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]wikzhawdiwli Bipbijl lz}l k|df}i iz}hlb fk akaiwli di}ldi}fjfik|b|f di}ldi}fjfik|b|f zf iz}wakad|b|f Pflwwl$Zfdak Bipbijl" jiwl i |}fbfi} ijliz}i }wikzhawdiwl fk b|jwiwli zi mlzpwl }lawfi pwagigfbf}i}fbaw+ ]wikzhawdiwli Bipbijl i |klf h|kj}ff h h|kj}ff h %} &" &" mlhfkf}i plk}w| }ai}l k|dlwlbl wlibl } ≬ 0" lz}l a h|kj} fl H %z&" mlhfkf}i pwfk lupwlzfi?
Bfdf}i fkhlwfaiwi 0 ∗ lz}l a ka}i}fl pwlzj|w}i}i jiwl fkzlidki
Piwidl}w|b z lz}l fk clklwib jadpblu? Ijliz}i }wikzhawdiwl fk}lcwibi iwl |k k|diw ml pwapwfl}i}f jiwl a hij |}fbi fk ikibfi bfkfiwi i zfz}ldlbaw mfkidfjl+ Jlb dif zldkfhfji}fx ixik}io lz}l ijlbi ji mlwfxiwli zf fk}lcwiwli mlxfk" wlzplj}fx" fkd|b}fwl j| z zf fdpiw}fwl bi z %zfdfbiw j| dam|b fk jiwl baciwf}dff }wikzhawdi a aplwiwl ml fkd|b}fwl i k|dlwlbaw fk im|kiwl i baciwf}dfbaw baw&+ Ijliz}i }wikzhawdi lj|i}ffbl fk}lcwibl zf mfhlwlk}fibl fk lj|i}ff pabfkadfibl" jiwl z|k} d|b} dif |zaw ml wlabxi}+ Ami}i wlabxi}l lj|i}ffbl" zl habazlz}l }wikzhawdiwli Bipbijl fkxlwzi plk}w| i im|jl wl|b}i}lbl fkipaf fk madlkf|b }fdp+ Madlkf| ml jakxlwclk}i Miji Ɨ lz}l a h|kj}fl bajib fk}lcwigfbi" fk }lcwigfbi" i}|kjf }wikzhawdiwli Bipbijl H %z& i b|f Ɨ jakxlwcl miji bfdf}i
lufz}i+ ]wikzhawdiwli Bipbijl jakxlwcl igzab|} miji fk}lcwibi
7
lufz}i %ji fk}lcwibi Blglzc|l&+ ]wikzhawdiwli Bipbijl lz}l fk}lblizi ml wlc|bi fk pwfd|b zlkz" jlb ib jakxlwclk}lf zfdpbl+ D|b}fdli xibawfbaw plk}w| jiwl H %z& lz}l igzab|} jakxlwclk}i lz}l hfl ml hawdi ~zrWl~z ~zrWl~zt ; it" hfl ml hawdi ~zrWl~z ~zrWl~z$ ≬ it" it" |kml i lz}l a jakz}ik}i wlibi lu}fkzi" $ ≭ i ≭ + Jakz}ik}i i lz}l j|kazj|}i ji igzjfzi ml igzab|} jakxlwclk}i" zf mlpfkml ml jwlz}lwli b|f Ɨ %} &+ &+ Ikibac" }wikzhawdi}i gfbi}lwibi jakxlwcl igzab|} pl a hizfl ml hawdi ~zri ~zri 2 Wl~z Wl~zt 2 gt" gt" fkjb|ikm pazfgfb zf bfkffbl Wl~z Wl~zt 4 i zi| Wl~z Wl~zt 4 g+ Z|gd|b}fdli xibawfbaw b|f z plk}w| jiwl }wikzhawdiwli Bipbijl lz}l igzab|} jakxlwclk}i zl k|dlz}l madlkf| ml igzab|} jakxlwclk}i+ jakxlwclk}i+ Fk ji|b gfbi}lwib" lb zl k|dlz}l |klawf hizfi |klawf hizfi ml igzab|} jakxlwclk}i+ jakxlwclk}i+ ]wikzhawdiwli Bipbijl lz}l ikibf}fji fk madlkf|b ml igzab|} jakxlwclk}i+ Zfdfbiw" d|bȞfdli xibawfbaw plk}w| jiwl H %z& jakxlwcl zl k|dlz}l madlkf| ml jakxlwclk}i+ Miji }wikzhawdi}i Bipbijl lz}l jakxlwclk}i bi z 4 z0" i}|kjf li lz}l jakxlwclk}i plk}w| awfjl z j| Wl~z Wl~zt ; Wl~z Wl~z0t+ Mljf" madlkf|b ml jakxlwclk}i lz}l |k zldfpbik ml hawdi ~zrWl~z ~zrWl~zt ; it" it" fkjb|ikm" lxlk}|ib" |klbl p|kj}l ml pl bfkfi Wl~z Wl~zt 4 i+ Fk wlcf|kli ml jakxlwclk}i Wl~z Wl~zt ; Wl~z Wl~z0t" }wikzhawdi}i Bipbijl i b|f Ɨ zl pai}l lupwfdi pwfk fk}lcwiwl pwfk piw}f" fk}lcwibi hffkm
Imfji" fk wlcf|kli ml jakxlwclk}i H %z& pai}l hf lupwfdi}i lhlj}fx ji }wikzhawdiwli Bipbijl igzab|} jakxlwclk}i i ib}lf h|kj}ff+ Fk piw}fj|biw" li l i lz}l ikibf}fji+ Lufz}i dif d|b}l }lawldl" ml hawdi }lawldlbaw Piblv$Sflklw" blci}l ml wlbi}fi mfk}wl pwapwfl}i}fbl b|f Ɨ zf pwapwfl}i}fbl }wikzhawdi}lf Bipbijl fk wlcf|kli ml jakxlwclk}i+
:
Jipf}ab|b F+ Pwapwfl}i}f ml gii ibl }wikzhawdi}lf Bipbijl ]wikzhawdiwli Bipbijl lz}l |k fkz}w|dlk} plwhawdik} jiwl lz}l habazf} i}i} fk di}ldi}fji p|wi ji} zf ipbfji}i+ Ml ijlli lz}l fdpaw}ik} zi fk}lblcld fdpaw}ik}i |}fbf}i}ff lf zf }lawfi lf ml gii+
Jakzfmlwid a h|kj}fl %wlibi zi| jadpblui& mlhfkf}i pl ^0" T+ ]wikzhawdiwli Bipbijl i h|kj}flf h%}& lz}l mlhfkf}i ji hffkm h|kj}fi h ml xiwfigfbl jadpblul " mi}i ml?
∮ ∅ℇ
xi hf mlhfkf}i pl ijlbl k|dlwl jad|kl plk}w| h z|k} B%h& zi| H+
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∅
ℇ
rh%}&r≭I
plk}w| awfjl }≬0+ J| ib}l j|xfk}l" h k| }wlg|fl zi jwlizji pwli wipfm+ Ml luldpb|" awfjl pabfkad zi}fzhijl jakmf}fi %7&+ ]ai}l h|kj}ffbl b|i}l fk jakzfmlwiwl fk wlz}|b jipf}ab|b|f xaw hf pwlz|p|zl ji hffkm ml awmfk lupaklk}fib+ Ml izldlkli zl xi pwlz|p|kl ji pl awfjl fk}lwxib hfkf} ^a"iT" h lz}l diwcfkf}i zf fk}lcwigfbi+
9+9 Igzjfzi ml jakxlwclk}i Pwfd|b wl|b}i} fdpaw}ik} fk ijlz} jipf}ab zl wlhlwi bi ki}|wi d|b}fdff fk jiwl h ˒%& lz}l mlhfkf}i zf ikibf}fji+
=
Hfc+ Zldfpbik|b ml jakxlwclk}i ib }wikzhawdi}lbaw Bipbijl
9+9+9 ]lwawldi ml jakxlwclk}i i }wikzhawdiwff Bipbijl
ℇ
Hfl h?^0"&
%zi| & ml awmfk lupaklk}fib zf hfl
∮ %&4
Miji lufz}i |k k|diw |kfj χ" $ ≭ χ 2 " iz}hlb fkji} ?
∮
lz}l ?
jakxlwclk}i" miji Wl ; χ
mfxlwclk}i" miji Wl 2 χ
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%:&
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∮
%=&
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∅ ∊ r
ℇ
I;0 i+f+ rh%}&r≭I
t" i}|kjf χ ≭ + %<&
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miji %& jakxlwcl pl bfkfi xlw}fjib Wl 4 χ+
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̊ ̊
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Dif pwljfz?
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|k
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Ijliz}i }lawldi k| lz}l i}i} ml zfdpbi pwlj|d piwl+ Fk ijliz}i b|jwiwl" k| z|k} z|hfjflk}l wl|b}i}l plk}w| i mi a mldakz}wi}fl jawlzp|ki}aiwl+ Habazfkm fmlf ml }lawfi fk}lcwiwff" p|}ld lu}fkml wl|b}i}|b }lawldlf ml |kfjf}i}l bi h|kj}ff mfzjak}fk|l" miw iw }wlg|f zi damfhfjid jl fkzlidki ‒lcibf}i}li h|kj}ffbaw—+ Ml luldpb|" miji h%}& zl damfhfji fk}w$a fk}w$a zfkc|wi xibaiwl }" i}|kjf k| zl damfhfji+
8
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6
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Bipbijl z$i jizi}awf} j| Diwfl$Jeiwba}}l ml Ja|w}v ml Wadikclz> j|pb|b i ix|} maf japff? a hi}i" Zapefl" zf |k gifi}" Jeiwblz$Ædfbl %kizj|} fk9356& fk9356&++ D|b}l maj|dlk}l awfcfkibl blci}l ml xfiȞi b|f Bipbijl z $i| pflwm|}" iz}hlb fkji} cab|wfbl gfacwihfjl i| haz} |klawf jadpbl}i}l ml df}|wf+ \klbl maj|dlk}l i| iwz fk fkjlkmf|b jiwl i j|pwfkz jiz}lb|b |k|fi mfk}wl |wdizff zif" fiw ib}lbl i| ji|} pwimi IbfiȞf— fk }fdp|b jlb|f ml$ib hbijiwfbaw fk }fdp|b gadgiwmiwff awiz|b|f Jilk ml ji}wl ‒IbfiȞf— ml$ib mafbli wigaf dakmfib+ Zpwl mlazlgfwl ml pwfl}lk|b zi| Ik}afkl Bixafzflw" Bipbijl k| i ix|} ml z|hlwf} mfk ji|i Wlxab|}flf hwikjll+ Fk }fdp|b ijlz}lfi" lb i jak}wfg|f} bi fk}wam|jlwli zfz}ld|b|f dl}wfj zf bi awcikfiwli pwlz}fcfaizlbaw fkz}f}|}ff ml fkxi}idik} ‛Æjabl pabv}ljekfy|l— zf ‛Æjabl kawdibl z|pæwfl|wl—+ Fk 936< i haz} iblz ji dldgw| ib ka|$jwli}|b|f Fkz}f}|} Hwikjl %b#Fkz}f}|} ml Hwikjl&" Hwikjl&" bi ji}lmwi ml di}ldi}fji" fkz}f}|} ib jiw|f pwlzlmfk}l mlxfkl fk 9597+ Fk 9598 Bipbijl i haz} iblz dldgw| }f}|biw ib Ijimldflf Hwikjll+ Fk 9579" ami}i j| hakmiwli ‛Zajfl}i}ff Clacwihfjl— %Zajfæ}æ ml Cæacwipefl&" Cæacwipefl&" mlxfkl zf pwfd|b pwlzlmfk}l ib ijlz}lfi+ Bipbijl i haz}" ml izldlkli" dldgw| ib pwfkjfpiblbaw ijimldff z}ffk}fhfjl mfk L|wapi ijlbaw }fdp|wf+ Fk 9508" Kipablak b$i fkxlz}f} j| }f}b|b ml jak}l ib Pwfd|b|f Fdplwf| Hwikjl+ M|pi wlz}i|wiȞfi Ga|wgakfbaw i mlxlkf} diwjef %mfk 9593&+ 9593&+ Pflwwl$Zfdak Bipbijl i d|wf} pl < diw}fl 9573" fk baj|fkȞi zi ml bi Iwj|lfb" mfk ipwapflwli Piwfz|b|f" j| maiw ji}lxi fbl fkifk}l ml i fdpbfkf xiwz}i ml 35 ml ikf+ Jak}wfg|}ff fk di}ldi}fji+
Pwfk}wl jak}wfg|}ffbl im|zl ml Bipbijl bi mlxab}iwli di}ldi}fjfbaw p|wl zf ipbfji}l zl pa} lk|dlwi?
]lawfi clklwibi i ml}lwdfkik}fbaw Mldakz}wi}fi }lawldlf jiwl ihfwdi ji awfjl lj|i}fl ibclgwfji ml cwim piw iwl jlb p|}fk |k hij}aw pi}wi}fj wlib> Zab|Ȟfi lj|i}flf j| mlwfxi}l piw}fibl bfkfiwl ml cwim|b maf> I haz} pwfd|b di}ldi}fjfik jiwl i b|i} fk f k jakzfmlwiwl pwagbldlbl mfhfjfbl fdpbfji}l fk lj|i}ffbl j| mfhlwlk}l hfkf}l dfu}l> i mldakz}wi} ji zab|}fi |klf 75
lj|i}ff fk mfhlwlk}l hfkf}l ml cwim|b F zi| ml cwim|b FF pai}l hf ag}fk|}i z|g hawdi |klf hwij}ff jak}fk|l+ Ml k|dlbl b|f Bipbijl lz}l z}wikz blci}i ka}f|kli ml aplwi}aw Bipbijl" Bipbijl" zi| bipbijfik+ bipbijfik+ Ijlz}i lz}l |k aplwi}aw mfhlwlk}fib ml awmfk|b ib mafbli" lbfp}fj" fk zpi}f|b l|jbfmfik k$mfdlkzfakib" mlhfkf} ji mfxlwclk}i cwimflk}|b|f+ Iwl k|dlwaizl ipbfji}ff" ml luldpb| fk hffji" |kml lz}l |}fbfi} bi damlbiwli pwapiciwff |kmlbaw zf pwapiciwff jibm|wff" z}ikm bi gii lj|iȞflf Elbdeab}+ Lz}l
lzlk}fib fk lblj}waz}i}fji zf fk dljikfji hb|fmlbaw" pwfk pwllk}i zi fk lj|i}fi Bipbijl zf fk lj|i}fi Pafzzak+ Fk di}ldi}fji" h|kj}ffbl ib jiwaw bipbijfik lz}l k|b zl k|dlzj h|kj}ff iwdakfjl+ Miji h Miji h lz}l a h|kjȞfl j| xibawf wlibl mlwfxigfbi ml ma|i awf" i}|kjf bipbijfik|b b|f h b|f h lz}l lz}l z|di }|}|waw mlwfxi}lbaw piw}fibl kldfu}l ml awmfk|b maf fk jaawmaki}l jiw}lflkl u jiw}lflkl u f f?
A ib}i jak}wfg|}fl fkzldki}i i b|f Bipbijl" fk ikibfi h|kj}fakibi" lz}l }wikzhawdi}i Bipbijl+ Bipbijl+ Ijliz}i" " lz}l |k aplwi}aw bfkfiw iz|pwi |klf h|kj}ff h h|kj}ff h %} &" &" k|df}i h|kj}fl k|df}i h|kj}fl awfcfkib " ml iwc|dlk} wlib } % } %} ≬ 0&+ Ijlz} aplwi}aw }wikzhawdi awfcfkib|b fk}w$a ib}i h|kj}fl H %z& ml iwc|dlk} jadpblu z" k|df}i h|kj}fl k|df}i h|kj}fl fdicfkl+ fdicfkl+ ]wikzhawdi}i Bipbijl iwl a pwapwfl}i}l lu}wld ml |}fbi" zf ik|dl ijlli ji d|b}l wlbi}ff zf aplwi}ff jl zl lhlj}|lii fk dam j|wlk} iz|pwi awfcfkib|b|f h awfcfkib|b|f h %} & jawlzp|km |kaw wlbi}ff zf aplwi}ff dif zfdpb| ml lhlj}|i} iz|pwi fdicfkff H %z&+ ]wikzhawdi}i Bipbijl iwl k|dlwaizl ipbfji}ff fdpaw}ik}l fk di}ldi}fji" hffji" ap}fji" fkcfklwfl lblj}wfji" i|}adi}fji" pwlb|jwiwli zldkiblbaw zf }lawfi pwagigfbf}i}fbaw+ Fk di}ldi}fji" lz}l habazf}i bi wlabxiwli lj|i}ffbaw mfhlwlk}fibl zf fk}lcwibl+ Fk hffji" lz}l l z}l habazf}i bi ikibfi zfz}ldlbaw bfkfiwl fkxiwfik}l fk }fdp" j|d iw hf jfwj|f}lbl lblj}wfjl" azjfbi}awff iwdakfjf" mfzpaf}fxl ap}fjl zf zfz}ldlbl dljikfjl+ ]wikzhawdi}i Bipbijl i |klf h|kj}ff h h|kj}ff h %} &" &" mlhfkf}i plk}w| }ai}l k|dlwlbl wlibl } ≬ 0" lz}l a h|kj}fl H %z&" mlhfkf}i pwfk lupwlzfi?
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Piwidl}w|b z lz}l fk clklwib k|diw jadpblu? Ijliz}i }wikzhawdiwl fk}lcwibi iwl |k k|diw ml pwapwfl}i}f jiwl a hij |}fbi fk ikibfi bfkfiwi i zfz}ldlbaw mfkidfjl+ Jlb dif zldkfhfji}fx ixik}io lz}l ijlbi ji mlwfxiwli zf fk}lcwiwli mlxfk" wlzplj}fx" fkd|b}fwl j| z zf fdpiw}fwl bi z %zfdfbiw j| dam|b fk jiwl baciwf}dff }wikzhawdi a aplwiwl ml fkd|b}fwl i k|dlwlbaw fk im|kiwl i baciwf}dfbaw baw&+ Ijliz}i }wikzhawdi lj|i}ffbl fk}lcwibl zf mfhlwlk}fibl fk lj|i}ff pabfkadfibl" jiwl z|k} d|b} dif |zaw ml wlabxi}+ Ami}i wlabxi}l lj|i}ffbl" zl habazlz}l }wikzhawdi}i Bipbijl fkxlwzi plk}w| i im|jl wl|b}i}lbl fkipaf fk madlkf|b }fdp+ Fk 9597" Bipbijl i p|gbfji} jlblgwi zi b|jwiwl ]eæawfl ikibv}fy|l mlz pwagigfbf}æz %‛]lawfi ikibf}fji i pwagigfbf}i}fbaw—&" fk jiwl i p|z fk xibaiwl k|dlwaizl wl|b}i}l h|kmidlk}ibl mfk madlkf|b }lawflf pwagigfbf}i}fbaw zf ib z}i}fz}fjff di}ldi}fjl+ Dif luij}" lb i pwllk}i} a ikibfi pwljfzi mfk p|kj} ml xlmlwl di}ldi}fj i fmlff pa}wfxf} jiwlfi pwagigfbf}i}li lz}l wipaw}|b mfk}wl k|diw|b lxlkfdlk}lbaw hixawigfbl zf ib jlbaw pazfgfbl" zf i ipbfji}$a bi pwagbldlbl hffjl jakjwl}l+ ]a} lb i fk}wam|z zf kaȞf|kli ml jawlbi}fl" jiwl xi hf }wi}i}i fk lu}lkza fk b|jwiwfbl b|f Hwikjfz Cib}ak+ Mlzf Jewfz}fiik E|vclkz h|zlzl pwfd|b zixik} jiwl igawmizl ijlz} z|gflj}" fk zljab|b ib UXFF$bli" fiw ib}f di}ldi}fjflkf i| jak}wfg|f} bi fkȞlblclwli hwljxlk}lf lxlkfdlk}lbaw" }lawfi jbizfji i pwagigfbf}i}fbaw i haz} mlzixiwzf}i ml Bipbijl+
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