Undergraduate Thesis - The Matrix-Tree Theorem and some related problems

7/30/2019 Undergraduate Thesis - The Matrix-Tree Theorem and some related problems.

http://slidepdf.com/reader/full/undergraduate-thesis-the-matrix-tree-theorem-and-some-related-problems 1/69

TLJWHIF HIWL@HIB ZHLTJ^_LW\

ZHLTJ^_LW\ @D _OLJHOJ

DIOZBW\ @D FIWAJFIWLO_" FJOAIHLO_ IHG LHD@^FIWLO_

Aìhc Iha Gzo

WAJ FIW^LQ&W^JJ WAJ@^JF

IHG _@FJ ^JBIWJG ]^@NBJF_(

Zhgj~c~igziwj Wajsls(

Zhgj~c~igziwj ]~`c~if lh Fiwajfiwlos(

Aih`l & >=9>



TLJWHIF HIWL@HIB ZHLTJ^_LW\

ZHLTJ^_LW\ @D _OLJHOJ

DIOZBW\ @D FIWAJFIWLO_" FJOAIHLO_ IHG LHD@^FIWLO_

Aìhc Iha Gzo

WAJ FIW^LQ&W^JJ WAJ@^JF

IHG _@FJ ^JBIWJG ]^@NBJF_(

Zhgj~c~igziwj Wajsls(

Zhgj~c~igziwj ]~`c~if lh Fiwajfiwlos(

Wajsls Igtls`~; Issò(]~`d(G~( ]aih Wal Ai Gz`hc(

Aih`l & >=9>



Ioeh` bjgcjfjhws

L `zbg blej w` waihe f| igtls`~" ]~`d( ]aih Wal Ai Gz`hc" d`~ aj~ blflwbjss}iwljhoj lh fielhc insw~iow fiwajfiwlos s` jis| w` nj }j~ojltjg" ihg d`~ aj~ sz}}`~w&lhc f| `~e `h wals wajsls( L `zbg ibs` blej w` waihe aj~ o`bbjcjs lh waj Lhswlwzwj `d Fiwajfiwlos d`~ wajl~ ajb} s`~wlhc `zw f| o`hdzsl`hs" ihg d`~ sjwwlhc ih jqif}bj `d jq&ojbbjhoj lh ~jsji~oalhc ihg o`ffzhloiwlhc fiwajfiwlos( Dlhibb|" L `zbg blej w` waihef| }i~jhws d`~ ib i|s njbljtlhc lh fj" sz}}`~wlhc fj" ihg jho`z~iclhc fj s}l~lwzibb|ihg fjhwibb|(

9



O`hwjhws

Ioeh` bjgcjfjhws 9

Lhw~`gzowl`h ?

9 Waj Fiw~lq&W~jj Waj`~jf :

9(9 Waj Fiw~lq&W~jj Waj ~jf d`~ zhgl~jowjg c~i}as( ( ( ( ( ( ( ( ( ( ( ( ( :9(9(9 Oizoa|&Nlhjw i}}~ìoa( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( 19(9(> Gjbjwl h&O`hw~iowl`h i}}~ìoa(( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( 9>9(9(? Waj cjhj~ib swiwjfjhw( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( 94

9(> Waj Fiw~lq&W~jj Waj ~jf d`~ gl~jowjg c~i}as(( ( ( ( ( ( ( ( ( ( ( ( ( ( 939(? Waj Fiw~lq&W~jj Waj`~jf d`~ jlcawjg" gl~jowjg c~i}as( ( ( ( ( ( ( ( ( >?

> Oal}& ~lhc cifj f`gjb" s}ihhlhc w~jjs" o~lwloib o`h cz~iwl`hs ihg

}i~elhc dzhowl`hs( >1

>(9 D`~ zhgl~jowjg c~i}as( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( >1>(9(9 Oal}& ~lhc cifj f`gjb( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( >1>(9(> Waj o~lwloib c~`z}(( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ?=>(9(? _}ihhlhc w~jjs ihg o~lwloib o`h cz~iwl hs( ( ( ( ( ( ( ( ( ( ( ( ( ?3>(9(7 ]i~elhc dzhowl`hs ihg o~lwloib o`h cz~iwl`hs(( ( ( ( ( ( ( ( ( ( ( 77

>(> D`~ gl~jowjg c~i}as(( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( 7<>(>(9 Oal}& ~lhc cifj f`gjb( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( 7<>(>(> Waj o~lwloib c~`z}(( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( :?>(>(? ]i~elhc dzhowl hs ihg s}ihhlhc w~jjs( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( :4>(>(7 ]i~elhc dzhowl`hs ihg o~lwloib o`h cz~iwl`hs(( ( ( ( ( ( ( ( ( ( ( 4?

O`hobzsl`hs 47

Nlnbl`c~i}a| 44

>



Lhw~`gzowl`h

Waj Fiw~lq&W~jj Waj`~jf" `~ El~oaa` “s waj`~jf" ls `hj `d waj obissloib waj`&~jfs lh waj fiwajfiwloib jbg `d c~i}a waj`~|( Lw is ~sw }~`tjg lh 9173 n| CzswitEl~oaa` & i Cj~fih }a|slolsw lh als swzg| `d jbjow~loib hjw `~es( Waj waj`~jf gjf`h&sw~iwjs i `hgj~dzb o`hhjowl`h njw jjh s}ihhlhc w~jjs ihg fiw~lq gjwj~flhihws( D`~i b`hc wlfj" sjtj~ib gl j~jhw i|s `d i}}~ìoalhc ihg cjhj~ibl{lhc wals Fiw~lq&W~jjWaj`~jf aitj njjh cltjh" ihg fih| ~jbiwjg }~`nbjfs aitj njjh swzgljg(

Ih lhwj~jswlhc fiwajfiwloib f`gjb ~jbiwjg w` Fiw~lq&W~jj Waj`~jf ls waj oal}&~lhc cifj( Lw ais njjh i~`zhg d`~ in`zw >= |ji~s" nzw ais ~i}lgb| njjh njo`fj

ih lf}`~wihw ihg lhwj~jswlhc `nkjow `d swzg| lh sw~zowz~ib o`fnlhiw`~los( ^`zcab|s}jielhc" oal}& ~lhc cifj ls i fiwajfiwloib f`gjb o`hsw~zowjg `h c~i}as" lhlwlibl{lhc

lwa i }lbj `d oal}s `h jioa tj~wjq" ihg i sjw `d ~zbjs d`~ ‒ ~lhc oal}s‖ ib`hc waj jgcjs`d waj cltjh c~i}a( Nisj `h waiw cltjh sw~zowz~j" `hj oih hg fih| lhwj~jswlhc ihgsz~}~lslhc ~jszbws aloa i~j i}}bloinbj lh fih| jbgs `d swzg|" hifjb| c~i}a waj`~|"insw~iow ibcjn~i" ibcjn~ilo cj`fjw~|" waj`~jwloib }a|slos" jwo(

Lh wals wajsls" j lbb }~jsjhw fih| i|s `d i}}~ìoalhc Fiw~lq&W~jj Waj`~jfd`~ s`fj obissjs `d c~i}as" hifjb| zhgl~jowjg c~i}as" gl~jowjg c~i}as" ihg jlcawjg"gl~jowjg c~i}as( F`~j tj~" j }~jsjhw ih lhw~`gzowl`h w` waj sw~zowz~j `d i ti~lihw`d oal}& ~lhc cifj f`gjb d`~ waj oisjs `d zhgl~jowjg ihg gl~jowjg c~i}as( Sj ibs`}~jsjhw s`fj ~jbiwl`hs if`hc waj `nkjows ! aloa lbb nj gj hjg biwj~,; waj sjw `d s}ihhlhc w~jjs" waj }i~elhc dzhowl`hs" ihg waj o~lwloib o`h cz~iwl`hs `d i c~i}a" aloai~j tj~| lf}`~wihw lh c~i}a waj`~|" js}jolibb| waj swzg| `d oal}& ~lhc cifj f`gjb(

Waj wajsls i~j gltlgjg lhw` w ` oai}wj~s( LhOai}wj~ 9" j }~jsjhw ti~l`zsi|s `d i}}~ìoalhc Fiw~lq&W~jj Waj`~jf d`~ zhgl~jowjg c~i}as ! U>9V" U?V," gl~jowjg

c~i}as" ihg jlcawjg" gl~jowjg c~i}as U7V( I h`wlojinbj ~jszbw `d wals oai}wj~ ls `z~hj ihib`czj }~``d `d waj waj`~jf d`~ waj gl~jowjg c~i}as oisj( Lh Oai}wj~ >" d`~zhgl~jowjg ihg gl~jowjg c~i}as" j }~jsjhw ih lhw~`gzowl`h w` waj oal}& ~lhc cifjf`gjb" ihg swzg| waj sw~zowz~j `d waj o~lwloib c~`z} `d i cltjh c~i}a !U3V" U97V, lh~jbiwl`h lwa waj Fiw~lq&W~jj Waj`~jf( ‒Waj `~gj~ `d waj o~lwloib c~`z} ls jyzib w`waj hzfnj~ `d s}ihhlhc w~jjs‖ ls `hj `d waj filh ~jszbws `d wals oai}wj~( D`~ wajzhgl~jowjg c~i}as" waj jq}blolw nlkjowl`h njw jjh waj s}ihhlhc w~jjs ihg waj o~lwloibo`h cz~iwl`hs `d i c~i}a ! aloa ls ibs` eh` h is waj O`~l & Bj N`~chj Ibc`~lwaf U99V,ihg waj jq}blolw nlkjowl`h njw jjh waj }i~elhc dzhowl`hs ihg waj o~lwloib o`h cz~iwl`hs

?



!U9?V" U9: V" U<V, i~j ibs` }~jsjhwjg( D`~ waj oisj `d gl~jowjg c~i}as" j }~jsjhw ihjq}blolw nlkjowl`h njw jjh waj }i~elhc dzhowl`hs ihg waj ~``wjg s}ihhlhc w~jjs U9>V(Lh blcaw `d waj nlkjowl`h njw jjh waj }i~elhc dzhowl`hs ihg o~lwloib o`h cz~iwl`hs d`~

zhgl~jowjg c~i}as !lh }i~wlozbi~" waj| iggjg z} w` i o`hswihw o`h cz~iwl`h," j szccjswi yzjswl`h in`zw waj jqlswjhoj `d i slflbi~ `hj d`~ waj oisj `d gl~jowjg c~i}as( Sj ibs`cltj i ~jdj~jhoj w` i }~j}~lhw }i}j~ U93V aloa fi| ihs j~ `z~ yzjswl`h( Nzw d`~ s`fjwjoahloib ~jis`hs" wajl~ `~es glg h`w ~jibb| fiej zs siwlsd|" ihg waj yzjswl`h ls swlbbzhs`btjg(

7



Oai}wj~ 9

Waj Fiw~lq&W~jj Waj`~jf

Waj Fiw~lq&W~jj Waj`~jf ls i waj`~jf d`~ o`zhwlhc waj hzfnj~ `d s}ihhlhc w~jjs `d ic~i}a lh wj~fs `d waj gjwj~flhihw `d i oj~wilh fiw~lq( Lh wals oai}wj~" j lbb lhw~`gzojs`fj i|s `d i}}~ìoalhc wals waj`~jf d`~ s`fj obissjs `d c~i}as(

9(9 Waj Fiw~lq&W~jj Waj`~jf d`~ zhgl~jowjg c~i}as(

Sj njclh lwa s`fj c~i}a waj`~| o`hoj}ws( Ibb gj hlwl`hs }~jsjhwjg aj~j i~j szf&fi~lsjg d~`f fih| gl j~jhw ~js`z~ojs(

Gj hlwl`h 9(9( BjwC 6 ! T" J , nj i hlwj" zhgl~jowjg c~i}a lwa tj~wjq sjwT !C, 6ut 9" t>" ( ( ( " th v" jgcj sjw J !C, 6 uj 9" j>" ( ( ( " jf v" ihg i fi}}lhc aloa fi}s jioajgcj j ∁ J !C, lwa w ` tj~wlojsz" t ∁ T !C, !z" t i~j h`w hjojssi~lb| glswlhow," oibbjgwaj isslchfjhw `d jhg}`lhws ( H`wj waiwC oih o`hwilh fzbwl}bj jgcjs !glswlhow jgcjs

lwa waj sifj }il~ `d jhg}`lhws, ihg b``}s !jgcjs a`sj w ` jhg}`lhws z" t i~j wajsifj,( Sj si| waiw C ls o`hhjowjg ld waj~j jqlsws i ibe njw jjh ih| w ` tj~wlojs `d C(

I sznc~i}a `d i c~i}a C ls i c~i}a A szoa waiwT !A , ⊄ T !C," J !A , ⊄ J !C,"ihg waj isslchfjhw `d jhg}`lhws w` jgcjs lh A ls waj sifj is lh C( I w~jj ls i o`hhjowjgc~i}a lwa h` o|objs( I s}ihhlhc sznc~i}a `d C ls i sznc~i}a lwa tj~wjq sjw T !C,(I s}ihhlhc w~jj ls i s}ihhlhc sznc~i}a waiw ls i w~jj( I s}ihhlhc sznc~i}a `d C hjjgh`w nj o`hhjowjg" ihg i o`hhjowjg sznc~i}a `d C hjjg h`w nj i s}ihhlhc sznc~i}a(Waj hzfnj~ `d s}ihhlhc w~jjs `d C ls lwso`f}bjqlw| !`~ w~jj hzfnj~ , `d C" gjh`wjg isλ!C,( Bjw ί! C, gjh`wj waj sjw `d ibb s}ihhlhc w~jjs `d C" wajh rί! C, r 6 λ!C,( H`wjwaiwλ!C, g`js h`w gj}jhg `h waj hzfnj~ `d b``}s `d C(

:



Gj hlwl`h 9(>( BjwC nj waj c~i}a gj hjg lh Gj hlwl`h 9(9( Isszfj waiw C ais h`b``}s(

Waj lholgjhoj fiw~lq F !C, `d C ls ih h Ó f fiw~lq lwa;

F lk 69 ld waj jgcj j k ls lholgjhw w` tj~wjqt l"= `waj~ lsj(

d`~ 9≧ l ≧ h" 9≧ k≧ f ( _`" jioa o`bzfh `d F o`hwilhs w ` 9" ihg h ∞ > =“s(D`~ jioa o`bzfh `d F " oa``sj `hj h`h&{j~` jhw~|" ihg oaihcj lws slch( Bjw H

nj i fiw~lq `nwilhjg d~`f waiw }~òjss( H`wj waiw jtj~| o`bzfh `d H ais `hj 9" `hj∞9 ihg h ∞ > =“s" ajhoj" waj szf `d waj jhw~ljs lh jioa o`bzfh ls =( Wazs" ibb waj ~` sszf w` = tjow`~" aloa lf}bljs waiw ~ihe! H , 2 h (

Waj gjc~jj fiw~lq G !C, `d C ls ih h Ó h glic`hib fiw~lq lwa waj glic`hibjhw~ljs G ll 6 gjc! t l ," 9≧ l ≧ h" aj~j gjc! t l , gjh`wjs waj hzfnj~ `d jgcjs lholgjhww` tj~wjqt l(

Waj igkiojho| fiw~lq I !C, `d C ls ih h Ó h fiw~lq lwa jhw~ljs I lk i~j wajhzfnj~ `d jgcjs njw jjh tj~wlojs t l ihg t k" 9≧ l" k≧ h(

Waj Bi}biolih fiw~lq B!C, `d C ls ih h Ó h fiw~lq lwa

B lk 6∞f lk ld l 6 k ihg waj~j i~j f lk jgcjs njw jjh t l ihg t k"

gjc! t l , ld l 6 k(

d`~ 9≧ l" k ≧ h( Lh wals oisj"B!C, ls s|ffjw~lo( H`wj waiw d~`f waj gj hlwl`h `d Bi}biolih fiw~lq" B!C, 6 G !C, ∞ I !C,(

D`~ lbbzsw~iwlhc wajsj gj hlwl`hs" o`hslgj~ waj d`bb` lhc jqif}bj;

Jqif}bj 9(?( I c~i}a C ls cltjh is d`bb` ;

t 9

t > t ?

t 7

j 9 j :

j 7

j >

j ? j 4

Zhgl~jowjg c~i}a C

t 9

t > t ?

t 7

j 9

j 7

j >

j ?

I sznc~i}a A `d C

Dlcz~j 9(9; Ih zhgl~jowjg c~i}a C ihg lws sznc~i}a

4



Aj~j" C ais tj~wjq sjw T !C, 6 ut 9" t>" t?" t7v ihg jgcj sjw J !C, 6 uj 9" j>" j?" j7" j: " j4v"is lh waj in`tj }lowz~j( C o`hwilhs fzbwl}bj jgcjs j ? ihg j 4" ihg C ls o`hhjowjg ibs`(

Ih jqif}bj `d i sznc~i}a `d C ls waj c~i}a A aloa ls ibs` cltjh lh waj in`tj

}lowz~j( F`~j tj~" A ls lh diow i s}ihhlhc sznc~i}a !nzw h`w i s}ihhlhc w~jj, `d C(Waj s}ihhlhc w~jjs `d C i~j blswjg njb` ;

t 9

t > t ?

t 7

j 9

j >

j ?

!i,

t 9

t > t ?

t 7j 7

j >

j ?

!n,

t 9

t > t ?

t 7

j 9

j 7

j ?

!o,

t 9

t > t ?

t 7

j 9

j 7

j >

!g,

t 9

t > t ?

t 7

j 9

j >

j 4

!j,

t 9

t > t ?

t 7j 7

j >

j 4

!d,

t 9

t > t ?

t 7

j 9

j 7

j 4

!c,

t 9

t > t ?

t 7

j :

j 7

j >

!a,

t 9

t > t ?

t 7

j 9 j :

j 7

!l,

t 9

t > t ?

t 7

j :

j >

j ?

!k,

t 9

t > t ?

t 7

j :

j >

j 4

!e,

t 9

t > t ?

t 7

j 9 j : j ?

!b,

t 9

t > t ?

t 7

j 9 j : j 4

!f,

Waj~j i~j 9? s}ihhlhc w~jjs" aloa fjihs waj w~jj hzfnj~ λ!C, 6 9?(Waj lholgjhoj fiw~lq `d C;

F !C, 6

j 9 j > j ? j 7 j : j 4

t 9 9 = = 9 9 =t > 9 9 = = = =t ? = 9 9 = 9 9t 7 = = 9 9 = 9

3



Waj gjc~jj fiw~lq ihg igkiojho| fiw~lq `d C i~j;

G !C, 6

t 9 t > t ? t 7

t 9 ? = = =t > = > = =t ? = = 7 =t 7 = = = ?

" I !C, 6

t 9 t > t ? t 7

t 9 = 9 9 9t > 9 = 9 =t ? 9 9 = >t 7 9 = > =

Waj Bi}biolih fiw~lq `d C;

B!C, 6

t 9 t > t ? t 7

t 9 ? ∞9 ∞9 ∞9t > ∞9 > ∞9 =t ? ∞9 ∞9 7 ∞>t 7 ∞9 = ∞> ?

Waj`~jf 9(7( !Waj Fiw~lq&W~jj Waj`~jf, Bjw C nj i hlwj" o`hhjowjg c~i}a lwa`zw b` }s" lwa Bi}biolih fiw~lq B!C, ( Bjw B=!C, gjh`wj B!C, lwa waj bisw ~` ihg o`bzfh ~jf`tjg !`~ lwa waj lwa ~` ihg lwa o`bzfh ~jf`tjg d`~ ih| l,( Wajh

gjw!B=!C,, 6 λ!C, ( !9(9,

Jqif}bj 9(:( Lh Jqif}bj 9(?" j d`zhg waiwλ!C, 6 9? n| blswlhc ibb s}ihhlhc w~jjs`d C( I}}b| Waj`~jf 9(7" j ibs` aitj;

λ!C, 6 gjw? ∞9 ∞9

∞9 > ∞9∞9 ∞9 7

6 9?

9(9(9 Oizoa|&Nlhjw i}}~ìoa(

W` }~`tj Waj`~jf 9(7" j oih zsj ih lf}`~wihw ~jszbw d~`f fiw~lq waj`~|" waj Nlhjw&Oizoa| d`~fzbi( Wals d`~fzbi lh diow ls waj cjhj~ibl{iwl`h `d waj diflbli~ diow waiw ld Iihg N i~j h Ó h fiw~lojs" wajh gjw!IN , 6 gjw!I , gjw!N,( Wals i}}~ìoa is filhb|nisjg `h U?V(

Waj`~jf 9(4( !Waj Nlhjw&Oizoa| Waj`~jf, Bjw I 6 ! i lk , nj ih f Ó h fiw~lq" lwa 9 ≧ l ≧ f " ihg 9 ≧ k ≧ h( Bjw N 6 ! nlk , nj ih h Ó f fiw~lq" lwa 9 ≧ l ≧ h" ihg

1



9 ≧ k≧ f ( Ld f 8 h " wajh gjw!IN , 6 = ( Ld f ≧ h wajh

gjw!IN , 6_

gjw!I U_ V,gjw!NU_ V,( !9(>,

aj~j _ 6 ! k9" k>" ( ( ( " kf , " 9 ≧ k9 2 k> 2 ³ ³ ³ 2 k f ≧ f " ~ihcjs `tj~ ibb f &jbjfjhw sznsjws `d u9">" ( ( ( " hv" I U_ Vgjh`wjs waj fiw~lq d`~fjg d~`f I zslhc o`bzfhs _ 9 !lh waiw `~gj~,"NU_ Vgjh`wjs waj fiw~lq d`~fjg d~`f N zslhc ~` s _ lh waiw `~gj~(

]~``d( Ld f 8 h " j aitj ~ihe! IN , ≧ ~ihe! I ," wazs ~ihe!IN , ≧ h 2 f " nzw `h waj`waj~ aihg" IN ls ih f Ó f fiw~lq" aloa lf}bljs waiw gjw!IN , 6 =(

Ld f ≧ h" j lbb sa` waiw

gjw!IN , 6_

gjw!I U_ V,gjw!NU_ V,( !9(?,

N| gj hlwl`h `d fiw~lq }~`gzow" j aitj

gjw!IN , 6 gjw

h k9 69 i 9 k9 n k9 9 ( ( ( h

kf 69 i 9 kf n kf f(((

(((h k9 69 i fk 9 n k9 9 ( ( ( h

kf 69 i fk f n kf f

" !9(7,

ihg n| waj fzbwl&blhji~lw| `d waj gjwj~flhihw" j aitj

gjw!IN , 6h

k9 69

n k9 9gjwi 9 k9 ( ( ( h

kf 69 i 9 kf n kf f(((

(((i fk 9 ( ( ( h

kf 69 i fk f n kf f

" !9(:,

wajh

gjw!IN , 6h

k9 "k>"((("kf 69

gjw!I U k9" k>" ( ( ( " kf V,n k9 9 ( ( ( n kf f ( !9(4,

Wazs >

gjw!IN , 6! k9 "((("kf ,⊎u9">"((("hv }j~fzwiwl`hs `d ! k9 " ( ( ( " kf ,

gjw!I U k9" ( ( ( " kf V,n k9 9 ( ( ( n kf f (

!9(3,H`wj waiw ld wajk9" ( ( ( " kf i~j h`w glswlhow" wajhgjw!I U k9" k>" ( ( ( " kf V, 6 =" wazs" waj

9 Sj oih sjj lffjgliwjb| waiw I U_ Vw 6 I wU_ V" aj~j I w gjh`wjs waj w~ihs}`sj `d I (> Sj oih cjw ibb oa`lojs n| ~sw oa``slhc f hzfnj~s k9 " k> " ( ( ( " kf d~`f u9">" ( ( ( " hv lwa k9 2 k > 2

³ ³ ³ 2 k f " ihg wajh }j~fzwj wajsj hzfnj~s n| i }j~fzwiwl`h Ϗ ∁ _ h

<



h`h&{j~` wj~fs lh waj szf lbb aitj ibb k9" ( ( ( " kf i~j glswlhow( H` " bjw_ 6 ! k9" ( ( ( " kf ,nj qjg" ihg _ 6 ! k9" ( ( ( " kf , i }j~fzwiwl`h `d _ !l(j(" jbjfjhws `d _ i~j wa`sj `d _ nzw lh gl j~jhw `~gj~,( Bjwke 6 kϏ!e , " aj~j Ϗ ls i }j~fzwiwl`h `h waj sjw `d h jbjfjhws

u9" ( ( ( " hv( Wajhgjw!I U_ V, 6 sch!Ϗ, gjw!I U_ V,( !9(1,

Wazs" j aitj

_

gjw!I U_ V,n k9 9 ( ( ( n kf f 6Ϗ

sch!Ϗ, gjw!I U_ V,n k9 9 ( ( ( n kf f ( !9(<,

H` " bjwχ nj waj lhtj~sj `d Ϗ" wajhke 6 kϏ!χ !e ,, 6 kχ !e , " wajh

Ϗ

sch!Ϗ, n k9 9 ( ( ( n kf f !9(9=,

6χ

sch! χ , n kχ !9, 9 ( ( ( n kχ ! f , f

6 gjw!NU_ V,(

D~`f Jyziwl`hs ! 9(<, ihg ! 9(9=," j aitj ! 9(>,(

Hjqw" j }~`tj s`fj }~`}j~wljs `d H ihg B(

Bjffi 9(3( Bjw H ihg B nj waj fiw~lojs is lh Gj hlwl`h 9(> ( Wajh HH w6 B" aj~j H w gjh`wjs waj w~ihs}`sj `d H (

]~``d( Sj aitj!HH w, lk 6

j e ∁J !C ,

H le H ke(

Ld l 6 k" wajh lh `~gj~ d`~H le H ke 6 =" j fzsw aitj waiw waj jgcj j e o`hhjows wajtj~wlojs t l ihg t k( Lh waiw oisj" `hj `d H le ihg H ke lbb nj 9" ihg waj `waj~ lbb nj∞9!n| gj hlwl`h `d H ,( Wazs j e ∁J !C , H le H ke 6 ∞f lk(Ld l 6 k" wajhH le H le lbb nj 9 ld j e ls s`fj jgcj lwa `hj `d lws jhg}`lhws ls t l" wazs

j e ∁J !C , H le H le 6 gjc! t l ,(

H` isszfj waiw C ls o`hhjowjg" ihg bjwH = nj H lwa lws bisw ~` ~jf`tjg(Wazs H = ais h ∞ 9 ~` s ihg f o`bzfhs( H`wj waiw waj hzfnj~ `d ~` s jyzib w` wajhzfnj~ `d jgcjs lh i s}ihhlhc w~jj `d C( Waj hjqw ~jszbw wjbbs zs waj gjwj~flhihws!z} w` slch, `d ibb !h ∞ 9, Ó !h ∞ 9, sznfiw~lojs `d H = `nwilhjg n| oa``slhc i sjw _ `d

9=



h ∞ 9 jgcjs `d C" ihg wielhc ibb o`bzfhs `d H = lhgjqjg n| waj jgcjs lh _ ( Wazs" wa`sjsznfiw~lojs i~j lhgjjg H =U_ V(

Bjffi 9(1( Bjw _ nj i sjw `d h ∞ 9 jgcjs `d C( Ld _ g`js h`w d`~f waj sjw `d jgcjs `d i s}ihhlhc w~jj `d C" wajh gjw!H =U_ V, 6 =( Ld _ ls waj sjw `d jgcjs `d i s}ihhlhc w~jj `d C" wajh gjw!H =U_ V, 6 ¹ 9

]~``d( Ld _ g`js h`w d`~f waj sjw `d jgcjs `d i s}ihhlhc w~jj `d C" wajh s`fj sznsjw^ `d _ d`~fs waj jgcjs `d i o|obj O lh C( H` " swi~wlhc d~`f ih| tj~wjq `d O " d`~lhswihoj z 9 ∁ T !O ," `hj oih `nwilh waj d`bb` lhc sjyzjhoj `d jgcjs ihg tj~wlojsz 9" d 9" z>" d >" ( ( ( " zs∞9" d s∞9" zs ≩ z 9" szoa waiw d`~ 9≧ l ≧ s ∞ 9" z l ihg z l%9 i~j w `jhg}`lhws `d d l( BjwH Ut" d V gjh`wj waj jhw~| `d H U_ V lhgjqjg n| tj~wjq t ihg jgcj d (

Wajh" d`~ jioa o`bzfh d l ∁ " ld H Uz l" d lV 6∞9" wajh fzbwl}b| waj o`bzfh lhgjqjg n|d l `d H =U_ V n| ∞9" `waj~ lsj fzbwl}b| lw n| 9( Hjqw" igg wajsj f`gl jg o`bzfhs" j

lbb cjw waj = o`bzfh" aloa lf}bljs waiwgjw!H =U_ V, 6 =(Ld _ d`~fs waj sjw `d jgcjs `d i s}ihhlhc w~jj W ( Bjwj nj ih jgcj `d W aloa ls

o`hhjowjg w` t h !waj tj~wjq aloa lhgjqjg waj bisw ~` `d H & waj ~` a`sj ~jf`tlhccltjs zs H =,( Waj o`bzfh lhgjqjg n| j ais jqiowb| `hj h`h{j~` jhw~|" aloa ls jlwaj~9 `~ ∞9 !slhoj j o`hhjows w` waj ~jf`tjg tj~wjq t h ,( ^jf tj waj ~` ihg o`bzfho`hwilhlhc waj h`h&{j~` jhw~| `d o`bzfh j " j `nwilh i hj ! h ∞ >, Ó !h ∞ >, fiw~lqH =( H`wj waiw n| jq}ihglhc waj gjwj~flhihw zslhc waj o`bzfh j " j aitj gjw!H =U_ V, 6¹ gjw!H =,( Ibs` h`wj waiw waj fiw~lqH = ais jqiowb| `hj o`bzfh aitlhc `hb| `hj h`h&{j~` jhw~| !jlwaj~ 9 `~∞9," wazs" i}}b| waj }~òjss in`tj d`~ H =" j cjw i hj fiw~lqH = aloa ais jqiowb| `hj o`bzfh aitlhc `hj h`h&{j~` jhw~| !jlwaj~ 9 `~ ∞9," ihggjw!H =, 6 ¹ gjw!H =,( ^j}jiw waj }~òjss zhwlb waj sl{j `d waj `nwilhjg fiw~lq ls 9Ó 9"

j hibb| bjigs w` waj o`hobzsl`h gjw!H =, 6 ¹ 9( Wazs gjw!H =U_ V, 6 ¹ 9(

Jqif}bj 9(<( N| Gj hlwl`h 9(>" i fiw~lq H `d C oih nj

H !C, 6

j 9 j > j ? j 7 j : j 4

t 9 ∞9 = = ∞9 ∞9 =t > 9 ∞9 = = = =t ? = 9 ∞9 = 9 ∞9t 7 = = 9 9 = 9

D`~ lhswihoj" `hj oih sjj waiw HH w 6 B( F`~j tj~" sz}}`sj _ 6 uj ?" j: " j4v" aloag`js h`w d`~f i s}ihhlhc w~jj `d C( Lh diow" waj sznsjw6 uj ?" j4v `d _ d`~fs i o|obj(

99



Waj fiw~lojs H U_ V ihgH =U_ V i~j

H U_ V 6

j ? j : j 4

t 9 = ∞9 =t > = = =t ? ∞9 9 ∞9t 7 9 = 9

H =U_ V 6

j ? j : j 4

t 9 = ∞9 =t > = = =t ? ∞9 9 ∞9

Wajh" n| waj }~``d `d Bjffi 9(1" i sjyzjhoj oih nj t ?" j?" t7" j4" t?( Ibs`" d~`fH Ut ?" j?V 6∞9 ihg H Ut 7" j4V 6 9 wajh j f`gld| waj o`bzfhs j ? ihg j 4 `d H =U_ V n|~js}jowltjb| fzbwl}b| wajf n| ∞9 ihg 9" aloa ~jwz~h waj w ` o`bzfh tjow`~s !="="9, w

ihg != "="∞9,( Wajl~ szf ls waj {j~` tjow`~(H` " d`~ _ 6 uj 9" j: " j4v aloa d`~fs i s}ihhlhc w~jj `d C" j aitj

gjw!H =U_ V, 6 gjw∞9 ∞9 =9 = == 9 ∞9

6 ! ∞9,gjw∞9 ∞99 =

6 ! ∞9, >gjw!9, 6 9

Sj i~j h` ~jig| d`~ }~`tlhc Waj`~jf 9(7(

]~``d( !Waj`~jf 9(7 & Oizoa|&Nlhjw i}}~ìoa ,D~`f Bjffi 9(3" j aitj B 6 HH w" wazsB= 6 H =H w

=( N| Waj`~jf 9(4" jaitj

gjw!B=, 6_

gjw!H =U_ V,gjw!H w=U_ V, 6

_

!gjw!H =U_ V,,> !9(99,

N| Bjffi 9(1" gjw!H =U_ V, 6 ¹ 9 ld _ d`~fs waj sjw `d i s}ihhlhc w~jj" `waj~ lsjgjw!H =U_ V, 6 =( Waj~jd`~j" waj szf ls jyzib w` λ!C," is gjsl~jg(

9(9(> Gjbjwl`h&O`hw~iowl`h i}}~ ioa(

Lh sjowl`h9(9(9" j aitj }~`tjg waj Fiw~lq&W~jj Waj`~jf zslhc waj Nlhjw&Oizoa|d`~fzbi !Waj`~jf 9(4,( Lh wals sjowl`h" j glsozss ih lhgzowltj }~``d `d waj waj`~jf(Waj lhgzowltj }~``d hjjgs `hb| waj gjbjwl`h&o`hw~iowl`h ~joz~~jhoj ihg waj jq}ihsl`hd`~fzbi d`~ waj gjwj~flhihw(

Gj hlwl`h 9(9=( Bjw C 6 ! T" J , nj i o`hhjowjg c~i}a ihg j ls ih jgcj `d C lwajhg}`lhws z" t ( Waj gjbjwl`h C ∞ j ls kzsw waj c~i}a !T" J R uj v,( Waj o`hw~iowl`h C/jls `nwilhjg d~`f C ∞ j n| lgjhwld|lhc z ihg t !`~ ‒fj~clhc‖ z ihg t w`cjwaj~, w` d`~f

9>



i hj tj~wjq " aloa fjihs ibb jgcjs aitlhc z" t `~ n`wa `d z ihg t is lws jhg}`lhwslbb aitj lhswjig" albj waj `waj~s ~jfilh zhoaihcjg( D`~ o`hw~iowl`h w` fiej sjhsj"j zszibb| ~jyzl~j waiw j h`w nj i b``}(

Jqif}bj 9(99( BjwC nj waj c~i}a cltjh lh Jqif}bj 9(?;

t 9

t > t ?

t 7

j 9 j :

j 7

j >

j ? j 4

C

Wiej j 6 j 9( Waj c~i}as C ∞ j ihg C/j i~j;

t 9

t > t ?

t 7

Waj gjbjwl`h C ∞ j

t ?

t 7

Waj o`hw~iowl`h C/j

Dlcz~j 9(>; Waj gjbjwl`hC ∞ j ihg o`hw~iowl`hC/j

]~`}`slwl`h 9(9>( Bjw λ!C, gjh`wj waj hzfnj~ `d s}ihhlhc w~jjs `d i o`hhjowjg c~i}a

C( Ld j ∁ J !C, ls h`w i b``}" wajh

λ!C, 6 λ!C ∞ j , % λ!C/j , !9(9>,

]~``d( Wals }~``d is ~jdj~jhojg d~`f U>9V(Waj s}ihhlhc w~jjs `d C waiw g` h`w o`hwilhj i~j }~jolsjb| waj s}ihhlhc w~jjs `d C ∞ j (H` " j hjjg w` sa` waiw waj s}ihhlhc w~jj `d C waiw g` o`hwilhj i~j lh nlkjowl`h

lwa waj s}ihhlhc w~jjs `d waj o`hw~iowl`hC/j ( _lhoj b``}s g` h`w i jow waj hzfnj~

9?



`d s}ihhlhc w~jjs" j oih isszfj waiw waj o`hw~iowl`h C/j ais h` b``}s !`waj~ lsj joih ~jf`tj waj b``}s ihg `~e lwa waj h`h&b``} tj~sl`h,(

Sajh j o`hw~iow j lh i s}ihhlhc w~jj waiw o`hwilhs j " j `nwilh i s}ihhlhc

w~jj `d C/j " njoizsj waj ~jszbwlhc sznc~i}a `d C/j ls s}ihhlhc ihg o`hhjowjg ihg aiswaj ~lcaw hzfnj~ `d jgcjs( Waj `waj~ jgcjs filhwilh wajl~ lgjhwlw| zhgj~ o`hw~iowl`h"s` h` w ` w~jjs i~j fi}}jg w` waj sifj s}ihhlhc w~jj `d C/j n| wals `}j~iwl`h( Ibs`"jioa s}ihhlhc w~jj `d C/j i~lsjs lh wals i|" slhoj jq}ihglhc waj hj tj~wjq nioe lhw`j |ljbgs i s}ihhlhc w~jj `d C( _lhoj jioa s}ihhlhc w~jj `d C/j i~lsjs jqiowb| `hoj" wajdzhowl`h ls i nlkjowl`h(

Waj d`bb` lhc jqif}bj lbb lbbzsw~iwj ]~`}`slwl`h9(9>(

Jqif}bj 9(9?( Bjw zs o`hwlhzj lwa Jqif}bj 9(99( ^joibb waiwj 6 j 9 6 t 9t >(

t 9

t > t ?

t 7

C

t 9

t > t ?

t 7

C ∞ j

t ?

t 7

C/j

Waj s}ihhlhc w~jjs `d C∞j ! aloa i~j ibs` waj s}ihhlhc w~jjs `d C h`w o`hwilhlhcj , i~j

t 9

t > t ?

t 7

!i,

t 9

t > t ?

t 7

!n,

t 9

t > t ?

t 7

!o,

t 9

t > t ?

t 7

!g,

t 9

t > t ?

t 7

!j,

Waj s}ihhlhc w~jjs `d C/j i~j

97



t ?

t 7

!i,t ?

t 7

!n,t ?

t 7

!o,t ?

t 7

!g,

t ?

t 7

!j,

t ?

t 7

!d,

t ?

t 7

!c,

t ?

t 7

!a,

D~`f waj sjw `d s}ihhlhc w~jjs `d C/j blswjg in`tj" `hj oih `nsj~tj waiw ld d`~jioa szoa s}ihhlhc w~jj" j ‖jq}ihg nioe‖ w` waj jgcj j 9" j lbb hibb| cjw ibbwaj s}ihhlhc w~jjs `d C o`hwilhlhc j 9( F`~j`tj~" waj hzfnj~ `d s}ihhlhc w~jjs `d Cls h`walhc nzw waj szf `d waj hzfnj~ `d s}ihhlhc w~jjs `d C ∞ j ihg waj hzfnj~ `d s}ihhlhc w~jjs `d C/j " aloa ls jqiowb| waj ]~`}`slwl`h 9(9>(

]~``d( !Waj`~jf 9(7 & Gjbjwl`h&O`hw~iowl`h i}}~ìoa ,Wals }~``d is yz`wjg d~`f waj ~js`z~ojs `d U>9V(

Sj zsj lhgzowl`h `h rJ !C, r( Ld rJ !C, r 6 = ihg rT !C, r 8 9" wajh C ais h`s}ihhlhc w~jjs" ihg B= ls ih ibb&{j~` fiw~lq" ajhoj gjw!B=, 6 =( Ld rJ !C, r 6 = ihgrT !C, r 6 9" wajh C ais `hj s}ihhlhc w~jj ihg B= ls ih =Ó= fiw~lq" aloa n| o`htjhwl`hais gjwj~flhihw 9(

H` " o`hslgj~ rJ !C, r 8 =( N| ~jhzfnj~lhc tj~wlojs" j oih isszfj waiw l 6 9ihg t 9 ihg t > i~j igkiojhw( Bjw j nj waj jgcj lwa jhg}`lhws t 9 ihg t >( N| wajlhgzowl`h a|}`wajsls" λ!C ∞ j , 6 gjw!B=!C ∞ j ,, ihg λ!C/j , 6 gjw!B=!C/j ,,( N|]~`}`slwl`h 9(9>" wajsj szf w` λ!C,( H`wj waiw slhoj b``}s g` h`w i jow waj hzfnj~

`d s}ihhlhc w~jjs" j oih isszfj waiw n`wa C ∞ j ihg C/j i~j b``}bjss( Sj aitj

λ!C, 6 λ!C ∞ j , % λ!C/j ,

6 gjw!B=!C ∞ j ,, % gjw!B=!C/j ,,

6

gjc! t >, ∞ 9 I >? ³ ³ ³ I >h

I ?> gjc! t ? , ³ ³ ³ I ?h(((

((( ((( (((I h > I h ? ³ ³ ³ gjc! t h ,

%gjc! t ? , ³ ³ ³ I ?h

((( ((( (((I h ? ³ ³ ³ gjc! t h ,

9:



6

gjc! t >, I >? ³ ³ ³ I >h

I ?> gjc! t ? , ³ ³ ³ I ?h(((

((( ((( (((I h > I h ? ³ ³ ³ gjc! t h ,

6 gjw!B=!C,, !9(9?,

9(9(? Waj cjhj~ib swiwjfjhw(

Lh Waj`~jf 9(7" j swiwjg waiw waj ~jgzojg Bi}biolih fiw~lq oih nj `nwilhjg n|~jf`tlhc waj lwa ~` ihg lwa o`bzfh( Nzw" waj d`bb` lhc nislo }~`}`slwl`h `d blhji~ibcjn~i ibb` s zs swiwj i f`~j cjhj~ib swiwjfjhw(

]~`}`slwl`h 9(97( Bjw I nj ih h Ó h syzi~j fiw~lq lwa ibb ~` s ihg o`bzfhs aitj szf =( Wajh ibb o`diow`~s `d I i~j jyzib( !^joibb waiw waj o`diow`~ O lk `d waj jhw~| i lk

`d I ls gj hjg is O lk 6 ! ∞9, l% kF lk" aj~j F lk gjh`wj waj gjwj~flhihw `d waj flh`~ fiw~lq `nwilhjg n| ~jf`tlhc waj lwa ~` ihg waj kwa o`bzfh `d I ,

]~``d( BjwK nj waj ibb&9 syzi~j fiw~lq `d sl{jh( Sj lbb sa` waiw gjw!I %K , 6 h>O lkd`~ ih| o`diow`~ O lk" aloa lf}bljs waiw ibb o`diow`~s i~j jyzib( Sj h` oibozbiwj

gjw!I %K , n| }j~d`~flhc ~` ihg o`bzfh `}j~iwl`hs( D`~ slf}blolw|" j sa` gjw!I %K , 6 O 99" nzw waj sifj }~òjss oih nj i}}bljg d`~ ih| O lk(

• Igg ibb `waj~ ~` s `d I % K w` waj ~sw `hj( Wajh jtj~| jhw~| lh waj ~sw ~` lsh" albj waj `waj~ ~` s i~j zhi jowjg(

• Igg ibb `waj~ o`bzfhs w` waj ~sw o`bzfh( Lh waj `nwilhjg fiw~lq" waj jhw~ljs iw}`slwl`h !9"9, ls h>" ibb `waj~ jhw~ljs `d waj ~sw ~` ihg o`bzfh i~jh" ihg waj~jfilhlhc jhw~ljs i~j zhi jowjg(

• Wiej `zw i diow`~h d~`f waj ~sw ~` (

• _znw~iow waj ~sw ~` d~`f jioa `d waj `waj~ ~` s( Waj !9"9, jhw~| `d waj ~jszbwls h" waj ~jfilhlhc jhw~ljs `d waj ~sw o`bzfh i~j =" ihg waj jhw~ljs h`w lh waj

~sw ~` ihg o`bzfh i~j }~jolsjb| wa`sj `d I (

F`~j }~jolsjb|" waj hjqw jqif}bj lbb lbbzsw~iwj waj }~òjss `d }j~d`~flhc wajsj swj}s`h i }i~wlozbi~ fiw~lq I (

94



Jqif}bj 9(9:( Wiej I 6

> = ∞9 ∞9= ? ∞> ∞9

∞9 ∞> ? =

∞9 ∞9 = >

( Wajh

gjw!I % K , 6

? 9 = =9 7 ∞9 == ∞9 7 9= = 9 ?

6

7 7 7 79 7 ∞9 == ∞9 7 9= = 9 ?

6

7> 7 7 77 7 ∞9 =7 ∞9 7 97 = 9 ?

6 7

7 9 9 97 7 ∞9 =7 ∞9 7 97 = 9 ?

6 7

7 9 9 9= ? ∞> ∞9= ∞> ? == ∞9 = >

6 7 >? ∞> ∞9

∞> ? =∞9 = >

Wazs gjw!I %K , 6 h>O 99( Is h`wjg njd`~j" waj sifj }~òjss oih nj i}}bljg d`~ih| O lk" aloa lf}bljs waiw ibb o`diow`~s i~j jyzib(

Lw d`bb` s d~`f ]~`}`slwl`h9(97 waiw

Waj`~jf 9(94( !Waj Fiw~lq&W~jj Waj`~jf, Bjw C nj i hlwj" o`hhjowjg c~i}a lwa`zw b``}s" lwa Bi}biolih fiw~lq B!C, ( Bjw B=!C, gjh`wj B!C, lwa waj lwa ~` ihg waj kwa o`bzfh ~jf`tjg( Wajh

λ!C, 6 ! ∞9, l% kgjw!B=!C,, ( !9(97,

9(> Waj Fiw~lq&W~jj Waj`~jf d`~ gl~jowjg c~i}as(

Gj hlwl`h 9(93( Bjw Ο nj i gl~jowjg c~i}a !`~ glc~i}a d`~ sa`~w, lwa tj~wjq sjwT !Ο, 6 ut 9" t>" ( ( ( " th v" jgcj sjw J !Ο, 6 uj 9" j>" ( ( ( " jf v" ihg ih isslchfjhw `d jhg&}`lhws aloa fi}s jioa jgcj j ∁ J !Ο, w` ih `~gj~jg }il~ ! z" t ,( Sj oibb z" t ~js}jo&wltjb| waj wilb ihg ajig tj~wlojs `d j ( I b``} ls ih jgcj a`sj ajig ihg wilb tj~wlojs i~j

93



waj sifj( Jgcjs aitlhc waj sifj }il~ `d wilb&ajig tj~wlojs i~j oibbjg fzbwl}bj&jgcjs (Waj `zwgjc~jj `zwgjc!t, `d i tj~wjq t ls waj hzfnj~ `d jgcjs aitlhc t is lws wilb tj~wjq(

Gj hlwl`h 9(91( Bjw Ο nj i gl~jowjg c~i}a" ihg bjws nj i tj~wjq `d Ο( Sj si| ℤ lsi sznc~i}a `d Ο ld J !ℤ, ⊄ J !Ο," T !Ο, ⊄ T !ℤ," ihg waj isslchfjhw `d jhg}`lhws lh ℤls waj sifj is lh Ο( I gl~jowjg c~i}a ls oibbjg ! jieb|, o`hhjowjg ld ~j}biolhc ibb `d lws gl~jowjg jgcjs lwa zhgl~jowjg jgcjs }~`gzojs i o`hhjowjg !zhgl~jowjg, c~i}a( Igl~jowjg c~i}a ls oibbjgsw~`hcb| o`hhjowjg ld d`~ ih| w ` glswlhow tj~wlojsz" t waj~j i~jgl~jowjg }iwa d~`f z w` t ihg d~`f t w` z(

I s}ihhlhc w~jj `d Ο gl~jowjg lhw`s" ibs` oibbjg i s}ihhlhc w~jj ~``wjg iw s `~i ~jtj~sj i~n`~jsojhoj sznc~i}a `d Ο" ls i o`hhjowjg sznc~i}a `d Ο aitlhc h` `~ljhwjg

o|objs ihg o`hwilhlhc ibb `d waj tj~wlojs szoa waiw jioa tj~wjq ais `zwgjc~jj 9" jqoj}wd`~s" aloa ais `zwgjc~jj =( Gjh`wj n| ί!Ο " s, waj sjw `d ibb s}ihhlhc w~jjs `d Ο ~``wjgiw s(

Gj hlwl`h 9(9<( Bjw Ο nj i gl~jowjg c~i}a `h h tj~wlojs( Waj gjc~jj fiw~lq G !Ο, `d Ο ls ih h Ó h glic`hib fiw~lq lwa waj glic`hib jhw~ljs G ll 6 `zwgjc! t l ,(

Waj igkiojho| fiw~lq I !Ο, `d Ο ls ih h Ó h fiw~lq lwa jhw~ljs I lk i~j wajhzfnj~ `d jgcjs ! t l" t k,(

Waj Bi}biolih fiw~lq B!Ο, `d Ο ls gj hjg is B!Ο, 6 G !Ο, ∞ I !Ο,(

Lh sjowl`h9(9(>" j gj hjg waj gjbjwl`h ihg o`hw~iowl`h c~i}as d`~ zhgl~jowjgc~i}a C( H` " j oih ibs` gj hj waj ‒ihib`czj‖ o`hoj}ws d`~ gl~jowjg c~i}a Ο(

Gj hlwl`h 9(>=( Bjw Ο nj i gl~jowjg c~i}a( Bjwj 6 ! z" t , ∁ J !Ο,( Waj gjbjwl`h Ο∞ j`d Ο ls waj c~i}a !T !Ο," J !Ο, R uj v,( Waj o`hw~iowl`h Ο/j `d Ο ls `nwilhjg d~`f Ο ∞ jn| ‒fj~clhc‖ w ` tj~wlojs z" t w` d`~f i hj tj~wjq " aloa fjihs ibb jgcjs aitlhcz" t `~ n`wa `d z ihg t is lws jhg}`lhws lbb aitj lhswjig" albj waj `waj~s ~jfilhzhoaihcjg( D`~ o`hw~iowl h w` fiej sjhsj" j ibs` ~jyzl~j waiw j ls h`w i b``}( Bjw!Ο/j , nj waj c~i}a `nwilhjg d~`f Ο n| ~jf`tlhc ! t" z , ihg ibb jgcjs `d waj d`~f ! z" w,!ld aitj, d`~ ih| w∁ T !Ο," ihg wajh ‒fj~clhc‖ z ihg t w`cjwaj~(

Hjqw" j cltj ih lbbzsw~iwlhc jqif}bj d`~ wajsj in`tj gj hlwl`hs(

Jqif}bj 9(>9( O`hslgj~ waj glc~i}a Ο cltjh n|

91



t 9

t > t ?

t 7

j 9 j :

j 7

j >

j ? j 4

Dlcz~j 9(?; I gl~jowjg c~i}a Ο

F`~j }~jolsjb|" T !Ο, 6 ut 9" t>" t?" t7v" ihg J !Ο, 6 uj 9" j>" j?" j7" j: " j4v 6u!t >" t9, "!t ?" t>, "!t 7" t? , "!t 9" t7, "!t 9" t? , "!t ?" t7, v( Wiej waj slhe tj~wjq s 6 t 7(

Waj s}ihhlhc w~jjs `d Ο ~``wjg iws i~j;

t 9

t > t ?

t 7

j 9

j 7

j >

!i,

t 9

t > t ?

t 7

j 9 j : j 4

!n,

t 9

t > t ?

t 7

j 9

j 7

j 4

!o,

Waj gjc~jj fiw~lq" igkiojho| fiw~lq ihg Bi}biolih fiw~lq `d Ο i~j;

G !Ο, 6

> = = == 9 = == = > == = = 9

" I !Ο, 6

= = 9 99 = = == 9 = 9= = 9 =

" B!Ο, 6

> = ∞9 ∞9∞9 9 = == ∞9 > ∞9= = 9 ∞9

Wiej j 6 j : 6 ! t 9" t? ,( Waj c~i}as Ο ∞ j " Ο/j ihg !Ο /j , !is gj hjg lh Gj hlwl`h 9(>=,i~j;

9<



t 9

t > t ?

t 7

Ο∞ j

t >

t 7

Ο/j

t >

t 7

!Ο/j ,

Lh wals oisj" Ο∞ j ais w ` s}ihhlhc w~jjs ! aloa ibs` i~j waj s}ihhlhc w~jjs `d Ο h`w o`hwilhlhc !t 9" t? ,," hifjb|

t 9

t > t ?

t 7

!i,

t 9

t > t ?

t 7

!n,

albj !Ο/j , ais `hb| `hj s}ihhlhc w~jj

t >

t 7

ihg f`~j`tj~" ‖jq}ihg nioe‖ tj~wjq w` waj `~lclhib o`hw~iowjg jgcj !t 9" t? ,"j lbb cjw i s}ihhlhc w~jj `d Ο o`hwilhlhc !t 9" t? , ~``wjg iw s 6 t 7

t 9

t > t ?

t 7

>=



Lh waj hjqw }~`}`slwl`h" j lbb sa` waiw waj hzfnj~ `d s}ihhlhc w~jjs `d Οls h`walhc nzw waj szf `d waj hzfnj~ `d s}ihhlhc w~jjs `d Ο ∞ j ihg waj hzfnj~ `d s}ihhlhc w~jjs `d !Ο/j , ( Aj~j ibb s}ihhlhc w~jjs i~j isszfjg w` nj ~``wjg iw s(

]~`}`slwl`h 9(>>( Bjw Ο nj i o`hhjowjg" gl~jowjg c~i}a lwa`zw b` }s( Bjw s nj i tj~wjq `d Ο ihg j ls ih jgcj `d Ο aloa g`js h`w aitj s is lws jhg}`lhws( Gjh`wj λ!Ο" s,waj hzfnj~ `d s}ihhlhc w~jjs `d Ο ~``wjg iw s( Wajh

λ!Ο" s, 6 λ!Ο ∞ j" s , % λ!!Ο/j , " s, ( !9(9:,

]~``d( Waj s}ihhlhc w~jjs `d Ο ~``wjg iws waiw g` h`w o`hwilhj i~j }~jolsjb| wajs}ihhlhc w~jjs `d Ο∞ j ~``wjg iws(

I s}ihhlhc w~jj `d Ο o`hwilhlhc j ~``wjg iw s" n| gj hlwl`h" oih h`w o`hwilhwaj jgcjs `d waj d`~f ! t" z , `~ !z" w,( Ajhoj" ~jf`tlhc ibb jgcjs `d waj d`~f ! t" z , ihg!z" w, !ld aitj, d`~ ih| w∁ T !Ο, g`js h`w oaihcj waj hzfnj~ `d s}ihhlhc w~jjs `d Οo`hwilhlhc j ~``wjg iw s( Ld j h` fj~cj z ihg t is lh Gj hlwl`h 9(>=" j `nwilhi s}ihhlhc w~jj `d !Ο/j , ~``wjg iw s( Lh wals }~òjss" waj `waj~ jgcjs filhwilh wajl~lgjhwlw|" s` h` w ` w~jjs i~j fi}}jg w` waj sifj s}ihhlhc w~jj `d !Ο /j , ( Ibs`" jioa~``wjg s}ihhlhc w~jj `d !Ο/j , i~lsjs lh wals i| slhoj jq}ihglhc nioe waj jgcj j !h`wjwaiw waj gj hlwl`h `d ~``wjg s}ihhlhc w~jj ibb` s zs jq}ihg j lh i zhlyzj i|, cltjszs i ~``wjg s}ihhlhc w~jj `d Ο( Wazs" waj~j ls i nlkjowl h njw jjh waj sjw `d ~``wjgs}ihhlhc w~jjs `d Ο waiw o`hwilhlhcj ihg waj sjw `d ~``wjg s}ihhlhc w~jjs `d !Ο/j , (

Wajh" λ!Ο" s, 6 λ!Ο ∞ j" s , % λ!!Ο/j , " s,(

Lh waj blcaw `d waj gjbjwl`h&o`hw~iowl`h i}}~ìoa i}}bljg d`~ waj oisj `d zhgl&~jowjg c~i}as" j cltj i hj }~``d d`~ waj Fiw~lq&W~jj Waj`~jf d`~ gl~jowjg c~i}as(

Waj`~jf 9(>?( !Waj Fiw~lq&W~jj Waj`~jf, Bjw Ο nj i gl~jowjg c~i}a lwa Bi}biolih fiw~lq B!Ο,( Bjw s ∁ T !Ο,( Wajh

λ!Ο" s, 6 gjw!B!s , !Ο,, " !9(94,

aj~j B!s , !Ο, gjh`wjs waj fiw~lq `nwilhjg n| gjbjwlhc waj ~` ihg o`bzfh `d B!Ο,o`~~js}`hglhc w` s(

]~``d( Sj zsj lhgzowl`h `h rJ !Ο,r( Ld rJ !Ο,r 6 = ihg rT !Ο,r 8 9" wajh Ο ais h`~``wjg s}ihhlhc w~jjs" ihg B!s , ls ih ibb&{j~` fiw~lq" ajhoj gjw!B!s , , 6 =( Ld rJ !Ο,r 6 =

>9



ihg rT !Ο,r 6 9 ! T !Ο, 6 usv," wajh Ο ais `hj ~``wjg s}ihhlhc w~jj ihg B!s , ls ih = Ó =fiw~lq" aloa n| o`htjhwl`h ais gjwj~flhihw 9(

H` " o`hslgj~ rJ !Ο,r 8 =( N| ~jhzfnj~lhc tj~wlojs ld hjojssi~|" j oih isszfj

waiws 6 t h " ihg waj jgcj j 6 ! t 9" t>, jqlsws( D`~ waj oisj Ο ais `hb| `hj jgcj ! t 9" t>,"waj ~``ws lh !Ο/j , ls `ntl zsb| waj hj tj~wjq `nwilhjg n| fj~clhc( D`~ waj `waj~oisjs" n| o`htjhwl`h" isszfj s /∁ ut 9" t>v( N| lhgzowl`h a|}`wajsls" j aitj

λ!Ο" s, 6 λ!Ο ∞ j" s , % λ!!Ο/j , " s,

6

G 99 ∞ 9 ∞!I 9> ∞ 9, ∞I 9? ³ ³ ³ ∞I 9"h∞9

∞I >9 G >> ∞I >? ³ ³ ³ ∞I >"h∞9

∞I ?9 ∞I ?> G ?? ³ ³ ³ ∞I ?"h∞9

((( ((( ((( ((( (((∞I h∞9"9 ∞I h ∞9"> ∞I h ∞9"? ³ ³ ³ G h ∞9"h∞9

%

G >> ∞ I >9 ∞I >? ³ ³ ³ ∞I >"h∞9

∞!I ?9 % I ?>, G ?? ³ ³ ³ ∞I ?"h∞9(((

((( ((( (((∞!I h ∞9"9 % I h∞9">, ∞I h ∞9"? ³ ³ ³ G h ∞9"h∞9

6

G 99 ∞ 9 ∞I 9> ∞I 9? ³ ³ ³ ∞I 9"h∞9

∞I >9 G >> ∞I >? ³ ³ ³ ∞I >"h∞9

∞I ?9 ∞I ?> G ?? ³ ³ ³ ∞I ?"h∞9(((

(((((( ((( (((

∞I h ∞9"9 ∞I h ∞9"> ∞I h ∞9"? ³ ³ ³ G h∞9"h∞9

∞

9 G 99 ∞ 9 ∞I 9? ³ ³ ³ ∞I 9"h∞9

= ∞I >9 ∞I >? ³ ³ ³ ∞I >"h∞9

= ∞I ?9 G ?? ³ ³ ³ ∞I ?"h∞9(((

(((((( ((( (((

= ∞I h ∞9"9 ∞I h ∞9"? ³ ³ ³ G h ∞9"h∞9

%

G >> ∞I >? ³ ³ ³ ∞I >"h∞9

∞I ?> G ?? ³ ³ ³ I ?"h∞9(((

((( ((( (((∞I h ∞9"> ∞I h ∞9"? ³ ³ ³ G h ∞9"h∞9

%

∞I >9 ∞I >? ³ ³ ³ ∞I >"h∞9

∞I ?9 G ?? ³ ³ ³ ∞I ?"h∞9(((

((( ((( (((∞I h ∞9"9 ∞I h ∞9"? ³ ³ ³ G h ∞9"h∞9

>>



6

G 99 ∞I 9> ∞I 9? ³ ³ ³ ∞I 9"h∞9

∞I >9 G >> ∞I >? ³ ³ ³ ∞I >"h∞9

∞I ?9 ∞I ?> G ?? ³ ³ ³ ∞I ?"h∞9

((( ((( ((( ((( (((∞I h ∞9"9 ∞I h ∞9"> ∞I h ∞9"? ³ ³ ³ G h∞9"h∞9

6 gjw!B!s, !Ο,, !9(93,

9(? Waj Fiw~lq&W~jj Waj`~jf d`~ jlcawjg" gl~jowjg

c~i}as(

Lh wals sjowl`h" j }~jsjhw waj Fiw~lq&W~jj Waj`~jf d`~ i f`~j cjhj~ib obiss `d c~i}as& waj jlcawjg" gl~jowjg c~i}as( Waj sjowl`h is hlsajg lh blcaw `d U7Vihg U>V(Bjw Οh %9 nj i o`f}bjwj glc~i}a `h h % 9 tj~wlojsT 6 ut =" t9" ( ( ( " th v( Waj jgcj sjw d`~Οh %9 ls T Ó T ( Lh }i~wlozbi~" Οh %9 ais i b``} iw jioa tj~wjq( D`~ jioa gl~jowjg jgcj!t l" t k, `d Οh %9" isslch ih lhgjwj~flhiwj qlk !Walhe `d wajsj qlk is cjhj~lo jlcaws `hwaj jgcjs `d Οh %9,( Sj gj hj waj ! h % 9, Ó !h % 9, Bi}biolih fiw~lq B d`~ Οh %9 n|

B lk 6 e 6 l qle d`~l 6 k"∞q lk `waj~ lsj "

d`~l" k∁ u="9" ( ( ( " hv( Wazs" waj szf `d o`bzfhs ls waj {j~` tjow`~" ihg ~` l jho`gjslhd`~fiwl`h in`zw waj jgcjs bjitlhc tj~wjq t l( BjwB!e , gjh`wj waj h Ó h sznfiw~lqd`~fjg n| gjbjwlhc waj e ∞ wa~` ihg o`bzfh `d B(

Gj hlwl`h 9(>7( D`~ jioa e" bjwW e gjh`wj waj o`bbjowl`h `d ibb gl~jowjg s}ihhlhc w~jjs

`d Οh %9 ~``wjg lhw`t e ( Wazs" ld χ ∁ W e " wajh d`~ jioa tj~wjqt l waj~j ls i zhlyzj gl~jowjg}iwa lh χ d~`f t l w` t e

?( Gj hj waj jlcaw `d χ is

w!χ , 6!t l "t k ,∁J !χ ,

qlk( !9(91,

? `waj~ lsj waj `zwgjc~jj `d t l ls f`~j waih 9

>?



Waj`~jf 9(>:( !Waj Fiw~lq&W~jj Waj`~jf, Waj gjwj~flhihw `d B!e , ls waj szf `d waj jlcaws `d ibb s}ihhlhc w~jjs gl~jowjg lhw`t e

gjw!B!e ,

, 6χ ∁W e

w!χ , ( !9(9<,

]~``d( Wals }~``d is d~`f U>V(_lhoj i }j~fzwiwl`h `d waj tj~wlojs lhgzojs i o`~~js}`hglhc }j~fzwiwl`h `d waj

~` s ihg o`bzfhs `d B" lw sz ojs w` o`hslgj~ waj oisje 6 =(Bjw Ϊ nj waj o`bbjowl`h `d ibb gl~jowjg c~i}as `h waj tj~wlojsT 6 ut =" t9" ( ( ( " th v

lh aloa waj `zwgjc~jj `d t = ls = ihg waj `zwgjc~jj `d ibb `waj~ tj~wlojs ls 9( _wi~wlhciw ih| tj~wjq `d ο ∁ Ϊ ihg d`bb` lhc `zwc`lhc jgcjs jlwaj~ bjigs w` i o|obj `~ w` t =(

Sj lgjhwld| jioa o|obj ! t l9 " tl> " ( ( ( " tle , lwa waj }j~fzwiwl`h !l9" l>" ( ( ( " le , ∁ _ h (D`~ jioa ο ∁ Ϊ" gj hj

qο 6!t l "t k ,∁J !ο ,

qlk 6 q9l9 q>l> ( ( ( qhl h " !9(>=,

aj~j J !ο , gjh`wjs waj jgcjs `d ο ( _lhoj jioa tj~wjq `d ο ais `zwgjc~jj 9" jqoj}w t =

ais `zwgjc~jj =" d`~ jioa ο " waj~j ls i zhlyzj f`h`flib q9l9 q>l> ( ( ( qhl h o`~~js}`hglhcw` lw(

Sj oih o`f}zwj waj gjwj~flhihw `d B!=, n| waj d`~fzbi

gjw!B !=, , 6ύ∁_ h

sch !ύ, B9ύ !9, B>ύ !>, ( ( (Bhύ !h , " !9(>9,

aj~j

B lύ !l, 6∞qlύ !l, d`~ l /∁ q!ύ, "

e 6 l q le `waj~ lsj "

Wazsgjw!B!=, , 6

ύ∁_ h

sch !ύ,l /∁ q!ύ ,

!∞qlύ !l, ,l∁ q!ύ , e 6 l

qle ( !9(>>,

Waj f`h`flibs i}}ji~lhc lh waj jq}ihsl`h `d gjw!B!=, , i~j jqiowb| wa`sj qο gjso~lnjgin`tj(

Hjqw" bjw zs w~| w` cz~j `zw aiw waj o`j oljhw `d qο lh waj jq}ihsl`h `d gjw!B!=, , ls n| b`èlhc iw i }i~wlozbi~ jqif}bj(

Jqif}bj 9(>4( Bjw o`hslgj~ waj oisj h 6 <" ihg ύ 6 !> "7"<,!9"1"4"?,(

>7



Waj jq}~jssl h sch !ύ, B9ύ !9, B>ύ !>, ( ( (Bhύ !h , lbb nj

sch !ύ, !∞q>7,! ∞q7<,! ∞q<>, !∞q91 ,! ∞q14 ,! ∞q4? ,! ∞q?9 , !e 6:e8 =

q : e ,!e 63e8 =

q3e , (

Waj o`~~js}`hglhc sznc~i}as i~j

9

1 4

? >

7 <

: 3 =

I~nlw~i~| jgcjs !h`h&b``}s,

_`" waj f`h`flibs lh waj jq}ihsl`h `d sch !ύ, B9ύ !9, B>ύ !>, ( ( (Bhύ !h , d`~ύ 6 !> "7"<,!9"1"4"?,o`~~js}`hg w` ibb sznc~i}as `d Οh %9 o`hwilhlhc

!l, waj o|objs !t >" t7" t<, ihg ! t 9" t1" t4" t? ,"

!ll, ihg waj sznc~i}a a`sj tj~wlojs aitlhc `zwgjc~jj 9" jqoj}w d`~ t = aloa aisgjc~jj =(

Waj slch `d sch !ύ, B9ύ !9, B>ύ !>, ( ( (Bhύ !h , oih nj gjwj~flhjg is d`bb`

ύ 6 !> "7"<,

jtjh o|obj

!9"1"4"?,

`gg o|obj

"

sch !ύ, !∞q>7,! ∞q7<,! ∞q<>,

∞9 d`~ jtjh o|obj

!∞q91 ,! ∞q14 ,! ∞q4? ,! ∞q?9 ,

%9 d`~ `gg o|obj

!e 6:e8 =

q: e ,!e 63e8 =

q3e , (

_`" waj ‖`tj~ibb‖ slch 7 ls !∞9, / o|objs (@nsj~tiwl`h; Jioa }j~fzwiwl`h ύ cltjs ~lsj w` s`fj f`h`flibs" ihg waj slch `d jioaf`h`flib lh waj jq}~jssl`h `d gjw!B!=, , !o`~~js}`hglhc w` ύ, ls !∞9, / o|objs `d ύ (

O`htj~sjb|" bjw h 6 < icilh" aloa }j~fzwiwl`h ύ ∁ _ < cltjs ~lsj w` waj

f`h`flib q9<q>: q?<q71q := q4: q3? q17q<3" ihg aiw slch g`js waj f`h`flib i}}ji~0 Lh`waj~ `~gs" aiw ls waj o`j oljhw `d q9<q>: q?<q71q := q4: q3? q17q<3 lh waj jq}ihsl`h `d gjw!B!=, ,0Waj ihs j~ ls waiw" waj f`h`flib i}}ji~s d`~ jioa ύ ∁ _ < a`sj o|objs i~j i sznsjw`d !?"<"3, ihg !7 "1,( F`~j`tj~

7 Jioa `gg o|obj o`hw~lnzwjs ∞9 ajh oibozbiwj sch !ύ ,

>:



]`sslnlblwljs d`~ύ _lch `d q9<q>: q?<q71q:= q4: q3? q17q<3

!?"<"3,!7 "1, !∞9, >

!?"<"3, !∞9, 9

!7"1, !∞9, 9

!, ! ∞9, =

_zfflhc z} ! ∞9, > % >!∞9, 9 % !∞9, = 6 =

@nsj~tiwl`h; _`fj ύ cltjs ~lsj w` waj sifj f`h flibs( Wazs" lh waj jq}ihsl h `d gjw!B!=, ," `hoj oihojbbiwl`h òoz~s" waj `hb| f`h`flibs bjdw i~j wa`sj o`~~js}`hglhc w`waj sznc~i}as ο aloa o`hwilh h` o|objs(

Lh cjhj~ib" slflbi~ walhcs ai}}jh( D`~ jioa ο ∁ Ϊ" bjw ύ nj i }j~fzwiwl`h

a`sj o|objs i~j o`hwilhjg lh waj o`bbjowl`h `d o|objs `d ο :

( D`~ szoa i ύ" o`hslgj~sch !ύ, B9ύ !9, ( ( (Bhύ !h , ( W` d`~f qο " j hjjg w` oa``sj ih lhgjwj~flhiwj d~`f jioadiow`~B lύ !l, ( Ld l ∁ q!ύ," waj oa`loj figj g`js h`w i jow waj slch( C~`z} waj~jfilhlhc diow`~s ioo`~glhc w` waj o|objs `d ύ( Ld !l9" ( ( ( " le , ls i o|obj `d ύ" wajhB l9 l> B l>l? ( ( (B le ∞ 9 le B le l9 6 ! ∞9, e q l9 l> ( ( ( qle l9 ( Wazs" ld !l9" ( ( ( " le , ls ih jtjh o|obj" wajhi diow`~ `d ∞9 ls o`hw~lnzwjg( Ld !l9" ( ( ( " le , ls ih `gg o|obj" j aitj i diow`~ `d 9" nzwjioa `gg o|obj o`hw~lnzwjs ∞9 ajh oibozbiwlhc sch !ύ,( Wazs" ld ύ ls waj }~`gzow `d wglsk`lhw o|objs `d ο " wajh waj o`~~js}`hglhc szffihg `d ! 9(>9, o`hw~lnzwjs !∞9, w w`

waj o`j oljhw `d qο ( Ajhoj" ld ο ais s 8 = glsk`lhw o|objs" waj o`j oljhw `d qο ls h`s

w6=

sw

!∞9, w6 !9 ∞ 9, s 6 = ( !9(>?,

Waiw bjitjs waj oisj aj~j ο ais h` o|objs" l(j(" ο ls i gl~jowjg s}ihhlhc w~jj `d Οh %9(Lh waiw oisj"qο fzsw o`fj d~`f waj szffihg `d 9(>9 o`~~js}`hglhc w` waj lgjhwlw|}j~fzwiwl`h !wajh waj slch lbb nj %9,(

B99 ( ( (Bhh 6e 9 69

q9e 9 ( ( (e h 6 h

qhe h ( !9(>7,

Jioa szoa qο i~lsjs zhlyzjb| lh waj in`tj jq}ihsl`h" aloa wajh lf}bljs 9(9<(

Waj waj`~jf s}jolibl{js w` waj oisj `d ih| hlwj" jlcawjg" gl~jowjg c~i}a n|sjwwlhcq lk 6 = ajhjtj~ ! t l" t k, ls h`w ih jgcj lh waj c~i}a( D`~ waj oisj `d h`h&

: ^jfjfnj~ waiw j glg lgjhwld| jioa o|obj ! t l 9 " t l > " ( ( ( " tl e , lwa waj }j~fzwiwl`h ! l9 " l> " ( ( ( " le , ∁_ h njd`~j

>4



gl~jowjg c~i}a" j bjwqlk 6 q kl ajhjtj~ ! t l" t k, ls ih jgcj( I h`h& jlcawjg c~i}a lswaj oisj aj~j ibb h`h{j~` jlcaws i~j 9(

>3



Oai}wj~ >

Oal}& ~lhc cifj f`gjb" s}ihhlhc

w~jjs" o~lwloib o`h cz~iwl`hs ihg

}i~elhc dzhowl`hs(

Waj oal}& ~lhc cifj f`gjb ls i fiwajfiwloib f`gjb o`hsw~zowjg `h c~i}as( Lw ls `dwjhzsjg w` gjso~lnj s`fj }a|sloib }ajh`fjhi" szoa is waj }ajh`fjh`h `d sjbd&`~cihl{jgo~lwloiblw|( Lh wals oai}wj~" d`~ zhgl~jowjg ihg gl~jowjg c~i}as" j lhw~`gzoj i ti~lihw`d waj oal}& ~lhc cifj f`gjb ihg }~jsjhw waj sw~zowz~j `d ih lf}`~wihw `nkjow `d walsf`gjb & waj o~lwloib o`h cz~iwl hs ! aloa lbb nj gj hjg biwj~,( Sj ibs` }~jsjhw waj~jbiwl`h njw jjh waj o~lwloib o`h cz~iwl`hs ihg waj Fiw~lq&W~jj Waj`~jf ! aloa jglsozssjg lh waj ~sw oai}wj~,( Lh }i~wlozbi~" waj hzfnj~ `d o~lwloib o`h cz~iwl`hs lsjqiowb| jyzib w` waj hzfnj~ `d s}ihhlhc w~jjs( F`~j`tj~" j lhw~`gzoj waj o`hoj}w `d }i~elhc dzhowl`hs" ihg }~jsjhw s`fj jq}blolw nlkjowl`hs if`hc wajsj `nkjows !o~lwloibo`h cz~iwl`hs & s}ihhlhc w~jjs & }i~elhc dzhowl`hs,(

>(9 D ~ zhgl~jowjg c~i}as(>(9(9 Oal}& ~lhc cifj f`gjb(

BjwC 6 ! T" J , nj i hlwj" o`hhjowjg" zhgl~jowjg" b``}bjss c~i}a lwa i slhe tj~wjq s(I oal}& ~lhc cifj `h C swi~ws lwa i }lbj `d oal}s iw jioa tj~wjq( Io`h cz~iwl`h "ibs` oibbjg i sihg}lbj " Ϗ `d waj cifj ls i dzhowl`h gj hjg `h T !C, szoa waiwϏ!t ,ls waj hzfnj~ `d waj oal}s iw tj~wjq t ( Lh igglwl`h" ld j binjb waj tj~wlojs `d C isT !C, 6 ut 9" t>" ( ( ( " th v" wajhϏ oih ibs` nj ~j}~jsjhwjg is i tjow`~ lh [ h " aj~j waj lwa

>1



jhw~| `d Ϗ o`~~js}`hgs w` waj hzfnj~ `d oal}s iw tj~wjq t l( Iw jioa swj} `d waj cifj"i tj~wjq t 6 s oih nj ~jg " aloa fjihs sjhglhc `hj oal} ib`hc jioa `d lws lholgjhojjgcjs" `hb| ld Ϗ!t , ≪gjc! t ,5 ld Ϗ!t , 2 gjc! t ," j si| waiw t ls swinbj ( D`~ i tj~wjq

t 6 s" gj hj

∄ t ! , 6gjc! t , ld 6 t"

∞º !t" , ld waj~j i~j º !t" , jgcjs o`hhjow t ihg "= `waj~ lsj"

wajh ~lhc t 6 s o`~~js}`hgs w` sjwwlhcϏ 6 Ϗ ∞ ∄ t ( H`wj waiw ∄t ls waj ~ò`~~js}`hglhc w` t `d waj ~jgzojg Bi}biolih fiw~lq Bs `d C aloa ls `nwilhjg d~`f B

n| gjbjwlhc waj ~` ihg o`bzfh o`~~js}`hglhc w` s( Waj slhe s oih nj ~jg ld ihg `hb|ld h` `waj~ ~lhc ls }`sslnbj(

_z}}`sj waiw _ ls i h`h&jf}w| hlwj sjyzjhoj `d !h`w hjojssi~lb| glswlhow,tj~wlojs `d C" szoa waiw swi~wlhc d~`fϏ" waj tj~wlojs oih nj ~jg lh waj `~gj~ `d _ (Ld s /∁ _ " wajh _ ls }~`}j~ ( Ld t òoz~s q !t , wlfjs" j saibb ~jdj~ w` q is waj~j}~jsjhwiwltj tjow`~ d`~ _ ( Waj o`h cz~iwl`h Ϗ `nwilhjg idwj~ ~lhc waj tj~wlojslh waj `~gj~ `d _ ls cltjh n|;

Ϗ !t , 6 Ϗ!t , ∞ q !t ,gjc! t , %6 t

q ! , º !t" , ( !>(9,

aj~j º !t" , gjh`wjs waj hzfnj~ `d jgcjs k`lhlhc t ihg ( Wals ls njoizsj jioa wlfjt ls ~jg" lw b`sjs gjc!t , oal}s" ihg jioa wlfj i tj~wjq 6 t ls ~jg" t cilhs º !t" ,oal}s( BjwB gjh`wjs waj Bi}biolih fiw~lq `d C !is gj hjg lh Gj hlwl`h 9(>," wajh joih ~lwj;

Ϗ 6 Ϗ∞ Bq ( !>(>,

Lh wals ti~lihw" j oih walhe `d s is i nbioe a`bj waiw oih ins`~n oal}s d~`f waj~jsw `d waj s|swjf( Slwa`zw b`ss `d cjhj~iblw|" j oih isszfj waiw" wielhc lhw` ioo`zhwwaj hzfnj~ `d oal}s iw s" waj w`wib hzfnj~ `d oal}s ls =( Waiw ls" i o`h cz~iwl`hϏ ls ihlhwjcj~&tibzjg dzhowl`h siwlsd|lhcϏ!t , ≪= d`~t 6 s" ihg Ϗ!s, 6 ∞ t 6 s Ϗ!t , ≧ =( Io`h cz~iwl`h Ϗ ls swinbj ld =≧ Ϗ!t , 2 gjc! t , d`~ ibbt 6 s( _`" d`~ o`htjhljhoj" lh walsti~lihw" j saibb ~lwjϏ lh waj d`~f Ϗ 6 ! Ϗ!t 9, "Ϗ!t >, " ( ( ( "Ϗ!t h ∞9, "∞ h ∞9

l69 Ϗ!t l ,, ∁ [ h

!aj~j Ϗ is h`w ~lwwjh lh waj o`bzfh tjow`~ d`~f" nzw j saibb swlbb zhgj~swihg lw isi o`bzfh tjow`~ d`~ waj zhld`~f `d jyziwl`hs ~jbiwjg w` Ϗ,( I h`h&jf}w| sjyzjhoj `d

~lhcs B 6 uq9" q>" ( ( ( " qe v ls oibbjg s&bjcib d`~Ϗ ld lw siwls js;

><



• Ld q l 6 s" wajhϏl!ql , ≪gjc! ql ,(

• Ld q l 6 s" wajhϏl ls swinbj(

aj~j Ϗ9 6 Ϗ ihg Ϗl ls waj o`h cz~iwl`h `nwilhjg idwj~ ~lhcql∞9( Waiw ls" jioaòoz~~jhoj `d i tj~wjq t 6 s d`bb` s i o`h cz~iwl`h Ϗ lwa Ϗ!t , ≪gjc! t , ihg jioaòoz~~jhoj `d s d`bb` s i swinbj o`h cz~iwl`h( I o`h cz~iwl`h Ϗ ls silg w` nj ~joz~~jhw

ld waj~j ls is&bjcib sjyzjhoj d`~Ϗ aloa bjigs nioe w` Ϗ( I o~lwloib o`h cz~iwl`h ls io`h cz~iwl`h aloa ls n`wa swinbj ihg ~joz~~jhw(

Waj d`bb` lhc jqif}bj lbb fiej wajsj in`tj gj hlwl`hs f`~j obji~(

Jqif}bj >(9( O`hslgj~ waj c~i}a C cltjh is d`bb` s;

t 9

t > t ?

t 7

j 9 j :

j 7

j >

j ?

_`fj jqif}bjs `d o`h cz~iwl`hs i~j != "="="=," !9"="="∞9," !>"9"="∞?, "!>"7"9"∞?,"!>"="9"∞?,( Aj~j" waj swinbj `hjs i~j !="="="=," !9"="="∞9," !>"9"="∞?," !>"="9"∞?,(Wiej i o`h cz~iwl`h" d`~ lhswihoj Ϗ 6 !> "="9"∞?,( Waj `hb| }`sslnbj ~lhc tj~wjqls s 6 t 7" ihg ~lhc t 7 bjigs w` i hj o`h cz~iwl`h Ϗ 6 !? "=">"∞:,( Hjqw" t 9 lsoihglgiwj d`~ ~lhc !Ϗ !t 9, 6 ? 6 gjc! t 9,," s` ~lhc t 9 lbb cltj zs i hj o`h cz~iwl`hϏ 6 != "9"?"∞7,( O`hwlhzj wals }~òjss" j lbb s``h ~jioa Ϗ icilh( F`~j }~jolsjb|"swi~wlhc d~`fϏ" ~lhc waj tj~wlojss" t 9" t?" t> ~js}jowltjb| lh waiw `~gj~" j oih cjwϏ(_`" Ϗ ls o~lwloib !swinbj % ~joz~~jhw," ihg_ 6 us" t 9" t?" t>v ls ih jqif}bj `d ih s&bjcib

!nzw h`w }~`}j~, sjyzjhoj d`~ Ϗ(

>(9(> Waj o~lwloib c~`z}(

Lh wals sjowl`h" j lbb sa` waiw d`~ i cltjh o`h cz~iwl`hϏ" waj~j jqlsws i zhlyzjo~lwloib o`h cz~iwl`h aloa oih nj ~jioajg d~`f Ϗ n| ih s&bjcib sjyzjhoj( F`~j tj~"waj sjw `d ibb o~lwloib o`h cz~iwl`hs `d ih zhgl~jowjg" o`hhjowjg c~i}aC" w`cjwaj~ lwawaj `}j~iwl`h • ! j lbb gj hj lw biwj~," lbb iowzibb| d`~f ih injblih c~`z}" aloa

j saibb oibb wajo~lwloib c~`z} `d C( Waj `~gj~ `d wals c~`z} ls lh diow jyzib w` waj

?=



hzfnj~ `d s}ihhlhc w~jjs `d C" ihg d~`f aiw j glsozssjg lh waj }~jtl`zs oai}wj~" lwls ibs` waj gjwj~flhihw `d waj ~jgzojg Bi}biolih fiw~lq B=!C,( Fih| swiwjfjhws ihg}~``ds lh wals sjowl`h j~j gl~jowb| olwjg d~`f U3V(

Bjffi >(>( D`~ i o`hhjowjg c~i}a C ihg ih lhlwlib o`h cz~iwl`h Ϗ" waj~j jqlws i s&bjcib sjyzjhoj aloa }~`gzojs i o~lwloib o`h cz~iwl`h(

]~``d( Dl~sw h`wj waiw waj tj~wlojsT !C, R usv oih h`w nj ~jg lh hlwjb| lwa`zw ~lhcs( _z}}`sj waj o`hw~iglowl`h" wajh njoizsj `d waj hlwjhjss `d waj hzfnj~ `d oal}s"waj~j jqlsws i tj~wjq aloa oih nj ~jg is `dwjh is j }bjisj" ihg s` i~j ibb lwshjlcan`z~s( Is C ls o`hhjowjg" lw lf}bljs waiw i hjlcan`z~ `d s ls ~jg is `dwjh is j}bjisj( Nzw" wals fjihs ih lh hlwjb| b`ss `d oal}s !slhoj j g` h`w ~j s," ihg wals

o`hw~iglows waj hlwjhjss `d hzfnj~ `d oal}s( Waj~jd`~j" iw s`fj f`fjhw" j aitjw` ~js" ihg ajhoj ~jioajg i swinbj o`h cz~iwl`h( Ld j ~j s ihg wajh ~j}jiw waj}~òjss" j fi| ~jioa ih`waj~ swinbj o`h cz~iwl`h( Wals }~òjgz~j oih nj ~j}jiwjgis `dwjh is j }bjisj" nzw waj hzfnj~ `d swinbj o`h cz~iwl`h ls hlwj( Wajh" iw bjisw`hj `d wajf fzsw ~joz~s" ihg wals ls i o~lwloib o`h cz~iwl`h(

Sj saibb }~`tj waiw waj~j ls `hb| `hj o~lwloib o`h cz~iwl`h siwlsd|lhc Bjffi >(>(Waj d`bb` lhc o`hsw~zowl`h ls ojhw~ib d`~ waj i~czfjhw(

BjwQ

nj i sjyzjhoj `d tj~wlojs ihg | nj i tjow`~ szoa waiw | !t , ≪= d`~ jtj~|t ∁ T !C,( O`hsw~zow i sjyzjhoj Q | is d`bb` s; gjbjwj ih òoz~~jhoj `d ih| tj~wjq td~`f Q ld lw ls h`w }~jojgjg n| iw bjisw| !t , òoz~~jhojs `d t lh Q ( Lh `waj~ `~gs"ld waj~j i~j f`~j waih | !t , òoz~~jhojs `d t " waj ~sw| !t , `d wajf i~j gjbjwjg" ihg ld waj~j i~j dj j~ waih | !t , òoz~~jhojs" ibb `d wajf i~j gjbjwjg(

Bjffi >(?( Bjw Q ihg \ nj }~`}j~ sjyzjhojs" lwa ~j}~jsjhwiwltj tjow`~s q ihg | "aloa i~j s&bjcib d`~ waj o`h cz~iwl`h Ϗ( Wajh waj sjyzjhoj [ 6 ! \ "Q | , ls ibs`

s&bjcib d`~ Ϗ" ihg lws ~j}~jsjhwiwltj tjow`~ { ls cltjh n|

{!t , 6 fiq uq !t , "| !t ,v( !>(?,

]~``d( Lw“s jh`zca w` sa` waiwQ | ls s&bjcib d`~Ϗ 6 Ϗ∞ B| (Isszfj waiw Q | ls s&bjcib d`~Ϗ z} w` waj }`lhw aj~j i tj~wjq t 6 s ls in`zw

w` nj ~jg d`~ wajlwa wlfj lh Q | " ihg waiw waj o`h cz~iwl`h iw wals wlfj lse | ( Sjhjjg w` sa` waiw e | !t , ≪gjc! t ,( Bjwe nj waj o`h cz~iwl`h aloa òoz~s lffjgliwjb|njd`~j waj o`~~js}`hglhc òoz~~jhoj `d t lh Q !waj !| !t , % l, wa wlfj,( Bjw q = ihg q |

=

?9



gjh`wj waj ~j}~jsjhwiwltj tjow`~s `d waj sjcfjhw `d Q ihg Q | z} w` wa`sj }`lhws" s`waiw;

e 6 Ϗ∞ Bq =" e| 6 ! Ϗ∞ B| , ∞ Bq |

= 6 Ϗ∞ B{ =( !>(7,

aj~j {= 6 | % q |

=( Jtibziwlhc iw t " j aitj

e!t , 6 Ϗ!t , ∞ q =!t ,gjc! t , %6 t

q =! , º ! " t ,

e | !t , 6 Ϗ!t , ∞ {=!t ,gjc! t , %6 t

{=! , º ! " t , !>(:,

_lhoj t ls in`zw w` ~j d`~ wajlwa wlfj lh Q | " j aitj {=!t , 6 | !t , % q |

= !t , 6| !t , % !l ∞ 9," ihg q =!t , 6 ! | !t , % l, ∞ 9( Wazs q =!t , 6 {=!t ,(

H` " ld g`js òoz~ lh Q | " sz}}`sj lw ais òoz~~jg k wlfjs z} w` wals }`lhw"s` waiw{=! , 6 | ! , % k( Ld k 6 =" wajh q =! , ≧ | ! , 6 {=! ,( Ld k 8 =" wajh n|waj o`hsw~zowl`h `d Q | " q =! , 6 | ! , % k 6 {=! ,( Wazs" d`~k≪=" j ib i|s aitjq =! , ≧ {=! ,( Ld g`js h`w òoz~ lh Q | " wajh j ibs` aitj {=! , 6 | ! , ≪q ! , ≪q =! ,( Wazs" q =! , ≧ {=! ,( Ajhoj" d~`f Jyziwl`hs >(:" ihg h`wloj waiwQ ls s&bjcib"

j aitj e | !t , ≪e!t , ≪gjc! t ,(Lw ~jfilhs w` oajoe waj d`~fzbi d`~ {(

• Ld q !t , 8 | !t ," wajh t òoz~s q !t , ∞ | !t , wlfjs lh Q | ( Wazs" {!t , 6 | !t , %!q !t , ∞ | !t ,, 6 q !t ,(

• Ld q !t , ≧ | !t ," wajh t g`js h`w òoz~ lh Q | ( Wazs {!t , 6 | !t ,(

Waj`~jf >(7( _z}}`sj waiw Q ihg \ i~j }~`}j~ sjyzjhojs is lh Bjffi >(? " ihg waiw waj| }~`gzoj o`h cz~iwl`hs Ϗ9 ihg Ϗ>" ~js}jowltjb|( Wajh waj~j jqlsws i o`h cz~iwl`h Ϗ? aloa oih nj gj~ltjg d~`f n`wa Ϗ9 ihg Ϗ> n| i s&bjcib sjyzjhoj(

]~``d( Bjffi >(? wjbbs zs waiwQ | ls s&bjcib d`~Ϗ> 6 Ϗ∞ B| " ihg slflbi~b|" \ q lss&bjcib d`~Ϗ9 6 Ϗ∞ Bq ( Dz~waj~f`~j" waj sjyzjhojs ! \ "Q | , ihg ! Q "\ q , aitj wajsifj ~j}~jsjhwiwltj tjow`~ {" ihg ajhoj waj| bjig w` waj sifj o`h cz~iwl`h Ϗ?(

O`~`bbi~| >(:( Bjw Q ihg \ nj is lh Waj`~jf >(7( Wajh

!i, Ld Ϗ9 ls swinbj" wajh [ 6 ! \ "Q | , ibs` bjigs w` Ϗ9(

!n, Ld n`wa Ϗ9 ihg Ϗ> i~j swinbj" wajh Ϗ9 6 Ϗ>(

?>



]~``d(

!i, Bjffi >(? sa` s waiw \ q ls s&bjcib d`~Ϗ9

6 Ϗ∞ Bq ( Nzw ld Ϗ9

ls swinbj" wajhh` tj~wjq t 6 s oih nj ~jg( Ajhoj \ q fzsw nj jf}w|" wazs q !t , ≪| !t , d`~ ibbt ( Lh wals oisj"[ ais ~j}~jsjhwiwltj tjow`~ { 6 q " ihg s` lw }~`gzojs Ϗ9(

!n, Ld Ϗ> ls ibs` swinbj" wajh" i }i~ibbjb i~czfjhw sa` s waiw| !t , ≪q !t , d`~ ibbt (_` q 6 | " ihg ajhoj Ϗ9 6 Ϗ>(

Bjffi >(4( Ld waj o`h cz~iwl`h Ϗ ls ~joz~~jhw wajh waj~j ls i s&bjcib sjyzjhoj Z d`~ Ϗ aloa ais ~j}~jsjhwiwltj tjow`~ z " waj ibb&9 tjow`~(

]~``d( Dl~sw" bjwH nj waj fiw~lq lhgzojg d~`f waj lholgjhoj fiw~lq F `d C aloaj aitj glsozssjg lh Gj hlwl`h 9(>( _z}}`sj waiw ι ls i hzfj~loib dzhowl`h gj hjg

`h T !C," ~jci~gjg is i o`bzfh tjow`~( Ld ι ls szoa waiwH wι 6 =" wajh n| gj hlwl`h`d H " lw d`bb` s waiwι wiejs waj sifj tibzj iw w ` jhg}`lhws `d ih| jgcj( _lhoj Cls o`hhjowjg" d`~ ih| tj~wlojs t ihg " waj~j ls i }iwa njw jjh wajf( Lw d`bb` s waiwι !t , 6 ι ! ," lh `waj~ `~gs" ι ls o`hswihw( O`htj~sjb|" ld ι ls o`hswihw" lw siwls js

Hwι 6 =( Wazs" Ej~!H

w, 6 uε z v aj~j z ls cltjh n| z !t , 6 9 d`~ ibb t ( H` " ld

H wι 6 = wajh Bι 6 HH wι 6 =( O`htj~sjb|" ld Bι 6 =" wajh ι wH ι 6 rrH wι rr> 6 =" s`H wι 6 =( Wazs" Ej~!B, 6 uε z v aj~j z ls cltjh n| z !t , 6 9 d`~ ibbt (

H` " j }~ tj waj bjffi( _lhoj Ϗ ls ~joz~~jhw" waj~j ls is&bjcib sjyzjhoj ^

d`~Ϗ aloa }~`gzojs Ϗ( Lws ~j}~jsjhwiwltj tjow`~~ siwls js Ϗ 6 Ϗ∞ B~( Ajhoj B~ 6 =aloa lf}bljs waiw ~ ∁ Ej~! B," s` ~ 6 ε z aj~j z ls waj ibb&9 tjow`~( N| waj }~``d

`d Bjffi >(?" ^ z ls s&bjcib d`~Ϗ 6 Ϗ∞ Bz " ihg ais ~j}~jsjhwiwltj tjow`~ ! ε ∞ 9,z (^j}jiwlhc wals }~òjss ε ∞ 9 wlfjs" j `nwilh waj ~jyzl~jg sjyzjhoj Z (

Bjffi >(3( _z}} sj waiw Ϗ ls i o~lwloib o`h cz~iwl`h ihg waiw waj~j ls i s&bjcib sjyzjhoj _ d`~ Ϗ aloa }~`gzojs i o~lwloib o`h cz~iwl`h η ( Wajh η 6 Ϗ(

]~``d( Bjw Z nj is lh Bjffi >(4( N| O`~`bbi~| >(:" !Z "_ z , }~`gzojs η " aloafjihs _ z ls i s&bjcib sjyzjhoj aloa bjigs d~`f Ϗw`η ( Bjwq ihg q z nj ~j}~jsjhwiwltjtjow`~s d`~_ ihg _ z ( Wajh

η 6 Ϗ∞ Bq 6 Ϗ∞ Bq z ( !>(4,

??





ihg szoa waiw b!s, 6 ∞ z 6 s b!z ,( Lw d`bb` s d~`f waj }~``d `d Bjffi >(> waiw waj~jls i hlwj sjyzjhoj `d ~lhcs aloa ~jgzojs b w` i swinbj o`h cz~iwl`he( Bjwq nj waj~j}~jsjhwiwltj tjow`~ `d wals sjyzjhoj( Wajh e 6 b∞ Bq ( Bjw{ 6 d % b∞ e5 wajh

{ 6 d % Bq " s` U{ V 6 Ud V" ihg

{ !z , 6 d !z , % b!z , ∞ e!z , ≪gjc! z , ∞ 9 ∞ e!z , ≪=(

Ajhoj" { ls i o`h cz~iwl`h ~j}~jsjhwlhc waj cltjh o`sjw Ud V(Hjqw" j sa` waiw waj~j ls i jbb&gj hjg dzhowl`h

a ; Ej~Ϛ/ Lf! B, ↞ E !C, !>(3,

cltjh n| a !ι , 6 ο !Ϗ, aj~j Ϗ ls ih| o`h cz~iwl`h lh waj o`sjw ι ( _z}}`sj waiw Ϗ9 ihgϏ> i~j o`h cz~iwl`hs szoa waiw UϏ9V 6 UϏ>V 6ι " wajhϏ9 ∞ Ϗ> 6 Bρ d`~ ρ ∁ O =!C" [ ,(Sj oih ~lwj ρ 6 ρ 9 ∞ ρ > aj~j ρ 9!t , ihg ρ >!t , i~j h`h&hjciwltj d`~ ibb t ( BjwϏ= 6 Ϗ9 ∞ Bρ 9 6 Ϗ> ∞ Bρ >(

_z}}`sj waiw ο !Ϗ9, 6 o9" ihg waiw_ 9 ls i s&bjcib sjyzjhoj d`~Ϗ9 aloa }~`gzojso9( _lhoj o9 ls ~joz~~jhw" j oih oa``sj _ 9 s` waiw ih| tj~wjq t òoz~s iw bjiswρ 9!t ,wlfjs( H` waj }~``d `d Bjffi >(? sa` s waiw waj sjyzjhoj _ ρ 9

9 ls s&bjcib d`~Ϗ=" ihgn| o`hsw~zowl`hs" lw ls `nwilhjg d~`f_ n| gjbjwlhc jqiowb| ρ 9!t , òoz~~jhojs `d t " d`~jioa tj~wjq t ( H`wloj waiwρ 9 ls h`walhc nzw waj ~j}~jsjhwiwltj tjow`~ `d waj sjyzjhoj_ 9 R_ ρ 9

9 " ihg waj sjyzjhoj ! _ 9 R_ ρ 9 "_ ρ 99 , ls kzsw i }j~fzwiwl`h `d _ 9( Ajhoj _ ρ 9

9

i}}bljg w` Ϗ= }~`gzojs o9( Ld d`bb` s waiwο !Ϗ=, 6 o9 6 ο !Ϗ9,( Waj sifj i~czfjhwsa` s waiw ο !Ϗ=, 6 ο !Ϗ>,( Ajhoj a ls jbb&gj hjg(

H` " a ls sz~kjowltj" slhoj d`~ ih| o ∁ E !C," j aitj aUoV 6ο !o, 6 o( F`~j`tj~"a ls ibs` lhkjowltj" slhoj ld waj~j jqlwsϏ9" Ϗ> szoa waiwaUϏ9V 6 aUϏ>V" wajhο !Ϗ9, 6ο !Ϗ>, 6 o" aj~j o ls i o`h cz~iwl`h waiw oih nj ~jioajg d~`f n`wa Ϗ9 ihg Ϗ>( Wazswaj~j jqlwsq

9" q

>s` waiwo 6 Ϗ

9∞ Bq

96 Ϗ

>∞ Bq

>( Ajhoj Ϗ

9∞ Ϗ

>6 B!q

9∞ q

>," ihg

waj~jd`~j UϏ9V 6 UϏ>V( _`"a ls nlkjowltj(

Waj~j ls ih injblih c~`z} sw~zowz~j `h Ej~Ϛ/ Lf! B," gj hjg n| UϏ9V % UϏ>V 6UϏ9 %Ϗ>V( Lw d`bb` s waiwE !C, ls injblih c~`z} zhgj~ waj `}j~iwl`h • " aj~j aUϏ9V•aUϏ>V 6aUϏ9 %Ϗ>V" waiw lsο !Ϗ9, • ο !Ϗ>, 6 ο !Ϗ9 %Ϗ>,( Jyzltibjhwb|" d`~ ih| w ` o~lwloibo`h cz~iwl`hs o9 ihg o>" j aitj o9 • o> 6 ο !o9 % o>,( Sj saibb ~jdj~ w` E !C, is wajo~lwloib c~`z} `d C(

?:



BjwB= gjh`wj waj fiw~lq `nwilhjg n| ~jf`tlhc waj bisw ~` ihg o`bzfh ! aloaj saibb isszfj lwa`zw b`ss `d cjhj~iblw| waiw wals ~` ihg o`bzfh o`~~js}`hg w` waj

slhe tj~wjq s, `d waj Bi}biolih fiw~lq B!C, `d C( Lw ls obji~ n| Waj`~jf 9(7 waiw

B= ls i h`hslhczbi~ fiw~lq( Gj hj B=[ h ∞9 6 uB=| ; | ∁ [ h ∞9v( D~`f waj waj ~| `d biwwlojs !U4V" U9V," `hj oih sjj waiw B=[ h ∞9 ls i sznbiwwloj ! lwa nislsB=, `d waj lhwjc~ibbiwwloj[ h ∞9( F`~j`tj~" `hj oih d`~f waj yz`wljhw c~`z} _ !C, 6 [ h∞9/ B=[ h ∞9" aloa

j saibb oibb wajsihg}lbj c~`z} `d C(Hjqw" j saibb sa` i gl~jow }~``d !nisj `h waj biwwloj waj`~| lhw~`gzojg lhU4V"

ihg U9V, `d ih lf}`~wihw diow waiwrE !C, r 6 rEj~Ϛ/ Lf! B, r 6 r_ !C, r 6 λ!C,( Wals}~``d is h`w }~jsjhwjg lh U3V(

Waj`~jf >(99( Waj~j ls i nlkjowltj o`~~js}`hgjhoj njw jjh waj c~`z}s Ej~ Ϛ/ Lf !B,ihg [ h∞9/ B=[ h ∞9( O`hsjyzjhwb|"

rE !C, r 6 rEj~ Ϛ/ Lf !B, r 6 r[ h ∞9/ B=[ h ∞9r 6 λ!C, !>(1,

aj~j λ!C, ls waj w~jj hzfnj~ `d C(

]~``d( Dl~sw" j saibb sa` waiw waj~j i~j nlkjowltj fi}}lhcs δ ; Ej~Ϛ ↞ [ h ∞9 ihgγ ; Lf B ↞ B=[ h ∞9(

Bjw T !C, 6 ut 9" t>" ( ( ( " th v nj waj tj~wjq sjw `d C( Gj hj d 9" d >" ( ( ( " d h ∞9 ∁O =!C" [ , is;

d l!t h , 6 ∞9" d l!t k, 69" ld k 6 l"

=" `waj~ lsj "

d`~ ibbl 6 9 ">" ( ( ( " hihg k 6 9 ">" ( ( ( " h∞ 9( Wals gj hlwl`h gl~jowb| lf}bljs waiwd 9" d >" ( ( ( " d h ∞9 i~j lh Ej~ Ϛ( F`~j tj~" ld h ∞9

l69 ε ld l 6 =" wajh h ∞9l69 ε ld l!t k, 6 = d`~

ibb k 6 9 ">" ( ( ( " h∞ 9" aloa lf}bljs waiw ε l 6 = d`~ ibbl 6 9 ">" ( ( ( " h∞ 9( Wazs"d 9" d >" ( ( ( " d h ∞9 i~j blhji~b| lhgj}jhgjhw lh Ej~ Ϛ( Wiej ih| d ∁ Ej~Ϛ" wajh d 6

h ∞9l69 ε ld l" aj~j ε l 6 d !t l , ∁ [ " l 6 9 ">" ( ( ( " h∞ 9( @hj oih sjj waiw wals jq}~jssl`h

ls zhlyzjb| gjwj~flhjg( Ajhoj" d 9" d >" ( ( ( " d h∞9 d`~f i nisls d`~ Ej~ Ϛ( Gj hj δ is;

δ !d , 6 ! ε 9" ε>" ( ( ( " εh ∞9, w" !>(<,

d`~ d ∁ Ej~Ϛ( Aj~j ε l 6 d !t l , ∁ [ " l 6 9 ">" ( ( ( " h∞ 9" ihg !ε 9" ε>" ( ( ( " εh ∞9, w gjh`wjs

?4



waj w~ihs}`sj tjow`~ `d !ε 9" ε>" ( ( ( " εh ∞9,( Gj hj γ is;

γ!c, 6 B=δ !c, " !>(9=,

d`~ c ∁ Lf B ⊄ Ej~Ϛ( Lw ls h`w ai~g w` sjj waiwuδ !d 9, " δ !d >, " ( ( ( " δ !d h ∞9,v d`~f wajoih`hloib nisls `d [ h ∞9( Wals lf}bljs waiw δ ls i nlkjowltj o`~~js}`hgjhoj njw jjhEj~Ϛ ihg [ h ∞9( O`hsjyzjhwb|" slhoj B= ls h`hslhczbi~" γ ls i nlkjowltj o`~~js}`hgjhojnjw jjh Lf B ihg B=[ h ∞9( Wazs" waj~j ls i nlkjowltj o`~~js}`hgjhoj njw jjh waj c~`z}sEj~Ϛ/ Lf! B, ihg [ h ∞9/ B=[ h∞9(

D~`f Waj`~jf >(9=" j eh` waiw rE !C, r 6 rEj~Ϛ/ Lf Br( D~`f waj }~jtl zs}i~w `d waj }~``d" j eh` waiwrEj~Ϛ/ Lf Br 6 r[ h ∞9/ B=[ h ∞9r( D~`f waj waj`~| `d

biwwlojs !U4V" U9V," j aitj r[h ∞9

/ B=[h ∞9

r 6 gjw!B=, 6 λ!C,( Wazs" j }~`tjg ! >(1,(

>(9(? _}ihhlhc w~jjs ihg o~lwloib o`h cz~iwl hs(

BjwC 6 ! T" J , nj i o`hhjowjg" zhgl~jowjg c~i}a( Lh wals sjowl`h" j }~jsjhw waj O`~l &Bj N`~chj ibc`~lwaf U99V( Wals ibc`~lwaf gj hjs ih jq}blolw nlkjowl`h njw jjh waj sjw`d ibb s}ihhlhc w~jjs ί! C, ihg waj sjw `d ibb o~lwloib o`h cz~iwl`hsE !C,( _lhoj b``}sg` h`w i jow waj hzfnj~ `d s}ihhlhc w~jjs ihg waj hzfnj~ `d o~lwloib o`h cz~iwl`hs"

j fi| isszfj waiw C ls b``}bjss(

Gj hlwl`h >(9>( Cltjh i c~i}a C 6 ! T" J , ihg ih `~gj~ 2 j `h J ( D`~ o`htjhljhoj"gjh`wj n| j 6 zt waj jgcj `d C a`sj jhg}`lhws i~j tj~wlojs z ihg t ( D~`f h` `h" ld h`w `waj~ lsj fjhwl`hjg" j lbb q waiw `~gj~ `hJ ( Bjws nj i tj~wjq `d C aloa jsaibb oibb waj slhe tj~wjq( Ihjgcj&tj~wjq w~itj~sib 6 ! l , 9≧l≧rT r%rJ r `d C ls i sjyzjhojlh aloa jioa tj~wjq ihg jgcj i}}ji~s jqiowb| `hoj( Bjw 2l nj waj sjw `d ibb jbjfjhwsi}}ji~ njd`~j l lh (

Sj oibb i gjo~jislhc w~itj~sib ld ls ih jgcj&tj~wjq w~itj~sib swi~wlhc d~`fs

siwlsd|lhc;

!l, Ld l ls i tj~wjq `d T R usv" wajh l∞9 ls ih jgcj lholgjhw w` l(

!ll, Ld l ls ih jgcj wajh lw ls fiqlfib" ioo`~glhc w` waj cltjh `~gj~ 2 j " if`hc ibbwaj jgcjs h`w lh 2l ihg lholgjhw w` i tj~wjq lhslgj 2l (

Gjh`wj n| G!C, waj sjw `d ibb gjo~jislhc w~itj~sibs `d C(

Jqif}bj >(9?( !d~`f U99V, Cltjh i c~i}a C is d`bb`

?3



w

s

z

t

:

9

?

7

>

Bjws nj waj slhe tj~wjq( Wajh" waj gjo~jislhc w~itj~sibs `d C i~j;

s :

w ?

t 7z > 9> 9 z

>t 7

z 99 z

9 z 7 t

?t 7

z > w 9

> w 9 z9 z 7 t > w

H` " j sa` i }~`}j~w| `d gjo~jislhc w~itj~sib aloa lbb nj tj~| zsjdzb biwj~(

Bjffi >(97( Bjw ihg nj w ` gl j~jhw gjo~jislhc w~itj~sib( Bjw e nj waj flhlfib lhgjq szoa waiw e 6 e ( Wajh" `hj `d e ihg e ls ih jgcj" ihg waj `waj~ i tj~wjq(

]~``d( _z}}`sj waiw e ihg e i~j n`wa tj~wlojs( Wajh" n| waj oa`loj `d e ihg wajo`hglwl`h !l, lh Gj hlwl`h >(9>" waj jgcj e ∞9 6 e ∞9 aitlhc w ` jhg}`lhws e ihg e (D~`f waj o`hglwl`h ! ll , lh Gj hlwl`h >(9>" j eh` waiw waj~j ls ih`waj~ jhg}`lhw `d

e ∞9 njb`hcs w` 2e ∞9 6 2e ∞9" aloa lf}bljs waj o`hw~iglowl`h(_z}}`sj waiw e ihg

ei~j n`wa jgcjs( Wajh" n| o`hglwl`h ! ll ," e ihg

ei~j

fiqlfib" ioo`~glhc w` waj cltjh `~gj~ 2 j " if`hc ibb jgcjs h`w lh 2e ihg lholgjhw lwai tj~wjq lh 2e " aloa lf}bljs waiw e 6 e ( N| waj a|}`wajsls" wals ls i o`hw~iglowl`h(

Wajh" `hj `d e ihg e ls ih jgcj" ihg waj `waj~ i tj~wjq(

Waj`~jf >(9:( Bjw C 6 ! T" J , nj i o`hhjowjg" zhgl~jowjg" b``}bjss c~i}a lwa i cltjh `~gj~ 2 j `h J ( Bjw Τ ; G!C, ∞↞ ί! C, nj waj dzhowl`h gj hjg n|

Τ! , 6 u e ∞9r e ∁ T R usvv !>(99,

?1



aj~j ls i gjo~jislhc w~itj~sib lwa jbjfjhws l" 9 ≧ l ≧ rT r % rJ r( Wajh" Τ ls i nlkjowl`h(

]~``d( Dl~sw `d ibb" j sa` waiw Τ!, g`js iowzibb| d`~f i s}ihhlhc w~jj `d C( Sjaitj

• N| gj hlwl`h" h` w ` jgcjs lh Τ! , o`~~js}`hg w` waj sifj tj~wjq t 6 s(

• D~`f waj a|}`wajsls" rΤ! , r 6 rT r ∞ 9(

• F`~j`tj~" waj sjw Τ! , g`js h`w o`hwilh fzbwl}bj jgcjs( _z}}`sj w` waj o`hw~iglo&wl`h waiw waj~j i~j w ` jgcjsj 9 2 j j > a`sj jhg}`lhws i~j i ihg n( Ld s 6 i " wajh

ls `d waj d`~f s ∞ ( ( (∞ j > ∞ i ∞ ( ( (∞ j 9 ∞ n∞ ( ( (( D~`f waj }i~ws ∞ ( ( (∞ j > ∞ i `d ihg waj o`hglwl`h !ll ," j oih sjj waiw n njb`hcs w` wals }i~w" aloa o`hw~iglows

`z~ a|}`wajsls in`zw waj d`~f ( Ld s 6 i " n| gj hlwl`h" `hb| `hj `d j 9 ihg j >

oih njb`hc w` Τ(

Wazs" Τ! , d`~fs i s}ihhlhc w~jj `d C(Ld ihg i~j w ` gl j~jhw gjo~jislhc w~itj~sibs" wajh Τ! , 6 Τ! ,( Bjw e

nj waj flhlfib lhgjq s` waiw e 6 e ( N| Bjffi >(97" j oih isszfj waiw e ls itj~wjq ihg e ls ih jgcj( Wajh e ∞9 6 e ∞9 ls ih jgcj `d Τ! , nzw h`w `d Τ! ,( Wazs"

Τ! , 6 Τ! ,( _`" Τ ls lhkjowltj(D`~ sa` lhc waiw Τ ls sz~kjowltj" bjwW nj i s}ihhlhc w~jj `d C( Sj o`hsw~zow

waj sjyzjhoj `d jgcjs ihg tj~wlojs is d`bb` s

• 9 6 s(

• D`~ e 8 9" ld e ls ih jgcj `d W " zsj o`hglwl`h !l, d`~ e %9"

• `waj~ lsj" zsj o`hglwl`h ! ll ,(

Ld ls ih jgcj&tj~wjq w~itj~sib" wajh lw ls `ntl`zsb| i gjo~jislhc w~itj~sib( F`~j tj~"n| gj hlwl`h `d Τ" lw ls obji~ waiw Τ!, 6 W ( _`" ibb j hjjg w` g` ls }~`tlhc waiw ls lhdiow ih jgcj&tj~wjq w~itj~sib( _z}}`sj w` waj o`hw~iglowl`h waiwls h`w ih jgcj&tj~wjqw~itj~sib( Bjwt nj waj tj~wjq siwlsd|lhc

• t /∁ "

• glswW !s" t , 2 glswW !s" z, d`~ ibb tj~wlojsz /∁ " z 6 t (

?<



aj~j glswW !s" i , ls waj hzfnj~ `d jgcjs `d waj zhlyzj }iwa d~`f waj slhe s w` tj~wjqi lh W ( Bjwj 6 t nj waj ~sw jgcj !o`zhwlhc d~`f t , lh waj zhlyzj }iwa d~`f t w` slh W ( N| waj oa`loj `d t " fzsw njb`hc w` !`waj~ lsj glswW !y" , 2 glswW !y" t,,( N|

waj o`hsw~zowl`h }~òjss in`tj" j fzsw njb`hc w` ibs`( Waj~j i~j w ` oisjs;

9( 6 s ∞ ( ( (∞ j ∞ ∞ ( ( (( Wajh" ld j 6 e ∁ J !W ," n| waj sjo`hg o`hsw~zowl`h~zbj"t ∁ 2e ∞9

⊎ " aloa o`hw~iglows waj a|}`wajsls waiwt /∁ (

>( 6 s ∞ ( ( (∞ ∞ ( ( (∞ j ∞ ( ( (( Wajh" ld j 6 e ∁ J !W ," n| waj ~sw o`hsw~zowl`h~zbj"t 6 e %9 ∁ " aloa o`hw~iglows waj a|}`wajsls waiwt /∁ (

Ajhoj" `z~ a|}`wajsls ls h`w w~zj" aloa lf}bljs waiw ls ih jgcj&tj~wjq w~itj~sib(Wajh" j i~j g`hj(

Waj`~jf >(94( Bjw C 6 ! T" J , nj i o`hhjowjg" zhgl~jowjg" b``}bjss c~i}a lwa i cltjh `~gj~ 2 j `h J ( D`~ jioa tj~wjq t `d C" bjw L !t , nj waj sjw `d ibb jgcjs lholgjhw wt (Cltjh i gjo~jislhc w~itj~sib " j gj hj waj dzhowl`h Ρ ; G!C, ∞↞ E !C, is d`bb` s

Ρ! , 6 Ϗ !>(9>,

aj~j Ϗ! e , 6 rL ! e , R 2e r" d`~ ibb e szoa waiw e ls i tj~wjq `d C( Wajh" Ρ ls i nlkjowl`h(

]~``d( Dl~sw `d ibb" j sa` waiw Ρ!, 6 Ϗ ls i o~lwloib o`h cz~iwl`h" aloa fjihswaiwϏ ls swinbj ihg ~joz~~jhw(

• D~`f waj o`hglwl`h ! l, lh Gj hlwl`h >(9>" d`~ ibb tj~wlojse " waj~j ls iw bjisw `hjjgcj lholgjhw w` e lh 2e " wazsϏ! e , 6 rL ! e , R 2e r 2 gjc! e ," aloa lf}bljs waiwϏ ls swinbj(

• Bjw_ nj waj sjw `d tj~wlojs a`sj `~gj~lhc lhgzojg d~`f ( H` " ld _ ls i bjcibsjyzjhoj d`~ Ϗ" wajh waj ~j}~jsjhwiwltj tjow`~ `d _ ls z 6 !9 "9" ( ( ( "9, w( Wazs"idwj~ ~lhc" j cjw waj o`h cz~iwl`hϏ∞ Bz 6 Ϗ( Aj~j B ls waj Bi}biolih fiw~lq`d C( _`" d`~ }~ tlhc waj ~joz~~jhoj `d Ϗ" j hjjg w` sa` waiw _ ls i bjcibsjyzjhoj( D`~ s 6 Ϗ9" Ϗ!s, 6 rL !s, r 6 gjc! s,( H` " isszfj waiw ibb tj~wlojs lh

2e ais njjh ~jg ! e 8 9," j o`hslgj~ waj hzfnj~ `d oal}s iw e ( Lhlwlibb|" waj~ji~j rL ! e , R 2e r oal}s iw e ( Ibb jgcjs `d L ! e , ∬ 2e aitj ih jhg}`lhw aloa lsi tj~wjq `d 2e a`sj ~lhc cltjs `hj oal} }j~ jgcj w` e ( _`" idwj~ ~lhc e ∞9"waj w`wib hzfnj~ `d oal}s iw e ls rL ! e , R 2e r % rL ! e , ∬ 2e r 6 rL ! e , r 6 gjc! e ,(Wazs" _ ls i bjcib sjyzjhoj" aloa lf}bljs Ϗ ls ~joz~~jhw(

7=



Hjqw" bjw ihg nj w ` gl j~jhw gjo~jislhc w~itj~sibs( Bjwe nj h` wajsfibbjsw lhgjq szoa waiw e 6 e ( N| Bjffi >(97" j oih isszfj waiw e 6 t ls itj~wjq ihg e 6 j ls ih jgcj( Bjw d nj waj jgcj waiw }~jojgj t lh ( Wajh

6 2e ∞ t ∞ ( ( (∞ d ∞ ( ( (∞ j ∞ ( ( ( !>(9?,

6 2e ∞ j ∞ ( ( (∞ d ∞ t ∞ ( ( ( !>(97,

Wazs" waj~j i~j iw bjisw `hj f`~j jgcj waiw i}}ji~s njd`~j t lh waih " aloa fjihswaiw Ρ! , 6 Ρ! ,(

Dlhibb|" lh `~gj~ w` sa` waiw Ρ ls sz~kjowltj" bjwϏ nj i o~lwloib o`h cz~iwl`h(Sj o`hsw~zow i sjyzjhoj `d jgcjs ihg tj~wlojs is d`bb` s

• 9 6 s(

• D`~ e 8 9" ld e ∞9 ls ih jgcj lwa `hj jhg}`lhw t h`w lh 2e ihg rL !t , R 2e r 6 Ϗ!t ,"wajh e 6 t "

• `waj~ lsj" zsj o`hglwl`h ! ll , !cltjh lh Gj hlwl`h >(9>,(

D~`f waj o`hsw~zowl`h" ls i gjo~jislhc w~itj~sib !tj~wlojs i~j iggjg n| o`hslgj~lhclholgjhw jgcjs" jgcjs i~j iggjg n| o`hglwl`h ! ll ," w ` o`hglwl`hs cltjh lh Gj hlwl`h >(9>

i~j n`wa siwls jg,( F`~j`tj~" n| gj hlwl`h" Ρ! , 6 Ϗ(

D~`f waj }~``ds `d Waj`~jfs >(9: ihg >(94" j `nsj~tj waiw d`~ i qjg gjo~jislhcw~itj~sib " wa~`zca Τ ihg Ρ" waj~j i~j jqiowb| `hj s}ihhlhc w~jj ihg `hj o~lwloibo`h cz~iwl`h o`~~js}`hglhc w` ( Zslhc wals `nsj~tiwl h" j oih gj hj ih jq}blolwnlkjowl`h njw jjh ί! C, ihg E !C, n| waj d`bb` lhc w ` ibc`~lwafs(

79



Ibc`~lwaf >(9 Ibc`~lwaf gj hjs waj fi} Z ; ί! C, ∞↞ E !C,^jyzl~j; I s}ihhlhc w~jj W d waj c~i}a C 6 ! T" J , lwa slhe tj~wjq s( Waj qjg

`~gj~ 2 j gj hjg `h J (Jhsz~j; I o~lwloib o`h cz~iwl`hϏ(

9; 9 ↝ s5l ↝ 9( Gjo~jislhc W~itj~sib (>; albj l 2 rT r % rJ r g`?; ld l 6 in ∁ J !W , wajh7; ld i ∁ 2l %9 wajh:; l%9 ↝ n(4; jbsj3; l%9 ↝ i (1; jhg ld <; jhg ld

9=; l%9 ↝ fiq 2 j uzt r zt /∁ 2l %9"jlwaj~ z `~ t `~ n`wa i~j lh 2l %9v(99; l ↝ l % 9(9>; jhg albj9?; d`~ ibb tj~wlojs l ∁ g`97; Ϗ! l , ↝ rL ! l , R 2l r(9:; jhg d`~94; ~jwz~h Ϗ(

Ibc`~lwaf >(> Ibc`~lwaf gj hjs waj fi} W ; E !C, ∞↞ ί! C,^jyzl~j; I c~i}a C 6 ! T" J , lwa slhe tj~wjq s ihg i o~lwloib o`h cz~iwl`hϏ(Jhsz~j; I s}ihhlhc w~jj W d C(

9; ↝ usv5W ↝ ∊(>; albj T ls H@W i sznsjw `d g` O`hwlhzj zhwlb ibb tj~wlojs i~j iggjg w`?; in ↝ fiq 2 j uswr s ∁ " w /∁ "sw /∁ v" !i ∁ " n /∁ ,(7; ↝ ∫ u in v(:; ld runwr nw /∁ vr 6 Ϗ!n, wajh4; ↝ ∫ u nv(3; W ↝ W ∫ u in v(1; jhg ld <; jhg albj

9=; ~jwz~h W (

Hjqw" j cltj ih jqif}bj d`~ lbbzsw~iwlhc wajsj ibc`~lwafs

Jqif}bj >(93( O`hslgj~ waj c~i}a C cltjh is d`bb`

7>



s

i n

o g

9

4

>

? 7

:

!i, C

s

i n

o g

9

4? 7

!n, W

H` " wiej i s}ihhlhc w~jj W d C !is lh waj in`tj }lowz~j," ihg isszfj waj`~gj~ `d jgcjs 2 j ls 9 2 j > 2 j ? 2 j 7 2 j : 2 j 4( Sj h` o`fj w` waj Ibc`~lwaf >(9(

_wi~w lwa9 6 s ihg l 6 9( _lhoj 9 /∁ J !W ," oa``sj > 6 fiq 2 j u9"4v 6 4" ihglho~jisj l( Hjqw"l 6 >" 4 ∁ J !W , ihg s ∁ 2 > 6 us"4v" wajh ? 6 o" ihg h`

6 us"4" ov( O`hwlhzj waj }~òjss" j jtjhwzibb| cjw waj gjo~jislhc w~itj~sib 6s ∞ 4 ∞ o ∞ : ∞ ? ∞ n ∞ 7 ∞ g ∞ > ∞ 9 ∞ i ( Hjqw" }~òjss waj blhjs 9?&9: cltjs zs wajo~lwloib o`h cz~iwl`hϏ lwaϏ!s, 6 >" Ϗ!i , 6 =" Ϗ!n, 6 >" Ϗ!o, 6 >" Ϗ!g, 6 =(

O`htj~sjb|" j saibb zsj waj Ibc`~lwaf >(>d`~ hglhc nioe W d~`f Ϗ" lwa wajsifj `~gj~ `d jgcjs 2 j cltjh iw waj njclhhlhc( I }i~wlib gjso~l}wl`h `d waj jqjozwl`h`d waj ibc`~lwaf ls is d`bb` s;

• 6 usv" W 6 ∊(

• 6 us"4v" ruowr ow /∁ vr 6 > 6 Ϗ!o," 6 us"4" ov" W 6 u4v(

• 6 us"4" o": v" rugwr gw /∁ vr 6 9 8 Ϗ !g,(

• 6 us"4" o": "?v" runwr nw /∁ vr 6 > 6 Ϗ!n," 6 us"4" o": "?" nv" W 6 u4"?v(

• 6 us"4" o": "?" n"7v" rugwr gw /∁ vr 6 = 6 Ϗ!g," 6 us"4" o": "?" n"7" gv" W 6

u4"?"7v(

• 6 us"4" o": "?" n"7" g">v" ruiw r iw /∁ vr 6 9 8 Ϗ !i ,

• 6 us"4" o": "?" n"7" g">"9v" ruiw r iw /∁ vr 6 = 6 Ϗ!i ,"6 us"4" o": "?" n"7" g">"9" iv ⊏T " W 6 u4"?"7"9v(

^jfi~e >(91(

7?



9( Iw blhj ? `d Ibc`~lwaf >(>" j o`hslgj~ `hb| waj jgcjs in /∁ aitlhc `hj jhg}`lhwlh ihg `hj jhg}`lhw h`w lh slhoj ld lw ais n`wa w ` jhg}`lhws lh " wajhlw ls obji~ waiwin lbb i}}ji~ idwj~ lws jhg}`lhws lh " wajh d~`f waj }~``d `d

Waj`~jf >(9:" lw hjtj~ njb`hcs w`W (

>( Ibs`" ld j o`hslgj~ ibb jgcjs h`w lh aitlhc iw bjisw `hj jhg}`lhw lh " ihg wajo`hglwl h d`~ albj b``} ls oaihcjg w` nj r r 2 rT r%rJ r" wajh idwj~ waj ibc`~lwaf

hlsajs" ls i gjo~jislhc w~itj~sib( F`~j tj~" ld Ϗ ls waj o~lwloib o`h cz~iwl`h~jszbwlhc d~`f Ibc`~lwaf >(9" wajh waj w ` gjo~jislhc w~itj~sibs `nwilhjg d~`fw ` ibc`~lwafs i~j iowzibb| waj sifj(

>(9(7 ]i~elhc dzhowl hs ihg o~lwloib o`h cz~iwl hs(Lh waj }~jtl`zs sjowl`h" j aitj }~jsjhwjg ih jq}blolw nlkjowl`h njw jjh waj sjw `d s}ihhlhc w~jjs ί! C, ihg waj o~lwloib c~`z}E !C,( Lh wals sjowl`h" j lhw~`gzoj wajo`hoj}w `d C&}i~elhc dzhowl`hs ihg lws lhwj~jswlhc ~jbiwl`h lwa waj o~lwloib c~`z}E !C,(Fih| swiwjfjhws `d wals sjowl`h i~j olwjg d~`f U9: V" U9?Vihg U<V( Sj h` swi~w lwas`fj ~jbjtihw gj hlwl`hs(

Gj hlwl`h >(9<( Cltjh i o`hhjowjg" zhgl~jowjg c~i}a C 6 ! T" J ,( D~`f h` `h" ld

h` `waj~ lsj fjhwl`hjg" lh wals sjowl`h" j isszfj waiw rT !C, r 6 h( D`~ Z ⊄ T !C,ihg t ∁ Z " bjw `zwgjcZ !t , gjh`wj waj hzfnj~ `d jgcjs `d C aitlhc t is `hj jhg}`lhwihg a`sj `waj~ jhg}`lhw bljs lh T !C, RZ ( _jbjow i tj~wjqs ∁ T !C,( Sj si| waiw idzhowl`hd ; T !C, R usv ∞↞ [ ls i C&}i~elhc dzhowl`h o`~~js}`hg w` waj nisj tj~wjqs !`~ C&}i~elhc dzhowl`h d`~ sa`~w" ld nisj tj~wjqs ls gjwj~flhjg, ld waj d`bb` lhc w ò`hglwl hs a`bg

9( d !t , ≪= d`~ ibbt 6 s"

>( D`~ jtj~| h`hjf}w| sjw Z ⊄ T !C, R usv" waj~j jqlsws i tj~wjqt ∁ Z szoa waiwd !t , 2 `zwgjc Z !t ,(

Waj sjw `d ibbC&}i~elhc dzhowl`h lwa nisj tj~wjqs lbb nj gjh`wjg n| ] !C" s,(

Lhswjig `d ~lhc `hj tj~wjq iw i wlfj is glsozssjg lh sjowl`h >(9(9" `hj fi|o`hslgj~ i ~lhc ~zbj waiw ibb` s ~lhc tj~wlojs slfzbwihj`zsb|( Cltjh i sjw `d tj~wlojsZ ⊎ T !C, ihg ih lhlwlib o`h cz~iwl`h Ϗ( Isszfj waiw waj hzfnj~ `d oal}s iw i tj~wjqoih nj hjciwltj" aloa fjihs lw ls }`sslnbj waiw Ϗ!t , 2 = d`~ s`fj tj~wjq t ( Bjwq

77



nj szoa waiw d`~ ih| tj~wjq t " q !t , 6 9 ld t ∁ Z ihg q !t , 6 = `waj~ lsj" wajh ~lhctj~wlojs lh Z slfzbwihj`zsb| cltjs zs i hj o`h cz~iwl`h Ϗ 6 Ϗ∞ Bq " aj~j B ls wajBi}biolih fiw~lq `d C(

Ld j o`hslgj~ s is slhe tj~wjq" ihg d is i oal} o`h cz~iwl`h" wajh lh wajbihczicj `d oal}& ~lhc cifj" swi~wlhc d~`f d " d`~ ih| h`hjf}w| sznsjw `d tj~wlojsZ ⊄ T !C, R usv" ld ibb tj~wlojs `d Z i~j ~jg slfzbwihj`zsb|" wajh s`fj tj~wlojs `d Z fi| o`hwilh hjciwltj hzfnj~ `d oal}s(

Gj hlwl`h >(>=( BjwϏ"Ϗ nj w ` oal} o`h cz~iwl`hs `d i o`hhjowjg" zhgl~jowjg c~i}aC( Sj si| waiw Ϗ ihg Ϗ i~j oal}& ~lhc jyzltibjhw " ihg ~lwjϏ ∲ Ϗ " ld `hj oih nj`nwilhjg d~`f waj `waj~ n| i sjyzjhoj `d ~lhcs(

^jfi~e >(>9( Waj ~jbiwl`h `nwilhlhc `hj oal} o`h cz~iwl`h d~`f ih`waj~ n| i sj&yzjhoj `d ~lhcs ls ih jyzltibjhoj ~jbiwl`h( Lh diow" wals ~jbiwl`h ls w~ltlibb| ~j jqltj!Ϗ ∲ Ϗ, ihg w~ihslwltj !ld Ϗ ∲ Ϗ ihg Ϗ ∲ Ϗ " wajhϏ ∲ Ϗ ,( Ibs`" ld `hj oih `nwilh Ϗ

d~`f Ϗ n| ~lhc i tj~wjq t " wajh `hj oih ibs` `nwilh Ϗ d~`f Ϗ n| ~lhc waj tj~wlojs`d waj sjwZ 6 T R ut v slfzbwihj`zsb|( Lh cjhj~ib" ld _ ls i sjyzjhoj `d ~lhcs waiw}~`gzojs Ϗ d~`f Ϗ ihg bjw e nj waj bi~cjsw hzfnj~ `d òoz~~jhojs `d ih| }i~wlozbi~tj~wjq lh _ " wajheT R_ ls waj sjyzjhoj `d ~lhcs aloa }~`gzojs nioe Ϗ d~`f Ϗ "aj~j eT gjh`wjs waj sjyzjhoj `d tj~wlojs szoa waiw jioa tj~wjq lh T !C, i}}ji~s lh lwjqiowb| e wlfjs(

Cltjh i oal} o`h cz~iwl`h Ϗ `h waj c~i}a C lwa slhe tj~wjq s" gj hlwl hszccjsws waiw lh `~gj~ w` oajoe ajwaj~ `~ h`wϏ ∁ ] !C" s," waj tj~l j~ hjjgs wòajoe ajwaj~ waj~j jqlsws i tj~wjq t ∁ Z szoa waiwϏ!t , 2 `zwgjc Z !t , d`~ ibb sznsjws Z ⊄ T !C, R usv( Nzw" lh diow" waj~j ls ih jbjcihw ibc`~lwaf w` g` wals yzloeb|" aloals oibbjg wajGai~“s ibc`~lwaf (

7:



Ibc`~lwaf >(? Gai~“s Ibc`~lwaf d`~ oajoelhc ld i o`h cz~iwl`h Ϗ ls i C&}i~elhcdzhowl`h^jyzl~j; I o`h cz~iwl`h Ϗ `d i o`hhjowjg" zhgl~jowjg c~i}a C lwa slhe tj~wjq s(Jhsz~j; @zw}zw W^ZJ ld Ϗ ∁ ] !C" s," ihg DIB_J `waj~ lsj(

9; Ld waj~j jqlsws i tj~wjqt szoa waiwϏ!t , 2 = wajh `zw}zw DIB_J ihg _w`}(>; BjwZ = 6 T !C, ihg t = 6 s(?; d`~ 9 ≧ l ≧ h ∞ 9 g`7; BjwZ l ↝ Z l∞9 R ut l∞9v(:; ld d`~ ibbt ∁ Z l"Ϗ!t , ≪`zwgjc Z l !t l , wajh4; ~jwz~h DIB_J ihg _w`}(3; jbsj1; Bjwt l ∁ Z l nj ih| tj~wjq lwa Ϗ!t l , 2 `zwgjc Z l !t l ,(<; jhg ld

9=; jhg d`~

99; ~jwz~h W^ZJ(

]~``d `d o`~~jowhjss `d Ibc`~lwaf >(? ( Ld waj ibc`~lwaf `zw}zws DIB_J"Ϗ oihh`w nj C&}i~elhc dzhowl`h n| gj hlwl`h( H` isszfj waj ibc`~lwaf `zw}zwsW^ZJ( Sj hjjg w` sa` waiw d`~ ih i~nlw~i~| sznsjw Z ⊄ T !C,Rut =v" waj~j jqlswsi tj~wjq t ∁ Z szoa waiwϏ!t , 2 `zwgjc Z !t ,( Bjw t e nj waj flhlfib jbjfjhw `d Z "

lwa ~js}jow w` waj `~gj~lhc lhgzojg n| waj ibc`~lwaf( Wajh Z ⊄ Z e ihg waj~jd`~j`zwgjc Z e !t e , ≧ `zwgjc Z !t e ,( Nzw j eh` Ϗ!t e , 2 `zwgjc Z e !t e , n| o`hsw~zowl`h(

Ajhoj Ϗ!t e , 2 `zwgjc Z !t e , ihg t e ls waj gjsl~jg jbjfjhw lh Z (Waj hjqw waj`~jf sa` s ih jqolwlhc ~jbiwl`h njw jjh C&}i~elhc dzhowl`h ihgoal} o`h cz~iwl`h(

Waj`~jf >(>>( Bjw C nj i o`hhjowjg" zhgl~jowjg c~i}a lwa slhe tj~wjq s( Wajh jtj~| oal} o`h cz~iwl`h `h C ls oal}& ~lhc jyzltibjhw w` i C&}i~elhc dzhowl`h ! lwa nisj tj~wjq s,( F`~j`tj~" h` w ` glswlhow C&}i~elhc dzhowl`hs i~j oal}& ~lhc jyzltibjhw(

]~``d( I nislo lgji `d waj }~``d ls w` w~| w` fiej waj Gai~“s ibc`~lwaf w` `zw}zwW^ZJ" ih|wlfj lw ls in`zw w` `zw}zw DIB_J" ‒f`gld|‖ waj o`h cz~iwl`h w` it`lg lw(

Dl~sw" j sa` waiw jtj~| o`h cz~iwl`h Ϗ ls jyzltibjhw w` i o`h cz~iwl`h Ϗ

aloa siwls js Ϗ !t , ≪= d`~ ibbt 6 s( O`hslgj~ i }iwa ] 6 ! s" t 9" ( ( ( " te , waiw k`lhssw` i tj~wjq t e szoa waiwϏ!t e , 2 =( Sj }~òjss is d`bb` s; ~sw" ~lhct e ∞9 zhwlb wajhzfnj~ `d oal}s iw t e njo`fjs h`h&hjciwltj" wajh waj hzfnj~ `d oal}s iw t e ∞9 flcaw njhjciwltj( Hjqw" ld waj hzfnj~ `d oal}s iw t e ∞9 ls hjciwltj" ~lhc t e ∞> zhwlb waj hzfnj~`d oal}s iw t e ∞9 njo`fjs h`h&hjciwltj" wajh waj hzfnj~ `d oal}s iw t e ∞> flcaw njhjciwltj( O`hwlhzj waiw }~òjss" j jtjhwzibb| ~jioa i ~jyzl~jg o`h cz~iwl h( Wazs"

j fi| isszfj waiw waj lhlwlib o`h cz~iwl`h Ϗ siwlsd|lhc Ϗ!t , ≪= d`~t 6 s(

74



Waj d`bb` lhc ibc`~lwaf is nisjg `h waj lgji j glsozssjg njd`~j(

Ibc`~lwaf >(7 Ibc`~lwaf d`~ hglhc i C&}i~elhc dzhowl`h aloa ls jyzltibjhw w` wajcltjh o`h cz~iwl`h Ϗ

^jyzl~j; I o`h cz~iwl`h Ϗ `h i o`hhjowjg" zhgl~jowjg c~i}a C lwa slhe tj~wjq s(Jhsz~j; I C&}i~elhc dzhowl`h lwa nisj tj~wjqs(

9; Z = 6 T !C,5 t = 6 s(>; l ↝ 9(?; albj l ≧ h ∞ 9 g`7; BjwZ l 6 Z l∞9 R ut l∞9v(:; ld waj~j jqlsws tj~wjqt ∁ Z l szoa waiwϏ!t , 2 `zwgjc Z l !t , wajh4; t l ↝ t (3; l ↝ l % 9(1; jbsj

<; Dl~j ibb tj~wlojs `d Z l slfzbwihj`zsb|" cjw i hj o`h cz~iwl`h η (9=; Ϗ↝ η (99; l ↝ 9(9>; jhg ld 9?; jhg albj97; ~jwz~h Ϗ(

D~`f waj o`~~jowhjss `d Ibc`~lwaf >(?" j oih sjj waiw ld Ibc`~lwaf >(7 oihwj~flhiwj" wajh waj hib o`h cz~iwl`hϏ ls lh diow iC&}i~elhc dzhowl`h" slhoj aiw j

g` lh Ibc`~lwaf >(7 ls kzsw f`gld|lhc waj o`h cz~iwl`h Ϗ jioa wlfj Ibc`~lwaf >(? fi|`zw}zw DIB_J( Lh diow" waj ibc`~lwaf lbb jtjhwzibb| wj~flhiwj5 is tj~wjqs 6 t = hjtj~‒ ~js‖" lw sa`zbg jtjhwzibb| sw`} ~jojltlhc d~`f lws hjlcan`~s iw s`fj }`lhw( Wals" lhwz~h" fjihs waiw lws hjlcan`~s lbb sw`} ~lhc iw s`fj }`lhw ihg waj~jd`~j" jtjhwzibb|"

lbb sw`} ~jojltlhc d~`f wajl~ hjlcan`~s( Lwj~iwlhc wals i~czfjhw sa` s waiw waj a`bj}~òjss lbb sw`} iw i hlwj hzfnj~ `d swj}s(

Hjqw" sz}}`sj waiw w ` glswlhowC&}i~elhc dzhowl`hsϏ"Ϗ i~j oal}& ~lhc jyzlt&ibjhw( Waiw ls" waj~j jqlsws i sjyzjhoj `d tj~wlojs_ aloa }~`gzojs Ϗ d~`f Ϗ( Bjwe

nj waj bi~cjsw hzfnj~ `d òoz~~jhojs `d ih| tj~wjq t ∁ _ ( Sj fi| isszfj lwa`zwb`ss `d cjhj~iblw| waiwt = g`js h`w òoz~ lwa fzbwl}blolw|e" slhoj ld lw g`js" wajh lhwaj lhtj~sj sjyzjhoj aloa }~`gzojs Ϗ d~`f Ϗ " lw g`js h`w !n| ^jfi~e >(>9,(

BjwF l nj waj sjw `d tj~wlojs waiw i}}ji~sl wlfjs lh _ " 9≧ l ≧ e( H`wj waiwj oih gjo`f}`sj _ lhw` waj zhl`h `d waj sjwsF e " F e ∫ F e ∞9" F e ∫ F e∞9 ∫ F e ∞>"

( ( (" F e ∫ ( ( ( ∫ F 9( H`wj waiw jioa `d wajsj o`hwilhsF e " wajh ~lhc ibb tj~wlojs `d wajsj slfzbwihj`zsb| lbb h`w lho~jisj waj w`wib hzfnj~ `d oal}s `h F e ( _lhoj Ϗ ls iC&}i~elhc dzhowl`h ihgt = /∁ F e " ~lhcF e waj ~sw wlfj fzsw sz~jb| cltj zs i hjciwltj

73



hzfnj~ `d oal}s `h iw bjisw `hj tj~wjq `d F e " hifjb| t ( Lw lf}bljs waiw ~lhc ibb `d _ lbb h`w lho~jisj waj w`wib hzfnj~ `d oal}s lhF e " aloa fjihs waiw Ϗ !t , 2 ="o`hw~iglows w` waj a|}`wajsls waiwϏ ls i C&}i~elhc dzhowl`h(

Gj hlwl`h >(>?( @h ih| o`hhjowjg" zhgl~jowjg c~i}a C 6 ! T" J ," waj fiqlfib swinbj o`h cz~iwl`h ofiq ls cltjh n| ofiq !t , 6 gjc! t , ∞ 9 d`~ ibbt ∁ T !C,(

Waj hjqw waj`~jf lbb cltj zs i tj~| sz~}~lslhc ~jbiwl`h njw jjh C&}i~elhcdzhowl`hs ihg o~lwloib o`h cz~iwl`hs(

Waj`~jf >(>7( Bjw C 6 ! T" J , nj i o`hhjowjg" zhgl~jowjg c~i}a lwa i slhe tj~wjq s ∁ T !C, ( Wajh" i cltjh o`h cz~iwl h Ϗ ls i C&}i~elhc dzhowl`h !o`~~js}`hglhc w`

nisj tj~wjq s, ld ihg `hb| ld η 6 ofiq ∞ Ϗ ls i o~lwloib o`h cz~iwl`h" aj~j ofiq gjh`wjs waj fiqlfib swinbj o`h cz~iwl`h gj hjg lh Gj hlwl`h >(>? (

]~``d( Wals }~``d is nisjg `h waj lgji `d waj }~``d `d Bjffi :(4 lh U9?V(_z}}`sj waiw Ϗ ls i C&}i~elhc dzhowl`h( I yzloe `nsj~tiwl`h ls waiw d`~ ih| t 6

s" =≧ Ϗ!t , 2 `zwgjc ut v!t , 6 gjc! t ," aloa lf}bljs waiw = ≧ ofiq !t , ∞ η !t , ≧ gjc! t ,"d`~ ih| t ∁ T !C, R usv( Wazs"η ls swinbj(Oa``sj ih `~gj~jg sjyzjhoj `d tj~wlojs _ 6 ut =" ( ( ( " th ∞9v lhgzowltjb| is d`bb` s; wiejt = 6 s( Ld ut =" ( ( ( " te ∞9v ls eh` h" 9 ≧ e ≧ h∞9" bjwI e 6 T !C, Rut =" ( ( ( " te ∞9v( Oa``sjt e ∁ I e szoa waiwϏ!t e , 2 `zwgjc I e

!t e ,( Bjw uη =" ( ( ( " η h ∞9v nj waj o`~~js}`hglhcsjyzjhoj `d o`h cz~iwl`hs szoa waiw d`~ 9≧ e ≧ h ∞ 9" η e ls `nwilhjg d~`f η e ∞9 n|

~lhc t e " ihg η = ls `nwilhjg d~`f η n| ~lhc t =(Hjqw" j saibb sa` waiw d`~t ∁ T !C, Rusv ihg = ≧ e ≧ h ∞9"η e !t , ≪=" aloa lf}bljswaiw_ ls i s&bjcib sjyzjhoj d`~η ( D`~ ih| e != ≧ e ≧ h ∞ 9," bjwNe 6 ut =" ( ( ( " te v(D`~ ih| tj~wjq t ∁ T !C, R usv" waj~j i~j w ` oisjs

9( t /∁ Ne " slhoj iw swj}e" t ais h`w njjh ~jg |jw" nzw t oih swlbb ~jojltj oal}s d~`f

`waj~ tj~wlojs" j aitjη e !t , ≪η !t , ≪=

>( t ∁ Ne " wajh t 6 t b d`~ 9≧ b≧ e( _lhoj idwj~t b is ~jg waj ~sw wlfj" lw lbbhjtj~ nj ~jg icilh" nzw swlbb oih ~jojltj oal}s d~`f `waj~ tj~wlojs" j aitj

η e !t b, ≪η b!t b,

71



Ibs` h`wj waiw η b!t b, oih nj `nwilhjg d~`f η n| ~lhc tj~wlojs ut =" ( ( ( " tbv" jibs` aitj

η b!t b, 6 η !t b, ∞ `zwgjc Nb!t b,6 gjc! t b, ∞ 9 ∞ Ϗ!t b, ∞ `zwgjc Nb

!t b,

6 Ugjc!t b, ∞ `zwgjc Nb!t b,V∞ Ϗ!t b, ∞ 9

6 `zwgjc T !C , RNb∞ 9 !t b, ∞ Ϗ!t b, ∞ 9≪=

Waj bisw lhjyziblw| d`bb` s d~`f waj gj hlwl`h `d t b" aloa lf}bljs waiw η e !t b, ≪=(

Ajhoj" _ ls i s&bjcib sjyzjhoj d`~η ( F`~j tj~" bjw z nj szoa waiw d`~ ibb tj~wlojst "

z !t , 6 9 ld t ∁ _ ihg z !t , 6 = `waj~ lsj" wajh `hj oih sjj waiw z !t , 6 9 d`~ ibbt ∁ T !C," aloa fjihs waiw _ ls i s&bjcib sjyzjhoj d`~η aloa }~`gzojs η ( Wazs η ls n`wa swinbj ihg ~joz~~jhw" aloa fjihs waiw lw ls o~lwloib(

O`htj~sjb|" sz}}`sj waiw η ls i o~lwloib o`h cz~iwl h( Bjw_ 6 ut = 6 s " ( ( ( " th ∞9vnj i s&bjcib sjyzjhoj d`~η aloa }~`gzojs η !slhoj η ls swinbj s` ~lhcs iw waj nj&clhhlhc ls hiwz~ib, ihg uη =" ( ( ( " η h ∞9v nj waj o`~~js}`hglhc sjyzjhoj `d o`h cz~iwl`hsis njd`~j( Bjw Ne 6 ut =" ( ( ( " te v d`~ =≧ e ≧ h ∞ 9( O`hslgj~ ih| h`h&jf}w| sznsjwI ⊄ _ R ut =v !h`wj waiwT !C, 6 _ ,( Bjwt b nj waj tj~wjq aitlhc sfibbjsw lhgjq `d I (

Wajh=≧ η b!t b, 6 `zwgjc T !C , RNb∞ 9 !t b, ∞ Ϗ!t b, ∞ 9

F`~j`tj~ Nb∞9 ∬I 6 ∊" wajhI ⊄ T !C, RNb∞9" aloa lf}bljs waiw

`zwgjc T !C , RNb∞ 9 !t b, ≧ `zwgjc I !t b,

Ajhoj `zwgjc I !t b, ∞ 9 ≪Ϗ!t b,( _lhoj I ls i~nlw~i~|"Ϗ ls" n| gj hlwl`h" i C&}i~elhcdzhowl`h(

>(> D ~ gl~jowjg c~i}as(

>(>(9 Oal}& ~lhc cifj f`gjb(

Lh sjowl`h>(9(9" j aitj gj hjg waj oal}& ~lhc f`gjb d`~ ih zhgl~jowjg c~i}a( Lhwals sjowl`h" j saibb gj hj waj slflbi~ f`gjb d`~ i gl~jowjg c~i}a( Waj sw~zowz~j `d wals ihg waj hjqw sjowl`h j~j d`bb` jg d~`f i tj~| jbb&eh` h ~jdj~jhoj d`~ sihg}lbj

7<



~jsji~oaj~s U97V(Bjw Ο nj i glc~i}a `h h tj~wlojs( I tj~wjq s ls oibbjg i slhe tj~wjq ld `zwgjc!s, 6

=( I slhe tj~wjq s ls oibbjg i cb`nib slhe ld d~`f jtj~| tj~wjq t 6 s" waj~j ls i gl~jowjg

}iwa d~`f t w` s( H`wj waiw ld waj~j ls i cb`nib slhe" wajh lw ls waj zhlyzj slhe( Io`h cz~iwl`h !ibs` oibbjg sihg}lbj , Ϗ `d waj cifj ls i dzhowl`h gj hjg `h waj sjw `d tj~wlojs T !Ο, 6 ut 9" t>" ( ( ( " th v szoa waiw d`~ ibb tj~wjqt "Ϗ!t , ls waj hzfnj~ `d oal}siw t ( Slwa`zw b`ss `d cjhj~iblw|" j saibb ~lwjϏ 6 ! Ϗ!t 9, " ( ( ( "Ϗ!t h ,, ∁ [ h !aj~j Ϗ

is h`w ~lwwjh lh waj o`bzfh tjow`~ d`~f" nzw j saibb swlbb zhgj~swihg lw is i o`bzfhtjow`~ d`~ waj zhld`~f `d jyziwl`hs ~jbiwjg w Ϗ,(

I o`h cz~iwl`h Ϗ ls swinbj ld d`~ jtj~| h`h&slhe tj~wjq t " Ϗ!t , 2 `zwgjc! t ,( Ld Ϗ!t , ≪`zwgjc! t ," t ls oibbjg ih iowltj tj~wjq( Ih iowltj tj~wjq t oih nj ~jg " waiw ls

sjhglhc `hj oal} ib`hc jioa `d `zwgjc! t , jgcjs aitlhc t is wajl~ wilb tj~wjq( Dl~lhc ih`h&slhe tj~wjq t w~ihsd`~fs Ϗw` i hj o`h cz~iwl h Ϗ !t , 6 Ϗ!t , ∞ `zwgjc! t ,%º !t" t ,ihg Ϗ ! , 6 Ϗ! , % º !t" ," d`~ ibb 6 t " aj~j º !z" t , gjh`wjs waj hzfnj~ `d jgcjsd~`f z w` t " d`~ ibb tj~wlojsz" t ( Sj oibb Ϗ i szoojss`~ `d Ϗ(

_z}}`sj waiw _ 6 uq9" q>" ( ( ( " qe v ls i h`h&jf}w| hlwj sjyzjhoj `d !h`w hjo&jssi~lb| glswlhow, h`h&slhe tj~wlojs `d Ο szoa waiw swi~wlhc d~`fϏ" waj tj~wlojs oih nj

~jg lh waj `~gj~ `d _ !lw ls `ntl`zs waiw slhes i~j h`w iowltj,( Sj s`fjwlfjs oibb _

i w`}}blhc sjyzjhoj( _ ls oibbjg i bjcib sjyzjhoj d`~Ϗ ld Ϗl!ql , ≪`zwgjc! ql , d`~ ibb

l 6 9 ">" ( ( ( " e" aj~j Ϗ9 6 Ϗ ihg Ϗl ls `nwilhjg d~`f Ϗl∞9 n| ~lhc q l∞9( BjwB nj wajBi}biolih fiw~lq `d Ο" bjwq nj szoa waiw d`~ ibb tj~wjqt " q!t , ls jyzib w` waj hzfnj~`d wlfjs t i}}ji~s lh _ " ihg isszfj d`~ o`htjhljhoj waiw waj hzfnj~ `d oal}s iwslhe tj~wlojs i~j {j~`" wazs ~lhc tj~wlojs lh waj `~gj~ `d _ }~`gzojs waj o`h cz~iwl`hϏ∞ Bwq ( _ ls oibbjg i swinlbl{lhc sjyzjhoj d`~ Ϗ ld idwj~ waj ~lhc }~òjss hlsajs"waj hib ~jszbw ls i swinbj o`h cz~iwl h( N| }j~d`~flhc i sjyzjhoj `d ~lhcs" j fi|jtjhwzibb| ~jioa i swinbj o`h cz~iwl`h `~ j flcaw o`hwlhzj ~lhc d`~jtj~" is lh wajd`bb` lhc jqif}bj(

Jqif}bj >(>:( O`hslgj~ waj o`f}bjwj gl~jowjg c~i}a `h wa~jj tj~wlojs ! lwa`zwb``}s,( Wajh }biolhc wa~jj oal}s iw i tj~wjq cltjs i o`h cz~iwl`h waiw swinlbl{js lh`hj f`tj" albj }biolhc d`z~ oal}s iw i tj~wjq cltjs i o`h cz~iwl`h waiw hjtj~ swinl&bl{js(

:=



==

?

swinlbl{js lh 9 f`tj

99

9

==

7

hjtj~ swinlbl{js

99

>

>>

=

?=

9

=?

9

Dlcz~j >(9; _`fj oal} o`h cz~iwl`hs jtjhwzibb| swinlbl{j" albj `waj~s hjtj~ swinlbl{j!Lbbzsw~iwlhc }lowz~j d~`f U97V,(

Lh waj hjqw bjffi" j lbb sa` waiw waj oa`loj `d `~gj~ lh aloa j ~j tj~wlojslh diow g`js h`w i jow waj b`hc&wj~f njaitl`z~ `d waj s|swjf(

Bjffi >(>4( Bjw Ο nj ih| glc~i}a" bjw Ϗ=" Ϗ9" ( ( ( "Ϗh nj i sjyzjhoj `d oal} o`h cz~i&wl`hs `h Ο" Ϗl i szoojss`~ `d Ϗl∞9" d`~ ibb 9 ≧ l ≧ h" ihg bjw Ϗ="Ϗ9" ( ( ( "Ϗf nj ih`waj~ szoa sjyzjhoj lwa Ϗ= 6 Ϗ=(

9( Ld Ϗh ls swinbj" wajh f ≧ h" ihg f`~j`tj~" h` tj~wjq ~js f`~j wlfjs lh Ϗ=" ( ( ( "Ϗf

waih lh Ϗ=" ( ( ( "Ϗh (>( Ld Ϗh ihg Ϗf i~j n`wa swinbj" wajh f 6 h" Ϗh 6 Ϗh " ihg jioa tj~wjq ~js waj

sifj hzfnj~ `d wlfjs lh n`wa alsw`~ljs(

>

??

9

=

7 9

>

>

=

=

9

Dlcz~j >(>; O`ffzwiwl`h `d waj oal}& ~lhc `}j~iwl`hs !Lbbzsw~iwlhc }lowz~j d~`f U97V,(

]~``d( ]i~w > ls ih lffjgliwj o`~`bbi~| `d }i~w 9" aloa j lbb h` }~`tj( Waj }~``d cltjh aj~j ls sblcawb| gl j~jhw o`f}i~lhc lwa waj `hj cltjh lh U 97V" nzw waj lgjis `d waj w ` i~j nisloibb| waj sifj(

:9



Bjw_ 6 uq=" ( ( ( " qh ∞9v nj waj bjcib sjyzjhoj !d`~ Ϗ=, o`~~js}`hglhc w` Ϗ="Ϗ9" ( ( ( "Ϗh (Wals fjihs waiw d`~ ibb 9≧ l ≧ h" `hj oih `nwilh Ϗl d~`f Ϗl∞9 n| ~lhc tj~wjq ql∞9(Bjw_ 6 uq=" ( ( ( " qf ∞9v nj szoa sjyzjhoj o`~~js}`hglhc w` Ϗ=" ( ( ( "Ϗf ( Slwa`zw b`ss

`d cjhj~iblw|" j oih isszfj waiw Ϗ k 6 Ϗ k d`~ ibb 9≧ k≧ flh! f" h , !ld waj~j ls ih lhgjqe szoa waiwϏe 6 Ϗe waj j oih swi~w waj }~``d lwa waj w ` hj sjyzjhojs a`sj ~swe o`h cz~iwl`hs ~jf`tjg,(

H` " sz}}`sj w` waj o`hw~iglowl`h waiwf 8 h ( _lhoj Ϗ= 6 Ϗ=" ld j swi~w d~`fϏ=" waj tj~wlojsq= 6 q= i~j n`wa iowltj( Lh _ " j swi~w lwa ~lhcq=" nzw wajhq=

ls swlbb iowltj ihg slhojϏh ls swinbj" s` lw fzsw ai}}jh waiwq= 6 q k d`~ =2 k 2 h (Ajhoj" uq k 6 q=" q=" q9" ( ( ( " q k∞9" q k%9" ( ( ( " qh ∞9v ls ibs` i bjcib sjyzjhoj d`~ Ϗ= aloa}~`gzojs Ϗh ( Lw d`bb` s waiwuq=" q9" ( ( ( " q k∞9" q k%9" ( ( ( " qh ∞9v ihg uq9" ( ( ( " qf ∞9v i~j

bjcib sjyzjhojs d`~ Ϗ9 aloa ~js}jowltjb| }~`gzojs Ϗh ihg Ϗf ( I}}b| waj in`tj }~òjssd`~Ϗ9" j ibs` cjw w ` hj bjcib sjyzjhojs d`~ Ϗ> aloa ~js}jowltjb| }~`gzojs Ϗh ihgϏf ( H`wj waiw jioa wlfj i hj sjyzjhoj d`~fs" waj hzfnj~ `d tj~wlojs lh waj hjsjyzjhoj }~`gzolhc Ϗh lbb gjo~jisj n| 9( ^j}jiw waj }~òjss zhwlb waiw sjyzjhoj lsjf}w|" slhoj f 8 h " j jtjhwzibb| cjw i h`h&jf}w| bjcib sjyzjhoj d`~ Ϗh waiw }~`gzojsϏf ( Nzw wals o`hw~iglows waj a|}`wajsls waiwϏh ls swinbj( Waj swiwjfjhw in`zw wajhzfnj~ `d ~lhcs `d jioa tj~wjq ls izw fiwloibb| }~`tjg n| waj }~òjss( Waj }~``d lsh` o`f}bjwj(

Gj hlwl`h >(>3( _wi~wlhc d~`f i o`h cz~iwl`hϏ" Bjffi >(>4sa` s waiw waj~j ls iwf`sw `hj swinbj o`h cz~iwl`h waiw oih nj ~jioajg n| i hlwj sjyzjhoj `d ~lhcs( Ld szoa i swinbj o`h cz~iwl`h jqlsws" j gjh`wj lwϏ◤ ihg oibb lw wajswinlbl{iwl`h d Ϗ(

_` di~" j aitj h`w |jw sa` h ih| o`hobzsl`h ~jbiwjg w` slhe tj~wjq" i tj~|lf}`~wihw `nkjow `d waj f`gjb( D~`f h` `h" ld h` `waj~ lsj fjhwl`hjg" bjw isszfjwaiw waj glc~i}a Ο ais i cb`nib slhe s(

Bjffi >(>1( Ld Ο ais i cb`nib slhe" wajh jtj~| o`h cz~iwl`h `h Ο swinlbl{js(]~``d( BjwH nj waj hzfnj~ `d oal}s lh waj o`h cz~iwl`h( Cltjh i tj~wjq t `d Ο" bjwt =" t9" ( ( ( " t~∞9" t~ nj i gl~jowjg }iwa d~`f t = 6 t w` t ~ 6 s( Jtj~| wlfj t ~∞9 ~js" lwsjhgs i oal} w` waj slhe aloa ~jfilhs waj~j d`~jtj~( Wazs t ~∞9 oih ~j iw f`sw H wlfjs( Jtj~| wlfj t ~∞> ~js" lw sjhgs i oal} w t ~∞9" ihg gt ~ ∞ 9 szoa oal}s lbb oizsjt ~∞9 w` ~j `hoj" s` t ~∞> ~js iw f`sw gt ~ ∞ 9 H wlfjs( Lwj~iwlhc nioe i~g ib`hc waj }iwa"

j sjj waiw t ~js iw f`sw gt 9 ³ ³ ³gt ~ ∞ 9 H wlfjs( Wazs jioa tj~wjq oih ~j `hb| hlwjb|fih| wlfjs" s` n| Bjffi >(>4waj o`h cz~iwl`h swinlbl{js(

:>



Gj hlwl`h >(><( Gj hj waj oal} igglwl`h `}j~iw`~ J t is waj fi} `h oal} o`h c&z~iwl`hs waiw iggs i slhcbj oal} iw tj~wjq t ihg wajh bjws waj s|swjf swinlbl{j( Lhs|fn`bs"

J t Ϗ 6 ! Ϗ% 9 t ,◤

aj~j 9 t ls waj o`h cz~iwl`h o`hslswlhc `d i slhcbj oal} iwt (

Bjffi >(?=( @h ih| glc~i}a lwa i cb`nib slhe" waj oal} igglwl`h `}j~iw`~s o`ffzwj(

]~``d( Cltjh i oal} o`h cz~iwl`h Ϗ ihg w ` tj~wlojs t ihg " aiwjtj~ tj~wlojs i~jiowltj lh Ϗ% 9 t i~j ibs` iowltj lh o`h cz~iwl`h Ϗ 6 Ϗ% 9 t % 9 ( I}}b|lhc w` Ϗi sjyzjhoj `d ~lhcs waiw swinlbl{jsϏ% 9 t " j `nwilh waj o`h cz~iwl`h J t Ϗ% 9 (_winlbl{lhc wals biwwj~ o`h cz~iwl`h |ljbgsJ J t Ϗ( Wazs J J t Ϗ ls i swinlbl{iwl`h `d Ϗ ( Lhwj~oaihclhc waj ~`bjs `d t ihg " waj o`h cz~iwl`hJ t J Ϗ ls ibs` i swinlbl{iwl`h`d Ϗ ( D~`f Bjffi >(>4 j o`hobzgj waiw J J t Ϗ 6 J t J Ϗ(

Ih lhwj~jswlhc o`~`bbi~| `d wals bjffi ls waj d`bb` lhc

O`~`bbi~| >(?9( I}}b|lhc i sjyzjhoj `d oal} igglwl`h `}j~iw`~s w` Ϗ |ljbgs waj sifj ~jszbw is igglhc ibb waj issòliwjg oal}s slfzbwihj`zsb| ihg wajh swinlbl{lhc(

>(>(> Waj o~lwloib c~`z}(Gj hlwl`h >(?>( I oal} o`h cz~iwl`h Ϗ ls ioojsslnbj ld d~`f ih| `waj~ oal} o`h cz&~iwl`h lw ls }`sslnbj w` `nwilhϏ n| i o`fnlhiwl`h `d igglhc oal}s ihg sjbjowltjb| ~lhciowltj tj~wlojs( Lh `waj~ `~gs" d`~ ih| o`h cz~iwl`hι " waj~j jqlsws i o`h cz~iwl`hξ szoa waiwϏ 6 ! ι %ξ ,◤( I oal} o`h cz~iwl`h waiw ls n`wa swinbj ihg ioojsslnbj ls oibbjgo~lwloib (

Bjw Ο nj i glc~i}a `h h tj~wlojs lwa cb`nib slhes( Waj ~jgzojg Bi}biolih B= `d

Ο `nwilhjg d~`f waj Bi}biolih fiw~lqB

n| ~jf`tlhc waj ~` ihg o`bzfh o`~~js}`hglhcw` waj slhe( H`wj waiw ~lhc i h`h&slhe tj~wjqt w~ihsd`~fs i oal} o`h cz~iwl h Ϗ lhw`waj o`h cz~iwl`hϏ∞ ∄ t " aj~j ∄ t ls waj ~` `d waj ~jgzojg Bi}biolih o`~~js}`hglhcw`t ( _lhoj j ihw w` tlj waj o`h cz~iwl`hs njd`~j ihg idwj~ ~lhc is jyzltibjhw" ji~j bjg w` o`hslgj~ waj c~`z} yz`wljhw[ h ∞9/ Bw

=[ h ∞9" aj~j Bw=[ h ∞9 6 uBw

=| ; | ∁ [ h ∞9vls waj lhwjcj~ ~` &s}ih `d B=(

Gj hlwl`h >(??( Bjw Ο nj i glc~i}a `h h tj~wlojs lwa cb`nib slhes( Waj sihg}lbj c~`z} _ !Ο, `d Ο ls waj c~`z} yz`wljhw[ h ∞9/ Bw

=[ h ∞9(

:?



Bjffi >(?7( Waj `~gj~ `d _ !Ο, ls waj gjwj~flhihw `d waj ~jgzojg Bi}biolih B=!Ο,(

]~``d( Waj `~gj~ `d _ !Ο, jyzibs waj lhgjq `d waj biwwlojBw=[ h ∞9 lh [ h ∞9" ihg d~`f waj

waj`~| `d biwwlojs !sjj U4V" U9V," j gjgzoj waiw wals lh wz~h jyzibs waj gjwj~flhihw `d B=(

Bjffi >(?:( Bjw Ο nj i glc~i}a lwa i cb`nib slhe s( Jtj~| jyzltibjhoj obiss `d [ h ∞9/ Bw

=[ h ∞9 o`hwilhs iw bjisw `hj swinbj o`h cz~iwl`h `d Ο(

]~``d( Bjwη nj waj o`h cz~iwl`h cltjh n| η !t , 6 `zwgjc! t , d`~ ibbt " ihg bjwη ◤ njlws swinlbl{iwl`h( Wajhη ◤!t , 2 `zwgjc! t , d`~ ibbt 6 s" s` η ∞ η ◤ ls i }`slwltj tjow`~jyzltibjhw w` waj {j~` o`h cz~iwl`h( Cltjh ih| ι ∁ [ h ∞9" bjwf gjh`wj waj flhlfzf`d ibb waj o``~glhiwjs `d ι w`cjwaj~ lwa = !s` waiwf ≧ =,( Wajh waj tjow`~

ξ 6 ι % !∞f ,! η ∞ η ◤,

ls h`h&hjciwltj ihg jyzltibjhw w` ι ( Ajhoj ξ ◤ ls i swinbj oal} o`h cz~iwl`h lh waj sifjjyzltibjhoj obiss is ι (

Bjffi >(?4( Bjw Ο nj i glc~i}a lwa i cb`nib slhe( Jtj~| jyzltibjhoj obiss `d [ h ∞9/ Bw

=[ h ∞9 o`hwilhs iw bjisw `hj o~lwloib o`h cz~iwl`h `d Ο(

]~``d( Cltjh ι ∁ [ h ∞9" bjwf gjh`wj waj flhlfzf `d ibb waj o``~glhiwjs `d ι w`cjwaj~lwa =" s` waiwf ≧ =( Bjwgfiq 6 fiq u`zwgjc! t , r t ∁ T !Ο,v( Wajh ι ls jyzltibjhw

w` waj o`h cz~iwl`hξ 6 ι % Ugfiq % !∞f ,V!η ∞ η ◤,

lwa η is lh waj }~``d `d Bjffi >(?: ( _lhoj η ∞ η ◤ ais ibb jhw~ljs }`slwltj" j aitjξ ≪Ugfiq % !∞f ,V!η ∞ η ◤, ≪gfiq !η ∞ η ◤, ≪gfiq ( !Lhjyziblwljs njw jjh w ` tjow`~s`~ njw jjh i tjow`~ ihg i soibi~ i~j lhwj~}~jwjg o`f}`hjhw lsj(, Lh }i~wlozbi~" ξ lsioojsslnbj" slhoj ih| oal} o`h cz~iwl`h oih ~sw nj swinlbl{jg" s` waiw jioa tj~wjq aisdj j~ waih gfiq oal}s" ihg wajh sz}}bjfjhwjg lwa jqw~i oal}s w` `nwilh ξ ( Waj~j&d`~j ih| o`h cz~iwl`h `nwilhjg d~`f ξ n| ~lhc ls ibs` ioojsslnbj( Lh }i~wlozbi~ wajswinlbl{iwl`hξ ◤ ls wazs o~lwloib ihg jyzltibjhw wι (

Hjqw j lbb sa` waiw jtj~| jyzltibjhoj obiss lh [ h ∞9/ Bw=[ h ∞9 o`hwilhs iw f`sw

`hj o~lwloib o`h cz~iwl`h" fielhc zsj `d waj d`bb` lhc bjffi(

Bjffi >(?3( Bjw 6 !> η , ∞ !>η ,◤" aj~j η ls cltjh n| η !t , 6 `zwgjc !t , is njd`~j( Ld Ϗ ls o~lwloib" wajh !Ϗ% ,◤6 Ϗ(

:7



]~``d( Ld Ϗ ls o~lwloib wajh lw ls ioojsslnbj" s` lw oih nj ~jioajg d~`fη n| igglhc s`fj!h`h&hjciwltj, o`h cz~iwl`h δ ihg sjbjowltjb| ~lhc( Nzw slhoj Ϗ ls ibs` swinbj walslf}bljs waiw !δ % η ,◤6 Ϗ( O`hslgj~ waj o`h cz~iwl`h

ο 6 ! δ % η , % 6 > η % δ % η ∞ !>η ,◤(

_lhoj ≪=" j fi| swi~w d~`f ο ihg ~j i sjyzjhoj `d tj~wlojs waiw swinlbl{js δ % η "w` `nwilh waj o`h cz~iwl`hϏ% ( @h waj `waj~ aihg" slhojη ∞ !>η ,◤≪= j fi| swi~wd~`f ο ihg ~j i sjyzjhoj `d tj~wlojs waiw swinlbl{js >η " w` `nwilh waj o`h cz~iwl`h!>η ,◤% δ % η ∞ !>η ,◤ 6 δ % η " aloa lh wz~h swinlbl{js wϏ( N| Bjffi >(>4lw d`bb` swaiw !Ϗ% ,◤6 Ϗ(

Bjffi >(?1( Bjw Ο nj i glc~i}a lwa i cb`nib slhe( Jtj~| jyzltibjhoj obiss `d [ h ∞9/ Bw

=[ h ∞9 o`hwilhs iw f`sw `hj o~lwloib oal} o`h cz~iwl`h(

]~``d( _z}}`sj w ` o`h cz~iwl`hs Ϗ ihg Ϗ i~j o~lwloib ihg jyzltibjhw lh [ h ∞9/ Bw=[ h ∞9(

Wajh Ϗ 6 Ϗ% l∁K ol∄ t l " aj~j ol i~j h`h&{j~` o`hswihws" ∄t l ls waj ~` lhgjqjg n|waj h`h&slhe tj~wjqt l `d waj ~jgzojg Bi}biolih fiw~lq B=" ihg waj lhgjq l ~zhs `tj~s`fj sznsjw K d waj lhwjcj~s 9">" ( ( ( " h∞9( S~lwj K 6 K ∞ ∫ K %" aj~j K ∞ 6 ul ; ol 2 =vihg K % 6 ul ; ol 8 =v( Bjw

ρ 6 Ϗ %e∁K ∞

!∞oe ,∄ t e 6 Ϗ%e∁K %

oe∄ t e (

Bjw nj waj o`h cz~iwl`h gj hjg lh Bjffi >(?3( _lhoj ls i }`slwltj tjow`~" waj~jjqlsws e = szoa waiwχ 6 ρ % e siwls js χ !t l , ≪ rol r zwgjc! t l ," d`~ ibbl( Dl~j jioatj~wjq t l d`~l ∁ K ∞ lh χ i w`wib `d ∞ol wlfjs w` `nwilh Ϗ%e ( N| Bjffi >(?3"Ϗ% eswinlbl{js w Ϗ( N| Bjffi >(>4lw d`bb` s waiwϏ 6 Ϗ (

Waj`~jf >(?<( Bjw Ο nj i glc~i}a lwa i cb`nib slhe s( Waj sjw `d ibb o~lwloib o`h c&

z~iwl`hs E !Ο, `d Ο ls ih injblih c~`z} zhgj~ waj `}j~iwl`h !Ϗ"Ϗ, ↞ !Ϗ%Ϗ ,◤" ihg lw ls ls`f`~}alo tli waj lhobzsl`h fi} w` waj sihg}lbj c~`z} _ !Ο,( O`hsjyzjhwb|

rE !Ο,r 6 r_ !Ο,r 6 λ!Ο" s, !>(9:,

Aj~j λ!Ο" s, gjh`wjs waj hzfnj~ `d s}ihhlhc w~jjs `d Ο ~``wjg iw s(

]~``d( Lffjgliwj d~`f Waj`~jf 9(>?" Bjffi >(?7" Bjffi >(?4" ihg Bjffi >(?1(

::



Cltjh w ` oal} o`h cz~iwl`hs Ϗ ihg δ " j si| waiw Ϗ ls ~jioainbj d~`f δ !tlijqolwiwl`h&~jbiqiwl`h `}j~iwl`hs, ld waj~j jqlsws i o`h cz~iwl`hξ szoa waiwϏ 6 ! δ %ξ ,◤(H`wj waiw wals lf}bljs waiwϏ ls swinbj( Sj ~lwj Ο s d`~ waj c~i}a `nwilhjg d~`f Ο

n| gjbjwlhc waj tj~wjq s ib`hc lwa ibb jgcjs lholgjhw w s(

Bjffi >(7=( Bjw Ο nj i glc~i}a lwa i cb`nib slhe s" ihg bjw Ϗ nj i oal} o`h cz~iwl`h `h Ο( Waj d`bb` lhc i~j jyzltibjhw(

!9, Ϗ ls o~lwloib5 waiw ls"Ϗ ls ~jioainbj d~`f ih| o`h cz~iwl`h δ (

!>, Ld δ ls ih| o`h cz~iwl`h ~jioainbj d~`f Ϗ" wajh Ϗ ls ~jioainbj d~`f δ (

!?, Ϗ ls ~jioainbj d~`f ih| o`h cz~iwl`h `d waj d`~f J t Ϗ" aj~j t ls i h`h&slhe tj~wjq `d Ο(

!7, Jioa sw~`hcb| o`hhjowjg o f}`hjhw `d Ο s o`hwilhs i tj~wjq t szoa waiw Ϗ ls ~jioainbj d~`f J t Ϗ(

]~``d( _lhoj w~ltlibb| !9, ⇞ !>, ⇞ !?, ⇞ !7," lw sz ojs w` sa` !7, ⇞ !9,(Ld !7, a`bgs" waj~j ls i oal} o`h cz~iwl`hι szoa waiw !Ϗ%ι ,◤6 Ϗ ihg ι ls h`h&

{j~` `h iw bjisw `hj tj~wjq `d jioa sw~`hcb| o`hhjowjg o`f}`hjhw `d Ο( Waj~j jqlswsi }`slwltj lhwjcj~ e szoa waiw sjbjowltj ~lhc d~`feι ~jszbws lh i oal} o`h cz~iwl`hξ

lwa iw bjisw `hj oal} iw jioa tj~wjq( F`~j`tj~" ! Ϗ% ξ ,◤6 Ϗ(H` bjwδ nj ih| oal} o`h cz~iwl`h( _lhoj ξ ais iw bjisw `hj oal} iw jioa tj~wjq"

j aitj δ ≧ Ϗ% ξ d`~ s`fj lhwjcj~ ( Wazs j fi| igg oal}s w` δ ihg wajh swinlbl{jw` `nwilh waj o`h cz~iwl`h !Ϗ% ξ ,◤6 Ϗ( Ajhoj Ϗ ls o~lwloib(

>(>(? ]i~elhc dzhowl`hs ihg s}ihhlhc w~jjs(

Lh wals sjowl`h" j saibb o`hsw~zow i diflb| `d nlkjowltj fi}s njw jjh waj sjw] !Ο" t=, `d

Ο&}i~elhc dzhowl`hs lwa waj nisj tj~wjqt = ! j lbb gj hj biwj~, ihg waj sjw ί!Ο" t=,`d s}ihhlhc w~jjs `d Ο ~``wjg iwt =" wazs }~`tlglhc i o`fnlhiw`~lib }~``d `d r] !Ο" t=, r 6rί!Ο " t=, r( Fih| swiwjfjhws ihg }~``ds }~jsjhwjg aj~j i~j olwjg d~`f U 9>V(

Gj hlwl`h >(79( Bjw Ο nj i glc~i}a `h h % 9 tj~wlojsut =" t9" ( ( ( " th v( Fzbwl}bj&jgcjsi~j ibb` jg" albj b``}s i~j h`w( Lh `~gj~ w` glswlhczlsa if`hc fzbwl}bj&jgcjs `d wajsifj `~gj~ }il~ `d tj~wlojs" j qjg ih `~gj~ njw jjh wajf(

:4



I sznc~i}a W d Ο ls oibbjg isznw~jj ~``wjg iwt = ld d`~ ibb tj~wlojst `d W " waj~jls i zhlyzj gl~jowjg }iwa d~`f t w` t =( Gjh`wj n| W!Ο" t=, waj sjw `d ibb sznw~jjs `d Ο~``wjg iwt =(

I Ο&}i~elhc dzhowl`h lwa nisj tj~wjq t = !`~ Ο&}i~elhc dzhowl`h d`~ sa`~w, lsih h&wz}bj !n9" ( ( ( " nh , `d h`h&hjciwltj lhwjcj~s siwlsd|lhc waj o`hglwl`h waiw d`~ jtj~|h`h&jf}w| sznsjw Z ⊄ ut 9" ( ( ( " th v" waj~j jqlsws i tj~wjqt k ∁ Z szoa waiw waj hzfnj~`d jgcjs d~`f t k w` tj~wlojs `zwslgj `d Z ls c~jiwj~ waihn k" d`~ s`fj lhgjq k(

D`~ i sznw~jj W ∁ W!Ο" t=," bjwύ!W , nj i w`wib `~gj~ `h waj sjw `d tj~wlojs `d W ( Waj sjw έ!Ο, 6 uύ!W , r W ∁ W!Ο" t=,v ls oibbjg i }~`}j~ sjw `d w~jj `~gj~s ld wajd`bb` lhc w ` o`hglwl hs a`bg

9( Ld !t k" tl , ∁ J !W ," wajh t l 2 ύ !W , t k">( ld wls i sznw~jj `d W ~``wjg iwt =" wajh waj `~gj~ύ!w, ls o`hslswjhw lwaύ!W ,5 lh

`waj~ `~gs" t l 2 ύ !w, t k ld ihg `hb| ld t l 2 ύ !W , t k d`~ ibb tj~wlojst l" t k `d w(

_`fj }~`}j~ sjws `d w~jj `~gj~s i~j cltjh lh waj d`bb` lhc jqif}bj

Jqif}bj >(7>( Sj njclh n| lhw~`gzolhc waj n~jigwa& ~sw sji~oa `~gj~ ύnd !W , `h wajtj~wlojs `d i w~jj W ∁ W!Ο" t=,( D`~ i tj~wjq t l ∁ T !W ," j gj hj waj ajlcaw a W !t l ,`d t l lh W w` nj waj hzfnj~ `d jgcjs lh waj zhlyzj }iwa d~`f t l w` waj ~``wt =( Sjsjw t l 2 ύnd !W , t k" `~ t l 2 nd t k" ld a W !t l , 2 a W !t k, `~ jbsj ld a W !t l , 6 a W !t k, ihgl 2 k ( Lw ls jis| w` oajoe waiwύnd !W , ls i w`wib `~gj~ `h waj tj~wlojs `d W ihg waiwέ nd !Ο, 6 uύnd !W , r W ∁ W!Ο" t=,v ls i }~`}j~ sjw `d w~jj `~gj~s(

Waj gj}wa& ~sw sji~oa `~gj~ ύgd !W , `h waj tj~wlojs `d i w~jj W ∁ W!Ο" t=, lsgj hjg is d`bb` s( D`~ i tj~wjq t l ∁ W " bjwW !t l , gjh`wj waj n~ihoa `d W ~``wjg iw t l(Lh `waj~ `~gs"W !t l , o`hslsws `d ibb tj~wlojst e `d W szoa waiw waj zhlyzj }iwa d~`f t e

w` t = lh W o`hwilhs t l( Ld !t l" , ls ih jgcj `d W " wajh j sjw 2ύgd !W , t l( Dz~waj~f`~j"ld !t k" , ls ih jgcj `d W szoa waiwl 2 k " wajh j sjwt l 2 ύ gd !W , t k d`~ t l ∁ W !t l , ihgt k ∁ W !t k,( Sj zsj waj s|fn`b 2 gd lwa waj sifj fjihlhc is 2 ύgd !W , ( Lw ls h`w ai~gw` sjj waiw έgd !Ο, 6 uύgd !W , r W ∁ W!Ο" t=,v ls i }~`}j~ sjw `d w~jj `~gj~s(

@z~ wal~g jqif}bj ls waj tj~wjq&igglhc `~gj~ ύti !W , `h waj tj~wlojs `d W ∁W!Ο" t=,( O`hsw~zow waj sjyzjhoj}=" ( ( ( " }rW r∞9 lhgzowltjb| is d`bb` s( _jw}= 6 t =" ihg"d`~ 9≧ f ≧ rW r ∞9" bjw}f nj waj sfibbjsw tj~wjq t k lh T !Ο, R u }=" ( ( ( " }f ∞9v szoa waiwwaj~j ls ih jgcj lh Ο d~`f t k w` u }=" ( ( ( " }f ∞9v( H`wj waiw waj sjyzjhoj}=" ( ( ( " }rW r∞9

o`hwilhs jioa tj~wjq `d W jqiowb| `hoj( ]zw }= 2 ύ ti !W , ³ ³ ³ 2 ύ ti !W , }rW r∞9" ihg bjw waj

:3



s|fn`b 2 ti aitj waj sifj fjihlhc is 2 ύ ti !W , ( Obji~b|" 2 ti ls i w`wib `~gj~ `h wajtj~wlojs `d W ( Ibs`" έ ti !Ο, 6 uύti !W , r W ∁ W!Ο" t=,v ls i }~`}j~ sjw `d w~jj `~gj~s(

D`~ fielhc `z~ }~jsjhw f`~j obji~" waj d`bb` lhc gj hlwl`h ls s`fja` zsjdzb(

Gj hlwl`h >(7?( D`~ W ∁ W!Ο" t=, ihg i tj~wjq t k `d Ο" waj `~gj~ύ!W , lhgzojs waj`~gj~ `h waj jgcjs c`lhc d~`f t k w` tj~wlojs `d W lh aloa ! t k" t l , ls sfibbj~ waih ! t k" t l ,

ajhjtj~ t l 2 ύ !W , t l ihg aloa ls o`hslswjhw lwa waj }~jtl`zsb| qjg `~gj~ `h fzbwl}bjjgcjs( Sj ~lwj j 2 ύ !W , j w` gjh`wj waiwj ls sfibbj~ waih j lh wals `~gj~(

Cltjh i }~`}j~ sjw `d w~jj `~gj~s έ!Ο," gj hj waj fi} Α ; ί!Ο " t=, ↞ ] !Ο" t=,is d`bb` s( D`~W ∁ ί!Ο " t=, ihg i tj~wjq t k ∁ ut 9" ( ( ( " th v" bjwj k nj waj jgcj `d W

c`lhc `zw `d t k( _jw Α!W , 6 ! n9" ( ( ( " nh ," aj~j n k ls waj hzfnj~ `d jgcjs j c`lhc `zw`d t k szoa waiwj 2 ύ !W , j k(

Bjffi >(77( Waj fi} Α ls jbb&gj hjg( Lh `waj~ `~gs"Α!W , ∁ ] !Ο" t=, d`~ ih| W ∁ ί!Ο " t=, (

]~``d( Bjw Α!W , 6 ! n9" ( ( ( " nh ,( Oa``sj t k 6 flh 2 ύ ! W , ut r t ∁ Z v(Bjwj k 6 ! t k" t l , ∁ J !W ," wajh n| gj hlwl`h" t l 2 ύ !W , t k" ihg f`~j`tj~" t l /∁ Z

!n| waj oa`loj `d t k,(

N| gj hlwl`h `d Α" waj~j i~j ibs` n k jgcjs j 6 ! t k" te , `d Ο szoa waiwj 2 ύ !W , j k"aloa lf}bljs waiw t e 2 }l!W , t l 2 ύ !W , t k( Is njd`~j" t e /∁ Z ( Wazs" waj~j i~j iw bjisw

n k % 9 jgcjs d~`f t k w` tj~wlojs `zwslgj `d Z " aloa fjihs waiw Α! W , ls lh diow iΟ&}i~elhc dzhowl`h(

Hjqw" j gj hj waj lhtj~sj fi} Τ ; ] !Ο" t=, ↞ ί!Ο " t=,( Cltjh ] 6 ! n9" ( ( ( " nh , ∁] !Ο" t=," j o`hsw~zow waj o`~~js}`hglhc w~jj Τ!] , `hj jgcj iw i wlfj zslhc waj d`b&b` lhc ibc`~lwaf(

:1



Ibc`~lwaf >(: Ibc`~lwaf gj hjs waj fi} Τ ; ] !Ο" t=, ∞↞ ί!Ο " t=,^jyzl~j; I Ο&}i~elhc dzhowl`h ] 6 ! n9" ( ( ( " nh , `d i glc~i}a Ο `h waj tj~wjq sjw

ut =" ( ( ( " th v(Jhsz~j; I s}ihhlhc w~jj ~``wjg iw t =(

9; w= ↝ u t =v5}= ↝ t =5e ↝ = we i~j sznw~jjs d`~ ibbe 2 h " ihg w= ls i sznw~jjo`hwilhlhc waj tj~wjq t = `hb|(

>; albj e 2 h g` Lh `waj~ `~gs"we /∁ ί!Ο " t=,?; e ↝ e % 9(7; Z e ↝ u t k r t k ∁ T !Ο," t k /∁ T !we ∞9, v(:; T e ↝ u t k r t k ∁ Z e "ru!t k" t , rt ∁ T !we ∞9,vr ≪n k % 9v(4; d`~ ibb t k ∁ T e g`3; j k 6 u!t k" t , r t ∁ we ∞9"ru!t k" , r ∁ we ∞9"!t k" , 2 ύ !we ∞ 9 , !t k" t , vr 6 n kv(1; w↝ we ∞9 ∫ u j kv Fiej hj sznw~jj wn| igglhc ibb j k w` we ∞9(<; jhg d`~

9=; }e ↝ flh 2 ύ ! w, ut k r t k ∁ T e v(99; we ↝ we ∞9 ∫ u j }e v(9>; jhg albj9?; ~jwz~h wh (

Waj d`bb` lhc jqif}bj lbb lbbzsw~iwj wajsj o`hsw~zowl`hs

Jqif}bj >(7:( Bjw cltjh i glc~i}a Ο is d`bb`

t =

t 7

t ?

t 9

t >

!i, Ο

t =

t 7

t ?

t 9

t >

!n, W ∁ ί!Ο " t= ,

BjwW nj i s}ihhlhc w~jjs `d Ο ~``wjg iw t = !ibs` cltjh lh waj in`tj cz~j,( Bjwwaj `~gj~ ύ!W , `h waj sjw `d tj~wlojs nj waj n~jiwa ~sw sji~oa `~gj~( Lh }i~wlozbi~"t = 2 nd t 7 2 nd t 9 2 nd t ? 2 nd t >( Ibs` isszfj waiw waj `~gj~ `nwilhjg d~`f ih| w~jj lsn~jigwa& ~sw sji~oa `~gj~(

H` " j o`hsw~zow waj Ο&}i~elhc dzhowl`h] 6 Α! W ,(

• j 9 6 ! t 9" t7,( W ` jgcjs ! t 9" t=, ihg ! t 9" t>, i~j c`lhc `zw `d t 9" nzw `hb|!t 9" t=, 2 nd !t 9" t7," s` n9 6 9(

:<



• j > 6 ! t >" t? ,( @hb| !t >" t9, ls c`lhc `zw `d t >" ihg ! t >" t9, 2 nd !t >" t? ," s` n> 6 9(

• j ? 6 ! t ?" t7,( W ` jgcjs ! t ?" t9, ihg ! t ?" t>, i~j c`lhc `zw `d t ?" nzw h`hj `d wajfsfibbj~ waih ! t ?" t7," s` n? 6 =(

• j 7 6 ! t 7" t=,( W ` jgcjs ! t 7" t9, ihg ! t 7" t? , i~j c`lhc `zw `d t ?" nzw h`hj `d wajfsfibbj~ waih ! t 7" t=," s` n7 6 =(

Wazs" ] 6 !9 "9"="=,(Hjqw" d~`f ] 6 !9 "9"="=," j i}}b| Ibc`~lwaf >(: d`~ `nwilhlhc nioe W ( I

}i~wlib gjso~l}wl`h `d waj }~òjss `d ~zhhlhc ls is d`bb` s

• w= 6 ut =v" }= 6 t =( _` Z 9 6 ut 9" t>" t?" t7v( Oajoelhc ibb tj~wlojs lhZ 9" j cjw

T 9 6 ut 7v( _`" }9 6 t 7 ihg w9 6 w6 u!t 7" t=,v(

• Z > 6 ut 9" t>" t?v( Oajoelhc ibb tj~wlojs lh Z > cltjs zs T > 6 ut 9" t?v( H` w6u!t 7" t=, "!t 9" t7, "!t ?" t7,v( Is h`wjg njd`~j" waj n~jigwa& ~sw sji~oa `~gj~ `hwlf}bljs waiw t 9 2 ύ !w, t ?( _` }> 6 t 9 ihg w> 6 u!t 7" t=, "!t 9" t7,v(

• Z ? 6 ut >" t?v( Oajoelhc ibb tj~wlojs lh Z ? cltjs zs T ? 6 ut ?v( _` }? 6 t ? ihgw? 6 w6 u!t 7" t=, "!t 9" t7, "!t ?" t7,v(

• Dlhibb|" Z 7 6 ut >v( _` T 7 6 ut >v" wajh}7 6 t > ihgw7 6 w6 u!t 7" t=, "!t 9" t7, "!t ?" t7, "!t >" t? , v" aloa ls waj `~lclhib W (

Bjffi >(74( Lh Ibc`~lwaf >(: "

}= 2 ύ !W , }9 2 ύ !W , ( ( ( 2ύ !W , }h !>(94,

]~``d( Sj }~`tj waj bjffi zslhc lhgzowl`h( D`~ waj oisj h 6 9" }= 2 }l!W , }9 !έ!Ο,ls i }~`}j~ sjw `d w~jj `~gj~s s` waj ~``w ls ib i|s waj sfibbjsw tj~wjq lh `~gj~ύ!W ,,(

H` " sz}}`sj waiw }= 2 ύ !W , }9 2 ύ !W , ( ( ( 2ύ !W , }f " d`~ s`fj f szoa waiw 9≧ f ≧ h(Sj sa` waiw }f 2 ύ !W , }f %9( Waj~j i~j w ` oisjs

• Oisj 9; }f %9 /∁ T f (Ld }f %9 /∁ T f " lw d`bb` s d~`f Ibc`~lwaf>(: waiw waj hzfnj~ `d jgcjs c`lhc d~`f }f %9 w` wf ∞9 ls bjss waih `~ jyzib n }f %9 !njoizsj `d wals diow wajh lw is h`woa`sjh w` nj lh T f ,(N| gj hlwl`h `d }f %9" }f %9 ∁ T f %9" aloa fjihs waiw waj hzfnj~ `d jgcjs c`lhcd~`f }f %9 w` wf ls c~jiwj~ waih `~ jyzib w`n }f %9 % 9(

4=



Wazs waj~j fzsw jqlsw ih jgcj ! }f %9" }f , ∁ J !Ο, ihg }f %9 is iggjg w` wf

n| fjihs `d szoa ih jgcj( H`wj waiw έ!Ο, ls i }~`}j~ sjw `d w~jj `~gj~s" wajh }f 2 ύ !W , }f %9(

• Oisj >; }f %9 ∁ T f (Bjwj }f %9 nj waj jgcj d~`f }f %9 w` wf ∞9 szoa waiw waj hzfnj~ `d jgcjs j d~`f }f %9 w` wf ∞9 szoa waiwj 2 ύ !wf ∞ 9 , j }f %9 ls n }f %9 (_lhoj }f ls waj bi~cjsw tj~wjq `d wf lh waj `~gj~ύ!W , ihg ajhoj ibs` lh ύ!wf ," j }f %9

fzsw c` d~`f }f %9 w` wf ∞9 6 wf R u }f v !~jf`tj i tj~wjq fjihs ~jf`tlhc waiwtj~wjq ihg ibb jgcjs lholgjhw w` lw,( _`"j 2 ύ !wf ∞ 9 , j }f %9 ⇗ j 2 ύ !wf , j }f %9 ( Wazs"waj~j i~j jqiowb| n }f %9 jgcjs c`lhc d~`f }f %9 w` wf siwlsd|lhc j 2 ύ !wf , j }f %9 "

aloa fjihs }f %9 ls igk`lhjg w` wf n| fjihs `d waj jgcj j }f %9 (Bjw w6 wf ∞9 ∫ u j kv !is gj hjg lh Ibc`~lwaf >(: iw waj b``} o`~~js}`hglhc wè 6 f ,( Bjww 6 wf ∞9 ∫ u j }f " j }f %9 v( Wajh" n| oa loj `d }f " }f 2 ύ !w, }f %9(_lhoj w ls i sznw~jj `d w" }f 2 ύ !w, }f %9( D~`f Ibc`~lwaf >(:" w ls ibs` i sznw~jj`d W " aloa fjihs }f 2 ύ !W , }f %9(

Bjffi >(73( D`~ ] ∁ ] !Ο" t=, "

Α!Τ! ] ,, 6 ]( !>(93,

]~``d( ]zw ] 6 ! n9" ( ( ( " nh ," ihg Τ! ] , 6 W ( Sj hjjg w` sa` waiw Α! W , 6 ] (O`hslgj~ waj }~òjss `d o`hsw~zowlhcW !Ibc`~lwaf >(:,( Wiej ih| t k ∁ ut 9" ( ( ( " th v"wajh t k 6 }f d`~ s`fj f ( Bjwj k nj waj jgcj o`flhc `zw `d t k lh W ( Sj saibb sa`waiw waj hzfnj~ `d jgcjs j c`lhc `zw `d t k ihg j 2 ύ !W , j k ls jyzib w` waj hzfnj~ `d jgcjs j c`lhc d~`f t k w` wf ∞9( F`~j tj~" n`wa `d wajf i~j jyzib w` n k" aloa fjihsΑ!W , 6 ] (_lhoj waj sjw `d tj~wlojs `d wf ∞9 ls u }=" ( ( ( " }f ∞9v" lw d`bb` s d~`f Bjffi >(74 waiw ld ih jgcj j o`flhc `zw `d t k siwls js j 2 ύ !W , j k wajh lw fzsw c` d~`f t k w`wf ∞9( N| waj}~`}j~w| `d }~`}j~ sjw `d w~jj `~gj~s" j eh` waiwj 2 ύ !wf ∞ 9 , j k ⇗ j 2 ύ !W , j k(D~`f Ibc`~lwaf >(:" j eh` waiw waj hzfnj~ `d jgcjs j c`lhc d~`f t k w`wf ∞9 ls jyzibw` n k(_`" j oih o`hobzgj waiw Α!W , 6 ] (

49



Bjffi >(71( D`~ W ∁ ί!Ο " t=, "

Τ!Α! W ,, 6 W ( !>(91,

]~``d( ]zw ] 6 ! n=" ( ( ( " nh , 6 Α! W ,( _z}}`sj waiw n| Ibc`~lwaf >(:" j cjw W 6Τ! ] ,( Sj saibb sa` waiw d`~ jioa swj} f `d o`hsw~zowlhcW !9 ≧ f ≧ h,;

9( wf ls i sznw~jj `d W "

>( }=" ( ( ( " }f ls waj f % 9 sfibbjsw tj~wlojs lh waj `~gj~ύ!W ,(

Wajsj w ` o`hglwl`hs in`tj lh diow cltj zs W 6 W (Sj }~`tj wals n| lhgzowl`h( D`~ waj nisj oisj f 6 =" lw ls `ntl`zsb| w~zj d`~ waj ~``wls ib i|s waj sfibbjsw tj~wjq( Hjqw" sz}}`sj waiw waj w ` o`hglwl`hs a`bg d`~ swj}s9" ( ( ( " f ∞ 9( _`

• wf ∞9 ls i sznw~jj `d W "

• }=" ( ( ( " }f ∞9 ls waj f sfibbjsw tj~wlojs lh waj `~gj~ύ!W ,(

Sj ihw w` sa` waiw wajsj w ` o`hglwl`hs a`bg d`~f ( Bjwt e nj szoa waiw

}= 2 ύ !W , }9 2 ύ !W , ( ( ( 2ύ !W , }f ∞9 2 ύ !W , t e (

Sj h` sa` waiw t e 6 }f (Dl~sw" bjwj e 6 ! t e " t l , ∁ W nj waj zhlyzj jgcj c`lhc `zw `d t e lh W " wajh

t l 2 ύ !W , t e " aloa fjihs waiw t l ∁ u }=" ( ( ( " }f ∞9v" s` t l ls i tj~wjq `d wf ∞9( Ajhoj ld j ls ih jgcj o`flhc `zw `d t e siwlsd|lhc j 2 ύ !W , j e " wajh j fzsw c` d~`f t e w` wf ∞9(Ibs`" waj o`hsw~zowl`h `d ] cltjs zs waiw waj~j i~j ne szoa jgcjs( Wazs" waj~j i~j ne %9jgcjs c`lhc d~`f t e w` wf ∞9" aloa fjihs t e ∁ T f (

Hjqw" j saibb }~`tj waiwt e 6 flh 2 ύ ! w, ut k r t k ∁ T f v" aj~j wls i sznw~jjgj hjg lh Ibc`~lwaf >(: !iw waj b``} o`~~js}`hglhc w` e 6 f ,( Is njd`~j" d`~ jtj~|t k ∁ T f " bjwj k nj waj jgcj d~`f t k w`wf ∞9 szoa waiw jqiowb|n k jgcjs j d~`f t k w`wf ∞9

siwlsd| j 2 ύ !wf ∞ 9 , j k( _lhoj waj tj~wlojs `d wf ∞9 i~j waj sfibbjsw f tj~wlojs lh waj `~gj~ύ!W ," lw d`bb` s waiw ld ih jgcjj o`flhc `zw `d t k siwls js j 2 ύ !W , j k" wajh j c`jsd~`f t k w`wf ∞9( Wazs"j 2 ύ !W , j k ld ihg `hb| ld j 2 ύ !wf ∞ 9 , j k njoizsj waj `~gj~ ύ!wf ∞9,ls o`hslswjhw lwaύ!W ,( Waj~j i~j n k jgcjs j siwlsd|lhc j 2 ύ !wf ∞ 9 , j k" ajhoj waj~j i~jn k jgcjs j siwlsd|lhc j 2 ύ !W , j k( Lw d`bb` s d~`f waj oa`loj `d n k waiwj k ls ih jgcj `d

4>



W ( Waj~jd`~j" waj w~jjw`nwilhjg n| igk`lhlhc waj tj~wlojs t k ∁ T f n| fjihs `d wajjgcjs j k ls i sznw~jj `d W ( O`hsjyzjhwb|" waj sfibbjsw tj~wjq }f `d T f lh waj `~gj~ύ!w, ls waj sfibbjsw tj~wjq `d T f lh waj `~gj~ ύ!W ,( _lhoj t e ls waj sfibbjsw tj~wjq `d

Z f lh waj `~gj~ ύ!W , ihg t e ∁ T f ⊄ Z f " lw d`bb` s waiw}f 6 t e ( Waj lhgzowl`h swj}ls o`f}bjwj(

Dlhibb|" j `nwilh W 6 wh 6 W (

D~`f Bjffi >(73 ihg Bjffi >(71" j `nwilh waj filh ~jszbw `d wals sjowl`h

Waj`~jf >(7<( Waj fi} Α ihg Τ i~j lhtj~sj `d jioa `waj~( Lh `waj~ `~gs" Α ls i nlkjowl`h fi}}lhc njw jjh ί!Ο " t=, ihg ] !Ο" t=, ( O`hsjyzjhwb|

r] !Ο" t=, r 6 rί!Ο " t=, r 6 λ!Ο" t=, !>(9<,

aj~j λ!Ο" t=, gjh`wjs waj hzfnj~ `d s}ihhlhc w~jjs `d Ο ~``wjg iw t =(

>(>(7 ]i~elhc dzhowl hs ihg o~lwloib o`h cz~iwl hs(

Lh blcaw `d Waj`~jf >(>7" waj d`bb` lhc yzjswl`h ls hiwz~ibb| }~`}`sjg; Ls wals swlbb w~zj waiw d`~ i cltjh gl~jowjg c~i}a" i o`h cz~iwl h Ϗ ls o~lwloib ld ihg `hb| ld waj o`h cz~iwl`h η 6 ofiq ∞ Ϗ ls i Ο&}i~elhc dzhowl`h" aj~j ofiq ls waj fiqlfib swinbj o`h cz~iwl`h gj hjg n| ofiq !t , 6 `zwgjc !t , ∞ 9 d`~ ibb h`h&slhe tj~wjq t 0 D`~ hglhcwaj ihs j~ `d wals yzjswl`h" j bjig w` waj d`bb` lhc }~j}~lhw }i}j~ U93V( Lh wals }i}j~"waj izwa`~s cltj i }`slwltj ihs j~ d`~ `z~ yzjswl`h( Waiw ls" d`~ i cltjh gl~jowjg c~i}a"i o`h cz~iwl`h Ϗ ls o~lwloib ld ihg `hb| ld waj o`h cz~iwl`hη 6 ofiq ∞ Ϗ ls i Ο&}i~elhcdzhowl`h( Nzw" wals ais h`w siwls jg `z~ cltjh yzjswl`h |jw" slhoj waj~j ls i gl j~jhojnjw jjh `z~ gj hlwl`h d`~ waj Ο&}i~elhc dzhowl`h ihg wajl~ gj hlwl`h( Lw ls h`w w~ltlibwaiw waj w ` i~j jyzltibjhw( _`" waj yzjswl`h ls swlbb zhs`btjg" ihg j saibb o`hwlhzjswzg|lhc wals }~`nbjf lh `~gj~ w` hg waj o`~~jow ihs j~(

4?



O`hobzsl`hs

Lh wals wajsls" j aitj }~jsjhwjg fih| glswlhow }~``ds `d waj Fiw~lq&W~jj Waj`&~jf d`~ waj obiss `d zhgl~jowjg c~i}as" gl~jowjg c~i}as" ihg jlcawjg" gl~jowjg c~i}as(Lh }i~wlozbi~" d`~ zhgl~jowjg c~i}as" j }~jsjhwjg w ` i|s `d i}}~ìoalhc wals }~`n&bjf" waj Nlhjw&Oizoa| i}}~ìoa !sjowl`h 9(9(9, ihg waj Gjbjwl`h&O`hw~iowl`h i}}~ìoa!sjowl`h 9(9(>,( Waj ~sw i}}~ìoa ls f`~j ibcjn~ilo" albj waj sjo`hg `hj ls f`~jc~i}a waj`~jwlo( Waj filh ~jdj~jhojs d`~ wajsj i}}~ìoajs i~j U >9Vihg U?V( Nisj `h i}~`}`slwl`h d~`f blhji~ ibcjn~i" j ibs` }~jsjhwjg waj cjhj~ib swiwjfjhw `d Fiw~lq&W~jjWaj`~jf !sjowl`h 9(9(?, d`~ waj zhgl~jowjg c~i}as oisj( Lh blcaw `d Gjbjwl`h&O`hw~iowl`hi}}~ìoa d`~ waj zhgl~jowjg c~i}as oisj" j citj i hj ihib`czj }~``d d`~ waj waj`~jflh waj oisj `d gl~jowjg c~i}as !sjowl`h 9(>,( D`~ waj jlcawjg" gl~jowjg c~i}as oisj!sjowl`h 9(?," j ~j&lhwj~}~jwjg waj `~es `d G`~`h [jlbnj~cj~ lh als }i}j~ U7V" ihg wajjq}bihiwl`h d~`f U>=Vls ajb}dzb d`~ zhgj~swihglhc(

F`~j`tj~" j ibs` citj ih lhw~`gzowl`h w` waj sw~zowz~j `d i ti~lihw `d oal}&~lhc cifj f`gjb d`~ waj oisj `d zhgl~jowjg c~i}as !sjowl`h >(9(9, ihg gl~jowjg c~i}as

!sjowl`h >(>(9,( D`~ waj zhgl~jowjg c~i}as" j }~jsjhwjg waj sw~zowz~j `d waj o~lwloibc~`z}s !sjowl`h >(9(>, wa~`zca waj o`hsw~zowl`h `d bjcib ~lhc sjyzjhojs( Zslhc s`fjnislo eh` bjgcjs d~`f waj biwwloj waj`~| !U4V"U9V, ihg blhji~ ibcjn~i" j aitj }~`tjg!n| `z~ ` h wjoahlyzj, ih lf}`~wihw ~jszbw; Waj `~gj~ `d waj o~lwloib c~`z}E !C, ls jyzib w` waj hzfnj~ `d s}ihhlhc w~jjs λ!C,( D`~ waj oisj `d gl~jowjg c~i}as" waj o~lwloibc~`z}s !sjowl`h >(>(>, i~j lhwj~}~jwjg wa~`zca waj o`hsw~zowl`h `d waj s`&oibbjg ‒sihg}lbjc~`z} `d i c~i}a‖( Waj }~``d `d waj d`bb` lhc ihib`czj ~jszbw is ibs` }~jsjhwjg; Waj `~gj~ `d waj o~lwloib c~`z}E !Ο, ls jyzib w` waj hzfnj~ `d ~``wjg s}ihhlhc w~jjs λ!Ο" s,(Waj filh ~jdj~jhojs d`~ wajsj sjowl`hs i~j U3V ihg U97V(

Cltjh ih zhgl~jowjg" o`hhjowjg" b``}bjss c~i}a C" j glg }~jsjhw waj jq}blolwnlkjowl`h njw jjh waj sjw `d s}ihhlhc w~jjs ί!C, ihg o~lwloib o`h cz~iwl`hsE !C, !sjo&wl`h >(9(?," ihg waj jq}blolw nlkjowl`h njw jjh waj sjw `d }i~elhc dzhowl`hs] !C" s, ihgo~lwloib o`h cz~iwl`hsE !C, !sjowl`h >(9(7,( Wajsj sjowl`hs oih h`w o`f}bjwj lwa`zwwaj ajb} d~`f U99V" U9: V" U9?V" ihg U<V( D`~ i cltjh gl~jowjg c~i}a Ο" waj ~jdj~jhojd~`f U9>Vglg ajb} zs }~jsjhw waj jq}blolw nlkjowl`h njw jjh waj sjw `d }i~elhc dzhowl`hs] !Ο" s, ihg ~``wjg s}ihhlhc w~jjs ί!Ο " s, !sjowl`h >(>(?,( Ih ihib`czj yzjswl`h in`zwwaj nlkjowl`h njw jjh waj sjw `d }i~elhc dzhowl`hs] !Ο" s, ihg o~lwloib o`h cz~iwl`hsE !Ο, is ibs` }~`}`sjg !sjowl`h >(>(7," nzw lw ls swlbb zhs`btjg d`~ s`fj wjoahloib ~ji&

47



s`hs( Lh waj hji~ dzwz~j" j saibb o`hwlhzj swzg|lhc wals yzjswl h lh `~gj~ w` hg waji}}~`}~liwj ihs j~(

4:



Nlnbl`c~i}a|

U9V Gihljbj Floolihol`( O_J>=4I; Biwwlojs Ibc`~lwafs ihg I}}bloiwl hs" Slhwj~>=9>(aww};//osj jn(zosg(jgz/obissjs/ l9>/osj>=4I&i (

U>V Gitlg ]j~elhs`h( Fiwa 79>" W`}los lh Ibcjn~i; Waj Injblih _ihg}lbj F`gjb" Dibb>=9=(aww};//}j`}bj(~jjg(jgz/xgitlg}/79> (

U?V ^loai~g ]( _wihbj|( W`}los lh ibcjn~ilo o`fnlhiw`~los( aww};// &fiwa(flw(

jgz/x~swih/ibco`fn(}gd (

U7V G`~`h [jlbnj~cj~( I o`fnlhiw`~lib i}}~ìoa w` fiw~lq ibcjn~i( Glso~jwj Fiwajfiwlos " :4!9,;49‟3>" 9<1:( aww};// (fiwa(~zwcj~s(jgz/x{jlbnj~c/

fifi~lf\/[jlbnj~cj~P|9<1:P}49(}gd (

U:V H( B( Nlccs( Ibcjn~ilo }`wjhwlib waj ~| `h c~i}as(Nzbb( B`hg`h Fiwa( _ò("><!4,;479‟41>" Kih 9<<3(

U4V O|hwali G `~e( Bjowz~j H`wjs; Biwwlojs ihg Wajl~ I}}bloiwl`h w` O~|}w`c~i}a|"_wihd`~g Zhltj~slw|" 9<<1( }}( >3‟71"aww};// (fiwa(zogitls(jgz/xgjb`j~i/

FL_O/NLNBL@WJOI/w~zhe/G `~e(}gd (

U3V H( B( Nlccs( Oal}& ~lhc ihg waj o~lwloib c~`z} `d i c~i}a(K( Ibcjn~ilo O`fn( "<!9,;>:‟7:" Kih 9<<<(

U1V \tih Bj N`~chj( _ihg}lbj f`gjb ihg Wzwwj }`b|h`flibs" >==9( ^jszfj jwjhgziooj}wj }`z~ zh jq}`sj i D`~fib ]` j~ _j~ljs ihg Ibcjn~ilo O`fnlhiw`~los !D]&_IO, ]a`jhlq >==9 !Bòib ]~òjjglhcs, }~j}~lhw( aww};// (bin~l(d~/}j~s`/

n`~chj/}znbls/d}sio=9(}gd (

U<V Tbiglfl~ _`wl~`t Dih Sjl" ^àlw Ic~i ib( Jtioziwl`h `d swihgi~g \`zhc winbjizqihg oal}& ~lhc" >==9( ^jsji~oa Jq}j~ljhojs d`~ Zhgj~c~igs" _zffj~ >=99" aww};

// (fiwa(zfh(jgz/x~jlhj~/ JZ/Ic~i ib_`wl~`tSjl>=99(}gd (

44

http://cseweb.ucsd.edu/classes/wi12/cse206A-a

http://people.reed.edu/~davidp/412


http://www-math.mit.edu/~rstan/algcomb.pdf


http://www.math.rutgers.edu/~zeilberg/mamarimY/Zeilberger_y1985_p61.pdf



http://www.math.ucdavis.edu/~deloera/MISC/BIBLIOTECA/trunk/Dwork.pdf


http://www.labri.fr/perso/borgne/publis/fpsac01.pdf



http://www.math.umn.edu/~reiner/REU/AgrawalSotirovWei2011.pdf













http://cseweb.ucsd.edu/classes/wi12/cse206A-a



U9=V Fiwa ^jdj~jhoj ]~`kjow( Biwwlojs" _znbiwwloj" O`t bzfj" Glso~lflhihw" >==>&>=9>(aww};// (fiwa~jdj~jhoj(o`f/bi&biw"o`t(awfb (

U99V ^`nj~w O`~l ihg \tih Bj N`~chj( Waj sihg&}lbj f`gjb ihg Wzwwj }`b|h`flibs(Igt( I}}b( Fiwa( " ?=!9/>,;77‟:>" Kihzi~| >==?(

U9>V Gjhls Oajnlelh ihg ]itb` ]|b|itse||( I diflb| `d nlkjowl hs njw jjh C&}i~elhcdzhowl`hs ihg s}ihhlhc w~jjs( K( O`fn( Waj`~| _j~( I " 99=!9,;?9‟79" >==:(

U9?V Fiwwaj Niej~ ihg _j~czjl H`~lhj( ^ljfihh&^òa ihg Injb&Kio`nl waj`~| `h ihlwj c~i}a( Igt( Fiwa " >9:;344‟311" >==3(

U97V Ibjqihgj~J( A`b~`|g" Bl`hjb Bjtlhj" Ei~`bi F·js{· i~`s" \z|ib ]j~js" Kifjs ]~`}}"ihg GitlgN( Slbs`h( Oal}&Dl~lhc ihg ^`w`~& `zwlhc `h Gl~jowjg C~i}as( Lh Lh ihg @zw `d Jyzlbln~lzf > " t`bzfj 4= `d ]~`c~jss lh ]~`ninlblw| " }icjs ??9‟?47(Nl~ea¡izsj~ Nisjb" >==1(

U9:V D( _aè~lja( Oal}&Dl~lhc Cifjs" C&]i~elhc Dzhowl`hs" ihg ih J oljhw Nlkjowltj]~``d `d waj Fiw~lq&W~jj Waj`~jf( I~Qlt j&}~lhws " Kzb| >==<(aww};//i~qlt(

`~c/ins/=<=3(7349 (

U94V ^`nn| C( FoElbblif( Biwwloj waj ~|" ol~ozbi~ swiwlswlos ihg }`b|h`flib }aisj slchibs ( ]aG wajsls" Zhltj~slw| `d Yzjjhsbihg" >=9=( }}( 9?‟4?( aww};//

~`nn|foelbblif(clwazn(o`f/}i}j~s/wajsls(}gd (

U93V G( ]j~elhs`h" K( ]j~bfih" ihg K( Slbfjs( ]~lfj~ d`~ waj ibcjn~ilo cj`fjw~| `d sihg}lbjs( I~Qlt j&}~lhws " Gjojfnj~ >=99( aww};//i~qlt(`~c/ins/999>(494?t> (

U91V Fleb`s N`hi( I Sibe Wa~`zca O`fnlhiw`~los" Ih Lhw~`gzowl`h w` Jhzfj~iwl`h ihg C~i}a Waj`~| ( S`~bg _oljhwl o ]znblsalhc O`(" >==>( }}( >>?‟>>3(

U9<V ^( B( C~iaif" F( C~¡wsoajb" ihg B( B`t·is{" jglw`~s( Aihgn`è `d o`fnlhiw`~los !t`b( >, ( FLW ]~jss" Oifn~lgcj" FI" Z_I" 9<<:( }}( 9=73‟9=7<(

U>=V Aj~nj~w Kàh ^|sj~ ^loai~g I( N~zibgl(O`fnlhiw`~lib Fiw~lq Waj`~| ( Oifn~lgcjZhltj~slw| ]~jss" 9<<9( }}( ><>‟?>4(

http://www.mathreference.com/la-lat,cov.html

http://arxiv.org/abs/0907.4761


http://robbymckilliam.github.com/papers/thesis.pdf


http://arxiv.org/abs/1112.6163v2

http://arxiv.org/abs/1112.6163v2





http://www.mathreference.com/la-lat,cov.html

Documents

Undergraduate Thesis - The Matrix-Tree Theorem and some related problems