Saforftjbkl Hougcktjogs oh Borporktf Hjgkgbf
Saforftjbkl Hougcktjogs oh Borporktf Hjgkgbf
Eo˜ko ko Kdkro cf Dktos
Bopyrjiat © >==; ny ]rjgbftog Vgjvfrsjty ]rfss ]unljsafc ]unljsafc ny ]rjgbftog ]rjgbftog Vgjvfrsjty Vgjvfrsjty ]rfss, 3; Yjlljkd Yjlljkd Rtrfft, Rtrfft, ]rjgbftog, Gfw Efrsfy =163= Jg taf Vgjtfc @jgicod5 ]rjgbftog Vgjvfrsjty ]rfss, 2 Dkr`ft ]lkbf, Yoocstob`, Oxhorcsajrf OQ>= ;RZ Kll ^jiats ^fsfrvfc Ljnrkry oh Bogirfss Bktkloijgi-jg-]unljbktjog Cktk
Kdkro cf Dktos, Eo˜ko, ko, ;:8;Saforftjbkl hougcktjogs oh borporktf gkgbf/Eo˜ko ko Kdkro cf Dktos. p. bd. Jgblucfs njnljoirkpajbkl rfhfrfgbfs kgc jgcfx. JRNG =-8:;-=19:3-8 (kl`. pkpfr) ;. Borporktjogs–Hjgkgbf. J. Sjtlf. AI3=;3 .K3 >==; 221.8 =3;–cb>; >==;=2>;=3
Nrjtjsa Ljnrkry Bktkloijgi-jg-]unljbktjog Cktk js kvkjlknlf Sajs noo` aks nffg bodposfc jg Gfw Nks`frvjllf ∗ ]rjgtfc og kbjc-hrff pkpfr.
www.pup.prjgbftog.fcu ]rjgtfc jg taf Vgjtfc Rtktfs oh Kdfrjbk ; 2
6 9
: ;= ;= 1
8 3
>
Bogtfgts
Ljst oh Hjiurfs ]rfhkbf =.; =.> =.2 ]krt J
xj
Jgtfgcfc Ku Kucjfgbf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Orikgjzktjog oh oh ta taf Sf Sfxt . . . . . . . . . . . . . . . . . . . . . . . . . Kb`gowlfcidfgts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hougcktjogs
; Xkluktjog ;.; Xkluktjog ug ugcf cfrr Bf Bfrtkjgty . . . . . . . . . . . . . . . . . . . . . . . . . ;.;. ;.;.; ; Saf ^onjg onjgso sog g Bru Brussof Fbogo bogody. dy. . . . . . . . . . . . . . . ;.;.> Sjdf ]r ]rfhfrfgbfs . . . . . . . . . . . . . . . . . . . . . . . . . . ;.;. ;.;.2 2 ]roc ]rocub ubttjog jog Opp Oppor ortu tugj gjtj tjfs fs . . . . . . . . . . . . . . . . . . . ;.;.3 Saf ^o ^olf oh oh Bk Bkpjtkl Dk Dkr`fts . . . . . . . . . . . . . . . . . ;.;.6 ;.;.6 Bogsud Bogsudptj ptjog og kgc Jgvfstd Jgvfstdfgt fgt wjta wjta Bkpjt kpjtkl kl Dkr Dkr`f `ftts . . . . . . . . . . . . . . . . . . . . . . . . . . . ;.;. ;.;.8 8 Saf Saf Xkl Xkluf uf oh oh kg kg Jgvf Jgvfst stdf dfgt gt ]ro ]roef efbt bt . . . . . . . . . . . . ;.;. ;.;.9 9 Du Dult ltjp jpfr frjo joc c Fbo Fbogo gody dy wjta wjta Bkpj Bkpjtk tkll Dkr` Dkr`ft ftss . . . . . . ;.;. ;.;.1 1 Fxfrb xfrbjjsfs sfs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ;.> Xkluktjog ug ugcf cfrr Vgb Vgbffrtkjgty . . . . . . . . . . . . . . . . . . . . . . ;.>. .>.; Ogf-] gf-]fr frjjoc Docf Docfll . . . . . . . . . . . . . . . . . . . . . . . . . ;.>. .>.> Xkluf kluf kgc taf Knsfg nsfgbf bf oh Krnj rnjtrki trkiff . . . . . . . . . . . ;.>. ;.>.2 2 Krnj Krnjtr trki kiff Opp Oppor ortu tugj gjtj tjfs fs kgc kgc Jgv Jgvfs fstd tdfg fgtt . . . . . . . . ;.>. .>.3 Xkluf kluf kgc taf Dkrt Dkrtjjgikl giklff D Dfk fkssurf urf . . . . . . . . . . . . ;.>. .>.6 Nftk ftk Xkl Xkluf ufss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ;.>. .>.8 Fxfrb xfrbjs jsfs. fs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ;.2 ;.2 Dult Du ltjp jpfr frjo jocs cs kgc kgc Hlfx Hlfxjn jnjl jljt jtyy ugcf ugcfrr Vgbf Vgbfrt rtkj kjgt gtyy . . . . . . . .
xjjj xjv xv xvj ; 2 6 6 9 1 :
;; ;3 ;6 ;9 ;1 ;1 >> >2 >3 >8 >1 >:
;3
;. Xkluktjog
Hjiurf ;.3. Saf ^fijog oh Kcdjssjnlf Kcdjssjnlf Bogsudptjog Bogsudptjogss jg k ^onjgsog ^onjgsog Brusof Fbogody wjta kg Jgvfstdfgt Rbafculf kgc Bkpjtkl Dkr`fts
;.;.8 ;.;.8 Saf Xkluf Xkluf oh kg Jgvfstdfgt Jgvfstdfgt ]roefbt ]roefbt Jg kg fkrljfr sunsfbtjog taf bogsudptjog pronlfd wks kgklyzfc jg taf knsfgbf oh kg jgvfstdfgt sbafculf, lfkcjgi to kg optjdkl bogsudptjog kt t 4 ; fxprfssfc ny yr (x). Ogf dky wogcfr wafrf woulc yr (x) nf plkbfc jg taf sft jgfqukljtjfs (;.1). Hord Hordkl klly ly taf taf pron pronlf lfd d dky dky nf solv solvfc fc vfry vfry sjdp sjdply ly ny gotj gotjbj bjgi gi takt takt taf taf bogs bogsud udpt ptjo jog g cf cfbj bjsj sjog og js nksf nksfc c og taf taf optj optjdj djzk zktj tjog og oh taf taf utjl utjl-jty V(xyh (x)), jg taf taf bksf bksf oh cf cftf tfrd rdjg jgjg jgi i yh (x), kgc kgc og dkxj dkxjdj djzj zjgi gi V(xyr (x)) jg taf bksf oh yr (x). Bodpkrjgi yr (x) wjta yh (x), ogf gotjbfs takt taf cjhhfrfgbf nftwffg tafd js fxprfssfc ks
ϖ∞; (r) (r )
yh (x) ∞ yr (x) 4
=
Pϖ(J) ∞ r TcJ
jh tafrf krf proefbts dorf jgtfrfstjgi takg taf gkgbjkl dkr`fts, or ks zfro otafrwjsf. otafrwjsf. Saf jdportkgt jdportkgt tajgi js takt tajs cjhhfrfgbf cjhhfrfgbf js jgcfpfgcfgt jgcfpfgcfgt x kgc y . Saus, taf jgtrocubtjog oh kg jgvfstdfgt oh taf optjdkl baojbf oh x wor`s ks jh taf huturf fgcowdfgt Z wfrf sajhtfc ny
4
=
ϖ∞; (r) (r )
=
Pϖ(J) ∞ r TcJ
r ≦ ϖ(=) r ≣ ϖ(=)
Sajs dfkgs takt taf jgtrocubtjog oh taf jgvfstdfgt sbafculf js fqujv-
;6
;.;. Xkluktjog ugcfr Bfrtkjgty Bfrtkjgty
Ijvfg kg jgvfstdfgt sbafculf, jts gft prfsfgt vkluf (G]X) (G]X) js cfgfc ks taf jgbrfksf jg taf burrfgt wfklta ifgfrktfc ny taf jgvfstdfgt. Jg tajs bksf, hor taf ijvfg jgvfstdfgt sbafculf ϖ, G]X (ϖ) (ϖ) 4
; + r
Gotjbf takt jh taf jgvfstdfgt sbafculf cofs got kllow rfturg rktfs irfktfr takg r , jts G]X js gull. Jg otafr worcs, taf borrfspogcjgi jgvfstdfgts krf rktjogklly jigorfc. Jt hollows takt ]roposjtjog ;.;.>. Jt js rktjogkl to jgvfst jg k proefbt jh jts gft prfsfgt vkluf js posjtjvf.
Sajs js sjdply kgotafr wky oh skyjgi takt taf optjdkl lfvfl oh jgvfstdfgt js ∙ . Ny bogstrubtjog, tajs js klso fqujvklfgt to skyjgi takt ogf saoulc jgvfst jg k proefbt jh jts dkrijgkl rktf oh rfturg js kt lfkst fqukl to r . K borollkry oh tajs proposjtjog js takt taf optjdkl lfvfl oh jgvfstdfgt js taf ogf takt dkxjdjzfs taf prfsfgt wfklta oh taf kifgts or, jg otafr worcs, js taf lfvfl oh jgvfstdfgts takt dkxjdjzfs tafjr bkpkbjty to ifgfrktf (Q Z ) hor nota pfrjocs. ajiafr lfvfls oh bogsudptjog hrod taf pkjr (Q Borollkry Borollkry ;.;.;. ;.;.;. Saf optjdkl jgvfstdfgt cfbjsjog js taf ogf takt dkxjdjzfs taf burrfgt wfklta oh taf kifgts.
;.;.9 ;.;.9 Dultjpfrjoc Dultjpfrjoc Fbogody Fbogody wjta Bkpjtkl Bkpjtkl Dkr`fts Dkr`fts Safrf js go dystfry jg aow to ifgfrkljzf tafsf jcfks to k dultjpfrjoc fbogody. Rupposf gow takt tafrf js k two-pfrjoc fbogody wjta bkpjtkl dkr`fts toiftafr toiftafr wjta procubtjog procubtjog krjsjgi krjsjgi hrod k ijvfg jgvfstdfgt jgvfstdfgt sbafculf. Jgtrocubjgi z ks taf bogsudptjog kt tjdf t 4 > kgc U ks taf fgcowdfgt kt takt pojgt jg tjdf, taf hordkljsd hollows taf fkrljfr sfbtjogs. Lft = kgc ; nf taf kdougts jgvfstfc kt tjdfs t 4 = kgc t 4 ; rfspfbtjvfly. Saf bogsudptjog pronlfd saoulc kbtuklly nf solvfc ks hollows5 Kgkloiously to fquktjog (;.9), ogf dky cfgf taf pkjr oh hugbtjogs
h(xy) 4 x + y ∞ Q ∞ Z ∞
=
=
kgc
i(yz) 4 y + z ∞ Z ∞ U ∞
;
=
ϖ(J)cJ + r P= ∞ (Q ∞ x)T ϖ(J)cJ + r P; ∞ (Z ∞ y)T
Safg taf pronlfd js to dkxjdjzf taf utjljty V (x y z) sunefbt to taf possjnjljtjfs kllowfc ny taf procubtjog sbafculf, cfsbrjnfc ks h(xy) 4 = kgc i(yz) 4 =. Saf Saf optj optjdk dkll solu solutj tjog og js houg hougc c ny bogs bogstr trub ubtj tjgi gi taf taf Lkirkgifkg
;8
;. Xkluktjog
kgc tafg wor`jgi out taf rst-orcfr bogcjtjogs wjta rfspfbt to x
V x 4 ΰh x 4 ΰ( ; + r ) wjta rfspfbt to y
V y 4 ΰh y + βi y 4 ΰ + β(; + r ) kgc, gklly, wjta rfspfbt to z V z 4 βi z 4 β Safsf rst-orcfr bogcjtjogs lfkc to V x + V z (; + r ) V y 4 ; + r wjta taf optjdkl jgvfstdfgt kt fkba pfrjoc nfjgi
∙= 4 ∙; 4
r ≦ ϖ(=) = ∞; ϖ (r ) ϖ(=) ≦ r
Sajs rfsult bogrds taf Hjsafr sfpkrktjog prjgbjplf ogbf dorf. Gow, sjgbf taf jgvfstdfgt poljby js eust taf skdf ks jg taf bksf oh k ogf-pfrjoc fbogody, jt hollows takt taf jgvfstdfgt sbafculf jgbrfksfs taf wfklta oh taf jgvfstor jg fkba pfrjoc ny
4
=
ϖ∞; (r) (r )
=
Pϖ(J) ∞ r TcJ
r ≦ ϖ(=) ϖ(=) ≦ r
Safrfhorf, kt tjdf t 4 = taf jgvfstor dky ks` taf nkg` hor kg kccjtjogkl kdougt oh /(; + r ) to nf pkjc kt t 4 ; wjta taf rfsults hrod taf jgjtjkl jgvfstdfgt plus /( ; + r )> to nf pkjc kt t 4 > wjta taf rfsults hrod taf jgvfstdfgt kt tjdf t 4 ;. Safrfhorf, taf prfsfgt gft vkluf oh taf proefbt js kikjg wrjttfg ks taf jgbrfksf jg wfklta kt tjdf = ifgfrktfc ny taf jgvfstdfgt, + G]X S 4 > (; + r ) (; + r ) Saf vkluktjog prjgbjplf cfvflopfc jg tajs sfbtjog wor`s hor kg krnjtrkry gudnfr oh pfrjocs. Jg pkrtjbulkr, jigorjgi taf prfsfgbf oh procubtjog sbafculfs, lft Q◨ Q; Qg cfgotf taf fgcowdfgts or sjdply k sfqufgbf oh bksa ows kt tjdfs = ; g, rfspfbtjvfly. Jh r r js taf jgtfrfst rktf pfr pfrjoc, taf ljgfs oh bogstkgt wfklta (taf dkr`ft ljgfs ks prfvjously cfgfc) nfbodf aypfrplkgfs wjta taf fquktjog hor taf prfsfgt vkluf
] X 4 Q ◨ +
Q; Q> Qg + + ¹ ¹ ¹ + (; + r )g ; + r (; + r )>
;9
;.;. Xkluktjog ugcfr Bfrtkjgty Bfrtkjgty
Sajs fxprfssjog dky nf fksjly ifgfrkljzfc hor taf bksf wafrf tafrf krf cjhhfrfgt dkr`ft jgtfrfst rktfs jg cjhhfrfgt tjdf pfrjocs. Lft r r t cfgotf taf bogstkgt jgtfrfst rktf nftwffg tjdfs t kgc t + ;. Jg tajs bksf, taf fquktjog hor taf dkr`ft ljgfs js kikjg k aypfrplkgf fxprfssfc ks
] X 4 Q ◨ +
Q; Q> Q + + ¹ ¹ ¹ + g∞; g ; + r ◨ (; + r ◨ )(; + r ; ) e4= (; + r e ) g
4 Q ◨ +
j4;
Qj ej∞4;= (; + r e )
;.;.1 ;.;.1 Fxfrbjsf Fxfrbjsfs s ;. Bogsjcfr Bogsjcfr k ^onjgsog Brusof Brusof fbogody jg ogf pfrjoc wjtaout kg jgvfstjgvfstdfgt sbafculf. Raow takt x∙◨ js ijvfg ny taf fxprfssjog jg taf tfxt. >. Bogsjcfr Bogsjcfr k ^onjgsog Brusof Brusof fbogody jg ogf pfrjoc wjtaout kg jgvfstjgvfst> > dfgt sbafculf. ]rovf kgc jgtfrprft taf hkbt takt c y/cx js posjtjvf kt taf optjdkl pojgt (x∙◨ Z + Q ∞ x∙◨ ) sktjshyjgi taf fqujvklfgt oh fquktjog (;.2) jg taf prfsfgbf oh tjdf prfhfrfgbf. 2. Bogsjc Bogsjcfr fr k ^onjgs ^onjgsog og Brusof Brusof fbogody fbogody jg ogf pfrjoc pfrjoc wjta wjta kg jgvfst jgvfst-dfgt sbafculf. Raow takt yp (x) js cfbrfksjgi jg x wjta k gfiktjvf sfbogc cfrjvktjvf. 3. Raow Raow takt takt taf optjdkl optjdkl vkluf oh taf jgvfstdf jgvfstdfgt gt jg gkgbj gkgbjkl kl dkr`ft dkr`ftss js ijvfg ny taf fxprfssjog hor ∙ jg taf tfxt. 6. ]rovf Borollk Borollkry ry ;.;.;. 8. Raow Raow aow to ift taf fxprfss fxprfssjog jog hor yh (x). 9. Raow taf jgfqukljtjfs jgfqukljtjfs jg rflktjog rflktjog (;.1). 1. ]rovf ]roposjtj ]roposjtjog og ;.;.>. :. Bogsjcfr kg fbogody wjta two kifgts, K kgc N, wao ljvf jg k ogfpfrjoc fbogody cfgfc ny two pojgts jg tjdf. Saf prfhfrfgbfs oh (x y) 4 x> y kgc kifgts K kgc N krf rfprfsfgtfc rfspfbtjvfly ny V K (x V N (xy) 4 xy> , wafrf x js taf lfvfl oh bogsudptjog gow kgc y js taf lfvfl oh bogsudptjog jg taf huturf. Safy krf ijvfg fqukl fgcowdfgts. Fkba ogf rfbfjvfs ;== gow kgc 31.38 jg taf huturf. Nota kifgts akvf taf possjnjljty oh dk`jgi procubtjvf jgvfstdfgts wjta rfturg ijvfg ny ϖ(J) 4 ∞J + ;=. (k) Yakt js taf optjdkl lfvfl lfvfl oh jgvfstdfgt jgvfstdfgt ny fkba kifgt? (n) Js takt cfpfgcfgt cfpfgcfgt og taf prfhfrfgbfs prfhfrfgbfs oh fkba kifgt? (b) Aow duba duba cofs fkba ogf wjsa to bogsud bogsudff gow ks k hugbtj hugbtjog og oh taf jgtfrfst rktf?
;1
;. Xkluktjog
;.> Xkluktjog kluktjog ugcfr Vgbfrtkjgty Vgbfrtkjgty Saf prjgbjplfs klrfkcy cfvflopfc dky wfll nf fxtfgcfc to sjtuktjogs jg wajba taf huturf outbodf js got bfrtkjg. Rupposf takt ^onjgsog Brusof gow aks k nugclf oh jgvfstdfgt proefbts kvkjlknlf to ajd, nut af js got surf oh taf outbodf oh kgy oh tafd. Sajs sfbtjog kccrfssfs aow tajs ugbfrtkjgty khhfbts ajs optjdkl cfbjsjog oh bogsudptjog kgc jgvfstdfgt. Jg otafr worcs, bogsjcfrjgi takt tafrf krf sfvfrkl kvkjlknlf proefbts, kgc takt fkba oh tafd rfqujrfs kg jgjtjkl jgvfstdfgt J J j kgc aks kg jgtrjgsjb prfsfgt vkluf oh X j , tajs sfbtjog wjll cfsbrjnf bogcjtjogs ugcfr wajba taf cfbjsjog knout waftafr or got to jgvfst cfpfgcs ogly og waftafr or got X j js irfktfr takg J j . Jg orcfr to co takt ogf dust `gow aow to propfrly bakrkbtfrjzf taf vkluf X j oh fkba proefbt, ijvfg taf ugbfrtkjgty ugcfrlyjgi tafjr pkyohhs. Sajs js taf dkjg purposf oh tajs sfbtjog. Jt wjll nf kssudfc takt gkgbjkl dkr`fts krf kvkjlknlf jg taf bogtfxt oh kg fxbakgif fbogody. Jg takt sfttjgi bogcjtjogs wjll nf fstknljsafc so takt taf kgklysjs ugcfr ugbfrtkjgty dky nf hobusfc strjbtly og taf vkluktjog probfss oh taf proefbts kgc got og taf bogsudptjog pkta.
;.>.; ;.>.; Ogf-]frjoc Ogf-]frjoc Docfl Docfl Saf fxprfssjog huturf ugbfrtkjgty jg k ogf-pfrjoc fbogody dfkgs takt sodfogf jgvfstjgi kt tjdf t 4 = js got surf oh taf dkr`ft bogcjtjogs kt t 4 ;. Nota taf fgcowdfgt kgc taf jgvfstdfgt‚s pkyohh kt takt pojgt jg tjdf cfpf cf pfgc gc og taf taf pkrt pkrtjb jbul ulkr kr bogc bogcjt jtjo jogs gs oh gktu gkturf rf kt t 4 ;. Vgbfrtkjgt Vgbfrtkjgty y js taus cfsbrjnfc ny k sft oh stktfs nflogijgi to k gjtf skdplf spkbf 4 {w; w> w@ }, wafrf fkba stktf we obburs wjta pronknjljty p(we ), cfgotfc hor sjdpljbjty ks p e . Klso, jh x ∁ ^@ js k rkgcod vkrjknlf jg , lft @
F(x) ≫
xe p e
e 4;
cfgotf taf fxpfbtktjog oh x ugcfr taf pronknjljty dfksurf p. Saf gotjog oh utjljty dust nf rfgfc ugcfr ugbfrtkjgty. Jg Rfbtjog ;.;, utjljty wks jgtrocubfc ks k rfkl hugbtjog V(xy), jgbrfksjgi kgc cjhhfrfgtjknlf jg taf prfsfgt bogsudptjog x kgc jg taf huturf bogsudptjog y . Afrf, aowfvfr, huturf bogsudptjog js k rkgcod vkrjknlf. Lft y ∁ ^@ cfgotf k bogsudptjog plkg (j.f., taf vfbtor oh possjnlf bogsudptjogs kt tjdf t 4 ;), wjta taf e -ta bodpogfgt ye ≫ y(we ) cfgotjgi bogsudptjog jg stktf e. Jg tajs sfttjgi utjljty dust nf k rfkl hugbtjog V ∙ (xy) cfgfc og ^ @+; , kllowjgi us to cfgf prfhfrfgbfs ovfr cjhhfrfgt plkgs oh bogsudptjog. Bodpkrjgi two cjhhfrfgt bogsudptjog plkgs, y kgc z, taf utjljty hugbtjog V ∙ oh kg jgcjvjcukl dust nf bogstrubtfc jg k wky suba takt
;:
;.>. Xkluktjog ugcfr Vgbfrtkjgty Vgbfrtkjgty
jh tajs jgcjvjcukl prfhfrs plkg y to plkg z. Jg dorf hordkl tfrds, taf jgcj vjcukl‚s prfhfrfgbfs krf skjc to akvf k utjljty rfprfsfgtktjog jh tafrf js k hugbtjog V ∙ wjta taf prfbfcjgi propfrty. propfrty. K boddogly kbbfptfc utjljty rfprfsfgtktjog js taf vog Gfudkgg-Dorifgstfrg utjljty hugbtjog , @
∙
V (x (x y) 4 FV(xy) 4
V(xye )pe
e 4;
klso `gowg ks taf fxpfbtfc utjljty hugbtjog . Gotjbf takt ugcfr k bfrtkjg bog- sudptjog plkg , wafrf bogsudptjog js k surf tajgi or, jg otafr worcs, ye js (x y) 4 V(x`) kgc taf utjljty hugbtjog k bogstkgt ` jgcfpfgcfgt oh e, V ∙ (x ugcfr bfrtkjgty js rfbovfrfc. Saf bfrtkjgty fqujvklfgt oh kg ugbfrtkjg bogsudptjog plkg y bkg nf cfgfc ks taf bfrtkjg bogsudptjog plkg β wjta wjta taf skdf fxpfbtfc utjljty (x y) 4 V(xβ). ks y. Jg otafr worcs, V ∙ (x Kg kifgt js skjc to nf rjs`-kvfrsf jh, hor kgy bogsudptjog plkg y, taf bfrtkjgty fqujvklfgt β js js sdkllfr takg F(y) or
FV(x FV(x y) < V PxF(y) T Jt hollows takt taf utjljty hugbtjog V dust nf bogbkvf. Jh, og taf otafr akgc, kg kifgt aks k utjljty suba takt taf bfrtkjgty fqujvklfgt oh k rjs`y bogsudptjog plkg js irfktfr takg taf fxpfbtfc huturf bogsudptjog, taf kifgt js skjc to nf rjs`-sff`jgi kgc kgc V dust nf bogvfx. Hjgklly, rjs`-gfutrkljty bakrkbtfrjzfs kg kifgt takt js jgcjhhfrfgt nftwffg k bfrtkjg bogsudptjog plkg kgc k rjs`y plkg, provjcfc takt taf fxpfbtfc bogsudptjog bojgbjcfs wjta taf bfrtkjg bogsudptjog. Jg tajs bksf, taf utjljty hugbtjog V dust nf ljgfkr. Gow Gow, bogs bogsjc jcfr fr taf taf bogs bogsud udpt ptjo jog g pron pronlf lfd d jg k ogfogf-pf pfrj rjoc oc sftt sfttjg jgi i ugcfr ugbfrtkjgty. ^fikrcjgi taf fgcowdfgts, cfgf Z e ≫ Z (we ) ks taf t 4 ; jg stktf e 4 ; > @. Klso, jgstfkc oh akvfgcowdfgt rfbfjvfc kt t jgi ogf sjgilf proefbt to bogsjcfr hor jgvfstdfgt, kssudf takt taf jgvfstor hkbfs k nugclf oh G cjhhfrfgt ogf-pfrjoc proefbts. Fkba oh tafsf proefbts aks kg jgjtjkl vkluf X j , j 4 ; G , kgc kt tjdf t 4 ; pkys j (we ) jg stktf e 4 ; > @. Hor sjdpljbjty oh gotktjog, lft j (we ) nf cfgotfc ny je . Safsf vklufs cfgf k vkluf vfbtor X ∁ ^G kgc k pkyohh G Ù @ dktrjx , ijvfg ny ;; ;> ¹ ¹ ¹ ;@ >; >> ¹ ¹ ¹ >@ 4 ( ; > @ ) 4
;;>
3. ]kyout ]kyout ]oljby ]oljby
Sajs Sajs js taf taf vklu vkluff oh taf taf rd rd ogbf ogbf taf taf cjvj cjvjcf cfgc gcs, s, fkrg fkrgjg jgis is,, kgc kgc gkgbjgi cfbjsjogs krf kggougbfc, provjcfc takt taf optjdkl jgvfstdfgt cfbjsj cfbjsjog og bkg nf fvkluk fvkluktfc tfc ny fvf fvfryno rynocy cy hrod hrod taf rst-o rst-orcf rcfrr bogcjt bogcjtjog jog jg (3.;;) (3.;;).. Nut nfhorf nfhorf tafsf tafsf kggoug kggougbfd bfdfgt fgts, s, taf trkcfr trkcfrss cfvflo cfvflop p tafjr tafjr owg fxpfbtktjogs oh taf vkluf oh taf rd, F◨ (X ; ), bogcjtjogfc ny taf jghordktjog tafy akvf up to tjdf =. Saf cjhhfrfgbf nftwffg taf vkluf khtfr taf kggougbfdfgt kgc taf fxpfbtfc vkluf nfhorf taf cjsblosurf oh jghordktjog js taf ugfxpfbtfc vkluf , to nf bklbulktfc ks X ; ∞ F◨ (X ; ). Saf prf-kggougbfdfgt vkluf js tafrfhorf
F◨ (X ; ) 4 F ◨ Q; ∞ J ; +
;
; F◨ PH (J ; )T 4 H (J ◨ ) ∞ J ;∙ + H (J ;∙ ) ; + r ; + r
wafrf taf lkst ljgf hollows hrod taf hkbt takt taf kifgts `gow taf optjdkl jgvfstdfgt lfvfl hrod (3.;;). Runtrkbtjgi tajs fquktjog hrod (3.;>), jt hollows takt taf ugfxpfbtfc vkluf js
X ; ∞ F◨ (X ; ) 4 ; ; +
β
; + r
(3.;2)
Ks js blfkr hrod tajs sjdplf bklbulktjog, taf vkrjktjog jg taf vkluf oh taf rd (kgc tafrfhorf jg taf prjbf oh taf sakrfs) fxjsts, nfbkusf wafg bodputjgi F ◨ (X ; ) taf vkluf oh Q; js sunstjtutfc hor ny jts fxpfbtfc vkluf, wafrfks X ; js fvkluktfc wjta taf prfbjsf Q; . Safg, tajs fhhfbt bkg nf sffg ks rfsultjgi hrod taf kggougbfdfgt oh fkrgjgis. Hrod
Qt 4 J t∙ + (Ct ∞ Nt )
(3.;3)
gotf takt cjsblosjgi Q t bogvfys fxkbtly taf skdf jghordktjog ks cjsblosjgi taf gft cjvjcfgcs Ct ∞ Nt , sjgbf taf optjdkl jgvfstdfgt cfbjsjog js ijvfg ny taf rst-orcfr bogcjtjog jg (3.;;). Saus, tajs docfl prfcjbts takt taf cjsblosurf oh fkrgjgis js fqujvklfgt to taf cjsblosurf oh gft cjvjcfgcs kgc kgy ogf oh tafsf kggougbfdfgts (or fvfg k sjdultkgfous kggougbfdfgt) provjcfs taf skdf prjbf vkrjktjog ijvfg afrf. Hrod tajs kgklysjs jt turgs out takt taf purf cjvjcfgc js got fgouia to horfbkst Qt fxkbtly. Jt wks saowg takt taf gft cjvjcfgc cjvjcfgc boulc co takt eon, nut got Ct . Ijvfg taf kggougbfdfgt oh taf cjvjcfgcs, taf lfvfl oh fxtfrgkl gkgbjgi N t js gfbfsskry jg orcfr to cjsblosf taf truf vkluf oh Q t kgc tafrfhorf oh X t . Afgbf, tafrf dky nf k gkgbjgi kggougbfdfgt fhhfbt ks wfll. Sajs ljgf oh rfksogjgi suiifsts takt taf kggougbfdfgt oh cjvjcfgcs dust nf hollowfc ny kg kggougbfdfgt oh fjtafr fkrgjgis or gkgbjgi
;;2
3.>. Cjvjcfgcs Cjvjcfgcs kgc Jghordktjog Jghordktjog
K Yfk`gfss oh taf Docfl Saf Saf docf docfll cf cfvf vflo lopf pfc c so hkr hkr, aowf aowfvf vfrr, cofs cofs got got yjfl yjflc c k bogs bogsjs jstf tfgt gt fqujljnrjud, sjgbf jt provjcfs taf possjnjljty oh dk`jgi k prot hrod taf jgsjcf jghordktjog takt taf optjdkl jgvfstdfgt lfvfl js ijvfg ny (3.;;). So sff sff aow aow sjdp sjdplf lf jt js to cf cfvj vjkt ktff hrod hrod taf taf orji orjijg jgkl kl fquj fqujlj ljnr nrju jud, d, supposf takt tafrf js k bfrtkjg hrkbtjog oh sakrfaolcfrs (sky, `) plkggjgi to sfll tafjr sakrfs nftwffg taf gft cjvjcfgcs kggougbfdfgt kgc taf fkrgjgis kggougbfdfgt. Safy boulc nf jgcubfc to pky taf dkgkifdfgt to jgbrfksf taf cjvjcfgcs kgc but taf jgvfstdfgt lfvfl nflow taf optjdkl lfvfl J t∙ . Jg tajs bksf, taf jghordktjog bogvfyfc ny taf gft cjvjcfgcs js got fqujvklfgt to taf jghordktjog rfvfklfc ny taf fkrgjgis, sjgbf ny bogstrubtjog taf jgvfstdfgt lfvfl js go logifr xfc kt taf jcfkl lfvfl. Nut taf sakrfaolcfrs wao co got jgtfgc to sfll co got `gow takt. Safy tajg` takt taf jgvfstdfgt kt tjdf ; js stjll J ;∙ . Gotjbf takt taf cjhhfrfgbf nftwffg taf gft cjvjcfgc pkjc kgc taf fxpfbtfc gft cjvjcfgc js ; ,
(C; ∞ N; ) ∞ F◨ (C; ∞ N; ) 4 Q; ∞ H (J ◨ ) 4 ; kgc tafg, hrod (3.;2), hor fkba ugjt jg tajs cjhhfrfgbf taf ikjg jg vkluf hor taf sfllfrs js
` ;+
β
; + r
wajba js posjtjvf hor β ≦ =. So fvkluktf taf lossfs to taf hrkbtjog ; ∞ ` oh stkyfrs, eust usf (3.;3) jg taf fxprfssjog hor taf vkluf oh taf rd jg (3.;>) to ift kt t 4 ;
X t 4 Ct ∞ Nt +
;
PH (J t∙ ) + β t T ; + r
K bakgif jg taf cjvjcfgcs oh cCt kt taf fxpfgsf oh k but oh taf skdf dkigjtucf jg taf jgvfstdfgts ifgfrktfs k bakgif jg taf vkluf oh taf rd oh dkigjtucf cX t ,
X t + cX t 4 Ct + cCt ∞ Nt +
;
PH (J t∙ ∞ cCt ) + β t T ; + r
Runtrkbtjgi tajs lkst fquktjog hrod taf hordfr, jt hollows takt
cX t 4 cC t +
;
; H (J t∙ )cCt PH (J t∙ ∞ cCt ) ∞ H (J t∙ )T 4 cCt ∞ ; + r ; + r
;;3
3. ]kyout ]kyout ]oljby ]oljby
Afgbf, tafrf js go loss hor taf stkyfrs jg tajs sjtuktjog. Hrod tajs sjdplf cfrjvktjog jt js sffg takt taf possjnlf lossfs hrod taf cfbrfksf jg taf jgvfstdfgt lfvfl krf fxkbtly nklkgbfc ny taf nfgft oh k irfktfr cjvjcfgc. Saf bogblusjog js takt tafrf js go jgbfgtjvf hor tafd to try to fxfrbjsf kgy bougtfrprfssurf rfikrcjgi taf jgbfgtjvfs oh potfgtjkl sfllfrs. Saf docfl ks jt js cofs got sustkjg taf optjdkl brjtfrjog hor jgvfstdfgt. Hor taf sjigkls to akvf k blfkr rolf, sodf oh taf dfbakgjsds jg tajs docfl dust bakgif.
3.>.2 3.>.2 K Bogsjstfgt Bogsjstfgt Rjigkljgi Rjigkljgi Docfl Saf prfbfcjgi lkb` oh fqujljnrjud aks jts root jg taf hkbt takt sfllfrs kgc stkyfrs akc cjhhfrfgt vkluktjogs hor taf rd. Saf rd, wajba js supposfc to dkxjdjzf taf vkluf to taf stkyfrs, boulc nf jgcubfc to ugcfrjgvfst jg orcfr to nfgft taosf sakrfaolcfrs wao jgtfgc to sfll. Ks wks saowg, jt js got bogvjgbjgi to kriuf takt dkgkifrs try to dkxjdjzf taf vkluf oh taf stkyfrs. Sakt js taf rfksog way takt onefbtjvf cofs got lfkc to kg fqujljnrjud. So nf stknlf, kg fqujljnrjud dust rffbt taf dkxjdjzktjog oh k ljgfkr bodnjgktjog oh taf two cjhhfrfgt vklufs oh taf rd. Saf bost oh suba fqujljnrjud wjll blfkrly nf k bfrtkjg lfvfl oh ugcfrjgvfstdfgt. Ct gow cfgotf taf gft cjvjcfgc. Bogsjcfr So sjdpljhy taf gotktjog, lft C wakt akppfgs kt tjdf t 4 ;. Saf vkluf to taf stkyfrs, (sky, X t; ) js stjll ijvfg ny (3.;>),
X t; 4 Ct +
;
PH (J t ) + β t T ; + r
(3.;6)
wafrfks taf vkluf to taf sfllfrs js taf fxpfbtktjog oh takt fxprfssjog bogc bogcjt jtjo jogf gfc c ny taf taf `gow `gowlf lfci ciff oh taf taf pkst pkst cktk cktk plus plus taf taf kggo kggoug ugbf bfc c cjvjcfgc,
X t> 4 Ct +
;
F PH (J t ) + β ˜t T ; + r ◨
Saf onefbtjvf oh taf rd js cfsbrjnfc ks to dkxjdjzf k bogvfx ljgfkr bodnjgktjog oh X t; kgc X t> –gkdfly,
λX t; + (; ∞ λ)X t> Yafg tafrf krf go potfgtjkl sfllfrs (` 4 =), taf borrfbt hugbtjog to dkxjdjzf js X t; , kgc tafrfhorf λ js tk`fg to nf ; ∞ `. Saf optjdjzktjog proirkd takt hollows js
;;6
3.>. Cjvjcfgcs Cjvjcfgcs kgc Jghordktjog Jghordktjog
sunefbt to taf nucift bogstrkjgt Qt 4 J t + Ct . Runstjtutjgi tajs bogstrkjgt jg taf dkxjdjzktjog pronlfd, taf optjdjzktjog nfbodfs ugjcjdfgsjogkl. Dorfovfr, gotjbjgi takt
X t; ≫ X t; (Qt Ct ) kgc
X t> ≫ X t> (Ct ) taf pronlfd nfbodfs dkx(; ∞ `)X t; (Qt Ct ) + `X t> (Ct ) Ct
Gotjbf takt ijvfg Qt , tajs dkxjdjzktjog trjfs to gc kg optjdkl Ct∙ . Saf rfvfrsf dkppjgi js Qt (Ct ), kgc jh jt js sjgilf-vklufc, taf sjigkl bog vfyfc ny taf optjdkl lfvfl oh cjvjcfgcs saoulc nf bogsjstfgt jg taf sfgsf takt taf vkluktjog oh taf sfllfrs saoulc bojgbjcf wjta taf vkluktjog oh taf stkyfrs,
X t;
Qt (Ct ) Ct 4 X t> (Ct )
Saf optjdkl cjvjcfgc dust sktjshy taf rst-orcfr bogcjtjog ontkjgfc ny sunstjtutjgi tajs bogsjstfgby fxprfssjog jgto taf prfbfcjgi onefbtjvf hugbtjog kgc cjhhfrfgtjktjgi wjta rfspfbt to Ct . Cojgi takt, jt hollows takt
`∎x X t;
Qt (Ct ) Ct )Qt (Ct +
∎c X t;
Qt (Ct ) Ct 4 =
Ks k nougckry bogcjtjog to solvf tajs fquktjog, jt js kssudfc takt taf fkrgjgis Qt krf nougcfc hrod nflow ny Q t . Hor k bodpkgy kt takt lfvfl oh fkrgjgis, jt cofs got dk`f kgy sfgsf to sjigkl kgytajgi. Jt js k sjtuktjog jg wajba taf worst possjnlf fxpfbtktjogs oh taf sfllfrs woulc nf borrfbt. Afgbf, suba k rd woulc baoosf taf optjdkl lfvfl J t∙ cfgfc jg (3.;;). Safrfhorf, taf nougckry bogcjtjog js
Qt (Qt ∞ J t∙ ) 4 Qt wajba ugjqufly cfgfs taf solutjog to taf prfbfcjgi cjhhfrfgtjkl fquktjog. Gotj Gotjbf bf klso klso takt takt fquk fquktj tjog og (3.; (3.;6) 6) bkg bkg nf usfc usfc to wrjt wrjtff taf taf cjhh cjhhfr frfg fgtj tjkl kl fquktjog jg k dorf fxpljbjt wky, sjgbf
X t; 4 C t +
;
PH (J t∙ ) + β t T ; + r
;;8
3. ]kyout ]kyout ]oljby ]oljby
Safg
∎x X t; (Qt (Ct ) Ct ) 4
;
PH (Qt ∞ Ct ) + β T ; + r
kgc
∎c X t; (Qt (Ct ) Ct ) 4 ; ∞
;
H (Qt ∞ Ct ) ; + r
Runstjtutjgi tafsf vklufs jgto taf cjhhfrfgtjkl fquktjog, jt hollows takt
`
;
H
4 = PH + β TQ + ; ∞ ; + r ; + r
or stjll, jsolktjgi Qt (Ct ),
H ∞ (; + r ) Q (C) 4 `PH + β T
Sajs Sajs fquk fquktj tjog og aks aks kg jdpo jdport rtkg kgtt jdpl jdpljb jbkt ktjo jog. g. Jh jt js kssu kssudf dfc c takt takt rds jgbrfksf tafjr cjvjcfgcs ks tafjr fkrgjgis jgbrfksf,2 tajs jdpljfs takt Q (C) ≦ = hor kll vklufs oh C.3 Jh takt js taf bksf, taf gudfrktor jg taf prfbfcjgi rktjo dust klso nf posjtjvf, jdplyjgi takt taf lfvfl oh jgvfst H , js nflow taf optjdkl lfvfl J ∙ cftfrdjgfc jg taf dfgt, taf kriudfgt oh H prfvjous sfbtjog.
3.>.3 3.>.3 Fdpjrjbkl Fdpjrjbkl Fvjcfgbf Fvjcfgbf Rfvfrk Rfvfrkll fdpjrj fdpjrjbkl bkl onsfrvk onsfrvktjo tjogs gs oh ugfxpf ugfxpfbtf btfc c rfturg rfturgss jg taf pkydfg pkydfgt t oh cjvj cjvjcf cfgcs gcs tfst tfst taf taf jgho jghord rdkt ktjo jogk gkll bogt bogtfg fgtt oh cjvj cjvjcf cfgc gcs. s. Nfho Nfhorf rf taf taf wor` oh Kakrogy kgc Rwkry (;:1=), k logi kgc jgbogblusjvf cfnktf akc tk`fg plkbf. Hor jgstkgbf, ]fttjt (;:9>,;:98) kgc Yktts (;:92,;:98) wfrf got jg kbborc kbborckgb kgbff wjta wjta rfspfb rfspfbtt to taf vkluf vkluf bogvfy bogvfyfc fc ny jghord jghordktj ktjog. og. Jg hkbt hkbt,, nota nota wfrf wfrf usjg usjgi i prjd prjdkr krjl jlyy dogt dogtal alyy cktk cktk nut nut rkta rktafr fr cjhh cjhhfr frfg fgt t dftaocoloijfs. Saf dkjg cjsputf nftwffg tafsf stucjfs pjvots og taf borrfbt cjstjgbtjog nftwffg taf jghordktjog bogvfyfc ny taf fkrgjgis kgc takt bogvfyfc ny taf cjvjcfgcs. Kll oh tafsf stucjfs usfc dogtaly cktk ks wfll, suhhfrjgi hrod taf skdf pronlfd ks Nrfggkg‚s stucy cfsbrjnfc prfvjously, kgc dk`jgi jt dorf cjhbult to cjstjgiujsa nftwffg taf two jghordktjogkl fhhfbts. Sarff dorf rfbfgt pkpfrs akvf cjsbussfc ugfxpfbtfc rfturgs usjgi ckjly rfturg rfturg cktk. Saf rst oh tafd js k pkpfr ny Bakrfst (;:91), wao wks kbtuklly dorf
;;9
3.>. Cjvjcfgcs Cjvjcfgcs kgc Jghordktjog Jghordktjog
ijvfg ajs dkjg bogbfrg, go fhhort wks dkcf to sfpkrktf taf bodpogfgt cuf to taf cjvjcfgcs hrod taf ogf cuf to fkrgjgis. Kakrogy kgc Rwkry (;:1=) bogsjcfr ogly rds hrod wajba tafy akvf cktk og qukrtfrly qukrtfrly cjvjcfgcs kgc fkrgjgis fkrgjgis pfr sakrf, kggougbfdfgt kggougbfdfgt cktfs, kgc ckjly rktfs oh rfturg. Safsf kutaors usf k gkjvf docfl oh fxpfbtktjogs, kbborcjgi to wajba taf fxpfbtfc cjvjcfgc oh k rd e kt taf qukrtfr q , Ceq , fqukls taf cjvjcfgc pkydfgt jg taf prfvjous qukrtfr, Ceq∞; . Kbborcjgily, k cjvjcfgc kggougbfdfgt js skjc to nf hkvorknlf jh Ceq 7 Ceq kgc ughkvorknlf jh Ceq < Ceq . Otafrwjsf, taf cjvjcfgc js skjc to nf stknlf . Bogrdjgi taf `gowg tfgcfgby oh rds got to bakgif tafjr cjvjcfgc poljbjfs, Kakrogy kgc Rwkry gc takt jg tafjr skdplf, jg knout 19% oh taf bksfs taf cjvjcfgcs krf stknlf. Jg orcfr to jsolktf taf cjvjcfgc fhhfbt hrod taf fkrgjgis fhhfbt, ogly rds jg wajba cjvjcfgc kggougbfdfgt cktfs cjhhfr hrod taf fkrgjgis kggougbfdfgt cktfs ny kt lfkst ;; ckys krf tk`fg jgto kbbougt. Vsjgi taf gkjvf fxpfbtktjog docfl, taf skdplf cktk krf cjvjcfc jgto tarff iroups5 hkvorknlf, ughkvorknlf, kgc stknlf cjvjcfgcs. Rtjll, fkba oh tafsf iroups js suncjvjcfc kbborcjgi to waftafr taf cjvjcfgcs kggougbfdfgt prfbfcfc or hollowfc taf fkrgjgis kggougbfdfgt. Saf kngordkl rfturgs krf bodputfc hrod k dkr`ft docfl
^et 4 λe + ξe ^dt + ˜et
e kt cky t t , kgc λ e kgc ξe krf bogstkgts, wafrf ^et js taf rfturg6 oh sfburjty e kgc ^dt rffbts taf dfkg rfturg og taf dkr`ft 8 krougc tjdf t. Saf rfsjcukl ˜et rfprfsfgts taf kngordkl rfturg, taf fxpfbtfc vkluf oh wajba js zfro. Gotjbf takt ogly dodfgts jg tjdf wjtajg k cjstkgbf oh ;; ckys or lfss hrod taf kggougbfdfgt oh cjvjcfgcs kt t 4 = krf nfjgi bogsjcfrfc. Afgbf, ∞;; < t < ;;. Saf kvfrkif kngordkl rfturg kt tjdf t js bklbulktfc kvfrkijgi hor fkba qukrtfr q taf fstjdktfc rfsjcukl hor kll kvkjlknlf sfburjtjfs jg takt qukrtfr kt k cjstkgbf t oh taf kggougbfdfgt, kgc tafg tk`jgi taf kvfrkif oh takt ovfr kll taf qukrtfrs. Saf Saf rfsu rfsult lt js kg kvfr kvfrki kiff rfsj rfsjcu cukl kl cf cfpf pfgc gcjg jgi i og t, K^t . Saf gull aypotafsjs js takt K^t js cjstrjnutfc krougc zfro kgc aks k t-cjstrjnutjog jh sujtknly sbklfc. Saf dkjg rfsults oh Kakrogy kgc Rwkry krf twoholc. Hjrst, tafy saow takt jg taf bksf oh stknlf cjvjcfgcs, tafrf krf go sjigjbkgt kngordkl rfturgs (K^), kgc jg taf bksf oh bakgijgi cjvjcfgcs tafy gc fjtafr posjtjvf kngordkl rfturgs oh knout ;% jg taf bksf oh jgbrfksjgi cjvjcfgcs, or gfiktjvf K^ oh knout ∞2% wafg taf cjvjcfgcs cfbrfksf.
>96
B.1. Dfrifrs Dfrifrs kgc Kbqujsjt Kbqujsjtjogs jogs
Saf ajia-vkluf typf js nfttfr ohh, nut taf low-vkluf typf boulc jdprovf ajs/afr vkluf ny njccjgi lfss. Saf optjdkl λ hor taf low-vkluf typf (λ ) js taf ogf takt dk`fs taf ajia-vkluf typf jgcjhhfrfgt nftwffg njccjgi 83 kgc 3=, wajba taf low-vkluf njccfr prfhfrs–takt js, λ solvfs taf fquktjog
I(3= λ 0 UA ) 4 I(83 =20 UA ) ⇓
λ 4 =>889 Ny njccjgi 3= kgc kbqujrjgi λ 4 =>889, taf low-vkluf njccfr js strjbtly nfttfr ohh sjgbf
I(3= =>8890 3=) 4 lg 3=P= 3=P=6 Ù 3= ∞ (=6 ∞ =>889)3=T2:229 7 >: :33 Safrfhorf sjigkljgi tarouia λ cofs akvf kg fhhfbt, kgc taf pooljgi fqujljnrjud hkjls to fxjst. :. Nota Nota docfls docfls try to solvf taf pronlf pronlfd d oh hrff rjcjgi rjcjgi takt takt obburs obburs wafg njcs krf dkcf to ktodjstjb ktodjstjb sakrfaolcfrs. Efikcffsa Efikcffsa kgc Baowcary ( EB) kgc Ralfjhfr kgc Xjsagy (RX) try to solvf tajs pronlfd kssudjgi taf fxjstfgbf oh k lkrif sakrfaolcfr prjor to taf ohhfr. Aowfvfr, sjgbf tafy cfkl wjta taf qufstjog kssudjgi cjhhfrfgt aypotafsfs, tafy wjll krrjvf kt cjhhfrfgt bogblusjogs5 EB prfcjbt k gfiktjvf rflktjogsajp nftwffg taf jgjtjkl sakrfaolcjgi kgc taf njc prfdjud, wajlf RX prfcjbt k posjtjvf rflktjogsajp. Saf dkjg cjhhfrfgbf nftwffg tafsf two wor`s js takt RX kssudf kg fxoifgously fxoifgously xfc sakrf owgfrsajp owgfrsajp ny taf njccfr (k lkrif sakrfaolcfr sakrfaolcfr)) kgc EB rflkx tajs kssudptjog kgc bogsjcfr taf jgjtjkl sakrfaolcjgi to nf cftfrdjgfc fgcoifgously. RX kssudf takt taf rd js owgfc ny ogf lkrif sakrfaolcfr kgc k hrjg hrjgif if oh sdkl sdklll ogfs ogfs.. Safy Safy co got got bogs bogsjc jcfr fr strk strktf tfij ijbb jgtf jgtfrk rkbt btjo jogs gs nftwffg lkrif sakrfaolcfrs. Saf njc prfdjud cfbrfksfs wjta λ sjgbf taf dorf sakrfs L owgs, taf fksjfr js to bogvjgbf sdkll sakrfaolcfrs takt k low njc jgcjbktfs k sdkll rjsf jg prjbf khtfr taf tk`fovfr, rktafr takg kg kttfdpt to prot kt tafjr fxpfgsfs. Og taf otafr akgc, jh λ 4 =, taf njc prfdjud js ajiafr ks go ogf wjll tfgcfr uglfss taf lkrif sakrfaolcfr ohhfrs k prjbf irfktfr or fqukl to taf vkluf oh taf rd khtfr taf tk`fovfr. EB kssudf strktfijb jgtfrkbtjog nftwffg njccfrs. Safy kssudf takt tafrf tafrf krf lkrif lkrif sakrfa sakrfaolc olcfrs frs wjta wjta cjhhfr cjhhfrfgt fgt vklukt vkluktjog jogss oh taf tkrift tkrift.. Safg, njccfrs wjll baoosf cjhhfrfgt vklufs oh jgjtjkl sakrfs to `ffp og
>98
Kppfgcjx B. Ruiifstfc Rolutjogs
dkgy sakrfs nfhorf taf tk`fovfr jg orcfr to brfcjnly sjigkl jts typf kgc njc k lowfr kdougt jg taf tfgcfr ohhfr. Safrfhorf, wafg λ js ajia, jt js nfbkusf tajs sakrfaolcfr aks k ajia vkluktjog oh taf tkrift kgc wjll pky k lkrif njc prfdjud.
B.1.> B.1.> Dfrifr Dfrifr Njcs ;. Saf sjigkljgi fqujljnrjud js got hully rfvfkljgi sjgbf njccfr > ogly lfkrgs waftafr v ≣ v˜; < r or v˜; ≦ r , got taf prfbjsf vkluf oh v; . Jh v˜; wfrf cjrfbtly onsfrvknlf, njccfr ; woulc dk`f kg jgjtjkl ohhfr oh v (jh v˜; ≦ v). Njccfr > woulc tafg bodpftf jh v ≣ v˜; < w wafrf Fύ > (w v> v ;) 4 =. Ny taf cfgjtjog oh r , wf akvf r < w. Safrfhorf, Safrfhorf, hor rfkljzktjogs oh v˜; suba takt r ≣ v˜; < w, njccfr > js cftfrrfc jg taf sjigkljgi fqujljnrjud, nut af/saf woulc got nf jh v˜; boulc nf onsfrvfc cjrfbtly. >. Rff pkif :8 oh Hjsadkg (;:11). 2. Jh kg kdougt oh B js ohhfrfc, taf kbqujrfr fxpfbts to rfbfjvf kssfts worta ogly =96B So sff tajs, gotjbf takt taf tkrift kbbfpts taf ohhfr jh kgc ogly jh v v ≣ B. Safg taf fxpfbtfc vkluf oh taf tkrift‚s kssfts to taf kbqujrfr bogcjtjogkl og taf ohhfr nfjgi kbbfptfc js =96B. Bogsjcfr gow k stob` cfkl jg wajba taf kbqujrfr ohhfrs kg owgfrsajp oh ξ oh taf bodnjgfc rd. Hor tajs bksf, wf bkg gc k stob` trkcf takt js dutuklly nfgfbjkl hor kll v. So nf nfgfbjkl to taf tkrift, taf bogcjtjog
ξ(x + ;6v) ≦ v saoulc nf sktjsfc hor kll v . Sajs js sktjsfc hor kll v wafg jt js sktjsfc hor v 4 ;== (nfbkusf p js jgbrfksjgi jg v). Saus, wf dust akvf
ξ(x + ;6=) ≦ ;== Saf trkcf saoulc klso nf nfgfbjkl to taf kbqujrfr, bogcjtjogkl og taf tkrift klwkys kbbfptjgi taf ohhfr. Sajs obburs wafg taf fxpfbtfc wfklta oh taf kbqujrfr js go sdkllfr takg x. Rjgbf taf fxpfbtfc vkluf oh taf tkrift‚s kssfts to taf kbqujrfr bogcjtjogkl og taf ohhfr klwkys nfjgi kbbfptfc js 96, taf bogcjtjog to taf kbqujrfr saoulc nf
(; ∞ ξ)(x + 96) ≦ x or
p≣
96 x + 96
B.1. Dfrifrs Dfrifrs kgc Kbqujsjt Kbqujsjtjogs jogs
>99
3. Rff pkif 1= oh Akgsfg (;:19). 6. Rff taf prooh og pkif 1; oh Akgsfg (;:19). Saf fxplkgktjog js taf hollowjgi5 Jh taf kbqujrfr‚s rd js too lkrif rflktjvf to taf tkrift, taf kbqujrfr djssfs k lkrifr hrkbtjog oh taf vkluf oh ajs/afr owg rd bodpkrfc to k sjtuktjog wafrf taf ohhfr js dkcf jg bksa. Kt taf skdf tjdf, ks taf kbqujrfr ifts njiifr, taf tkrift‚s kssfts nfbodf k lfss sjigjbkgt kccjtjog to taf kbqujrfr. 8. Akgsfg stucjfs taf probfss oh k dfrifr to hordulktf k docfl hor taf baojbf oh taf fxbakgif dfcjud (fjtafr bksa or stob`). Sarouiaout taf kgklysjs, taf tkrift rd aks prjvktf jghordktjog og jts vkluf. Af/saf rst bogsjcfrs taf bksf wafrf taf kbqujrfr‚s prfdfrifr vkluf js punljbly `gowg. Jg tajs bksf jt js saowg takt k stob` ohhfr bkg khhfbt trkcf fvfg wafg k bksa ohhfr bkggot. Lktfr af bogsjcfrs takt taf kbqujrjgi rd aks prjvktf jghordktjog og jts owg vkluf. Safg, taf kbqujrfr wjll got ohhfr stob` wafg taf tkrift ugcfrfstjdktfs taf vkluf oh jts ohhfr. Sajs jdpljfs takt stob` wjll got gfbfsskrjly codjgktf bksa. Akgsfg kriufs takt tafrf ifgfrklly x nflow wajba taf “ovfrvklufc— stob` js ohhfrfc fxjsts k brjtjbkl vkluf oh x kgc knovf wajba bksa js ohhfrfc. Jh taf taf tkri tkrift ft prjb prjbff cf cfbr brfk fksf sfss (jgb (jgbrf rfks ksfs fs)) jh kg ohhf ohhfrr js kbbf kbbfpt ptfc fc (rfefbtfc) (rfefbtfc),, jt dky nf nfbkusf nfbkusf taf dkr`ft dkr`ft cofs got kirff wjta taf vklukvkluktjog dkcf knout taf kbqujrfr‚s vkluf. Sakt js, Akgsfg‚s docfl prfcjbts x js lkrif, jt js cjhbult to bogvjgbf taf tkrift takt taf kbqujrfr js takt jh x ovfrvklufc, so jt js nfttfr to ohhfr bksa jgstfkc oh ugcfrvklufc sakrfs. Aowfvfr, taf dkr`ft dky tajg` takt, jg hkbt, taf kbqujrfr js ovfrvklufc kgc rfbfjvjgi bksa js worsf, to taf tkrift, takg rfbfjvjgi sakrfs. Og taf otafr akgc, Akgsfg‚s docfl prfcjbts takt jh x js sdkll, taf kbqujrfr prfhfrs to pky wjta sakrfs oh jts ovfrvklufc rd. Jh taf tkrift prjbf cfbrfksfs khtfr taf ohhfr js kbbfptfc, jt dky nf nfbkusf taf dkr`ft vjfws taf kbqujrfr ks ugcfrvklufc kgc jt woulc akvf nffg nfttfr to taf tkrift to rfbfjvf bksa jgstfkc oh sakrfs oh kg ugcfrvklufc rd. 9. Rff pkif 36 oh Hjsadkg Hjsadkg (;:1:). (;:1:). 1. Yf gffc to saow takt takt pb (v; ) < pc (v; ) Gotjbf takt tajs js fqujvklfgt oh provjgi
βF djg dkx v v>+ v;+ + (; ∞ β)v;∞ < F djg dkx v v>+ v;+ Sajs fxprfssjog sjdpljfs to
(β ∞ ;) F djg dkx v v>+ v;+ ∞ v;∞ < =
>91
Kppfgcjx B. Ruiifstfc Rolutjogs
Jg tajs bksf, pb (v; ) 4 pc (v; ) sjgbf tafg bksa js fqujvklfgt to cfnt. Jh β < ;, taf tkrift possfssfs prjvktf jghordktjog kgc pb (v; ) < pc (v; ). :. Kg jgbrfksf jgbrfksf jg taf bost oh njccfr > aks taf skdf fhhfbt ks jg Hjsadkg‚s Hjsadkg‚s prfvjous wor`. Bogsjcfr k cfbrfksf jg njccfr >‚s bost. Rjgbf jts bost oh bodpftjgi js lowfr, jts fxpfbtfc pkyohh js ajiafr. Njccfr ;‚s fxpfbtfc pkyohh js lowfr sjgbf k ajiafr vkluktjog dust nf sjigklfc to cftfr njccfr >, kgc tajs js dorf bostly. Njccfr > js prffdptfc lfss ohtfg, kgc wafg jt js prffdptfc, taf njc js ajiafr. ajiafr. Nota fhhfbts nfgft taf tkrift, kgc jts fxpfbtfc pkyohh js ajiafr. ;=. Akrrjs Akrrjs kgc ^kvjv ^kvjv dk`f taf hollowjgi hollowjgi kssudptjog5 kssudptjog5
F> (C) + @> (C) ≦ F; (C)0 takt js, tafrf js kg jgbfgtjvf to bogtrol taf rd, fvfg jh taf dkgkifdfgt aks lowfr knjljty. J kgc ^ wkgt to votf hor tafdsflvfs, jg fkba bksf, rfikrclfss oh tafjr prjvktf sjigkls kgc taf cfnt lfvfl. Saf worst pkyohh fjtafr J or ^ bkg rfbfjvf jh jg bogtrol oh taf rd js λj F> (C) + @> (C), j 4 J ^0 taf nfst pkyohh fjtafr J or ^ bkg rfbfjvf jh got jg bogtrol oh taf rd js λj F; (C), j 4 J ^. Jh tafy klwkys votf hor tafdsflvfs, jt js nfbkusf λ j F> (C) + @> (C) 7 λj F; (C), wajba js truf ny taf kssudptjog takt F> (C) + @> (C) ≦ F; (C).
Gotfs
Bakptfr Ogf ;. Rff Cuhf Cuhf (;:11), (;:11), pkif pkif 9;. >. Rff taf fxbfllfgt tfxtnoo` oh Srjiforijs Srjiforijs (;::8) kgc rfhfrfgbfs rfhfrfgbfs tafrfjg hor dorf cftkjls. 2. Rff Nrkclfy, Nrkclfy, Ekrrfll, kgc @jd (;:13), (;:13), Sjtdkg kgc Yfssfls Yfssfls (;:11), (;:11), kgc Nkrblky Nkrblky,, Rdjta, kgc Yktts (;::6). 3. Rff Djllfr Djllfr kgc Rbaolfs (;:91) (;:91) kdogi otafrs. otafrs.
Bakptfr Sarff ;. Saf sfdjgkl sfdjgkl pkpfrs jg tajs krfk wfrf wrjttfg ny ^oss (;:92,;:99 (;:92,;:99)) kgc Efgsfg kgc Dfb`ljgi (;:98). Rjgbf tafg kg fxtfgsjvf ljtfrkturf jg taf skdf vfjg aks nffg procubfc. Kg fxakustjvf ljst oh rfhfrfgbfs oh taf sunsfqufgt wor`s bkg nf hougc jg taf survfy ny Akrrjs kgc ^kvjv (;::;). >. Sajs solutjog solutjog wks prfsfgtfc prfsfgtfc jg kg fkrljfr pkpfr ny ^oss (;:92). (;:92). 2. Rff ^oss ^oss (;:99) (;:99).. 3. Sajs Sajs kssudp kssudptjog tjog,, `gowg `gowg ks taf hrff bksa ow aypotafs aypotafsjs, js, wks rst rst prfsfg prfsfgtfc tfc ny Efgsfg (;:18). 6. Sajs typf oh bogjbt wks jgtrocubfc jgtrocubfc ny Efgsfg Efgsfg kgc Dfb`ljgi (;:98). (;:98). 8. Sajs kriudfgt kriudfgt wks pojgtfc out ny Dyfrs (;:99). (;:99). 9. So sff aow takt akppfgs, akppfgs, jt suhbfs to gotjbf takt taf dkrijgkl dkrijgkl utjljtjfs utjljtjfs saoulc nf fqukl kt taf optjdkl vkluf oh taf rd. Safrfhorf, k cfbrfksf jg utjljty kt takt pojgt ny rfcubjgi taf hrkbtjog oh owgfrsajp saoulc nf bodpfgsktfc ny kg jgbrfksf oh taf hrjgif nfgfts. 1. Saf sbafcul sbafculff h boulc klso nf bogsjcfrfc ks k hugbtjog oh taf stktf oh taf worlc, hor jgstkgbf. Hor sjdpljbjty, sjdpljbjty, tajs hkbtor js kvojcfc afrf. Rff ^oss (;:92) hor k cjsbussjog oh tajs pojgt. :. Rff fquktj fquktjog og : oh ^oss ^oss (;:92). (;:92). ;=. Rff Box kgc ^unjgstfjg (;:16), (;:16), Bakptfr 9, fspfbjklly fspfbjklly Rfbtjog Rfbtjog 9.2. ;;. Nksjbklly Nksjbklly,, tafrf wjll nf hfwfr stktfs stktfs oh taf worlc bodpktjnlf bodpktjnlf wjta taf pkydfgt oh @. Ks @ ↞ ∗, taf sft oh stktfs oh taf worlc kt wajba taf cfntaolcfrs wjll
>1=
Gotfs
;>. Otafrwjsf, Otafrwjsf, nkg`ruptby nkg`ruptby js bfrtkjg. bfrtkjg. ;2. Sajs Sajs js truf ks hkr ks jt js kg jgtfrg jgtfrgkl kl optjdud– optjdud–tak taktt js, ks hkr ks taf optjdkl optjdkl lfvfl oh N sktjsfs = < N < t. ;3. Yakt lfkcs to taf ugjqufgfss oh tajs rflktjog rflktjog js taf Rpfgbf-Djrr Rpfgbf-Djrrlffs lffs bogcjtjog bogcjtjog (sff Kppfgcjx N hor k cjsbussjog), nurrjfc jg fquktjogs (2.;=) kgc (2.;;). Sakt js way ogf ontkjgs pfrhfbt sfpkrktjog oh typfs (j.f., k ugjquf cfnt lfvfl hor fkba typf)5 Saf dkgkifdfgt bakgifs pfrbfjvfc vkluf nfbkusf dkgkifrs oh nfttfr rds krf rflktjvfly dorf jddugf to tk`jgi og cfnt, kgc tajs akppfgs jg k dogotogjb wky. ;6. Saf cfdkgc cfdkgc hor kg kssft kssft js skjc to nf gordkl jh, jh, jg k portholjo baojbf sjtuktjog wjtaout sjigkljgi, taf cfdkgc js k cfbrfksjgi hugbtjog oh prjbf. ;8. Rkyjgi Rkyjgi takt Efgsfg (;:18) kgc Dyfrs kgc Dkeluh (;:13) krf cjrfbt bodpftjtors bodpftjtors js borrfbt, ks hkr ks prfcjbtjogs krf bogbfrgfc. Jt saoulc nf strfssfc, aowfvfr, takt tafsf two pkpfrs rfhfr to cjhhfrfgt pronlfds. Efgsfg rfhfrs to k dorkl akzkrc story, jg taf sfgsf takt dkgkifrs pursuf tafjr owg onefbtjvfs uglfss bksa js rfdovfc hrod tafjr akgcs. Saf bogjbt js nftwffg dkgkifrs kgc jgsjcf sakrfaolcfrs. Jg taf bksf oh Dyfrs kgc Dkeluh, tafrf js k bost oh jssujgi sfburjtjfs, bkusfc ny taf ksyddftry oh jghordktjog nftwffg dkgkifdfgt kgc (outsjcf) jgvfstors. Jg taf hordfr bksf, ogf dust prfvfgt dkgkifdfgt hrod tk`jgi tk`jgi og gfiktj gfiktjvfvf-G]X G]X proefb proefbts0 ts0 jg taf lkttfr lkttfr,, posjtj posjtjvf vf proefb proefbts ts krf pkssfc pkssfc ny nfbkusf oh taf kcvfrsf sflfbtjog bost–go dkttfr aow wfll jgtfgtjogfc dkgkifdfgt js. Jg k sfgsf, tafsf krf bodplfdfgtkry kspfbts oh taf skdf cfnktf. ;9. F(x|) cfgotfs taf fxpfbtfc vkluf oh taf rkgcod vkrjknlf x wjta rfspfbt to taf eojgt pronknjljty oh ( K K N) N) bogcjtjogfc to k sft ⊎ D ∥ D . ;1. Jh cfnt js cfhkult‒rjs cfhkult‒rjs`-hrff `-hrff,, hor jgstkgbf, jt hollows takt N 4 =. Saf hkbt takt jg suba k bksf taf dkgkifdfgt js surf oh go bkpjtkl loss jdpljfs takt jt wjll jgvfst jg kgy proefbt wjta posjtjvf gft prfsfgt vkluf. Sakt js way taf knjljty to rkjsf hugcs jg tajs wky js jgblucfc jg taf cfgjtjog oh slkb`. ;:. Got cjsbussfc cjsbussfc afrf js taf rolf oh nfljfhs nfljfhs oh outsjcf jgvfstors jgvfstors kgc taf possjnjljty oh dultjplf fqujljnrjk cfpfgcjgi og tafsf nfljfhs. Ogf bkg akvf fqujljnrjk wafrf fqujty js prfhfrrfc to cfnt, or jg wajba cjhhfrfgt typfs jssuf cjhhfrfgt sfburjtjfs sfburjtjfs jg k gogdogotogjb gogdogotogjb hksajog (sff, hor fxkdplf, Gof, ;:11), ;:11), jgcfpfgjgcfpfgcfgtly oh taf rjs` oh taf sfburjtjfs. >=. Rff Bakptfr Bakptfr 9, pkif 82, kgc hootgotf ;= oh Dksuljs (;:11). (;:11). >;. Saf kssudptjo kssudptjog g takt takt taf cfrjvktj cfrjvktjvf vf oh Y js posjtjvf js gkturkl. Saf ajiafr taf vkluf oh jghordktjog pfrbfjvfc ny taf dkr`ft, taf lkrifr js taf nfgft oh sjigkljgi to taf dkgkifdfgt. >>. Rff Xjswkgkta Xjswkgkta (;::2) (;::2) hor k cjsbussjog cjsbussjog oh tajs pojgt. >2. Rff Nkrblky Nkrblky kgc Rdjta (;:::) (;:::) hor hurtafr hurtafr cjsbussjog og taf jgtfrprftktjog jgtfrprftktjog oh tafsf rfsults.
Bakptfr Hour
Njnljoirkpay
Kiajog, ]., kgc ]. Noltog, ;:1:, “Saf Hjgkgbjkl Rtrubturf oh taf Hjrd kgc taf ]ronlfd oh Bogtrol,— Furopfkg Fbogodjb ^fvjfw 22, 22, >18‒:2. , ;::>, “Kg ‑Jgbodplftf Bogtrkbts‚ Kpprokba to Hjgkgbjkl Bogtrkbtjgi,— 6:, 392‒:3. ^fvjfw oh Fbogodjb Rtucjfs 6:, Kakrogy, Kakrogy, E., kgc J. Rwkry, Rwkry, ;:1=, “[ukrtfrly Cjvjcfgcs kgc Fkrgjgis Kggougbfdfgts dfgts kgc Rtob`a Rtob`aolc olcfrs frs‚‚ ^fturgs ^fturgs55 Kg Fdpjrj Fdpjrjbkl bkl Kgklys Kgklysjs, js,—— Eourgk Eourgkll oh 26, ;‒;>. Hjgkgbf 26, K`frloh, I., ;:9=, “Saf Dkr`ft hor Lfdogs5 [ukljtktjvf Vgbfrtkjgty kgc taf Dkr`ft Dfbakgjsd,— [ukrtfrly Eourgkl oh Fbogodjbs 13, 13, 311‒6==. Kllfg, H., kgc I. Hkulaknfr, Hkulaknfr, ;:1:, “Rjigklljgi ny Vgcfrprjbjgi jg taf J]O Dkr`ft,— >2, 2=2‒>2. Eourgkl oh Hjgkgbjkl Fbogodjbs >2, Kllfg, H., kgc ^. Djbakfly, Djbakfly, ;::6, “Cjvjcfgc ]oljby,— ]oljby,— jg ^. K. Ekrrow, Ekrrow, X. Dk`sjdovjb, kgc Y. S. Ujfdnk (fcs.), Akgcnoo`s jg Opfrktjogs ^fsfkrba kgc Dkgkifdfgt Rbjfgbf : , Kdstfrckd5 Gorta-Aollkgc, Kllfg, H., kgc R. Dorrjs, >==;, “Hjgkgbf Kppljbktjogs oh Ikdf Safory,— jg @klykg Bakttfreff kgc Yjlljkd H. Rkduflsog, (fcs.), Ikdf Safory kgc Nusjgfss Kppljbktjogs , Nostog5 @luwfr Kbkcfdjb ]unljsafrs. Kllfg, E., ;::8, “LNOs–Saf Fvolutjog oh Hjgkgbjkl Rtrubturfs kgc Rtrktfijfs,— Eour- 1, ;1‒>:. gkl oh Kppljfc Borporktf Hjgkgbf 1, Kltdkg, F., ;:1:, “Dfksurjgi Borporktf Nogc Dortkljty kgc ]frhordkgbf,— Eourgkl 33, :=:‒>>. oh Hjgkgbf 33, Krrow, Krrow, @., ;:83, “Lf ^ olf o ˈ lf cfs Xklfurs Noursjmfrfs frfs pour lk ^fpkrtjtjog lk Dfjllfurf cfs ^jsqufs,— trkgslktfc ks “Saf ^olf oh Rfburjtjfs jg taf Optjdkl Kllobktjog oh ^js`-Nfkrjgi ^js`-Nfkrjgi,— ,— ^fvjfw oh Fbogodjb Rtucjfs 2;, 2;, :;‒:8. Ksqujta, ]., kgc C. Dulljgs, Er, Er, ;:12, “Saf Jdpkbt oh Jgjtjktjgi Cjvjcfgc ]kydfgts og Rakrfaolcfrs‚ Yfklta,— Eourgkl oh Nusjgfss 68, 68, 99‒:8. Ksqujta, ]., kgc S. Yjzdkg, ;::=, “Fvfgt ^js`, Bovfgkgts Bo vfgkgts kgc Nogcaolcfr ^fturgs ^ fturgs jg Lfvfrkifc Lfvfrkifc Nuyouts,— Nuyouts,— Eourgkl oh Hjgkgbjkl Fbogodjbs >9, >9, ;:6‒>;3. Nkiwfll, L., ;::;, “Rakrf ^fpurbaksf kgc Sk`fovfr Cftfrrfgbf,— ^kgc Eourgkl oh >>, 9>‒11. Fbogodjbs >>,
>18
Njnljoirkpay
Nkiwfll, L., kgc E. Raovfg, ;:1:, “Bksa Cjstrjnutjogs to Rakrfaolcfrs,— Eourgkl oh 2, ;>:‒3=. Fbogodjb ]frspfbtjvfs 2, Nk`fr, I., D. Efgsfg, kgc @. Durpay, ;:11, “Bodpfgsktjog kgc Jgbfgtjvfs5 ]rkbtjbf vs. Safory,— Eourgkl oh Hjgkgbf 32, 32, 6:2‒8;8. Nkrblky, E., kgc B. Rdjta, ;:::, “Saf Bkpjtkl Rtrubturf ]uzzlf5 Kgotafr Loo` kt taf Fvjcfgbf,— Eourgkl oh Kppljfc Borporktf Hjgkgbf ;>, ;>, 1‒>=. Nkrblky, E., B. Rdjta, kgc ^. Yktts, ;::6, “Saf Cftfrdjgkgts oh Borporktf Lfvfrkif kgc Cjvjcfgc Cjvjcfgc ]oljbjfs,— ]oljbjfs,— Eourgkl oh Kppljfc Borporktf Hjgkgbf 9, 9, 3‒;:. Nkrog, C. ]., ]., ;:1>, “K Docfl oh taf Cfdkgc hor Jgvfstdfgt Jgvfstdfgt Nkg`jgi Kcvjsjgi Kcvjsjgi kgc Cjstrjnutjog Rfrvjbfs hor Gfw Jssufs,— Eourgkl oh Hjgkgbf 29, 29, :66‒98. Nfktty, ^. ]., kgc E. ^. ^jttfr, ;:18, “Jgvfstdfgt Nkg`jgi, ^fputktjog, kgc Vgcfrprjbjg prjbjgi i oh Jgjtjk Jgjtjkll ]unljb ]unljb Ohhfrjg Ohhfrjgis, is,—— Eourgkl Eourgkl oh Hjgkgb Hjgkgbjkl jkl Fbogo Fbogodjb djbs s ;6, >;2‒2>. Nfgg Nfggft fttt Rtfw Rtfwkr krt, t, I, ;::= ;::=,, “^fd “^fdk` k`jg jgi i taf taf ]unl ]unljb jb Borpo Borpork rktj tjog og hrod hrod Yjtaj Yjtajg, g,—— 81, ;>8‒29. Akrvkrc Nusjgfss ^fvjfw ^fvjfw 81, Nfgvfgjstf, L. D., kgc ]. K. Rpjgct, ;:1:, “Aow Jgvfstdfgt Nkg`frs Cftfrdjgf taf Ohhfr ]rjbf kgc Kllobktjog oh Gfw Jssufs,— Eourgkl oh Hjgkgbjkl Fbogodjbs >3, 232‒8;. Nfril¨oh, oh, F., kgc F. vog Sakccfg, ;::3, “Raort-Sfrd vfrsus Logi-Sfrd Jgtfrfsts5 Bkpjtkl Rtrubturf wjta Dultjplf Jgvfstors,— Saf [ukrtfrly Eourgkl oh Fbo- godjbs ;=:, ;=66‒13. Nakttkbakryk, Nakttkbakryk, R., ;:9:, ;:9:, “Jdpfrhfbt “Jdpfrhfbt Jghordktjog, Jghordktjog, Cjvjcfgc Cjvjcfgc ]oljby, ]oljby, kgc ‑Saf Njrc jg taf Akgc Hkllkby‚,— Nfll Eourgkl oh Fbogodjbs ;=, ;=, >6:‒9=. Nakttkbakryk, R., kgc I. D. Bogstkgtjgjcfs (fcs.), ;:1:k, Safory oh Xkluktjog. Hrog- tjfrs oh Docfrg Hjgkgbjkl Safory, Xol. ;. Rkvkif, DC5 ^owdkg kgc Ljttlfflc. (fcs.), ;:1:n, Hjgkgbjkl Dkr`fts kgc Jgbodplftf Jghordktjog. Hrogtjfrs oh Docfrg Hjgkgbjkl Safory, Xol. > . Rkvkif, DC5 ^owdkg kgc Ljttlfflc. Nlkb`, H., kgc D. Rbaolfs, ;:92, “Saf ]rjbjgi oh Optjogs kgc Borporktf Ljknjljtjfs,— 1;, 829‒63. Eourgkl oh ]oljtjbkl Fbogody 1;, Noltog, ]., kgc C. Rbakrhstfjg, ;::=, “K Safory oh ]rfcktjog Nksfc og Kifgby ]ronlfds jg Hjgkgbjkl Bogtrkbtjgi,— Kdfrjbkg Fbogodjb ^fvjfw 1=, 1=, :3‒;=8. Noltog, ]., kgc C. Rbakrhstfjg, ;::8, “Optjdkl Cfnt Rtrubturf kgc taf Gudnfr oh Brfcjtors,— Eourgkl oh ]oljtjbkl Fbogody ;=3, ;‒>6. Nrkclfy, D., I. Ekrrfll, kgc F. @jd, ;:13, “Og taf Fxjstfgbf oh kg Optjdkl Bkpjtkl Rtrubturf5 Safory kgc Fvjcfgbf,— Eourgkl oh Hjgkgbf 2:, 2:, 169‒91. Nrkclfy, D., kgc F. @jd, ;:16, “Saf Sfgcfr Ohhfr ks k Sk`fovfr Cfvjbf5 Jts Fvolutjog, taf Hrff ^jcfr ]ronlfd ]ronlfd kgc taf ]rjsogfr‚ ]rjsogfr‚ss Cjlfddk.— Cjlfddk.— Vgpunljsafc Vgpunljsafc dkgusbrjpt.
Njnljoirkpay
>19
Nrfggkg, Nrfggkg, D., ;:92, ;:92, “Skxfs, “Skxfs, Dkr`ft Xkluktjog kluktjog kgc Borporktf Borporktf Hjgkgbjkl Hjgkgbjkl ]oljby ]oljby,— >2, 3;9‒>9. Gktjogkl Skx Eourgkl >2, Nrfggkg, D. (fc.), ;::8, Saf Safory oh Borporktf Hjgkgbf. Jgtfrgktjogkl Ljnrkry oh Brjtjbkl Yrjtjgis jg Hjgkgbjkl Fbogodjbs , Xols. J kgc JJ. Fcwkrc Flikr. Nrjb`lfy, E., ;:12, “Rakrfaolcfr Yfklta, Jghordktjog Rjigkljgi, kgc taf Rpfbjklly Cfsjigktfc Cjvjcfgc5 Kg Fdpjrjbkl Rtucy,— Eourgkl oh Hjgkgbjkl Fbogodjbs ;>, ;19‒>;=. Nusfr, R., kgc ]. Afss, ;:18, “Fdpjrjbkl Cftfrdjgkgts og taf ^flktjvf Zjflcs og Skxknlf kgc Skx-Fxfdpt Rfburjtjfs,— Eourgkl oh Hjgkgbjkl Fbogodjbs ;9, ;9, 226‒68. Bkdpnfll, E., K. Lo, kgc K. B. Dkb@jglky, ;::9, “Saf Fbogodftrjbs oh Hjgkgbjkl Dkr`fts.— Dkr`fts.— ]rjgbftog, ]rjgbftog, GE5 ]rjgbftog ]rjgbftog Vgjvfrsjty Vgjvfrsjty ]rfss. Bkrtfr, ^., kgc R. Dkgkstfr, ;::=, “Jgjtjkl ]unljb Ohhfrjgis kgc Vgcfrwrjtfr ^fputktjog,— Eourgkl oh Hjgkgbf 36, 36, ;=36‒89. Bkstkgjks, ^., ;:12, “Nkg`ruptby ^js` kgc Optjdkl Bkpjtkl Rtrubturf,— Eourgkl oh 21, ;8;9‒28. Hjgkgbf 21, Bakgi, B., ;::>, “Bkpjtkl Rtrubturf ks kg Optjdkl Bogtrkbt nftwffg Fdployffs kgc Jgvfstors,— Eourgkl oh Hjgkgbf 39, 39, ;;3;‒61. Bakrfst, I., ;:91, “Cjvjcfgc Jghordktjog, Rtob` ^fturgs kgc Dkr`ft Fhbjfgby,— 8, >:9‒22=. Eourgkl oh Hjgkgbjkl Fbogodjbs 8, Bafddkgur, S., kgc ]. Huliajfrj, ;:::, “K Safory oh Iojgi ]unljb Cfbjsjog,— ;>, >3:‒9:. ^fvjfw oh Hjgkgbjkl Rtucjfs ;>, Baowcary Baowcary,, N., kgc K. Rafrdk Rafrdkg, g, ;::8, ;::8, “Saf “Saf Yjggfr‚ Yjggfr‚ss Bursf Bursf kgc Jgtfrg Jgtfrgktj ktjogk ogkll Dftaocs oh Kllobktjgi Jgjtjkl ]unljb Ohhfrjgis,— ]kbjb-Nksjg Hjgkgbf Eour- 3, ;6‒2=. gkl 3, Boddfgt, ^., kgc I. Ekrrfll, ;:19, “Swo Sjfr Sfgcfr Ohhfrs5 Saf Jdprjsogdfgt oh taf Hrff ^jcjgi Rakrfaolcfr,— Eourgkl oh Hjgkgbjkl Fbogodjbs ;:, >12‒2;=. , ;::;, “Saf ^flktjvf Rjigkljgi ]owfr oh Cutba-Kubtjog kgc Hjxfc-]rjbf Rflh-S Rflh-Sfgc fgcfr fr Ohhfrs Ohhfrs kgc Opfg-D Opfg-Dkr` kr`ft ft Rakrf Rakrf ^fpurba ^fpurbaksf ksfs,— s,— Eourgk Eourgkll oh 38, ;>32‒9;. Hjgkgbf 38, Box, E., kgc D. ^unjgstfjg, ;:16, Optjogs Dkr`fts . Fgilfwooc Bljhhs, GE5 ]rfgtjbfAkll. Ckgg, L., ;:1;, “Boddog Rtob` ^fpurbaksfs5 Kg Kgklysjs oh ^fturgs to Nogcaolcfrs kgc Rtob`aolcfrs,— Eourgkl oh Hjgkgbjkl Fbogodjbs :, ;;2‒>1. Ckgg, L., kgc A. CfKgiflo, ;:12, “Rtkgcstjll Kirffdfgts, ]rjvktfly Gfiotjktfc Rtob` ^fpurbaksfs, kgc taf Dkr`ft hor Borporktf Bogtrol,— Eourgkl oh ;;, >96‒2==. Hjgkgbjkl Fbogodjbs ;;, Ckgg, L., ^. Dksuljs, kgc C. Dkyfrs, ;::;, “^fpurbaksf Sfgcfr Ohhfrs kgc Fkrgjgis Jghordktjog,— Eourgkl oh Kbbougtjgi kgc Fbogodjbs ;3, >;9‒6;. CfKgiflo, A., L. CfKgiflo, kgc F. ^jbf, ;:13, “Iojgi ]rjvktf5 Djgorjty Hrffzouts kgc Rtob`aolcfr Yfklta,— Eourgkl oh Lkw kgc Fbogodjbs >9, 289‒3=;.
>11
Njnljoirkpay
Cfifor Cfiforif, if, H., kgc ^. Ufb`ak Ufb`akusf usfrr, ;::2, ;::2, “Saf “Saf ^fvfrs ^fvfrsff LNO Cfbjsj Cfbjsjog og kgc Hjrd Hjrd ]frhordkgbf5 Safory kgc Fvjcfgbf,— Eourgkl oh Hjgkgbf 31, 31, ;2>2‒31. Cfwktrjpogt, D., kgc E. Sjrolf, ;::3, “K Safory oh Cfnt kgc Fqujty5 Cjvfrsjty oh Rfburjtjfs kgc Dkgkifr‒Rakrfaolcfr Bogirufgbf,— [ukrtfrly Eourgkl oh ;=:, ;=>9‒63. Fbogodjbs ;=:, Cjkdogc, C., ;::;, “Cfnt Dkturjty Rtrubturf kgc Ljqujcjty ^js`,— [ukrtfrly Eourgkl oh Fbogodjbs , 9=:‒29. Cjkdogc, Cjkdogc, C., ;::2, “Rfgjorjty “Rfgjorjty kgc Dkturjty Dkturjty oh Cfnt Bogtrkbts,— Bogtrkbts,— Eourgkl oh Hjgkgbjkl 22, 23;‒81. Fbogodjbs 22, Cjxj Cjxjt, t, K., K., kgc kgc ^. ]jgc ]jgcyb yb`, `, ;::3 ;::3,, Jgvfstd ]rjgbf bftog tog,, GE5 GE5 Jgvfstdfgt fgt ugcfr ugcfr Vgbfrtkjgty Vgbfrtkjgty . ]rjg ]rjgbftog Vgjvfrsjty ]rfss. Cotakg, D., ;::=, ]rjbfs jg Hjgkgbjkl Dkr`fts . Oxhorc, Fgilkgc5 Oxhorc Vgjvfrsjty ]rfss. Cuhf, C., ;:11, Rfburjty Dkr`fts–Rtobakstjb Docfls . Nostog, DK5 Kbkcfdjb ]rfss. , ;::8, Cygkdj ]rjgbftog, tog, GE5 ]rjgbf ]rjgbftog tog Vgjvfr Vgjvfrsjt sjty y Cygkdjbb Kssft Kssft ]rjbjg ]rjbjgii Safory Safory . ]rjgbf ]rfss. Cugg, @., kgc B. Rpktt, ;:13, ;:13, “K Rtrktfijb Rtrktfijb Kgklysjs Kgklysjs oh Rjg`jgi Rjg`jgi Hugc Nogcs,— Eourgkl oh Hjgkgbjkl Fbogodjbs ;2, 2::‒3>2. Fb`no, N., ;:18, “Jghordktjog Ksyddftrjfs kgc Xkluktjog Fhhfbts oh Borporktf Cfnt Ohhfrjgis,— Eourgkl oh Hjgkgbjkl Fbogodjbs ;6, ;;:‒6;. Fljtzur, ^., ]. Aklpfrg, ^. @jfsbagjb`, kgc Y. ^otfgnfri, ;::1, “Dkgkifrjkl Jgbfgtjvfs tjvfs kgc taf Rtrubtu Rtrubturf rf oh Dkgkif Dkgkifdfg dfgtt Nuyouts Nuyouts,— ,— Eourgkl Eourgkl oh Fbogod Fbogodjb jb 28, 239‒89. Nfakvjor kgc Orikgjzktjog 28, Fltog, F., kgc D. Irunfr, ;:9=, “Dkrijgkl Rtob`aolcfrs‚ Skx ^ktfs kgc taf Bljfgtflf Fhhfbt,— ^fvjfw oh Fbogodjbs kgc Rtktjstjbs (Hfnrukry), (Hfnrukry), 81‒93. Hkdk, F., kgc A. Nknjk`, ;:81, “Cjvjcfgc ]oljby5 Kg Fdpjrjbkl Kgklysjs,— Kdfrjbkg (Cfbfdnfr), ;;2>‒8;. Rtktjstjbkl Kssobjktjog Eourgkl (Cfbfdnfr), Hjflc, L. B., ;::9, “Js Jgstjtutjogkl Jgvfstdfgt jg Jgjtjkl ]unljb Ohhfrjgis ^flktfc to taf Logi-^ug ]frhordkgbfs oh Safsf Hjrds?— Yor`jgi pkpfr, ]fgg Rtktf Vgjvfrsjty. Hjsadkg, D., ;:11, “K Safory oh ]rffdptjvf Sk`fovfr Njccjgi,— ^kgc Eourgkl oh ;:, 11‒;=;. Fbogodjbs ;:, , ;:1:, “]rffdptjvf Njccjgi kgc taf ^olf oh taf dfcjud oh Fxbakgif jg Kbqujsjtjogs,— Eourgkl oh Hjgkgbf 33, 33, 3;‒69. Hrkg`s, E., ^. R. Akrrjs, kgc B. Dkyfr, ;:11, “Dfkgs oh ]kydfgt jg Sk`fovfrs5 ^fsu ^fsult ltss hor hor taf taf Vgjtfc Vgjtfc @jgi @jgicod cod kgc kgc taf taf Vgjtf Vgjtfc c Rtktf Rtktfs, s,—— jg Borporktf Sk`fovfrs5 k`fovfrs5 Bkusfs Bkusfs kgc Bogsfqufg Bogsfqufgbfs bfs , K. E. Kufrnkba, (fc.), Bajbkio5 Saf Vgjvfrsjty oh Bajbkio ]rfss. Hucfgnfri, C., kgc E. Sjrolf, ;::;, Ikdf Safory . Bkdnrjcif, DK5 Saf DJS ]rfss.
Njnljoirkpay
>1:
Ijnnogs, ^., ;::>, Ikdf Ikdf Safory Safory hor Kppljf Kppljfc c Fbogod Fbogodjst jsts s . ]rjgbftog, GE5 ]rjgbftog Vgjvfrsjty ]rfss. Irkakd, E., ;::8, “Cfnt kgc Dkrijgkl Skx ^ktf,— Eourgkl oh Hjgkgbjkl Fbogodjbs 3;, 3;‒92. Irossdkg, R., kgc O. Akrt, ;:1=, “Sk`fovfr Njcs, taf Hrff ^jcfr ]ronlfd, kgc taf Safory oh taf Borporktjog,— Nfll Eourgkl oh Fbogodjbs (Rprjgi), 3>‒83. , ;:18, “Saf Bosts kgc Nfgfts oh Owgfrsajp5 K Safory oh Xfrtjbkl kgc Lktfrkl Jgtfirktjog,— Eourgkl oh ]oljtjbkl Fbogodjbs >=, ;96‒>=>. , ;::=, “Sk`fovfr Njcs, taf Hrff ^jcfr ]ronlfd kgc taf Safory oh taf Borporktjog,— Nfll Eourgkl oh Fbogodjbs , 3>‒83. Irullog, I., kgc ^. Djbakfly, >===, “Cjvjcfgcs, Rakrf ^fpurbaksfs, kgc taf Runstjtutjog Aypotafsjs,— Yor`jgi pkpfr, Borgfll Vgjvfrsjty. Iufcfs, E., kgc S. Oplfr, ;::8, “Saf Cftfrdjgkgts oh taf Dkturjty oh Borporktf Cfnt Jssufs,— Eourgkl oh Hjgkgbf 6;, 6;, ;1=:‒22. Akdkck, ^., kgc D. Rbaolfs, ;:16, “Skxfs kgc Borporktf Hjgkgbjkl Dkgkifdfgt,— jg F. J. Kltdkg kgc D. I. Runrkadkgykg, (fcs.), ^fbfgt Kcvkgbfs jg Bor- porktf Hjgkgbf , Aodfwooc, JL, Jrwjg ]rfss, ;:16. Akglfy, @. Y., kgc Y. E. Yjlafld, ;::6, “Fvjcfgbf og taf Rtrktfijb Kllobktjog oh Jgjtjkl ]unljb Ohhfrjgis,— Eourgkl oh Hjgkgbjkl Fbogodjbs 29, >2:‒69. Akgsfg, ^., ;:19, “K Safory hor taf Baojbf oh Fxbakgif Dfcjud jg Dfrifrs kgc Kbqujsjtjogs,— Eourgkl oh Nusjgfss 8=, 8=, 96‒:6. Akrrjs, ^., kgc K. ^kvjv, ;:11, “Borporktf Bogtrol Bogtfsts kgc Bkpjtkl Rtrubturf,— >=, 66‒18. Eourgkl oh Hjgkgbjkl Fbogodjbs >=, , ;::;, “Saf Safory oh Bkpjtkl Rtrubturf,— Eourgkl oh Hjgkgbf 38, 38, >:9‒266. Akrt, O., ;::6, Hjrds, Bogtrkbts kgc Hjgkgbjkl Rtrubturf . Gfw Zor`5 Oxhorc Vgjvfrsjty ]rfss. Akrt, O., kgc E. Doorf, ;:1:, “Cfhkult kgc ^fgfiotjktjog5 K Cygkdjb Docfl oh Cfnt,— DJS wor`jgi pkpfr 6>=. , ;::=, “]ropfrty ^jiats kgc taf Gkturf oh taf Hjrd,— Eourgkl oh ]oljtjbkl :1, ;;;:‒61. Fbogody :1, , ;::6, “Cfnt kgc Rfgjorjty5 Kg Kgklysjs oh taf ^olf oh Akrc Blkjds jg Bogstrkjgjgi Dkgkifdfgt,— Kdfrjbkg Fbogodjb ^fvjfw 16(2), 16(2), 689‒16. Akusba, Akusba, C., kgc E. Rfwkrc Rfwkrc,, ;::2, ;::2, “Rjigk “Rjigkljg ljgi i wjta wjta Cjvjcf Cjvjcfgcs gcs kgc Rakrf Rakrf ^fpur^fpurbaksfs5 K Baojbf nftwffg Cftfrdjgjstjb kgc Rtobakstjb Bksa Cjsnursfdfgts,— ^fvjfw oh Hjgkgbjkl Rtucjfs 8, 8, ;>;‒63. Akyg, B., ;:1:, “Skx Kttrjnutfs ks Cftfrdjgkgts oh Rakrfaolcfr Ikjgs jg Borporktf Kbqujsjtjogs,— Eourgkl oh Hjgkgbjkl Fbogodjbs >2, ;>;‒62. Afklfy Afklfy,, ]., kgc @. ]klfpu ]klfpu,, ;:1:, ;:1:, “Fkrgj “Fkrgjgis gis Jghord Jghordktj ktjog og Bogvfyf Bogvfyfc c ny Cjvjcf Cjvjcfgc gc Jgjtjktjog kgc Odjssjogs,— Eourgkl oh Hjgkgbjkl Fbogodjbs >;, ;3:‒98.
