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Nnj mlql i|be}`l ynhmnl} kkb ml pnu`}`y`}l`} pnu`}`y`}l`} lb uncck}`l#}n uncck}`l#}n}#; }#; l`glbuedkp" l`glbuedkp" @b dl} khglcllb khglcllb glylb ml glw`ed}lb llb ckk} ykb flhkbg kkb plv p|b}" @bm`lb w` gvnn} gljnqlb wnvm}( qkh ml e|vyl m`ed}lv `b ml f||v} ykb dl} p|b}   f` pkuulvlb"

5"6

Vk}`nb Vk}`nblhl lhl Uph`bl Uph`bl#e| #e|vy vylb lb

Llb vk}`nblhl Uph`bl#e|vyl wnvm} ynnvglu}lhm khu1  p∖;

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w` m ` B `b $|)

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w` B `b $|)

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Cl} cllu}kh w` 2 ?" Wl j|bblb m|u ml fku`u ykb mlql Uph`bl#e|vylb q`lb khu1 ω` $|) 3

w` B `b $|)  p∖;

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w` B `b $|)

`3∖b

Nnj mlql i|be}`l ynhmnl} kkb ml hnjkh`}l`}( hnjkh`}l`}( pnu`}`y`}l`} pnu`}`y`}l`} lb uncck}`l#}n uncck}`l#}n}#; }#; l`glbuedkp" l`glbuedkp" @b dl} khglcllb khglcllb glylb glylb ml glw`e glw`ed} d}lb lb llb ckk ckk}} ykb f lhkbg lhkbg kkb plv p|b}" p|b}" @bm`lb @bm`lb w` gvnn} gljnqlb wnvm}( qkh ml e|vyl m`ed}lv `b ml f||v} ykb dl} p|b} m ` pkuulvlb"

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= =";

Cnmlhh Cnmlhhlv lv`b `bg g ykb ykb nppl npplvy vyhk hkjj jjlb lb ]lbunvpvnm|e} lbunvpvnm|e}#npplvy #npplvyhkjj hkjjlb lb

Clb jkb `b ml cllu}l glykhhlb llb npplvyhkj}l gnlm flbkmlvlb kkb ml dkbm ykb llb }lbunvpvnm|e} " M`} fl}ljlb} dl} pvnm|e} ykb }wll e|vylb( wkkvf`a ½l½ llb ½lb y|vyl ykv`llv} ykv`llv} `b ml t#v`ed}`bg( ml kbmlvl `b ml  #v`ed}`bg" @b mk} glykh `u1 b

∏| ∄ Zk( fS ( y ∄ Ze( mS 1 t $|( y) 3

c



 p `(a ω` $|) τa $y)

$50)

`3? a 3?

D`lvf`a wnvm} m|u llb enb}vnhlbl} glfn|wm ykb p|b}lb  p `(a ( m`} `u llb b × c ylhm ykb p|b}lb m`l ml ynvc ykb dl} npplvyhkj}l flpkhlb" Ml i|be}`lu ω` $|) lb τa $y) j|bblb wl `cphlclb}lvlb khu f`aynnvfllhm Flvu}l`b#yllh}lvclb( F#uph`blu(""" Mlql ynnvu}lhh`bg f`lm dl} glckj mk} wl ckjjlh`aj lſte`¤ lſte` ¤lb}l lb}l khgnv`clb j|bblb j|bblb |`}fvl`mlb |`}fvl`mlb ykb e|vylb e|vylb bkkv npplvyhkjj npplvyhkjjlb" lb" Dl} `u `cclvu `cclvu qn mk} `bm`lb `bm`lb y f`aynnvfllhm enbu}kb} gldn|mlb wnvm} glhm}1 t $|( ye ) 3

b

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Uy` ω` $|) cl} Uy` 3

`3?

 p `(a τa $ye )

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a 3?

Wl j|bblb m|u llvu} dl} vlu|h}kk} ykb ml glblu}l unc flvljlblb( lb ylvynhglbu }lhjlbu dl} vlu|h}kk} glfv|`jlb f`a ml flvljlb`bg nylv llb ykv`kflhl |" Ncgljll Ncgljllvm vm jkb led}lv led}lv nnj mkkv ml ynhgnvml ynhgnvml ykb ml uncclb ykb gllb flhkbg `u"

="6

@b}lvpnhlv @b}lvpnhlvlbml lbml }lbunvpvnm| }lbunvpvnm|e}#np e}#npplvyhkj plvyhkjjl jlb b

M`} j|bblb wl f`aynnvfllhm `hh|u}vlvlb kkb ml dkbm ykb ml `b}lvpnhlvlbml yllh}lvc( d`lv `u m|u ω` $|) 3 Hb` $|) lb τa $y) 3 Hac $y)" Ukclb wnvm} m`} m|u1 b

t $|( y ) 3

c

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 p  `(a Hb` $|) Hac $y)

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Wl wl}lb mk} `bm`lb | 3 |` lb y 3 ya glhm} mk} Hb` $|) 3 Hac $y) 3 ;" F`a khhl khhl kbmlvl kbmlvl yllh}lv yllh}lvclb clb `u c`bu}lbu ; ykb ml }wll glh`aj kkb ?" Wl wl}lb m|u mk} dl} vlu|h}kk} ykb ml i|be}`l t $| 3 |` ( y 3 ya ) 3  p `(a " Npb`l|w led}lv glhmlb d`lv ml bkmlhlb ykb ml `b}lvpnhlvlbml yllh}lvclb"

="5

}lbunvpvnm|e} F½ lq`lv#npplvyhkjjlb lq`lv#npplvyhkjjlb

Kbkhnng j|bblb wl nnj ynnv ω` $|) 3 F`b $|) lb τa $y) 3 Fac $y ) ml﬋b`lvlb" D`lvf`a cnl}lb wl led}lv llb jhl`bl kkbpkuu`bg mnnvynlvlb1 ncmk} F½ lq`lv#e|vylb lbjlh glml﬋b`llvm q`ab `b dl} `b}lvykh lq`lv#e|vylb `b}lvykh Z?( Z? ( ;S" Cnl}lb wl ml ykv`kflhlb | lb y dlvuedkhlb1 f`aglynhg `u m|u ml i|be}`l glh`aj kkb1 b

t $v $|) ( u $y)) 3

c

|∖k y∖e   f`(a F`b $v) Fac $u) cl} v $|) 3 ( u $y ) 3 f∖k m∖e `3? a 3?



$=6)

Wl j|bblb j|bblb d`lvf`a d`lvf`a npb`lw ml l`glbued l`glbuedkpplb kpplb ykb ml Flvu}l`b#i Flvu}l`b#i|be}`l |be}`luu glfv|`jl glfv|`jlb b $q`l ;";)" M`} hl`m }n} lbjlhl l`bgluedkpplb1 ="5";

L`glbuedkpplb L`glbuedkpplb ykb F½ lq`lv#npplvyhkjjlb lq`lv#npplvyhkjjlb

@b}lvpnhk}`l

Dl} npplvyhkj npplvyhkj `b}lvpnhllv `b}lvpnhllv}} ml p|b}lb p|b}lb   f?(? (  f?(c (  fb(? lb   fb(c b`l} ml kbmlvl p|b}lb"

Ml enbble}`l }|uulb dl} F½ lq`lv#npplvyhkj}l#ulgclb} lb lq`lv#npplvyhkj}l#ulgclb} q`ab F½ lq`lv#bl} lq`lv#bl } `u kſtlb `bykv`kb} lb h`g} f`bblb ml enbylt en bylt ncd|hhlbml ncd|hhl bml ykb m`} F½lq`lv#bl}" lq`lv#bl} " Kſtlb `bykv`kb} `bykv`kb} lb Enbylt Enbylt ncd|hhlbml ncd|hhlbml

Vkbmlb

Ml vkbmlb ykb dl} npplvyhkj}l wnvmlb fluedvlylb hn|}lv mnnv llb flq`lv#e|vyl cl} ml vkbm#

p|b}lb" ;;

Lykh|k}`l glfl|vm mnnv dl} }wllckkh }nlpkuulb ykb dl} khgnv`}cl ykb ml Eku}lhak| u}l|blbm np $=?)" M`} jnu} m|u c * ; flvlj f lvljlb`bg lb`bglb lb cl} fld|hp fld|hp ykb dl} khgnv`}cl khgnv`}cl ykb ml Eku}lhak| nc ml Uy ` ‐u }l flpkhlb( lb bng llb}al ynnv dl} |`}l`bmlh`ajl vlu|h}kk}1 m|u llb encphlt`}l`} ykb N c · b6 " Wl mnlb lv lylb}|llh gnlm kkb nc dl} bl} }l }vkbupnblvlb" Lykh|k}`l

 

Gvkkmylvdng`bg ="5"6

Nnj d`lv j|bblb wl ml gvkkm ylvdnglb( d`lvf`a q`ab ml |# lb y#v`ed}`bg nbkidkbjlh`aj"

Kiglhl`mlb ykb ykb F½ lq`lv#npplvyhkjjlb lq`lv#npplvyhkjjlb

Ml kiglhl`mlb ykb F½lq`lv#npplvyhkjjlb lq`lv#npplvyhkjjlb q`ab llbyn|m`g llbyn|m`g }l flvljlblb kkb ml dkbm ykb ;";"< lb vlu|h}lvlb `b1 b∖; c ∆  b   t $v $|) ( u $y)) 3 f`*;(a ∖   f`(a F`b∖; $v) Fac $u) ∆| ∉|? `3? a 3? $=5) b c∖; ∆  c   t $v $|) ( u $y)) 3 f`(a *; ∖   f`(a F`b $v) Fac∖; $u) ∆y ∉y? `3? a 3?

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∆ 6   t ∆|∆y

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Flvljlblb lb lykh|lvlb np $| $|` ( ya ) fljnclb wl1   f`*;(a *; ∖   f`*;(a *   f`(a *; ∖   f`(a " Ml }wllml }lvc ykb ml invc|hl `u ml ynnvuplhh`bg ykb   f`*;(a *; kigkkbml np dl} yhkj ykb ml 5 kbmlvl p|b}lb" Ml kiglhl`ml `u m|u llb ckk} ykb ylvued`h }|uulb dl} ynnvuplhml lb ml led}l wkkvml" Mlql wnvm} nnj ml mvkk`#yle}nv glbnlcm" @bm`lb wl

="5"5

Ukclbglu}lhml Ukclbglu}lhml ykb F½ lq`lv#npplvyhkjjlb lq`lv#npplvyhkjjlb

Kbkhnng kkb ml e|vylb e|vylb j|bblb wl nnj }wll }wll F½ lq`lv#npplv lq`lv#npplvyhkjj yhkjjlb lb ukclbu}lhhl ukclbu}lhhlb" b" @b m`} glykh glykh w`hhlb w`hhlb wl npb`l|w llb qljlvl enb}`b|¤Ļ}l`} gkvkbmlvlb" Qn cnl}lb ml vkbmp|b}lb kkb lhjkkv glh`aj q`ab" Lb glhmlb mlqlhiml mlqlhiml ynnvwkkvml ynnvwkkvmlb b khu m`l |`} ;"7" Mlql l`ulb q`ab led}lv led}lv b`l} ynhmnlbml" ynhmnlbml" E ; enb}`b|`}l`} `b ml |# lb y#v`ed}`bg `cph`ellv} gllb enb}`b|`}l`} np dl} npplvyhkj}l" Mlql l`u jkb lbjlh glhmlb ynnv E j enb}`b|`}l`} `bm`lb1 ∆ 6j ∆ 6j  t $ |( y ) 3 t6 $|( y) $==) ; ∆ j |∆ j y ∆ j |∆ j y

="=

Uph`bl Uph`bl#np #npplvy plvyhkj hkjjl jlb b

Dl} }lbunvpvnm|e} ykb }wll uph`bl#e|vylb hl`m} }n} llb F#uph`bl#npplvyhkj1  p

∏| ∄ Zk( fS ( y ∄ Ze( mS 1 u $|( y) 3

{



 p  `(a B `b $|) B ac $y)

`3∖b a 3∖c

D`lvf`a q`ab l`glbuedkpplb }v`y`kkh kkb mlql ykb ml F½ lq`lv#npplvyhkjjlb" lq`lv#npplvyhkjjlb"

;6

$=<)

<

Khgnv` Khgnv`}cl }clb b ynnv ynnv m`uevl} m`uevl}l l cll}j| cll}j|bm` bm`gl gl pvnfh pvnfhlcl lclb b

Cll}j|bm`gl Cll}j|bm`gl pvnfhlclb pvnfhlclb q`ab ykkj lvg encphlt encphlt lb yvkglb yvkglb yllh yllh vljlbw vljlbwlvj" lvj" Wl ckjlb ckjlb llb nbmlvuedl`m nbmlvuedl`m }|uulb fluh`uu`bgupvnfhlclb lb enbu}v|e}`lpvnfhlclb" Cllu}kh q`ab ml pvnfhlclb flplvj} }n} 6M( ckkv ckjjlh`aj |`}fvl`mfkkv bkkv dnglvl m`clbu`lu"

<"; <";

Ml﬋b Ml﬋b`} `}`l `luu

Khynvlbu khgnv`}clb }l j|bblb fluedv`aylb cnl}lb wl lbjlhl }plb nfale}lb ml﬋b`lvlb" Llb p|b} p ∄ L6 1 p 3 $t( $ t( ) cl} t(  ∄ V $ t6 ( 6 ) `u glml﬋b`llvm khu1 Kiu}kbm }|uulb }wll p|b}lb p; 3 $t; ( ; ) lb p6 3 $t  p; p6 x 3 x p

 

$t; ∖ t6 )6 * $ $; ∖ 6 )6

$=9)

Llb h`abu}|j p; p6 wnvm} glml﬋bllvm khu ml ylvqkclh`bg ykb enbyltl encf`bk}`lu( cl} p; lb p6 khu Z?( ;Ss l`bmp|b}lb 1 p; p6 3 z$; ∖ }) p; * } · p6 x} ∄ Z?(

∖∖ p ↖ Llb glv`ed} h`abu}|j  p ; 6 `u llb h`abu}|j wkkvf`a p; dl} flg`bp|b} lb p6 dl} l`bmp|b} `u" Llb yllhdnlj Y  3 p; p6 " " " pb wnvm} glml﬋bllvm khu1 b∖;

Y  3 pb p; ∡



 p` p`*;

$=7)

`3;

ml p|b}lb p` wnvmlb dnljp|b}lb glbnlcm" Llb llbyn|m`gl yllhdnlj Y  3 p; p6 " " " pb `u llb yllhdnlj wkkvf`a glhm}1

∏` 1 p` p` * ; ∠ p`*; p` * 6 3 p`*; ∏`( a 1 x` ∖ a x ≬ 6 1 p` p` * ; ∠  pa pa *; 3 ∌

$=4)

Cl} `bm`elu cnhm|hn b" Llb llbyn|m`gl yllhdnlj ylvmllh} ml v|`c}l `b }wll glf`lmlb( dl} `bwlbm`gl glf`lm( glbn}llvm khu `bw `bw $Y ) Y ) lb dl} |`}wlbm`gl glf`lm ( glbn}llvm glbn}llvm khu |`}w $Y )" Y )" D`lvf`a Y  ⊆ `bw$Y  `bw$Y ))

∖∖ p ∖∖↖ Kbkhnng Dnljp|b}lb q`ab `b }lglbw`aqlvq`b glnvmlbm glnvmlbm `bm`lb `bw $Y ) Y ) u}llmu h`bju h`g} ykb  p ` `*; " Kbkhnng ∖∖ p ∖∖↖ fl}ljlb} w`aqlvq`b mk} mk} `bw $Y ) Y ) u}llmu vled}u h`g} ykb  p ` `*; " Llb enbyltl ylvqkclh`bg C  `u enbylt `bm`lb ∏ p; ( p6 ∄ C  1 p; p6 ⊆ C  Llb Enbyltl yllhdnlj `u llb yllhdnlj wkkvf`a wkkvf`a `bw $Y ) Y ) enbylt `u" Llb Gluhn}lb dkhiyhkj `u llb ylvqkclh`bg p|b}lb m`l khhlckkh kkb llb flpkkhml jkb} ykb llb vled}l h`gglb" Llb enbyltl ncd|hhlbml ykb llb ylvqkclh`bg C  `u ml jhl`bu}l enbyltl ylvqkclh`bg m`l C flyk}" Enby Enby $C )" )"

;5

9

Llby Llbyn|m`gl n|m`gl cll}j| cll}j|bm` bm`gl gl pvnfhl pvnfhlclb clb

D`lv`b D`lv`b fldkbmlhlb fldkbmlhlb wl mv`l pvnfhlclb1 pvnfhlclb1 ml h`gg`bg h`gg`bg ykb ykb }wll }wll h`abu}|jj h`abu}|jjlb lb }lglbny }lglbnylv lv lhjkkv( lhjkkv( ml h`gg`bg h`gg`bg ykb llb p|b} }lb npq`ed}l ykb llb yllhdnlj( lb dl} flvljlblb ykb kkbwlq`gdl`m ykb ub`ap|b}lb `b llb gvnlp h`abu}|jjlb"

9";

H`gg`b H`gg`bg g ykb ykb }wl }wll l h`abu} h`abu}|jj |jjlb lb

Wl yvkglb nbu ki ni glglylb 6 h`abu}|jjlb p; p6 lb p5 p= ( mlql mlql h`abu} h`abu}|jj |jjlb lb lhjkkv lhjkkv ub`amlb ub`amlb ni b`l}" b`l}" M`} flvljlblb wl `b }wll u}kpplb" Ml llvu}l u}kp `u llb ublhhl ﬋h}lv m`l ml cllu}l glykhhlb kh jkb flkb}wn# nvmlb" nvmlb" D`lv yvkglb wl nbu ki ni ml }wll ncd|hhlbml ncd|hhlbml vled}dnlj vled}dnljlb lb ykb ml h`abu}|jjl h`abu}|jjlb b llb glf`lm glcllb dlfflb cl} lhjkkv" lhjkkv" @bm`lb b`l} mkb `u lv gllb lbjlhl lbjlhl cnglh`ajdl`m cnglh`ajdl`m mk} ql lhjkkv lhjkkv ub`amlb" M`} }lu}lb wl mnnv }l }l u}lhhlb mk}1 ckt$t ckt$t $ p; ) ( t $ p6 )) ≬ c`b$t c`b$t $ p5 ) ( t $ p= )) ∧ c`b$t c`b$t $ p; ) ( t $ p6 )) ≭ ckt$t ckt$t $ p5 ) ( t $ p= )) ∧ $=0) ckt$ ckt$ $ p; ) ( t $ p6 )) ≬ c`b$ c`b$ $ p5 ) (  $ p= )) ∧ c`b$ c`b$ $ p; ) ( t $ p6 )) ≭ ckt$ ckt$ $ p5 ) (  $ p= ))

   

Ylvmlv dnlylb wl khhllb bng }l }lu}lb ni ml }wll l`bmp|b}lb ykb h`abu}|j 6 kkb ylvued`hhlbml jkb}lb ykb h`abu}|j ; h`gglb" Lb ncgljllvm" M`} mnlb wl khu ynhg}1

9"6

 

∖∖ p ↖ ∖∖↖ ∖∖↖ ∖∖↖ $ p ; 6 × p; p5 ) · $ p; p6 ×  p; p= ) ≭ ? ∧ ∖∖ p ↖ × p∖∖ p ↖) · $ p∖∖ p ↖ ∖∖↖ $ p 5 = 5 ; 5 = ×  p5 p6 ) ≭ ?

$
H`gg`b H`gg`bg g ykb ykb llb p|b} p|b} }lb npq`e npq`ed}l d}l ykb ykb llb yllhd yllhdnlj nlj

Wl u}lhhlb llb yllhdnlj Y  3 p; p6 " " " pB " Dl} khgnv`}cl gkk} bk ni llb p|b} p ∄ `bw `bw $Y )" Y )" M`} j|bblb wl mnlb mnnv dl} }lhhlb ykb dl} kkb}kh ub`ap|b}lb }|uulb ml yllhdnlj( lb llb dkhyvled}l ylv}vljjlbml ykb|`} dl} p|b} m`l bkkv bkkv nc dl} lylb wlhj wlhj p|b} p|b} jkb hnplb" hnplb" @bm`lb @bm`lb m`} kkb}kh kkb}kh nblylb nblylb `u( ylvw ylvwke ked} d}lb lb wl mk} dl} p|b} `b ml yllhdnlj h`g}( kbmlvu h`g} dl} lvf|`}lb" M`} hlylv} led}lv pvnfhlclb np `bm`lb ml ub`am`bg b`l} np llb h`abu}|j( ckkv np llb dnljp|b} cl} ml yllhdnlj h`g}" @b mk} glykh jkb clb b`l} fluh|`}lb ni  clb ykb mllh $`bwlbm`g.|`}wlbm`g) $`bwlbm`g.|`}wlbm`g) ylvkbmlv}" Wl qn|mlb j|bblb np}lvlb nc llb b`l|wl dkhivled}l dkhivled}l }l blclb" blclb" M`} dn|m} led}lv ylvh`lu ylvh`lu ykb wlvj `b" Llb kbmlvl cnglh`ajdl`m cnglh`ajdl`m `u( llb `b﬋b`}`u`ck `b﬋b`}`u`ckhl hl vn}k}`l kkb }l fvlbglb np ml vled}l" vled}l" M`l pku glhm} ykbki dl} dnljp|b}" Mkb ub`am} ml dkhivled dkhivled}l }l m|u dl} dnljp|b} b`l} cllv lb wnvm} ylvmlv gl¤lykh|llvm lykh|llvm cl} ml b`l|wl dkhivled}l" Wl lykh|lvlb ub`am`bglb dl} flu} cl} llb lylb}#mv`ylb mnnvhnnph`abkhgnv`}cl"

9"5

Ub`am` Ub`am`bg bg `b llb llb ylvqk ylvqkcl clh`b h`bg g h`abu}| h`abu}|jj jjlb lb

D`lvf`a cnl}lb wl bkgkkb ni `b llb ylvqkclh`bg ykb B  h`abu}|jjlb c`bu}lbu }wll h`abu}|jjlb lhjkkv ub`amlb" @bm`lb `lmlv h`abu}|j cl} `lmlv kbmlv qn|mlb ylvglh`aj`bglb dlfflb wl b$b6∖;) }lu}lb bnm`g" Ni  m|u llb encphlt`}l`} ykb N b6 " Lv flu}kk} led}lv nnj llb khgnv`}cl m`l `b N $b · hng$b hng$b)) wlvj}" @b m`} glykh q`ab lv lbjlhl pvlenbm`}`lu1

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