζ
99z z  ¼ ¼ ϖϖ((z z )),,
6666
  
` quf c`s bfvm m o`cobunr quf` quf c`s bfvm m o`cobunr quf   ϖϖ((z z )) ≧ ≧ 5. O`j` 5. O`j`   z z  · ·f f uj uj vftvft`r `r c˒cm˒`m`-cu-cub ̀b ̀ mrmrinintrtr··mrn` fjmrn` fj   F F ,,
nss` vn`bm m s`irfkftnvngmgf gfnss` vn`bm m s`irfkftnvngmgf gf   ϖϖ, f p`rtmct`, f p`rtmct`   m m //∁∁  ϖ ϖ((MM). ). Mcmb`lmMcmb`lmjfctf pr`jfctf pr`vvm-sf quf sfm-sf quf sf
ϖϖ((MM) d`r bnjntmg` ncdfrn`rjfctf fct˒) d`r bnjntmg` ncdfrn`rjfctf fctm˒`m`  ϖ ϖ((MM) ) c˒cm˒` m` o`ct·o`ct·fj fj sum sum fxtrffxtrfjngmgf jngmgf ncdfrncdfrn`r. n`r. Fst·Fst·mm
o`jpbftm m gfj`cstrm±o`jpbftm m gfj`cstrm±o˒om˒` gf quf ` nctfrvmb`m` gf quf ` nctfrvmb`   ϖϖ((MM) ) ··f f mimifrfrt`t`..
Umrm snjpbn﬍omr m c`tm±Umrm snjpbn﬍omr m c`tm±o˒om˒`, m`, ` ` anpanpfrpbmfrpbmc`c`   A A    :: { {xx ∁ ∁   F F    ;;   ϖϖ((xx) ) ::   mm}}   sfr· sfr·m gfc`tmg`m gfc`tmg`
pfpfb` b` ss··ĽjiĽji`b` `b` XXϖϖ : :  m mZ.Z.
Wf`rfjm ?.7.1Wf`rfjm ?.7.1    (Urnjfnrm (Urnjfnrm d`rjm d`rjm lf`j·lf`j·ftrnom ftrnom g` g` WWf`rfjm f`rfjm gf gf Amac‟Imcmoa)Amac‟Imcmoa) _fkmj  _fkmj  M M  f  f 
II  suio`ckuct`s o`cvfx`s, c˒  suio`ckuct`s o`cvfx`s, c˒ m`-vmzn`s f gnskuct`s g` fspm±m`-vmzn`s f gnskuct`s g` fspm±o` c`rjmg`o` c`rjmg`   F F . . _f _f   MM · ·f f mmifrifrt̀t̀ ,,
fct˒ fct˒ m` fxnstfj uj ducon`cmb m` fxnstfj uj ducon`cmb   ϖϖ ∁ ∁ F F     f f   mm ∁∁ YY  tmns quf  tmns quf 
ϖϖ((xx)) 2  2 mm  ≧≧ ϖϖ((yy) ) pmpmrm rm t`gt`g`s`s  xx ∁∁ MM   ff   yy ∁ ∁ I.I.
Cfstf oms` gnz-sf quf ` anpfrpbmc` dfoamg`Cfstf oms` gnz-sf quf ` anpfrpbmc` dfoamg`    XXϖϖ : :  m mZZ  sfpmrm  sfpmrm   MM   f f   II..
Gfj`cstrm±Gfj`cstrm±oo˒˒mm`̀..   Oamjf  Oamjf   O O   :: M M ∙∙ I.I.  C`tf quf C`tf quf
(n)(n)  O  O  · ·f f mimifrtfrt`, `, p`p`nsns   O O   ::
  ii∁∁II
((MM
∙∙ {{
ii
}}
))..
(nn)(nn)   O O  · ·f f o`cvfx`, o`cvfx`, p`p`ns ns pmrmpmrm   xx==,, xx99 ∁ ∁ MM,,  yy==,, yy99 ∁ ∁ II   f f 55 2  2 t t 22 = =,,
tt((xx== ∙∙ yy==) + (=) + (= ∙∙ tt)()(xx99 ∙∙ yy99) ) : : ((txtx== + ( + (== ∙∙ tt))xx99)) ∙∙ ((tyty== + (= + (= ∙∙ tt))yy99)) ∁∁ MM ∙∙ I.I.
(nnn)(nnn) ∆ ∆ :: O  O   :: F  F , p`ns, p`ns  MM  f f  I I   s˒sm˒` gnskuct`s f c˒m` gnskuct`s f cm˒`-vmzn`s.m`-vmzn`s.
O`j` 5O`j` 5   //∁∁ O,O, p`ns p`ns M M ∮∮II  :: ∆∆,, pfb` Bfjm ?.7.4 fxnstf pfb` Bfjm ?.7.4 fxnstf ϖ ϖ ∁ ∁ F F   tmb quf tmb quf ϖ ϖ((z z )) 2  2 ϖϖ(5) : 5(5) : 5
pmrm t`g`pmrm t`g`   z z  ∁ ∁ O O . B`l`. B`l`
ϖϖ((xx) ) :: ϖ ϖ((xx ∙∙ yy) +) + ϖϖ((yy)) 2  2 ϖϖ((yy) ) ((??..44))
pmrm t`g`spmrm t`g`s    xx ∁ ∁    MM    ff    yy  ∁∁    II. . UU`̀rr    II   sfr c˒  sfr cm˒`-vmzn`,m`-vmzn`,    mm   ;: sup  ;: sup
xx∁∁MM
ϖϖ((xx) ) ··f f uj uj c·c·ujfr` rfmb.ujfr` rfmb.
@ivnmjfct@ivnmjfctf f ··f f vfrgmgvfrgmgf f qufquf   ϖϖ((MM)) ⊄ ⊄   ((∙∓∙∓,, mmZ f gf (?.4) sfluf qufZ f gf (?.4) sfluf quf   mm ≧ ≧   ϖϖ((yy) pmrm t`g`) pmrm t`g`
yy ∁ ∁ II. G` Bfjm ?.7.< smifj`s quf. G` Bfjm ?.7.< smifj`s quf   ϖϖ((MM) ·) ·f f uj uj nctfrvmb` nctfrvmb` mifmifrt`, rt`, b`l`b`l`   ϖϖ((MM)) ⊄⊄ ((∙∓∙∓,, mm),),
nnsstt ̀̀ ··ff,,  ϖϖ((xx)) 2  2 mm pmrm t`g` pmrm t`g`   xx
∁∁
MM..
Wf`rfjm ?.7.>Wf`rfjm ?.7.>    (_flucgm (_flucgm d`rjm d`rjm lf`j·lf`j·ftrnom ftrnom g` g` WWf`rfjm f`rfjm gf gf Amac‟Imcmoa)Amac‟Imcmoa)  _fkmj  _fkmj  M M  f  f 
II  suio`ckuct`s o`cvfx`s, c˒  suio`ckuct`s o`cvfx`s, c˒ m`-vmzn`s f gnskuct`s g` fspm±m`-vmzn`s f gnskuct`s g` fspm±o` c`rjmg`o` c`rjmg`  F  F . . _f _f  M M  · ·f f dfodfoamamgg`̀
f f   II  · ·f f o`jpmo`jpmot`ot`, , fcfct  ˒t  ˒m` fxnstfj uj ducon`cmb m` fxnstfj uj ducon`cmb   ϖϖ  ∁∁ F F     f f   m,m, ii  ∁∁ YY  tmns quf  tmns quf 
ϖϖ((xx))  ≧≧ m m 2 2 ii  ≧≧ ϖϖ((yy) ) pmpmrm rm t`gt`g`s`s  xx ∁∁ MM   ff   yy ∁ ∁ I.I.
Umrm Umrm   oo  ∁∁ ((m,m, ii))  gnz-sf quf ` anpfrpbmc` dfoamg` gnz-sf quf ` anpfrpbmc` dfoamg`    XXϖϖ : :  o oZZ  sfpmrm  sfpmrm   MM   f f   II  fstrntmjfctf. fstrntmjfctf.
Gfj`cstrm±Gfj`cstrm±oo˒˒mm`̀..    TTfkmfkmj`s j`s quf quf ··f f p`p`ssss··Ľvfb Ľvfb fsofso`ba`bafrfr    ζ ζ 33   5 gf j`g` quf  5 gf j`g` quf   MM + +  I I(58(58 ζζ) ) ff
II + + II(58(58 ζζ) sfkmj mifrt`s, o`cvfx`s f gnskuct`s;) sfkmj mifrt`s, o`cvfx`s f gnskuct`s;
(n) Umrm qumbqufr fso`bam gf(n) Umrm qumbqufr fso`bam gf  ζ  ζ 33 5, `s o`ckuct`s 5, `s o`ckuct`s   MM ++ II(58(58 ζζ) ) ff  II + + II(58(58 ζζ) ) s˒sm˒` mifrt`s.m` mifrt`s.
Gf dmt`,Gf dmt`,
MM ++ II(58(58 ζζ) ) ::  νν∁∁MM ((νν + + II(58(58 ζζ)) )) ::  νν∁∁MMII((νν88 ζζ)),,
6464
  
f ` jfsj` `o`rrf pmrmf ` jfsj` `o`rrf pmrm   II + + II(58(58 ζζ))..
(nn) Umrm qumbqufr fso`bam gf(nn) Umrm qumbqufr fso`bam gf   ζ ζ 33 5, o`j` 5, o`j`   MM,,   II    ff   II(58(58 ζζ) ) s˒sm˒` m` o`cvfx`s, o`cvfx`s, ··f f njfgnmnjfgnmt` t` qufquf
`s o`ckuct`s`s o`ckuct`s   MM ++ II(58(58 ζζ) ) ff   II + + II(58(58 ζζ) ) s˒sm˒ ̀m ̀ o`o`cvfxcvfx`s`s..
(nnn) Imstm fct˒(nnn) Imstm fctm˒` m` pr`vmr pr`vmr quf quf ··f f p`p`ssss··Ľvfb Ľvfb fsofso`ba`bafrfr  ζ ζ  gf j`g` quf gf j`g` quf M M ++ II(58(58 ζζ) ) ff I I ++ II(58(58 ζζ))
sfkmsfkmj gnskuj gnskuctct`s. `s. _up`c_up`cam quf nss` c˒am quf nss` cm˒` m` sfkm psfkm p`ss`ss··ĽvfĽvfb. b. Cfssf oms`, Cfssf oms`, pmrm omgmpmrm omgm    cc ∁ ∁   CC,,
tfj`stfj`s 
MM ++ II

5858    ==
cc

II + + II

5858    ==
cc

:: ∆∆..
B`lB`l` ` fxfxnsnstfj tfj ((xxcc))
∓∓
cc:=:=   ⊄⊄    MM, , ((yycc))∓∓cc:=:=   ⊄⊄    II    f f ((z z cc))∓∓cc:=:=,, ((wwcc))∓∓cc:=:=   ⊄⊄    II

5858    ==
cc

  tmn  tmns s qufquf
xxcc + + z z cc  :: y ycc + + wwcc  pmrm t`g` pmrm t`g`   cc  ∁∁ CC. Mssnj. Mssnj
xxcc ∙∙ yycc :: z z cc ∙∙ wwcc 22
   99
cc
  pmrm t`g`  pmrm t`g`   cc  ∁∁ CC..   (?.<)  (?.<)
O`j`O`j` I I  · ·f f o`o`jpjpmomot`t`, , m m sfsfqu˄quf˄cfcononm m ((yycc))
∓∓
cc:=:= tf tfj sj suisfquisfqu˄uf˄conm fconm o`cvfrlfo`cvfrlfctf ctf fjfj I I , gnlmj`s, gnlmj`s
yycchh ∙ ∙↙↙ ι ι  ∁ ∁ II. . U`U`r (?.<) sflur (?.<) sfluf quff quf x xcchh ∙ ∙↙↙ ι ι . O`j`. O`j` M M  · ·f f dfoamgdfoamg`, `, smismifj`s fj`s qufquf  ι  ι  ∁ ∁ MM,,
f mssnjf mssnj   ι ι  ∁ ∁ MM ∮∮ I,I,  ` quf o`ctrmgnz ` dmt` gf sfrfj ` quf o`ctrmgnz ` dmt` gf sfrfj   MM  ff   II  gnskuct`s. gnskuct`s.
Dnxfj`s fct˒Dnxfj`s fctm˒`m` ζ  ζ 33 5 tmb quf 5 tmb quf M M ++II(58(58 ζζ) ) ff I I ++II(58(58 ζζ) sfkmj gnskuct`s. Ufb` Wf`rfjm) sfkmj gnskuct`s. Ufb` Wf`rfjm
?.7.1 fxnstf uj anpfrpbmc` dfoamg` X?.7.1 fxnstf uj anpfrpbmc` dfoamg` Xϖϖ : :  o oZ Z gfgf   F F  quf sfpmrm quf sfpmrm   MM + + I I(58(58 ζζ) ) ff   II + + I I(58(58 ζζ))..B`l`, pmrmB`l`, pmrm   xx ∁∁ MM  ff  y y ∁ ∁ II,,
ϖϖ((xx) ) + + susupp
zz==2ζ2ζ
ϖϖ((z z ==)) ≧≧ oo  ≧≧ ϖϖ((yy) ) + + nnccd d  
zz992ζ2ζ
ϖϖ((z z 99)),,
f, gm bncfmrngmgf gff, gm bncfmrngmgf gf   ϖϖ,,
ϖϖ((xx) +) + ζζ ¼¼    supsup
 zz==ζζ 22==
ϖϖ
z z ==
ζζ

≧≧ oo ≧≧ ϖϖ((yy) +) + ζζ ¼¼    ncd ncd 
 zz99ζζ 22==
ϖϖ
z z 99
ζζ

..
Gnss` sfluf qufGnss` sfluf quf   ϖϖ((xx) ) ++ ζ ζ ϖϖ ≧ ≧   oo ≧ ≧   ϖϖ((yy)) ∙∙ ζζ ϖϖ  pmrm t`g`s pmrm t`g`s   xx ∁ ∁   MM    ff   yy ∁ ∁   II. . ImImststmm
t`jmrt`jmr   mm : :  o o ∙∙    ζζϖϖ
99
   ff   ii : :  o o ++    ζζϖϖ
99
   ..
M sflunctf vmrnm±M sflunctf vmrnm±o˒om˒` gm Ur`p`sn±m` gm Ur`p`sn±o˒om˒` ?.?.= sfluf junt` dmonbjfctf;m` ?.?.= sfluf junt` dmonbjfctf;
O`O`r`br`b··mrnmrn ̀̀ ?.?.7.7.=5=5   _fkm   _fkm   J J   uj suifspm±  uj suifspm±o` dfoamg` g` fspm±o` dfoamg` g` fspm±o` c`rjmg`o` c`rjmg`    F F . . FcFctt˒ ˒ m` pmrm m` pmrm 
t`g`t`g`   xx55 ∁ ∁   F F  ∙ ∙ J J   fxnstf uj ducon`cmb  fxnstf uj ducon`cmb   ϖϖ ∁ ∁   F F    tmb quf   tmb quf   ϖϖ((xx55) ) : : ==   f f   ϖϖ((xx) ) : : 55  pmrm t`g` pmrm t`g`
xx ∁∁ J J ..
Gfj`cstrm±Gfj`cstrm±oo˒˒mm`̀..Mpbnomcg` ` Wf`rfjm ?.7.> pmrm  Mpbnomcg` ` Wf`rfjm ?.7.> pmrm   MM   ::   J J    ff    II    :: { {xx55}}, fxnstfj uj, fxnstfj uj
ducon`cmb Τducon`cmb Τ ∁∁ F F  f f  o o ∁∁ YY tmns quf Τ(tmns quf Τ(xx)) 2  2 o o 22 Τ( Τ(xx55) pmrm t`g`) pmrm t`g` x x ∁∁ J J . Jms m njmlfj gf. Jms m njmlfj gf
uj suifspm±uj suifspm±o` o` p`r p`r uj `uj `pfrmg`r pfrmg`r bncfmr ·bncfmr ·f uj f uj suifspm±suifspm±o` o` g` g` o`ctrmo`ctrmg`jg`j··Ľcn`, Ľcn`, b`l` b`l` Τ(Τ(xx) ) : : 55
pmrm t`g`pmrm t`g`   xx ∁∁ J J . Mssnj Τ(. Mssnj Τ(xx55)) 3 3  5 f p`rtmct` imstm t`jmr 5 f p`rtmct` imstm t`jmr   ϖϖ : :    ==Τ(Τ(xx55))Τ.Τ.
O`O`r`br`b··mrnmrn ̀̀ ?.?.7.7.====   _fkm   _fkm    J J   uj   uj suisuifspfspm±m±o` g` fspm±o` g` fspm±o` o` c`rc`rjmgjmg`̀    F F . . FFcctt˒ ˒  m` m` ppmrmrm m t`t g̀`g`
xx55  ∁∁    F F ,,    xx55  ∁∁    J J   sf, f s`jfctf, sf   sf, f s`jfctf, sf    ϖϖ((xx55) ) : : 55   pmr  pmrm m t`g`t`g`    ϖϖ ∁ ∁    F F    tmb quf   tmb quf    ϖϖ((xx) ) : : 55
pmrm t`g`pmrm t`g`   xx ∁∁ J J . @u sfkm,. @u sfkm,
J J   :: {{hfr(hfr(ϖϖ) ) ;; ϖ ϖ ∁ ∁ F F     ff   J J  ⊄ ⊄ hfr(hfr(ϖϖ))}}..
6<6<
  
Gfj`cstrm±Gfj`cstrm±oo˒˒mm`̀..   _up`cam  _up`cam    xx55    ∁∁    J J . . UUmmrrm tm t`g`g`̀    ϖϖ   ∁∁    F F    tmb   tmb ququff    J J    ⊄⊄   hfr(  hfr(ϖϖ),),
J J  ⊄ ⊄ hfr(hfr(ϖϖ) : hfr() : hfr(ϖϖ), p`ns hfr(), p`ns hfr(ϖϖ) ) ··f f dfdfoamoamg`g`..
Yfonpr`omjfctf, sup`cam qufYfonpr`omjfctf, sup`cam quf    xx55    //∁∁    J J . . O`O`j`j`  J J   ·  ·f f uj uj suisuifspfspm±m±o` dfoamg` gfo` dfoamg` gf   F F ,,
pfb` O`r`b·pfb` O`r`b·mrn` ?.7.=5 fxnstf uj ducon`cmbmrn` ?.7.=5 fxnstf uj ducon`cmb   ϖϖ55
 ∁ ∁
 F  F    tmb quf tmb quf   ϖϖ55((xx55) : = f) : = f   ϖϖ55((xx) : 5 pmrm) : 5 pmrm
t`g`t`g`   xx ∁∁ J J . B`l`. B`l`  x x55    //∁∁ {{hfr(hfr(ϖϖ) ) ;; ϖ ϖ  ∁∁ F F    ff  J  J  ⊄ ⊄ hfr(hfr(ϖϖ))}}..
?.6 ?.6 O`jfcO`jfct·t·mrn`s f mrn`s f c`tms anst·c`tms anst·̀rnoms`rnoms
D`rjms jmns snjpbfs, `u jfc`s lfrmns, g` Wf`rfjm gf Amac‟Imcmoa d`rmj pr`vmgmsD`rjms jmns snjpbfs, `u jfc`s lfrmns, g` Wf`rfjm gf Amac‟Imcmoa d`rmj pr`vmgms
p`r Afbp`r Afbby fj by fj =>=>=9 f =9 f p`r Amap`r Amac c fj =>9fj =>99. 9. AmAmac jfbac jfba`a`r`r`u u sfsfu u rfrfsusubtbtmgmg` ` f f fj =>9fj =>9<<
puibno`u ujm d`rjm g` tf`rfjm pmrm ducon`cmns gf﬍cng`s fj fspm±puibno`u ujm d`rjm g` tf`rfjm pmrm ducon`cmns gf﬍cng`s fj fspm±o`s go`s gf Imcf Imcmomoa. a. MM
d`rjm gf﬍cntnvm g` Wf`rfjm gf Amac‟Imcmoa, pmrm ducon`cmns gf﬍cng`s fj fspm±d`rjm gf﬍cntnvm g` Wf`rfjm gf Amac‟Imcmoa, pmrm ducon`cmns gf﬍cng`s fj fspm±o`so`s
c`rjmg`sc`rjmg`s, d`n puibnomgm p`r , d`n puibnomgm p`r ImcmoImcmoa fj a fj =>9>. =>9>. Fssms prnjfnrFssms prnjfnrms gfj`cstrm±ms gfj`cstrm±o˒o˒̀fs usmvmj`fs usmvmj
ncgu±ncgu±o˒om˒` trmcs﬍cm` trmcs﬍cntm8 ` ntm8 ` Bfjm gf Bfjm gf [`rc mpmrf[`rc mpmrfofu mpfcms fj ofu mpfcms fj =>?6=>?6. . GumGums s d`cd`ctfs anst·tfs anst·̀rnoms`rnoms
o`c﬍·o`c﬍·mvfns, X41Z f X<?Z, fcdmtnzmj quf Afbby tfvf ujm njp`rt˄mvfns, X41Z f X<?Z, fcdmtnzmj quf Afbby tfvf ujm njp`rtm˄conm c` cmsonjfct` gmmconm c` cmsonjfct` gm
Mc·Mc·mbnsf Ducon`cmb junt` mbnsf Ducon`cmb junt` jmn`r g` jmn`r g` quf quf m quf m quf c`rjmbjfctf baf c`rjmbjfctf baf ··f of orfgntmgm.rfgntmgm.
@ @ omsoms` ` o`jpo`jpbfxbfx` ` g` g` WWf`rff`rfjm jm gm gm AmacAmac‟Im‟Imcmocmoa a d`n d`n gfj`gfj`cstrcstrmg` mg` prnjprnjfnrfnrmjfcmjfctftf
pmrm `s fspm±pmrm `s fspm±o`so`s    BB p p,,    p p 33   =,   =, p`r D. p`r D. JuJurrrrmmy y fj =>?fj =>?4. 4. @ @ omoms` lfrs` lfrmb d`n pr`mb d`n pr`vvmgmg` ` gfgf
d`rjm ncgfpfcgfctf fj g`ns trmimba`s, uj gf L. M. _`uha`jbnc`v f `utr` gf A. D.d`rjm ncgfpfcgfctf fj g`ns trmimba`s, uj gf L. M. _`uha`jbnc`v f `utr` gf A. D.
I`acfcibust f M. _`iozyh, mji`s ourn`smjfctf puibnomg`s c` jfsj` mc` gf =>?1. FjI`acfcibust f M. _`iozyh, mji`s ourn`smjfctf puibnomg`s c` jfsj` mc` gf =>?1. Fj
=><6 K. A`bir``h pr`v`u ujm vfrs˒=><6 K. A`bir``h pr`v`u ujm vfrsm˒` quf vmbf fj o`rp`s mrqunjfgnmc`s qumnsqufr.m` quf vmbf fj o`rp`s mrqunjfgnmc`s qumnsqufr.
Cm Cm gfj`gfj`cstrcstrm±m±o˒om˒` m` g` g` WWf`f`rfjrfjm m gf gf AmAmac‟ac‟ImImcmocmoa a usmusmj`j`s s ` ` MxMxn`n`jm jm gm gm FsoFso`b`bamam
cm cm d`d`rjm g` rjm g` BfjBfjm m gf gf [`r[`rc. c. @ @ bfbfntnt`r `r p`gf p`gf sf sf pfrpfrlucluctmr sf, tmr sf, m m fxffxfjpjpb` b` g` g` prpr··̀p`prnrn`̀
Bfjm Bfjm gf gf [`rc[`rc, , g` g` WWf`rff`rfjm jm gf gf [frj[frjfb` fb` (Urn(Urncoco··ĽpnĽpn` ` gm gm I`m I`m @rgf@rgfcm±cm±o˒om˒`) f gf v·m`) f gf v·mrn`smrn`s
`utr`s `utr`s fcuconmg`s ofcuconmg`s o`cafong`s, `cafong`s, ` ` WWf`rfjm f`rfjm gf Amagf Amac‟Imcmoa ·c‟Imcmoa ·f fquf fqunvnvmbfctf m` mbfctf m` Mxn`jm gmMxn`jm gm
Fso`bam. Fso`bam. M rfspM rfsp`stm ·`stm ·f cff cflmtnvm f lmtnvm f m gfj`m gfj`cstrm±cstrm±o˒om˒` p`gf sfr fco`ctrmgm fj X41, <.==Z.m` p`gf sfr fco`ctrmgm fj X41, <.==Z.
M iusom gf vfrs˒M iusom gf vfrs˒̀fs `fs vft`rnmns g` vft`rnmns g` WWf`rfjm f`rfjm gf gf Amac‟Imcmoa, c` Amac‟Imcmoa, c` sfctng` gm sfctng` gm _f±_f±o˒om˒` ?.9,m` ?.9,
sf t`rc`u ujm vfrtfctf njp`rtmctf gm tf`rnm g`s fspm±sf t`rc`u ujm vfrtfctf njp`rtmctf gm tf`rnm g`s fspm±o`s gf Imcmoo`s gf Imcmoa. a. O`jO`j` vnj`s c`` vnj`s c`
Wf`rfjm ?.9.<, `pfrmg`rfs m vmb`rfs fjWf`rfjm ?.9.<, `pfrmg`rfs m vmb`rfs fj    
∓∓
  p`gfj sfjp  p`gfj sfjprf sfr fstfcgrf sfr fstfcgng`ng`s. s. @ jfsj`@ jfsj`
vvmbf pmrm mbf pmrm `utr`s fspm±`utr`s fspm±o`s, p`r fxfjpb`, ` fspm±o`s, p`r fxfjpb`, ` fspm±o` gf duc±o` gf duc±o˒o˒̀fs bnjntmgms`fs bnjntmgms I I((Q Q ) g` Fxfjpb`) g` Fxfjpb`
=.=.9 =.=.9 (vfkm (vfkm FxfroFxfro··Ľon` Ľon` ?.4.=1) ?.4.=1) f f ` ` fspm±fspm±o`o`   BB∓∓((¶¶), `cgf), `cgf   ¶¶ · ·f f ujm ujm jfgngm jfgngm gf gf pr`impr`iminbngminbngmgfgf
(vfkm X95, Waf`rfj 7.=7Z). Cfssf o`ctfxt` gnz-sf quf uj fspm±(vfkm X95, Waf`rfj 7.=7Z). Cfssf o`ctfxt` gnz-sf quf uj fspm±o` gf Imcmoao` gf Imcmoa  F  F   ··ff nckftnv` nckftnv`
sf vmbf m tfsf g` Wf`rfjm gf Amac‟Imcmoa pmrm `pfrmg`rfs m vmb`rfs fjsf vmbf m tfsf g` Wf`rfjm gf Amac‟Imcmoa pmrm `pfrmg`rfs m vmb`rfs fj   F F , , nsnst ̀t ̀ ··f, f, sfsf
pmrm t`g`s fspm±pmrm t`g`s fspm±o`s gf Imcmoao`s gf Imcmoa   D D    ff   LL    o`jo`j   D D   ⊄⊄   LL, , t`g` t`g` `pfrmg`r `pfrmg`r bncfmr bncfmr o`cto`ct··Ľcu` gfĽcu` gf
LL    fjfj    F F   mgjntf ujm fxtfcs˒  mgjntf ujm fxtfcsm˒` m` bncfmr bncfmr o`cto`ct··Ľcum Ľcum mm    D D   gf jfsj  gf jfsjm c`rjm. m c`rjm. U`U`r fcqumr fcqumct`ct`
smifj`s fct˒smifj`s fctm˒` qufm` quf   ∓∓,,   II((Q Q ) ) ff   BB∓∓((¶¶), o`j), o`j   ¶¶  jfgngm gf pr`iminbngmgf, s˒ jfgngm gf pr`iminbngmgf, sm˒` nckftnv`s.m` nckftnv`s.
M M jfsjm jfsjm gfj`cgfj`cstrm±strm±o˒om˒` g` O`r`b·m` g` O`r`b·mrn` ?.9.1 c`s j`strm quf t`g` fspm±mrn` ?.9.1 c`s j`strm quf t`g` fspm±o` o` nckftnnckftnv` v` ··f f =-=-
o`jpbfjfctmg` fj qumbqufr fspm±o`jpbfjfctmg` fj qumbqufr fspm±o` quf ` o`ctfcam. M. Lr`tafcgnfoh pr`v`u fj =>6?o` quf ` o`ctfcam. M. Lr`tafcgnfoh pr`v`u fj =>6?
quf t`g` fspm±quf t`g` fspm±o` o` ncknckftnftnv` ·v` ·f f c˒cm˒`-sfpmr·m`-sfpmr·mvfb. mvfb. Nckftnvngmgf ·Nckftnvngmgf ·f uf ujm pjm pr`prnfgmgf r`prnfgmgf fso`rrflmgnmfso`rrflmgnm
f f eems ms vfzfs vfzfs surprffcgfctf, surprffcgfctf, p`r p`r fxfjpb`, fxfjpb`, c˒cm˒` m` ··f f afrfgnt·afrfgnt·mrnmrnm; m; smifjsmifj`s`s   ∓∓  · ·f f ncknckftnftnv` v` jmsjms
sfu sfu suisuifspm±fspm±o` o` dfoamg`dfoamg`    oo55    c˒cm˒` m` ··f f nckftnv` nckftnv` (p`r (p`r sfr sfr sfpmr·sfpmr·mvmvfb). fb). M fstrutM fstruturm g`s fspmurm g`s fspm±±o`so`s
nckftnv`s d`n gfsvfcgmgm pfb` trmimba` gf v·nckftnv`s d`n gfsvfcgmgm pfb` trmimba` gf v·mrn`s jmtfj·mrn`s jmtfj·mtno`s mtno`s c` c` pfrpfr··Ľ`g` Ľ`g` =>65-=>61,=>65-=>61,
6161
  
ncobuncg` ` fjncfctf jmtfj·ncobuncg` ` fjncfctf jmtfj·mtno` irmsnbfnr` Bf`p`bg` Cmoainc (=>99-=>>?);mtno` irmsnbfnr` Bf`p`bg` Cmoainc (=>99-=>>?);
Wf`rfjm ?.6.=Wf`rfjm ?.6.=   (Cm  (Cmoaoaincinc, , L``gcfL``gcfr, r, HfbHfbbfybfy, , AmsAmsujnujn))   Ms   Ms sflunctfs sflunctfs m﬍rjm±m﬍rjm±o˒ o˒ `f`fs s s˒ s˒ m`m`
fqunvmbfctfs s`irf uj fspm±fqunvmbfctfs s`irf uj fspm±o` gf Imcmoa (rfmb `u o`jpbfx`)o` gf Imcmoa (rfmb `u o`jpbfx`)   F F ;;
(m)(m)   F F  · ·f f ncnckfkftntnv`v`..
(i)(i)F F   ·  ·f f =-o=-ò jpbf`jpbfjfctmjfctmg` g` fj fj qumbqqumbqufr ufr fspfspm±m±o` o` gf gf ImImcmcmoa oa ququf f ` ` oo`c`ctftfcacam m oo`j`j`̀
suifspm±suifspm±o`.o`.
(o)(o)   F F   tfj m pr`prnfgmgf gm nctfrsf±  tfj m pr`prnfgmgf gm nctfrsf±o˒ o˒ m` inc· m` inc· mrnm, mrnm, nst` nst` ··f, f, t`gm t`gm o`bf±o`bf±o˒ o˒ m` gf i`bms dfoamgms m` gf i`bms dfoamgms 
fj fj   F F  o`j nctfrsf± o`j nctfrsf±o˒ o˒ m` j· m` j· utum tfj nctfrsf±utum tfj nctfrsf±o˒ o˒ m` c˒ m` c˒ m`-vmznm.m`-vmznm.
(g)(g)    F F   ns`jft  ns`jftrnornomjfctmjfctf f ns`j`rns`j`rd` d` m` m` fspfspm±m±o`o`    O O ((H H ))   gms   gms duduc±c±o˒ o˒ `fs `fs o`cto`ct··Ľcums Ľcums m m vmb`rfs vmb`rfs 
fsombmrfs gf﬍cngms fj uj fspm±fsombmrfs gf﬍cngms fj uj fspm±o` t`p`b· o` t`p`b· `lno` o`jpmot` fxtrfjmjfctf gfso`cfx``lno` o`jpmot` fxtrfjmjfctf gfso`cfx`   H H    o`j o`j 
m c`rjm g` suprfj`.m c`rjm g` suprfj`.
Vj fspm±Vj fspm±o` t`p`b·o` t`p`b·̀lno``lno` H  H  · ·f ff fxtrfjmjfctf gxtrfjmjfctf gfso`cfx` fso`cfx` sf sf ` df` dfoa` gf oa` gf qumbqufr qumbqufr o`ckuct`o`ckuct`
mifrt` fjmifrt` fj H  H  · ·f mif mifrt`. frt`. B. CmoaincB. Cmoainc, fj , fj =>65, =>65, tmji·tmji·fj rffj rfs`bvfu gs`bvfu gf﬍cntf﬍cntnvmjfctf m qnvmjfctf m qufst˒ufstm˒`m`
g`s g`s fspm±fspm±o`s o`s nckftnv`s nckftnv`s rfmns rfmns gf gf gnjfcs˒gnjfcsm˒` ﬍cntm;m` ﬍cntm;
Wf`rfjm ?.6.9Wf`rfjm ?.6.9   (Cmoainc X44Z) (Cmoainc X44Z) Vj fspm± Vj fspm±o` gf Imcmoa rfmb o` gf Imcmoa rfmb  F  F  gf gnjfcs˒  gf gnjfcs˒ m` ﬍cntm m` ﬍cntm  c c  ∁∁ CC
··f f nckftnv` nckftnv` sf, sf, f f s`jfctf s`jfctf sf,sf,   F F  · ·f f ns`j`ns`j`rd` rd` ns`jftns`jftrnomjfctf rnomjfctf m m   ((YYcc,,  ¼¼ ∓∓))..
Fxninr Fxninr uj uj suifspm±suifspm±o` o` dfoamdfoamg` g` c˒cm˒`-o`jpbfjfctmg` gf uj fspm±m`-o`jpbfjfctmg` gf uj fspm±o` o` gf gf ImImcmcmooa, a, `u`u
jfsj` jfsj` pr`vpr`vmr mr sum sum fxnst˄fxnstf˄conm, fconm, cucom cucom ··f f ujm ujm tmrfdm tmrfdm junt` junt` d·d·monb. monb. Mb·Mb·fj fj g`s g`s fxfjpbfxfjpb̀ s`s
jfcon`cmjfcon`cmg`s g`s c` c` tfxt`, tfxt`, smif-sf smif-sf tmjitmji··fj fj qufquf    ==   o`ct·  o`ct·fj fj uj uj suisuifspm±fspm±o` dfoamg` c˒o` dfoamg` cm˒`-m`-
o`jpo`jpbfjbfjfctfctmg` mg` (v(vfkm fkm X9, X9, O`r`O`r b̀bmbbmry ry 9.?9.?.?Z.?Z), ), qufquf    BB==X5X5,, =Z =Z o`ct·o`ct·fj fj uj uj suisuifspm±fspm±o` o` c˒cm˒`-m`-
o`jpbfjfctmg` ns`j`rd` mo`jpbfjfctmg` ns`j`rd` m    99   (vfkm X95, p·  (vfkm X95, p·mlncm =7Z) f quf pmrm =mlncm =7Z) f quf pmrm =    2 2 p p 22 ∓ ∓,,    pp  : : 9,9,
BB p pX5X5,, =Z =Z o`ct·o`ct·fj fj uj uj suisuifspm±fspm±o` c˒o` cm˒`-o`jpbm`-o`jpbfjfctfjfctmg` mg` ns`j`rdns`j`rd̀ ` mm   p p  (vfkm X95, p· (vfkm X95, p·mlncm 94Z).mlncm 94Z).
M prnjfnrm vfrs˒M prnjfnrm vfrsm˒` g` m` g` WWf`rfjm ?.7.1 d`n pr`vf`rfjm ?.7.1 d`n pr`vmgm p`r mgm p`r Jmzur fj =>??. Jmzur fj =>??. Umrm fcucoUmrm fcucon·n·m-bmm-bm
prfonsmj`s g` sflunctf o`cofnt`; ujprfonsmj`s g` sflunctf o`cofnt`; uj  suifspm± suifspm±o` m﬍j o` m﬍j  (`u (`u  vmrnfgmgf bncfmr  vmrnfgmgf bncfmr ) gf uj fspm±) gf uj fspm±o`o`
vft`rnmbvft`rnmb  T T    ··f f ` ` trmcsbmgmg` trmcsbmgmg` gf gf uj uj suifspm±suifspm±o` o` vft`rnmb, `u vft`rnmb, `u sfkm, ·sfkm, ·f f uj uj o`ckuct` go`ckuct` gm d`m d`rjmrjm
xx55 + + \ \   `cgf  `cgf  x x55 ∁ ∁ T T    ff   \ \  · ·f f uj uj suisuifspfspm±m±o` vft`rnmb gfo` vft`rnmb gf   T T ..
Wf`rfjm ?.6.?Wf`rfjm ?.6.?   X41, Waf`rfj <.<.?Z X41, Waf`rfj <.<.?Z  _fkmj  _fkmj   MM  uj suio`ckuct` mifrt` f o`cvfx` gf uj  uj suio`ckuct` mifrt` f o`cvfx` gf uj 
fspm±fspm±o` c`rjmg`o` c`rjmg`   F F    f f   J J  uj suifspm± uj suifspm±o` m﬍j gf o` m﬍j gf   F F  tmns quf  tmns quf   J J  ∮ ∮MM : : ∆∆. Fct˒ . Fct˒ m` fxnstf uj m` fxnstf uj 
anpfrpbmc` dfoamg`anpfrpbmc` dfoamg`   A A    gf gf   F F   o`ctfcg`  o`ctfcg`   J J   tmb quf   tmb quf   A A  ∮ ∮ MM : : ∆∆..
Fstf rfsubtmg` gf Jmzur Fstf rfsubtmg` gf Jmzur d`n oamjmg` gfd`n oamjmg` gf d`rjm  d`rjm lf`j·lf`j·ftrnom ftrnom g` g` WWf`rf`rfjm fjm gf gf Amac‟Amac‟Imcmoa Imcmoa 
c`s bnvr`s gm o`bf±c`s bnvr`s gm o`bf±o˒om˒` m` I`I`uriurimhmhn. n. @ @ j`tj`tnvnv` ` gfsgfstm tm gfcgfc`j`jncncm±m±o˒om˒` m` ··f f quf quf fstf fstf rfsubtmg`rfsubtmg`
lfcfrmbnzm ` dmt` gf quf t`gm rftm fjlfcfrmbnzm ` dmt` gf quf t`gm rftm fj   YY?? quf quf c˒cm˒` m` durm durm ujm ujm gmgm gmgm fsdfrm fsdfrm fst·fst·m o`ctngmm o`ctngm
fj uj pbmc` quf c˒fj uj pbmc` quf cm˒` m` nctfnctfrsfotm ` rsfotm ` nctfrnctfrn`r gm n`r gm fsdfrm. fsdfrm. M M pmrtnr gmpmrtnr gm··Ľ Ľ rfsubtmgrfsubtmg`s `s gfstf tnp`gfstf tnp`
pmssmrmj pmssmrmj m sfr m sfr oamjmg`s gf oamjmg`s gf d`rjms d`rjms lf`j·lf`j·ftrnoms ftrnoms g` Wf`rfjm gf g` Wf`rfjm gf Amac‟Imcmoa.Amac‟Imcmoa.
?.4 ?.4 FxfroFxfro··Ľon`sĽon`s
FxFxfro·fro·ĽoĽon ̀n ̀ ?.?.4.4.==   _fkm  _fkm   F F  u uj j fspm±fspm±o` o` c`rjmc`rjmg`,g`,   F F   :: {{55}}. J`strf quf. J`strf quf  F  F   :: {{55}}..
6>6>
  
FxFxfro·fro·ĽoĽon ̀n ̀ ?.?.4.4.99   _fkm  _fkm   F F  uj fspm± uj fspm±o` c`rjmg`. J`strf qufo` c`rjmg`. J`strf quf   F F    sfpmrm p`ct`s   sfpmrm p`ct`s   gfgf   F F , nst`, nst`
··ff, , ssff   x,x, yy ∁ ∁ F F    ff   xx :: y y, fct˒, fctm˒` fxnstfm` fxnstf   ϖϖ  ∁∁ F F   tmb quf tmb quf   ϖϖ((xx))  :: ϖ ϖ((yy).).
FxFxfro·fro·ĽoĽon ̀n ̀ ?.?.4.4.??   _fkmj  _fkmj    xx==, . . . , x, . . . , xcc   vf  vft`rt`rfs fs bnbncfcfmrjmrjfcfctf tf ncncgfgfpfcgpfcgfcfctfs tfs g` g` fsfspm±pm±o`o`
c`rjmg`c`rjmg`  F  F   ff m m==, . . . , m, . . . , mcc fsombmrfs gmg`s. J`strf quf fxnstf uj ducon`cmb fsombmrfs gmg`s. J`strf quf fxnstf uj ducon`cmb  ϖ ϖ ∁ ∁ F F   tmb quf tmb quf
ϖϖ((xx k k) ) :: m m k k  pmrm t`g` pmrm t`g`   kk   : : ==, . . . , c, . . . , c..
FxFxfro·fro·ĽoĽon ̀n ̀ ?.?.4.4.77    J`strf J`strf quf, fj quf, fj lfrmb, m lfrmb, m fxtfcs˒fxtfcsm˒` gf m` gf Amac‟Imcmoa c˒Amac‟Imcmoa cm˒m ̀̀ ··f f ··ucnom.ucnom.
FxFxfro·fro·ĽoĽon ̀n ̀ ?.?.4.4.66  C` Wf`rfjm ?.=.9, sf C` Wf`rfjm ?.=.9, sf   F F  · ·f f uj uj fspfspm±m±o` o` c`rc`rjmgjmg` ` f f fxnfxnstfstf   O O 33 5 tmb quf 5 tmb quf
 p p((xx)) ≧≧ O O xx  pmrm t`g` pmrm t`g`   xx ∁∁ F F , pr`vf quf, pr`vf quf  ϖϖ ≧≧ O  O ..
FxFxfro·fro·ĽoĽon ̀n ̀ ?.?.4.4.44    Gfj`cstGfj`cstrf ` rf ` WWf`rfjm gf Amac‟Imcmof`rfjm gf Amac‟Imcmoa, cm a, cm d`rjm g` O`r`b·d`rjm g` O`r`b·mrn` ?.=.?,mrn` ?.=.?,
c` oms` fj qufc` oms` fj quf  F  F  · ·f f uj uj fsfspmpm±±o` sfpmr·o` sfpmr·mvfb sfj mvfb sfj usmr ` usmr ` Bfjm gf Bfjm gf [`rc, nst` [`rc, nst` ··f, f, sfj usmr sfj usmr `s`s
Wf`rfjms ?.=.= f ?.=.9.Wf`rfjms ?.=.= f ?.=.9.
FxFxfro·fro·ĽoĽon ̀n ̀ ?.?.4.4.<<   _fkm  _fkm  D D  uj fspm± uj fspm±o` c`rjmg`. Ur`vf quf sf fxnstf uj fspm±o` c`rjmg`. Ur`vf quf sf fxnstf uj fspm±o` c`rjmg`o` c`rjmg`
F F   :: { {55}}   tmb quf tmb quf B B((F,F, D D ) ) ··f f uj uj fspm±fspm±o` gf Imcmoa, fct˒o` gf Imcmoa, fctm˒`m`   D D    tmtmji·ji·fj fj ··f f uj uj fsfspmpm±±o` gfo` gf
Imcmoa.Imcmoa.
FxFxfro·fro·ĽoĽon ̀n ̀ ?.?.4.4.11   _fkmj  _fkmj   F F    ff   D D   fspm±  fspm±o`s c`rjmg`s fo`s c`rjmg`s f   W W   ∁ ∁ BB((F,F, D D ). ). Ur`Ur v̀vf quff quf  W W    ::
supsup{|{|ϖϖ((W W ((xx))))|| ;; ϖ ϖ ∁∁ IID D     ff   xx ∁∁ IIF F }}..
FxFxfro·fro·ĽoĽon ̀n ̀ ?.?.4.4.>>   _fkm  _fkm   D D   uj suifspm±  uj suifspm±o` gfcs` g` fspm±o` gfcs` g` fspm±o` c`rjmg`o` c`rjmg`   F F . . Ur`Ur v̀vf quff quf  D D     ff
F F    s˒sm˒` m` ns`j`rdns`j`rd̀ s `s ns`jftrns`jftrnomjfcnomjfctf.tf.
FxFxfro·fro·ĽoĽon ̀n ̀ ?.?.4.4.=5=5   _fkmj  _fkmj F F,, D D   fspm±  fspm±o`s o`s c`rjmg`sc`rjmg`s,, L L  uj suifspm± uj suifspm±o` gfo` gf F  F   ff W  W  ∁ ∁ BB((L,L, D D ))
uj `puj `pfrmg`r frmg`r gf pgf p`st` ﬍cnt` `st` ﬍cnt` (vfkm Fxfro(vfkm Fxfro··Ľon` 9.<.95). Ľon` 9.<.95). Ur`vf quf fxnstfUr`vf quf fxnstf  W W  ∁ ∁ BB((F,F, D D ) quf) quf
fstfcgffstfcgf   W W  f tfj m jfsjm njmlfj quf f tfj m jfsjm njmlfj quf   W W ..
FxFxfro·fro·ĽoĽon ̀n ̀ ?.?.4.4.====   _fkm  _fkm    F F   uj   uj fspfspm±m±o` o` c`rc`rjmgjmg` ` sfpsfpmr·mr·mvmvfbfb. . UrUr`̀vvf f ququf f fxfxnsnstf ujtf ujmm
sfsfqu˄quf˄cfcononm m ((ϖϖcc))
∓∓
cc:=:=  gf ducon`cmns fj gf ducon`cmns fj    F F 
   tmns quf  tmns quf  ϖϖcc   : = pmrm t`g`: = pmrm t`g`    cc   f, pmrm t`g`  f, pmrm t`g`
xx ∁∁ F F ,,  xx : sup: sup
cc
||ϖϖcc((xx))||  c` oms` o`jpbfx` f c` oms` o`jpbfx` f  xx : sup: sup
cc
ϖϖcc((xx) c` oms` rfmb.) c` oms` rfmb.
FxFxfro·fro·ĽoĽon ̀n ̀ ?.?.4.4.=9=9   _fkmj  _fkmj   F F  uj fspm± uj fspm±o` c`rjmg` fo` c`rjmg` f   D D   uj suifspm±  uj suifspm±o` gfo` gf   F F . J`strf quf. J`strf quf
fxnstf ujm mpbnom±fxnstf ujm mpbnom±o˒om˒`m`   W W  ; ;   D D  ∙ ∙↙↙ F  F    nckft`rm tmb quf nckft`rm tmb quf  W W ((ϖϖ)) : :  ϖϖ  pmrm t`g` ducon`cmb pmrm t`g` ducon`cmb
ϖϖ ∁∁ D D . Nss` snlcn﬍om quf. Nss` snlcn﬍om quf   F F   o`ct· o`ct·fj fj uj uj suisuifspfspm±m±o` ns`j`rd` ns`jftrnomjfctf mo` ns`j`rd` ns`jftrnomjfctf m   D D 00
FxFxfro·fro·ĽoĽon ̀n ̀ ?.?.4.4.=?=?   _fkm  _fkm   II  uj suio`ckuct` gf uj fspm± uj suio`ckuct` gf uj fspm±o` c`rjmg`o` c`rjmg`   F F   tmb quf  tmb quf   ϖϖ((II) ) ··ff
bnjntmg` pmrm omgmbnjntmg` pmrm omgm   ϖϖ ∁ ∁ F F . J`strf quf. J`strf quf   II  · ·f f bnjbnjntntmgmg`.`.
FxFxfro·fro·ĽoĽon ̀n ̀ ?.?.4.4.=7=7* (Mcubmg`r) _fkmj* (Mcubmg`r) _fkmj    F F   uj fspm±  uj fspm±o` c`rjmg` fo` c`rjmg` f    J J   uj suio`ckuct`  uj suio`ckuct`
c˒cm˒`-vmzn` gfm`-vmzn` gf  F  F . . @@  mcubmg`r gf  mcubmg`r gf   J J  · ·f f gf﬍gf﬍cngcng` ` p`p`rr
J J ⊦⊦ : : {{ϖϖ  ∁∁ F F   ; ;  ϖ ϖ((xx) : 5 pmrm t`g`) : 5 pmrm t`g`   xx ∁∁ J J }}..
Ur`vf quf;Ur`vf quf;
(m)(m)   J J ⊦⊦  · ·f f uj uj suisuifspfspm±m±o` dfoamg` gfo` dfoamg` gf   F F ..
(i) _f(i) _f  J  J   ·  ·f f susuiifsfspmpm±±o` gfo` gf   F F , fct˒, fctm˒`m`  J  J   · ·f f ns`j`ns`j`rd` rd` ns`jfns`jftrnomtrnomjfctf jfctf mm   F F /J /J ⊦⊦..
(o) _f(o) _f   J J  · ·f f susuififspspm±m±o` dfoamg` gfo` dfoamg` gf   F F , fct˒, fctm˒`m`   J J ⊦⊦  · ·f nsf ns`j`rd`j`rd` n` ns`jfts`jftrnomjrnomjfctf fctf m (m (F/J F/J ))..
4545
  
FxFxfro·fro·ĽoĽon ̀n ̀ ?.?.4.4.=6=6   _fkmj  _fkmj D  D    ff L L  suifspm± suifspm±o`s dfoamg`s f gnstnct`s g` fspm±o`s dfoamg`s f gnstnct`s g` fspm±o` c`rjmg`o` c`rjmg` F  F ..
Ur`vf qufUr`vf quf   D D ⊦⊦  :: L L⊦⊦..
FxFxfro·fro·ĽoĽon ̀n ̀ ?.?.4.4.=4=4   _fkm  _fkm    D D   uj   uj suifssuifspm±pm±o` o` g` g` fsfspmpm±±o` o` gf gf ImImcmcmooaa    F F   ouk  ouk` ` gugumbmb    F F     ··ff
sfpmr·sfpmr·mvfb. Ur`vf qufmvfb. Ur`vf quf   D D   · ·f f sfsfpmpmr·r·mvfb.mvfb.
FxFxfro·fro·ĽoĽon ̀n ̀ ?.?.4.4.=<=<   Gmg`s uj fspm±  Gmg`s uj fspm±o` c`rjmg`o` c`rjmg`    F F   f uj vft`r  f uj vft`r    xx55   ∁∁    F F , pr`vf quf `, pr`vf quf `
o`ckuct`o`ckuct` 
ϖϖ  ∁∁ F F   ;; ϖϖ :: xx55   ff   ϖϖ((xx55) ) :: xx5599

..
··f f c˒cm˒`-m`-vmzn`vmzn`, , o`cvfo`cvfx` x` f f dfoadfoamg` mg` fjfj   F F ..
FxFxfro·fro·ĽoĽon ̀n ̀ ?.?.4.4.=1=1   _fkmj  _fkmj   D D    uj uj suifspm±suifspm±o` o` g` g` fspm±fspm±o` c`rjmg`o` c`rjmg`   F F    ff   W W   ∁ B∁ B((D,D, II((Q Q )))),,
`cgf`cgf   II((Q Q ) ) ··f f ` ` fspfspm±m±o` g` o` g` FxfFxfjpbjpb` ` =.=.=.=.9. 9. Ur`Ur`vf quf fxnstvf quf fxnstff  W W   ∁ B∁ B((F,F, II((Q Q )) fxtfcs˒)) fxtfcsm˒`m`
gfgf   W W    mm  F  F   tmb quf tmb quf  W W  :: W W ..
FxFxfro·fro·ĽoĽon ̀n ̀ ?.?.4.4.=>=>   _fkmj  _fkmj   F F  uj fspm± uj fspm±o` gf Imcmoa fo` gf Imcmoa f   D D   uj suifspm±  uj suifspm±o` o`jpbfjfctmg`o` o`jpbfjfctmg`
gfgf   F F . Ur`vf quf sf. Ur`vf quf sf   F F   :: D  D  ⊖ ⊖ J J    ff   F F   :: D  D  ⊖ ⊖ C C , fct˒, fctm˒`m`  J  J  · ·f f nsns`j`j`r`rd ̀d ̀ mm   C C ..
FxFxfro·fro·ĽoĽon ̀n ̀ ?.?.4.4.9595   @   @ WWf`rf`rfjm gf fjm gf _`_`iozyiozyh h gngnz z ` ` sflsflununctctf; f; _f_fkmjkmj    D D   uj   uj suifssuifspm±pm±o`o`
dfoamg` g` fspm±dfoamg` g` fspm±o` c`rjmg` sfpmr·o` c`rjmg` sfpmr·mvfbmvfb    F F    ff    W W   ∁ ∁ BB((D,D, oo55))..  Fct˒ Fctm˒` fxnstfm` fxnstf  W W   ∁ ∁ BB((F F 88 oo55))
quf fstfcgfquf fstfcgf   W W    ff  W W  ≧≧ 9 9 W W . Umrm ujm gfj`cstrm±. Umrm ujm gfj`cstrm±o˒om˒`, m`, vfkm vfkm X9, X9, Waf`Waf`rfj rfj 9.6.1Z9.6.1Z..
Vsmcg` ` Wf`rfjm gf _`iozyh, j`strf quf sfVsmcg` ` Wf`rfjm gf _`iozyh, j`strf quf sf o o55 · ·f f susuiifsfspmpm±±o` o`jpbfjfctmg` gf ujo` o`jpbfjfctmg` gf uj
fspm±fspm±o` c`rjmg` sfpmr·o` c`rjmg` sfpmr·mvfbmvfb   F F , fct˒, fctm˒` m` ··f f 9-o`jp9-o`jpbfjfctmbfjfctmg` g` fjfj   F F ..
FxFxfro·fro·ĽoĽon ̀n ̀ ?.?.4.4.9=9=   _fkm  _fkm   D D  uj suifspm± uj suifspm±o` dfoamg` g` fspm±o` dfoamg` g` fspm±o` gf Imcmoao` gf Imcmoa   F F  tmb quf tmb quf   F/D F/D 
tfj gnjfcs˒tfj gnjfcsm˒` ﬍cntm. Ur`vf qufm` ﬍cntm. Ur`vf quf   D D  · ·f f o`jo`jpbfpbfjfctjfctmg` mg` fjfj   F F ..
FxFxfro·fro·ĽoĽon ̀n ̀ ?.?.4.4.9999   _fkmj  _fkmj    D D   suifspm±  suifspm±o` o` g` g` fsfspmpm±±o` o` c`rjc`rjmg`mg`    F F ,,    U U  ; ;    F F    ∙∙↙↙    F F    ujmujm
pr`kf±pr`kf±o˒om˒` s`irfm` s`irf    D D    ff    ]] ∁  ∁ BB((F,F, F F )). . __ff    U U  ◤ ◤  ] ]    ::    ]]    ff    ]]  ◤◤ U  U    ::    U U , fct˒, fctm˒`m`   ]]   t  tmmjjii··ffj j ··ff
ujm pr`kf±ujm pr`kf±o˒om˒` s`irfm` s`irf   D D ..
FxFxfro·fro·ĽoĽon ̀n ̀ ?.?.4.4.9?9?  O`csngfrf ` sflunctf `pfrmg`r gf trmcsbm± O`csngfrf ` sflunctf `pfrmg`r gf trmcsbm±o˒om˒`m`
W W  ; ;   ∓∓ ∙ ∙↙↙ ∓∓    , , W W ((((mm==,, mm99,, mm??, . . ., . . .)) : ()) : (mm99,, mm??,, mm77, . . ., . . .))..
(m) Ur`vf quf(m) Ur`vf quf  W  W  ∁ ∁ BB((∓∓,, ∓∓) f omboubf m c`rjm gf) f omboubf m c`rjm gf   W W ..
(i) Ur`vf quf fxnstf uj ducon`cmb(i) Ur`vf quf fxnstf uj ducon`cmb   ϖϖ  ∁∁ ((∓∓))  tmb quf tmb quf
ϖϖ ◤◤ W W   :: ϖ  ϖ ff   bn  bnjj ncncd d 
cc
mmcc ≧ ≧ ϖϖ((((mm k k))∓∓ k k:=:=)) ≧≧ bnbnjj susupp
cc
mmcc   pmrm t`gm (  pmrm t`gm (mm k k))
∓∓
 k k:=:= ∁ ∁ ∓∓..
FxFxfro·fro·ĽoĽon ̀n ̀ ?.?.4.4.9797   _fkm  _fkm   O O  uj  uj suio`ckucsuio`ckuct` t` o`cvo`cvfx`, mifx`, mifrt` frt` f f quf quf o`cto`ct··fj fj m m `rnlfj g``rnlfj g`
fspm±fspm±o` c`rjmg` rfmbo` c`rjmg` rfmb  F  F . Ur`vf quf sf. Ur`vf quf sf  O  O  · ·f f bnbnjnjntmtmg ̀g ̀ f f snsnj·j·ftftrnrno ̀o ̀ (n(nstst ̀· ̀·f,f,  O  O   :: ∙∙O O ), fct˒), fctm˒`m`
` ducon`cmb gf Jnch`wshn` ducon`cmb gf Jnch`wshn   ppO O    gfgf   O O  · ·f f ujm ujm c`rjm c`rjm fqunvmbfctf fqunvmbfctf eem c`rjm `rnlncmb gfm c`rjm `rnlncmb gf   F F ..
4=4=
  
FxFxfro·fro·ĽoĽon ̀n ̀ ?.?.4.4.9696   _fkm (  _fkm (F,F,  ¼¼ ) uj fspm±) uj fspm±o` c`rjmgo` c`rjmg` rfmb. ` rfmb. Ur`Ur`vf quf ` vf quf ` ducoducon`cn`cmb gfmb gf
Jnch`wshn gm i`bm mifrtmJnch`wshn gm i`bm mifrtm   II(5(5,, =) =) :: {{xx ∁∁ F F    ;; xx 22 = =}}  o`ncongf o`j o`ncongf o`j   ¼¼ ..
FxFxfro·fro·ĽoĽon ̀n ̀ ?.?.4.4.9494* _fkmj* _fkmj M M  f f  I I  suio`ckuct`s o`cvf suio`ckuct`s o`cvfx`s g` x`s g` fspm±fspm±o` c`rjmg` rfmbo` c`rjmg` rfmb F  F  tmns tmns
qufquf
η η  ;: gnst( ;: gnst(M,M, II) : ncd ) : ncd {{xx ∙∙ yy ;; x x ∁∁ MM  ff   yy ∁ ∁ II}} 33 5 5..
Ur`vf quf fxnstf uj ducon`cmbUr`vf quf fxnstf uj ducon`cmb   ϖϖ  ∁∁ F F   tmb quf tmb quf  ϖϖ : = f ncd : = f ncd  ϖ ϖ((MM) : sup) : sup ϖϖ((II) +) + η η ..
FxFxfro·fro·ĽoĽon ̀n ̀ ?.?.4.4.9<9<   (Om  (Oms` s` o`jo`jpbfxpbfx` ` gms gms d`rjd`rjms ms lf`jlf`j··ftrnftrnoms oms g` g` WWf`rff`rfjm jm gf gf AmaAmac‟c‟
Imcmoa) _fkmjImcmoa) _fkmj  F F  uj fspm± uj fspm±o` c`rjmg` o`jpbfx` fo` c`rjmg` o`jpbfx` f   MM  f f   II  suio`ckuct`s gnskuct`s, c˒ suio`ckuct`s gnskuct`s, cm˒`-m`-
vmzn`s f o`cvfx`s gfvmzn`s f o`cvfx`s gf   F F ..
(m) _f(m) _f   MM · ·f f mimifrtfrt`, `, fct˒fctm˒` m` fxnfxnstfstfj uj uj j ducducon`on`cmbcmb   ϖϖ  ∁∁ F F   f uj c· f uj c·ujfr` rfmbujfr` rfmb  mm  tmns quf tmns quf
Yf(Yf(ϖϖ((xx)))) ≧≧ mm ≧≧ Yf(Yf(ϖϖ((yy)) )) pmrpmrm tm t`g``g`ss   xx ∁∁ M M f f yy ∁ ∁ I.I.
(i) _f(i) _f M M  · ·f f dfdfoaoamgmg ̀̀ ff I I  · ·f f o`jo`jpmopmot`, t`, fct˒fctm˒` fxnstfj uj ducon`cmbm` fxnstfj uj ducon`cmb ϖ ϖ ∁ ∁ F F    f f c·c·ujfr`s rfmnsujfr`s rfmns
mm  ff  i i  tmns quf tmns quf
Yf(Yf(ϖϖ((xx)))) 2 m 2 i 2 2 m 2 i 2 Yf( Yf(ϖϖ((yy)) )) pmpmrm rm t`gt`g`s`s  xx ∁∁ M M f f yy ∁ ∁ I.I.
FxFxfro·fro·ĽoĽon ̀n ̀ ?.?.4.4.9191    VjVj   sfjn-fspm±  sfjn-fspm±o` o` dfdfoamgoamg`̀   fj   fj uj uj fsfspmpm±±o` o` c`c`rjmrjmg` g` rfmrfmbb    F F   ·  ·f f uujj
o`ckuct` gm d`rjmo`ckuct`gm d`rjm
ϖϖ∙∙==(X(Xm,m, ∓∓)) )) :: { {xx ∁∁ F F   ;; ϖ ϖ((xx)) ≦≦ mm}},,
`cgf`cgf   ϖϖ ∁ ∁   F F     ff   mm ∁ ∁   YY. . Ur`Ur`vf quf t`g` vf quf t`g` suio`suio`ckucckuct` dfoamt` dfoamg` f g` f o`co`cvfvfx` gf x` gf uj fspm±uj fspm±o`o`
c`rjmc`rjmg` g` rfmb rfmb ··f f m m nctfrsfnctfrsf±±o˒om˒` gf t`g`s `s sfjn-fspm±m` gf t`g`s `s sfjn-fspm±o`s o`s dfoamgdfoamg`s `s quf quf ` ` o`ct·o`ct·fj.fj.
FxFxfro·fro·ĽoĽon ̀n ̀ ?.?.4.4.9>9>   (m) Ur`vf quf ` `pfrmg`r (m) Ur`vf quf ` `pfrmg`r
W W  ; ;   ∓∓ ∙ ∙↙↙ BB((99,, 99))   , , W W ((((mm k k))∓∓ k k:=:=)(()((ii k k))∓∓ k k:=:=)) : ()) : (mm k kii k k))∓∓ k k:=:=
··f f ujm ujm ns`jfns`jftrnm trnm bncfmbncfmr.r.
(i) O`cobum quf(i) O`cobum quf B B((99,, 99) ) c˒cm˒ ̀m ̀ ··f f sfsfppmrmr··mvmvffb.b.
FxFxfro·fro·ĽoĽon ̀n ̀ ?.?.4.4.?5?5  Ur`vf quf uj suifspm± Ur`vf quf uj suifspm±o` vft`rnmbo` vft`rnmb   J J   g` fspm±  g` fspm±o` c`rjmg`o` c`rjmg`   F F  · ·f f ggffccss`̀
fjfj F  F  sf, f s`jfctf sf, ` · sf, f s`jfctf sf, ` ·ucno` ucno` ducon`cmducon`cmbb ϖ ϖ  ∁∁ F F   quf sf mcubm fj quf sf mcubm fj J  J  · ·f ` f ` ducducon`on`cmb cmb cub`cub`..
FxFxfro·fro·ĽoĽon ̀n ̀ ?.?.4.4.?=?=   _fkmj  _fkmj    xx    f f ((xxnn))nn∁∁N N   vft`rfs gf uj fspm±  vft`rfs gf uj fspm±o` c`rjmg`o` c`rjmg`    F F . . UrUr`̀vvf qf qufuf
xx ∁∁ XXxxnn  ; ;  n n ∁∁ N N Z sf, f s`jfctf sf,Z sf, f s`jfctf sf,  ϖ ϖ((xx) : 5 pmrm t`g`) : 5 pmrm t`g`  ϖ ϖ  ∁∁ F F   tmb quf tmb quf  ϖ ϖ((xxnn) : 5 pmrm t`g`) : 5 pmrm t`g`
nn ∁∁ N N ..
FxFxfro·fro·ĽoĽon ̀n ̀ ?.?.4.4.?9?9   _fkmj  _fkmj    H H ==, . . . , H  , . . . , H  cc   suio`  suio`ckckucuct`s t`s dfodfoamamg`s g`s f f o`o`cvcvfxfx`s `s g` g` fsfspm±pm±o`o`
c`rjmg` rfmbc`rjmg` rfmb F  F , om, omgm gm uj gfuj gfbfs obfs o`ctfcg` m `ctfcg` m `rnlfj, `rnlfj, f f sfkmj sfkmj tmji·tmji·fjfj o o==, . . . , o, . . . , occ   c·c·ujfr`sujfr`s
rfmns p`sntnv`s. Ur`vf quf sfrfmns p`sntnv`s. Ur`vf quf sf  x x ∁∁ F F   ff  x x  c˒ cm˒` p`gf sfr fsornt` gm d`rjmm` p`gf sfr fsornt` gm d`rjm  o o==xx== + + ¼¼ ¼¼ ¼¼ ++ ooccxxcc
o`jo`j   xxnn ∁ ∁  H  H nn  pmrm t`g` pmrm t`g`   nn :  : ==, . . . , c, . . . , c, fct˒, fctm˒` fxnstfm` fxnstf   ϖϖ ∁ ∁ F  F    tmb quf tmb quf   ϖϖ((xx))  3 3  =  = ff   ϖϖ((yy)) ≧ ≧    ==oonn
pmrm t`g`spmrm t`g`s   nn :  : ==, . . . , c, . . . , c  ff   yy
 ∁ ∁
H H nn..
4949
  
OOmmpp··ĽĽttuubb ̀̀ 77
GumbGumbngmgngmgf f f f FspmFspm±±o`s o`s Yf﬉fYf﬉fxnv`sxnv`s
O`jf±O`jf±omrfj`s omrfj`s fstf fstf ompomp··Ľtub` Ľtub` j`strmcg` j`strmcg` quf quf ··f f p`ssp`ss··Ľvfb Ľvfb gfsorfvfr gfsorfvfr t`g`s t`g`s `s `s ducon`cmnsducon`cmns
bncfmrfs bncfmrfs o`cto`ct··Ľcu`s Ľcu`s fj fj mblucs mblucs g`s g`s fspm±fspm±o`s ob·o`s ob·mssno`s o`j `s qumns vnj`s trmimbamcg`.mssno`s o`j `s qumns vnj`s trmimbamcg`.
M M omrmoomrmotfrtfr··Ľstnom Ľstnom rfbfvmctf rfbfvmctf gfssm gfssm gfsorgfsorn±n±o˒om˒` g`s fspm±m` g`s fspm±o`s o`s gumns ·gumns ·f f quf quf `s `s ducon`cducon`cmns mns s˒sm˒`m`
gfsornt`s o`j` `ikft`s pfrtfcofctfs egfsornt`s o`j` `ikft`s pfrtfcofctfs em jfsjm obmssf g`s fbfjfct`s g` fspm±m jfsjm obmssf g`s fbfjfct`s g` fspm±o`, m smifr,o`, m smifr,
ducon`cmns ducon`cmns bncfmrfs bncfmrfs o`cto`ct··Ľcu`s Ľcu`s fj fj fspm±fspm±o`s gf duc±o`s gf duc±o˒o˒̀fs `fs sfr˒sfrm˒` m` gfsornt`s gfsornt`s o`j` o`j` duc±duc±o˒o˒̀fs f`fs f
duconducon`cmns `cmns bncfmrbncfmrfs ofs o`ct`ct··Ľcu`s fĽcu`s fj fsj fspm±pm±o`s o`s gf gf sfqsfqu˄uf˄cofconms nms sfrsfr˒˒m` m` gfsorgfsornt`s nt`s o`j` o`j` sfqu˄sfquf˄conmfconms.s.
Mncgm c` o`ctfxt` gm gumbngmgf, m` t`jmrj`s ` gumb g` gumb gf uj fspm±Mncgm c` o`ctfxt` gm gumbngmgf, m` t`jmrj`s ` gumb g` gumb gf uj fspm±o` c`rjmg`,o` c`rjmg`,
eems ms vvfzfzfs fs c`c`s s vvfjfj`s `s gf gf vv`b`btm tm m` m` fsfspmpm±±o` `o` `rnrnlnlncmcmb. b. FsFssfsfs s˒s sm˒` m` `s `s fsfspmpm±±o`s o`s rf﬉fxnvrf﬉fxnv`s,`s,
tfjm gm _f±tfjm gm _f±o˒om˒` 7.?m` 7.?. . @ fstu@ fstug` g`s fspg` g`s fspm±m±o`s rf﬉fxnv`s c`s pr`p`ron`cmr·o`s rf﬉fxnv`s c`s pr`p`ron`cmr·m ujm fxofbfctfm ujm fxofbfctf
`p`rtucngmgf pmrm nctr`guznr f mpbnomr m njp`rtmctf c`±`p`rtucngmgf pmrm nctr`guznr f mpbnomr m njp`rtmctf c`±o˒om˒` gf mgkuct` gf uj `pfrmg`rm` gf mgkuct` gf uj `pfrmg`r
bnbncfcfmr mr o`o`ct·ct·ĽcuĽcu`.`.
7.= 7.= @ gum@ gumb g`s b g`s fspmfspm±±o`so`s  BB p p
M pmrtnr gf ml`rm, gmg` =M pmrtnr gf ml`rm, gmg` =  2  2 p p 22  ∓∓, p`r, p`r  p p∜∜  gfc`tmrfj`s ` (· gfc`tmrfj`s ` (·ucno`) c·ucno`) c·ujfr` jmn`r quf =ujfr` jmn`r quf =
tmb quftmb quf    ==
 p p
 + +    ==
 p p∜∜
   : : ==.. Umrm Umrm   pp : = t`jmrfj`s : = t`jmrfj`s   pp∜∜  ::  ∓∓. Gnz-sf quf. Gnz-sf quf   pp∜∜   ··f f `̀   o`ckulmg` o`ckulmg`   gfgf  p p   ff
vnof-vfrsm.vnof-vfrsm.
TTfkmj`s fkmj`s prnjfnrprnjfnrmjfctmjfctf f o`j` o`j` fbfjfcfbfjfct`s t`s gfgf  BB p p∜∜  lfrmj ducon`cmns bncfmrfs fj lfrmj ducon`cmns bncfmrfs fj   BB p p;;
Fxfjpb` 7.=.=Fxfjpb` 7.=.=  _fkmj = _fkmj = ≧≧  p  p 22 ∓∓ f (f (Q,Q, ίί,, ¶¶) uj ) uj fspm±fspm±o` gf o` gf jfgngm. jfgngm. Gmgm ujm Gmgm ujm duc±duc±o˒om˒`m`
ll ∁ ∁ BB p p∜∜((Q,Q, ίί,, ¶¶), o`csngfrf ` ducon`cmb), o`csngfrf ` ducon`cmb
ϖϖll ;;   BB p p((Q,Q, ίί,, ¶¶)) ∙∙↙↙ HH  , , ϖϖll((d d ) ) ::
  
Q Q 
dlg¶.dlg¶.
Gm gfsnlumbgmgf gf A¨Gm gfsnlumbgmgf gf A῭bgfr =.9.= sfluf quf`bgfr =.9.= sfluf quf
||ϖϖll((d d ))|| ::

  
Q Q 
d l g¶d l g¶
 ≧≧
  
Q Q 
||dd ll|| g¶g¶ ≧≧ ll p p∜∜ ¼ ¼ d d  p p
pmrm t`gmpmrm t`gm   d d 
 ∁ ∁
BB p p((Q,Q, ίί,, ¶¶). Nss` lmrmctf quf). Nss` lmrmctf quf  ϖ ϖll   fst·  fst·m ifj gf﬍cngm f, o`j` cfssf oms` mm ifj gf﬍cngm f, o`j` cfssf oms` m
bncbncfmrfmrngmngmgf gf ··f f obmobmrm, rm, sflsfluf uf tmji·tmji·fj fj qufquf   ϖϖll ∁ ∁ BB p p((Q,Q, ίί,, ¶¶))   ff  ϖϖll ≧≧ ll p p∜∜ ..
4?4?
  
Umrm pr`vmr m nlumbgmgfUmrm pr`vmr m nlumbgmgf  ϖϖll   ::  ll p p∜∜  prfonsmj`s gnvngnr m gfj`cstrm± prfonsmj`s gnvngnr m gfj`cstrm±o˒om˒` fj g`nsm` fj g`ns
oms`s. _foms`s. _f   ll  : 5 fj : 5 fj   BB p p∜∜((Q,Q, ίί,, ¶¶), c˒), cm˒` a·m` a·m ` quf dmzfr. U`gfj`s fct˒m ` quf dmzfr. U`gfj`s fctm˒` sup`rm` sup`r  ll p p∜∜  : : 5.5.
_up`camj`s prnjfnr` quf_up`camj`s prnjfnr` quf   p p 33  =.  =. CfCfssssf oms`f oms`  pp∜∜   22 ∓ ∓  f p`rtmct` fst· f p`rtmct` fst·m ifj gf﬍cngm mm ifj gf﬍cngm m
duc±duc±o˒om˒`m`
d d ((xx) ) ::
 | |ll((xx))|| p p∜∜∙∙== ¼¼ ll((xx))
ll p p∜∜∙∙== p p∜∜    ¼¼ ||ll((xx))||
sf sf ll((xx)) : : 55    f f d d ((xx) ) : : 55   sf sf ll((xx) ) : : 55..
O`j` (O`j` ( p p∜∜ ∙∙ =)=) p p : :  p p∜∜  sfluf quf sfluf quf   d d  ∁ ∁  B B p p((Q,Q, ίί,, ¶¶) ) ff  d d  p p : =. Vjm o`ctm r· : =. Vjm o`ctm r·mpngm d`rcfofmpngm d`rcfof
||ϖϖll((d d ))|| :: ll p p∜∜, f p`rtmct`, f p`rtmct`  ϖϖll :: ll p p∜∜ ..
Umrm ` oms`Umrm ` oms` p p :  : =, sup`camj`s quf m jfgngm=, sup`camj`s quf m jfgngm ¶¶ sfkmsfkm ϏϏ-﬍cntm-﬍cntm, nst`, nst` ··f, fxnstf, fxnstfj o`ckuctfj o`ckuct`s`s
((MMcc))
∓∓
cc:=:=   fj ί, t`g`s gf jfgngm ﬍cntm, tmns quf  fj ί, t`g`s gf jfgngm ﬍cntm, tmns quf    Q Q    ::
∓∓
cc:=:=
MMcc. . __ffkkm 5m 5    2 2 o o 22  ll∓∓..
W`jmcg`W`jmcg`   MM   :: { {xx ∁ ∁   Q Q    ;; | |ll((xx))||   3 3 oo}}, gm gf﬍cn±, gm gf﬍cn±o˒om˒` gfm` gf   ¼¼ ∓∓   sfluf quf  sfluf quf   ¶¶((MM))   33   5  5. . GGff
Q Q    ::
∓∓
cc:=:=
MMcc   rfsubtm quf rfsubtm quf   ¶¶((MM ∮∮ MMcc))   33  5 pmrm mbluj 5 pmrm mbluj   cc. . OamjOamjmcg`mcg`  II   ::   MM ∮∮ MMcc   sfluf  sfluf
quf 5quf 5 2  2 ¶¶((II)) 2 2 ∓∓  ff | |ll((xx))|| 3 3 oo  pmrm t`g` pmrm t`g`   xx ∁∁ II. Gf﬍cncg`. Gf﬍cncg`
d d ((xx) ) ::    ll((xx))
¶¶((II))||ll((xx))||    sf sf xx  ∁∁ I I f f d d ((xx) ) : : 55  sf x / sf x /∁∁ I,I,
tfj`stfj`s  d d ==  : = f p`rtmct` : = f p`rtmct`  ϖϖll ≦≦ ||ϖϖll((d d )) ::
  
II
||ll||
¶¶((II))
 g¶ g¶ ≦≦ oo. O`j` nss` vmbf pmrm t`g`. O`j` nss` vmbf pmrm t`g`
55 2  2 o o 22 ll∓∓  sfluf quf sfluf quf  ϖϖll ≦≦ ll∓∓..
_mifj`s fct˒_mifj`s fctm˒` m` quf quf m m o`rro`rrfspfsp`cg˄`cgf˄conmfconm
ll ∁ ∁ BB p p∜∜((Q,Q, ίί,, ¶¶)) ↙↙ ϖϖll ∁ ∁ BB p p((Q,Q, ίί,, ¶¶))    , , ϖϖll((d d ) ) ::
  
Q Q 
dldl gg¶¶,,
··f ujm f ujm ns`jftrnm bncfmr ns`jftrnm bncfmr (b`l`nckft`(b`l` nckft`rm)8 o`j rm)8 o`j m rfssmbvm gf m rfssmbvm gf quf m quf m jfgngm gfvf sfjfgngm gfvf sfrr Ϗ Ϗ-﬍cntm-﬍cntm
c` oms`c` oms`   pp : =. Umrm quf `s fspm± : =. Umrm quf `s fspm±o`so`s   BB p p∜∜((Q,Q, ίί,, ¶¶) ) ff   BB p p((Q,Q, ίί,, ¶¶))
   sfkmj ns`jftrnomjfctf  sfkmj ns`jftrnomjfctf
ns`j`rdns`j`rd̀ s `s dmbtm dmbtm vfrnvfrn﬍omr ﬍omr quf quf fssm fssm o`rrfsp`cgo`rrfsp`cg˄˄fconm fconm ··f f s`irfkft`s`irfkft r̀m, rm, nst` nst` ··f, f, quf quf t`g`t`g`
ducon`cmb fjducon`cmb fj   BB p p((Q,Q, ίί,, ¶¶) ) pr`v·pr`v·fj fj gf gf ujm ujm duc±duc±o˒om˒` gfm` gf    BB p p∜∜((Q,Q, ίί,, ¶¶).).    ··F fxmtmjfctf nss`F fxmtmjfctf nss`
quf gnz ` pr·quf gnz ` pr·̀xnj` tf`rfjm, uj g`s v·`xnj` tf`rfjm, uj g`s v·mrn`s rfsubtmg`s o`cafong`s o`j`mrn`s rfsubtmg`s o`cafong`s o`j`   W  Wf`rf`rfjm fjm gm gm 
Yfprfsfctm±Yfprfsfctm±o˒ o˒ m` gf Ynfsz m` gf Ynfsz . . C` omsC` oms` gf ` gf fsfsombombmrfmrfs rfmns rfmns f s f ==   2 2 p p 22 ∓ ∓, ujm gfj`cstrm±, ujm gfj`cstrm±o˒om˒`m`
fssfconmbjfctf ncgfpfcgfctf gm Wf`rnm gm Jfgngm sfr·fssfconmbjfctf ncgfpfcgfctf gm Wf`rnm gm Jfgngm sfr·m mprfsfctmgm c` Wf`rfjm 4.4.=?.m mprfsfctmgm c` Wf`rfjm 4.4.=?.
Wf`rfjm 7.=.9Wf`rfjm 7.=.9   _fkmj   _fkmj   == ≧ ≧   p p 22 ∓ ∓    f f    ((Q,Q, ίί,, ¶¶))  uj fspm± uj fspm±o` gf jfgo` gf jfgngm. ngm. C` omC` oms`s`   pp   : : ==
sup`j`s quf sup`j`s quf  ¶ ¶ sfkm ujm jfgngm  sfkm ujm jfgngm  Ϗ Ϗ-﬍cntm. Fct˒ -﬍cntm. Fct˒ m ̀m ̀ m m o`rrfso`rrfsp`cgp`cg˄˄fcfcononm m  l l  ↙↙ ϖ ϖll fstmifbfof uj  fstmifbfof uj 
ns`ns`j`r﬍j`r﬍sj` sj` ns`ns`j·j·ftrnftrno` fco` fctrf trf  B B p p∜∜((Q,Q, ίί,, ¶¶))  f  f  B B p p((Q,Q, ίί,, ¶¶))
  fj quf m rfbm± fj quf m rfbm±o˒ o˒ m` gf gumbngmgf m` gf gumbngmgf 
··f f gmgmgm gm p`r p`r 
ϖϖll((d d ) ) ::
  
Q Q 
d l g¶d l g¶  pmrm t`gm pmrm t`gm   d d  ∁ ∁ BB p p((Q,Q, ίί,, ¶¶))..
Gfj`cstrm±Gfj`cstrm±oo˒˒mm`̀..  Gfj`cstrmrfj`s cfstf j`jfct` mpfcms ` oms` fj quf m jfgngm Gfj`cstrmrfj`s cfstf j`jfct` mpfcms ` oms` fj quf m jfgngm   ¶¶
··ff   ϏϏ-﬍cntm f =-﬍cntm f = ≧ ≧ p  p 22 ∓ ∓. Nss` fclb`im `s fspm±. Nss` fclb`im `s fspm±o`s gf jfgngm jmns usmg`s, ncobusnvf cmso`s gf jfgngm jmns usmg`s, ncobusnvf cms
4747
  
mpbnom±mpbnom±o˒o˒̀f`fs. s. Gf t`gGf t`gm m d`d`rjrjm, ` m, ` omoms` fj qufs` fj quf    p p 33    = = ff    ¶¶  · ·f f ujm ujm jfgngm jfgngm mrintr·mrintr·mrnm sfr·mrnm sfr·mm
gfj`cgfj`cstrmgstrmg` ` c` c` Mp˄Mpf˄cgnofcgnof f O.O.
Gf c`ssm gnsouss˒Gf c`ssm gnsoussm˒` m` pr·pr·fvnm fvnm imstm imstm pr`vpr`vmr mr quf quf gmg` gmg` uj uj ducon`cducon`cmbmb    ϖϖ ∁ ∁    BB p p((Q,Q, ίί,, ¶¶)),,
fxnstffxnstf l l
 ∁ ∁
BB p p∜∜((Q,Q, ίί,, ¶¶) tmb quf) tmb quf ϖ ϖll   :: ϖ ϖ. . _up`camj`s prnj_up`camj`s prnjfnrmjfcfnrmjfctf quf m jfgngmtf quf m jfgngm ¶ ¶ sfkm sfkm
﬍cntm. C˒﬍cntm. Cm˒` m` ··f f gnd·gnd·Ľonb Ľonb oafooafomr mr quf quf m m o`ro`rrfsrfsp`p`cg˄cgf˄cofconmnm  M M ∁∁ ίί ↙↙ ϖϖ((Q Q MM), `cgf), `cgf Q  Q MM  · ·f f m m duduc±c±o˒om˒`m`
omromrmotmotfr·fr·ĽstĽstnom nom gfgf   MM, , ··f f ujm ujm jfgngm jfgngm (c`tf (c`tf qufquf Q  Q MM ∁ ∁   BB p p((Q,Q, ίί,, ¶¶) p`ns fstmj`s sup`cg`) p`ns fstmj`s sup`cg`
¶¶   ﬍c  ﬍cntmntm). ). O`jO`j`̀   ϖϖ(5) : 5 tfj`s(5) : 5 tfj`s   ϖϖ((Q Q MM) : 5 pmrm t`g`) : 5 pmrm t`g`   MM ∁ ∁  ί tmb quf ί tmb quf   ¶¶((MM) : 5. ) : 5. UUfbfb`̀
Wf`rfjm gf Ymg`c‟Cnh`g·Wf`rfjm gf Ymg`c‟Cnh`g·yj (vfkm Wf`rfjm O.5.?=) fxnstf ujm duc±yj (vfkm Wf`rfjm O.5.?=) fxnstf ujm duc±o˒om˒`m` l l ∁ ∁ BB==((Q,Q, ίί,, ¶¶))
tmb quftmb quf ϖ ϖ((Q Q MM) ) ::
  
MM
ll g¶g¶ pmrm t`g` pmrm t`g` M M  ∁∁ ί. U`rtmct`,ί. U`rtmct`, ϖ ϖ((d d ) ) ::
  
Q Q 
 d l d l gg¶¶ pmrm t`gm duc± pmrm t`gm duc±o˒om˒`m`
snjpbfssnjpbfs   d d . Mssnj, pmrm t`gm duc±. Mssnj, pmrm t`gm duc±o˒om˒` snjpbfsm` snjpbfs   d d   tfj`s  tfj`s   dd ll ∁ ∁ BB==((Q,Q, ίί,, ¶¶) ) ff
  
Q Q 
d l g¶d l g¶
 :: ||ϖϖ((d d ))|| ≧≧ d d  p p ¼¼ ϖϖ..
_fluf _fluf gm gm rforfo··Ľpr`om Ľpr`om gm gm gfsnlumbgmgf gfsnlumbgmgf gf gf A¨A῭bgfr `bgfr (Fxfr(Fxfroo··Ľon` Ľon` =.1.?>=.1.?>) ) qufquf   ll ∁ ∁ B B p p∜∜((Q,Q, ίί,, ¶¶).).
O`j` O`j` ` ` o`ckuct` o`ckuct` gms gms duc±duc±o˒o˒̀fs `fs snjpbfs snjpbfs ··f f gfcs` gfcs` fjfj    BB p p((Q,Q, ίί,, ¶¶) X?=, Ur`p`sntn`c 4.<Z,) X?=, Ur`p`sntn`c 4.<Z,
gm bncfmrngmgf f gm o`ctncungmgf gfgm bncfmrngmgf f gm o`ctncungmgf gf    ϖϖ   sfluf quf  sfluf quf    ϖϖ((d d ) ) ::
  
Q Q 
 d l d l gg¶¶   pmrm t`gm  pmrm t`gm    d d    ∁∁
BB p p((Q,Q, ίί,, ¶¶), ), nsnst ̀t ̀ ··f,f,  ϖ ϖll   :: ϖ ϖ..
DDm±m±omj`s ml`rm ` oms` fj qufomj`s ml`rm ` oms` fj quf    ¶¶   ··ff    ϏϏ-﬍c-﬍cntm. ntm. CfsCfssf oms` p`gfj`s o`csnsf oms` p`gfj`s o`csngfrmgfrmr ujmr ujm
sfqu˄sfquf˄conm fconm orfsoorfsofctf fctf ((MMcc))
∓∓
cc:=:=   gf o`ckuct`s jfcsur·  gf o`ckuct`s jfcsur·mvfns tmb quf 5mvfns tmb quf 5    2 2 ¶¶((MMcc))    22 ∓ ∓   pmrm  pmrm
t`g`t`g` c c  f f  Q  Q   ::
∓∓
cc:=:=
MMcc. Umrm omgm. Umrm omgm c c ∁∁ CC, oamjmcg` ί, oamjmcg` ίcc  : : {{MM∮∮ MMcc ; ;  M M ∁∁ ίί}} f f  ¶ ¶cc  :: ¶ ¶||ίίcc,,
rfsubtm quf (rfsubtm quf (MMcc,, ίίcc,, ¶¶cc) ) ··f f uj uj fsfspmpm±±o` gf jfgngm ﬍cntm. Gf﬍cncg`o` gf jfgngm ﬍cntm. Gf﬍cncg`   ϖϖcc   p`rp`r
d d cc ∁ ∁ BB p p((MMcc,, ίίcc,, ¶¶cc)) ∙∙↙↙ ϖϖcc((d d cc) ) :: ϖ ϖ((d d ccQ Q MMcc)),,
tfj`stfj`s    ϖϖcc  ∁∁    BB p p((MMcc,, ίίcc,, ¶¶cc)). . UfUfb` oms` mcb` oms` mctfrtfrn`n`r r fxfxnsnstftf    llcc  ∁∁    BB p p∜∜((MMcc,, ίίcc,, ¶¶cc) tmb quf) tmb quf
ϖϖcc((d d cc) ) ::
  
MMcc
d d ccllcc g¶g¶cc  pmrm t`gm pmrm t`gm   d d cc ∁ ∁ BB p p((MMcc,, ίίcc,, ¶¶cc). Jmns mncgm,). Jmns mncgm,
llccBBpp∜∜ ((MMcc,,ίίcc,¶,¶cc))  : : ϖϖcc((BBpp((MMcc,,ίίcc,¶,¶cc)))) ≧ ≧ ϖϖBBpp((Q,Q,ίί,¶,¶))..
WWmbmb l lcc  ··f f ··ucnom (sulfrnj`s quf ` bfnt`r vfrn﬍quf nss`. _f tnvfr fj jfctfucnom (sulfrnj`s quf ` bfnt`r vfrn﬍quf nss`. _f tnvfr fj jfctf B B p p∜∜  m m ̀̀ ncncv·v·fs fs gfgf
BB p p∜∜, ` bfnt`r p`gfr·, ` bfnt`r p`gfr·m nctfrprftmr qufm nctfrprftmr quf   llcc   ··f f ··ucnom m jfc`s gf o`ckuct`s gf jfgngm cubm,ucnom m jfc`s gf o`ckuct`s gf jfgngm cubm,
nst` nst` ··f, f, `utr`utrm m duc±duc±o˒om˒` o`j ms jfsjms pr`prnfgmgfs gfm` o`j ms jfsjms pr`prnfgmgfs gf   llcc   o`ncongf o`j  o`ncongf o`j   llcc   m jfc`s gf  m jfc`s gfuj o`ckuct` gf jfgngm cubm). U`gfj`s fstfcgfruj o`ckuct` gf jfgngm cubm). U`gfj`s fstfcgfr   llcc   mm   Q Q   dmzfcg`  dmzfcg`   llcc((xx) : 5 pmrm t`g`) : 5 pmrm t`g`
x x //∁∁   MMcc. . O`j` m O`j` m sfqu˄sfquf˄conm fconm ((MMcc))∓∓cc:=:=  · ·f f orfsofctf, orfsofctf, sfluf sfluf quf quf sfsf    c c 2 2 jj  fct˒ fctm˒`m`   llcc    ::   lljj
¶¶-qumsf sfjprf fj-qumsf sfjprf fj   MMcc. Fst·. Fst·m fct˒m fctm˒` m` ifj ifj gf﬍cngm gf﬍cngm f f ··f f jfcsur·jfcsur·mvfb m duc±mvfb m duc±o˒om˒`m`   ll  gmgm p`r gmgm p`r
ll((xx) ) :: l lcc((xx) ) sfsf   xx ∁∁ MMcc. . Ufb` Ufb` Wf`rfjWf`rfjm m gm gm O`cvfO`cvfrl˄rlf˄cofconm nm J`cJ`c··̀t``t`cm cm o`co`cobu·obu·Ľj`Ľj`s s qufquf
llBBpp∜∜((Q,Q,ίί,¶,¶))  : bnj  : bnjcc↙∓↙∓ llccBBpp∜∜ ((MMcc,,ίίcc,¶,¶cc)) ≧ ≧ ϖϖBBpp((Q,Q,ίί,¶,¶)),,
` quf c`s lmrmctf quf` quf c`s lmrmctf quf    ll  ∁∁    BB p p∜∜((Q,Q, ίί,, ¶¶). ). UUmrmrm omm omgmgm    d d   ∁∁    BB p p((Q,Q, ίί,, ¶¶), pfb` Wf`rfjm), pfb` Wf`rfjm
gm gm O`cvfrlO`cvfrl˄˄fconm fconm G`jncmgm G`jncmgm smifj`s smifj`s qufquf  d d Q Q MMcc ∙ ∙ d d  p p ∙ ∙↙↙  5,  5, nstnst ̀̀ ··f,f,   d d Q Q MMcc ∙ ∙↙↙   d d    fjfj
BB p p((Q,Q, ίί,, ¶¶). M o`ctncungmgf gf). M o`ctncungmgf gf   ϖϖ njpbnom quf njpbnom quf
ϖϖ((d d ) : bnj) : bnj
cc↙∓↙∓
ϖϖ((d d Q Q MMcc) : bnj) : bnj
cc↙∓↙∓
ϖϖcc((d d ||MMcc))
: bnj: bnj
cc↙∓↙∓   MMcc
dd llccg¶g¶cc : bnj : bnj
cc↙∓↙∓  MMcc
dlg¶dlg¶ : :
  Q Q 
dldl gg¶¶,,
4646
  
`cgf m pfc·`cgf m pfc·ubtnjm ubtnjm gfsnlumbgmgf gfsnlumbgmgf ··f f o`csfqu˄o`csfquf˄conm fconm gm gm gf﬍cn±gf﬍cn±o˒om˒` gfm` gf    ¶¶cc    ff    ll    f f m m ··ubtnjmubtnjm
nlumbgmgf nlumbgmgf ··f f jmns jmns ujm ujm mpbnom±mpbnom±o˒om˒` gm` g` Wf`rfjm gm ` Wf`rfjm gm O`cvfrl˄O`cvfrlf˄conm fconm G`jncmgm. G`jncmgm. Nss` pr`vmNss` pr`vm
qufquf  ϖ ϖll   :: ϖ ϖ..
7.7.9 9 @s @s ggumumns ns gg`s `s fsfspmpm±±o`so`s    p p    ff   oo55
_fkm =_fkm = ≧≧  p  p 22 ∓∓.Yfbfjirf quf. Yfbfjirf quf   p p  :: B B p p((Q,Q, ίί,, ¶¶) pmrm) pmrm Q  Q   :: CC, ί : pmrtfs g`s cmturmns,, ί : pmrtfs g`s cmturmns,
¶¶  :  : jfgjfgngm ngm gf gf o`co`ctmlftmlfj, j, f f quf quf m m jfgnjfgngm gm gf gf o`co`ctmlftmlfj j cms cms pmrtpmrtfs fs g`s g`s cmtucmturmns ·rmns ·ff   ϏϏ--
﬍cn﬍cntm. tm. _fcg_fcg` ` mssnmssnj, j, p`gfj`p`gfj`s s mpbmpbnomnomr r ` ` WWf`rff`rfjm 7.=.9 m jm 7.=.9 m fssfssf f oms`oms .̀.    ··F njfgnmt` quf mF njfgnmt` quf m
rfbm±rfbm±o˒om˒` gf gumbngmgf, mpbnomgm m fssf oms`, sf trmcsd`rjm fjm` gf gumbngmgf, mpbnomgm m fssf oms`, sf trmcsd`rjm fj
ii :  : ((ii k k))
∓∓
 k k:=:= ∁ ∁  p p∜∜  ↙↙ ϖϖii ∁ ∁ (( p p))    , , ϖϖii((((mm k k))∓∓ k k:=:=) ) ::
∓∓
 k k:=:=
mm k kii k k  pmrm t`gm ( pmrm t`gm (mm k k))
∓∓
 k k:=:= ∁ ∁  p p..   (7.=)  (7.=)
@ Wf`rfjm 7.=.9 sf trmcsd`rjm fct˒@ Wf`rfjm 7.=.9 sf trmcsd`rjm fctm˒` fjm` fj
Ur`p`sn±Ur`p`sn±o˒om˒ ̀m ̀ 7.7.9.9.==   Umrm   Umrm   == ≧ ≧ p  p 22 ∓ ∓ m  m o`rrfso`rrfsp`cgp`cg˄˄fcfcononm m   ii ∁ ∁   p p
∜∜
  ↙↙ ϖ ϖii ∁ ∁  ( ( p p))   fstmifbfof   fstmifbfof uj uj ns`j`rns`j`r﬍sj` ﬍sj` ns`j·ns`j·ftrno` ftrno` fctrf fctrf    p p∜∜    f f    (( p p))  fj quf m rfbm± fj quf m rfbm±o˒ o˒ m` m` gf gf gumbngmgf gumbngmgf ··f f gmgm gmgm p`r p`r 
(7.=).(7.=).
@isfrvm±@isfrvm±o˒om˒ ̀m ̀ 7.7.9.9.99    eeM buz gm M buz gm ncd`rjmncd`rjm±±o˒om˒` gf quf ` gumb gfm` gf quf ` gumb gf  == · ·f f ns`j`rns`j`rd` d` ns`jftns`jftrnomjrnomjfctffctf
mm  ∓∓, , o`cobo`cobuu··Ľj`s Ľj`s quf quf m m rfo·rfo·Ľpr`Ľpr`om om g` g` WWf`rff`rfjm jm ?.?.9 ?.?.9 c˒cm˒` ·m` ·f vff vfrgmrgmgfngfnrm, rm, p`p`nsns  == · ·f f sfsfpmpmr·r·mvfbmvfb
fcqumct` quffcqumct` quf   ∓∓  · ·f f cc˒˒m`-sfpmr·m`-sfpmr·mvfb.mvfb.
M M jfsjfsjm jm rfbrfbm±m±o˒om˒` m` gf gf gumbngmgf gumbngmgf monjm monjm gfsorfvf tgfsorfvf tmji·mji·fj fj ` ` gumb gumb gfgf   oo55;;
Ur`p`sn±Ur`p`sn±o˒om˒ ̀m ̀ 7.7.9.9.??   @s fspm±  @s fspm±o`s o`s    ==    f f    ((oo55))
    s˒ s˒ m` ns`j`rd`s ns`jftrnomjfctf p`r jfn` gm m` ns`j`rd`s ns`jftrnomjfctf p`r jfn` gm 
rfbm±rfbm±o˒ o˒ m` gf gumbngmgf m` gf gumbngmgf 
ii :  : ((ii k k))
∓∓
 k k:=:=
 ∁ ∁
==
  ↙↙
ϖϖii
 ∁ ∁
((oo55))
    , , ϖϖii((((mm k k))
∓∓
 k k:=:=) ) ::
∓∓
 k k:=:=
mm k kii k k  pmrm t`gm ( pmrm t`gm (mm k k))
∓∓
 k k:=:=
 ∁ ∁
oo55..   (7.9)  (7.9)
Gfj`cstrm±Gfj`cstrm±oo˒˒mm`̀..  Gfnxmj`s m` bfnt`r m vfrn﬍om± Gfnxmj`s m` bfnt`r m vfrn﬍om±o˒om˒` m` gf gf quf quf m m o`rrfsp`o`rrfsp`cg˄cgf˄conm fconm fst·fst·m ifjm ifj
gf﬍cgf﬍cngm, ngm, ··f f bncfmbncfmr r ff  ϖϖii∓∓ ≧  ≧ ii==. . TTfkmj`s fkmj`s quf ·quf ·f f s`irfkft`rm s`irfkft`rm ff  ϖϖii∓∓ ≦  ≦ ii==. . _f_fkmkm
ϖϖ ∁ ∁   ((oo55)). . O`csnO`csngfrf gfrf m sfm sfqu˄quf˄conmfconm   ii  : : ((ϖϖ((ff k k))))∓∓ k k:=:=, `cgf (, `cgf (ffcc))∓∓cc:=:=    s˒sm˒` `s vft`rfs ucnt·m` `s vft`rfs ucnt·mrn`smrn`s
omc˄omc˄̀cno`cno`s `s (v(vfkm Fxfjpbfkm Fxfjpb` ` =.4=.4.7). .7). GfvGfvfj`fj`s s j`sj`strmr quftrmr quf   ii ∁ ∁   ==,,   ϖϖii    ::   ϖϖ   ff  ii== ≧  ≧ ϖϖ..
Umrm nss`, pmrm omgmUmrm nss`, pmrm omgm   cc ∁∁ CC o`cs o`csngfrf ngfrf m m sfqu˄sfquf˄conm fconm ((νν k k))∓∓ k k:=:=  `cgf `cgf
νν k k   ::
   ϖϖ((ff k k))
||ϖϖ((ff k k))||
   sfsf   ϖϖ((ff k k)) : : 5 5 ff   kk ≧ ≧ c c , , νν k k  : 5 oms` o`ctr· : 5 oms` o`ctr·mrn`mrn`..
··F obmr` qufF obmr` quf | |νν k k|| ≧≧ = pmrm t`g` = pmrm t`g`   kk. _fkm. _fkm   kk ≧ ≧ cc  tmb quf tmb quf   ϖϖ((ff k k)) : 5. Fct˒: 5. Fctm˒`m`
νν k kϖϖ((ff k k) ) ::
   ϖϖ((ff k k))
||ϖϖ((ff k k))|| ¼ ¼ ϖϖ((ff k k) ) :: ||ϖϖ((ff k k))||..
4444
  
··F F obmobmr` r` tmji·tmji·fj fj qufquf  ν ν k kϖϖ((ff k k) ) :: ||ϖϖ((ff k k))|| sf sf  ϖ ϖ((ff k k) : 5. Fct˒) : 5. Fctm˒`m`  ν ν k kϖϖ((ff k k) ) :: ||ϖϖ((ff k k))|| pmrm t`g` pmrm t`g`
 k k ≧ ≧ cc. U`rtmct`. U`rtmct`
cc

 k k:=:=
||ϖϖ((ff k k))|| ::
cc

 k k:=:=
νν k kϖϖ((ff k k) ) :: ϖ ϖ
   cc

 k k:=:=
νν k kff k k ≧≧ ϖϖ ¼¼ 
cc

 k k:=:=
νν k kff k k
:: ϖϖ ¼¼ jmxjmx{|{|νν==||, . . . ,, . . . , ||ννcc| } ≧ | } ≧ ϖϖ..
O`j` nss` vmbf pmrm t`g`O`j` nss` vmbf pmrm t`g`   cc, dmzfcg`, dmzfcg`   cc  ∙∙↙↙ ∓∓ `itfj`s `itfj`s
ii==  ::
∓∓
 k k:=:=
||ϖϖ((ff k k))|| ≧≧ ϖϖ..
` quf j`strm quf` quf j`strm quf  i i ∁∁ ==   ff  ii== ≧ ≧ ϖϖ. _fkm. _fkm  x x  :  : ((mm k k))∓∓ k k:=:= ∁ ∁ oo55. C` Fxfjpb` =.4.7 vnj`s,. C` Fxfjpb` =.4.7 vnj`s,
fctrf `utrms o`nsms, quffctrf `utrms o`nsms, quf   xx  : bnj  : bnj
cc↙∓↙∓
cc
 k k:=:=
mm k kff k k. _fluf fct˒. _fluf fctm˒` qufm` quf
ϖϖ((xx) ) :: ϖ ϖ

bnjbnj
cc↙∓↙∓
cc
 k k:=:=
mm k kff k k

: bnj: bnj
cc↙∓↙∓
ϖϖ
    cc
 k k:=:=
mm k kff k k

: bnj: bnj
cc↙∓↙∓
cc
 k k:=:=
mm k kϖϖ((ff k k) ) ::
∓∓
 k k:=:=
mm k kϖϖ((ff k k) ) :: ϖ ϖii((((mm k k))
∓∓
 k k:=:=) ) :: ϖ ϖii((xx)),,
pr`vmcg` qufpr`vmcg` quf   ϖϖii : :  ϖ ϖ..
Fj vnstm g` quf pr`vmj`s cfstm sf±Fj vnstm g` quf pr`vmj`s cfstm sf±o˒om˒` f m` f cm mcm mctfrn`rctfrn`r, ·, ·f usuf usumb fsmb fsorfvfr orfvfr ((BB p p((Q,Q, ίί,, ¶¶))))
  ::
BB p p∜∜((Q,Q, ίί,, ¶¶), ), (( p p))
   ::   p p∜∜  pmrm = pmrm = ≧ ≧ p  p 22 ∓ ∓, , f f ((oo55))   ::   ==, sfcg` quf omgm nlumbgmgf gfvf, sfcg` quf omgm nlumbgmgf gfvf
sfr sfr fctfcgngm m fctfcgngm m jfc`s jfc`s g` ns`j`rg` ns`j`r﬍sj` ns`j·﬍sj` ns`j·ftrno` ftrno` o`rrfsp`o`rrfsp`cgfctf.cgfctf.
77..? ? IInngguummb b f f ffssppmm±±o`s rf﬉fxnv`so`s rf﬉fxnv`s
Umrm t`g` fspm±Umrm t`g` fspm±o` c`rjmg`o` c`rjmg`  F F , p`gfj`s o`csngfrmr sfu gumb, p`gfj`s o`csngfrmr sfu gumb   F F , quf k·, quf k·m smifj`s sfr ujm smifj`s sfr uj
fspm±fspm±o` gf o` gf ImcImcmomoa. a. U`gfU`gfj`s fctj`s fct˒˒m` o`csngfrmr ` gumb gfm` o`csngfrmr ` gumb gf   F F , oamjmg` gf, oamjmg` gf   ingumb gf   ingumb gf   F F 
f gfc`tmg` p`rf gfc`tmg` p`r   F F . . @u s@u sfkmfkm,,   F F    : : ((F F )). . O`jO`jf±f±omj`s fstm sf±omj`s fstm sf±o˒om˒` j`strmcg` quf t`g`m` j`strmcg` quf t`g`
fspm±fspm±o` c`rjmg`o` c`rjmg` F  F  p`gf sfr fco`ctrmg` gfctr` g` sfu ingumb p`gf sfr fco`ctrmg` gfctr` g` sfu ingumb F  F   gm d`rjm jmns cmturmb gm d`rjm jmns cmturmb
pp s̀̀ss·s·ĽvĽvffb;b;
Ur`p`sn±Ur`p`sn±o˒om˒ ̀m ̀ 7.7.?.?.==  Umrm t`g` fspm± Umrm t`g` fspm±o` c`rjmg`o` c`rjmg`   F F , ` `pfrmg`r bncfmr , ` `pfrmg`r bncfmr 
K K F F ;;  F F  ∙ ∙↙↙ F F     , , K K F F ((xx)()(ϖϖ) ) :: ϖ ϖ((xx) pmrm t`g`s) pmrm t`g`s   xx ∁∁ F F    ff   ϖϖ  ∁∁ F F ,,
··f f ujm ujm ns`jftrnm ns`jftrnm bncfmrbncfmr, , oamjmg` oamjmg` gf gf jfrljfrluba` uba` omc˄ omc˄ `cno` gf `cno` gf   F F    fj fj   F F ..
4<4<
  
Gfj`cstrm±Gfj`cstrm±oo˒˒mm`̀..  Gfnxmj`s m omrl` g` bfnt`r m vfrn﬍om± Gfnxmj`s m omrl` g` bfnt`r m vfrn﬍om±o˒om˒` gf qufm` gf quf K  K F F   fst· fst·m m ifj gf﬍cngifj gf﬍cng`̀
((nsnstt ̀̀ ··f,f,  K  K F F ((xx)) ∁∁ F F   pmrm t`g` pmrm t`g`   xx ∁∁ F F ) ) f ·f ·f f bncfmr. bncfmr. Umrm Umrm omgmomgm   xx ∁∁ F F , , gfgf

K K F F ((xx))

: : ssuupp
ϖϖ∁∁IIFF ||
K K ((xx)()(ϖϖ))
||
: : ssuupp
ϖϖ∁∁IIFF ||
ϖϖ((xx))
||
::

xx

,,
`cgf m ·`cgf m ·ubtnjm nlumbgmgf sfluf g` Wf`rfjm gf Amac‟Imcmoa cm d`rjm g` O`r`b·ubtnjm nlumbgmgf sfluf g` Wf`rfjm gf Amac‟Imcmoa cm d`rjm g` O`r`b·mrn`mrn`
?.=?.=.6, .6, o`co`cobu·obu·Ľj`Ľj`s qs qufuf   K K F F  · ·f f ujm ujm ns`jftns`jftrnm rnm bncfmrbncfmr..
@isfrvm±@isfrvm±o˒om˒ ̀m ̀ 7.7.?.?.99   (m)   (m) _f _f ` ` fspm±fspm±o` o` c`rjmg`c`rjmg`    F F   o`ct·  o`ct·fj fj uj uj suisuifspm±fspm±o` ns`j`rd` m`o` ns`j`rd` m`
fspm±fspm±o` c`rjmg`o` c`rjmg` D  D , gnzfj`s quf, gnzfj`s quf F  F  o o`c`ct·t·fj fj ujujmm o·  o· `pnm ns`j`rdm gf `pnm ns`j`rdm gf  D  D . . F sF sff F  F  o o`c`ct·t·ffj j ujuj
suifspm±suifspm±o` ns`j`rd` ns`jftrnomjfctf mo` ns`j`rd` ns`jftrnomjfctf m  D  D , gnzfj`s quf, gnzfj`s quf  F  F  o o`c`ct·t·fj fj ujujmm  o·  o· ̀p`pnm nm nsns`j·`j·ftrftrnom nom 
gf gf  D  D . Gm Ur`p`sn±. Gm Ur`p`sn±o˒om˒` 7.?.= tfj`s, fj pmrtnoubmr, quf pmrm t`g` fspm±m` 7.?.= tfj`s, fj pmrtnoubmr, quf pmrm t`g` fspm±o` c`rjmg`o` c`rjmg` F  F , sfu, sfu
ingumbingumb  F  F   o o`c`ct·t·fj fj ujujm m o·o·̀p`pnmnm nsns`j`j··ftftrnrnom om gfgf  F  F . Cfstm bnclumlfj, m Ur`p`sn±. Cfstm bnclumlfj, m Ur`p`sn±o˒om˒` ?.?.? gnzm` ?.?.? gnz
qufquf   ∓∓  o` o`ct·ct·fj fj ujujm m o·o·̀pnm `pnm ns`j·ns`j·ftrnoftrnom gm gf tf t`g` `g` fspm±fspm±o` sfpmr·o` sfpmr·mvfb.mvfb.
(i) O`j`(i) O`j` F  F  f f  K  K F F ((F F ) s˒) sm˒` m` ns`jftrnomjfctf ns`jftrnomjfctf ns`j`rd`s, ·ns`j`rd`s, ·f f o`juj ngfctn﬍omr o`juj ngfctn﬍omr `s `s g`ns fspg`ns fspm±m±o`so`s
f m mji`s sf rfdfrnr snjpbfsjfctf o`j`f m mji`s sf rfdfrnr snjpbfsjfctf o`j`   F F . . @u sfk@u sfkm, fj uj m, fj uj o`co`ctfxtfxt` fj quft` fj quf   F F   sfkm  sfkm
vnst` o`j` suifspm±vnst` o`j` suifspm±o` gfo` gf   F F , fstmrfj`s o`jftfcg` ` mius` gf c`tm±, fstmrfj`s o`jftfcg` ` mius` gf c`tm±o˒om˒` gf fsorfvfrm` gf fsorfvfr   F F 
`cgf `cgf gfvfrgfvfr··Ľmj`s Ľmj`s tfr tfr fsornfsornt`t`   K K F F ((F F ).).
O`j` ujm prnjfnrm mpbnom±O`j` ujm prnjfnrm mpbnom±o˒om˒` gm Ur`p`sn±m` gm Ur`p`sn±o˒om˒` 7.?.=, gmj`s mimnx` ujm m` 7.?.=, gmj`s mimnx` ujm gfj`cstrgfj`cstrm±m±o˒om˒`m`
ourtourt··ĽssnjĽssnjm m gf gf quf quf t`g` t`g` fspm±fspm±o` o` c`rc`rjmjmg` g` mgmgjnjntf tf uj uj o`jo`jpbpbftmftmjfcjfct`t`. . @ @ bfbfnt`nt r̀ r gfvgfvff
bfjirmr-sf quf m gfj`cstrm±bfjirmr-sf quf m gfj`cstrm±o˒om˒` g` tf`rfjm g` o`jpbftmjfct` fj fspm±m` g` tf`rfjm g` o`jpbftmjfct` fj fspm±o`o`s s j·j·fftrtrnono s̀ s̀ ··ff
b`clm f tb`clm f trmimrmimba`sm. ba`sm. Mb·Mb·fj gnss`fj gnss`, mgmp, mgmpt·t·m-bm pmrm fspm±m-bm pmrm fspm±o`s c`rjmg`s, c` sfctng` gf j`strmro`s c`rjmg`s, c` sfctng` gf j`strmr
quf ` o`jpbftmjfct` gf uj fspm±quf ` o`jpbftmjfct` gf uj fspm±o` o` c`rc`rjmgjmg` ` ··f f tmji·tmji·fj fj uj uj fspfspm±m±o` c`rjmg` (b`l` ujo` c`rjmg` (b`l` uj
fspm±fspm±o` gf Imcmoa), mlrflm jmns trmimba` mncgm eo` gf Imcmoa), mlrflm jmns trmimba` mncgm em tmrfdm.m tmrfdm.
Ur`p`sn±Ur`p`sn±o˒om˒ ̀m ̀ 7.7.?.?.??  Umrm t`g` fspm± Umrm t`g` fspm±o` c`rjmg`o` c`rjmg`    F F   fxnstf uj uj fspm± fxnstf uj uj fspm±o` gf Imcmoa o` gf Imcmoa   F F 
quf quf o`ct·o`ct·fj fj ujm ujm o· o· ̀pn`pnm m ns`ns`j·j·ftrnftrnom om gf gf   F F  qu quf f ··f f gfcgfcsm sm fj fj   F F ..
Gfj`cstrm±Gfj`cstrm±oo˒˒mm`̀..   Gmg`  Gmg`  F  F , t`jf, t`jf
  
F F   :: K  K F F ((F F )) ⊄⊄ F F ..
@ @ fsfspmpm±±o`o`  F F  · ·f f oamoamjmjmg ̀g ̀ gfgf   o`jpbftmjfct`  o`jpbftmjfct`  g` fspm± g` fspm±o` c`rjmg`o` c`rjmg`  F  F . @ bfnt`r c˒. @ bfnt`r cm˒` gfvfm` gfvf
sf nbugnr o`j m snjpbnongmgf gm gfj`cstrm±sf nbugnr o`j m snjpbnongmgf gm gfj`cstrm±o˒om˒` monjm, p`ns cfbm fjiutngm fco`ctrm-sf `m` monjm, p`ns cfbm fjiutngm fco`ctrm-sf `
Wf`rfjm gf Amac‟Imcmoa, usmg` cm gfj`cstrm±Wf`rfjm gf Amac‟Imcmoa, usmg` cm gfj`cstrm±o˒om˒` gm Ur`p`sn±m` gm Ur`p`sn±o˒om˒` 7.?.=.m` 7.?.=.
Umrm quf ` jfrluba` omc˄Umrm quf ` jfrluba` omc˄̀cno``cno`    K K F F    gfgf    F F   c` sfu ingumb  c` sfu ingumb    F F 
   sfkm uj ns`j`r﬍sj`  sfkm uj ns`j`r﬍sj`
ns`jns`j··ftrno` dmbtm ftrno` dmbtm mpfcms mpfcms quf quf sfkm s`irfkft`r. sfkm s`irfkft`r. Fssm pr`prnfgmgf omrmotfrnzm ujm Fssm pr`prnfgmgf omrmotfrnzm ujm obmssfobmssf
njp`rtmctf gf fspm±njp`rtmctf gf fspm±o`s;o`s;
Gf﬍cn±Gf﬍cn±o˒om˒ ̀m ̀ 7.7.?.?.77   Vj   Vj fspfspm±m±o` o` c`rjc`rjmg`mg`    F F   ·  ·f f gngntt`̀   rf﬉fxnv`  rf﬉fxnv`   sf   sf ` ` jfjfrlrlububa` a` omomc˄c˄̀cno``cno`
K K F F ;;   F F  ∙ ∙↙↙   F F    d`r s`irfkft`r, `u sfkm, d`r s`irfkft`r, `u sfkm,   K K F F ((F F ) ) ::   F F ..   Cfstf oms`  Cfstf oms`   K K F F   ·  ·f f uj uj ns`ns`j`rj`r﬍sj﬍sj`̀
ns`j·ns`j·ftrno`, ftrno`, f f ` ` mius` mius` gf gf c`tm±c`tm±o˒om˒`m`   F F   :: K  K F F ((F F ) sf trmcsd`rjm fj) sf trmcsd`rjm fj   F F   :: F  F 
..
@ @ pprr··̀x`xnjnj` ` rfrfsusubtbtmgmg` ` j`j`ststrm rm ququf f c˒cm˒m` ` aa··m m pfpfrgrgm m gf gf lflfcfcfrmrmbnbngmgmgf gf fj fj gfgf﬍c﬍cnrnr
rf﬉fxnvngmgf fxobusnvmjfctf fj fspm±rf﬉fxnvngmgf fxobusnvmjfctf fj fspm±o`s gf Imcmoa;o`s gf Imcmoa;
4141
  
Ur`p`sn±Ur`p`sn±o˒om˒ ̀m ̀ 7.7.?.?.66   W`g` fspm±  W`g` fspm±o` o` c`rjmg` c`rjmg` rf﬉fxnv` rf﬉fxnv` ··f f uj uj fspm±fspm±o` gf Imcmoa.o` gf Imcmoa.
Gfj`cstrm±Gfj`cstrm±oo˒˒mm`̀..   Vj fspm±  Vj fspm±o` o` c`rjmg` c`rjmg` rf﬉fxnvrf﬉fxnv` ` ··f f ns`j`rdns`j`rd̀ ` ns`jftrnns`jftrnomjfcomjfctf tf m` m` sfusfu
ingumb, ingumb, quf ·quf ·f f Imcmoa pImcmoa p`r `r sfr sfr uj uj fspm±fspm±o` gumb (O`r`b·o` gumb (O`r`b·mrn` 9.=.6).mrn` 9.=.6).
Fxfjpb` 7.?.4Fxfjpb` 7.?.4
(m) Fspm±(m) Fspm±o`s c`rjmg`s gf gnjfcs˒o`s c`rjmg`s gf gnjfcsm˒` ﬍cntm s˒m` ﬍cntm sm˒` rf﬉fxm` rf﬉fxnvnv`s. `s. Gf dmt`Gf dmt ,̀ sf, sf   F F   tfj gnjfcs˒ tfj gnjfcsm˒`m`
﬍cntm﬍cntm   cc, smifj`s gm, smifj`s gm    ··Mblfirm Mblfirm Bncfmr Bncfmr quf quf sfu sfu gumb gumb mbl·mbl·firno`firno`   F F ∜∜  tm tmji·ji·fj fj tfj tfj gnjgnjfcsfcs˒˒m`m`
cc   f,  f, pfb` pfb` FxfroFxfro··Ľon` Ľon` 9.<.9, 9.<.9, ` ` gumb gumb mbl·mbl·firno` firno` o`ncongf o`ncongf o`j o`j ` ` gumb gumb t`p`b·t`p`b·̀lno`lno`. `. YfpftYfpftncgncg`̀
` mrlujfct` rfsubtm quf` mrlujfct` rfsubtm quf    F F    tm  tmji·ji·fj fj tfj tfj gnjfcgnjfcs˒sm˒`m`    cc   f sfluf dmonbjfctf quf  f sfluf dmonbjfctf quf    K K F F    ··ff
s`irfkft`rm, f p`rtmct`s`irfkft`rm, f p`rtmct`   F F  · ·f f rfrf﬉f﬉fxnxnv`v`..
(i) Fspm±(i) Fspm±o`s c`rjmg`s nco`jpbft`s, p`r fxfjpb`o`s c`rjmg`s nco`jpbft`s, p`r fxfjpb`    oo5555, , c˒cm˒ ̀m ̀ s˒sm˒` rf﬉fxnv`s (Ur`p`sn±m` rf﬉fxnv`s (Ur`p`sn±o˒om˒`m`
7.?.6).7.?.6).
(o)(o)   oo55   c˒cm˒` m` ··f f rf﬉frf﬉fxnv`. xnv`. K·K·m sm smifmifj`s j`s gms gms Ur`pUr`p`sn±`sn±o˒o˒̀fs 7.9.= f 7.9.? quf`fs 7.9.= f 7.9.? quf
((oo55))
  :  : ((((oo55))
))  :  : ((==))
   ::  ∓∓..
··F F d·d·monb monb vfrn﬍omvfrn﬍omr qur quf ` f ` jfrljfrluba` uba` omc˄omc˄̀cno``cno`  K  K oo55 ;;   oo55 ∙ ∙↙↙ ((oo55))  : :   ∓∓ · ·f f cm cm vfrgmgvfrgmgf f m m ncobusncobus˒˒m`m`
gfgf   oo55   fjfj   ∓∓, quf `ivnmjfctf c˒, quf `ivnmjfctf cm˒` m` ··f f s`is`irfkrfkft`ft`rm.rm.
@ p@ pmss` cmss` cmturmb mturmb ml`rm ·ml`rm ·f ncvfstnlmr f ncvfstnlmr m rm rf﬉fxnvngmgf f﬉fxnvngmgf g`s g`s fspm±fspm±o`so`s    p p. Gnvngnrfj`s fssm. Gnvngnrfj`s fssm
mc·mc·mbnsf mbnsf fj fj tr˄trf˄s fs ftmpms;ftmpms;    pp   : =, =  : =, =    2 2 p p 22  ∓∓    ff    pp    :: ∓ ∓. . Omgm ujm Omgm ujm gfbms smnr·gfbms smnr·m m o`j`o`j`
o`csfqu˄o`csfquf˄conm fconm gf gf rfsubtmg`s rfsubtmg`s quf quf s˒sm˒` nctfrfssmctfs p`r sn jfsj`s f quf tfr˒m` nctfrfssmctfs p`r sn jfsj`s f quf tfrm˒` `utrmsm` `utrms
mpbnom±mpbnom±o˒o˒̀fs m` b`cl` g` tfxt`. O`jf±`fs m` b`cl` g` tfxt`. O`jf±omj`s o`j ` oms`omj`s o`j ` oms`   pp :  : =;=;
Ur`p`sn±Ur`p`sn±o˒om˒ ̀m ̀ 7.7.?.?.<<    _f _f   F F  · ·f f ssffpmpmr· r· mvfb f rf﬉fxnv`, fct˒ mvfb f rf﬉fxnv`, fct˒ m`m`   F F   · ·f f ssffpmpmrr· · mvfb.mvfb.
Gfj`cstrm±Gfj`cstrm±oo˒˒mm`̀..   O`j`  O`j`   F F  · ·f f ns`j`rns`j`rd` d` ns`jfns`jftrnomtrnomjfctf jfctf mm   F F ,,  smifj`s quf smifj`s quf   F F   t tmmjiji··ffjj
··f f sfsfpmpmrr··mvfb. Ufb` Wf`rfjm ?.?.9 sfluf fct˒mvfb. Ufb` Wf`rfjm ?.?.9 sfluf fctm˒` qufm` quf   F F   · ·f f sfsfppmrmr··mvfb.mvfb.
Fxfjpb` 7.?.1Fxfjpb` 7.?.1    ==    c˒cm˒` ·m` ·f f rf﬉fxnvrf﬉fxnv`. `. Gf Gf dmt`, dmt`, o`j`o`j`    ==   ·  ·f f sfsfpmpmr·r·mvfb (Fxfjpb` =.4.7) fmvfb (Fxfjpb` =.4.7) fsfu gumbsfu gumb  ∓∓   c˒cm˒ ̀m ̀ ··f f sfsfpmpmr·r·mvfb mvfb (Fxfjpb` (Fxfjpb` =.4.6), =.4.6), pfbm pfbm Ur`p`sn±Ur`p`sn±o˒om˒` 7.?.< sfluf qufm` 7.?.< sfluf quf   ==   c˒cm˒m ̀̀ ··ff
rf﬉fxnv`.rf﬉fxnv`.
Mpr`vMpr`vfntmj`s m fntmj`s m `p`rtucngmg`p`rtucngmgf f g` fstug` g` fstug` gm rf﬉fxnvngmgf g`s fspm±gm rf﬉fxnvngmgf g`s fspm±o`so`s   p p, , == 2  2 p p 22  ∓∓,,
pmrm nctr`guznr m c`±pmrm nctr`guznr m c`±o˒om˒` gf `pfrmg`r mgkuct`;m` gf `pfrmg`r mgkuct`;
Gf﬍cn±Gf﬍cn±o˒om˒ ̀m ̀ 7.7.?.?.>>   _fkmj  _fkmj   F,F, D D   fspm±  fspm±o`s vft`rnmns c`rjmg`s fo`s vft`rnmns c`rjmg`s f   W W  ∁ B ∁ B((F F 88 D D ) uj `pfrmg`r) uj `pfrmg`r
bncfmr bncfmr o`cto`ct··Ľcu`. Ľcu`. Gf﬍cnj`s Gf﬍cnj`s ` ` `pfrmg`r`pfrmg`r  W  W  ;;   D D  ∙ ∙↙↙ F F    p`rp`r
W W ((ϖϖ)()(xx) ) :: ϖ ϖ((W W ((xx)) pmrm t`g`s)) pmrm t`g`s  xx ∁∁ F F   ff   ϖϖ  ∁∁ D D ..
@ `pfrmg`r@ `pfrmg`rW W   · ·f f oaoammjmjmg`g`  mgkuct` mgkuct`   gfgf   W W ..
4>4>
  
Fxfjpb` 7.?.=5Fxfjpb` 7.?.=5  Gmg` = Gmg` =  ≧≧  p  p 22 ∓ ∓, o`csngfrf ` `pfrmg`r, o`csngfrf ` `pfrmg`r   ‑sandt pmrm tr·   ‑sandt pmrm tr· ms„ ms„ 
W W  ; ;    p p ∙ ∙↙↙  p p    , , W W ((((mm==,, mm99,, mm??, . . . ,, . . . , )) : ()) : (mm99,, mm??, . . ., . . .))..
··F F d·d·monb vfr qufmonb vfr quf   W W  · ·f f bncbncfmr fmr f f o`ct·o`ct·Ľcu`Ľcu`. . U`r U`r gf﬍gf﬍cn±cn±o˒om˒`, ` `pfrmg`r mgkuct`m`, ` `pfrmg`r mgkuct`   W W   vmn gf ( vmn gf ( p p))
fj fj (( p p))
. . DDm±m±omj`s m ngfctn﬍om±omj`s m ngfctn﬍om±o˒om˒` gf (m` gf ( p p))
    o`jo`j    p p, , nst` nst` ··f, f, fcxfrlmrfj`s fcxfrlmrfj`s uj uj ducon`cmbducon`cmb
ϖϖ ∁∁ (( p p))  o o`j` `j` ujm ujm sfqu˄sfquf˄conm fconm ((iicc))∓∓cc:=:= ∁ ∁  p p  f m m± f m m±o˒om˒` g` ducon`cmbm` g` ducon`cmb  ϖϖ  s`irf `s fbfjfct`s s`irf `s fbfjfct`s
gfgf    p p  sf g· sf g·m pfbm rfbm±m pfbm rfbm±o˒om˒` gf gumbngmgf (7.=);m` gf gumbngmgf (7.=);
ϖϖ ((((mmcc))
∓∓
cc:=:=) ) ::
∓∓
 k k:=:=
mmcciicc  pmrm t`g` ( pmrm t`g` (mmcc))
∓∓
cc:=:= ∁ ∁  p p..
Mssnj pr`ofgfcg`, gmg`sMssnj pr`ofgfcg`, gmg`s   ϖϖ :  : ((iicc))
∓∓
cc:=:= ∁ ∁  p p    ff   xx :  : ((mmcc))∓∓cc:=:= ∁ ∁  p p,,
W W ((ϖϖ)()(xx) ) :: ϖ ϖ((W W ((xx)) )) :: ϖ ϖ ((((mm99,, mm??, . . . ,, . . . , )) )) :: i i==mm99 + + ii99mm?? + + ¼¼ ¼¼ ¼¼
: : 55mm== + + ii==mm99 + + ii99mm?? + + ¼¼ ¼¼ ¼¼ : :  ρ ρ ((xx)) ,,
`cgf`cgf   ρρ   ;: (5 ;: (5,, ii==,, ii99, . . ., . . .)) ∁ ∁    p p. . NssNss` rfvf` rfvfbm qufbm quf  W W ((ϖϖ) ) ::   ρρ, `u sfkm, ` `pfrmg`r mgkuct`, `u sfkm, ` `pfrmg`r mgkuct`
W W    ··f f `̀  ‑sandt pmrm drfctf„  ‑sandt pmrm drfctf„ 
W W  ;;    p p ∙ ∙↙↙  p p    , , W W  (( ((ii==,, ii99,, ii??, . . . ,, . . . , )) : (5)) : (5,, ii==,, ii99, . . ., . . .))..
··F nctfrfssmctf `isfrvmr quf c` oms`F nctfrfssmctf `isfrvmr quf c` oms`   pp : 9 tfj-sf : 9 tfj-sf   W W  ◤ ◤ W W   :  : ngng99   jmsjms   W W  ◤◤ W W   : : ngng99..
Ur`p`sn±Ur`p`sn±o˒om˒ ̀m ̀ 7.7.?.?.====   _fkm   _fkm    W W   ∁ ∁ BB((F,F, D D )). . FcFctt˒ ˒ m`m`    W W   ∁ ∁ BB((D D ,, F F ))    f f   W W     ::  W W . . JJmnmns s 
mncmncgmgm, , sf sf    W W   ·  ·f f uj uj ns`j`rns`j`r﬍sj` ﬍sj` (ns`j·(ns`j·ftrno`), ftrno`), fct˒ fct˒ m`m`    W W    tm  tmji·ji·fj fj ··f f uj uj ns`ns`j`r﬍j`r﬍sj`sj`
(n(ns`s`j·j·ftftrnrno`)o`)..
Gfj`cstrm±Gfj`cstrm±oo˒˒mm`̀..   @ bfnt`r gfvf vfrn﬍omr quf  @ bfnt`r gfvf vfrn﬍omr quf   W W    fst·  fst·m m ifj ifj gf﬍cng` gf﬍cng` (nst` (nst` ··f,f,   W W ((ϖϖ)) ∁ ∁   F F 
pmrm omgmpmrm omgm  ϖ ϖ ∁ ∁ D D ) ) f ·f ·f f bncfmr. bncfmr. Umrm Umrm omgmomgm   ϖϖ  ∁∁ D D   tfj`s tfj`s
W W ((ϖϖ)) : sup: sup
xx∁∁IIFF
||W W ((ϖϖ)()(xx))|| : sup: sup
xx∁∁IIFF
||ϖϖ((((W W ((xx))))|| ≧≧ ϖϖ ¼¼    supsup
xx∁∁IIFF
W W ((xx)) :: ϖϖ ¼¼ W W  ,,
` quf j`strm quf` quf j`strm quf  W W  ≧  ≧ W W . . U`U`r `utr` bmg`r `utr` bmg ,̀ pmrm omgm, pmrm omgm  xx ∁ ∁  F F , , pfbpfb` ` O`r`b·O`r`b·mrn` mrn` ?.=.6?.=.6
tfj`stfj`s
W W ((xx)) : : ssuupp
ϖϖ∁∁IID D 
||ϖϖ((W W ((xx))))|| : : ssuupp
ϖϖ∁∁IID D 
||W W ((ϖϖ)()(xx))|| ≧≧ xx ¼¼    supsup
ϖϖ∁∁IID D 
W W ((ϖϖ)) :: xx ¼¼ W W  ,,
pr`vmcg` qufpr`vmcg` quf  W W  ≧  ≧ W W . . _up`_up`cam mlcam ml`r`rm qufm quf   W W   sfkm uj ns`j`r﬍sj  sfkm uj ns`j`r﬍sj`. `. TTfkmj`s quffkmj`s quf
W W   · ·f f s`irfkft`r. s`irfkft`r. Umrm Umrm nss` nss` sfkmsfkm   ϖϖ ∁ ∁   F F . . O`csO`csngfngfrfrf   ττ ∁ ∁   D D   gmg` p`r gmg` p`r   ττ((z z ) ) ::   ϖϖ((W W ∙∙==((z z ))))
f pfrofim quff pfrofim quf   W W ((ττ) ) ::   ϖϖ, pr`vmcg` quf, pr`vmcg` quf   W W    ·  ·f sf s`irf`irfkft`rkft`r. . TTfkmj`fkmj`s qs qufuf   W W    ·  ·f f ncknckft`ft`r. r. _f_f
ρρ ∁ ∁ hfr(hfr(W W ), fct˒), fctm˒`m`
ρρ((W W ((xx)) )) :: W  W ((ρρ)()(xx) : 5 pmrm t`g`) : 5 pmrm t`g`   xx ∁∁ F.F.
<5<5
  
O`j`O`j` W  W    ··f f s`irfs`irfkft`r kft`r sfluf sfluf qufquf  ρ ρ  : 5. _up`cg` quf : 5. _up`cg` quf  W  W  sf sfkm ukm uj nsj ns`j`r﬍`j`r﬍sj` sj` ns`j·ns`j·ftrnoftrno`,`,
imstm j`strmr qufimstm j`strmr quf  W W ((ϖϖ)) :: ϖϖ pmrm t`g` pmrm t`g` ϖ ϖ ∁∁ D D . O`j`. O`j` W  W   ··f f nsns`j`j`r`r﬍s﬍sj ̀j ̀ nsns`j·`j·ftftrnrno`o`,,
xx ∁∁ IIF F  sf, f s`jfctf sf, sf, f s`jfctf sf,  W W ((xx)) ∁∁ IID D , f p`rtmct`, f p`rtmct`
W W ((ϖϖ)) : sup: sup
xx∁∁IIFF
||W W ((ϖϖ)()(xx))|| : sup: sup
xx∁∁IIFF
||ϖϖ((W W ((xx))))|| : : ssuupp
W W ((xx))∁∁IID D 
||ϖϖ((W W ((xx))))|| : sup: sup
zz∁∁IID D 
||ϖϖ((z z ))|| :: ϖϖ ..
Ur`p`sn±Ur`p`sn±o˒om˒ ̀m ̀ 7.7.?.?.=9=9   Umrm   Umrm   == 2  2 p p 22  ∓∓, `s fspm±, `s fspm±o`s o`s   BB p p((Q,Q, ίί,, ¶¶))   f f    p p    s˒ s˒ m` rf﬉fxnv`s.m` rf﬉fxnv`s.
Gfj`cstrm±Gfj`cstrm±oo˒˒mm`̀..   Imstm dmzfr ` oms` gf  Imstm dmzfr ` oms` gf   BB p p((Q,Q, ίί,, ¶¶). ). OaOamjmjf gff gf   W W  ; ;   BB p p∜∜((Q,Q, ίί,, ¶¶)) ∙ ∙↙↙
((BB p p((Q,Q, ίί,, ¶¶))))
    ff    _ _ ;;   BB p p((Q,Q, ίί,, ¶¶))   ∙∙↙↙    ((BB p p∜∜((Q,Q, ίί,, ¶¶))))   `s   `s ns`j`r﬍sj`s ns`j`r﬍sj`s ns`j·ns`j·ftrno`s ftrno`s g`g`
WWf`rfjm 7.=.9. f`rfjm 7.=.9. B`l`B`l` W  W ∙∙== ··f f tmji·tmji·fj fj uj uj ns`ns`j`rj`r﬍sj﬍sj` ` ns`ns`j·j·ftrftrno` no` f f pfpfbm bm Ur`Ur`p`p`sn±sn±o˒om˒` 7.?.==m` 7.?.==
sfluf quf sfu mgkuct` (sfluf quf sfu mgkuct` (W W ∙∙==)) ; ; ((BB p p∜∜((Q,Q, ίί,, ¶¶))))
   ∙∙↙↙    ((BB p p((Q,Q, ίί,, ¶¶))))   ·  ·f f uj uj ns`j`rns`j`r﬍sj`﬍sj`
ns`j·ns`j·ftrnoftrno` ` tmji·tmji·fj. fj. B`l` B`l` ` ` `pf`pfrmg`rmg`r r ((W W ∙∙==))  ◤◤ _ _ ;;   BB p p((Q,Q, ίί,, ¶¶)) ∙ ∙↙↙  ((BB p p((Q,Q, ίί,, ¶¶))))  · ·f f uujj
ns`j`ns`j`r﬍sj` r﬍sj` ns`j·ns`j·ftrnoftrno`. `. Imstm Imstm fct˒fctm˒` pr`vmr quf (m` pr`vmr quf (W W ∙∙==)) ◤◤ _ _   :: K  K BBpp((Q,Q,ίί,¶,¶)).. Umrm nss` sfkmj Umrm nss` sfkmj
d d  ∁ ∁ BB p p((Q,Q, ίί,, ¶¶),),   ϖϖ ∁∁ ((BB p p((Q,Q, ίί,, ¶¶))))   ff   ll  :: W  W ∙∙==((ϖϖ)) ∁∁ BB p p∜∜((Q,Q, ίί,, ¶¶). Fct˒). Fctm˒`m`
((W W ∙∙==)) ◤◤ _ _  ((d d )()(ϖϖ) ) : : ((W W ∙∙==))((_ _ ((d d ))())(ϖϖ) ) :: _  _ ((d d )(()((W W ∙∙==)()(ϖϖ)) )) :: _  _ ((d d )()(ll))
::
  
Q Q 
d l g¶d l g¶ : :  W  W ((ll)()(d d ) ) :: ϖ ϖ((d d ) ) :: K  K BBpp((Q,Q,ίί,¶,¶))((d d )()(ϖϖ)),,
o`jpbftmcg` m gfj`cstrm±o`jpbftmcg` m gfj`cstrm±o˒om˒`.m`.
Dncmbjfctf, pmrm ` oms`Dncmbjfctf, pmrm ` oms`   pp : : ∓∓  usmrfj`s ` sflunctf rfsubtmg`; usmrfj`s ` sflunctf rfsubtmg`;
Ur`p`sn±Ur`p`sn±o˒om˒ ̀m ̀ 7.7.?.?.=?=?   Vj fspm±  Vj fspm±o` gf Imcmoa o` gf Imcmoa   F F  · ·f f rf﬉frf﬉fxnv` xnv` sf, sf, f f s`jfctf s`jfctf sf, sf, sfu sfu gumb gumb   F F 
tmtmjiji··fj fj ··f f rf﬉rf﬉fxfxnvnv .̀̀.
Gfj`cstrm±Gfj`cstrm±oo˒˒mm`̀..  _up`cam quf _up`cam quf F  F  sfkm r sfkm rf﬉fxnf﬉fxnv`. v`. Fct˒Fctm˒` ` jfm` ` jfrlubarluba` omc` omc˄̀˄cno``cno` K  K F F ;;   F F  ∙ ∙↙↙
F F    ·  ·f f s`irs`irfkffkft`r. t`r. J`sJ`strfjtrfj`s `s quf quf ` ` jfrljfrlubauba` ` omc˄omc˄̀cno``cno`    K K F F  ;;   F F 
   ∙∙↙↙    F F    t  tmmjjii··ffj j ··ff
s`irs`irfkftfkft`r. `r. _fk_fkmm    xx
  ∁∁
   F F . . UfUfbm Ubm Ur`p`r`p s̀nsn±±o˒om˒` 7.?.== smifj`s quf (m` 7.?.== smifj`s quf (K K F F ))
 ;;   F F 
  ∙∙↙↙
   F F 
··f f uj uj ns`ns`j`rj`r﬍sj﬍sj` n` ns`j·s`j·ftrftrno`no`. . OamOamjfjf  x x  :  : ((K K F F ))((xx)) ∁∁ F F .. Ur`vfj`s quf Ur`vfj`s quf  K  K F F ((xx) ) :: x x..
Umrm omgmUmrm omgm   xx ∁∁ F F , tfj`s, tfj`s
K K F F ((xx
)()(K K F F ((xx)) )) :: K  K F F ((xx)()(xx
) ) :: x x((xx) ) : : ((K K F F ))
((xx)()(xx) ) :: x x((K K F F ((xx))))..
Gm·Gm·Ľ, Ľ, oo`j`j`̀   K K F F ((F F ) ) :: F  F 
,, o` o`cocobu·bu·Ľj`Ľj`s s ququff  K  K F F ((xx
) ) :: x x..
_up`camj`s ml`rm quf_up`camj`s ml`rm quf  F  F   sfkm rf﬉fxnv`. U`r misurg`, sup`camj`s quf sfkm rf﬉fxnv`. U`r misurg`, sup`camj`s quf   F F   c˒cm˒` sfkmm` sfkm
rf﬉frf﬉fxnvxnv`. `. FcFct˒tm˒`m`    K K F F ;;   F F   ∙∙↙↙    F F     c˒cm˒` ·m` ·f f s`irfkft`s`irfkft r̀. r. O`j`O`j`    K K F F ;;   F F   ∙∙↙↙K K F F ((F F ) ) ··f f ujujmm
ns`jftrnm bncfmr, rfsubtm qufns`jftrnm bncfmr, rfsubtm quf    K K F F ((F F ) ) ··f f dfoamgdfoamg` ` fjfj    F F 
    f f c˒cm˒` o`ncongf o`jm` o`ncongf o`j    F F . . UUffbbmm
Ur`p`sn±Ur`p`sn±o˒om˒` ?.?.=, fxnstf 5m` ?.?.=, fxnstf 5  ::   ϖϖ ∁ ∁   F F    ::   K K F F ((F F ) tmb quf) tmb quf   ϖϖ((K K F F ((F F )) )) :: { {55}}. . O`O`j`j`  F F    ··ff
rf﬉fxnv`,rf﬉fxnv`,  ϖϖ : :  K  K F F ((ττ) pmrm mbluj) pmrm mbluj   ττ ∁∁ F F . Mssnj, pmrm t`g`. Mssnj, pmrm t`g`   xx ∁∁ F F ,,
5 5 :: ϖ ϖ((K K F F ((xx)) : ()) : (K K F F ((ττ))())(K K F F ((xx)) )) :: K  K F F ((xx)()(ττ) ) :: τ τ((xx))..
Nss` pr`vm qufNss` pr`vm quf   ττ : 5 f p`rtmct` : 5 f p`rtmct`  ϖ ϖ  : :  K  K F F ((ττ) : ) : 5, 5, ` ` quf ·quf ·f f ujm ujm o`ctrmgn±o`ctrmgn±o˒om˒`.m`.
<=<=
  
Fxfjpb` 7.?.=7Fxfjpb` 7.?.=7    ∓∓ c c˒˒m` ·m` ·f rff rf﬉fxnv`. ﬉fxnv`. _flu_fluf gm f gm c˒cm˒` rf﬉fxnvngmgf gfm` rf﬉fxnvngmgf gf  == (Fxfjpb` 7.?.1), (Fxfjpb` 7.?.1),
gm nlumbgmgf (gm nlumbgmgf (==))
  : :   ∓∓  (Ur`p`sn± (Ur`p`sn±o˒om˒` 7.9.=) f gm Ur`p`sn±m` 7.9.=) f gm Ur`p`sn±o˒om˒` 7.?.=?.m` 7.?.=?.
@utr`s fxfjpb`s f pr`prnfgmgfs g`s fspm±@utr`s fxfjpb`s f pr`prnfgmgfs g`s fspm±o`s rf﬉fxnv`s sfr˒o`s rf﬉fxnv`s sfrm˒` vnst`s cms _f±m` vnst`s cms _f±o˒o˒̀fs 4.7`fs 4.7
f 4.6. f 4.6. J`sJ`strmrtrmrfj`sfj`s, fj , fj pmrtpmrtnounoubmr, quf `s fspm±bmr, quf `s fspm±o`so`s   O O XXm,m, iiZ,Z,   BB==XXm,m, iiZ Z ff   BB∓∓XXm,m, iiZ Z c˒cm˒` s˒m` sm˒`m`
rf﬉fxnv`s.rf﬉fxnv`s.
7.7 7.7 O`jfcO`jfct·t·mrn`s f mrn`s f c`tms anst·c`tms anst·̀rnoms`rnoms
Fxnstf Fxnstf tmji·tmji·fj fj ujm ujm vfrs˒vfrsm˒` g` tf`rfjm gm rfprfsfctm±m` g` tf`rfjm gm rfprfsfctm±o˒om˒` gf Ynfsz quf gfsorfvf `sm` gf Ynfsz quf gfsorfvf `s
duconducon`cmns `cmns bncfmrbncfmrfs ofs o`ct`ct··Ľcu`s Ľcu`s c` fc` fspm±spm±o` (o` (O O ((H H )),,  ¼¼∓∓), fj quf), fj quf H  H  · ·f f uj uj fsfspmpm±±o` t`p`b·o` t`p`b·̀lno``lno`
o`jpo`jpmot` gf mot` gf AmuAmusg`rsg`rst, st, gm gm sflusfluncnctf tf d`rjd`rjm; m; m m omgm duconomgm ducon`cmb`cmb  ϖϖ ∁ ∁   O O ((H H ))  o`rrfsp`cgf o`rrfsp`cgf
ujm jfgngm gf Ymg`cujm jfgngm gf Ymg`c  ¶ ¶  c`s i`rfbnmc`s gf c`s i`rfbnmc`s gf   H H   tmb quf tmb quf
ϖϖ((d d ) ) ::
  
H H 
d d g¶g¶  pmrm t`gm pmrm t`gm   d d  ∁ ∁ O O ((H H ))..
Gftmbafs p`gfj sfr fco`ctrmg`s fj X?=, <.?Z. C` oms` gfGftmbafs p`gfj sfr fco`ctrmg`s fj X?=, <.?Z. C` oms` gf   O O XXm,m, iiZ, `s ducon`cmns bncfmrfsZ, `s ducon`cmns bncfmrfs
s˒sm˒` rfprfsfctmg`s p`r duc±m` rfprfsfctmg`s p`r duc±o˒o˒̀fs gf vmrnm±`fs gf vmrnm±o˒om˒` bnjntmgm, f m rfbm±m` bnjntmgm, f m rfbm±o˒om˒` m` gf gf gumbngmgf gumbngmgf ··f f gmgmgmgm
pfbm nctflrmb gf Ynfjmcc‟_tnpfbm nctflrmb gf Ynfjmcc‟_tnfbtkfs. fbtkfs. Umrm jmn`rfs gftmbUmrm jmn`rfs gftmbafs vfkm X75, _f±afs vfkm X75, _f±o˒om˒` NN.>Z `u X41,m` NN.>Z `u X41,
Waf`rfj Waf`rfj <.7.7Z<.7.7Z..
M c`±M c`±o˒om˒` gf rf﬉fxnvngmgf surlnu o`j Afbby, quf fj =>9= o`cstmt`u quf ` jfrluba`m` gf rf﬉fxnvngmgf surlnu o`j Afbby, quf fj =>9= o`cstmt`u quf ` jfrluba`
omc˄omc˄̀cno` g` fspm±`cno` g` fspm±o`o` o o  gms s gms sfqu˄fquf˄conmfconms o`s o`cvfrlfctfs cvfrlfctfs o`j o`j m c`m c`rjmrjm  ¼¼ ∓∓  c` sfu ingumb c˒ c` sfu ingumb cm˒`m`
··f f s`irfkft`s`irfkft r̀. r. GnmcGnmctf tf gfssf gfssf fxfjpbfxfjpb̀ ` t`rc`u-st`rc`u-sf f cfofss·cfofss·mrn` sfpmrmr m obmssf g`s fspm±mrn` sfpmrmr m obmssf g`s fspm±o`so`s
pmrm `s qumns ` jfrluba` omc˄pmrm `s qumns ` jfrluba` omc˄̀cn`cno` o` ··f f s`irs`irfkftfkft`r, `r, f f nssnss` ` d`n d`n dfndfnt` t` p`r p`r AmaAmac c fj fj =>9<=>9<..
Amac oamj`u tmns fspm±Amac oamj`u tmns fspm±o`s gfo`s gf  rflubmrfs  rflubmrfs , uj tfrj` ncdfbnz p`ns mpmrfof nc·, uj tfrj` ncdfbnz p`ns mpmrfof nc·ujfrms vfzfsujfrms vfzfs
fj `utr`s o`ctfxt`s. Dfbnzjfctf F. B`roa ouca`u ` tfrj`fj `utr`s o`ctfxt`s. Dfbnzjfctf F. B`roa ouca`u ` tfrj`  fspm± fspm±o` rf﬉fxnv`o` rf﬉fxnv`  fj =>?>. fj =>?>.
@ @ fstug` quf fstug` quf ﬍zfj`s gm ﬍zfj`s gm rf﬉fxnvrf﬉fxnvngmgf ·ngmgf ·f f obmrmjfcobmrmjfctf tf nco`jpbnco`jpbft`, ft`, p`r p`r fxfjpbfxfjpb̀ , `, c˒cm˒`m`
t`omj`s cm qufst˒t`omj`s cm qufstm˒` gf sf suifspm±m` gf sf suifspm±o` dfoamg` gf fspm±o` dfoamg` gf fspm±o` o` rf﬉rf﬉fxnfxnv` v` ··f f tmji·tmji·fj fj rf﬉rf﬉fxnfxnv`v`
(c` (c` Fxfro·Fxfro·Ľon` Ľon` 7.6.?5 7.6.?5 ` bf` bfnt`r nt`r pr`vmr·pr`vmr·m fstf dmt` usmcg` fspm±m fstf dmt` usmcg` fspm±o`s qu`onfctfs). @ fstug` gmo`s qu`onfctfs). @ fstug` gmrf﬉fxnvrf﬉fxnvngmgf ngmgf sfr·sfr·m m rft`jmg` rft`jmg` c` c` OmpOmp··Ľtub` Ľtub` 4 4 f m f m qufst˒qufstm˒` monjm, ifj o`j` v·m` monjm, ifj o`j` v·mrnms `utrms,mrnms `utrms,
sfr˒sfrm˒` m` rfsp`cgngms rfsp`cgngms smtnsdsmtnsdmt`rnmjfcmt`rnmjfctf.tf.
Tnj`s quf sf uj fspm±Tnj`s quf sf uj fspm±o` c`rjmg`o` c`rjmg` F  F  · ·f rf rf﬉f﬉fxfxnv`nv`, f, fct˒ctm˒`m` F  F  · ·f f ns`j`rns`j`rd` d` ns`jftns`jftrnomjrnomjfctffctf
m` m` sfsfu u iningugumbmb    F F    p`r   p`r jfjfnn` ` g` g` jjfrfrlulubaba` ` omomc˄c˄̀cno``cno`    K K F F . . _f_f    F F    d`  d`r r ns`j·ns`j·ftrnoftrno`̀
ns`jftrnns`jftrnomjfcomjfctf tf mm    F F    p`r   p`r jfnjfn` ` gf gf `utr`utr` ` ns`ns`j`r﬍j`r﬍sj` sj` ns`ns`j·j·ftrnftrno`, o`, sfr·sfr·m m fcfct˒tm˒`m`    F F 
cfofssmrncfofssmrnmjfctmjfctf f rf﬉fxnvrf﬉fxnv̀ 0 `0 M M rfsp`stm ·rfsp`stm ·f f c˒cm˒`; m`; fj =>65fj =>65, , Y. O. Y. O. KmKmjfjfs s X7X79Z o`cs9Z o`cstrutrununu
uj uj fsfspm±pm±o`o ,̀ , m m pmpmrtrtnr nr gf gf fcfct˒tm˒` m` o`o`cafcafonong` g` o`o`j`j`   fspm±  fspm±o` o` gf gf KmKmjfjfs s , , quf quf ··f f ns`j`rdns`j`rd̀̀
ns`jftrnomjfctf m` sfu ingumb jms c˒ns`jftrnomjfctf m` sfu ingumb jms cm˒` ·m` ·f f rf﬉frf﬉fxnvxnv`. `. @ @ fspmfspm±±o` gf Kmjfs fco`ctr`uo` gf Kmjfs fco`ctr`u
juntjuntms ms `utrms mpbnom±`utrms mpbnom±o˒o˒̀fs f junt`s `utr`s fspm±`fs f junt`s `utr`s fspm±o`s rfbmon`cmg`s, o`j o`s rfbmon`cmg`s, o`j pr`prnfgpr`prnfgmgfs junt`mgfs junt`
fspfspfonmnfonmns, ds, d`rmj `rmj o`csto`cstruru··Ľg`s. Ľg`s. M j`cM j`c`lrm﬍`lrm﬍m X9m X91Z ·1Z ·f ujf ujm fxm fxofbfctf ofbfctf rfdfr˄rfdfrf˄conm fconm pmrm pmrm ` fs` fspm±pm±o`o`
gf Kmjfs, sums pr`prnfgmgfs, sums mpbnom±gf Kmjfs, sums pr`prnfgmgfs, sums mpbnom±o˒o˒̀fs f sfus fspm±`fs f sfus fspm±o`s o`s rfbmon`crfbmon`cmg`s.mg`s.
Cm _f±Cm _f±o˒om˒` m` 9.4 o`jfctmj`s s`irf 9.4 o`jfctmj`s s`irf `pfrmg`rfs quf `pfrmg`rfs quf mtnclfj m mtnclfj m c`rjm, f c`rjm, f c` c` FxfroFxfro··Ľon`Ľon`
9.<.== vnj`s 9.<.== vnj`s quf quf cfj cfj jfsj` ducon`cmns jfsj` ducon`cmns bncfmrbncfmrfs fs o`cto`ct··Ľcu`Ľcu s̀ s sfjprf mtnclfj sfjprf mtnclfj m m c`rjm.c`rjm.
<9<9
  
U`r `uU`r `utr` btr` bmg`, mg`, c` Fc` Fxfro·xfro·Ľon` 7Ľon` 7.6.9< .6.9< vfrfj`s vfrfj`s quf dquf ducon`ucon`cmns bcmns bncfmrncfmrfs o`fs o`ctct··Ľcu`s fĽcu`s fj fspmj fspm±±o`so`s
rf﬉fxnvrf﬉fxnv`s `s mtnclfj m mtnclfj m c`rjm. c`rjm. M M rforfo··Ľpr`om Ľpr`om d`n pr`vd`n pr`vmgm mgm pfb` pfb` jfsj` Y. jfsj` Y. O. O. Kmjfs fj Kmjfs fj X7?Z;X7?Z;
uj fspm±uj fspm±o` gf Imcmoao` gf Imcmoa    F F   ·  ·f f rf﬉fxnv` rf﬉fxnv` sf, sf, f f s`jfctf s`jfctf sf, sf, t`g` t`g` ducon`cmb ducon`cmb bncfmr bncfmr o`cto`ct··Ľcu`Ľcu`
ϖϖ
∁∁
F F 

 mtnclf m c`rjm. Umrm ujm gfj`cstrm± mtnclf m c`rjm. Umrm ujm gfj`cstrm±o˒om˒` vfkm X94, O`r`bbmry ?.64Z.m` vfkm X94, O`r`bbmry ?.64Z.
7.6 7.6 FxfroFxfro··Ľon`sĽon`s
FxFxfro·fro·ĽoĽon ̀n ̀ 7.7.6.6.==  _fj pmssmr pfb`s fspm± _fj pmssmr pfb`s fspm±o`so`s   BB p p, pr`vf quf, pmrm =, pr`vf quf, pmrm =   2 2 p p 22 ∓ ∓, m rfbm±, m rfbm±o˒om˒`m`
gf gumbngmggf gumbngmgf (7.=) f (7.=) gf dmt` gf dmt` fstmifbfof fstmifbfof uj ns`j`r﬍sj` uj ns`j`r﬍sj` ns`j·ns`j·ftrno` ftrno` fctrf (fctrf ( p p))
   ff    p p..
FxFxfro·fro·ĽoĽon ̀n ̀ 7.7.6.6.99   _fkm =  _fkm =   2 2 p p 22 ∓ ∓. . Ur`Ur`vfvf, , sfj usmr rf﬉fxnvsfj usmr rf﬉fxnvngmgngmgf, f, quf t`g` quf t`g` ducoducon`cmn`cmbb
bncbncfmr fmr o`ct·o`ct·Ľcu` Ľcu` fjfj    p p  mtnclf  mtnclf m m c`rjm, c`rjm, nst` ·nst` ·f, f, pmrm pmrm t`g`t`g`   ϖϖ ∁ ∁ ( ( p p))   fxnstf fxnstf   xx ∁ ∁   p p  tmb quf tmb quf
xx p p  : : = = ff  ϖϖ :: ||ϖϖ((xx))||..
FxFxfro·fro·ĽoĽon ̀n ̀ 7.7.6.6.??   _fkmj  _fkmj m m  :  :((mmcc))
∓∓
cc:=:=  ujm  ujm sfqu˄sfquf˄conm fconm gf gf fsombmfsombmrfs rfs ff p p  ≦≦ =. _up`cam quf,=. _up`cam quf,
pmrm pmrm t`gt`gm m sfqu˄sfquf˄conm fconm ((iicc))
∓∓
cc:=:=
 ∁ ∁
   p p, , m m s·s·frfrnfnf
∓∓
cc:=:=
mmcciicc   sfkm o`cvf sfkm o`cvfrlfctfrlfctf. . Ur`vUr`vf quff quf   mm
 ∁ ∁
  ∓∓
sfsf   pp :  : = = ff   mm ∁∁  p p    sfsf   p p 33 =. =.
FxFxfro·fro·ĽoĽon ̀n ̀ 7.7.6.6.77  Oamjf gf Oamjf gf   oo  `  ` fspm±fspm±o` o` gms gms sfqu˄sfquf˄conmfconms s fsombfsombmrfs mrfs quf quf s˒sm˒` o`cvfrlfctfsm` o`cvfrlfctfs
jucng` gm c`rjm g` suprfj`jucng` gm c`rjm g` suprfj`   ¼¼ ∓∓  (vfkm  (vfkm FxfroFxfro··Ľon` Ľon` =.1.9<). =.1.9<). Ur`vf qufUr`vf quf   oo · ·f f ns`ns`j`j`rdrd ̀̀ mm
oo55, o`cobum quf, o`cobum quf   oo
  · ·f f nsns`j`j`r`rd ̀d ̀ mm   ==  f gfsorfvm m rfbm± f gfsorfvm m rfbm±o˒om˒` gf gumbngmgf `itngm.m` gf gumbngmgf `itngm.
FxFxfro·fro·ĽoĽon ̀n ̀ 7.7.6.6.66   _fkm  _fkm   ϖϖ ∁ ∁ ( (∓∓)). . Ur`vf Ur`vf quf quf m m s·s·frnffrnf
∓∓
cc:=:=
||ϖϖ((ffcc))|| · ·f f o`cvfrlfo`cvfrlfctf ctf f f quf quf sumsum
s`jm s`jm ··f f jfc`r jfc`r `u `u nlumb nlumb mm  ϖϖ..
FxFxfro·fro·ĽoĽon ̀n ̀ 7.7.6.6.44   Ur`vf, sfj usmr rf﬉fxnvngmgf, quf  Ur`vf, sfj usmr rf﬉fxnvngmgf, quf   oo55    ff     p p, , ==   2 2 p p 22 ∓ ∓, , c˒cm˒ ̀m ̀ o`o`ct·ct·fjfj
suifspm±suifspm±o` ns`j`rd` mo` ns`j`rd` m   ==..
FxFxfro·fro·ĽoĽon ̀n ̀ 7.7.6.6.<<   _fkmj =  _fkmj =   2 2 p p 22
 ∓ ∓
  f f ((F F cc))
∓∓
cc:=:=
  ujm   ujm sfqu˄sfquf˄conm fconm gf gf fspm±fspm±o`s gf Imcmoa.o`s gf Imcmoa.
Gf﬍cmGf﬍cm
((

F F cc)) p p  ::


((xxcc))
∓∓
cc:=:=  ; ;  x xcc ∁ ∁ F F cc  pmrm t`g` pmrm t`g`   cc  ∁∁ CC  ff  ((xxcc))∓∓cc:=:= p p : :
   ∓∓
cc:=:=
xxcc p p
 ==
pp
22 ∓∓


 ..
UrUr`̀vvf f ququff

((

F F cc)) p p ,,  ¼¼  p p

  ·  ·f f uj uj fspfspm±m±oo` ` ggf f IImmccmmooa a f f qquuff

((

F F cc)) p p

··ff
ns`jftrnns`jftrnomjfcomjfctf tf ns`j`rns`j`rd` d` mm

((

F F cc)) p p

..
FxFxfro·fro·ĽoĽon ̀n ̀ 7.7.6.6.11   _fkmj =  _fkmj = ≧ ≧ p p  ≧≧ ∓∓   ff   F F  uj fspm± uj fspm±o` gf Imcmoa. Gf﬍cmo` gf Imcmoa. Gf﬍cm
 p p((F F ) ) ::

((xxcc))
∓∓
cc:=:=  ;; x xcc
 ∁ ∁
F F  pmrm t`g` pmrm t`g`   cc
  ∁∁
CC  ff
  
((xxcc))
∓∓
cc:=:=

 p p : :

  ∓∓
cc:=:= 
xxcc

 p p

==
pp
22
∓∓
,,
<?<?
  
o`j m j`gn﬍om±o`j m j`gn﬍om±o˒om˒` m` ··̀ivnm c` oms``ivnm c` oms`   pp : : ∓∓. Ur`vf quf;. Ur`vf quf;
(m) ((m) ( p p((F F )),,  ¼¼  p p) ) ··f f uj uj fsfspmpm±±o` gf Imcmoa.o` gf Imcmoa.
(i) Umrm =(i) Umrm =  ≧≧  p  p 22  ∓∓,,   p p((F F ))  · ·f f ns`jftns`jftrnomjrnomjfctf fctf ns`j`ns`j`rd` rd` mm    p p((F F ) ) ..
FxFxfro·fro·ĽoĽon ̀n ̀ 7.7.6.6.>>* J`strf quf pmrm omgm* J`strf quf pmrm omgm  c c ∁∁ CC fxn fxnstf stf ujm ujm ducduc±±o˒om˒`m` l lcc ∁ ∁ BB∓∓X5X5,, =Z tmb quf=Z tmb quf
 p p(5) +(5) + p p(5)(5) ∙∙ ?? p p(=) :(=) :
     ==
55
 p p((tt))llcc((tt))gt,gt,
pmrm pmrm t`t`g` g` p`bp`bnc˄nc˄̀jn``jn`  p p  gf lrmu jfc`r `u nlumb m gf lrmu jfc`r `u nlumb m   cc. Ur`vf mncgm quf. Ur`vf mncgm quf
bnjbnj
cc↙∓↙∓
llccBB==X5X5,,=Z=Z  : : ++∓∓..
FxFxfro·fro·ĽoĽon ̀n ̀ 7.7.6.6.=5=5* O`csngfrf `s o`ckuct`s* O`csngfrf `s o`ckuct`s
BB ;: ;: {{d d ; ; X5X5,, =Z=Z ∙∙↙↙ YY ;; d  d (5) : 5(5) : 5, , d d (=) : = f(=) : = f | |d d ((xx)) ∙∙ d d ((yy))|| ≧≧ 99||xx ∙∙ yy||,,  ∅∅x,x, yy ∁ ∁ X5X5,, =Z=Z}} ,,
_ _   ;:;:

ll ; ; X5X5,, =Z=Z ∙∙↙↙ YY    ;; l l((tt) ) ::
=55=55

nn:9:9
θθnntt
==/n/n , , θθnn ≦ ≦ 55 ∅ ∅nn   ff
=55=55

nn:9:9
θθnn  : : ==

..
J`strf quf fxnstf ujm duc±J`strf quf fxnstf ujm duc±o˒om˒`m`   ττ ∁∁ BB∓∓X5X5,, =Z tmb quf=Z tmb quf
supsup
d d ∁∁BB
     ==
55
d d ((tt))ττ((tt))gt gt 22   ncd ncd 
ll∁∁_ _ 
     ==
55
ll((tt))ττ((tt))gt.gt.
FxFxfro·fro·ĽoĽon ̀n ̀ 7.7.6.6.====   _fkm  _fkm    F F   uj fspm±  uj fspm±o` c`rjmo` c`rjmg`. g`. Ur`Ur`vf quf ` o`ckuvf quf ` o`ckuctct`̀    K K F F ((F F ) sfpmrm) sfpmrm
p`cp`ct`s t`s gfgf    F F , , nst` nst` ··f, f, gmg`sgmg`s    ϖϖ==,,    ϖϖ99    ∁∁    F F ,,    ϖϖ==    ::    ϖϖ99, , fxnfxnstfstf    d d    ∁∁    K K F F ((F F ) tmb quf) tmb quf
d d ((ϖϖ==)) :: d  d ((ϖϖ99).).
FxFxfro·fro·ĽoĽon ̀n ̀ 7.7.6.6.=9=9* _fkmj* _fkmj  D  D    ff   LL suifspm± suifspm±o`s dfoamg`s g` fspm±o`s dfoamg`s g` fspm±o` gf Imcmoao` gf Imcmoa   F F  tmns quf tmns quf
F F   :: D  D  ⊖ ⊖ LL. Ur`vf quf;. Ur`vf quf;
(m)(m)   F F    :: D  D ⊦⊦ ⊖⊖ LL⊦⊦..
(i)(i)   F F   o o`̀ctct··fj fj o·o·̀pnms `pnms ns`j·ns`j·ftrnoftrnoms oms o`jpbf`jpbfjfctmgmjfctmgms gfs gf   D D    f f gfgf   LL..
(o)(o)   F F   · ·f nf ns`j`rs`j`rd` d` ns`jftns`jftrnomjrnomjfctf fctf m (m (D D  ÜÜ LL,,  ¼¼ ==).).
FxFxfro·fro·ĽoĽon ̀n ̀ 7.7.6.6.=?=?   _fkmj  _fkmj F  F  uj fspm± uj fspm±o` c`rjmg`,o` c`rjmg`,  F F  sfu o`jpbftmjfct` f sfu o`jpbftmjfct` f  D  D  uj fspm± uj fspm±o`o`
gf Imcmogf Imcmoa. a. Ur`Ur`vf quf `s fspmvf quf `s fspm±±o`so`s B B((F,F, D D ) ) ff B B((  F,F, D D ) ) s˒sm˒` m` ns`j`rd`s ns`j`rd`s ns`jftrnomjfctf.ns`jftrnomjfctf.
Fj pmrtnoubmr, (Fj pmrtnoubmr, (  F F ))  : :  F  F ..
FxFxfro·fro·ĽoĽon ̀n ̀ 7.7.6.6.=7=7   _fkm  _fkm J  J  uj suifspm± uj suifspm±o` dfoamg` g` fspm±o` dfoamg` g` fspm±o` gf c`rjmg`o` gf c`rjmg` F  F . Ur`vf quf;. Ur`vf quf;
(m)(m)   J J  ⊄ ⊄ ((J J ⊦⊦))⊦⊦. (Tfkm m gf﬍cn±. (Tfkm m gf﬍cn±o˒om˒` gfm` gf   J J ⊦⊦  c`  c` FxfFxfro·ro·Ľon` Ľon` ?.4?.4.=7.=7))
(i) _f(i) _f  F  F  · ·f f rf﬉rf﬉fxnfxnv`, v`, fct˒fctm˒` m` ((J J ⊦⊦))⊦⊦  : :  J  J ..
FxFxfro·fro·ĽoĽon ̀n ̀ 7.7.6.6.=6=6   Gftf  Gftfrjnrjncf cf `s `s mgkumgkuct`ct s̀s    W W  ;;    ∓∓   ∙∙↙↙    ∓∓   g`s   g`s sflusfluncnctfs tfs `pfrm`pfrmg`rfg`rfss
W W  ; ;   == ∙ ∙↙↙ ==;;
(m)(m)   W W ((((mm==,, mm99, . . . ,, . . . , )) : ()) : (mm==, . . . , m, . . . , mcc,, 55,, 55, . . ., . . .), `cgf), `cgf   cc  ∁∁ CC fst· fst·m ﬍xmg`.m ﬍xmg`.
(i)(i)   W W ((((mmcc))∓∓cc:=:=) ) : : ((iiccmmcc))∓∓cc:=:=, `cgf (, `cgf (iicc))∓∓cc:=:= ∁ ∁ II∓∓   fst·  fst·m ﬍xmgm.m ﬍xmgm.
<7<7
  
FxFxfro·fro·ĽoĽon ̀n ̀ 7.7.6.6.=4=4   _fkmj =  _fkmj = ≧≧  p  p 22 ∓∓  ff   ll ∁ ∁ BB∓∓X5X5,, =Z. Gf﬍cm=Z. Gf﬍cm
W W ll ;;   BB p pX5X5,, =Z=Z ∙ ∙↙↙ BB p pX5X5,, =Z=Z   , , W W ll((d d ) ) :: d  d  ¼ ¼ l.l.
(m) Ur`vf quf(m) Ur`vf quf  W  W ll ∁ ∁ BB((BB p pX5X5,, =Z=Z,, BB p pX5X5,, =Z) f=Z) f  W W ll :: ll∓∓..(i) Ur`vf quf, fcxfrlmcg`(i) Ur`vf quf, fcxfrlmcg`   W W ll ;;   BB p pX5X5,, =Z=Z ∙ ∙↙↙   BB p pX5X5,, =Z,=Z,   W W ll   tfj m jfsjm bfn gf d`rjm± tfj m jfsjm bfn gf d`rjm±o˒om˒`m`
gfgf   W W , , nsnst ̀t ̀ ··f,f,   W W ll((d d ) ) :: d  d  ¼ ¼ ll  pmrm t`gm pmrm t`gm   d d  ∁ ∁ BB p pX5X5,, =Z.=Z.
(o) O`cobum quf(o) O`cobum quf  W  W ll   :: W  W ll  c` oms` c` oms`   pp :  : 9.9.
FxFxfro·fro·ĽoĽon ̀n ̀ 7.7.6.6.=<=<   _fkmj  _fkmj    F F    ff    D D   fspm±  fspm±o`s o`s c`rjc`rjmg`smg`s. . Ur`Ur`vf vf quf quf m m o`rro`rrfsp`cfsp`cg˄gf˄confconmm
W W  ∁ ∁ BB((F,F, D D )) ↙↙ W W  ∁ ∁ BB((D D ,, F F ) ) ··f f uj uj ns`j`ns`j`r﬍sj` r﬍sj` ns`j·ns`j·ftrnoftrno` ` s`irf s`irf sum sum njmlfnjmlfj.j.
FxFxfro·fro·ĽoĽon ̀n ̀ 7.7.6.6.=1=1   _fkmj  _fkmj   F, D , L  F, D , L   fspm±  fspm±o`o`s c`rs c`rjmjmg`g`s. s. GmGmg`g`ss    W W    ∁ ∁ BB((F,F, D D ) ) ff    V V    ∁∁
BB((D,D, LL), j`strf quf (), j`strf quf (V V  ◤ ◤ W W ))  :: W  W  ◤◤ V V ..
FxFxfro·fro·ĽoĽon ̀n ̀ 7.7.6.6.=>=>   _fkmj  _fkmj   F,F, D D   fspm±  fspm±o`s gf Imco`s gf Imcmoamoa. . GmgGmg`̀   W W   ∁ B∁ B((F,F, D D ), pr`vf quf), pr`vf quf   W W 
··f f ujm ujm fxtfcfxtfcs˒sm˒` gfm` gf    W W    mm    F F    c` sfctng` gf quf  c` sfctng` gf quf    W W  ◤ ◤   K K F F     ::    K K D D  ◤ ◤   W W . . Fj pmFj pmrtrtnonoububmrmr,,
W W ((K K F F ((F F )))) ⊄⊄ K K D D ((D D ) ) ff   W W   :: K  K ∙∙==D D    ◤◤ W W  ◤◤ K K F F   `cgf  `cgf   K K ∙∙==D D     ;;   K K D D ((D D )) ∙∙↙↙ D D ..
FxFxfro·fro·ĽoĽon ̀n ̀ 7.7.6.6.9595   _fkmj  _fkmj    F F    ff    D D   fspm±  fspm±o`s c`rjmg`s o`jo`s c`rjmg`s o`j    F Frf﬉fxnv` f  rf﬉fxnv` f    W W   ∁ ∁ BB((F,F, D D ).).
Ur`vf qufUr`vf quf   W W    :: W  W ..
FxFxfro·fro·ĽoĽon ̀n ̀ 7.7.6.6.9=9=   _fkmj  _fkmj   F,F, D D   fspm±  fspm±o`s c`rjmg`s fo`s c`rjmg`s f   W W   ∁ B∁ B((F,F, D D ). ). Ur`Ur`vf quvf quf sff sf   W W    tfjtfj
ncvfrsmncvfrsm    W W ∙∙== ;;   D D   ∙∙↙↙    F F   o`  o`ctct··ĽcuĽcum, m, fctfct˒˒m`m`    W W    tfj ncvfrsm ( tfj ncvfrsm (W W ))∙∙== ;;   F F   ∙∙↙↙    D D    o  o`c`ct·t·ĽcĽcum um ff
((W W ))∙∙== : : ((W W ∙∙==))..
FxFxfro·fro·ĽoĽon ̀n ̀ 7.7.6.6.9999   _fkm  _fkm F  F  uj fspm± uj fspm±o` c`rjmg`. Ur`vf qufo` c`rjmg`. Ur`vf quf  W  W  ∁ ∁ BB((F,F, F F ) ·) ·f f ujm ujm pr`pr` kf±kf±o˒om˒`m`
sf, f s`jfctf sf,sf, f s`jfctf sf,   W W  ∁ ∁ BB((F F ,, F F ) ·) ·f f ujm ujm pr`pr` kf±kf±o˒om˒`.m`.
FxFxfro·fro·ĽoĽon ̀n ̀ 7.7.6.6.9?9?* (m) _fkm* (m) _fkm I I  uj  uj suio`ckucsuio`ckuct` g` gumbt` g` gumb F  F  g` fspm± g` fspm±o` c`rjmg`o` c`rjmg` F  F . Ur`vf. Ur`vf
qufquf
⊦⊦II  ;:;: {{xx ∁∁ F F    ;; ϖ ϖ((xx) : 5 ) : 5 pmrpmrm m t`g`t`g`  ϖϖ  ∁∁ II}}
··f f uj uj suisuifspfspm±m±o` dfoamg` gfo` dfoamg` gf   F F ..(i) _fkmj(i) _fkmj   F F    ff    D D   fspm±  fspm±o`s c`rjmg`s fo`s c`rjmg`s f   W W   ∁ ∁ BB((F,F, D D ). ). Ur`Ur v̀vf quf hfrf quf hfr((W W ) ) ::⊦⊦((W W ((D D )) )) ff
hfr(hfr(W W ) ) : : ((W W ((F F ))))⊦⊦..
FxFxfro·fro·ĽoĽon ̀n ̀ 7.7.6.6.9797* _fkm* _fkm   F F  uj fspm± uj fspm±o` c`rjmg`. Ur`vf quf;o` c`rjmg`. Ur`vf quf;
(m) ((m) (K K F F ))
 ◤◤ K K F F    : : ngngF F ..
(i) ((i) (K K F F ))
 ;;  F F  ∙ ∙↙↙ F F   · ·f f s`is`irfkrfkft`ft`rm.rm.
(o) (Wf`rfjm gf Gnxjnfr)(o) (Wf`rfjm gf Gnxjnfr)   F F   · ·f f o`jo`jpbfpbfjfctjfctmg` mg` fjfj   F F ..
FxFxfro·fro·ĽoĽon ̀n ̀ 7.7.6.6.9696   _fkm  _fkm   F F  uj fspm± uj fspm±o` c`rjmg`. Ur`vf quf;o` c`rjmg`. Ur`vf quf;
(m) Fxnstf uj fspm±(m) Fxnstf uj fspm±o` c`rjmg`o` c`rjmg`    D D   tmb quf  tmb quf    F F   ·  ·f f ns`j`rd` ns`j`rd` ns`jftrnomjfctf ns`jftrnomjfctf mm    D D    sf, f  sf, f
s`jfctf, sf fxnstf uj fspm±s`jfctf, sf fxnstf uj fspm±o` gf Imcmoao` gf Imcmoa  L L  tmb quf tmb quf  F  F  · ·f f ns`j`ns`j`rd` rd` ns`jfns`jftrnomtrnomjfctf jfctf mm  L L..
Cfssf oms` gnz-sf qufCfssf oms` gnz-sf quf   F F  · ·f f uujj  fspm± fspm±o` gumb o` gumb ..
(i) _f(i) _f   F F  · ·f f uj uj fsfspmpm±±o` gumb, fct˒o` gumb, fctm˒`m`   F F  · ·f f o`jpbfjfctmg` o`jpbfjfctmg` c` c` sfu sfu ingumbingumb   F F . . VsVsf quf quff   oo55
c˒cm˒` m` ··f f o`jpbfo`jpbfjfctmg` jfctmg` fjfj   ∓∓  pmrm o`cobunr quf pmrm o`cobunr quf  o o55   c˒cm˒ ̀m ̀ ··f f uj uj fsfspmpm±±o ̀o ̀ gugumbmb..
<6<6
  
FxFxfro·fro·ĽoĽon ̀n ̀ 7.7.6.6.9494  Ur`vf quf t`g` fspm± Ur`vf quf t`g` fspm±o` co` c`rjmg`rjmg` quf ·` quf ·f ns`f ns`j`rd` j`rd` m uj m uj fspmfspm±±o` rf﬉fxnv`o` rf﬉fxnv`
··f f tmtmji·ji·fj fj rfrf﬉f﬉fxnxnv`v`..
FxFxfro·fro·ĽoĽon ̀n ̀ 7.7.6.6.9<9<   _fkm  _fkm F  F  uj fspm± uj fspm±o` rf﬉fo` rf﬉fxnv`. xnv`. Ur`vf quf t`Ur`vf quf t`g` ducong` ducon`cmb bncf`cmb bncfmr o`ctmr o`ct··Ľcu`Ľcu`
fjfj    F F   mtncl  mtnclf f m m c`rjc`rjm, m, nstnst` ` ··f, f, pmrm pmrm t`g`t`g`    ϖϖ  ∁∁    F F    fxnstf  fxnstf    xx  ∁∁    F F ,,  xx   : =, tmb quf  : =, tmb quf
ϖϖ :: ||ϖϖ((xx))||..
FxFxfro·fro·ĽoĽon ̀n ̀ 7.7.6.6.9191   Ur`vf quf ` suifspm±  Ur`vf quf ` suifspm±o` dfoamg`o` dfoamg`    F F    :: { {d d   ∁∁    O O X5X5,, =Z =Z ;;    d d (5) : 5(5) : 5}}    gfgf
O O X5X5,, =Z, =Z, fstugmg` fstugmg` c` c` FxfroFxfro··Ľon` Ľon` =.1.?7, =.1.?7, c˒cm˒ ̀m ̀ ··f f rfrf﬉f﬉fxnvxnv`.`.
FxFxfro·fro·ĽoĽon ̀n ̀ 7.7.6.6.9>9>   _fkmj  _fkmj   F F    ff   D D   fspm±  fspm±o`s gf o`s gf ImcImcmoamoa. . C` C` pr`gupr`gut` t` omrtomrtfsnfsnmc` o`csngmc` o`csngfrffrf
qumbqufr qumbqufr ujm ujm gms gms c`rjms c`rjms usumns usumns (vfkm Fxfr(vfkm Fxfroo··Ľon` Ľon` =.1.=9).=.1.=9).
(m) Ur`vf quf(m) Ur`vf quf  F  F  ÜÜ D D   · ·f f ns`ns`j`rj`rd` d` m m ((F F   ÜÜ D D ))..
(i) Ur`vf quf sf(i) Ur`vf quf sf   F F    ff   D D    s˒sm˒` rf﬉fxnv`s fct˒m` rf﬉fxnv`s fctm˒`m`   F F   ÜÜ D D   ·  ·f f rfrf﬉f﬉fxnxnv`v`..
FxFxfro·fro·ĽoĽon ̀n ̀ 7.7.6.6.?5?5* _fkm* _fkm   J J  uj suifspm± uj suifspm±o` dfoamg` g` fspm±o` dfoamg` g` fspm±o` gf Imcmoao` gf Imcmoa   F F ..
(m) Umrm omgm(m) Umrm omgm   ϖϖ ∁ ∁  J J   ` Wf`rfjm gf Amac‟Imcmoa fstfcgf ` Wf`rfjm gf Amac‟Imcmoa fstfcgf   ϖϖ  m uj ducon`cmb m uj ducon`cmb
  
ϖϖ ∁ ∁  F F ..
Ur`vf qufUr`vf quf
W W J J  ; ;   J J  ∙ ∙↙↙ F F /J /J ⊦⊦    , , W W J J ((ϖϖ) ) : : XXϖϖZZ,,
··f f uj uj ns`j`r﬍ns`j`r﬍sj` sj` ns`jns`j··ftrno` ftrno` (o`j` (o`j` m m fxtfcs˒fxtfcsm˒` m` gf gf AmAmac‟ac‟ImImcmocmoa a c˒cm˒` m` ··f f ··ucucnonom, m, ··ff
cfofss·cfofss·mrn` pr`vmr qufmrn` pr`vmr quf  W  W J J   fst·  fst·m m ifj ifj gf﬍cgf﬍cngmngm).).
(i) _fkm(i) _fkm   υυJ J  ; ;   F F  ∙ ∙↙↙ F/J F/J  m mpbnom± m mpbnom±o˒om˒` qu`onfctf. Ur`vf qufm` qu`onfctf. Ur`vf quf
W W J J  ; ; ((F/J F/J )) ∙ ∙↙↙ J J ⊦⊦    , , W W J J ((d d ) ) :: d  d  ◤ ◤ υυJ J ,,
··f f uj uj ns`ns`j`rj`r﬍sj﬍sj` ` ns`ns`j·j·ftrftrno` no` (c˒(cm˒` m` sf sf fsquf±fsquf±om om gf gf pr`vpr`vmr mr qufquf   W W J J  gf dmt` t`jm vmb`rfsgf dmt` t`jm vmb`rfs
fjfj   J J ⊦⊦).).
(o) O`cobum quf suifspm±(o) O`cobum quf suifspm±o` o` dfoamg` dfoamg` gf gf fspm±fspm±o` o` rf﬉frf﬉fxnv` xnv` ··f f rf﬉frf﬉fxnv`.xnv`.
FxFxfro·fro·ĽoĽon ̀n ̀ 7.7.6.6.?=?=   _fkmj  _fkmj   F F   ff   D D   fspm±  fspm±o`s c`rjmg`s. Ur`vf quf;o`s c`rjmg`s. Ur`vf quf;
(m)(m) B B((F,F, D D ) ) o`ct·o`ct·fj fj ujm ujm o·o·̀pn`pnm m ns`ns`j·j·ftrftrnom nom gfgf  F  F ..
(i) ((i) (BB((F,F, D D ))))  o` o`ct·ct·fj fj ujujm m o·o·̀pn`pnm m ns`ns`j·j·ftrftrnom nom gfgf   D D ..
FxFxfro·fro·ĽoĽon ̀n ̀ 7.7.6.6.?9?9   _fkmj  _fkmj   F F   ff   D D   fspm±  fspm±o`s gf Imcmoa. Ur`vf quf;o`s gf Imcmoa. Ur`vf quf;
(m) _f(m) _f B B((F,F, D D ) ) ··f f rf﬉rf﬉fxnfxnv`, v`, fct˒fctm˒`m`   F F  · ·f f rfrf﬉f﬉fxnxnv`v`..
(i) _f(i) _f B B((F,F, D D ) ) ··f f rf﬉rf﬉fxnfxnv`, v`, fct˒fctm˒`m`  D  D   ·  ·f f rfrf﬉f﬉fxnxnv`v`..
FxFxfro·fro·ĽoĽon ̀n ̀ 7.7.6.6.????   _fkm  _fkm   F F  uj fspm± uj fspm±o` o` c`rjmg`. c`rjmg`. Mssnj oMssnj o`j` `j` amvamv··Ľmj`s Ľmj`s gf﬍cng`gf﬍cng`   F F    ::
((F F )), gf﬍cnj`s, gf﬍cnj`s   F F (?)(?) : : ((F F )),,  F F (7)(7) : : ((F F (?)(?))), . . ., . . .,,  F F ((cc)) : : ((F F ((cc∙∙=)=))), . . ., . . ...
(m) Ur`vf quf ms sflunctfs m﬍rjm±(m) Ur`vf quf ms sflunctfs m﬍rjm±o˒o˒̀fs s˒`fs sm˒` fqunvmbfctfs;m` fqunvmbfctfs;
(n)(n)   F F  · ·f f rfrf﬉f﬉fxnxnv`v`..
(nn)(nn)   F F ((cc)) ··f f rf﬉frf﬉fxnv` xnv` pmrm pmrm t`gt`g`̀   cc ∁∁ CC..
(nnn)(nnn)   F F ((cc)) ··f f rf﬉frf﬉fxnv` xnv` pmrm pmrm mblujmbluj   cc  ∁∁ CC..
(i) _f(i) _f  F  F   c˒cm˒` m` ··f f rf﬉rf﬉fxnfxnv`, v`, fct˒fctm˒` t`gms ms ncobus˒m` t`gms ms ncobus˒̀fs cmturmns`fs cmturmns
F F  ⊄ ⊄ F F  ⊄ ⊄ F F (7)(7) ⊄ ¼ ¼ ¼⊄ ¼ ¼ ¼    ff    F F  ⊄ ⊄ F F (?)(?) ⊄⊄ F F (6)(6) ⊄ ¼ ¼ ¼⊄ ¼ ¼ ¼
s˒sm˒` fstrntms.m` fstrntms.
<4<4
  
OOmmpp··ĽĽttuubb ̀̀ 66
Fspm±Fspm±o`s gf Anbifrto`s gf Anbifrt
G`s ours`s gf O·G`s ours`s gf O·mboub` f mboub` f Lf`jftrnLf`jftrnm m McmbMcmb··Ľtnom ` Ľtnom ` bfnt`r smif bfnt`r smif quf quf ` ` pr`gut` pr`gut` nctfrnctfrc` c` usumbusumb
g`g`  Y Ycc
¼¼,, ¼¼ ;;   YYcc ÜÜ YYcc ∙∙↙↙ YY    ,,    ((x,x, yy)) ↙↙ x,x, yy ;:;:
cc
 k k:=:=
xx k kyy k k,,
`cgf`cgf   xx :  : ((xx==, . . . , x, . . . , xcc) ) ff   yy   : : ((yy==, . . . , y, . . . , ycc), ·), ·f f ujm ujm dfrrmjfctm dfrrmjfctm fssfconmb fssfconmb cm cm o`cstru±o`cstru±o˒om˒` gmm` gm
tf`rnm. tf`rnm. _mif _mif tmji·tmji·fj fj quf m quf m c`rjm c`rjm fuobngnmcm ·fuobngnmcm ·f ncgf ncguzngm puzngm pfb` fb` pr`gut` pr`gut` nctfrc` c` nctfrc` c` sfctng`sfctng`
gf qufgf quf
xx99  : :
  
x,x, xx  pmrm t`g` pmrm t`g`   xx ∁∁ YYcc..
F g` F g` oursours` gf` gf   ··Mblfirm Bncfmr ofrtmjfctf sf rfo`rgm gm utnbngmgf gf sf jucnr uj fspm±Mblfirm Bncfmr ofrtmjfctf sf rfo`rgm gm utnbngmgf gf sf jucnr uj fspm±o`o`
vft`rnmb gf uj pr`gut` nctfrc`, prnconpmbjfctf c` sfctng` gf quf ` pr`gut` nctfrc`vft`rnmb gf uj pr`gut` nctfrc`,prnconpmbjfctf c` sfctng` gf quf ` pr`gut` nctfrc`
pfrjnpfrjntf tf ‑lf`‑lf`jftrjftrnzmrnzmr„ „ ` ` fstufstug` g` g`s g`s fspmfspm±±o`s o`s vfvft`rnt`rnmns mns f f gms gms trmctrmcsd`rsd`rjm±jm±o˒o˒̀fs `fs bncfmrfbncfmrfs.s.
TTfrfj`s frfj`s cfstf cfstf ompomp··Ľtub` Ľtub` quf quf cm cm Mc·Mc·mbnsf mbnsf DDucon`ucon`cmb cmb tmji·tmji·fj fj c˒cm˒` m` ··f f pfqpfqufc` ufc` ` ` lmca`lmca`
mudfrng` qumcg` sf trmimbam o`j fspm±mudfrng` qumcg` sf trmimbam o`j fspm±o`s o`j pr`gut` nctfrc`.o`s o`j pr`gut` nctfrc`.
U`gf-sf gnzfr U`gf-sf gnzfr quf, quf, c` c` c`ss` omjnca` c`ss` omjnca` cfstf bnvr` cfstf bnvr` mt·mt·f f ml`rm, tfj`s ml`rm, tfj`s c`s gnstmconmg`c`s gnstmconmg`
omomgm gm vvfz fz jmjmns ns g`g`s s fsfspmpm±±o`s o`s fuobngnfuobngnmc`smc`s    YYcc. . MM` o`c` o`csnsngfgfrmrmr fsr fspmpm±±o`s o`s o`j o`j pr`pr`gutgut̀̀
nctfrc`, fstmj`s dmzfcg` ` omjnca` ncvfrs` f c`s mpr`xnjmcg` c`vmjfctf g`s fspm±nctfrc`, fstmj`s dmzfcg` ` omjnca` ncvfrs` f c`s mpr`xnjmcg` c`vmjfctf g`s fspm±o`so`sfuobngnfuobngnmc`s. mc`s. BfjirmBfjirmcg` quf fssfs fspm±cg` quf fssfs fspm±o`s s˒o`s sm˒` o`jpbft`s, pmrm c`s mpr`xnjmr mncgm jmnsm` o`jpbft`s, pmrm c`s mpr`xnjmr mncgm jmns
gfvfjgfvfj`s `s o`csngfro`csngfrmr mr fspm±fspm±o`s o`j pr`gut` nctfrc` quf s˒o`s o`j pr`gut` nctfrc` quf sm˒` o`jpbft`s cm c`rjm ncguzngmm` o`jpbft`s cm c`rjm ncguzngm
pfb` pr`gut` nctfrc`. Fstfs s˒pfb` pr`gut` nctfrc`. Fstfs sm˒` `sm` `s  fspm± fspm±o`s gf Anbifrt o`s gf Anbifrt . Gfvng` m fssm pr`xnjngmgf o`j. Gfvng` m fssm pr`xnjngmgf o`j
`s fspm±`s fspm±o`s o`s fuobngfuobngnmc`s, nmc`s, `s `s fspm±fspm±o`s o`s gf gf Anbifrt Anbifrt s˒sm˒`, m`, p`r p`r bmrlm bmrlm jmrlfj, jmrlfj, `s `s fspm±fspm±o`s gfo`s gf
Imcmoa jmns utnbnzmg`s cms mpbnom±Imcmoa jmns utnbnzmg`s cms mpbnom±oo˒˒̀̀ffss..
66..= = FFssppmm±±o`s o`j pr`gut` nctfrc`o`s o`j pr`gut` nctfrc`
_fkm_fkm F  F  uj fspm± uj fspm±o` vft`rnmb s`irf ` o`rp`o` vft`rnmb s`irf ` o`rp` HH :: YY `u`u OO. . VjVj pr`gut` nctfrc` pr`gut` nctfrc`   fjfj F  F  · ·f f uujjmm
mpbnom±mpbnom±o˒om˒`m`
¼¼,, ¼¼ ;;  F F  Ü Ü F F  ∙ ∙↙↙ HH  ,,    ((x,x, yy)) ↙↙ x,x, yy,,
tmb quf pmrm qumnsqufrtmb quf pmrm qumnsqufr   x,x, xx==,, xx99,, yy ∁ ∁ F F   ff  θ θ ∁∁ HH;;
<<<<
  
(U=)(U=)  xx== + + xx99,, yy :: xx==,, yy ++ xx99,, yy,,
(U9)(U9)  θx,yθx,y :: θ θx,x, yy,,
(U?)(U?)  x,x, yy :: y,y, xx,,
(U7) Umrm t`g`(U7) Umrm t`g`   xx : : 5,5,  x,x, xx · ·f f uj uj cc··ujfr` rfmb fstrntmjfctf p`sntnv`.ujfr` rfmb fstrntmjfctf p`sntnv`.C`tf quf gms pr`prnfgmgfs monjm sfluf quf, pmrm t`g`sC`tf quf gms pr`prnfgmgfs monjm sfluf quf, pmrm t`g`s  x x,, y,yy,y==,, yy99 ∁ ∁ F F    ff   θθ ∁∁ HH;;
x,x, 55 :: 55,, yy : : 5858
x,x, xx : 5 sf: 5 sf,,   f s`jfctf sf f s`jfctf sf, , xx :  : 5858
x,x, yy== + + yy99 :: x,x, yy== ++ x,x, yy99,,    x,x, yy== ∙∙ yy99 :: x,x, yy== ∙∙ x,x, yy998 8 ((66..==))
x,θyx,θy :: θ θx,x, yy    ff
_f_f  z,z, yy== :: z,z, yy99  pmrm t`g` pmrm t`g`   z z  ∁ ∁ F,F,   fct˒  fctm˒`m`   yy==  :: y y99..   (6.9)  (6.9)
Mb·Mb·fj fj gnss` gnss` ··f nf njp`jp`rtmctf rtmctf tfr tfr fj fj jfctf jfctf quf quf c` c` oms` oms` rfmb rfmb ` p` pr`r`gut` gut` nctfrc` nctfrc` ··f uf ujm jm d`rjmd`rjm
inbinbncfncfmr mr snj·snj·ftrftrnomnom..
@ pmr (@ pmr (F,F, ¼¼,, ¼¼) ) ··f f oamjoamjmg` mg` gfgf  fspm± fspm±o` o`j pr`gut` nctfrc`o` o`j pr`gut` nctfrc` . Cfstf oms` gnzfj`s quf. Cfstf oms` gnzfj`s quf
m duc±m duc±o˒om˒`m`
 ¼¼  ;;  F F  ∙ ∙↙↙ YY  ,,  xx ::  x,x, xx,,
··f f mm c`rjm ncguzngm pfb` pr`gut` nctfrc` c`rjm ncguzngm pfb` pr`gut` nctfrc` ¼ ¼,, ¼¼. Mctfs gf pr`vmr quf m duc±. Mctfs gf pr`vmr quf m duc±o˒om˒ ̀m ̀ momonjm njm ··f f gfgf
dmt` ujm c`rjm, vfkmj`s mblucs fxfjpb`s;dmt` ujm c`rjm, vfkmj`s mblucs fxfjpb`s;
Fxfjpb` 6.=.=Fxfjpb` 6.=.=   (m)   (m) Gmgms Gmgms ms ms sfqu˄sfquf˄conms fconms cuj·cuj·frnomsfrnoms    xx    : : ((xx k k))
∓∓
 k k:=:=,, yy    : : ((yy k k))
∓∓
 k k:=:=, , mm
fxprfss˒fxprfssm˒`m`
x,x, yy ::
∓∓
 k k:=:=
xx k kyy k k
gf﬍cf uj pr`gut` gf﬍cf uj pr`gut` nctfnctfrc` fjrc` fj  99. . M o`M o`cvfrl˄cvfrlf˄conm fconm gm s·gm s·frnf frnf monjm monjm gfo`gfo`rrf rrf gm ggm gfsnlumfsnlumbgmgfbgmgf
gf A¨gf A῭bgfr =.7.=. M c`rjm ncguzngm `ivnmjfctf o`ncongf o`j m c`rjm `rnlncmb`bgfr =.7.=. M c`rjm ncguzngm `ivnmjfctf o`ncongf o`j m c`rjm `rnlncmb   ¼¼ 99   gfgf
99..
(i) M fxprfss˒(i) M fxprfssm˒`m`
d,d, ll ::   
Q Q 
d l g¶d l g¶
gf﬍cf uj pr`gut` nctfrc` fjgf﬍cf uj pr`gut` nctfrc` fj   BB99((Q,Q, ίί,, ¶¶). ). M M fxnstfxnst˄˄fconm gm fconm gm nctflnctflrmb rmb monjm gfo`rrf monjm gfo`rrf gmgm
gfsnlumbgmgf gf A¨gfsnlumbgmgf gf A῭bgf`bgfr r =.9=.9.=. .=. M M c`rjc`rjm m ncgncguznguzngm m `i`ivnmvnmjfcjfctf o`ncontf o`ncongf o`j gf o`j m m c`rjc`rjmm
`rnlncmb`rnlncmb   ¼¼ 99   gfgf  B B99((Q,Q, ίί,, ¶¶).).
UrfoUrfonsmnsmj`s j`s gm gm gfsngfsnlumblumbgmgf gmgf mimnmimnx` x` pmrm pmrm pr`pr`vvmr mr quf quf gf gf dmt` dmt` m m o`rro`rrfsp`cfsp`cg˄gf˄confconmm
xx ∁∁ F F   ↙↙
  
x,x, xx · ·f f ujm ujm c`rc`rjm jm fjfj   F F ..
Ur`p`sn±Ur`p`sn±o˒om˒ ̀m ̀ 6.6.=.=.99  (Gfsnlumbgmgf gf Omuoay‟_oawmrz) (Gfsnlumbgmgf gf Omuoay‟_oawmrz)  _fkm  _fkm  F  F  uj fspm± uj fspm±o` vft`rnmb o`j o` vft`rnmb o`j 
pr`gut` pr`gut` nctnctfrc`frc`. . Fct˒Fctm˒`m`
||x,x, yy | ≧  | ≧ xx ¼¼ yy
pmrm qumnsqufr pmrm qumnsqufr   x,x, yy    fj fj   F F . . Mb·Mb·fj gnssfj gnss`, m `, m nlumbgmgf `o`rrnlumbgmgf `o`rrf sf, f sf, f s`jfctf, f s`jfctf, sf `s sf `s vft`rfvft`rfs s 
xx   f f   yy    s˒ s˒ m` bncfmrjfctf gfpfcgfctfs.m` bncfmrjfctf gfpfcgfctfs.
<1<1
  
Gfj`cstrm±Gfj`cstrm±oo˒˒mm`̀..    @ @ rfsubtmg` rfsubtmg` ··f f njfgnmt` njfgnmt` sfsf    xx  : 5 `u : 5 `u    yy   : 5.   : 5. U`gU`gfj`fj s̀ fcts fct˒˒m` sup`rm` sup`r
xx : : 5 5 ff   yy  : 5. Oamjf: 5. Oamjf   mm : : y,y, yy   ff   ii : : x,x, yy  pmrm o`cobunr quf pmrm o`cobunr quf
mimi

x,x, yy

::

y,y, yy

iiii : :
||
ii
||
99

y,y, yy
··f f uj uj cc··ujfr` rfmb p`sntnv`. Bfjirmcg` qufujfr` rfmb p`sntnv`. Bfjirmcg` quf   mm t tmmjiji··ffj j ··f f uuj j cc··ujfr` rfmb p`sntnv`,ujfr` rfmb p`sntnv`,
55 ≧≧ mxmx ∙∙ iy,mxiy,mx ∙∙ iyiy
:: mm mmx,x, xx ∙∙ imimy,y, xx ∙∙ mimix,x, yy ++ iiiiy,y, yy
:: m m99x,x, xx ∙∙ 99 YfYf

mimix,x, yy

++ ||ii||99 y,y, yy
:: m m99x,x, xx ∙∙ 99 ||ii||99 y,y, yy ++ ||ii||99 y,y, yy
:: m m99x,x, xx ∙∙ ||ii||99 y,y, yy   (6.?)  (6.?)
:: y,y, yy

y,y, yyx,x, xx ∙∙ ||ii||99

:: y,y, yy

y,y, yyx,x, xx∙ | ∙ | x,x, yy||99

..
O`j`O`j`  y,y, yy 33 5 sfluf quf 5 sfluf quf
yy99xx99 ∙∙ ||x,x, yy||99 :: y,y, yyx,x, xx∙ | ∙ | x,x, yy||99 ≦≦ 55,,
` ` ququf f prpr`v`vm m m m gfsgfsnlnlumbumbgmggmgf f gfsgfsfkmfkmgm. gm. UmUmssssfjfj`s `s ml`ml r̀m rm eem sflucgm m﬍rjm±m sflucgm m﬍rjm±o˒om˒m`̀. . __f `f `ss
vft`rfsvft`rfs   xx   ff   yy    s˒sm˒` m` bncfmrjfctf bncfmrjfctf gfpfcgfctfs gfpfcgfctfs ··f f d·d·monmonb b vfvfr r quf vmquf vmbf m bf m nlunlumbgmbgmgf. mgf. U`U`r r `utr`utr̀̀
bmg`, sup`cam quf m nlumbgmgf sfkm vfrgmgfnrm, `u sfkm,bmg`, sup`cam quf m nlumbgmgf sfkm vfrgmgfnrm, `u sfkm, | |x,x, yy|| :: xx ¼¼ yy. Cfstf oms`,. Cfstf oms`,
t`jmcg` c`vmjfctft`jmcg` c`vmjfctf   mm  ::  y,y, yy   ff   ii  ::  x,x, yy,,  `itfj`s `itfj`s | |ii||99 ::  x,x, xxmm. . Gf (6.Gf (6.?) p`gfj?) p`gfj`s`s
o`cobunr qufo`cobunr quf
mxmx ∙∙ iy,mxiy,mx ∙∙ iyiy :: m m99x,x, xx ∙∙ ||ii||99 y,y, yy : : 55..
G` mxn`jm (U7) `itfj`sG` mxn`jm (U7) `itfj`s  mx mx ∙∙ iyiy : 5. _f : 5. _f y y  : 5, fct˒ : 5, fctm˒` `s vft`rfsm` `s vft`rfs x x  f f  y y   s˒sm˒` `ivnmjfctfm` `ivnmjfctf
bncbncfmrjfmrjfcfctf tf gfpfcgfpfcgfcgfctfs. tfs. _f_f    yy  : 5, fct˒: 5, fctm˒`m`    m m 33  5 f p`rtmct` 5 f p`rtmct`    yy    ::    ii
mm
xx, pr`vmcg` quf, pr`vmcg` quf
tmji·tmji·fj fj cfstf cfstf oms` oms` `s `s vft`rfsvft`rfs   xx  ff   yy   s˒sm˒` bncfmrjfctf gfpfcgfctfs.m` bncfmrjfctf gfpfcgfctfs.Ml`rm snj p`gfj`s pr`vmr quf ` pr`gut` nctfrc` ncguz ujm c`rjm c` fspm±Ml`rm snj p`gfj`s pr`vmr quf ` pr`gut` nctfrc` ncguz ujm c`rjm c` fspm±o`;o`;
O`O`r`br`b··mrnmrn ̀̀ 6.6.=.=.??   _fkm   _fkm   F F  uj fspm± uj fspm±o` o`j pr`gut` nctfrc`. M duc±o` o`j pr`gut` nctfrc`. M duc±o˒ o˒ m`m`
 ¼¼  ;;  F F  ∙ ∙↙↙ YY    ,, xx ::
  
x,x, xx
··f f ujm ujm c`rjc`rjm m fj fj   F F ..
Gfj`cstrm±Gfj`cstrm±oo˒˒mm`̀..    Ur`Ur v̀vmrfmrfj`j`s s mpfmpfcms cms m m gfsgfsnlnlumbumbgmggmgf f trntrnmclmclububmr, mr, ujm ujm vfvfz z ququf f mm
vfrn﬍om±vfrn﬍om±o˒om˒` g`s m` g`s gfjmns mxn`jms gfjmns mxn`jms ··f njfgnmtm. f njfgnmtm. Umrm t`Umrm t`g`sg`s x, x, yy ∁ ∁ F F , usmcg` m gfsnlumbgmgf, usmcg` m gfsnlumbgmgf
gf Omuoay‟_oawmrz `itfj`sgf Omuoay‟_oawmrz `itfj`s
xx + + yy99 :: xx + + y,y, xx + + yy :: xx99 ++ x,x, yy ++ y,y, xx ++ yy99
:: xx99 ++ x,x, yy ++ x,x, yy ++ yy99
:: xx99 + 9Yf(+ 9Yf(x,x, yy) +) + yy99 ≧≧ xx99 + 9+ 9||x,x, yy|| ++ yy99
≧≧ xx
99
+ 9+ 9xx¼¼yy ++ yy
99
: : ((xx ++ yy))
99
..
<><>
  
Fxtrmncg` m rmnz qumgrmgm m gfsnlumbgmgf trnmclubmr sfluf.Fxtrmncg` m rmnz qumgrmgm m gfsnlumbgmgf trnmclubmr sfluf.
_f_fjpjprf rf ququf f c`c`s s rfrfdfdfrnrnrjrj`s `s m m uj uj fsfspmpm±±o` o` o`o`j j prpr`gu`gut` t` ncnctftfrcrc`̀    F F , , fstfstmrfmrfj`j`ss
mut`jmtnmut`jmtnomjfcomjfctf tf o`csngfro`csngfrmcg` mcg` fjfj    F F   m   m c`rc`rjm jm ncncguguznzngm gm f f m m o`co`csfsfququfcfctf tf fstfstrutrutururmm
t`p`b·t`p`b·̀lnom. Vjm sflucgm mpbnom±`lnom. Vjm sflucgm mpbnom±o˒om˒` gm gfsnlumbgmgf gf Omuoay‟_oawmrz lmrmctf quf `m` gm gfsnlumbgmgf gf Omuoay‟_oawmrz lmrmctf quf `
pr`gut` nctfrc` o`cvnvf ifj o`j fssm fstruturm t`p`b·pr`gut` nctfrc` o`cvnvf ifj o`j fssm fstruturm t`p`b·̀lnom;`lnom;
Ur`p`sn±Ur`p`sn±o˒om˒ ̀m ̀ 6.6.=.=.77   _fkmj   _fkmj   F F  uj fspm± uj fspm±o` o`j pr`gut` nctfrc` f o` o`j pr`gut` nctfrc` f   yy ∁ ∁ F F  uj vft`r ﬍xmg`. uj vft`r ﬍xmg`.
Fct˒ Fct˒ m` ms duc±m` ms duc±o˒ o˒ `fs `fs 
xx ∁∁ F F   ↙↙ x,x, yy ∁∁ HH    ff    xx ∁∁ F F   ↙↙ y,y, xx ∁∁ HH
s˒ s˒ m ̀m ̀ o`ct·o`ct·ĽcĽcumums.s.
Gfj`cstrm±Gfj`cstrm±oo˒˒mm`̀..   _fkm  _fkm   xx55 ∁ ∁ F F . . GfGf
||x,x, yy ∙∙ xx55,, yy|| :: ||xx ∙∙ xx55,, yy | ≧  | ≧ xx ∙∙ xx55 ¼¼ yy,,
sfluf m o`ctncungmgf gm prnjfnrm duc±sfluf m o`ctncungmgf gm prnjfnrm duc±o˒om˒`. m`. Fj vnstm Fj vnstm gf (6.=), gf (6.=), ` ` oms` gm oms` gm sflucgm duc±sflucgm duc±o˒om˒`m`
··f f mmcc··mb`l`.mb`l`.
Gf﬍cn±Gf﬍cn±o˒om˒ ̀m ̀ 6.6.=.=.66   Vj fspm±  Vj fspm±o` o` o`j o`j pr`gut` pr`gut` nctfrnctfrc` c` quf ·quf ·f f o`jpbft` cm o`jpbft` cm c`rjm ncguzngmc`rjm ncguzngm
pfb` pfb` pr`gut` pr`gut` nctfrc` ·nctfrc` ·f f oamjmg` oamjmg` gfgf   fspm±  fspm±o` gf Anbifrt o` gf Anbifrt . . Fj pmrFj pmrtntnouboubmr, uj fspmmr, uj fspm±±o` gfo` gf
AnbiAnbifrt frt ··f f uj uj fspm±fspm±o` gf Imcmoa o`j m c`rjm ncguzngm pfb` pr`gut` nctfrc`.o` gf Imcmoa o`j m c`rjm ncguzngm pfb` pr`gut` nctfrc`.
Fxfjpb` 6.=.4Fxfjpb` 6.=.4   O  O`j`j` ` k·k·m m smsmifjifj`s `s ququff    99    ff    BB99((Q,Q, ίί,, ¶¶) ) s˒sm˒` m` o`o`jpjpbfbft`t`s s cmcms s susumsms
rfspfotnvrfspfotnvms ms c`rjms, c`rjms, g` g` Fxfjpb` 6.=.= Fxfjpb` 6.=.= sfluf sfluf quf quf fssfs fspm±fssfs fspm±o`s o`s s˒sm˒` fspm±m` fspm±o`s gf Anbifrto`s gf Anbifrt
o`j sfus rfspfotnv`s pr`gut`s nctfrc`s.o`j sfus rfspfotnv`s pr`gut`s nctfrc`s.
Tfkmj`s mblucs rfsubtmg`s ·Tfkmj`s mblucs rfsubtmg`s ·utfns quf rfbmon`cmj ` pr`gut` nctfrc` o`j m c`rjm p`rutfns quf rfbmon`cmj ` pr`gut` nctfrc` o`j m c`rjm p`r
fbf ncguzngm;fbf ncguzngm;
Ur`p`sn±Ur`p`sn±o˒om˒ ̀m ̀ 6.6.=.=.<<   _fkm   _fkm    F F   uj fspm±  uj fspm±o` vft`rnmb oo` vft`rnmb o`j uj `j uj prpr`g`gut` nctfrcut` nctfrc`. `. Fct˒ Fct˒ m`, pmrm m`, pmrm 
qumnsqufr qumnsqufr   x,x, yy ∁ ∁ F F ,,
(n) (Bfn g` Umrmbfb`lrmj`)(n) (Bfn g` Umrmbfb`lrmj`)
xx + + yy99 ++ xx ∙∙ yy99 : : 99

xx99 ++ yy99

..
(nn) (D·(nn) (D·̀rjubm gf U`bmrnzm±`rjubm gf U`bmrnzm±o˒om˒`, oms` rfmb)m`, oms` rfmb)
x,x, yy ::  ==
77

xx + + yy99 ∙∙ xx ∙∙ yy99

..
(nnn) (D·(nnn) (D·̀rjubm gf U`bmrnzm±`rjubm gf U`bmrnzm±o˒om˒`, oms` o`jpbfx`)m`, oms` o`jpbfx`)
x,x, yy ::
  ==
77 xx ++ yy
99
∙∙ xx ∙∙ yy
99
++ nn xx + + nyny
99
∙∙ xx ∙∙ nyny
99
 ..
1515
  
Gfj`cstrm±Gfj`cstrm±oo˒˒mm`̀..  _`jmcg` ms nlumbgmgfs _`jmcg` ms nlumbgmgfs

  xx ++ yy99 :: xx99 ++ x,x, yy ++ y,y, xx ++ yy99

xx
∙∙
yy

99 ::

xx

99
∙∙ 
x,x, yy
 ∙∙ 
y,y, xx

++

yy

99    ,,
`itfj`s (n)8 f, c` oms` rfmb, suitrmncg`-ms `itfj`s (nn).`itfj`s (n)8 f, c` oms` rfmb, suitrmncg`-ms `itfj`s (nn).
Ur`vfj`s (nnn). _uitrmncg` ms nlumbgmgfsUr`vfj`s (nnn). _uitrmncg` ms nlumbgmgfs

   xx ++ nyny99 :: xx99 ++ x,nyx,ny ++ ny,xny,x ++ yy99 :: xx99 ∙∙ nnx,x, yy ++ nny,y, xx ++ yy99
xx ∙∙ nyny99 :: xx99 ++ x,x, ∙∙nyny ++ ∙∙ny,xny,x ++ yy99 :: xx99 ++ nnx,x, yy ∙∙ nny,y, xx ++ yy99    ,,
`itfj`s`itfj`s
xx + + nyny99 ∙∙ xx ∙∙ nyny99 :: ∙∙99nnx,x, yy + 9+ 9nny,y, xx..
_uitrmncg` _uitrmncg` c`vc`vmjfctf ms mjfctf ms gums gums nlumbgmgfs nlumbgmgfs usmgms usmgms cm cm gfj`cstrm±gfj`cstrm±o˒om˒` g`s ntfcs (n) f (nn),m` g`s ntfcs (n) f (nn),
sfluf qufsfluf quf
xx + + yy99 ∙∙ xx ∙∙ yy99 : : 99x,x, yy + 9+ 9y,y, xx..
O`jincmcg` fssms gums ·O`jincmcg` fssms gums ·ubtnjms nlumbgmgfs,ubtnjms nlumbgmgfs,
xx + + yy99 ∙∙ xx ∙∙ yy99 ++ nn xx + + nyny99 ∙∙ xx ∙∙ nyny99 : : 99x,x, yy + 9+ 9y,y, xx + 9+ 9x,x, yy ∙∙ 99y,y, xx
: : 77x,x, yy,,
o`jpbftmcg` m gfj`cstrm±o`jpbftmcg` m gfj`cstrm±o˒om˒`.m`.
6.6.9 9 @r@rt`t`l`l`cmcmbnbngmgmgfgf
Gnz-sf quf ` pr`gut` nctfrc` lf`jftrnzm ` fstug` g`s fspm±Gnz-sf quf ` pr`gut` nctfrc` lf`jftrnzm ` fstug` g`s fspm±o`s o`s vft`rnmvft`rnmns pns p`ns ·`ns ·f pf p`r j`r jfn`fn`
gfbf quf gf﬍cnj`s ` o`cofnt` gf vft`rfs `rt`l`cmns. Ncnonfj`s rfo`rgmcg` quf fjgfbf quf gf﬍cnj`s ` o`cofnt` gf vft`rfs `rt`l`cmns. Ncnonfj`s rfo`rgmcg` quf fj  Y Y99 mm
`rt`l`cmbngmgf `rt`l`cmbngmgf ··f f uj uj o`cofnt` o`cofnt` lf`j·lf`j·ftrno` ftrno` f f quf quf g`ns g`ns vft`rfsvft`rfs   xx :  : ((xx==,, xx99) ) ff   yy   : : ((yy==,, yy99))
gfgf   YY99 s˒sm˒` m` `rt`l`cmns `rt`l`cmns sf, sf, f f s`jfctf s`jfctf sf,sf,
xx==yy== + + xx99yy99  :  : 55..
Fj `utrms pmbmvrms, g`ns vft`rfs g`Fj `utrms pmbmvrms, g`ns vft`rfs g`   YY99 s˒sm˒` `rt`l`cmns sf, f s`jfctf sf, ` pr`gut`m` `rt`l`cmns sf, f s`jfctf sf, ` pr`gut`
ncnctfrctfrc` ` fctfctrf rf fbffbfs s ··f f nlunlumb mb m m zfr`zfr .̀ . Fssm ·Fssm ·f f m m oaoamvmvf f pmrm pmrm sf sf gf﬍cgf﬍cnr nr `rt``rt`l`cml`cmbngbngmgf mgf fjfj
fspm±fspm±o`s vft`rnmns o`j pr`gut` nctfrc`;o`s vft`rnmns o`j pr`gut` nctfrc`;
Gf﬍cn±Gf﬍cn±o˒om˒ ̀m ̀ 6.6.9.9.==   _fkm  _fkm   F F  uj fspm± uj fspm±o` o`j pr`gut` nctfrc`. Gnzfj`s quf `s vft`rfso` o`j pr`gut` nctfrc`. Gnzfj`s quf `s vft`rfs   xx  ff
yy   gfgf   F F   s˒sm˒`m`  `rt`l`cmns  `rt`l`cmns   sfsf  x,x, yy : 5. Cfstf oms` fsorfvfj`s: 5. Cfstf oms` fsorfvfj`s   xx ⊦⊦ yy..
Umrm sf Umrm sf o`cvfcofr gf o`cvfcofr gf quf quf ` ` pr`gut` pr`gut` nctfrc` nctfrc` nctr`guz nctr`guz gf gf dmt` dmt` uj uj vn·vn·fs fs lf`j·lf`j·ftrno` ftrno` c`c`
fstug`, ` bfnt`r p`gf dmonbjfctf gfj`cstrmr `fstug`, ` bfnt`r p`gf dmonbjfctf gfj`cstrmr `  Wf`rfjm  Wf`rfjm gf gf Unt·Unt·ml`ml`rmsrms; ; _f_f   xx   ff   yy    s˒sm˒`m`
vft`rfs `rt`l`cmns fj uj fspm±vft`rfs `rt`l`cmns fj uj fspm±o` o`j pr`gut` nctfrc`, fct˒o` o`j pr`gut` nctfrc`, fctm˒`m`
xx + + yy99 :: xx99 ++ yy99 ..
@ @ tf`rfjm mimnx` tf`rfjm mimnx` ··f f ofctrmb c` ofctrmb c` fstug` fstug` g`s g`s fspm±fspm±o`s c`ro`s c`rjmgjmg`s`s. . Fj irfvFj irfvf f ` ` bfbfntnt`r`r
fctfcgfr·fctfcgfr·m m p`r p`r quf quf fbf fbf ··f, f, eems vfzfs, oamjmg` gf tf`rfjm gm pr`kf±ms vfzfs, oamjmg` gf tf`rfjm gm pr`kf±o˒om˒` `rt`l`cmb.m` `rt`l`cmb.
1=1=
  
Wf`rfjm 6.9.9Wf`rfjm 6.9.9_fkm   _fkm    F F   uj   uj fspfspm±m±o` o` oo`j `j prpr̀̀ gugut` t` ncnctftfrcrc` ` f f sfsfkm km    J J   uj   uj suisuifspfspm±m±o`o`
o`jpbft` gf o`jpbft` gf   F F . Umrm t`g`. Umrm t`g`   xx ∁∁ F F  fxnstf uj ·  fxnstf uj · ucno`ucno`   pp ∁∁ J J   tmb quf   tmb quf 

xx
∙∙
 p p

: gnst(: gnst(x,x, J J ) ;: ncd ) ;: ncd 
yy∁∁J J 
xx
∙∙
yy

..
Gfj`cstrm±Gfj`cstrm±oo˒˒mm`̀..   Oamjfj`s  Oamjfj`s   gg  : gnst( : gnst(x,x, J J ). ). Gm gfGm gf﬍c﬍cn±n±o˒om˒` m` gf ·gf ·Ľc﬍j` Ľc﬍j` p`gp`gfj`s fj`s t`jmrt`jmr
ujm ujm sfqsfqu˄uf˄cofconm nm ((yycc))
∓∓
cc:=:=  gf vft`rfs gf gf vft`rfs gf   J J   tmb quf tmb quf
gg ≧≧ xx ∙∙ yycc 2 2 gg ++
  ==
cc
  (6.7)  (6.7)
pmrm t`g`pmrm t`g` c c. _fkmj. _fkmj j, j, cc ∁ ∁CC. Mpbnomj`s m Bfn g` Umrmbfb`lrmj` pmrm `s vft`rfs. Mpbnomj`s m Bfn g` Umrmbfb`lrmj` pmrm `s vft`rfs x x ∙∙yycc
ff   xx ∙∙ yyjj  pmrm `itfr pmrm `itfr
99 xx ∙∙ yyjj99 + 9+ 9 xx ∙∙ yycc99 :: xx ∙∙ yycc + + xx ∙∙ yyjj99 ++ xx ∙∙ yycc ∙∙ xx ++ yyjj99 ..
O`j`O`j`   J J  · ·f f susuififspspm±m±o` o` vft`rvft`rnmb,nmb,    yycc++yyjj
99
   ∁∁ J J , gm nlumbgmgf monjm f gf (6.7) sfluf quf, gm nlumbgmgf monjm f gf (6.7) sfluf quf
yycc ∙∙ yyjj99 : : 99 xx ∙∙ yyjj99 + 9+ 9 xx ∙∙ yycc99 ∙∙ 99xx ∙∙ ((yycc + + yyjj))99
: : 99 xx ∙∙ yyjj99 + 9+ 9 xx ∙∙ yycc99 ∙∙ 77 xx ∙∙
  yycc + + yyjj
99

99
≧≧ 99

gg ++
   ==
jj
99
+ 9+ 9

gg ++
  ==
cc
99
∙∙ 77gg99 ∙∙↙↙ 5 5 sfsf   j,j, cc  ∙∙↙↙ ∓∓..
NssNss` p` pr`vm qur`vm quf m f m sfqsfqu˄uf˄cofconm (nm (yycc))
∓∓
cc:=:= · ·f gf gf f OmuOmuoay coay c` ` fspfspm±m±o` o`jpbft`o` o`jpbft` J  J , b`l` o`cvfrlfctf, b`l` o`cvfrlfctf
pmrm uj ofrt`pmrm uj ofrt`    pp ∁ ∁    J J . . OO`j`j`̀   yycc  ∙∙↙↙    pp, dmzfcg`, dmzfcg`   cc ∙ ∙↙ ↙ ∓∓   fj   fj (6.7) (6.7) o`cobo`cobuu··Ľj`s Ľj`s qufquf
xx ∙∙ p p :: g g..
Umrm pr`vmr m ucnongmgf, sfkmUmrm pr`vmr m ucnongmgf, sfkm    q q   ∁∁    J J   tmb quf  tmb quf  xx ∙∙ q q     ::    gg. . MpMpbnbnomomcgcg` m ` m BfBfn g`n g`
Umrmbfb`lrmj` pmrm `s vft`rfsUmrmbfb`lrmj` pmrm `s vft`rfs   xx ∙∙ p p  ff   xx ∙∙ q q  `itfj`s `itfj`s
77gg99 : : 99gg99 + 9+ 9gg99 : : 99xx ∙∙ p p99 + 9+ 9xx ∙∙ q q 99
::

xx
∙∙
 p p + + xx
∙∙
q q 

99 ++

xx
∙∙
 p p
∙∙
xx ++ q q 

99
:: 99xx ∙∙ p p ∙∙ q q 99 ++  p p ∙∙ q q 99 : : 77xx ∙∙
  pp + + q q 
99

99
++  p p ∙∙ q q 99 ..
O`j`O`j`    pp++qq
99
   ∁∁ J J , gnss` sfluf quf, gnss` sfluf quf
55 ≧≧  p p ∙∙ q q 99 : : 77gg99 ∙∙ 77
xx ∙∙
  pp ++ q q 
99

99
≧≧ 77gg99 ∙∙ 77gg99 : : 55,,
` ` quf ·quf ·f f su﬍onfctf su﬍onfctf pmrm pmrm o`cobunr o`cobunr qufquf   pp : :  q  q ..
Gf﬍cn±Gf﬍cn±o˒om˒ ̀m ̀ 6.6.9.9.??   _fkmj  _fkmj  F  F  uj  uj fspm±fspm±o` o` o`j o`j pr`gut` pr`gut` nctfrc` fnctfrc` f   MM uj suio`ckuct` gf uj suio`ckuct` gf  F  F ..
Gfc`jncmj`s ` suio`ckuct`Gfc`jncmj`s ` suio`ckuct`
MM⊦⊦ : : {{yy ∁ ∁ F F   ;; x,x, yy : 5 pmrm t`g`: 5 pmrm t`g`   xx ∁∁ MM}}
gfgf  o`jpbfjfct` `rt`l`cmb gf  o`jpbfjfct` `rt`l`cmb gf   M.M.
1919
  
Ms sflunctfs pr`prnfgmgfs s˒Ms sflunctfs pr`prnfgmgfs sm˒` m` dmonbjdmonbjfctf fctf vfrnvfrn﬍omgms;﬍omgms;
••   MM  ⊄⊄ ((MM⊦⊦))⊦⊦,,   F F ⊦⊦ : : {{55}}   ff { {55}}⊦⊦  : :  F  F ..
••   MM⊦⊦  · ·f f uj uj suisuifspfspm±m±o` dfoamg` gfo` dfoamg` gf   F F , jfsj` quf, jfsj` quf   MM   c˒cm˒` sfkm cfj suifspm±m` sfkm cfj suifspm±o` (` dmt` gfo` (` dmt` gf
sfr dfoamg` gfo`rrf gm Ur`p`sn±sfr dfoamg` gfo`rrf gm Ur`p`sn±o˒om˒` 6.=.7).m` 6.=.7).
••   MM ∮∮ MM⊦⊦  ::
 { {55}},,    sf sf 55 ∁∁ M,M,
∆∆,,    sf sf 55   //∁∁ M.M.
@isfrvm±@isfrvm±o˒om˒ ̀m ̀ 6.6.9.9.77   M c`tm±  M c`tm±o˒om˒`m`    MM⊦⊦   pmrfof fstmr fj o`c﬉nt` o`j m c`tm±  pmrfof fstmr fj o`c﬉nt` o`j m c`tm±o˒om˒`m`    J J ⊦⊦    g`sg`s
FxfroFxfro··Ľon`s ?.4.=7 Ľon`s ?.4.=7 f 7.6.=7f 7.6.=7. . Fj irfFj irfvf ` vf ` WWf`rfjm f`rfjm 6.6.9 j`st6.6.9 j`strmr·rmr·m quf fssms c`tm±m quf fssms c`tm±o˒o˒̀fs s˒`fs sm˒`m`
o`csnstfctfs.o`csnstfctfs.
Wf`rfjm 6.9.6Wf`rfjm 6.9.6   _fkmj   _fkmj   F F  uj fspm± uj fspm±o` o`j pr`gut` nctfrc` f o` o`j pr`gut` nctfrc` f   J J   uj suifspm±  uj suifspm±o` o`jpbft`o` o`jpbft`
gf gf   F F . Fct˒ . Fct˒ m`m`
(m)(m)   F F   :: J  J  ⊖ ⊖ J J ⊦⊦, , nsnst ̀t ̀ ··f, f, omgm omgm   xx ∁∁ F F  mgjntf ujm ·  mgjntf ujm · ucnom rfprfsfctm±ucnom rfprfsfctm±o˒ o˒ m` cm d`rjm m` cm d`rjm 
xx : :  p p + + q q   o`j o`j   pp ∁∁ J J    f f   q q  ∁ ∁ J J ⊦⊦..
Mb·Mb·fj fj gngnssss ,̀̀, xx ∙∙ p p : gnst(: gnst(x,x, J J ))..
@ vft`r @ vft`r   pp · ·f f oamoamjmgjmg` ` gf gf pr`kfpr`kf±±o˒ o˒ m` `rt`l`cmb gf m` `rt`l`cmb gf   xx  s`irf  s`irf   J.J.
(i)(i)  @s `pfrmg`rfs  @s `pfrmg`rfs 
U,U, ]] ;;  F F  ∙ ∙↙↙ F, F, U U ((xx) ) :: p p    ff    ]]((xx) ) :: q  q 
s˒ s˒ m` m` prpr`kf±`kf±o˒ o˒ `fs`fs, , nstnst` ` ··f, f, s˒sm˒` m` bncfmrfsbncfmrfs, , o`cto`ct··ĽcuĽcu`s `s f f    U U 99 ::    U U    f f    ]]99 ::    ]]. . JJmnmns mns mncgcgm,m,
U U ((F F ) ) :: J  J ,,   ]]((F F ) ) :: J  J ⊦⊦,,  U U  : : ==   sf sf   J J   :: {{55}}   f f   ]] : : ==   sf sf   J J   :: F  F . @ `pfrmg`r . @ `pfrmg`r   U U    ··f f 
oamjmg` gf pr`kf±oamjmg` gf pr`kf±o˒ o˒ m` `rt`l`cmb gf m` `rt`l`cmb gf   F F  s`irf  s`irf   J J ..
(o)(o)   U U  ◤ ◤ ]] : :  ] ] ◤◤ U U    : : 55..
Gfj`cstrm±Gfj`cstrm±oo˒˒mm`̀..  (m) Gmg` (m) Gmg`  x x ∁∁ F F , pfb` Wf`rfjm 6.9.9 fxnstf uj ·, pfb` Wf`rfjm 6.9.9 fxnstf uj ·ucno` vft`rucno` vft`r p p ∁∁ J J    tmbtmb
qufquf
xx ∙∙ p p : gnst(: gnst(x,x, J J ))..
W`jmcg`W`jmcg`    q q    ::    xx ∙ ∙ p p   sfluf njfgnmtmjfctf quf  sfluf njfgnmtmjfctf quf    xx    ::    pp + +  q  q . . ImImststm fcm fct˒tm˒` pr`vmr qufm` pr`vmr quf
q q  ∁ ∁ J J ⊦⊦. Umrm t`g`. Umrm t`g`   yy ∁ ∁ J J  f t`g` fsombmr f t`g` fsombmr   θθ, ` vft`r, ` vft`r   pp + + θyθy  pfrtfcof m pfrtfcof m   J J , b`l`, b`l`
q q 99 :: xx ∙∙ p p99 : gnst(: gnst(x,x, J J ))99 ≧≧ xx ∙∙ (( p p + + θyθy))99 :: q q ∙∙ θyθy99
:: q q ∙∙ θy,q θy,q ∙∙ θyθy :: q q 99 ∙∙ θθy,y, q q  ∙∙ θθq,q, yy ++ θθθθ yy99 ..
GnsGnss` s` o`co`cobu·obu·Ľj`Ľj`s qs qufuf
55 ≧≧ ||θθ||99 yy99 ∙∙ 9 Y f (9 Y f (θθy,y, q q )) ..
FsorfvmFsorfvm  y,y, q q    cm d`rjm p`bmr  cm d`rjm p`bmr  ||y,y, q q || ffnαnα f pmrm omgmf pmrm omgm    tt  ∁∁    YY   oamjf  oamjf    θθ    ::    tftf∙∙nαnα. . GmGm
gfsnlumbgmgf monjm sfluf qufgfsnlumbgmgf monjm sfluf quf
55 ≧≧ tt99 yy99 ∙∙ 99tt ||y,y, q q ||  pmrm t`g` pmrm t`g`   tt ∁∁ YY,,
1?1?
  
f o`csfqufctfjfctf ` gnsornjncmctf g` inc˄f o`csfqufctfjfctf ` gnsornjncmctf g` inc˄̀jn` ·`jn` ·f f jfc`r `u jfc`r `u nlumb m nlumb m zfr`. zfr`. Nss` c`s Nss` c`s g·g·mm
||y,y, q q || : 5, f p`rtmct`: 5, f p`rtmct`   q q  ∁ ∁ J J ⊦⊦..
Umrm pr`vmr m ucnongmgf, sup`cam qufUmrm pr`vmr m ucnongmgf, sup`cam quf   pp + + q  q  : :  p p== + + q  q ==   o`jo`j   p,p, pp== ∁ ∁ J  J    ff   q,q, q q == ∁ ∁ J  J ⊦⊦..
O`j`O`j`   J J    ff  J  J ⊦⊦   s˒sm˒` m` suisuifspfspm±m±o`so`s,,
 p p ∙∙ p p==  :: q  q == ∙∙ q q  ∁ ∁ J J  ∮ ∮ J J ⊦⊦ : : {{55}}..
_fluf quf_fluf quf   pp : :  p p==   ff   q q  : :  q  q ==..
(i) _fluf gf (m) f gm Ur`p`sn±(i) _fluf gf (m) f gm Ur`p`sn±o˒om˒` ?.9.9 qufm` ?.9.9 quf   U U    ff   ]]   s˒sm˒` m` bncfmrfs, bncfmrfs, o`cto`ct··Ľcu`s Ľcu`s f f pr`pr`kf±kf±oo˒˒̀̀ffss..
Umrm omgmUmrm omgm   xx ∁ ∁ F  F , pfb` ntfj (m) p`gfj`s fsorfvfr, pfb` ntfj (m) p`gfj`s fsorfvfr   xx : :  p p + + q  q   o`jo`j   pp ⊦ ⊦ q  q . Ufb` Wf`rfjm. Ufb` Wf`rfjm
gf Unt·gf Unt·ml`rms,ml`rms,
xx99 ::  p p + + q q 99 ::  p p99 ++ q q 99 ,,
f p`rtmct`f p`rtmct`
U U ((xx))99 ::  p p99 ≧≧ xx99 ff  ]]((xx))99 :: q q 99 ≧≧ xx99 ..
Gm·Gm·Ľ Ľ o`o`cocobu·bu·ĽjĽj s̀ s̀ ququff  U U  ≧≧ =  = ff  ]] ≧≧ =. Ms gfsnlumbgmgfs ncvfrsms sflufj g` dmt`, p`r =. Ms gfsnlumbgmgfs ncvfrsms sflufj g` dmt`, p`r
c·c·̀s `s k·k·m m o`co`cafoafong` ng` (vfk(vfkm m Gf﬍Gf﬍cn±cn±o˒om˒` ?.9.=) gf quf pr`kf±m` ?.9.=) gf quf pr`kf±o˒o˒̀fs c˒`fs cm˒`-cubms m`-cubms t˄tf˄j fj c`rjmc`rjms s jmn`rjmn`rfsfs`u nlumns m =.`u nlumnsm =.
(o) Fssms m﬍rjm±(o) Fssms m﬍rjm±o˒o˒̀fs s˒`fs sm˒` gf d·m` gf d·monb vfrn﬍om±monb vfrn﬍om±o˒om˒`, p`r nss` s˒m`, p`r nss` sm˒` gfnxmgms m omrl` g` bfnt`r.m` gfnxmgms m omrl` g` bfnt`r.
@isfrvm±@isfrvm±o˒om˒ ̀m ̀ 6.6.9.9.44    ··F nctfrfssmctf `isfrvmr quf jfsj` ` ntfj (m) g` tf`rfjm mctfrn`rF nctfrfssmctf `isfrvmr quf jfsj` ` ntfj (m) g` tf`rfjm mctfrn`r
c˒cm˒m ̀̀ ··f f vv··mbng`, mbng`, fj lffj lfrmb, rmb, sfj m sfj m anp·anp·̀tfsf `tfsf gfgf J  J  sfr uj suifspm± sfr uj suifspm±o` o`jpbft`. O`j` fxfjpb`o` o`jpbft`. O`j` fxfjpb`
t`jft`jf F  F   ::  99  ff J  J   : : XXff k k    ;; k k ∁ ∁ CCZ. Z. C` C` FxfroFxfro··Ľon` 6Ľon` 6.<.7 .<.7 ` ` bfnt`r bfnt`r o`jpr`vmr·o`jpr`vmr·m qufm quf J  J ⊦⊦  : : {{55}},,
f mssnj fst·f mssnj fst·m obmr` quf cfstf oms`m obmr` quf cfstf oms`   F F   :: J  J  ⊖ ⊖ J J ⊦⊦..
@ o`r`b·@ o`r`b·mrn` m mrn` m sflunr ·sflunr ·f f njfgnmnjfgnmt` t` m m pmrtnr g` pmrtnr g` WWf`rfjm 6.9.6. f`rfjm 6.9.6. @ptmj`s p`r @ptmj`s p`r fcucofcucon·n·m-b`m-b`
p`r sf trmtmr gf ujm gms pr`prnfgmgfs ducgmjfctmns g`s fspm±p`r sf trmtmr gf ujm gms pr`prnfgmgfs ducgmjfctmns g`s fspm±o`s gf Anbifrt.o`s gf Anbifrt.
O`O`r`br`b··mrnmrn ̀̀ 6.6.9.9.<<   W  W`g` `g` suifspsuifspm±m±o` o` dfdfoaoamgmg` ` c˒ c˒ m`-m`-cucub` b` gf gf uj uj fsfsppm±m±o` o` gf gf AnbAnbiifrt frt ··f f =-=-
o`jpbfjfctmg`.o`jpbfjfctmg`.
6.6.? ? O`O`ckckucuct`t`s `s `rtrt`c`c`r`rjmjmns fns fj fj fspspm±m±o`s gf Anbifrto`s gf Anbifrt
@ dmt` gf t`g` fspm±@ dmt` gf t`g` fspm±o` o` vvft`ft r̀nmrnmb b tfr tfr imsimsf f (Ur(Ur`p``p`snsn±±o˒om˒` m` M.M.7) 7) lmrlmrmcmctf tf ququf f t`g` t`g` vfvft`rt`r
gf uj fspm±gf uj fspm±o` vft`rnmb p`gf sfr fsornt` o`j` o`jincm±o` vft`rnmb p`gf sfr fsornt` o`j` o`jincm±o˒om˒` bncfmr g`s vft`rfs gm imsf.m` bncfmr g`s vft`rfs gm imsf.
Fj gnjfcs˒Fj gnjfcsm˒` ﬍cntm ` bfnt`r fst·m` ﬍cntm ` bfnt`r fst·m mo`stujmg` o`j nss` f sflurmjfctf o`cvfcong` gmsm mo`stujmg` o`j nss` f sflurmjfctf o`cvfcong` gms
vmctmlfcs gf sf tfr tmb tnp` gf rfprfsfctm±vmctmlfcs gf sf tfr tmb tnp` gf rfprfsfctm±o˒om˒`. m`. Fctrftmct`, fj Fctrftmct`, fj gnjfcs˒gnjfcsm˒` nc﬍cntm m` nc﬍cntm fssmfssm
rfprfsfctm±rfprfsfctm±o˒om˒` fj lfrmb c˒m` fj lfrmb cm˒` tfj m` tfj utnbutnbngmgngmgf mblujmf mblujm. . CfsCfstm sf±tm sf±o˒om˒` vfrfj`s ujm jmcfnrmm` vfrfj`s ujm jmcfnrm
fxtrfjmjfxtrfjmjfctf fctf ··utnutnb b gf gf rfprrfprfsfcfsfctmr tmr vfvft`rft`rfs s gf gf uj uj fspmfspm±±o` o` AnbAnbifrt ifrt o`j` o`j` s`js`jms ms (cfj(cfj
sfjprf ﬍cntms) gf j·sfjprf ﬍cntms) gf j·ubtnpbubtnpb̀ s gf `s gf vft`rfvft`rfs o`cvfcns o`cvfcnfctfjfctfjfctf fso`banfctf fso`bang`s. g`s. Nss` mpr`xnNss` mpr`xnjmjm
mncgm jmns `s fspm±mncgm jmns `s fspm±o`s gf Anbifrt g`s fspm±o`s gf Anbifrt g`s fspm±o`s o`s fuobngnfuobngnmc`s.mc`s.
1717
  
Gf﬍cn±Gf﬍cn±o˒om˒ ̀m ̀ 6.6.?.?.==   _fkm  _fkm   F F  uj fspm± uj fspm±o` o`j o` o`j pr`gupr`gut` nctfrt` nctfrc`. c`. Vj o`ckucVj o`ckuct`t`  _ _  ⊄ ⊄ F  F  · ·f f ggnntt`̀
`rt`c`rjmb `rt`c`rjmb  sf pmrm t`g`s sf pmrm t`g`s   x,x, yy ∁ ∁ _ _ ,,
x,x, yy :: 
   55,,    sfsf   xx

:: y y,,
==,,    sfsf   xx : :  y y..
Vj Vj o`cko`ckucuct` t` `rt``rt`c`rjc`rjmbmb    _ _   tmb   tmb ququff    _ _ ⊦⊦    ::    {{55}}   ·  ·f f oamjmg` oamjmg` gfgf   snstfjm   snstfjm `rt`c`rt`c`rjmb `rjmb 
o`jpbft`o`jpbft`..
Fxfjpb` 6.?.9Fxfjpb` 6.?.9   M imsf omc˄  M imsf omc˄̀cnom`cnom { {(=(=,, 55, . . . ,, . . . , 5)5), . . . ,, . . . , (5(5, . . . ,, . . . , 55,, =)=)}}   g`g`   HHcc ··f f uj uj snssnstfjtfjmm
`rt`c`rjm`rt`c`rjmb o`jpbft`b o`jpbft .̀ . @ o`ckuc@ o`ckuct`t` { {ffcc    ;;   cc ∁ ∁   CC}}  g`s vft`rfs ucnt· g`s vft`rfs ucnt·mrn`s omc˄mrn`s omc˄̀c`cnono`s `s ··f f ujuj
snstfjm `rt`c`rjmb o`jpbft` fjsnstfjm `rt`c`rjmb o`jpbft` fj   99..
@ bfnt`r c˒@ bfnt`r cm˒` m` tfr·tfr·m m gn﬍gn﬍ouboubgmgf gmgf fj fj vfvfrn﬍rn﬍omr omr quf quf t`g` t`g` o`cko`ckucuct` t` `rt``rt`c`rjc`rjmb mb fj fj ujuj
fspm±fspm±o` o` o`j o`j pr`gut` pr`gut` nctfnctfrc` ·rc` ·f f bncfmrjbncfmrjfctf ncgfpfcgfctf. fctf ncgfpfcgfctf. Cm Cm vfrgvfrgmgf mgf morfgntmjmorfgntmj`s `s qufquf
nss` k·nss` k·m tfcam sng` dfnt` c` ours` gfm tfcam sng` dfnt` c` ours` gf    ··MblfirMblfirm m Bncfmr.Bncfmr.
Umrm snjpbn﬍omr m c`tm±Umrm snjpbn﬍omr m c`tm±o˒om˒` nctr`guznj`s ml`rm `m` nctr`guznj`s ml`rm `   gfbtm   gfbtm gf gf HrHr`cf`cfohfr ohfr ; ; gmgmg`g`s ujs uj
o`ckuct` qumbqufro`ckuct` qumbqufr   N N    ff   n,n, kk
 ∁ ∁
N N , gf﬍cnj`s, gf﬍cnj`s
η η nknk   ::

   55,,    sfsf   nn  :: k, k,
==,,    sfsf   nn : :  k. k.
Gfssm d`rjm, uj o`ckuct`Gfssm d`rjm, uj o`ckuct` { {xxnn  ;; n n  ∁∁ N  N }}  gf vft`rfs gf uj fspm± gf vft`rfs gf uj fspm±o` o` o`j o`j pr`pr`gut` gut` nctfrc` nctfrc` ··ff
`rt`c`rjmb sf, f s`jfctf sf,`rt`c`rjmb sf, f s`jfctf sf,  xxnn,, xx k k :: η  η nknk  pmrm t`g`s pmrm t`g`s   n,n, kk ∁ ∁ N N ..
K·K·m vnj`s c` Wf`rfjm 6.9.9 quf qumcg`m vnj`s c` Wf`rfjm 6.9.9 quf qumcg`  J  J  d`r uj suifspm± d`r uj suifspm±o` o`jpbft` gf uj fspm±o` o`jpbft` gf uj fspm±o`o`
o`j pr`gut` nctfrc`o`j pr`gut` nctfrc` F  F , t`g` vft`r, t`g` vft`r x x  gf gf  F  F  mgjntf ujm (· mgjntf ujm (·ucnom) jfba`r mpr`xnjm±ucnom) jfba`r mpr`xnjm±o˒om˒` fjm` fj
J J . @ rfsubtmg` sflunctf d`rcfof ujm gfsorn±. @ rfsubtmg` sflunctf d`rcfof ujm gfsorn±o˒om˒` gfssm jfba`r mpr`xnjm±m` gfssm jfba`r mpr`xnjm±o˒om˒` gfm` gf x x  fj uj fj uj
suifspm±suifspm±o` gf gnjfcs˒o` gf gnjfcsm˒` ﬍cntm gf uj fspm±m` ﬍cntm gf uj fspm±o` o`j pr`gut` nctfrc`.o` o`j pr`gut` nctfrc`.
Ur`p`sn±Ur`p`sn±o˒om˒ ̀m ̀ 6.6.?.?.??   _fkmj   _fkmj    F F    uj   uj fsfsppm±m±o` o` oo`j `j prpr̀̀ gugut` t` ncnctftfrcrc` ` f f    {{xx==, . . . , x, . . . , xcc}}    uj uj 
o`ckuct` `rt`c`rjmb ﬍cnt` fj o`ckuct` `rt`c`rjmb ﬍cnt` fj   F F ..
(m)(m)   _f _f   J J   : : XXxx==, . . . , x, . . . , xccZZ   f f   xx ∁∁ F F , fct˒ , fct˒ m`m`xx ∙∙
cc
nn:=:=
x,x, xxnnxxnn
 : gnst(: gnst(x,x, J J ))..
(i)(i)  Umrm t`g` Umrm t`g`   xx   fj fj   F F ,,
cc
nn:=:=
||x,x, xxnn||99 ≧≧ xx99..
Gfj`cstrm±Gfj`cstrm±oo˒˒mm`̀..  (m) Ufb` Wf`rfjm 6.9.6 p`gfj`s t`jmr (m) Ufb` Wf`rfjm 6.9.6 p`gfj`s t`jmr   pp ∁∁ J J    ff   q q  ∁ ∁ J J ⊦⊦  tmns quf tmns quf
xx : :  p p + + q q    ff    xx ∙∙ p p : gnst(: gnst(x,x, J J ))..
O`j`O`j`   pp ∁ ∁   J J , fxnstfj fsombmrfs, fxnstfj fsombmrfs    νν==, . . . , ν, . . . , νcc  tmns quf tmns quf   pp    ::   νν==xx==  ++ ¼¼ ¼¼ ¼¼ ++ ν νccxxcc. . F oF o`j`j`̀
xx ∙∙ p p : :  q  q  ∁ ∁ J J ⊦⊦, tfj`s (, tfj`s (xx ∙∙ p p)) ⊦⊦ xx k k  pmrm t`g` pmrm t`g`   kk   : : ==, . . . , c, . . . , c. _fluf fct˒. _fluf fctm˒` qufm` quf
5 5 :: xx ∙∙ p, p, xx k k :: x,x, xx k k ∙∙ νν k k ,,
1616
  
nnsstt ̀̀ ··ff,,  νν k k   :: x,x, xx k k, pmrm t`g`, pmrm t`g`   kk   : : ==, . . . , c, . . . , c. @ ntfj (m) fst·. @ ntfj (m) fst·m pr`vmg`.m pr`vmg`.
(i) Gmg`(i) Gmg`   xx ∁ ∁ F  F , sfkmj, sfkmj   νν==, . . . , ν, . . . , νcc  o`j` fj (m). Mssnj, o`j` o`j` fj (m). Mssnj, o`j`  xxnn,, xx k k : :  η  η nknk  pmrm t`g`s pmrm t`g`s
n,n, kk   : : ==, . . . , c, . . . , c,,
55 ≧≧

xx ∙∙
cc
nn:=:=
x,x, xxnnxxnn,, xx ∙∙
cc
nn:=:=
x,x, xxnnxxnn

:: xx99 ∙∙
cc
nn:=:=
||x,x, xxnn||99 ∙∙
cc
nn:=:=
||x,x, xxnn||99 ++
cc
nn:=:=
||x,x, xxnn||99 ,,
f ` rfsubtmg` sfluf.f ` rfsubtmg` sfluf.
@isfrvm±@isfrvm±o˒om˒ ̀m ̀ 6.6.?.?.77  M fxprfss˒ M fxprfssm˒`m`
cc
nn:=:=
x,x, xxnnxxnn  · ·f, f, p`p`r r j`tj`tnv`s nv`s ··̀ivn`s, oamjmgm gf`ivn`s, oamjmgm gf  jfba`r  jfba`r 
mpr`xnjm±mpr`xnjm±o˒ o˒ m`m`   gfgf   xx  fjfj   J J   : : XXxx==, . . . , x, . . . , xccZZ..
OmjncamOmjncamrfj`s ml`rm cm rfj`s ml`rm cm gnrf±gnrf±o˒om˒` gf fstmifbfofr gums rfbm±m` gf fstmifbfofr gums rfbm±o˒o˒̀fs ducgmjfctmns gm tf`rnm`fs ducgmjfctmns gm tf`rnm
g`s fspm±g`s fspm±o`s gf Anbifrt, m smifr, m gfsnlumbgmgf gf Ifssfb f m ngfctngmgf gf Umrsfvmb.o`s gf Anbifrt, m smifr, m gfsnlumbgmgf gf Ifssfb f m ngfctngmgf gf Umrsfvmb.
Umrm nss` prfonsmj`s g`;Umrm nss` prfonsmj`s g`;
Bfjm 6.?.6Bfjm 6.?.6   _fkm   _fkm    _ _    :: { {xxnn    ;;    nn ∁ ∁    N N }}   uj o`ckuct` `rt`c`rjmb